Nucleation Theory and Applications ¨ rn W. P. Schmelzer, Gerd Ro ¨ pke, and Ju Vyatcheslav B. Priezzhev (Editors)
Dubna JINR 2006
ii
vch 4 Okt 2005 10:43
Preface Es ist nicht genug damit, einen gesunden Geist zu haben, sondern die Hauptsache ist, ihn in gesunder Weise anzuwenden. Rene Descartes The present book consists of contributions devoted to the analysis of first-order phase transitions both from experimental and theoretical points of view. They have been presented and discussed in the course of the research workshops Nucleation Theory and Applications held in Dubna, Russia, in June/July 2005 and in preceding similar meetings carried out also with great success in 2003 and 2004 at the same place. Hopefully, the results will be of use for other colleagues engaged in similar problems. The programs of the workshops are given in the appendix. In the appendix, the content of the first two volumes of the proceedings, covering the years from 1997 till 1999 and 2000 till 2002, is reprinted as well for interested colleagues. A comparison gives some interesting insight into in the topics of discussion for about a period of nine years, now. The general aim of these meetings was and is [i.] to bring together a number of leading scientists in the field of the theoretical description and experimental investigations of first- and second-order phase transformations of the member countries of the Joint Institute for Nuclear Research, of Germany and beyond; [ii.] to discuss recent developments in this field with particular emphasis on the work done in the different groups invited; [iii.] to establish and/or tighten direct cooperation links; [iv.] to carry out research on common research projects; [v.] and to check whether the experimental facilities available at the JINR in Dubna can be utilized for an experimental investigation of the kinetics of phase transformation processes in different systems of interest. These aims could be fully realized, again. The workshops could not have been organized without the financial support of a number of organizations and institutions. Further funding was required, of course,
iv to obtain the results which are presented and discussed in the workshop. Some of the sponsoring organizations, we would like to express our particular gratitude, are (in alphabetic order): • Brazilian State of S˜ ao Paulo Research Foundation; • Bundesministerium f¨ ur Bildung, Wissenschaft, Forschung und Technologie (BMBF) Germany (via Research projects, the TRANSFORM and HeisenbergLandau programs); • Deutsche Forschungsgemeinschaft (DFG) (via Research projects, travel, conference and publication grants); • Deutscher Akademischer Austauschdienst (DAAD); • Helmholtz-Gemeinschaft Deutscher Forschungszentren; • Russian Foundation for Basic Research (RFBR); • UNESCO (Venice Office). To all above cited organizations and those not mentioned explicitely, we would like to express our sincere thanks, as well as to all colleagues who helped us in the organization of the workshops. In particular, we would like to express our thanks here to V. I. Zhuravlev, G. G. Sandukovskaya, and E. N. Rusakovich. It is planned to continue the research workshops also in the next years. The respective information will be given at the homepage http://thsun1.jinr.ru of the Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research and via Email. For questions, please contact us [preferably via Email:
[email protected] (J. W. P. Schmelzer)].
J¨ urn W. P. Schmelzer
vch 4 Okt 2005 10:43
Gerd R¨ opke
Vyatcheslav B. Priezzhev
Contents 1
Introductory Remarks
1
2
Thermodynamics and the Kinetics of First-Order Phase Transitions 3
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Thermodynamics and Nucleation Phenomena . . . . . . . . . . . .
7
2.3
Trajectories of Cluster Evolution
2.4
Nucleation versus Spinodal Decomposition . . . . . . . . . . . . . . 15
2.5
Saddle Point versus Ridge Crossing . . . . . . . . . . . . . . . . . . 19
2.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3
Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1
Identification of Crystals . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2
Crystal Growth in the Glass Bulk . . . . . . . . . . . . . . . . . . . 28
3.3.3
Crystal Growth on the Surface . . . . . . . . . . . . . . . . . . . . . 29
3.3.4
Comparison of Crystal Growth in the Bulk and on Polished Surfaces 32
3.3.5
Some Peculiarities of Crystal Growth on the Glass Surface . . . . . 37
3.4
Main Results and Conclusions . . . . . . . . . . . . . . . . . . . . . 42
3.5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
. . . . . . . . . . . . . . . . . . . 12
23
vi Contents 4
Metastable Extensions of Phase Equilibrium Curves
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2
Models and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3
Limits of Thermodynamic Stability and Phase-Equilibrium . . . . . 55
4.4
Terminal Critical Points . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5
Dynamical Clustering in Chains of Atoms with Exponential Repulsion 65
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2
Models of Interaction with Exponential Repulsion . . . . . . . . . . 66
5.3
Nonlinear Dynamics of Morse Chains . . . . . . . . . . . . . . . . . 68
5.4
Excitation of Running Local Compressions: Dynamical Clusters . . 73
5.4.1
Driving Solitons by External Forcing . . . . . . . . . . . . . . . . . 73
5.4.2
Excitation of Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5
Dynamics of Electrons Coupled to Running Local Compressions . . 77
5.5.1
Semiclassical Model of Electron Dynamics . . . . . . . . . . . . . . 77
5.5.2
Coupling Between Soliton Modes and Electron Dynamics . . . . . . 80
5.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6
Solid State of Repelling Particles
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2
Ensembles of Repelling Particles at Low Temperatures . . . . . . . 88
6.3
Model of Hard Disks for an Ensemble of Repelling Atoms . . . . . . 91
6.4
Equation of State for Disks . . . . . . . . . . . . . . . . . . . . . . . 95
6.5
The Cell Model for Disk Particles . . . . . . . . . . . . . . . . . . . 97
6.6
Diffusion Coefficient of Vacancies for Cell Model . . . . . . . . . . . 100
6.7
Peculiarities of the Solid State of an Ensemble of Repulsing Particles104
6.8
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
vch 4 Okt 2005 10:43
45
86
Contents
vii
7
Microcanonical Thermostatistics as Foundation of Thermodynamics 107
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.2
What is Entropy? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.3
The Zeroth Law in Conventional Extensive Thermodynamics . . . . 113
7.4
Phase Separation and S(E)-Dependence . . . . . . . . . . . . . . . 114
7.5
The Origin of the Convexities of S(E) and of Phase-Separation . . 117
7.5.1
Liquid-Gas Phase Transition . . . . . . . . . . . . . . . . . . . . . . 118
7.5.2
Solid-Liquid Phase Transition . . . . . . . . . . . . . . . . . . . . . 121
7.5.3
Summary of Section 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.6
Application in Astrophysics . . . . . . . . . . . . . . . . . . . . . . 122
7.7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8
A Comprehensive Gibbs Potential of Ice Ih
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2
Gibbs Potential Function . . . . . . . . . . . . . . . . . . . . . . . . 129
8.3
Regression Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.4
Comparison with Experiments . . . . . . . . . . . . . . . . . . . . . 133
8.4.1
Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.4.2
Thermal Expansion Coefficient . . . . . . . . . . . . . . . . . . . . . 135
8.4.3
Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8.4.4
Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.4.5
Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.4.6
Sublimation Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.4.7
Melting Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
126
vch 4 Okt 2005 10:43
viii
Contents
9
Thermodynamics of Amorphous Solids and Disordered Crystals 153
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.2
Thermal, Mechanical and Electrical Properties of Disordered Solids 157
9.3
Alternative Ways of Determining Caloric Properties of Glasses . . . 175
9.4
Consequences of Simon’s Classical Approximation . . . . . . . . . . 176
9.5
Change of Kinetic Properties at Tg and Vitrification Kinetics . . . . 178
9.6
Liquid Crystals and Frozen-in Orientational Modes in Crystals . . . 181
9.7
”Spectroscopic” Determination of Zero-point Entropies . . . . . . . 187
9.8
Entropy of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.9
Caloric Properties of Typical Glass-forming Systems
9.10
General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10
Fundamental Equations and Third Law of Thermodynamics209
10.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
10.2
The Thermodynamic Method . . . . . . . . . . . . . . . . . . . . . 210
. . . . . . . . 192
10.2.1 General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 10.3
Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
10.3.1 A: Thermal Equation of State Given . . . . . . . . . . . . . . . . . 214 10.3.2 B: Caloric Equation of State Given . . . . . . . . . . . . . . . . . . 214 10.3.3 Completion of the Method . . . . . . . . . . . . . . . . . . . . . . . 215 10.4
Demonstration of the Method . . . . . . . . . . . . . . . . . . . . . 215
10.4.1 A: Thermal Equation of State Given . . . . . . . . . . . . . . . . . 215 10.4.2 B: Caloric Equation of State Given . . . . . . . . . . . . . . . . . . 217 10.5
The Third Law of Thermodynamics . . . . . . . . . . . . . . . . . . 217
10.5.1 Basic Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 10.5.2 Differential Form of the Third Law . . . . . . . . . . . . . . . . . . 218 10.6
State of Fundamental Equations in Use . . . . . . . . . . . . . . . . 219
10.7
Modifying Thermal Equations of State to Fulfill the Third Law . . 221
10.8
Further Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
vch 4 Okt 2005 10:43
Contents
ix
10.9
Discussion of Literature . . . . . . . . . . . . . . . . . . . . . . . . . 227
10.10
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
10.11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
11
Gas-Filled Bubbles in Low-Viscosity Liquids
11.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
11.2
Nucleation (tlag < t < tN ) . . . . . . . . . . . . . . . . . . . . . . . 233
230
11.2.1 Reduced Equations describing the Process of Bubble Nucleation in a Low-Viscosity Liquid . . . . . . . . . . . . . . . . . . . . . . . . . 233 11.2.2 Time of Establishment of Steady-state Conditions of Nucleation . . 237 11.2.3 Quasi-Stationary Distribution of Sub-critical Bubbles . . . . . . . . 242 ˜ . . . . 243 11.2.4 Distribution Function of Bubbles in the Range Nc < N < N ˜ . . . . . 246 11.2.5 Distribution Function of Bubbles within the Range N > N 11.3
The Intermediate Stage (tN < t < tf ) . . . . . . . . . . . . . . . . . 252
11.4
The Late Stage (t > tf ) . . . . . . . . . . . . . . . . . . . . . . . . . 261
11.5
Results of Numerical Computations . . . . . . . . . . . . . . . . . . 273
11.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
11.7
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
11.7.1 Some Mathematical Transformations . . . . . . . . . . . . . . . . . 274 11.7.2 Estimation of the Conditions when Merging of Colliding Bubbles can be Neglected . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 11.8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
12
Deliquescence of Small Soluble Particles
12.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
12.2
Work of Droplet Formation on Condensation Nuclei . . . . . . . . . 281
12.3
The Generating Properties of the Work of Droplet Formation . . . 286
12.4
Two- and One-Dimensional Theories of Deliquescence . . . . . . . . 289
12.5
Kinetics of Droplet Growth over the Deliquescence Barrier . . . . . 296
12.6
Some Approximate Formulas and Conclusions . . . . . . . . . . . . 300
12.7
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
279
vch 4 Okt 2005 10:43
x
Contents
13
Kinetics of Nucleation with Specific Regimes of Droplet Growth 305
13.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
13.2
Kinetics of Nucleation for a Decreasing Rate of Embryos Growth . 308
13.2.1 Evolution Equations
. . . . . . . . . . . . . . . . . . . . . . . . . . 308
13.2.2 Decay of the Metastable State . . . . . . . . . . . . . . . . . . . . . 310 13.2.3 Dynamic Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 313 13.3
Kinetics of Nucleation at Growth Termination . . . . . . . . . . . . 324
13.3.1 Decay of the Metastable State . . . . . . . . . . . . . . . . . . . . . 324 13.3.2 Nucleation at Smooth Behavior of External Conditions . . . . . . . 338 13.4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
14
Laminar Flow Diffusion Chamber in Nucleation
14.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
14.2
Mathematical Model of LFDC Performance . . . . . . . . . . . . . 352
14.3
Qualitative Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 355
14.4
Methods of Numerical Analysis . . . . . . . . . . . . . . . . . . . . 360
14.5
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
14.6
Comparison with Experiment . . . . . . . . . . . . . . . . . . . . . 368
14.7
Nonuniform Fields of Supersaturation . . . . . . . . . . . . . . . . . 368
14.8
Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 372
14.9
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
vch 4 Okt 2005 10:43
349
Contents
xi
15
Condensation-Relaxation of Supersaturated Vapor
376
15.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
15.2
Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . 377
15.3
Results of Simulation and Their Analysis . . . . . . . . . . . . . . . 381
15.3.1 Discussion of the Results of Numerical Simulation . . . . . . . . . . 381 15.3.2 Scaling Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 15.3.3 On the Possibility of Experimental Determination of the Nucleation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 15.4
Static and Dynamic Conditions . . . . . . . . . . . . . . . . . . . . 386
15.5
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 392
15.6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
16
Diffusion Equation and Multiple Light Scattering
16.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
16.2
Dispersed Nonabsorbing Turbid Media . . . . . . . . . . . . . . . . 396
16.3
”Critical” Medium Heating by Probing Radiation . . . . . . . . . . 400
16.4
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
17
Cluster Evolution in Neutron Irradiated Reactor Steels
17.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
17.2
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
395
408
17.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 17.2.2 System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 412 17.2.3 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 17.2.4 Cluster-Matrix Interactions . . . . . . . . . . . . . . . . . . . . . . . 414 17.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
17.3.1 The Ideal Solution Model (W (n) = 0) . . . . . . . . . . . . . . . . . 417 17.3.2 Account of Cluster-Matrix Interaction, W (n) . . . . . . . . . . . . . 417 17.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
17.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
17.6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
vch 4 Okt 2005 10:43
xii Contents 18
Statistical Theory of Electrolytic Skin Effects
18.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
18.2
Electrolytic Skin Effects . . . . . . . . . . . . . . . . . . . . . . . . 424
18.3
BBGKY-Hierarchy Equations . . . . . . . . . . . . . . . . . . . . . 427
18.4
Analytical Solution for the Pair Distribution Function . . . . . . . . 430
18.5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
19
On the η 4 -Model in the Displacive Limit
19.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
19.2
Critical Temperature and Landau Effective Hamiltonian (LEH) . . 443
19.3
Specific Heat Calculation in the η 4 -Model . . . . . . . . . . . . . . 447
19.4
Critical Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
19.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
19.6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
20
A Model Explaining Necking in Polymer Spinning
20.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
20.2
Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
20.3
Chain in the Frame of Classical Statistics . . . . . . . . . . . . . . . 460
20.3.1 Theoretical Background
420
440
456
. . . . . . . . . . . . . . . . . . . . . . . . 460
20.3.2 Chain Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 20.3.3 Random Chain Generation: Trivial Approach . . . . . . . . . . . . 465 20.3.4 Simulation with a Biased Generator . . . . . . . . . . . . . . . . . . 465 20.3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469 20.4
Simplified Model for Longer Chains . . . . . . . . . . . . . . . . . . 469
20.4.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . . 469 20.4.2 Justification of the Simplified Model
. . . . . . . . . . . . . . . . . 471
20.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 20.5
Understanding the Physics . . . . . . . . . . . . . . . . . . . . . . . 473
20.6
Dumpbell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
20.7
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
20.8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
vch 4 Okt 2005 10:43
Contents
xiii
21
Appendices
479
21.1
Programs of the Research Workshops 2003, 2004 and 2005 . . . . . 479
21.1.1 Research Workshop 2003 . . . . . . . . . . . . . . . . . . . . . . . . 479 21.1.2 Research Workshop 2004 . . . . . . . . . . . . . . . . . . . . . . . . 483 21.1.3 Research Workshop 2005 . . . . . . . . . . . . . . . . . . . . . . . . 486 21.2
Content of Proceedings . . . . . . . . . . . . . . . . . . . . . . . . . 490
21.3
List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
vch 4 Okt 2005 10:43
xiv
Contents
vch 4 Okt 2005 10:43
1
Introductory Remarks
Freiheit, Friede und elektrischen Strom sch¨ atzt man mehr, wenn man sie gelegentlich nicht hat. . . . bekanntlich ist das, was den Menschen freut, in der Regel nicht moralisch, sondern moralisch sein heißt, etwas zu tun, was einen nicht freut . . . Manfred Rommel The present book contains overviews on lectures which have been presented at the Research workshops Nucleation Theory and Applications in Dubna in the last three years (2003-2005). In accordance with the general line of organization of these workshops, they represent either an account of the work which has been performed independently by colleagues from the different groups invited or they contain results worked out in the last years during the research workshops and in between them in the framework of different common projects. Thus, the presence of the same authors in several contributions is obviously unavoidable in order to give some account of the results of the common work. Of course, neither all contributions presented nor all of the results obtained in the common research can be included in one book. Some others are contained in the list of references of the papers presented here, but also with such supplement the list is rather incomplete. Thus, in the appendix the programs of the workshops are given in order to allow some more general overview and to give also the possibility to establish direct contacts to the respective colleagues. In addition, also the content of the first and second volume of the proceedings is reprinted giving an account on the work performed in the three years-periods from 19971999 and 2000-2002. Another volume of overview lectures with the same title (Nucleation Theory and Applications) has been published by WILEY-VCH in 2005. The content is given in the appendix as well. It is one of the most pleasing results of the workshops that the number of direct contacts in the network of cooperation links has been again increased considerably. Hopefully, this book can facilitate further advances in this respect.
2 1 Introductory Remarks In the present proceedings, the spectrum of different applications includes direct experimental investigations both of thermodynamic properties of matter (in thermodynamic equilibrium and non-equilibrium states including glasses) and nucleation-growth phenomena and their interpretation, the theoretical analysis of the course of first- and second-order phase transitions, the discussion of principal problems of the thermodynamic description of clusters and a variety of applications. Hereby, contributions are included (at the status of preprints), again, which have been submitted in an appropriate form till the end of the workshop or some finite time afterwards. In the absolute majority of cases (but not in all), the results reported, and presented now here, have been approved by the participants. Thus, it was decided, as at the last times, not to establish some kind of final refereing system in order to allow a rapid publication of the proceedings. This way, except some minor editorial changes, to bring the contributions into the same form and to remove obvious misprints in the original manuscripts, no changes have been made in the course of preparation of this book for publication. It is believed that the presentation of even different points of view may stimulate the further discussion (and will be more useful) until, finally, some agreement can be reached. Thus, the responsibility for the content of the papers is retained, again, totally with the authors. They have to bear up both the honour for their results and the possible risk. Due to the support from the host institution, the Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research (JINR), the workshops could be carried out under nearly perfect conditions, again (and slight fluctuations like one temporary switch-off of electricity – due to extremal weather conditions – could not cause serious troubles). Finally, I would like to note that it was a real pleasure, again, to prepare this volume for publication and I acknowledge herewith the perfect cooperation with Alexander S. Abyzov (in addition to the common work on one of the chapters, for his help in the preparation of the figures, if required), Irina G. Polyakova (for preparing the cover pages) and the colleagues of the Publishing Department of the JINR for the final completion of the work. ”Aller guten Dinge sind drei” – as it is well-known – nevertheless, we are in good hope that the work can be continued as successfully as in the past also in the coming years.
Dubna & Rostock, Summer 2005
vch 4 Okt 2005 10:43
Dr. J¨ urn W. P. Schmelzer
2
Generalized Gibbs’ Approach to the Thermodynamics of Heterogeneous Systems and the Kinetics of First-Order Phase Transitions J¨ urn W. P. Schmelzer(1) and Alexander S. Abyzov(2) (1)
Institute of Physics, University of Rostock, 18051 Rostock, Germany (2)
National Scientific Center, Kharkov Institute of
Physics and Technology, 61108 Kharkov, Ukraine For many years, Gibbs’ work has served as a source of ideas and inspiration for new generations of investigators. . . Its significance and range of applicability continue to widen, and it will for long time to come serve the interests of science and technology. It is certain that, together with the development of new trends, Gibbs’ theory will continue to develop in the future. Anatoli I. Rusanov It does not prove a thing to be right because the majority says it is so. Friedrich Schiller . . . wenig Witz geh¨ ort dazu, so zu schreiben, wie es Mode ist, aber sehr viel, so zu schreiben, wie es Mode werden kann . . . Georg Ch. Lichtenberg
4 2 Thermodynamics and the Kinetics of First-Order Phase Transitions
Abstract In the theoretical interpretation of the kinetics of first-order phase transitions, thermodynamic concepts are widely employed developed long ago by Gibbs and van der Waals. However, the results of such analysis are partly unsatisfactory and internally contradictory. By generalizing Gibbs’ approach, existing deficiencies and internal contradictions of these two well-established theories can be removed and a new generally applicable tool for the interpretation of these processes can be developed. The basic ideas of the generalized Gibbs approach and a variety of consequences for the understanding of the general features of the kinetics of first-order phase transitions are outlined.
2.1
Introduction
Nucleation-growth and spinodal decomposition processes are two basic mechanisms first-order phase transitions – like condensation and boiling, segregation in solid and liquid solutions or crystallisation and melting – may proceed. They determine the kinetics of self-structuring processes of matter from nanoscale up to galactic dimensions with a wide spectrum of applications in both fundamental and applied research (physics, astronomy, chemistry, biology, meteorology, medicine, materials science) and technology. In the interpretation of experimental results on the dynamics of first-order phase transitions starting from metastable (stable with respect to small and unstable with respect to sufficiently large fluctuations exceeding some critical sizes, the so-called critical cluster sizes) initial states, up to now predominantly the classical nucleation theory is employed treating the respective processes in terms of cluster formation and growth [1-9]. In the specification of the cluster properties, thermodynamic methods are intensively employed based in the majority of cases on the thermodynamic description of heterogeneous systems developed by Gibbs [10]. As one additional simplifying assumption it is assumed hereby frequently that the bulk properties of the clusters are widely similar to the properties of the newly evolving macroscopic phases. This or similar assumptions, underlying the classical approach to the description of cluster formation and growth, are supported by the results of Gibbs’ classical theory of heterogeneous systems applied to processes of critical cluster formation. Indeed, following Gibbs thermodynamic treatment one comes to the conclusion that the critical clusters have properties widely similar to the properties of the newly evolving macroscopic phases. Treating clusters of arbitrary sizes as small
vch 4 Okt 2005 10:43
2.1 Introduction
5
particles with bulk and interfacial properties of the macroscopic phase, the process of cluster growth and dissolution is considered then to proceed basically via addition or emission of single units (atoms, molecules) [6-9]. As a second additional thermodynamic assumption (the so-called capillarity approximation), the interfacial specific energy of critical clusters is supposed in a first approximation to be equal to the respective value for an equilibrium coexistence of both phases at planar interfaces. In order to come to an agreement between experimental and theoretical results on nucleation-growth processes, this second assumption often has to be released by introducing a curvature dependence of the surface tension. However, such assumption leads to other internal contradictions in the theory which cannot be resolved remaining inside the concepts of Gibbs’ thermodynamic treatment of cluster properties [11, 12]. This way, Gibbs’ classical treatment of surface phenomena is confronted with serious principal difficulties in application to nucleation. Gibbs employed in his approach a simplified model considering the cluster as a homogeneous body divided from the otherwise homogeneous ambient phase by a sharp interface of zero thickness. The alternative continuum’s concept of a thermodynamic description of heterogeneous systems was developed by van der Waals [13]. It has been applied for the first time to an analysis of nucleation by Cahn and Hilliard [14, 15]. In application to nucleation-growth processes, Cahn and Hilliard came to the conclusion that the bulk state parameters of the critical clusters may deviate considerably from the respective values of the evolving macroscopic phases and from the predictions of Gibbs’ theory. Such deviations occur, in particular, in the vicinity of the classical spinodal curve dividing thermodynamically metastable and thermodynamically unstable initial states of the systems under consideration. These results of the van der Waals’ approach were reconfirmed later-on by more advanced density functional computations [16]. Moreover, Cahn and Hilliard developed also the alternative to the nucleationgrowth model theoretical description of spinodal decomposition. According to the common believe (having again its origin in the classical analysis of Gibbs), the nucleation-growth model works well for the description of phase formation starting from metastable initial states, while thermodynamically unstable states are believed to decay via spinodal decomposition. As one consequence, the problem arises how one kinetic mode of transition (nucleation-growth) goes over into the alternative one (spinodal decomposition) if the state of the ambient phase
vch 4 Okt 2005 10:43
6 2 Thermodynamics and the Kinetics of First-Order Phase Transitions is changed continuously from metastable to unstable states, i.e., how the transition proceeds in the vicinity of the classical spinodal curve. The classical Gibbs’ approach predicts here some kind of singular behavior, which is, however, not confirmed by the Cahn-Hilliard description, statistical-mechanical model analyses [17, 18] and experiment [19]. From a more general point of view, we are confronted here with an internal contradiction in the predictions of two well-established theories which has to be, hopefully, resolved. The resolution of this contradiction is possible in the framework of a generalization of Gibbs’ classical thermodynamic method developed by us in recent years. While Gibbs’ thermodynamic theory is restricted in its applicability to equilibrium states exclusively, the generalized Gibbs’ approach is aimed from the very beginning at a description of thermodynamic non-equilibrium states consisting of clusters of arbitrary sizes and composition in the otherwise homogeneous ambient phase [20-22]. It was demonstrated that, by developing such generalization of Gibbs’ thermodynamic approach, Gibbs’ and van der Waals’ methods of description of critical cluster formation can be reconciled. The generalized Gibbs’ approach was shown to lead for model systems to qualitatively and partly even quantitatively similar results as compared with density functional approaches. In particular, it leads to a significant dependence of the bulk and surface properties of the critical clusters on supersaturation and – in contrast to the classical Gibbs’ approach when the capillarity approximation is employed – to a vanishing of the work of critical cluster formation for initial states in the vicinity of the spinodal curve. The generalized Gibbs’ approach has, however, one additional advantage as compared with existing alternative approaches to the description of cluster formation. Similarly to the classical Gibbs’ approach, the van der Waals’ method as well as modern density functional analyses of the description of heterogeneous systems have the same common limitation: they are restricted in their applicability to thermodynamic equilibrium states exclusively. As a consequence, the mentioned theories can supply us with information on the properties of critical clusters, governing nucleation. However, they cannot supply us with any theoretically founded description of the properties of single clusters or ensembles of clusters (required for a description of their further evolution) being not in equilibrium with the ambient phase. By this reason, in order to describe the evolution of ensembles of clusters in first-order phase transitions, evolving either as the result of nucleation or of spinodal decomposition, additional assumptions have to be made concerning their properties and the evolution of their properties with the changes in cluster size and supersaturation in the system. However, as far as one remains inside
vch 4 Okt 2005 10:43
2.2 Thermodynamics and Nucleation Phenomena 7 mentioned approaches, one has no theoretical tool to check the degree of validity of these assumptions. As will be demonstrated here, the generalized Gibbs’ approach supplies – in addition to its advantages in the specification of the properties of critical clusters – the basis for determining the change of the composition of the clusters in dependence on their sizes and supersaturation allowing us in this way a detailed description not only of nucleation but also of growth and dissolution processes [23, 24]. In addition, a variety of additional general conclusions can be derived. In particular, it turns out that the classical picture of nucleation does not give, in general, a correct description of the initial stages of cluster formation and growth. In contrast, nucleation proceeds via a scenario widely similar to spinodal decomposition. Vice versa, even in unstable initial states, not spinodal decomposition but ridge-crossing nucleation (in a generalized interpretation) may represent the basic mechanism of evolution of the new phase. The generalized Gibbs approach allows one also the understanding of some other typical features of spinodal decomposition. These and further consequences of the generalized Gibbs approach will be discussed below.
2.2
Thermodynamics and Nucleation Phenomena
In his fundamental papers [10], published first in the period of 1875-78, J. W. Gibbs extended classical thermodynamics to the description of heterogeneous systems consisting of several macroscopic phases in a thermodynamic equilibrium and gave first a theoretical interpretation of the physical origin of metastability and instability. As one of the applications, he analysed thermodynamic aspects of nucleation phenomena and the dependence of the properties of critical clusters – aggregates being in unstable equilibrium with the otherwise homogeneous ambient phase (corresponding, in general, to saddle points of the appropriate thermodynamic potential) – on supersaturation. Such critical clusters have to be formed by fluctuations in nucleation processes in order to allow their subsequent deterministic growth to macroscopic sizes. Regardless of existing impressive advances of computer simulation techniques and density functional computations [16-18], the method developed by Gibbs is predominantly employed till now in the theoretical interpretation of experimental data on nucleation phenomena. It is utilized either in order to estimate the so-called work of critical cluster formation, or, in cases when this quantity
vch 4 Okt 2005 10:43
8 2 Thermodynamics and the Kinetics of First-Order Phase Transitions is determined by density functional computations or other methods, to determine the properties of the critical clusters employing Gibbs’ model assumptions. Hereby it is often supposed that the properties of Gibbs’ model clusters give a correct description of the real critical clusters evolving in the systems under consideration. The thermodynamic state parameters – size and composition – of the critical clusters (being of essential significance for the determination of the nucleation rate) are determined in Gibbs’ classical approach via a subset of the well-known thermodynamic equilibrium conditions (equality of temperature and chemical potentials of the different components) identical to those obtained for the description of phase equilibria of macroscopic systems. Employing these relations, the bulk properties of the critical clusters turn out to be widely the same as those of the newly evolving macroscopic phases. However, the above mentioned result of Gibbs’ theory is in contradiction to predictions of molecular dynamics and density functional computations of the respective parameters as demonstrated first by Cahn and Hilliard [14, 15]. In such alternative approaches it is shown that the properties of critical clusters deviate, in general, considerably from the properties of the newly evolving macroscopic phases the deviations being particularly significant for large supersaturations, i.e., states in the vicinity of the classical spinodal curve. This way, the question arises what the origin of such discrepancies is and how they can be resolved eventually. In order to arrive at a solution of this problem, we have to remember first that Gibbs restricted his analysis from the very beginning to ”equilibrium states of heterogeneous substances” (the title of his analysis), exclusively. He never even posed the problem to determine thermodynamic potentials for heterogeneous systems in non-equilibrium states. In application to the analysis of phase equilibria of macroscopic systems, Gibbs’ theory served so well that it is considered frequently as being equivalent to the basic laws of thermodynamics or even as being a consequence of them. Such point of view is not correct as can be traced easily following Gibbs’ derivations. In addition, such interpretation contradicts Gibbs’ own point of view considering his theory merely as one of the possible methods of description of heterogeneous systems but, of course, a good one. He wrote (cf. [25]): Although my results were in a large measure such as had been previously obtained by other methods, yet, as I readily obtained those which were to me before unknown or vaguely known, I was confirmed in the suitableness of the method adopted. Mentioned point of view about the equivalence of Gibbs’ approach to the basic laws of thermodynamics is also in contrast to different attempts to modify or replace Gibbs’ treatment as developed, for example, by Guggenheim, Prigogine,
vch 4 Okt 2005 10:43
2.2 Thermodynamics and Nucleation Phenomena 9 Defay et al. or Hill (cf. also [20-27]). However, these alternative approaches have their own limitations as mentioned partly by the authors themselves or as it turned out in their further discussion in the scientific community. In most cases, these and further alternative approaches (if correct) turned out to be widely equivalent in their consequences to the results of Gibbs’ theory. However, Gibbs’ theory has – in application to the description of cluster formation – one grave limitation which has not been noticed so far. Restricting the analysis to equilibrium states, Gibbs considers exclusively variations of the state of heterogeneous systems proceeding via sequences of equilibrium states. For such quasi-stationary reversible changes of the states of a heterogeneous system, Gibbs’ theory leads to the consequence that the surface tension depends on the state parameters of one of the coexisting phases merely. This limitation is not restrictive with respect to the analysis of macroscopic equilibrium states and quasi-stationary processes proceeding between them. For such cases, the properties of one of the phases are uniquely determined via the equilibrium conditions by the properties of the alternative coexisting phase. However, the situation is very different if Gibbs’ theory is applied to the description of cluster nucleation and growth. Critical clusters, determining the rate of nucleation processes, correspond to a saddle point of the appropriate thermodynamic potential. In order to search for saddle or other singular points of any potential surface, we have to know the values of the potential function first for any possible states of the system. In application to cluster formation and growth, we have to know also the thermodynamic functions of a cluster or an ensemble of clusters not being, in general, in equilibrium with the otherwise homogeneous ambient phase. Only having this information, we can search then for singular points by well-established rules. Since Gibbs restricts his analysis from the very beginning to equilibrium states, his theory does not allow us – strictly speaking – to apply the common methods of search for saddle points. And here we come to the basic limitation of Gibbs’ theory in application to cluster formation and growth processes: In the search for the critical cluster we have to compare not different equilibrium states but different non-equilibrium states of the heterogeneous system under consideration. For the different non-equilibrium states considered, the surface tension has to depend, in general, on the state parameters of both coexisting phases. Gibbs’ classical approach does not allow, in principle, to account for such dependence and has to be generalized to allow us to incorporate this essential new ingredient into the thermodynamic description. An extension of Gibbs’ thermodynamic treatment of heterogeneous systems to include non-equilibrium states along the lines as discussed above was performed
vch 4 Okt 2005 10:43
10
2 Thermodynamics and the Kinetics of First-Order Phase Transitions
0.7 0.6
7 1
3
2
5
P, MPa
xgas
0.5 0.4 0.3 0.2
a
0 4
0 100 80
4
DGc /kBT
Rc, nm
0 20
2
3 1
3
1 1
4 2
0.1
2
b
1
6
5 2
3 21
60 40 20
c 22 23 24 rliq, kmol/m3
25
0 20
3
2
1
d 21
22 23 24 rliq, kmol/m3
25
Fig. 2.1 Composition of the critical cluster xgas (here a bubble of critical size), thermodynamic driving force of nucleation Π, the radius, Rc , of the critical cluster for different definitions of this parameter (full curves refer to the surface of tension while the dashed curves refer to the equimolecular dividing surfaces in Gibbs’ classical approach) and the work of critical cluster formation, ∆Gc , for bubble formation in a binary liquid-gas solution [21] (see text)
recently by one of us in cooperation with Ivan Gutzow (Sofia, Bulgaria), Vladimir G. Baidakov and Grey Sh. Boltachev (both Ekaterinburg, Russia) ([20-22]; cf. also [26, 27]). This generalised Gibbs’ approach employs Gibbs’ model as well. However, Gibbs’ fundamental equation for the superficial or surface quantities is extended (assuming certain well-defined constraints to prevent irreversible flow processes) allowing one to introduce into the description the essential dependence of the surface state parameters (including the surface tension) on the bulk state
vch 4 Okt 2005 10:43
2.2 Thermodynamics and Nucleation Phenomena 11 parameters of both coexisting phases. Then the thermodynamic potentials for the respective non-equilibrium states are formulated. After this task is performed, the equilibrium conditions are derived. Similarly to the Gibbs’ classical theory, the critical cluster corresponds to a saddle point of the characteristic thermodynamic potential (a maximum with respect to variations of the cluster size at fixed intensive state parameters of both cluster and ambient phase and a minimum with respect to variations of the intensive bulk state parameters of the cluster). The equilibrium conditions, derived via the generalized Gibbs’ approach, coincide with Gibbs’ expressions for the limiting case of phase coexistence at planar interfaces; they are, however, of a different form when applied to the determination of the properties of finite size critical clusters. These different equilibrium conditions lead, consequently, also to different results for the work of critical cluster formation as compared to the Gibbs’ classical treatment. In the majority of cases, the generalized Gibbs’ approach leads to lower values of the work of critical cluster formation as compared with the classical treatment utilizing the capillarity approximation. Some examples of the resulting differences between the predictions of the classical and generalised Gibbs’ approaches and their relation to density functional studies are given in Fig. 2.1. In Fig. 2.1, parameters of the critical clusters are shown in dependence on supersaturation (here, as an example, boiling in helium-nitrogen solutions is considered [21]). For the chosen temperature, the density ρliq = 24.7 kmol/m [3] corresponds to the binodal curve (representing the boundary between thermodynamically stable and metastable homogeneous states of the system), while the density ρliq = 22.48 kmol/m [3] refers to the spinodal curve. In Fig. 2.1, x is the molar fraction of helium in the critical bubble, Π is a measure of the thermodynamic driving force of critical bubble formation, Rc is a well-defined measure (surface of tension (full curves) and equimolecular dividing surfaces for both components (dashed curves 4 and 5, correspondingly)) of the size of the critical bubble, ∆Gc is the work of critical bubble formation. The dependencies, obtained via Gibbs’ classical approach employing the capillarity approximation, are given by curves 1, the results obtained via the generalised Gibbs’ approach by curves 2 and the results of density-functional computations are given by curves 3 (for more details see [21]). It is evident that the generalised Gibbs’ approach leads to different values of the work of critical cluster formation as compared to the classical Gibbs’ approach employing the capillarity approximation and – similarly to density functional computations – to vanishing values of the work of critical cluster formation for
vch 4 Okt 2005 10:43
12
2 Thermodynamics and the Kinetics of First-Order Phase Transitions
initial states near the spinodal curve. Note as well that, according to the generalised Gibbs’ approach, the driving force of nucleation is not a monotonously increasing function of supersaturation. Moreover, the radius of the surface of tension in the generalised Gibbs approach behaves similarly as the characteristic size parameters of the critical clusters obtained via density functional computations [14, 15]. It diverges for initial states both near the binodal and spinodal curves. In the intermediate range of moderate supersaturations, where nucleation processes may occur, its size is of the same order of magnitude as the respective parameter obtained via Gibbs’ classical method employing the capillarity approximation. Moreover, in this range of moderate supersaturations this size parameter varies only slightly. It turns out that, for moderate supersaturations where nucleation phenomena can be observed commonly, the size of the critical clusters is widely independent on supersaturation. Qualitatively similar results have been also obtained already for segregation processes in solid or liquid solutions [28] and condensation and boiling in one-component van der Waals’ fluids [29]. The results of above given analyses show that the generalised Gibbs’ approach allows us to describe the parameters of the critical clusters both in one- and multi-component systems in a way which is qualitatively in agreement with density functional computations. As compared to the latter ones, it has the advantage that only the thermodynamic properties of both bulk phases and the dependence of the surface tension on the state parameters of both coexisting phases for planar interfaces have to be known. Methods to determine such dependencies are discussed in Refs. 21 and 22.
2.3
Trajectories of Cluster Evolution
Since – in the classical Gibbs’ approach – the bulk properties of the critical clusters turn out to be widely identical to the properties of the newly evolving macroscopic phases, one can assume then with some sound foundation that suband supercritical clusters have similar properties as well. This assumption is commonly employed so far in the theoretical description of growth and dissolution processes [4, 9, 30]. However, the above performed analysis – based on the generalised Gibbs’ approach – leads to the consequence that clusters of critical sizes have properties which are different, in general, from the properties of the newly evolving macroscopic phases. By this reason, also the properties of sub- and supercritical clusters have to depend, as a rule, both on supersaturation and cluster size. In order to
vch 4 Okt 2005 10:43
n2
DG/DGc
2.3 Trajectories of Cluster Evolution
13
n1
Fig. 2.2 Illustration of the method of determination of the preferred trajectory for cluster growth and dissolution processes. The trajectory is determined by the solution of the deterministic growth equations – representing generally the most probable course of evolution of a stochastic process – starting with initial states slightly above or below the critical point corresponding to the saddle point of the thermodynamic potential surface
develop an appropriate description of the course of the phase transition, one has to establish, consequently, the dependence of the composition of arbitrarily sized clusters on mentioned parameters. In order to solve this task, we proposed recently that the preferred path of cluster evolution is defined by the deterministic equations of cluster growth and dissolution starting with initial states slightly above and below the critical cluster size [23, 24]. In other words, we identify the deterministic trajectory with the most probable trajectory in the stochastic realisations of the respective processes. The behavior of the system in the space of cluster variables resembles then the motion of a body in a viscous fluid in some force field determined by the shape of the thermodynamic potential surface. The method of determination of the most probable evolution path is illustrated in Fig. 2.2. It is applicable regardless of the particular kind of phase transformation considered. In application to segregation in binary solutions, the change of the state of the clusters in dependence on their sizes (both for sub- and supercritical
vch 4 Okt 2005 10:43
14
2 Thermodynamics and the Kinetics of First-Order Phase Transitions
Fig. 2.3 Change of composition, xα , of the cluster in dependence on its size in reduced units, R/Rc (left), and path of evolution in the space of particle numbers in the clusters, n1 and n2 (right), for metastable initial states. The state of the ambient phase is assumed to be unchanged by cluster formation processes and determined by the molar fraction, x, of one of the components of the solution. Here Rc and nc are the radius (referred to the generalised surface of tension) and the total number of particles in the critical cluster. The computations are performed for segregation in a regular solution for different values of the ratio of the diffusion coefficients of both components: (1) D1 /D2 = 10, (2) D1 /D2 = 1, (3) D1 /D2 = 0.1 [24, 33]
cluster sizes) is illustrated in Fig. 2.3 for a metastable initial state and different values of the ratio of the diffusion coefficients, D1 and D2 , of both components (for details see [24]).
vch 4 Okt 2005 10:43
2.4 Nucleation versus Spinodal Decomposition
15
10
Cp [at%P]
8 6 4 2 0 0
2
4
6 8 R, nm
10
12
Fig. 2.4 Size-dependence of the cluster composition (Cp is here the content of phosphorus) for the case of primary crystallisation of Ni(P) particles in a hypoeutectic Ni-P amorphous alloy as obtained via small-angle scattering of polarized neutrons [31]
The change of the composition of the clusters in dependence on their sizes leads to a size-dependence of almost all thermodynamic (in particular, driving force of cluster growth and surface tension) and kinetic (diffusion coefficients and growth rates) parameters determining the dynamics of the phase transition [23, 24]. Some first results of experimental analyses confirming these theoretical predictions are given in Refs. 11, 12, 23 and 31 (see also Fig. 2.4). Taking into account such size dependence, it can be also easily explained, in particular, why thermodynamic and kinetic parameters obtained from nucleation experiments may not be appropriate for the description of growth or dissolution and vice versa (cf. [32]).
2.4
Nucleation versus Spinodal Decomposition
Following the analysis given above, we come to the conclusion that the kinetics of nucleation and growth in solutions exhibits features typical for spinodal decomposition [23, 24]. Indeed, according to the results illustrated in Fig. 2.3,
vch 4 Okt 2005 10:43
16
2 Thermodynamics and the Kinetics of First-Order Phase Transitions
nucleation proceeds as follows (cf. also Fig. 2.5): in a certain part of the ambient
DG/DGc
1.5
1.0
0.5
0
0
0.5
1.0
1.5
Reduced radius, R/Rc
Cluster composition, xa
0.8 0.6
0.4 0.2
0 0
0.5
1.0
1.5
2.0
2.5
Reduced radius, R/Rc
Fig. 2.5 Comparison of the classical model of phase separation in multi-component solutions (top) with the scenario as developed based on the generalised Gibbs’ approach (bottom)
phase with a radius close to the critical one, the composition is changed until the properties of the newly evolving macroscopic phase are nearly reached. Only
vch 4 Okt 2005 10:43
2.4 Nucleation versus Spinodal Decomposition
17
afterwards, the classical picture – change in size of aggregates with nearly constant composition – reflects the situation correctly. The classical model does not supply us, consequently, with a correct description of nucleation. Note that this result is reconfirmed by statistical mechanical analyses of model systems [17, 18] giving thus an additional confirmation of the validity of the generalised Gibbs’ approach.
Cluster composition, xa
1.0
50
a
DGc /kBT
0.8 0.6 0.4
xb 0.2
0.3 x
0.4
20
c
Rc
15 10 xb
xsp
0 0.1
0.2
20
0.3 x
0.4
0
0.5
0.5
-1 Comp. amplification, ta
0.1
30
10 x b
xsp
0.2
5
b
40
0.1
xsp 0.3 x
0.2
0.6
0.4
0.5
d
1
0.4 0.2
2 0
3
0 0.2 0.4 0.6 0.8 1.0 Reduced wave number, k/kc
Fig. 2.6 Composition (a), work of formation (b) and size (c) of critical clusters obtained via the generalised Gibbs’ approach (full curves) for segregation in solutions in a wider range of the initial supersaturation [24]. For the considered system and temperature, the left side value of the binodal curve corresponds to x = 0.086 while the respective branch of the spinodal curve is located at x = 0.226. By dashed curves, size, composition and work of formation of a particular ridge cluster are shown having the same size as the critical cluster (referred to the surface of tension) in Gibbs’ original method employing the capillarity approximation. In Fig. 2.6d, the dependence of the rate of composition amplification on the wave number for x = 0.4 and different values of the ratio of the diffusion coefficients of both components are shown: (1)D1 /D2 = 10, (2)D1 /D2 = 1, (3)D1 /D2 = 0.1)
vch 4 Okt 2005 10:43
18
2 Thermodynamics and the Kinetics of First-Order Phase Transitions
Fig. 2.7 Cluster evolution for unstable initial states (all notations are the same as in Fig. 2.3). (1) (2) Here xmacro and xmacro are the values of the concentration for an equilibrium coexistence of the two different phases of the solution at planar interfaces, i.e., the values at the binodal curve
vch 4 Okt 2005 10:43
2.5 Saddle Point versus Ridge Crossing 19 So far we have restricted the analysis to thermodynamically metastable states located between classical binodal and spinodal curves. The generalised Gibbs’ approach allows us to get also some insight into the kinetics of phase formation processes in solutions starting from thermodynamically unstable initial states. Indeed, in Fig. 2.6 the composition and the size of the critical clusters are shown for a broader range of initial supersaturations including both metastable and unstable initial states. For initial states in the unstable region, the generalised Gibbs approach predicts (full curves on Fig. 2.6a–c): a) values of the composition of the critical cluster equal to the composition of the ambient phase; b) a value of the work of critical cluster formation equal to zero; c) a dependence of the critical cluster size on supersaturation similar to the characteristic sizes of the spatial regions of highest composition amplification [24] as obtained in the classical Cahn-Hilliard theory of spinodal decomposition [14, 15]; and d) a dependence of the rate of composition amplification on the wave number ((1) D1 /D2 = 10, (2) D1 /D2 = 1, (3) D1 /D2 = 0.1) similar to the growth increment as derived in the classical Cahn-Hilliard theory of spinodal decomposition. This way, the generalised Gibbs approach allows us to assign a definite meaning – in terms of critical cluster parameters – to some of the well-known features of spinodal decomposition. In Fig. 2.7, similar dependencies are shown as in Fig. 2.3 but here for the description of the evolution of the new phase starting with unstable initial states. In contrast to the results shown in Fig. 2.3, once the critical cluster size is reached, here the cluster evolution may follow two different trajectories specified by full and dashed curves, respectively (for details see [33]). Summarizing the results, we conclude that nucleation and spinodal decomposition are not qualitatively different but very similar in their nature modes of first-order phase transitions. The only qualitative difference consists in the existence (for nucleation) or absence (for spinodal decomposition) of an activation barrier for the evolution to the new phase.
2.5
Saddle Point versus Ridge Crossing
Another important question is whether the system will always select the thermodynamically favoured evolution path through the saddle-point of the landscape of the thermodynamic potential or pass via the ridge of the potential well (cf.
vch 4 Okt 2005 10:43
20
2 Thermodynamics and the Kinetics of First-Order Phase Transitions
Fig. 2.2). As demonstrated, for example, in Fig. 2.6a–c by full curves, for states in the vicinity of the classical spinodal curve, saddle-point crossing requires the formation of very large in size aggregates. Although being favourable from an energetic point of view, such path of transition is unfavourable by kinetic reasons and alternative trajectories of evolution gain in importance. Indeed, as shown in Fig. 2.6a-c with dashed curves [24], in the vicinity of the spinodal curve both for metastable and unstable initial states ridge crossing is a possible path of evolution to the new phase with relatively low values of the activation energy and characteristic sizes of the ridge clusters comparable in size to the critical clusters as determined via Gibbs’ classical approach employing the capillarity approximation. Consequently, ridge-crossing nucleation (overcoming a finite potential barrier) may be the dominating mechanism of formation of viable units of the newly evolving phase not only for metastable but also for unstable initial states in the vicinity of the spinodal curve. Provided the system follows such ridge crossing channels of evolution to the new phase, no peculiarities have to be expected in the kinetics of the transformation in the vicinity of the spinodal curve as it is exemplified also both by computer simulations of model systems [17, 18] and by direct experimental analyses [19].
2.6
Conclusions
The fascination of a growing science lies in the work of the pioneers at the very borderland of the unknown, but to reach this frontier one must pass over welltravelled roads; of these one of the safest and surest is the broad highway of thermodynamics. These words, expressed about a century ago by Lewis and Randall in their well-known book on chemical thermodynamics, retain their validity till now. Nonetheless, even the highway of thermodynamics is shown here to be in need of improvement when applied to the description processes of self-structuring of matter at nanoscale dimensions. Performing these corrections, the generalised Gibbs’ approach leads, in addition to the already discussed conclusions, to a variety of further new insights into the course of first-order phase transformations. It allows, for example, a new interpretation of the problem of existence or nonexistence of metastable phases in crystallisation of different glass-forming melts – considering them not as metastable but as transient phases of cluster evolution – and the formation of bimodal cluster size distributions for intermediate stages of segregation processes in solutions (cf. [23]). Moreover, a detailed analysis allows us to suggest that the temperature of the critical clusters is, in general, different from the temperature of the ambient phase [20-22].
vch 4 Okt 2005 10:43
2.7 References
21
This way, we believe that the further development of the generalised Gibbs’ approach may serve in future – combining the simplicity of the classical Gibbs’ approach in nucleation with the accuracy of density functional approaches and computer simulation methods and giving, in addition, a safe basis for the description of cluster growth and dissolution – as a quite powerful new and generally applicable tool in order to resolve problems in the comparison of experimental results on processes of self-structuring of matter and theoretical predictions which have not found a satisfactory solution so far.
2.7
References
1. M. Volmer, Kinetik der Phasenbildung (Th. Steinkopff, Dresden, Leipzig, 1939). 2. J. P. Hirth and G. M. Pound, Condensation and Evaporation (Pergamon, London, 1963). 3. V. P. Skripov, Metastable Liquids (WILEY, New York, 1974). 4. A. E. Nielsen, Kinetics of Precipitation (Pergamon, Oxford, 1964). 5. J. W. Christian, The Theory of Transformations in Metals and Alloys (Oxford University Press, Oxford, 1975). 6. I. Gutzow and J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization (Springer, Berlin, 1995). 7. H. A. Stanley, Introduction to Phase Transitions and Critical Phenomena (Clarendon, Oxford, 1971). 8. J. W. P. Schmelzer: Phases, Phase Transitions, and Nucleation. In: A. Hubbard (Ed.), Encyclopedia of Surface and Colloid Science (Marcel Dekker, New York, 2002; pp. 40174029). 9. D. T. Wu: Nucleation Theory. In: Solid State Physics: Advances in Research and Applications, vol. 50; Ehrenreich, H., Spaepen, F. (Eds.) (Academic Press, New York, 1997). 10. J. W. Gibbs, The Collected Works, vol. 1, Thermodynamics (Longmans & Green, New York, 1928). 11. E. D. Zanotto and V. M. Fokin, Phil. Trans. Royal Society London. A 361, 591 (2003). 12. V. M. Fokin, N. S. Yuritsyn, and E. D. Zanotto: Nucleation and Crystallization Kinetics in Silicate Glasses: Theory and Experiment. In: Nucleation Theory and Applications, Schmelzer, J. W. P. (Ed.) (WILEY-VCH, Berlin-Weinheim, 2005, pp. 74-125).
vch 4 Okt 2005 10:43
22
2 Thermodynamics and the Kinetics of First-Order Phase Transitions
13. J. D. van der Waals and Ph. Kohnstamm, Lehrbuch der Thermodynamik (JohannAmbrosius-Barth Verlag, Leipzig und Amsterdam, 1908). 14. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). 15. J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 31, 688 (1959). 16. D. W. Oxtoby, Accounts of Chemical Research 31, 91 (1998). 17. J. D. Gunton, M. San Miguel, and P. S. Sahni: The Dynamics of First-Order Phase Transitions. In: Phase Transitions and Critical Phenomena, vol. 8, Domb, C., Lebowitz, J. L. (Eds.) (Academic Press, London-New York, 1983). 18. K. Binder, Rep. Progr. Phys. 50, 783 (1987). 19. M. Shah, O. Galkin, and P. Vekilov, J. Chem. Phys. 121, 7505 (2004). 20. J. W. P. Schmelzer, Physics and Chemistry of Glasses 45, 116 (2004). 21. J. W. P. Schmelzer, V. G. Baidakov, and G. Sh. Boltachev, J. Chem. Phys. 119, 6166 (2003). 22. J. W. P. Schmelzer, G. Sh. Boltachev, and V. G. Baidakov: Is Gibbs Theory of Heterogeneous Systems Really Perfect?. In: Nucleation Theory and Applications, Schmelzer, J. W. P. (Ed.) (WILEY-VCH, Berlin-Weinheim, 2005, pp. 418-446). 23. J. W. P. Schmelzer, A. R. Gokhman, and V. M. Fokin, J. Colloid Interface Sci. 272, 109 (2004). 24. J. W. P. Schmelzer, A. S. Abyzov, and J. M¨ oller, J. Chem. Phys. 121, 6900 (2004). 25. M. Rukeyser, Willard Gibbs (Doubleday, Doran & Company, New York, 1942). 26. P. G. Debenedetti and H. Reiss, J. Chem. Phys. 108, 5498 (1998). 27. D. Reguera and H. Reiss, J. Chem. Phys. 119, 1533 (2003). 28. V. G. Baidakov, G. Sh. Boltachev, and J. W. P. Schmelzer, J. Colloid Interface Science 231, 312 (2000). 29. J. W. P. Schmelzer and V. G. Baidakov, J. Chem. Phys. 105, 11595 (2001). 30. K. F. Kelton, Solid State Physics 45, 75 (1991). 31. D. Tatchev, A. Hoell, R. Kranold, and S. Armyanov, S., Size Distribution and Composition of Magnetic Precipitates in Amorphous Ni-P Alloys, J. Applied Crystallography, submitted for publication. 32. L. Granasy and P. James, J. Chem. Phys. 113, 9810 (2000). 33. A. S. Abyzov and J. W. P. Schmelzer: Thermodynamic Analysis of the Transition from Metastability to Instability in Phase Separating Solutions, submitted for publication.
vch 4 Okt 2005 10:43
3
Crystal Growth on the Surface and in the Bulk of Na2 O · 2CaO · 3SiO2-Glass Nikolay S. Yuritsyn Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24-2, 199 155 St. Petersburg, Russia Many people believe that thermodynamics is simple, but this is obviously not true. Werner Ebeling
Abstract The growth rate of crystals nucleated on polished surfaces as well as in the volume was measured in a wide temperature range for the glass of composition close to Na2 O·2CaO·3SiO2. It was found that the crystal growth rate on the surface is higher than the one in the bulk and the ratio between the growth rates decreases with increasing temperature. A comparative analysis of surface and volume growth is presented. The crystals on the surface have commonly the shape of circles with a radial ribbed structure. After some heat treatment two types of crystal growth on the surface were observed: first crystals grow as smooth circles (or rings) and then form a ring with ribbed structure. The peculiarities of crystal growth on the glass surface are considered.
3.1
Introduction
In recent years considerable attention has been devoted to the study of crystal nucleation on the glass surface [1-8]. In these analyses it turned out that surface nucleation is mainly determined by heterogeneous nucleation on some, often
24
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
unknown, surface defects. The following two experimental facts give a strong evidence for this statement: i.) the number density of crystals formed on the glass surfaces does not depend on the heat treatment time or rapidly achieves some saturation [8]; ii.) the saturation level can vary by many orders of magnitude depending on the way of surface preparation [8]. The peculiar features of phase formation at or near glass-vapor interfaces may be caused also by the potential effect of elastic strains, arising due to the differences between crystal and glass densities, which can diminish the effective thermodynamic driving force for crystallization [9]. As was shown in Ref. 10 this effect decreases when the critical crystal position approaches the interface. Moreover, generally, the structure of surface layers can differ from that of glass bulk. The mentioned factors can affect both surface nucleation and surface growth. This work presents the results of the comparative study of crystal growth in the bulk and on the glass surface.
3.2
Experiment
The glass was synthesized from analytical grade carbonates of sodium and calcium and anhydrous SiO2 . Melting was performed in a platinum crucible at about 1450 o C in an electric furnace in air for 3 hours. The melt was cast onto a steel plate. The glass composition by analysis (17.2Na2 O·33.0CaO·49.8SiO2 mol%) was close to the stoichiometric one (16.7Na2 O · 33.3CaO · 50.0SiO2 mol%). The glass specimens were prepared in the shape of plates having sizes of about 0.5 × 5 × 5 mm with surfaces polished by CeO2 water suspension. The growth rates of the crystals in the bulk and on the glass surface were determined from the crystal size measurements after a set of heat treatments at the given temperature. The heat treatments were performed in a vertical electric furnace in the temperature range 600-750 o C. Optical microscope Jenaval was used to determine the crystal sizes. The crystals in the bulk of glass had a shape of a cube with slightly rounded corners. The distance, l, between two parallel faces was measured for crystals of maximal size. The crystals on the glass surface had the shape close to a circle. The fixed crystals on the glass surface were chosen to measure the evolution of their radii R. The crystal growth rates on the surface, US , and in the volume,
vch 4 Okt 2005 10:43
3.2 Experiment 25
Fig. 3.1 Crystals on the polished glass surface after different numbers n of 3 min heat treatments at 720 o C: n = 1 (a), 2 (b), 3 (c), 4 (d), and 5 (e)
vch 4 Okt 2005 10:43
26
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
UV , of the glass were determined from the initial linear parts of dependencies R(t) and l(t) as US =
∆R , ∆t
UV =
1 ∆l , 2 ∆t
(3.1)
where ∆R and ∆l are increments of the crystal sizes for the time ∆t. To measure US the following procedure was used. The first short heat treatment was prolonged several times by the given period of time. Each prolongation resulted in the formation of a well-pronounced ring around pre-existing crystals (crystals formed before the last heat treatment). Since the times of the heat treatments are known one can easily plot the size, R, of the given crystal versus time. An example of such crystals is shown in Fig. 3.1.
3.3
Results and Discussion
The growth rate of crystals in the bulk, UV , and on the surface, US , was investigated in the temperature range 600-750 o C located above the glass transition temperature, Tg = 585 o C. This glass reveals homogeneous volume nucleation with the maximum of the steady-state nucleation rate at about 595 o C [11] (Fig. 3.2). At temperatures higher than about 690-700 o C the nucleation rate is negligibly small. Hence the growth of crystals nucleating during melt cooling and glass heating (so-called ”athermal” crystals) mainly occurs in the bulk of glass at these temperatures. The crystal nucleation on the surface is a heterogeneous one since the crystal number density depends only on the quality of polished surfaces and does not vary with heat treatment time. It means that some local surface defects serve as active centers for nucleation. The number of such centers after thorough polishing with cerium oxide amounts to about 100 per mm2 . This value allows measuring crystal sizes R on average up to 100 µm.
3.3.1 Identification of Crystals X-ray diffraction (XRD) analysis was performed with a diffractometer DRON2.0 using CuKα radiation. To obtain XRD-patterns of crystals formed in the bulk, the powder of the fully crystallised glass at 720 o C for 12 h was used. The XRD-patterns of the crystals on the glass surface were obtained directly from the
vch 4 Okt 2005 10:43
3.3 Results and Discussion
100000
Ist
27
10
US
0,1
0,01 1000
U , Pm/min
10000
-3
Ist , mm min
-1
1
UV
1E-3
100
1E-4 600
650
700
750
T , °C
Fig. 3.2 Steady-state nucleation rate (data from Ref. 11) and growth rates of crystals in the bulk and on the polished surface versus temperature for the glass of composition close to stoichiometry Na2 O · 2CaO · 3SiO2
surface fully crystallised at 720 o C for 7 min. It should be noted that in latter case the glass bulk was crystallised only partially. The XRD-analysis shows that in the bulk and on the surface of the glass the Na2 O · 2CaO · 3SiO2 crystals (JCPDS card N22-1455) are formed. According to the phase diagram of the Na2 O − CaO − SiO2 -system [12], between Na2 O · 2CaO · 3SiO2 and Na2 O · CaO · 2SiO2 -compounds there exists a range of solid solutions with a hexagonal unit cell. The lattice parameters are changing remarkably with the Na2 O content [13-15]. Employing an internal standard (germanium) the lattice parameters were determined for the crystals in the bulk (a = 10.476 ± 0.007˚ A, c = 13.157 ± 0.017˚ A) and on the glass surface (a = 10.489 ± 0.007˚ A , c = 13.154 ± 0.017˚ A ). Hence under mentioned above conditions in the bulk and on the surface the crystals were formed with equal parameters of the unit cell, corresponding to the known values for the Na2 O · 2CaO · 3SiO2 -compound [16].
vch 4 Okt 2005 10:43
28
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
3.3.2 Crystal Growth in the Glass Bulk The results of crystal size measurements at different temperatures are shown in Fig. 3.3. At the advanced stage of phase transition the l/2 vs t dependence deviates remarkably from a linear one. This deviation is caused by the difference
2,0
5
a d/2 , Pm
b
4
1,5
3 1,0 2 0,5 0,0
600°C 0
10
20
30
630°C
1
40
0
0
5
10
t,h
15
20
t,h 20
d
c
15
d/2 , Pm
15 10 10 5
5
660°C 0
0
50
690° 100
0
0
20
80
60
80
e d/2 , Pm
40
t , min
t , min
60
60
40
40
20
f
20
750°C
720°C 0
0
50
100
t , min
0
0
10
20
30
t , min
Fig. 3.3 Sizes of the crystals in the bulk of glass versus time of heat treatment at 600 (a), 630 (b), 660 (c), 690 (d), 720 (e), and 750 o C (f)
vch 4 Okt 2005 10:43
3.3 Results and Discussion
29
between crystal and glass compositions and was widely discussed in Refs. 13, 14 and 17. The compositional shifts of critical nuclei relative to the composition of fully crystallized glass (or parent glass) were observed in both stoichiometric Na2 O·2CaO·3SiO2 glass and glasses belonging to the solid solution region between Na2 O · 2CaO · 3SiO2 and Na2 O · CaO · 2SiO2 [13, 14, 17]. In particular it was shown that the formation of stoichiometric crystals in glass of Na2 O · 2CaO · 3SiO2 -composition occurs via nucleation of the solid solution whose composition is enriched with sodium. Then crystal composition continuously approaches the stoichiometric one (Na2 O · 2CaO · 3SiO2 ) and arrives that at full crystallization. The exhaustion of sodium in the glassy matrix during crystallization leads to an inhibition of both nucleation and crystal growth. Tab. 3.1 Crystal growth rates on the surface (US ) and in the bulk (UV ) of the 17.2Na2 O· 33.0CaO · 49.8SiO2 glass (in mol%) at different temperatures
T , oC US , µm/min UV , µm/min US / UV
600 0.153 0.00137 111.7
630 0.474 0.00576 82.3
660 0.390 0.0859 4.54
690 1.12 0.327 3.42
720 3.49 1.61 2.17
750 7.40 4.01 1.84
The crystal growth rates UV presented by Table 3.1 and Fig. 3.2 were estimated from linear approximations of the l/2 vs t dependencies (Fig. 3.3) at the early stage of the phase transformation e.g. when the distance between crystals considerable exceeds their sizes.
3.3.3 Crystal Growth on the Surface Fig. 3.1 presents photos of crystals on the surface of specimens subjected by a number of heat treatments at 720 o C for 3 min. After the first heat treatment, Fig. 3.1a, rounded crystals are formed. Then, after each following heat treatment a new ring with a radial ribbed structure is formed around already existing crystals. Hence the number of rings is equal to (n − 1) where n is the number of heat treatments. However, sometimes the crystals with the number of rings less than (n − 1) are observed (Fig. 3.1). It means that these crystals nucleated on some new active centers that accidentally appeared between heat treatments. Crystal size versus time is shown in Fig. 3.4 for several temperatures. Different kinds of points refer to different crystals. White and black points relate to different
vch 4 Okt 2005 10:43
30
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
40
a
R , Pm
10
600°C
b 30 20
5
630°C
10 0
0
50
0
100
0
20
R , Pm
25
c
20
40
60
t , min
t , min
d
20 15
10
10
660°C 0
0 0
10
20
30
40
690°C
5
50
0
5
10
15
20
t , min
t , min 80
e R , Pm
60 40
720°C
20 0
0
5
10
15
20
t , min
Fig. 3.4 Sizes of the crystals on the polished glass surface versus time at temperatures 600 (a), 630 (b), 660 (c), 690 (d), 720 o C (e). The lines demonstrate the growth of separate crystals or the mean growth for a few crystals (marked by different points) with close sizes. White and black points relate to different sides of the specimen
vch 4 Okt 2005 10:43
3.3 Results and Discussion
31
surfaces of a glass plate. The growth rates had near values for different crystals at 630-720 o C. Only at a temperature 600 o C the growth rates measured for crystals on different sides of a glass specimen differed noticeably being equal to 0.19 and 0.11 µm/min, respectively. To estimate the growth rates the mean values were used (Table 3.1, Fig. 3.2).
T, °C 700
750
650
600
2
ln(U , Pm/min)
0
surface -2
-4
volume
-6
-8 0,95
1,00
1,05 3
1/T x 10 , K
1,10
1,15
-1
Fig. 3.5 Logarithm of growth rates for the bulk (white circles) and surface (black circles) versus inverse temperature
A remarkable induction period was observed only at 600 o C. The straight lines R(t) at 600 o C intersect the time-axis at t ∼ 38 min (Fig. 3.4a). It seems likely that this time is determined mainly by the time-lag for nucleation. At 750 o C the growth rate of the crystals is so high that after a second heat treatment all crystals impinged themselves. That is why at this temperature the growth rate was estimated from the size of the crystals measured after only single heat treatment for 3 min (Table 3.1).
vch 4 Okt 2005 10:43
32
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
3.3.4 Comparison of Crystal Growth in the Bulk and on Polished Surfaces The temperature dependencies in Arrhenius coordinates of the crystal growth rate in the glass bulk and on the polished surface are presented in Fig. 3.5. At low temperatures the growth rate of the crystals on the glass surface, US , significantly exceeds the one in the volume, UV . For example the ratio (US /UV ) at the temperature 600 o C is equal to 107 (Table 3.1). With increasing temperature this ratio decreases and at the temperature 750 o C the rates have near values (US /UV is here equal to 1.8). The ln U versus (1/T ) plots on Fig. 3.5 are linear in the temperature interval 600720 o C for the bulk growth and in the range 600-750 o C for the surface growth. Using Eq. (3.2) H = −R
d(ln U ) , d(1/T )
(3.2)
where R is the gas constant, the activation enthalpies HS and HV for crystal growth on the glass surface and in the bulk were determined as 185 ± 25 and 440±30 kJ/mol, respectively. It is naturally to suppose that at high temperatures (above ∼ 750o C) the growth rates UV and US will have equal or near values. 3.3.4.1 Estimation of the Effective Diffusion Coefficient from Crystal Growth Rates To analyze the crystal growth rates, U , we will employ the following equation for interface kinetic limited growth [18] ∆µ D ef , (3.3) 1 − exp − U =f do kB T where Def is the self-diffusion coefficient of the ”structure” units of size do in the melt, ∆µ is the difference of chemical potentials per ”structure” unit in melt and crystal, kB is the Boltzmann constant, and f is a dimensionless parameter reflecting the growth mechanism. In the case of normal growth f = 1 holds while for screw dislocation growth f can be written as f=
do ∆G , 4πσVm
vch 4 Okt 2005 10:43
(3.4)
3.3 Results and Discussion
Na Ca
-12
33
2s 2v 1s 1v
2
lg(D, m /s)
-14
-16
-18
400
500
600
700
800
900
1000
T, K Fig. 3.6 Temperature dependences of diffusion coefficients for sodium and calcium ions [21] and effective diffusion coefficients for structure units in glass bulk (curves 1v, 2v) and on glass surface (curves 1s, 2s) calculated from the crystal growth rate data for the cases of normal (curves 1v, 1s) and screw dislocations (curves 2v, 2s) growth mechanisms
where ∆G is the thermodynamic driving force per mole, Vm is the molar volume, and σ is the specific energy of crystal/melt interface. Using the Turnbull/Scapski equation (Eq. (3.5)) [17] σ=α
∆Hm 2/3 1/3 Vm NA
,
(3.5)
where ∆Hm is melting enthalpy per mole, NA is Avagadro’s number, α is an empirical dimensionless coefficient equal to about 0.5, and Hoffman’s approximation for the thermodynamic driving force [19] ∆G = ∆Hm (1 − Tr ) Tr ,
Tr ≡
T , Tm
(3.6)
vch 4 Okt 2005 10:43
34
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
where Tm is the melting temperature, one can rewrite Eq. (3.4) as
f=
(1 − Tr ) Tr . 2π
(3.7)
In our case the crystal compositions differ from the one of the parent glass [13, 14, 17]. Nevertheless we used as a first approximation Eq. (3.3) since the experimental data (R vs t) refer to the initial stage of the phase transformation before impingement of diffusion fields around growing crystals and are well extrapolated by straight lines. The crystal growth rates, estimated from the linear approximations of R(t) dependencies, were fitted to Eq. (3.3) and the effective diffusion coefficients vs temperature were calculated (see Fig. 3.6). Hereby the experimental values for the thermodynamic driving force [20] and d0 = (2M/NA )1/3 = 7.5 · 10−10 m (M is molar volume of the crystal) were used. This procedure was performed for normal and screw dislocation mechanisms. Since experimental values of ∆G(T ) are close to Hoffman’s approximation Eq. (3.7) was employed in the case of screw dislocation mechanism. As was expected the calculations in the framework of the normal growth mechanism (f = 1) lead to values of the diffusion coefficient lower than that under the assumption of screw dislocation one (f < 1) (see Fig. 3.6, curves 1 and 2). Moreover the real thermodynamic driving force, ∆G, could be lower than the one for the stoichiometric compound Na2 O · 2CaO · 3SiO2 , since according to Refs. 13 and 14 at first stages the crystals, being solid solutions, are enriched by sodium relative to the stoichiometry Na2 O · 2CaO · 3SiO2 . A decrease in the thermodynamic driving force has to increase the Def value calculated from Eq. (3.3) at given U . Hence curves 1 and 2 (Fig. 3.6) should be considered as lower estimates of D ef .
3.3.4.2 Diffusion in the Bulk of the Glass For comparison, the experimental diffusion coefficients for Na and Ca are presented in Fig. 3.6 [21]. In spite of the assumptions made above one can conclude that the effective diffusion coefficients determining crystal growth in the bulk of the glass are closer to the diffusion coefficient for Ca than the one for Na (see
vch 4 Okt 2005 10:43
3.3 Results and Discussion
35
Fig. 3.6, curves 1v and 2v). To estimate the expected effective diffusion coefficients, Eq. (3.8) proposed in Ref. 22 can be used 1 D ef = 2 , νi xi Di
(3.8)
i
where Di is the partial diffusion coefficient of different components in the system, xi and νi are the molar fractions of each component in parent phase and evolving crystal respectively (in our case νi ≈ xi ). Unfortunately we do not know the data about diffusion of Si and O in glasses with compositions close to that of the glass under study. Nevertheless employing the diffusion coefficients for Si and O lower than that for Ca by 2-3 orders of magnitude we can roughly estimate Def as about 10−17 m2 /s. This value does not strongly differ from that calculated from growth rates (Fig. 3.6, curves 1v, 2v). 3.3.4.3 Volume and Surface Diffusion Comparing DVef and DSef it should be remembered that their estimation was performed under the assumption of the same thermodynamic driving force for crystal growth on the surface and in the bulk of the glass. This assumption is quite reasonable since surface and bulk crystals have the same structure and composition. Also the potential effect of elastic stresses due to the difference between glass and crystal densities which could diminish the thermodynamic driving force for crystallization on the surface to a lesser degree than in the bulk of glass does not exceed about 4% for the glass under study. However, since in Eq. (3.3) ∆µ/(kB T ) 1 one can write to a first approximation U ∼ ∆µ. Hence the variation of ∆µ by 4% cannot lead to a strong difference between volume and surface growth rates observed in experiment (see Table 3.1). Thus the difference in crystal growth rates in the bulk and on the surface of the glass and its temperature dependencies is mainly determined by the difference in diffusion coefficients. Recall that the surface was prepared via polishing with water suspension of cerium oxide. Since the chemical resistance of the investigated glass is not high, resulting from polishment changes in the surface layer structure or composition have naturally to be assumed. In connection with this assumption it should be mentioned that in the case of glasses of cordierite composition, having very high chemical resistance, crystal growth rates in the bulk and on the surface have the
vch 4 Okt 2005 10:43
36
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
Fig. 3.7 Crystals on the polished glass surface after different numbers n of heat treatment: 720 o C, 4 min, n=1 (a,b); 720 o C, 2 min, n=3 (c,d); 690 o C, 4 min, n=3 (e); 690 o C, 4 min, n=4 (f)
vch 4 Okt 2005 10:43
3.3 Results and Discussion
37
50
690°C 40
R , Pm
30
20
10
0 0
5
10
15
20
t , min Fig. 3.8 Sizes of the crystals on the glass surface versus time at 690 o C. The data for three different crystals are marked by three species of points
same values. The potential defects in the surface layer of Na2 O·2CaO·3SiO2 -glass that could be caused by the polishing procedure may lead to an increase of the diffusion coefficient. With increasing temperature the difference between surface and bulk diffusion decreases as it is the case e.g. with the viscosity of ”dry” and water containing glasses [23].
3.3.5
Some Peculiarities of Crystal Growth on the Glass Surface
3.3.5.1 Crystal Growth SG2 Let us denote crystal growth on the glass surface described in above sections as growth SG1 (surface growth 1). It is characterised by the radial ribbed structure. However in some experiments another type of crystal growth was detected on the polished glass surface. In Figs. 3.7a and b crystals formed after heat treatment at 720 o C for 4 min are shown. They have a smooth rounded central part surrounded by petals of radially growing crystals. The crystals after a few heat treatments at 720 and 690 o C are
vch 4 Okt 2005 10:43
38
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
40
630°C
c
R , Pm
30
20
USG2= 7.02 Pm/min surface
10
USG1= US = 0.474 Pm/min surface
0 0
100
200
300
t,s
volume UV= 0.00576 Pm/min
Fig. 3.9 Crystal growth (SG2) on glass surface at T = 630 o C: (a,b) sketch and photo of circles formed by condensed small droplets around crystals that have been grown in the bulk (see text); (c) size of circle radii versus time for different crystals (different points; for comparison the crystal growth SG1 on the surface (dashed line) and in the bulk (dotted line) is shown)
vch 4 Okt 2005 10:43
3.3 Results and Discussion
39
shown in Figs. 3.7c and d and Figs. 3.7 e and f, respectively. An appearance of double rings consisting of the inner smooth ring and the outer ribbed one was observed after the second and third heat treatments at 720 o C (Figs. 3.7 c and d) and after the third and fourth heat treatments at 690 o C (Figs. 3.7 e and f). Thus, in the course of these heat treatments first crystal growth occurred as a smooth growth SG2 (surface growth 2) during the time tSG2 , then it was transformed into the usual ribbed growth SG1 that was in progress till finishing of the heat treatment (the duration of SG1 was tSG1 ). The change in size of three crystals at 690 o C is given in Fig. 3.8. Only SG1 was observed during the first two heat treatments (Figs. 3.7e,f). But after third and fourth heat treatments SG2 and SG1 were distinctly observed. As a result, the rate of crystal growth increased from 1.5 to 4.4 µm/min. In all experiments in which the SG2 and SG1 were observed the width of the ribbed ring always was about equal to the one measured in experiments in which only SG1 was observed (after a similar time of heat treatment). During a heat treatment the ribbed ring always had sufficient time for its appearance. Thus, it is possible to conclude that the growth time of the smooth ring, tSG2 , is much less than the growth time of the ribbed ring, tSG1 , and the growth rate of the smooth ring USG2 is significantly greater than the one of the ribbed ring, USG1 ≡ US . The following experiment confirms the high rate of SG2. Preliminarily the glass sample was heat treated at 630 o C for 18 min for crystal nucleation and growth in the bulk. Then the surfaces of the sample were grinded and polished. After polishing on the glass surface sections of crystals grown in the bulk could be observed. Then a set of short heat treatments (1-1.25 min) has been carried out at 630 o C. The time of each heat treatment was comparable with the time of sample heating until the temperature 630 o C after its deposition in a furnace. Really the sample was heat treated in some interval of temperatures with some mean temperature Tmean < 630 o C. After such short heat treatments we did not observe SG1. But it was possible to observe with microscope circles around the crystals formed by condensed small droplets (Figs. 3.9a,b). The number of circles was equal to the number of short heat treatments minus one (the circles of condensed droplets were formed only between rings of SG2). We propose that in this case the circles characterize SG2. The growth of circle radii around three different crystals is shown in Fig. 3.9c. These data give the opportunity to estimate the rate of SG2, USG2 . The value USG2 = 7.02 µm/min is considerably higher than USG1 ≡ US = 0.474 µm/min for SG1 at 630 o C (USG2 /USG1 = 14.8). It is worth to remember that it is
vch 4 Okt 2005 10:43
40
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
150
720°C
R , Pm
100
50
b 0 0
10
20
30
40
t , min Fig. 3.10 Crystal growth on polished glass surface at 720 o C: (a) photo of crystals after 14 heat treatments for 2-5 min; (b) sizes of four crystals versus time
vch 4 Okt 2005 10:43
3.3 Results and Discussion
41
more correct to relate the value USG2 to Tmean < 630 o C. Thus the real value of USG2 /USG1 at 630 o C (or at Tmean ) can exceed the one indicated above. The results of the experiments give the opportunity to suppose that SG2 observed on Fig. 3.7 occurs during the heating of the sample in some interval T1 − T2 of relatively low temperatures less than the temperature of heat treatment, Tht . At some temperature T2 < Tht , SG2 stops and SG1 starts. In this process the time of SG2 tSG2 is much less than the time of SG1 tSG1 because the mean rate of SG2 in the interval T1 − T2 is considerably higher than the rate of SG1 at temperature Tht . The high rate of SG2 explains why the width of its rings in Figs. 3.7c-f (or the radius of the inner circle in Figs. 3.7a,b) have the same order as the width of SG1 rings. Thus one can conclude that the smooth and ribbed rings are formed via different growth mechanisms. To clarify the nature of SG2 and the reasons for its transition to SG1 further investigations are needed. 3.3.5.2 Influence of Glass Composition Change on Crystal Growth Because the rate of crystal growth on the surface considerably exceeds the rate in the bulk, the polished glass surface fully crystallises long before full crystallisation of the glass bulk. But sometimes, if on the thoroughly polished surface there are only a small number of crystallisation centers, it is possible to observe some areas of the glass surface unoccupied by crystals after relatively extended heat treatments. In Fig. 3.10a crystals are shown grown on the glass surface subjected to heat treatments at 720 o C for 2-5 min 14 times. As can be seen the crystals do not impinge themselves in some parts of the glass surface. The change of the radii R for four partly unimpinged crystals is shown in Fig. 3.10b. After five 2 min heat treatments a delay of crystal growth was observed, and at the subsequent heat treatments the crystal growth practically stops. The decrease of the growth rate can be explained by depletion of Na2 O in the surface layer of the glass after crystallization of a considerable part of this layer. After the first five heat treatments a sufficiently large number of crystals is formed in the glass bulk. The glass around the crystals has a reduced content of Na2 O. The crystals growing in the bulk close to the glass surface can diminish the Na2 O content in the surface layer of the glass. Thus, the same cause as in the bulk can lead to a decrease of the growth rate on the glass surface. The effect of depletion of the Na2 O content in the glass on the crystal growth rate was observed also in a following special experiment. First a small piece of
vch 4 Okt 2005 10:43
42
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
Fig. 3.11 Crystals on the polished glass surface after two heat treatments (690 o C, 18 min and 720 o C, 3 min (see text)): (a,b) photos of two different regions on glass surface; (c) scheme of crystal layers (cross-hatched regions) grown on the glass surface after heat treatment at 720 o C, 3 min
glass was heat treated at 690 o C for 18 min to form some number of crystals in the bulk. After polishing the sections of crystals grown in the bulk could be observed on the glass surface. This specimen was again heat treated at 720 o C for 3 min. After that a new crystal layer was observed around above-mentioned crystal sections (Figs. 3.11a,b). The thickness of the layer is not constant: the less the distance between crystals, the less the thickness of the new crystal layer (see also the scheme in Fig. 3.11c). This effect can be explained by different degrees of depletion of Na2 O in glass around crystals: the less the distance between crystals the less the Na2 O content in this region.
3.4
Main Results and Conclusions
1. The growth of crystals was investigated on the polished glass surface and in the bulk for the glass close to the stoichiometric Na2 O · 2CaO · 3SiO2 composition in the range 600-750 o C. 2. The crystals in the bulk had a shape of a cube with rounded corners. The crystals on the glass surface had a shape close to a circle with a radial ribbed structure.
vch 4 Okt 2005 10:43
3.5 References
43
3. The crystal growth rate on the glass surface is higher than the one in the bulk. The ratio between the growth rates on the surface and in the bulk decreases with increasing temperature. 4. The effective diffusion coefficient on the glass surface estimated from crystal growth rates at relatively low temperatures is higher than that in the bulk of the glass. This difference decreases with increasing temperature. 5. After some heat treatment the smooth and ribbed growth is observed reflecting two different growth mechanisms. The smooth growth occurs with higher rate than ribbed growth. 6. The decrease of the crystal growth rate on the glass surface with time was observed in some experiments. This effect is similar to the one in glass bulk, which occurs as the result of composition difference between crystal and glass matrix.
Acknowledgment The author is grateful to V. M. Fokin for his participation in the discussion of the results.
3.5
References
1. Surface Nucleation. Collection of Selected Papers, Ed. W. Pannhorst (International Commission on Glass, ICG, Chaleroi, Belgium, 2000). 2. J. Deubener, R. Br¨ uckner, and H. Hessnkemper, Glastech. Ber. 65, 256 (1992). 3. N. S. Yuritsyn, V. M. Fokin, A. M. Kalinina, and V. N. Filipovich, Glass Physics and Chemistry 20, 116, 125 (1994). 4. R. M¨ uller, S. Reinsch, and W. Pannhorst, Glastech. Ber. Glass Sci. Technol. 69, 12 (1996). 5. V. N. Filipovich, V. M. Fokin, N. S. Yuritsyn, and A. M. Kalinina, Thermochimica Acta 280/281, 205 (1996). 6. V. M. Fokin, N. S. Yuritsyn, V. N. Filipovich, and A. M. Kalinina, J. Non-Crystalline Solids 219, 37 (1997). 7. V. N. Filipovich, A. M. Kalinina, V. M. Fokin, N. S. Yuritsyn, and G. A. Sycheva, Glass Physics and Chemistry 25, 246 (1999). 8. R. M¨ uller, E. D. Zanotto, and V. M. Fokin, J. Non-Crystalline Solids 274, 208 (2000).
vch 4 Okt 2005 10:43
44
3 Crystal Growth of Na2 O · 2CaO · 3SiO2 -Glass
9. J. W. P. Schmelzer, R. Pascova, J. M¨ oller, and I. Gutzow, J. Non-Crystalline Solids 162, 26 (1993). 10. J. W. P. Schmelzer, J. M¨ oller, I. Gutzow, R. Pascova, R. M¨ uller, and W. Pannhorst, J. Non-Crystalline Solids 183, 215 (1995). 11. O. V. Potapov, V. M. Fokin, V. L. Ugolkov, L. Ya. Suslova, and V. N. Filipovich, Glass Physics and Chemistry 26, 27 (2000). 12. G. K. Moir and F. P. Glasser, Phys. Chem. Glasses 15, 6 (1974). 13. V. M. Fokin, O. V. Potapov, C. R. Chinaglia, and E. D. Zanotto, J. Non-Crystalline Solids 258, 180 (1999). 14. V. M. Fokin, O. V. Potapov, E. D. Zanotto, F. M. Spiandorello, V. L. Ugolkov, and B. Z. Pevzner, J. Non-Crystalline Solids 331, 240 (2003). 15. E. N. Soboleva, N. S. Yuritsyn, and V. L. Ugolkov, Glass Physics and Chemistry 30, 481 (2004). 16. N. A. Toropov, V. P. Barzakovskiy, V. V. Lapin, N. N. Kurtseva, and A. I. Boykova, Diagrammy Sostoyaniya Silikatnykh Sistem (Leningrad, Nauka, vol. 3, 1972 (in Russian)). 17. V. M. Fokin, N. S. Yuritsyn, and E. E. Zanotto: E.D. Nucleation and Crystallization Kinetics in Silicate Glasses: Theory and Experiment. In: Nucleation Theory and Applications, J. W. P. Schmelzer (Ed.) (WILEY-VCH, Berlin-Weinheim, 2005, pgs. 74-125). 18. I. Gutzow and J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Cystallization (Springer, Berlin, 1995). 19. K. F. Kelton, Solid State Physics 45, 75 (1991). 20. V. I. Babushkin, G. M. Matveyev, and O. P. Mchedlov-Petrosyan, Thermodynamics of Silicates (Springer, Berlin, 1985). 21. G. H. Frischat and H. J. Oel, Glastech. Ber. 39, 50, 524 (1966). 22. J. W. P. Schmelzer, R. M¨ uller, J. M¨ oller, and I. Gutzow, J. Non-Crystalline Solids 315, 144 (2003). 23. O. V. Potapov, V. M. Fokin, and V. N. Filipovich, J. Non-Crystalline Solids 247, 74 (1999).
vch 4 Okt 2005 10:43
4
Metastable Extensions of Phase Equilibrium Curves in the Lennard-Jones System Vladimir G. Baidakov and Sergey P. Protsenko Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, GSP-828, Amundsen Str. 106, Russia Commotion here has reached its apogee. William Shakespeare
Abstract The method of molecular dynamics has been used to calculate the pressure, the internal energy and the isochoric heat capacity in a system of 2048 Lennard-Jones particles in the intervals of reduced temperatures and and densities 0.1 - 2.0 and 0.001 - 1.15, respectively. Local equations of state describing stable and metastable states have been built for the liquid, the gas and the crystal phase. Spinodals for a stretced liquid, crystal and supersaturated vapor have been calculated. The equations of state have been used to determine the lines of liquid-gas, liquid-crystal and crystal-gas phase equilibrium, which are in good agreement with te results of independent calculations of two-phase systems of 4096 particles. The metastable extensions of the equilibrium coexistence curves have been examined. It has been shown that, distinct in comparison with the saturation line, the curves of melting and sublimation terminate at finite temperatures on the spinodals of liquid and crystal, respectively. The points, where the metastable extensions of the lines of liquid-crystal and crystal-gas phase equilibrium come in contact with the liquid and the crystal spinodals are singular points of the thermodynamic surface of states.
46
4.1
4 Metastable Extensions of Phase Equilibrium Curves
Introduction
First-order phase transitions presuppose the existence of metastable states [1]. In a simple one-component system the number of phases co-existing in equilibrium does not exceed three. Each of the phases - crystal, liquid, and gas - may exist in a metastable state. The decay of the metastable state is initiated by the formation of a viable nucleus of a new phase. Since a spontaneous formation of a sufficiently great mass of a new phase has an infinitesimal probability, a metastable system, in the absence of external actions, is capable of a long existence. Two phases (denoted by the subscripts α and β) metastable with respect to a third phase (specified by γ) may co-exist for a finite time in equilibrium with each other [2, 3]. In this case, the conventional equilibrium conditions are fulfilled, equality of chemical potentials at equality of temperature and pressure, i.e. µα (p, T ) = µβ (p, T ) .
(4.1)
Hence it follows that the lines of liquid-gas (LV ), liquid-crystal (LS) and crystalgas (SV ) phase transitions may be extended beyond the triple point into the region of metastable states (see Fig. 4.1). The slopes of these lines are determined by the Clapeyron-Clausius equation, and for normally melting substances (∆vLS = vL − vS > 0) in the vicinity if the triple point we have dp dp dp > > . dTLS dTSV dTLV
(4.2)
Modern techniques of shock wave experiments make it possible to investigate substance properties at negative pressures up to -(15-20) GPa and more [4]. The metastable extension of the ice melting line was observed experimentally up to -24 MPa [5]. Pioneering attempts to measure the compressibility of solids [6] and liquids [7] at tensile stress have been made, but till now the methods suggested have a limited range of applicability. In Ref. 8, results of measurements of the surface tension for a supercooled liquidsupersaturated vapor interface are measured well away from the triple point. Less evident is the possibility of existence of the metastable crystal-gas equilibrium, when both phases are metastable with respect to a liquid. In connection with complete wetting of the crystal with its own melt [3], the prolongation of the
vch 4 Okt 2005 10:43
4.1 Introduction 47
p
C SV
D LV
t
M
T
SL E F
Fig. 4.1 Phase diagram of a simple substance: LV, SL, SV are saturation, melting and sublimation lines and their metastable extensions; CF is the spinodal of a stretched liquid, CD is the spinodal of the supersaturated vapor, BE is the spinodal of a superheated crystal; C: critical point, t: triple point
crystal-gas equilibrium line for a considerable distance beyond the triple point will be a complicated task. One of the most efficient approaches to determine the properties of metastable systems is the computer experiment (Monte-Carlo and molecular dynamics methods). The small dimensions of computer model systems make it possible to realize high supersaturations of homogeneous phases and obtain information on the properties of a model sustance in the regions of state variables that for the time being are not accessible for full-scale real experiments. This property allows us a simultaneous solution of problems on the determination of both the limits
vch 4 Okt 2005 10:43
48
4 Metastable Extensions of Phase Equilibrium Curves
of essential instability of a homogeneous metastable phase (the location of the spinodal) and the parameters of phase equilibrium of metastable phases. In the present paper the problems mentioned above are solved by the method of molecular dynamics applied to a Lennard-Jones system, which represents the properties of all aggregate states of a simple substance of argon type in a qualitatively correct way.
4.2
Models and Results
The interaction of particles was described by the Lennard-Jones potential with a cut-off at r = rc ⎧ σ 12 σ 6 ⎪ ⎪ , 4ε − ⎨ r r φ(r) = ⎪ ⎪ ⎩ 0,
r ≤ rc , (4.3) r > rc .
The particle mass is m = 66.336 · 10−27 kg, the potential parameters are σ = 0.3405 nm, ε/kB = 119.8 K, where kB is the Boltzmann constant. Hereafter m, σ, and ε are used as parameters of reduction of thermodynamic quantities. Reduced (dimensionless) quantities are marked with the sign ∗ . Two models were used in the present analyses denoted as model A and model B. Model A consists of a cubic cell with N = 2048 Lennard-Jones particles. This model was used to investigate one-phase states. In model B, the basic cell has the form of a parallelepiped L∗x × L∗y × L∗z = 13.56 × 13.56 × 58 and contains N = 4096 particles. The model was used for investigating phase equilibria. For this purpose a film-like layer of a dense phase was formed at the center of the cell. On both sides, it was surrounded by a phase of lower density, therefore, the interfaces were parallel to the plane XOY . With respect to all calculated state variables, the film thickness ensured the isolation of a bulk phase in it. For calculating the distribution of density ρ(z) and the normal component of the pressure tensor pN (z) the cell was divided into 5800 layers parallel to the plane XOY . As in model A, surface effects at the boundaries of the basic cell were eliminated by periodic boundary conditions. The intermolecular potential was cut-off at a distance rc∗ = rc /σ = 6.78 at densities ρ∗ = ρσ 3 ≤ 0.82. The cut-off radius of the potential was taken equal to
vch 4 Okt 2005 10:43
4.2 Models and Results
49
half the cell length at ρ∗ > 0.82. The integration step of the classical equations of particle motion was 0.01 ps for a liquid and 0.005 ps for a crystal. Thermodynamic properties (pressure, internal energy, isochoric heat capacity) were calculated in the (N, V, E)-ensemble along 11 isotherms from the interval T ∗ = kB T /ε = 0.1 − 2.0. Calculations were always started from a stable region (model A). In gas and liquid simulations, a random packing was chosen as the initial configuration of particles in the cell, in crystal simulations it was a face-centered cubic lattice.
(a)
20
15
- 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 -10 -11
25
(b)
20
15
10
p*
p*
10 5
5 0
0 -5
-5 0.8
0.9
1.0 ρ*
1.1
1.2
0.7
0.8
0.9
1.0
1.1
ρ*
Fig. 4.2 Isotherms of crystal (a) and liquid (b): 1 - T ∗ = 0.1, 2 - 0.2, 3 - 0.3, 4 - 0.4, 5 - 0.55, 6 -0.7, 7 - 0.85, 8 - 1.0, 9 - 1.15, 10 - 1.5, 11 - 2.0. Dashed line shows line of liquid-crystal phase equilibrium
In calculations of the crystalline phase properties, the initial state on an isotherm is always the state with the highest density. Calculations were made up to levels of stretching at which the loss of crystal order of particles in the model took place. As this point is approached, the thermodynamic stability of the crystalline phase decreases (Fig. 4.2a). The dependence p(ρ) is in agreement with the concept of
vch 4 Okt 2005 10:43
50
4 Metastable Extensions of Phase Equilibrium Curves
the existence of points on isotherms at which the derivative (∂p/∂ρ)T equals zero. The line of such points, the spinodal, determines the boundary of thermodynamic stability for a homogeneous phase. It follows from Fig. 4.2a that at high temperatures the disruption of a crystalline structure is observed well away from the spinodal. With decreasing temperature the value of the derivative (∂p/∂ρ)T at the boundary of spontaneous disorder decreases, and at T ∗ ≤ 0.4 the disruption of a crystalline structure occurs at stretches close to spinodal ones. This result is in agreement with the general concepts of the thermo-activated nucleation theory. At temperatures T ∗ = 0.7−2.0, penetration into the metastable region of a liquid was realized by a successive isothermal compression of particles in the cell. The final configuration of particles at every given value of density was taken as the initial one in calculating the states with a higher density. A balanced irregular packing of particles at T ∗ = 0.7 and ρ∗ = 0.85 was taken as the initial particle configuration when thermodynamic properties of a liquid were calculated in a region of negative pressures. All the states on the isotherms T ∗ = 0.4 and 0.55 have been obtained by an isochoric cooling of this configuration of particles with its subsequent stretching and compression. The characteristic time in which the system reaches equilibrium in a stable region is ∼ 0.5 ns, in a metastable region it is prolonged up to 1 ns. The time of averaging for the parameters computed varied from 5 ns to 10 ns. The results of calculations of the pressure in the liquid phase are presented in Fig. 4.2b. The last points on the isotherms on the highdensity side determine the maximum supercompression (supercooling) of a liquid in the model. The extremal points on the isotherms T ∗ = 0.4, 0.55, 0.7 of the low-density side show the maximum stretching preceding the liquid boiling-up. As it follows from Fig. 4.2b, an increase in the density (pressure) is accompanied by an increase in the isothermal elasticity of an irregular structure. Thus, to the point of maximum supercooling the liquid phase does not tend to decrease thermodynamic stability against infinitesimal long-wave perturbations of density. This result agrees with the statement that there is no spinodal in a supercooled one-component liquid [9]. This fact is also corroborated by the type of density dependences of the internal energy, u, and the isochoric heat capacity, cV . Fig. 4.3 presents isotherms of internal energy for a liquid (dark dots) and crystal (light dots). At high temperatures the internal energy of the liquid decreases with increasing density, passes through the minimum and increases then, again. The points of the minima of the isotherms u(ρ) at T ∗ > 1.15 are located in the region
vch 4 Okt 2005 10:43
4.2 Models and Results
51
0
-2
-4
u* - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11
-6
-8
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
ρ*
Fig. 4.3 Internal energy of the liquid (dark dots) and the crystal (light dots) phase along isotherms: 1 - T ∗ = 0.1, 2 - 0.2, 3 - 0.3, 4 - 0.4, 5 - 0.55, 6 -0.7, 7 - 0.85, 8 - 1.0, 9 - 1.15, 10 - 1.5, 11 - 2.0. Dashed lines show lines of liquid-crystal phase equilibrium
of stable states, and at lower temperatures are shifted into the metastable region. Isotherms of the internal energy of the crystalline phase also have a clearly defined minimum, which in the whole temperature range investigated approximately coincides with the isochore ρ∗ = 1.1. A penetration into the metastable region of the crystalline phase (T ∗ ≤ 2.0) is accompanied by an increase in the internal energy. The thermodynamic relation ρ
2
∂u ∂ρ
=p−T T
∂p ∂T
(4.4) ρ
vch 4 Okt 2005 10:43
52
4 Metastable Extensions of Phase Equilibrium Curves
correlates the thermal and the caloric equations of state. The quantity in the left-hand side of Eq. (4.4) is, by definition, the internal pressure pi taken with the opposite sign, i.e. [2] pi = −ρ
2
∂u ∂ρ
.
(4.5)
T
The character of the dependence of pi on thermodynamic state variables reflects the change in the relation between attractive, pi > 0, and repulsive, pi < 0, forces in averaging over the positions of all particles. The fact that in approaching the melting line the liquid internal pressure at high temperatures becomes negative reflects the fact that a liquid-crystal phase transition, as distinct from a liquid-gas transition, is connected with the predominance of repulsive over attractive forces. In the region of liquid-gas phase transition, particular attention was given to the vicinity of the critical point, where calculations were made along lines close to the isotherms T ∗ = 1.2; 1.25; 1.3; 1.35. The approximation of the data obtained by polynomial equation yielded Tc∗ = 1.331, ρ∗c = 0.3111, p∗c = 0.1371. The penetration of a superheated (stretched) liquid into the metastable region was realized by a successive increase of the volume of the base cell. The results of calculating the pressure in the region adjacent to the critical point are shown in Fig. 4.4a. A certain scattering of points in the metastable region is connected with the absence of strict isothermality in the model. The properties of the gas phase were calculated with the same integration step as for a liquid. The lowest density value at which the calculations were made was ρ∗ = 0.001. For such rarefied systems the acquisition of reliable data on thermodynamic properties required averaging over no less than 106 steps. As in the case of the liquid phase, an isothermal penetration into the metastable region is connected with a decrease in the elasticity of the gas phase (Fig. 4.4b). Such a behavior of isotherms points to the presence of a boundary of essential instability for the liquid and the vapor phase. The density dependence of the internal energy along isotherms in the region of a liquid-gas phase transition is given in Fig. 4.5. For the liquid and the vapor phase, the derivative (∂u/∂ρ)T is less than zero. At the critical point, we find the result u∗c = −0.2727. As already mentioned earlier, the lines of liquid-gas, liquid-crystal and crystalgas phase equilibria were calculated in the framework of a two-phase model [10]. For this purpose the more dense phase was generated in the form of a film at
vch 4 Okt 2005 10:43
4.2 Models and Results
0.4
0.20 (a)
0.2
53
(b)
C
0.15
C 0.0
p*
p*
-0.2
-0.4
-0.6
0.10
- 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11
0.4
0.05
0.6 ρ*
0.8
0.00 0.0
0.1
ρ*
0.2
0.3
Fig. 4.4 Isotherms of liquid (a) and gas (b): 1 - T ∗ = 0.55, 2 - 0.7, 3 - 0.85, 4 - 1.0, 5 - 1.15, 6 - 1.2, 7 - 1.25, 8 - 1.3, 9 - 1.35, 10 - 1.5, 11 - 2.0. Dashed lines show lines of liquid-gas phase equilibrium
the center of the base cell surrounded on both sides by a phase with a lower density. After the balancing of a two-phase system the density of the phases and the phase-equilibrium pressure were calculated. Calculations always began in the regions removed from the triple point. Then by a successive decrease of the temperature in the case of liquid-gas, liquid-crystal equilibrium and its increase at crystal-gas equilibrium the system was brought to the temperature corresponding to the triple point. The criteria of attainment of the triple point were equality of pressures in all the three phases at a certain isolated temperature (the temperature of the triple point Tt∗ ), and equality of the liquid and the gas densities, respectively, at the points of intersection of the saturation-melting
vch 4 Okt 2005 10:43
54
4 Metastable Extensions of Phase Equilibrium Curves
2
0
u* -2
-4
- 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12
-6 0.0
0.4
0.8
1.2
ρ∗
Fig. 4.5 Internal energy of the liquid and the gas phase along isotherms: 1 - T ∗ = 0.4, 2 - 0.55, 3 - 0.7, 4 - 0.85, 5 - 1.0, 6 - 1.15, 7 - 1.2, 8 - 1.25, 9 - 1.3, 10 - 1.35, 11 - 1.5, 12 - 2.0. Dashed line shows line of liquid-gas phase equilibrium
and the melting-sublimation lines. The following values were obtained for the thermodynamic parameters of the triple point Tt∗ = 0.692 ,
p∗t = 0.0012 ,
ρ∗t,l = 0.847 ,
ρ∗t,cr = 0.962 .
(4.6)
Two projections of the phase diagram in (T ∗ , ρ∗ ) and (p∗ , T ∗ )-coordinates are presented in Figs. 4.6a and b. A metastable extension of the saturation line has been obtained in simulating liquid-gas phase equilibrium (see Fig. 4.6). Attempts to obtain such extensions
vch 4 Okt 2005 10:43
4.3 Limits of Thermodynamic Stability and Phase-Equilibrium 55
1.2 (a)
(b)
0.4
1.0
T
p*
ρ*t,l
0.8
*
0.2
Tt*
C ρ*t,cr
0.6
0.0
0.4 0.6
0.7
0.8
0.9
1.0
ρ*
1.1
0.4
T t*
0.8
1.2
T*
Fig. 4.6 Lines of liquid-gas (), liquid-crystal (◦) and crystal-gas (∇) phase equilibrium in the region of the triple point at the planes (T ∗ , ρ∗ ) (a) and (p∗ , T ∗ ) (b)
for melting and sublimation curves have failed. This failure is connected with the specific character of the model under investigation. The model is so constructed that the crystal lattice has no possibility to change the number of particles in planes parallel to the directions of OX- and OY of the basic cell. A lattice deformation at the stage of attaining a given temperature gives rise to tensile (compressing) stress, which relieve phase metastability and prevent such a twophase system from advancing beyond the triple point.
4.3
Limits of Thermodynamic Stability and Phase-Equilibrium
The (p, ρ, T )-data of the liquid, gas and the crystalline phase of the Lennard-Jones system have been approximated by local equations of state of the type p∗ =
n m
aij ρ∗j T ∗i .
(4.7)
j=0 i=0
vch 4 Okt 2005 10:43
56
4 Metastable Extensions of Phase Equilibrium Curves
The coefficients of Eq. (4.7) and also the maximum values of the exponents n and m were determined by the method of regression analysis. The result of approximation of the obtained (p, ρ, T )-data by Eq. (4.7) are shown in Figs. 4.2 and 4.4. The spinodal for the liquid, the gas and the crystal phase have been determined from the condition (∂p/∂ρ)T = 0 and Eq. (4.7). The data obtained are presented in Fig. 4.7a, b, and c. The spinodal of the stretched liquid begins at the critical point and ends on the zero isotherms at p∗sp,l (0) −5.5 and ρ∗sp,l (0) 0.85. The value of the density ρ∗sp,l (0) is close to its value at the triple point. The spinodals of a stretched liquid and the supersaturated vapor have in (p∗ , T ∗ )-coordinates opposite signs of curvature and converge at the critical point. At T → 0 the pressure p∗sp,V and the density ρ∗sp,V on the spinodal of the supersaturated vapor take zero values. ∗ The crystal spinodal intersects the isobar p∗ = 0 at Tsp,cr = 0.895 and extends into the region of negative pressures. In the (p, T )-plane, there is a point at which the condition (∂p/∂ρ)T = 0 is fulfilled for both condensed phases (see Fig. 4.7a). This state corresponds to the point of intersection of the boundaries of thermodynamic stability for a liquid and a crystal. Naturally there is no such intersection on other projections of the spinodals. At T → 0 the value p∗sp,cr (0) −6.0. Thus, at T = 0 the spinodal of the Lennard-Jones crystal in (p, T )-coordinates is found somewhat lower than the spinodal of the Lennard-Jones liquid, though these differences are not too cardinal. The values of the liquid and the crystal densities at T = 0 are also closely analogous in spinodal states (see Fig. 4.7b, c).
The method of molecular dynamics gives no way of calculating directly the chemical potential and the free energy. However, in the framework of this method it is quite easy to determine the values of the internal energy, which is related to the thermal equation of state via Eq. (4.4). Going over in Eq. (4.4) to a new variable β = 1/T and integrating Eqs. (4.4) and (4.7) with respect to density we obtain values of u with an accuracy up to a certain function of temperature u0 (T ) determined from the results of molecular-dynamics calculations of the internal energy. Using the obtained function u(ρ, T ) and integrating the Gibbs-Helmholtz equation
∂(βf ) ∂β
=u,
(4.8)
ρ
we find the free energy of the liquid, the gas and the crystalline phase with an accuracy of f0 =const. The value of f0 was determined then from data on the
vch 4 Okt 2005 10:43
4.3 Limits of Thermodynamic Stability and Phase-Equilibrium 57
1
C
(b)
(a)
p*sp,v
t
0
0
C
p*sp,v
p*sp,cr
t
-1
p*sp,l
-1
p*sp,l
p*sp,cr -2
-2
p*
p* -3
-3
-4
-4
-5 -5
-6 0.0
0.4
T*
0.8
0.0
1.2
C
(c)
0.3
ρ*
0.6
0.9
T*sp,cr
1.2
T*sp,v
T*sp,l
0.8
t
T*
0.4
0.0 0.0
0.3
ρ*
0.6
0.9
Fig. 4.7 Binodals (dashed lines) and spinodals (solid lines) at the surfaces (p∗ , T ∗ ) (a), (p∗ , ρ∗ ) (b), (T ∗ , ρ∗ ) (c). C: critical point, t: triple point
vch 4 Okt 2005 10:43
58
4 Metastable Extensions of Phase Equilibrium Curves
20
(a)
(b)
K
0.015
p*m
15
p*sub
p*m p*sp,cr
10
0.010
p* p
p*sp,cr
*
5 0.005
C
t 0
p*sp,l
K t 0.000 0.0
-5 0.0
0.4
0.8
1.2
1.6
0.2
0.4
0.6
T
2.0
T*
0.8
1.0
*
(c)
0.015
p*m p*s 0.010
p*
0.005
t 0.000 0.0
0.2
0.4
0.6
0.8
1.0
T*
Fig. 4.8 Lines of liquid-crystal (a), crystal-gas (b) and liquid-gas (c) phase equilibria at the surface (p∗ , T ∗ ). c: critical point, t: triple point, K: end points of melting and sublimation curves
vch 4 Okt 2005 10:43
4.4 Terminal Critical Points
59
parameters of the triple point of the Lennard-Jones system, which were calculated in a separate computer experiment. The results of calculating the lines of liquid-crystal, crystal-gas, and liquid-gas phase equilibria in (p, T )-coordinates are presented in Fig. 4.7 and 4.8. All the lines of phase equilibria have metastable extensions beyond the triple point. As it follows from Fig. 4.8a, the melting line in the region of negative pressures is butted up against the spinodal of a stretched liquid. The metastable extension of the sublimation curve ends on the spinodal of a superheated crystal (Fig. 4.8b). As distinct from the melting and the sublimation curves, the line of liquid-vapor phase equilibrium extends to the absolute zero of temperature (Fig. 4.8c).
4.4
Terminal Critical Points of the Melting Line and the Sublimation Curve
The spinodal is a separate line at the thermodynamic surface of states. Isothermal compressibility and isobaric heat capacity increase on it in an unlimited way, therefore the point of contact of the spinodal with the phase-equilibrium line (point K, Fig. 4.8a, b) should also be separated. As at the point K the derivative (∂ 2 p/∂ρ2 )T = 0 is not equal to zero and, evidently, the adiabatic compressibility and isochoric heat capacity are finite, the singularity isochoric heat capacity of this point is not so strong as that of the critical one. Considering the pressure as a function of temperature and specific volume v = 1/ρ, for the spinodal and the phase-equilibrium line we have dp = dTsp dp = dTb
∂p ∂T
∂p ∂T
+ v
+ v
∂p ∂v
∂p ∂v
T
T
dv , dTsp dv . dTb
(4.9)
(4.10)
For any point of the spinodal, including the critical one, the second term at the right-hand side of Eq. (4.9) is equal to zero, therefore, we arrive at [1] dp = dTsp
∂p ∂T
.
(4.11)
v
vch 4 Okt 2005 10:43
60
4 Metastable Extensions of Phase Equilibrium Curves
A relation similar to Eq. (4.11) may also be written for the binodal at the point of its contact with the spinodal, if only the second term in Eq. (4.10) is equal to zero. This is the case at the liquid-vapor critical point, where ∂p dp dp = = . (4.12) dTsp c ∂Tv c dTb c Eq. (4.12) implies that the critical point is the point of tangency of the spinodal, binodal and critical isochore. As evident from Fig. 4.8a and b, the point of contact of the melting and the sublimation curve with the liquid and the crystal spinodal, respectively, does not possess such a property, i.e. ∂p dv = 0 . (4.13) ∂v T dTb k Since at the point of contact of the spinodal and of the binodal (∂p/∂v)T = 0, the inequality Eq. (4.13) and the conditions (dp/dT )b,k > (dp/dT )sp,k or (dp/dT )b,k < (dp/dT )sp,k will be true if (dT /dv)m,k = 0 and (d2 T /dv 2 )m,k = 0. For the melting curve of the Lennard-Jones system we get ∗ = 0.5286 , Tm,k
p∗m,k = −1.7128 ,
ρ∗ml ,k = 0.7374 ,
(4.14)
ρ∗mcr ,k = 0.9423 . For the derivatives, we obtain [11]
dp∗ dT ∗
= 4.748 , sp,k
dp∗ dT ∗
= 8.480 .
(4.15)
m,k
For the sublimation curve, we have ∗ = 0.8874 , Tsub,k
p∗sub,k = 0.01624 ,
ρ∗subcr ,k = 0.8469 ,
(4.16)
ρ∗subv ,k = 0.02125 . The derivatives read ∗ dp = 8.1771 , dT ∗ sp,k
vch 4 Okt 2005 10:43
dp∗ dT ∗
= 0.1661 . sub,k
(4.17)
4.4 Terminal Critical Points
-1.2
61
-1.2 (a)
(b)
B
B'
B
C -1.4
C -1.6
K -1.6
p*
K'
p* 0.940
0.944
0.948
K -1.8
D -2.0
D -2.0
0.50
0.52
0.54
0.56
0.72
0.58
0.74
0.76
0.78
0.80
ρ*
T* 0.60
C
(c)
B'
B
T*
0.55
K K' 0.940 0.944 0.948
D 0.50 0.72
0.74
0.76 ρ*
0.78
0.80
Fig. 4.9 Vicinity of the point of contact of the melting curve (KB, K B ) and the spinodal of a stretched liquid (DC) in coordinates (p∗ , T ∗ ) (a), (p∗ , ρ∗ ) (b), (T ∗ , ρ∗ ) (c)
vch 4 Okt 2005 10:43
62
4 Metastable Extensions of Phase Equilibrium Curves
0.017
0.017
(a)
C
(b)
C
0.018 0.019 0.020 0.021
K
K 0.016
K'
0.016
p*
p*
0.015
0.015
D
D
B
B' B
0.014
0.87
0.014 0.88
0.84
0.89
0.86
ρ*
T*
(c)
0.88
C 0.018 0.019 0.020 0.021
0.89
K
K'
T*
0.88
B' D 0.87 0.84
B 0.86
ρ*
0.88
Fig. 4.10 Vicinity of the point of contact of the sublimation curve (KB, K B ) and the spinodal of a superheated crystal (DC) in coordinates (p∗ , T ∗ ) (a), (p∗ , ρ∗ ) (b), (T ∗ , ρ∗ ) (c)
vch 4 Okt 2005 10:43
4.5 Conclusions 63 In this case the point K in the planes (T, v) and (p, v) is the point of minimum for the liquid branch of the line of crystal-liquid phase equilibrium and maximum for the crystal branch of the line of gas-crystal phase equilibrium (Figs. 4.9 and 4.10). Thus, the points of cessation of liquid-crystal and crystal-gas phase equilibria are certain singular points at the thermodynamic surface of states of a one-component system. At these points for one of the branches the lines of phase equilibrium come in contact with the spinodal. At the point of contact dp dT =0, =0 (4.18) dvb k dvb k for the liquid branch of the melting curve and the crystal branch of the sublimation curve. For the other branch the mentioned derivatives are finite (see Figs. 4.9 and 4.10).
4.5
Conclusions
The computer experiment performed has shown that in a simple one-component system each of the phases has a spinodal only on one side: for a liquid and crystal the spinodal is achieved by heating or stretching, for vapor, by cooling or compression. The curves of liquid-gas, liquid-crystal and crystal-gas phase equilibria have analytic extensions beyond the triple point in the regions of metastable states. As distinct from the line of liquid-gas phase equilibrium, which can be determined down to the temperature of absolute zero, the lines of melting and sublimation have points of cessation of phase equilibrium. It is observed at the intersection of the melting line in the region of negative pressures with the spinodal of a stretched liquid and the sublimation curve with the spinodal of a superheated crystal. The terminal points on melting and sublimation curves are singular points of the thermodynamic surface of states. However, the singularity is weaker than that at the liquid-vapor critical point.
Acknowledgment The work has been supported by the program of the Presidium of the Russian Academy of Science ”Thermal Physics and Mechanics of Intensive Energy Actions”, the project of the Russian Foundation for Basic Research No. 05-02-16251, and State contract No. 10104-71/DEMMPC-06/067-350/200605-053.
vch 4 Okt 2005 10:43
64
4.6
4 Metastable Extensions of Phase Equilibrium Curves
References
1. V. P. Skripov, Metastable Liquids (Wiley, New York, 1974). 2. V. P. Skripov and M. Z. Faizullin, Solid-Liquid-Gas Phase Transitions and Thermodynamic Similarity (Fizmatlit, Moscow, 2003 (in Russian); WILEY-VCH, Berlin-Weinheim, 2006). 3. V. P. Skripov. In: Non-Equilibrium Phase Transitions and Thermal Properties of Matter (Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia, 1996, pgs. 3 ff). 4. T. Antoun, L. Seaman, D. R. Curran et al., Spall Fracture (Springer, New York, 2003). 5. S. J. Henderson and R. J. Speedy, J. Phys. Chem. 91, 13069 (1978). 6. G. S. Bezrouchko, G. I. Kanel, and S. V. Razorenov, High Temperature Physics 10, 1226 (1974), (in Russian). 7. A. D. Alvarenga, M. Grimsdith, and R. J. Bodnar, J. Chem. Phys. 98, 8392 (1993). 8. V. G. Baidakov and I. J. Sulla, Zhurnal Fiz. Khim. 59, 955 (1985), (in Russian). 9. V. P. Skripov and V. G. Baidakov, High Temperature Physics 10, 1226 (1974). 10. V. G. Baidakov, G. G. Chernykh, and S. P. Protsenko, Zhurnal Fiz. Khim. 74, 1382 (2000), (in Russian). 11. V. G. Baidakov and S. P. Protsenko, Phys. Rev. Lett. 95, 015701 (2005).
vch 4 Okt 2005 10:43
5
Dynamical Clustering in Chains of Atoms with Exponential Repulsion Alexander P. Chetverikov(1,2), Werner Ebeling(1,3) and Manuel G. Velarde(1) (1)
Instituto Pluridisciplinar, Universidad Complutense, Paseo Juan XXIII, 1, 28040 Madrid, Spain (2) Physical Faculty, Chernychevsky State University, 410012 Saratov, Russia (3)
Institut f¨ ur Physik, Humboldt Universit¨ at,
Newtonstr. 15, 12489 Berlin, Germany Personally I’m always ready to learn, although I do not always like being taught. Sir Winston Churchill
Abstract We investigate the dynamical clustering in chains of atoms with Morse-type interactions at higher densities, where the exponential repulsion dominates and the structure is latticelike. First we study several mechanisms to generate and stabilize soliton-like dynamical clusters. Although, generally, clusters are unstable, yet, there exist dynamical clusters with finite lifetime, which are due to local compressions running along the chain. We show that these dynamical, metastable clusters may give rise to significant physical effects. In order to study the effects of dynamical clusters on electrical transport we assume that each atom may generate a free electron which is able to move on the lattice. Their motion is described in a classical approximation. The dynamical clusters (localized compressions)
66
5 Dynamical Clustering in Chains of Atoms with Exponential Repulsion
running along the chain may interact with the electron system and influence their motion creating some extra electronic current.
5.1
Introduction
For the case of one dimension (1D) several problems of statistical physics were solved exactly [1, 2]. In several recent papers we developed the statistical thermodynamics of chains with Morse-type interactions [3, 4, 5]. In particular we investigated the clustering problem in low density Morse systems. A cluster was defined as a local density peak, which is relatively stable with respect to collisions. Here we concentrate on higher densities where the exponential repulsion dominates and the configuration is lattice-like. The usual equilibrium clusters are unstable due to the strong interactions. However as we will show, there exist dynamic clusters, representing running local compressions, which are soliton-like. We will investigate here the generation and the properties of these dynamic clusters moving along the lattice. Further we discuss the influence of dynamic clustering on electrical transport. In recent communications [6, 7] the present authors have already discussed, albeit in a sketchy way, the possibility of soliton-mediated electric conductance phenomena in a lattice with Toda interactions. Here we will study also the influence of dynamic clustering on electron dynamics.
5.2
Models of Interaction with Exponential Repulsion
The Morse potential was introduced in 1929 by P. Morse in a paper with the title ”Diatomic molecules according to wave mechanics” [8]. In this fundamental work Morse treated the problem of atomic interactions at small distances and derived a simple expression for the quantum interactions between atoms. The potential depends on two parameters B, D and reads in its general form
U M (r) = D (exp(−B(r − σ)) − 1)2 − 1 .
(5.1)
The potential has a minimum at r = σ and may be considered as a good alternative to the well-known model of Lennard-Jones U L−J (r) =
vch 4 Okt 2005 10:43
B A − . r 12 r 6
(5.2)
5.2 Models of Interaction with Exponential Repulsion 67 The long range part of the Morse potential is less realistic than the 1/r 6 term in the Lennard-Jones potential, however the exponential repulsion term in the Morse potential is well founded on quantum-mechanical calculations. The predominant cause of exponential repulsion between atoms is the wave functions overlapping of the valence electrons and is created mostly in the region close to the axis between the atoms [9]. The repulsive part of the Morse potential is identical to the exponential potential studied by Toda [10] U T (r) =
a exp(−b(r − σ)) . b
(5.3)
This potential, as well as a modification with an additional (unphysical) linear term, was treated in great detail by Toda [10]. The characteristic frequency connected with the Toda potential is ω02 =
ab , m
(5.4)
where m is the mass of the particles [10]. A useful combination between the Toda- and the Lennard-Jones potentials is the Buckingham (exp6) potential
σ 6 D B 6 exp(−b(r − σ)) − 16 (5.5) U (r) = 10 r that describes well realistic cases [11]. Since simulations with an r −6 -tail may give rise to numerical difficulties (due to the long range which often is avoided by setting a cut-off at some finite distance) we work here with the Morse potential which has an exponentially decaying attracting tail. In order to be consistent with the notation for the exponential potential, we introduce here a (generalized) Morse potential in the following form b a M exp(−b(r − σ)) − 2α exp − (r − σ) . (5.6) U (r) = b 2 In the region of higher densities the first term dominates and the second one, which is proportinal to a parameter α, will give only a small correction. The potential has a minimum at r = rmin = σ −
2 ln α . b
(5.7)
vch 4 Okt 2005 10:43
68
5 Dynamical Clustering in Chains of Atoms with Exponential Repulsion
The potential crosses zero at the distance rc = σ −
2 ln(2α) . b
(5.8)
For r < rc , the repulsive part grows exponentially with the stiffness b. In the simplest case α = 1 we recover the standard Morse potential in the form of Eq. (5.1) with B = 2b and D = a/b. In this case we get rmin = σ and the characteristic frequency of oscillations around the minimum is given by ω12 =
ab . 2m
(5.9)
The exponential repulsive part of the interaction forces models the Pauli repulsion between the overlapping valence shells of the atoms. The Morse potential may be considered as a quite realistic model for atomic interactions for both repulsive as well as attractive interactions. The evolution of anharmonic lattices with Morse interactions, in short Morse chains, has been studied [3, 4, 5] including particle clustering, thermodynamical and kinetic transitions. Most of the mentioned work was devoted to chains with relative low density, where the attractive part of the Morse interactions dominates the dynamics. Here we will focus attention on the opposite case of relatively high density. We shall discuss the excitations and their possible influence on electrical transport. In particular we will study the coupling between dynamical clusters and electrical charges in ionized chains.
5.3
Nonlinear Dynamics of Morse Chains
First we will study the dynamics of a 1D Morse-type atomic lattice consisting of N atoms with periodic boundary conditions. We will disregard any ionization phenomena. Let us assume that the mean distance between the Morse particles is r0 = L/N where L and N are, respectively, the length and the number of particles in the chain. We will study now the forces acting on a particle. Take a Morse particle placed midway between two other nearest neighbor Morse particles separated by the average distance 2r0 . If the displacement from the center is denoted by x the effective potential felt by the central particle is (see Fig. 5.1) 2a b b M exp(b(σ − r0 )) cosh(bx) − 2α exp (σ − r0 ) cosh x . (5.10) Uef f = b 2 2
vch 4 Okt 2005 10:43
5.3 Nonlinear Dynamics of Morse Chains 69
9 8
U(x).(b2/m w02)
7 6 5 4 3 2 -1
-0.5
0 x/r0 = elongation
0.5
1
Fig. 5.1 Morse lattice: Effective potential acting on a given particle placed between two other Morse particles at distance 2r0 . We compare with the case of a pure exponential potential α = 0, corresponding to the density r0 = σ; the two curves for the exponential case corresponding to the stiffness bσ = 3 or bσ = 4 respectively touch the abscissa. The two other curves correspond to a Morse chain with α = 1 and to a two times higher density of particles r0 = σ/2, the stiffness is bσ = 3 or bσ = 4 respectively.
In the following we shall omit the superscript M. We see that an exponential potential with the density r0 = σ leads to quite similar effective interactions as the proper Morse potential with the double density r0 = σ/2. Clearly the inner particle is bound to experience quasilinear oscillations around the minimum for small excitations or large elongations depending on the nonlinear terms. In the presence of random forces and also external forces the dynamics of particles with mass m in the chain is described by the Langevin equations (k = 1, 2, . . . , N ) d xk = vk , dt
√ d ∂U 1 vk + γ0 vk + = Fk (rk , vk )) + 2D ξk (t) , dt m∂xk m
(5.11)
governing √ the stochastic motion of the k-th particle on the ring. The stochastic forces 2D ξk (t) model a surrounding heat bath (Gaussian white noise).
vch 4 Okt 2005 10:43
70
5 Dynamical Clustering in Chains of Atoms with Exponential Repulsion
The potential energy stored in the ring reads
U=
N
Uk (rk ) .
(5.12)
k=1
The term γ0 describes the standard friction frequency acting on the atoms in the chain from the side of the surrounding heat bath. The validity of an Einstein relation is assumed [3] D = kB T γ0 /m .
(5.13)
Here T is the temperature of the heat bath. Note that there exist several other temperature concepts [5]. The force Fk acting on the chain of particles may include external driving as well as interactions with host particles (like the electrons) imbedded into the chain. First we studied the basic excitations when Fk = 0, γ0 = 0, and D = 0. We consider a linear chain of N = 10 masses m located on a ring; this is equivalent to a chain with periodic boundary conditions. The particles are described by coordinates xj (t) and velocities vj (t), j = 1, . . . , N , i.e. xk+N = xk + L .
(5.14)
First we assumed that the mean density of the particles r0 = N/L is near to the equilibrium distance rmin . Then we may linearize the potential, hence approximating the potential Eq. (5.6) by linear springs. We introduce the deviations from it rj = xj+1 − xj − rmin (relative mutual displacements) and find in the case of small amplitudes Ui (rj ) =
m 2 2 ω r , 2 1 j
(5.15)
corresponding to a harmonic pair interaction potential. For N masses connected by linear springs without external noise and friction we get the following linear system of dynamical equations for the displacements from equilibrium positions uj d2 uj + ω12 (uj+1 + uj−1 − 2uj ) = 0 . dt2
vch 4 Okt 2005 10:43
(5.16)
5.3 Nonlinear Dynamics of Morse Chains 71 The basic solution of this system reads (n)
uj (t) = A cos(ωn t − jkn σ) .
(5.17)
As well known there exist N different excitations corresponding to different wave lengths and the corresponding wave numbers −
N N 0 the point v = 0 becomes unstable but there are now two additional zeros at √ v = ±v0 = v1 δ − 1 .
(5.33)
These two velocities are the new attractors of the free deterministic motion if δ > 0. In recent work [6, 7] it was shown that electrons may be coupled to the driven solitons and form rather stable dynamic bound states with the solitons (”solectrons”). This effect will be studied again in the next section.
5.5
Dynamics of Electrons Coupled to Running Local Compressions
5.5.1
Semiclassical Model of Electron Dynamics
In order to study the possible influence of dynamical clusters of the type described above on electric conduction we will assume now that the atoms can be ionized
vch 4 Okt 2005 10:43
78
5 Dynamical Clustering in Chains of Atoms with Exponential Repulsion
emitting one free electron to a kind of ”band” and leaving a negative ion. In other words, in the modified model we imbedded N electrons between the N ionic masses on the chain. To describe the dynamics of the electrons we stay on a classical level similar as the early conductance theories of Drude, Lorentz and Debye. We mention quantum-meachanical approaches to related problems by Davydov, Hennig and others [14, 15, 16]. For simplicity we start here with Langevin equations for N electrons (mass me , charge −e) and N ions (mass mi , charge +e) moving on a lattice of length L = N σ with me mi and periodic boundary conditions. Take the N electrons located at the positions yj moving in the nonuniform, and, in general, time-dependent electric field generated by the positive chain particles located at xk . The electron-electron interaction, which results from Coulomb repulsion, Heisenberg uncertainty, and Pauli’s exclusion principle, is modelled here in a rather crude way. We take into account that at small distances the effective potential is linear before it approaches at larger distances the classical Coulomb interaction [7] Uee (r) = Uee (0) −
e2 r + O(r 2 ) , λ2 (5.34)
Uee (r) = 0
if
r>
Uee
(0)λ2 e2
,
where λ= √ mkB T
(5.35)
is the de Broglie thermal wave length of the electrons. This leads to a rather weak constant repulsive force at small distances Fee = F0 =
e2 = const , λ2
(5.36)
which is much weaker than a purely classical Coulomb repulsion. The repulsive force Fee acts between any pair of nearest neighbor electrons and keeps them away from clustering. Due to the weak influence of the electron-electron repulsion we neglected it in most calculations. We assume that the chain particles are atomic ions or atoms with an ionic core. In order to simplify the description, we decribe the electron-ion interaction by a
vch 4 Okt 2005 10:43
5.5 Dynamics of Electrons Coupled to Running Local Compressions 79 Coulomb potential with an appropriate cut-off as often used e.g. in plasma theory [7, 22] Uek (rjk ) = (eek κ) −
k
eek 2 rjk
if
rjk < r1
+ h2
(5.37)
and Uek (rjk ) = 0
if
rjk > r1 ,
(5.38)
where rjk = yj − xk is the distance between the electron and its neighbors in the chain and 1/κ as well as r1 play the role of an appropriate ”screening length” [7]. Here our choice is r1 = 3σ/4 and κ = 2/σ. Further −e is the electron charge and ek the charge of the ion core of the chain particles. We introduced h as a free parameter which determines the short-range cut-off of the Coulombic pole, an appropriate choice is h 0.3σ.
-11 -12
U(x)
-13 -14 -15 -16 -17 -18
-4
-2
0 x
2
4
Fig. 5.6 Typical configuration of the local electric field created by the solitonic excitation. The minimum corresponds to a local compression of ions which means an enhanced charge density
Similar pseudo-potentials were introduced first by Hellman and are of current use in solid state theory [13]. The choice of the concrete value of ”height” of the pole is made such that the electrons are only weakly bound to the ion cores and may
vch 4 Okt 2005 10:43
80
5 Dynamical Clustering in Chains of Atoms with Exponential Repulsion
tunnel from one side of an ion to the other one. Accordingly the electrons are able to transit from one to the other side of an ion and yield an electron current. For the electron dynamics we take a classical ”Drude-Lorentz-Debye dynamics” dvj ∂Ue (yj ) + = −γe0 vj + 2De ξk (t) . dt me ∂yj
(5.39)
k
The evolution of the electrons is assumed to be passive √ (i.e. damping for all velocities), including white noise. The stochastic forces, 2De ξj (t), model a surrounding heat bath (Gaussian white noise), obeying a fluctuation-dissipation theorem. Note that the friction acting on the electron is small me γe mγ0 . The character of the electron dynamics depends on h and on the positions of the ions. Our choice h 0.3σ allows to generate strong local minima at the positions of strong (soliton-like) compressions (see Fig. 5.6). Several of the assumptions made here with respect to the electrons are not very realistic. However what matters here only is the principal effect. We wanted to show, how the dynamical clusters created by solitonic excitations act on the electrons. A quantum-mechanical tratment of the electron dynamics within the tight-binding approximation is presented elsewhere [16]; it has been shown there that the essential effects described in the present work are not influenced by the approximations.
5.5.2
Coupling Between Soliton Modes and Electron Dynamics
In order to study the coupling of electrons to the lattice vibrations we will consider long trajectories of the electronic positions and velocities, vje = y˙ j . We measure the energy (temperature) in units U0 = mω02 σ 2 , fixed bσ = 1 and taking e2 /(mσ) = 0.2U0 . All computations start with the initial state of equal distances between ions. The initial velocities of the ions were randomly taken from a Gaussian distribution with amplitude vin . Initially each electron is placed midway between two ions at rest, ve,l = 0. Differential equations Eqs. (5.39) have been integrated by means of a fourth-order Runge-Kutta algorithm adapted for solving stochastic problems [12]. We used l0 = σ as the length unit and t0 = 1/ω0 = (mi /ab)1/2 as the time unit. Our assumption that the initial velocities of the ions were randomly taken from a normal distribution corresponds to an initial Maxwell distribution and therefore to an initial temperature. We mention that such conditions may be reached experimentally by a heat shock applied to the lattice. The motions of ions
vch 4 Okt 2005 10:43
5.5 Dynamics of Electrons Coupled to Running Local Compressions 81 and electrons occur in different time scales. Heavy ions are not affected practically by light electrons and electrons move on the background of the Coulomb potential profile created by the ions. The dynamics of the ion ring leads to soliton-like excitations. Typical solitonic excitations correspond to local compressions moving on the ring. Fig. 5.6 shows a characteristic profile of the electric field created by the ion ring at certain time moment. We see a rather deep potential well moving around the ring. The light electron may be captured in this dynamic potential well and eventually may follow the soliton dynamics. In our simulations the integration step is chosen to describe correctly the fastest component of the process, the oscillations of electrons in the potential well.
20 10
xi(t), y(t)
0 -10 -20 -30 -40 -50
0
5
10
15
20
25 t
30
35
40
45
50
Fig. 5.7 Metastable ”solectron”: As a result of quenching of an initial state with T (0) 0.1 the trajectories of ten Toda particles (ions) generate solitons. A soliton forms a bound state with an electron captured by the soliton (a ”solectron”). During this time interval the electronic trajectory is parallel to the ”tangent” representing the solitonic √ velocity. (Parameter values: γi0 = γe0 = (0.0002)/t0 , unit of time on the abscissa, t0 / 5, unit of length on the ordinate, l0 = σ)
In the computer experiments demonstrated in Fig. 5.7 we used an initial Gaussian distribution of the ion velocities corresponding to a high-temperature Maxwellian with kB Tin 0.1 (in units of the energy of harmonic oscillations with amplitude σ). As in the case earlier studied, this is near to the critical temperature kB Tcr 0.16, where we are in the soliton-generating region [17]. Besides other excitations many solitons are generated. However they are difficult to recognize
vch 4 Okt 2005 10:43
82
5 Dynamical Clustering in Chains of Atoms with Exponential Repulsion
due to the random motions of the particles. Then we quenched to a temperature near to zero. The solitons survive since they have a higher lifetime than most other excitations. Looking at the trajectories we observe the expected nonlinear soliton-like excitations that decay after a time of the order trel 1/γ0 (Fig. 5.7). These excitations exist also under equilibrium conditions [17].
40 20 0 xi(t), y(t)
-20 -40 -60 -80 -100 -120 -140 0
10
20
30
40
50 t
60
70
80
90
100
Fig. 5.8 Trajectories of 10 ions moving clockwise creating one fast dissipative soliton moving in opposite direction to the motion of the ions, and trajectories of 10 electrons captured in part by the soliton which is made stable due to the energy input (δ = 2)
In order to sustain the solitonic excitations for a longer time interval we applied an active Rayleigh friction in the period after heating and quenching as described in the previous section. Then the soliton regime becomes a stable attractor [4, 18, 19]. The simulations presented below correspond to the Rayleigh approximation with δ = 2, vd = 1, m/me = 1000, γ0 = γe0 = 0.2. A soliton corresponds to a local compression of the lattice which is running opposite to the mean ion motion. This creates a charge density wave. Snapshots show that the electrons are captured by local concentrations of the ionic charge. Since the electrons search for the deepest nearby minimum of the potential, they will be most of the time located near to local ion clouds. The soliton is a dynamic phenomenon, the ions participating in the local compression are changing all the time. Hence, the electrons have always new partners for forming the ”solectron”. Three stages are found: In the first one the initial state tends to one of N + 1
vch 4 Okt 2005 10:43
5.6 Discussion
83
for N odd (or N in the other case) attractors [18]. The maximal average velocity among the running waves has the excitation with one local compression on the ring. The attractor, reached by the system without noise and external field, is defined mainly by the value of vin given by the initial conditions. Rotations appear at very small initial velocity values. Then, upon increasing vin , k-solitonic waves may be excited with increasing k. There exists always a target attractor for a given value vin . For our case the initial conditions lead preferentially to the onesoliton attractor. In the absence of the external field both directions have equal probability, the field breaks the symmetry. As mentioned above our choice for the value of a cut-off distance in the electronion interaction Eq. (5.36) is h = 0.3σ. In this case the difference between the maximum in the electron-ion interaction force and the corresponding value for an electron and an ion being away from the electron more than 1.5σ, is an order of magnitude lower, and hence interaction of the electron with such ions is not taken into account in the simulations. Thus we consider the interaction of each electron with the ions placed, e.g., about one third of the ring near that electron only (N = 10, ne = ni = 1). To simplify, the parameters of the potentials, of the Rayleigh formula, the friction coefficients, both masses and charges of particles were held fixed. The initial velocities vin , the values of the external field and the electronic temperature Te are varied in different runs. In Fig. 5.8 we show a simulation for the trajectories (left to right) of 10 ions creating 1 dissipative soliton which moves in opposite direction (right to left). After a transient regime, the electron is coupled to the soliton and moves approximately with the soliton velocity opposite to the motion of the ions. In the driven case (δ = 2) the ions perform a constant drift. After a transient regime, solitonic excitations of the ions are formed moving with velocity vs opposite to the average drift of the ions. Most of the electrons are captured by these dynamical clusters.
5.6
Discussion
We have shown that in dense lattices of particles with exponential repulsion, special nonlinear waves may be excited which may be interpreted as dynamical clusters - running local compressions. These dynamical clusters are similar to the cnoidal waves in Toda’s theory. We have shown that these strongly localized excitations corresponding to local compressions of the chain may be generated also in dissipative lattices by external forcing, stochastic initial conditions or
vch 4 Okt 2005 10:43
84
5 Dynamical Clustering in Chains of Atoms with Exponential Repulsion
negative friction. In order to measure the solitonic strength of the excitations, we introduced a special function. We studied the properties of ionized atomic, hence electrically conducting, chains. Each atom was assumed to provide one electron moving along the chain in the field of the remaining ionic lattice. The electrons prefer positions near to the deep (electrostatic) potential wells formed by the local compression connected with the soliton. We have shown that, as time proceeds, most of the electrons are captured by such local compressions and move with the soliton velocity opposite to the ion drift. This way we have shown how significant for electric conduction is the role of localized running excitations in lattices with exponential repulsion and, in particular, the role of bound states between dynamical clusters and electrons (”solectrons”).
Acknowledgment The authors thank Dirk Hennig, J¨ urn Schmelzer and Boris Smirnov for useful discussions and several suggestions.
5.7
References
1. N. Bernasconi and N. D. Schneider, Physics in One Dimension (Winston, Philadelphia, 1976). 2. A. Scott, Nonlinear Science (Oxford University Press, Oxford, 1999). 3. J. Dunkel, W. Ebeling, and U. Erdmann, Eur. Phys. B 24, 511 (2001). 4. J. Dunkel, W. Ebeling, U. Erdmann, and V. Makarov, Int. J. Bifurc. & Chaos 12, 2359 (2002). 5. A. Chetverikov, W. Ebeling, and M. G. Velarde, Eur. Phys. J. B, 44, 509(2005). 6. M. G. Velarde, W. Ebeling, and A. P. Chetverikov, Int. J. Bifurc. & Chaos 15, 245 (2005). 7. A. Chetverikov, W. Ebeling, and M. G. Velarde, Contr. Plasma Phys. 45, 275 (2005). 8. P. Morse, Phys. Rev. 34, 57 (1929). 9. R. S. Berry and B. M. Smirnov, Phys. Rev. B 71, 144105 (2005). 10. M. Toda, Nonlinear Waves and Solitons (Kluwer, Dordrecht, 1983). 11. E. M. Apfelbaum, V. S. Vorob’ev, and E. M. Martynov, Chem. Phys. Lett., in press, (2005).
vch 4 Okt 2005 10:43
5.7 References
85
12. N. N. Nikitin and V. D. Razevich, J. Comput. Math & Meth. Phys. 18, 108 (1978). 13. N. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehardt (Eds.), Winston, Philadelphia, 1976). 14. A. S. Davydov, Solitons in Molecular Systems (Naukova Dumka, Kiev, 1984). 15. D. Hennig et al., Physica D 180, 256 (2003), J. Biol. Physics 30, 227 (2004). 16. M. G. Velarde, W. Ebeling, D. Hennig, and C. Neissner, Int. J. Bifurc. & Chaos, in press (2005). 17. W. Ebeling, A. Chetverikov, and M. Jenssen, Ukrain. J. Phys. 45, 479 (2000). 18. V. Makarov, W. Ebeling, and M. G. Velarde, Int. J. Bifurc. & Chaos 10, 1075 (2000). 19. V. Makarov, E. del Rio, W. Ebeling, and M. G. Velarde, Phys. Rev. E 64, 036601 (2001). 20. E. del Rio, V. A. Makarov, M. G. Velarde, and W. Ebeling, Phys. Rev. E 67, 056208 (2003). 21. U. Erdmann, W. Ebeling, L. Schimansky-Geier, and F. Schweitzer, Eur. Phys. J. B 15, 105 (2000). 22. T. Pohl, U. Feudel, and W. Ebeling, Phys. Rev. E 65, 046228 (2002).
vch 4 Okt 2005 10:43
6
Solid State of Repelling Particles Boris M. Smirnov(1) and R. Stephen Berry(2) (1)
Institute for High Temperatures, Izhorskaya 13-19, Moscow 127412, Russia (2)
Department of Chemistry, University of Chicago,
5735 South Ellis Avenue, Chicago, IL 60637, USA Better be wise by the misfortunes of others than by your own. Aesop Abstract The peculiarity of an ensemble of repelling particles at low temperatures, where interaction between nearest neighbors dominates its properties, is its polycrystalline structure with a density of particles lower than that of the crystal lattice. This result follows from computer simulations, model experiments with filling a container with hard balls, and the virial theorem. Moreover a polycrystalline structure is observed directly in a colloid solution. The question arises of whether such a polycrystalline state of an ensemble of repulsed atoms at low temperatures is thermodynamically stable, or is a metastable, glassy state that results from the kinetics of relaxation of this ensemble to low temperatures and high densities. The disk model for repelling particles and the cell model for the configurations of these particles show that the glassy state of this system under indicated conditions is thermodynamically favorable.
6.1
Introduction
We consider ensembles of particles with a sharply varying repulsive potential between them, and suppose that these ensembles are supported by external pressure
6.1 Introduction 87 or other external forces. Table 6.1 gives examples of such particle ensembles; we include in this list a dusty plasma with Yukawa interaction potentials between particles for a screening length of this interaction potential less than the mean distance between nearest particles. All the above systems are simple and, like systems of bound atoms, have two aggregate states, solid and liquid, as follows from computer simulations and observations [1, 2, 3, 4]. One can expect that the liquid state of these systems corresponds to a random spatial distribution of particles, but in the solid, the particles form a face-centered cubic (fcc) or hexagonal structure, in which each internal particle has 12 nearest neighbors. But in reality an ensemble of repelling particles in the solid state has a polycrystal structure, i.e. a long-range order is not realized for such particle ensembles. This follows from model experiments with filling a container with hard balls [4, 5, 6, 7], from computer simulations within the framework of hard sphere model [4, 8, 9, 10], and from the virial theorem for the crystal state of an ensemble of repelling atoms under an external pressure [11]. Moreover, the polycrystalline structure of a colloid solution was observed directly by light scattering [13]. Thus, repelling atoms do not form an infinite crystal lattice. Tab. 6.1 Ensembles of repelling particles and boundary conditions that allow one to concentrate the particles in a restricted spatial region
Ensemble of particles Inert gases under high pressure Hard balls in a box Colloid solutions Dusty plasma
Boundary conditions External pressure Pressure due to weight of upper particles External pressure Electric traps
The crystal structure of repelling atoms at high pressure and low temperatures is unstable thermodynamically according to the virial theorem [11, 12]. But the virial theorem deals with uniform particle distributions, and this prohibition relates to an infinite crystal rather than to the domain structure of an ensemble of repelling atoms when this ensemble includes individual crystallites. This distribution in colloid solutions is considered as a glassy state [14, 15, 16, 17] because of the long times for equilibrium establishment when the density of monomer particles is high. This means that the kinetics of relaxation processes to low temperatures and high densities does not allow a system to reach a state close to the crystal, under these conditions. We also show, with a simple model, that the
vch 4 Okt 2005 10:43
88
6 Solid State of Repelling Particles
kinetics of particle displacements involves obstacles along the way to establishing a favorable particle distribution in a space within the framework of simple model. Specifically, we introduce a disk model for a system of repelling particles, and the cell model for their spatial distribution when this distribution is close to that in crystals.
6.2
Ensembles of Repelling Particles at Low Temperatures
We consider ensembles of identical classical particles with a pair interaction between them, so that the pair interaction potential U (r) at a distance r between particles can be approximated by U (r) = U (Ro )
Ro r
k =
A , rk
(6.1)
Moreover, we take this interaction potential to vary sharply at a critical distance, i.e. k=
d ln U 1, d ln r
(6.2)
so a test particle interacts only with nearest neighbors. In this limit one can use the model of hard spheres for interaction of particles, so that the pair interaction potential can be changed to the following one [18] U (r) =
∞, 0,
for for
r ≤ 2a , r > 2a ,
(6.3)
where a is the effective particle radius. In spite of identical interactions between particles for different ensembles of repelling particles of Table 6.1, these ensembles due to different external conditions all exhibit a particular behavior at low temperatures where they have a polycrystalline structure. An individual crystallite probably contains less than hundreds of balls in the case of a container filled with hard balls [5, 6, 7]. A typical size of an individual crystallite of inert gas under high pressure is estimated to be ∼ 103 [11, 12]. In the case of colloid solutions, an individual crystallite contains ∼ 106 − 107 monomers [13]. In a dusty plasma the particles form a unit crystal
vch 4 Okt 2005 10:43
6.2 Ensembles of Repelling Particles at Low Temperatures 89 in the solid state [19], but the absence of the polycrystalline structure in this case may be explained by a restricted number of particles that does not exceed 104 − 105 . This difference can be explained by the weakness of the processes which provide the formation and existence of a group of particles-monomers with crystal structure, and therefore these processes depend on external conditions. In particular, gravity is of importance for colloidal crystals [20]. Nevertheless, the tendency to reach the polycrystalline structure is identical for the various ensembles of repelling particles which are given in Table 6.1. This tendency and the character of formation of the polycrystalline structure can be described by the hard sphere model for particle interactions. This allows one to study the character of evolution of ensembles of repelling particles using the hard sphere model with standard molecular dynamics [21]. Hence we will use the results of molecular dynamics based on the hard sphere model for various properties of particle ensembles under consideration. If the hard sphere model is applicable for particle interactions, it is convenient to characterize the distribution of the hard spheres in space by the packing parameter [4], given by ϕ=
4π 3 r N, 3n
(6.4)
where r is the sphere’s radius, N is the number density of spherical particles, n is the number of particles inside a given sphere, and the packing parameter ϕ is the fraction of the space occupied by hard particles. Evidently, the maximum value of this parameter for hard spheres corresponds to a close-packed crystal lattice for which the packing parameter is √ π 2 = 0.74 . (6.5) ϕcr = 6 The packing parameter ϕ of an ensemble of hard spheres follows from simple experiments based on filling a container with hard balls [5, 6, 7] and on simulations with hard spheres [8, 9, 10]. The observed value ϕd = 0.64 [6] is in accord with a more precise value from computer simulations for the packing density of this system [9] ϕd = 0.644 ± 0.005 .
(6.6)
This means that an ensemble of hard spheres does not form an infinite closepacked crystal lattice. Using the connection of the mean coordination number q
vch 4 Okt 2005 10:43
90
6 Solid State of Repelling Particles
(an average number of nearest neighbors for a test particle) with the corresponding atom density ρ [22], we find
q = 12
ρ , ρcr
(6.7)
where ρcr is the crystal density. With q = 12 for the close packed structure, we have, on the basis of Eqs. (6.4) and (6.5), for an average number of nearest neighbors at the maximum packing parameter Eq. (6.6)
q = 12
ϕ = 16.2ϕ , ϕcr
(6.8)
Eqs. (6.5) and (6.7) give q = 10.4± 0.1, close to the coordination number of liquid inert gases at low pressures, in which atoms are bonded by attractive forces, and for which q = 10.1 ± 0.1 [22, 23]. Note that although the ensemble of repelling particles does not form an infinite crystal lattice, it exhibits short-range order. In particular, the correlation function Q6 [24], that is zero for an amorphous structure, is not zero for an ensemble of hard spheres [9]. Therefore this ensemble is separated into individual crystallites. Meanwhile, the time of establishment of an equilibrium spatial distribution of particles in such a system increases sharply with an increase of the packing parameter from ϕm = 0.545 for the solid state at the melting point up to its maximum value ϕd . For example, a typical time to establish an equilibrium distribution in colloids in experiment [20] varies from 1 hour up to several days, with the packing parameter varying from ϕm up to ϕ = 0.62. Thus, in spite of the simplicity, the structure of solid systems of repelling particles is not as simple as one might expect at a first sight, and an ensemble of repelling particles includes clusters, i.e. crystallites. Hexagonal structure of these crystallites is preferred for colloid solutions, although they may also contain an admixture of the fcc structure [20, 25, 26]. For compressed inert gases, only the hexagonal structure is observed in the limit of high pressures [27]. The surface of an individual crystallite is nonspherical and is similar to that of a fractal aggregate. The fractal dimension for the crystallite in the colloid case is 2.35 ± 0.15 [26].
vch 4 Okt 2005 10:43
6.3 Model of Hard Disks for an Ensemble of Repelling Atoms
6.3
91
Model of Hard Disks for an Ensemble of Repelling Atoms
In order to understand the properties of a classical ensemble of strongly repelling particles and the character of its relaxation as a result of displacements, it is convenient to use the model of hard disks, the simplest model for such an ensemble of particles locked in a box. We consider a two-dimensional case to simplify the problem of repelling particles; this allows us to extract the principal properties of the ensemble in a simple manner. To characterize the particle density in a box, we introduce the packing parameter like that of Eq. (6.4), for the three-dimensional case ϕ=
πa2 n , S
(6.9)
where a is the radius of a particle-disk, n is the number of particles inside the box, and S is the area of a square inside the box. Evidently, the maximum of the packing parameter corresponds to the crystal lattice of the hexagonal structure that provides the most packing of particles. Then the distance between neighboring particles of √ the same line is 2a, and the distance between neighboring lines 3, i.e. the density of a bulk crystal of the hexagonal structure of particles is a √ −1 2 is (2a 3) . Correspondingly, the packing parameter for the bulk hexagonal crystal π ϕhex = √ = 0.907 2 3
(6.10)
is the maximum of the packing parameter that is possible for particle-disks. We will operate with an element of this lattice that contains 16 particles. This element may be cut out of the crystal lattice. We place it in a rhombic box (Fig. 6.1) where these particles are packed densely. Then the √ angles between rhombus sides are π/3 and 2π/3, and the side length is (6 + 4/ 3)a = 8.309a. The packing parameter for this system is less than that for a bulk hexagonal lattice and is equal to √ 8π 3 = 0.841 . (6.11) ϕ16 = √ (3 3 + 2)2 One can construct ensembles of particles in a box with one vacancy (Fig. 6.2) and two vacancies (Fig. 6.3), by removal of one and two atoms from an element
vch 4 Okt 2005 10:43
92
6 Solid State of Repelling Particles
Fig. 6.1 An element of the hexagonal structure consisting of 16 particle-disks in a box
B 4 2 1 7
3 6 5 A
Fig. 6.2 An element of the hexagonal structure of Fig. 6.1 in a box with one vacancy
of Fig. 6.1. The packing parameters for these ensembles in these cases are given by ϕ15 =
15 ϕ16 = 0.788 , 16
ϕ14 =
14 ϕ16 = 0.736 . 16
(6.12)
In the latter case two vacancies may be transformed into two voids, as shown in Fig. 6.4 for a symmetric configuration of the particles and voids. In contrast to vacancies, voids can change their shape and size in the course of evolution. But the dense packing of Fig. 6.2 does not allow for particles and vacancy to change their positions. Indeed, a particle 1 of Fig. 6.2 can transfer to the position
vch 4 Okt 2005 10:43
6.3 Model of Hard Disks for an Ensemble of Repelling Atoms
93
Fig. 6.3 An element of the hexagonal structure of Fig. 6.1 in a box with two vacancies
Fig. 6.4 An element of the hexagonal structure of Fig. 6.1 in a box with two voids
2 if the distance between two neighboring particles 3 and 4 exceeds 4a, whereas √ it is 2a + a 3 for close packing of the particles. This particle transition becomes possible, if we increase the box size.
vch 4 Okt 2005 10:43
94
6 Solid State of Repelling Particles
2 1
Fig. 6.5 Particle-disks in a box at the density that ensures transitions of particles and voids to other positions
Let us introduce the parameter r, so that a rhombus side for the box is 4 r. l = 6+ √ 3
(6.13)
Under conditions of Fig. 6.1 we have r = a, and we consider below the case ∆r = r − a a when relative positions of particles-disks in a box change slightly. √ √ The distance AB of Fig. 6.2 is equal to (3 3 + 2)a and becomes (3 3 + 2)r after a boxwidening. For passing of a particle 1 into a √ new position it is necessary 3)a, i.e. this value exceeds to increase the initial value AB at least by (2 − √ (4 + 2 3)a. Thus, a particle 1 of Fig. 6.2 may transfer to a position 2, if the criterion r − a ≥ 0.037a is fulfilled, as takes place under conditions of Fig. 6.5, if √ √ (6.14) (3 3 + 2)r ≥ (4 + 2 3)a or r ≥ 1.037a. This corresponds to the packing parameter
a 2 ϕ15 = 0.732 . ϕ≥ r
(6.15)
This value is approximately that of the packing parameter for formation of two vacancies as given in Fig. 6.3. Note that the critical packing parameter for particle transition depends on the box shape.
vch 4 Okt 2005 10:43
6.4 Equation of State for Disks
95
In the same manner one can find the minimum packing parameter required for a particle transition between positions 6 and 1 of Fig. 6.2. On the basis of the above consideration, we obtain Eq. (6.15) for the packing parameter in this case. Next, let us consider a transition between positions 5 and 6 of Fig. 6.2 or between positions 5 and 3, when one of these positions is located in a box angle and, of course, one of transition positions is free. This transition is possible if the length √ AB is increased by (2 − 3)a, which allows transfer of an angle particle or of another particle to an angle position. We obtain the minimal packing parameter for particle transfer in this case also in accordance with Eq. 6.7. Thus, we find the critical value of the packing parameter when transitions of particles and vacancies are possible in a box under the conditions given above. This consideration is based on conservation of the particle configuration in the course of box widening, i.e. it requires that the criterion ∆r = r − a a holds true. This criterion gives ∆r ≥ 0.037a; this requirement may be fulfilled. At the critical value of the packing parameter, the diffusion coefficient for particles in a box due to their thermal motion is zero; it increases with a decrease of the packing parameter. For s small number of particles the diffusion coefficient in a box depends also on a box shape.
6.4
Equation of State for Disks
A disk model for particles confined in a finite space is a simplified description of an ensemble of repelling particles. First, this simplification uses a two-dimensional space that allows us to study geometric properties of the ensemble in an obvious manner. Second, assuming within the framework of this model, that particles interact only during their contact, we reduce the problem of a condensed-phase system to that of a dense gas, since a time of particle contact is brief in comparison with the time between two subsequent contacts of these particles with one another. Therefore, though the mean free path of particles may be small compared to their size, particles move freely most of the time. Thus, the behavior of particles in a condensed system within the framework of the disk model is rather like that in a gaseous system. Below we derive the equation of state for a bulk ensemble of such particles. Characterizing the particle density in a box by the packing parameter ϕ, we assume the particle’s spatial distribution to be random, i.e. ϕ < ϕhex , although these values have the same order of magnitude. Fig. 6.6 gives the trajectory of the center of a test particle located near the plane boundary-wall and encounters
vch 4 Okt 2005 10:43
96
6 Solid State of Repelling Particles
0
ya
y0
wall
Fig. 6.6 Trajectory of the center of a surface particle. The particle-disk with center O is shown by a dashed circle
it. The particle center of course cannot approach to the wall closer than its radius a. This particle reflects elastically from the wall at each contact. Then it collides with surrounding particles and returns to the wall. Let us estimate the force on the wall due to collisions of a test surface particle. Assuming a Maxwell distribution of particle velocities, we have for a typical particle momentum P ∼ (mT )1/2 , where m is the particle mass, T is the temperature expressed in energetic units. Since collisions of a test particle with the wall occur in a mean time τ ∼ λ/vT , where λ is the mean free path and vT is a typical particle velocity (vT ∼ (T /m)1/2 , one can use this to estimate the mean force transmitted to the wall by a test particle at the surface F ∼
T . λ
(6.16)
From this expression, one can derive the equation of state for an ensemble of particles by introducing the pressure p as the force per unit area that acts on the walls. Let us consider first the gaseous case in which the particle number density N is small, and particles that reach the walls are located in a layer of a width ∼ λ.
vch 4 Okt 2005 10:43
6.5 The Cell Model for Disk Particles
97
Therefore the number of particles per unit area of the surface is ∼ N λ, and the equation of state takes the form p ∼ N T . More specifically, it has the form p = NT .
(6.17)
In the case under consideration, in which we deal with a dense ensemble of particles, we obtain for ϕ ∼ 1 and N ∼ a−3 , instead of Eq. (6.17), p = cN T
a , λ
(6.18)
where the numerical coefficient c ∼ 1. Thus, applying the gaseous criterion leads to another equation of state for a dense ensemble of particles [28, 29, 30]. In the case of a high density of disks, it is convenient to introduce a small parameter α=
S − So , So
(6.19)
where S is the area per individual particle-disk, and So is its limit for the closepacked structure. Then the equation of state for this ensemble of particles-disks can be given in the form p = N T F (α)
(6.20)
and the function F (α) is given by [30] F (α) =
6.5
2 + 1.90 + 0.67α + 0(α2 ) . α
(6.21)
The Cell Model for Disk Particles
We consider now the cell model of particles [8, 31, 32] in which the particles can be located in certain cells. In particular, within the framework of the disk model, these cells may be circles as shown in Fig. 6.7. Let us determine the partition function for this system at a specified value of the packing parameter. Starting from the hexagonal structure of disks with the packing parameter ϕhex = 0.907, we decrease the packing parameter. Within the framework of the cell model this can proceed in two (nonexclusive) ways, either by an increase of the radius r of an individual cell that exceeds the disk radius a, or by formation of vacancies. Let the initial area contain n + v cells, of which
vch 4 Okt 2005 10:43
98
6 Solid State of Repelling Particles
Fig. 6.7 The cell model, in which disk particles (solid circles) are located in their cells (open circles)
n are occupied by particles (i.e. v empty cells are vacancies). Then the packing parameter is ϕ = ϕhex
n a 2 . n+v r
(6.22)
The partition function of this system with an accuracy up to a constant factor is equal to Z=
n Cn+v
r 2 − a2 a2
n .
(6.23)
We take the partition function of an individual particle in its cell to be proportional to the area π(r 2 − a2 ) that can be occupied by this particle in its cell; the first factor of this formula takes into account permutations of particles and cells. In order to obtain a thermodynamically stable state, it is necessary to optimize the entropy S = ln Z with respect to the number of vacancies at a given value of the packing parameter ϕ. For large values of n and v we have
r 2 − a2 + n ln S = ln Z = a2 2
n r − a2 v + v ln 1 + + n ln . = n ln 1 + n v a2 n ln Cn+v
vch 4 Okt 2005 10:43
(6.24)
6.5 The Cell Model for Disk Particles
99
Expressing the cell radius from Eq. (6.22) and substituting it into Eq. (6.25), we find for the entropy per particle (s = S/n) the result
1 s = c ln 1 + c
+ ln(cmax − c) ,
(6.25)
where the vacancy concentration is introduced as c = v/n, and its maximum value at a given packing parameter ϕ is attained when the cell radius is equal to the particle radius (r = a). This is cmax =
ϕhex − ϕ ϕ
.
(6.26)
To deduce Eq. (6.25), we take the cell radius r at a given packing parameter ϕ from Eq. (6.14) and substitute it into Eq. (6.15). Note that the entropy per particle s is defined with accuracy up to a constant. For optimization of cell and particle distributions we find the vacancy concentration co that corresponds to the maximum entropy. If ϕ ∼ ϕhex , this corresponds to a low value of the optimal vacancy concentration given approximately by
1 + cmax co = exp − cmax
.
(6.27)
This value varies from 0.012 up to 0.12, if cmax varies from 0.2 up to 0.6, corresponding to a variation of the packing parameter from 0.76 to 0.57. Thus, within the framework of the disk model for particles, a decrease of the packing parameter leads mostly to an increase of the cell radius, while only a small part of the increase of area per particle goes into formation of new vacancies. Note that an ensemble of disk particles can be considered as a gas because the interactions between particles are assumed to occur only when they touch each other, making the interaction time of a test particle with a neighbor brief compared with the time when this particle is free. Evidently, in a real condensed system of repelling particles, a test particle interacts with nearest neighbors strongly, i.e. at each time the interaction potential energy of a test particle with surrounding particles is comparable to its kinetic energy.
vch 4 Okt 2005 10:43
100
6 Solid State of Repelling Particles
In order to understand the role of interactions between particles in establishing their optimal space distribution, we analyze below the contribution of these interparticle interactions to the entropy of the entire ensemble. The entropy for the cell model when particles are found in cell centers, on average, has the form S = ln Z =
n ln Cn+v
+ n ln
r 2 − a2 a2
− qn
U (r) , T
(6.28)
where q = 6 − c is the average number of nearest neighbors of a test particle, U (r) is the pair interaction potential at a distance r between particles, and we change the interaction potential between nearest particles to that at the average interparticle distance. The temperature T is expressed in energetic units. This gives the entropy variation
1 − cmax U (r) qk U (r) 1 1 + − ds = dc ln(1 + ) − c (1 + c)(cmax − c) T 2 T 1+c
.
(6.29)
From this formula it follows that including terms proportional to U (r)/T decrease the entropy as the cell radius decreases. Hence the optimal conditions correspond to a lower vacancy concentration c than that for the hard disk model.
6.6
Diffusion Coefficient of Vacancies for Cell Model
We now estimate the diffusion coefficient of particles or vacancies in a dense ensemble of disk-particles within the framework of the cell model. Transitions of particles between cells are shown in Fig. 6.8; these lead to reverse transitions of vacancies. Hence the diffusion coefficients of particles Dp and vacancies Dv are connected by a simple relationship Dp = cDv ,
(6.30)
because they result from the same process. We first estimate a typical time of particle transition to a neighboring cell (see Fig. 6.9). For simplicity, we place a transferring particle-disk in the center of its cell. The transitions are possible with small angles with respect to the arrow of Fig. 6.9 if a transferring disk does not touch its neighbors in the course of the transition. If a test particle is moving along the arrow and a neighboring particle
vch 4 Okt 2005 10:43
6.6 Diffusion Coefficient of Vacancies for Cell Model
101
Fig. 6.8 Transition of particle-disks to neighboring positions within the framework of the cell model. The initial configuration of particles is left and the final configuration is given right; the arrows indicate the transferring particles
occupies a favorable position in its cell being touching the cell circle, the gap between particles is equal to √ (6.31) ∆ = (1 + 3)r − 3a , From this relation, on the basis of the condition ∆ > 0, we have the packing density of particle-disks when the diffusion coefficient for particles or vacancies is nonzero, and displacement of vacancies and particles is possible in the interior of the particle ensemble √ 2 1+ 3 = 0.75 . (6.32) ϕ < ϕhex 3 The diffusion process for particles inside their ensemble ceases at values above this packing parameter. To estimate the vacancy diffusion coefficient, we recognize that a transferring particle collides elastically many times with its nearest neighbors as a result of its thermal motion, and that this goes on until the angle between its velocity direction and an arrow like that of Fig. 6.9 would be below θ, so that θ∼
∆ . r
(6.33)
vch 4 Okt 2005 10:43
102
6 Solid State of Repelling Particles
Fig. 6.9 Transition of a particle-disk to a neighboring cell through a line separating neighboring cells. The transition proceeds along an arrow or at small angles to it
Correspondingly, a typical transition time of a test particle to a free neighboring cell is τ∼
(r − a)∆ r−a . θ∼ vt vT r
(6.34)
We consider a range of the packing parameter values when R ∼ a and r − a a. Note that r − a ≈ 0.1a at ∆ = 0. From this we obtain an estimate of the vacancy diffusion coefficient in a dense ensemble of particle-disks as Dv ∼
a∆vT r2 ∼ . τ r−a
(6.35)
Hence the vacancy diffusion coefficient Dv → 0 if ∆ → 0. We again take into account that the disk model for an ensemble of repelling particles models this ensemble as a gas of particles and therefore cannot describe some properties of this system. In particular, this model gives a general character of particle diffusion inside this system, but does not automatically reveal the temperature dependence of this quantity. We now estimate this dependence within the framework of the cell model.
vch 4 Okt 2005 10:43
6.6 Diffusion Coefficient of Vacancies for Cell Model
103
The diffusion process has an activation character, and the temperature dependence for the diffusion coefficient has the form
Ea Dv ∼ exp − T
,
(6.36)
where Ea is the activation energy, and T is the temperature expressed in energy units. Let us locate particles in the centers of their cells and determine the activation energy of the diffusion process for this particle configuration. Then the activation energy is the difference of the interaction potentials for a transferring particle located midway between the initial and final positions and in its initial (or final) position. Accounting for interactions with nearest neighbors in those initial, midway and final positions of a transferring particle, we obtain in this case √ √ r 7 3r r 3 + 4U + 2U Ea = 2U 2 2 2
√ − 5U (r) − 2U r 30 − U (2r) .
(6.37)
In particular, approximating the pair interaction potential of particles by the dependence U (r) ∼ r −k and taking k = 8, we obtain Ea = 0.36Uo ,
(6.38)
where Uo = 5U (r). Accounting for displacements of particles inside their cells would decrease the barrier energy. As follows from this formula, within the framework of the hard disk model, the diffusion coefficient of vacancies becomes zero for values of the packing parameter Eq. (6.32) because a high particle density does not allow a particle to transfer to a neighboring free cell. Repulsive interactions between particles can reinforce this effect at low temperatures, and then diffusion of vacancies and particles cease at even lower densities.
vch 4 Okt 2005 10:43
104
6.7
6 Solid State of Repelling Particles
Peculiarities of the Solid State of an Ensemble of Repulsing Particles
Simple models considered here confirm once more that repelling particles do not form a crystal lattice in the solid state. The polycrystalline atomic structure may be considered as a glassy state as it is done for colloid solutions [14, 15, 16, 17]. One might think that this is not a major issue, but in reality it is important. Indeed, the uniform crystal structure is not thermodynamically stable according to the virial theorem [11] alone, but a polycrystalline state does not submit to the virial theorem and of course it can be realized. But since this state is not a unique, thermodynamically stable form, different values of the parameters of this structure are possible. In particular, the number of monomers, i.e. particles not nested in crystalline structures, in a typical assembly of microcrystalline clusters depends both on the physical object and on its method of preparation, including the rate at which it makes its transition into its final state. As with other glassy systems, one can expect that a typical size of a crystalline cluster in the system will be larger, the slower is the process of preparation of the solid state. Evidently, this explains (or at least rationalizes) the different sizes of individual crystalline particles for different objects, as discussed in the introduction to the present chapter. In particular, let us consider the solid state of inert gases at low temperatures and high pressures. Such a state can be prepared by two methods. One can fix the temperature and increase the pressure, starting from the crystal structure at the triple-point pressure, or one can start from the liquid state at high pressure and decrease the temperature at a constant pressure. In the first case the stacking instability that corresponds to displacement of layers, will lead to transition from an fcc lattice to the hexagonal structure for atoms of nearest layers. Since the displacement of layers becomes difficult after layers in other directions are displaced, this method can lead to a higher density of atoms than that from the second method, in which we start from an amorphous atomic distribution, and creation of small crystallites results from diffusion of vacancies and voids to the outside of the system. In the same manner, one can conclude that the final density for the second method depends on the rate of cooling since the final density is determined by diffusion of atoms and voids. Thus, we conclude that the glassy nature of the solid state of an ensemble of repelling atoms can lead to different internal parameters of these objects depending on their nature and preparation method. Nevertheless, all these objects have
vch 4 Okt 2005 10:43
6.8 Conclusion
105
polycrystalline structure, although the sizes of individual crystallites can depend both on the substance and the method of its production.
6.8
Conclusion
This analysis of an ensemble of repelling classical particles on the basis of simplest models exhibits the nature of the structure and diffusion of this particle ensemble. The domain structure established by a temperature decrease or a density increase may become frozen at some particle density because displacement of particles becomes impossible. Therefore, these solid states of the particle-disk ensemble have features of a glassy state.
Acknowledgments R.S.B. would like to acknowledge support from the National Science Foundation. B.M.S. thanks the Russian Foundation for Basic Research (Grant 03-02-16059) for partial support.
6.9
References
1. B. J. Alder and T. E. Wainwright, J. Chem. Phys. 27, 208 (1957). 2. W. G. Hoover, S. G. Gray, and K. W. Johnson, J. Chem. Phys. 55, 128 (1971). 3. S. M. Stishov, Sov. Phys. Uspekhi 17, 625 (1974); UFN 114, 3 (1974). 4. I. Gutzow and J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization (Springer, Berlin, 1995). 5. J. D. Bernal, Nature 183, 141 (1959). 6. G. D. Scott, Nature 178, 908 (1960). 7. J. D. Bernal and J. Mason, Nature 188, 908 (1964). 8. W. G. Hoover and F. H. Ree, J. Chem. Phys. 49, 3609 (1968). 9. M. D. Rintoul and S. Torquato, Phys. Rev. Lett. 77 4198 (1996). 10. M. D. Rintoul and S. Torquato, Phys. Rev. 58E, 533 (1998). 11. R. S. Berry and B. M. Smirnov, Phys. Rev. 71B, 051510 (2005). 12. R. S. Berry and B. M. Smirnov, Phys. Uspekhi 48, 331 (2005); UFN, 175, 367 (2005).
vch 4 Okt 2005 10:43
106
6 Solid State of Repelling Particles
13. P. N. Pusey et al., Phys. Rev. Lett. 63, 2753 (1989). 14. L. V. Woodcock, J. Chem. Soc. Faraday II 72, 1667 (1976). 15. W. van Megen and S. M. Underwood, Phys. Rev. Lett. 70, 2766 (1993); Phys. Rev. E 49, 4206 (1994). 16. R. J. Speedy, J. Chem. Phys. 100, 6684 (1994). 17. J. Yeo, Phys. Rev. E52, 853 (1995). 18. B. M. Smirnov, Physics of Ionized Gases (Wiley, New York, 2001). 19. V. E. Fortov et al., Phys. Uspekhi 47, 447 (2004). 20. J. Zhu et al., Nature 387, 883 (1997). 21. B. J. Alder and T. E. Wainwright, J. Chem. Phys. 33, 1439 (1960). 22. B. M. Smirnov, Phys. Uspekhi 37, 1079 (1994). 23. B. M. Smirnov, Clusters and Small Particles in Gases and Plasmas (Springer, New York, 2000). 24. P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys.Rev. B28, 784 (1983). 25. S. Auer and D. Frenkel, Nature 409, 1020 (2001). 26. U. Gasser et al., Science 292, 258 (2001). 27. H. Cynn et al., Phys. Rev. Lett. 86, 4552 (2001). 28. Z. W. Salsburg and W. W. Wood, J. Chem. Phys. 37, 798 (1962). 29. W. G. Hoover, J. Chem. Phys. 44, 221 (1966). 30. B. J. Alder, W. G. Hoover, and D. A. Yang, J. Chem. Phys. 49, 3688 (1968). 31. J. G. Kirkwood, J. Chem. Phys. 18, 380 (1950). 32. C. H. Bennet and B. J. Alder, J. Chem. Phys. 54, 4796 (1971).
vch 4 Okt 2005 10:43
7
Microcanonical Thermostatistics as Foundation of Thermodynamics: The Microscopic Origin of Condensation and Phase Separation Dieter H. E. Gross Hahn-Meitner Institut and Freie Universit¨ at Berlin, Fachbereich Physik, Glienicker Str. 100, 14109 Berlin, Germany
Jedermann weiß, daß die W¨ arme die Ursache der Bewegung sein kann, daß sie sogar eine bedeutende bewegende Kraft besitzt: die heute so verbreiteten Dampfmaschinen beweisen dies f¨ ur jedermann sichtbar . . . Das Studium dieser Maschinen ist von h¨ ochstem Interesse, denn ihre Wichtigkeit ist ungeheuer, und ihre Anwendung steigert sich von Tag zu Tag. Sie scheinen bestimmt zu sein, eine große Umw¨ alzung in der Kulturwelt zu bewirken . . . Die Erzeugung von Bewegung ist bei den Dampfmaschinen stets an einen Umstand gekn¨ upft . . . Dieser Umstand ist die Wiederherstellung des Gleichgewichtes des W¨ armestoffes, ¨ d.h. ein Ubergang von einem K¨ orper mit mehr oder weniger erh¨ ohter Temperatur auf einen anderen, wo sie niedriger ist . . .
108
7 Microcanonical Thermostatistics as Foundation of Thermodynamics
¨ Uberall, wo ein Temperaturunterschied besteht, und wo daher die Wiederherstellung des Gleichgewichtes des W¨ armestoffes eintreten kann, kann auch die Erzeugung von bewegender Kraft stattfinden. Der Wasserdampf ist ein Mittel zur Erlangung dieser Kraft, aber er ist nicht das einzige, alle Stoffe der Natur k¨ onnen zu diesem Zweck benutzt werden . . . Umgekehrt ist es stets m¨ oglich, wo man solche Kraft anwenden kann, Temperaturunterschiede entstehen zu lassen. Sadi Carnot (1796-1832) Abstract Conventional thermo-statistics addresses infinite homogeneous systems within the canonical ensemble. However, some 150 years ago the original motivation of thermodynamics was the description of steam engines, i.e. boiling water. Its essential physics is the separation of the gas phase from the liquid. Of course, boiling water is inhomogeneous and as such cannot be treated by canonical thermo-statistics. Then it is not astonishing, that a phase transition of first order is signaled canonically by a Yang-Lee singularity. Thus it is only treated correctly by microcanonical Boltzmann-Planck statistics. This is elaborated in the present article. It turns out that the Boltzmann-Planck statistics is much richer and gives fundamental insight into statistical mechanics and especially into entropy. The respective procedure can even be performed to some extend rigorously and analytically. The microcanonical entropy has a very simple physical meaning: It measures the microscopic uncertainty that we have about the system, i.e. the number of points in 6N -dimensional phase, which are consistent with our information about the system. It can rigorously be split into an ideal-gas part and a configuration part which contains all the physics and especially is responsible for all phase transitions. The deep and essential difference between ”extensive” and ”intensive” control parameters, i.e. microcanonical and canonical statistics, is exemplified by rotating, self-gravitating systems.
7.1
Introduction
Since the beginning of thermodynamics in the middle of the 19th century its main motivation was the description of steam engines and the liquid to gas transition of water. Here water prefers to become inhomogeneous and develop a separation of
vch 4 Okt 2005 10:43
7.2 What is Entropy?
109
the gas phase from the liquid, i.e. water boils. As conventional canonical statistics works only for homogeneous, infinite systems, phase separation processes remain outside of standard Boltzmann-Gibbs thermo-statistics, which, consequently, signals phase-transitions of first order by Yang-Lee singularities. It is amusing that this fact that is essential for the original purpose of thermodynamics to describe steam engines was treated incompletely in the past 150 years. The system must be somewhat artificially split into (still macroscopic) pieces for each individual phase [1]. For this purpose, and also to describe small systems or non-extensive ones like self-gravitating very large systems, we need a new and deeper definition of statistics and as the heart of it: of entropy. Also the second law can rigorously be formulated only microcanonically. Already Clausius [2, 3, 4] and Prigogine [5] distinguished between external and internal entropy generating mechanisms. Canonical Boltzmann-Gibbs statistics is not sensitive to this important difference.
7.2
What is Entropy?
Entropy, S, is the fundamental entity of thermodynamics which distinguishes thermodynamics from all other physics. Therefore, its proper understanding is essential. The understanding of entropy is sometimes obscured by frequent use of the Boltzmann-Gibbs canonical ensemble, and the thermodynamic limit. Also its relationship to the second law is often beset with confusion between external transfers of entropy, de S, and its internal production, di S. The main source of the confusion is of course the lack of a clear microscopic and mechanical understanding of the fundamental quantities of thermodynamics like heat, external vs. internal work, temperature, and last not least entropy, at the times of Clausius and possibly even today. Clausius [2, 3] defined a quantity which he first called the ”value of metamorphosis” in Ref. [3]. Eleven years later he [4] gave it the name ”entropy”, S, defined by b Sb − Sa =
dE , T
(7.1)
a
where T is the absolute temperature of the body when the momentary change is done, and dE is the increment (positive respectively negative) of all different
vch 4 Okt 2005 10:43
110
7 Microcanonical Thermostatistics as Foundation of Thermodynamics
forms of energy (heat and potential) put into respectively taken out of the system (later, however, we will learn that care must be taken of additional constraints on other control parameters like e.g. the volume, see below). From the observation that heat does not flow from cold to hot (see Section 7.3, however Section 7.4) he went on to enunciate the second law as ∆S =
dE ≥0, T
(7.2)
which Clausius called the ”uncompensated metamorphosis”. As will be worked out in Section 7.4 the second law as presented by Eq. (7.2) remains valid even in cases where heat flows from low to higher temperatures. Prigogine ([5], c.f. also [1]) quite clearly stated that the variation of S with time is determined by two distinct mechanisms: the flow of entropy, de S, to or from the system under consideration; and its internal production, di S. While the first type of entropy change de S (that effected by exchange of heat with its surroundings) can be positive, negative or zero, the second type of entropy change di S (that caused by the internal creation of entropy) can be only positive in any spontaneous transformation. Clausius gives an illuminating example in Ref. [3]: When an ideal gas suddenly streams under insulating conditions from a small vessel with volume V1 into a larger one (V2 > V1 ), neither its internal energy U , nor its temperature changes, nor external work done, but its internal (Boltzmann-)entropy Si rises, by ∆S = N ln (V2 /V1 ) (c.f. Eq. (7.18)). Only by compressing the gas (e.g. isentropically) and creating heat ∆E = E1 [(V2 /V1 )2/3 − 1] (which must be finally drained) it can be brought back into its initial state. Then, however, the entropy change in the cycle, as expressed by the integral Eq. (7.2), is positive (= N ln (V2 /V1 )). This is also a clear example for a microcanonical situation where the entropy change by an irreversible metamorphosis of the system is absolutely internal. It occurs during the first part of the cycle, the expansion, where there is no heat exchange with the environment, and consequently no contribution to the integral Eq. (7.2). The construction by Eq. (7.2) is correct though artificial. After completing the cycle the Boltzmann-entropy of the gas is of course the same as initially. All this will become much more clear by Boltzmann’s microscopic definition of entropy, which will moreover clarify its real statistical nature. Boltzmann [6] later defined the entropy of an isolated system (for which the energy exchange with the environment dQ = 0) in terms of the sum of possible
vch 4 Okt 2005 10:43
7.2 What is Entropy?
111
configurations, W , which the system can assume consistent with its constraints of given energy and volume S = kB · ln W
(7.3)
as written on Boltzmann’s tomb-stone, with W (E, N, V ) =
→
→
→ → d3N p d3N q 0 δ(E − H{ q , p }) 3N N !(2π)
(7.4)
in semi-classical approximation. E is the total energy, N is the number of particles and V the volume. Or, more appropriate for a finite quantum-mechanical system W (E, N, V ) = T r[PE ] all eigenstates n of H with given N,V , = and E < En ≤ E + 0
(7.5)
and 0 is an energy parameter being a measure of the macroscopic energy resolution. This is still up to day the deepest, most fundamental, and most simple definition of entropy. There is no need of the thermodynamic limit, no need of concavity, extensivity and homogeneity. In its semi-classical approximation, Eq. (7.4), W (E, N, V, . . .) simply measures the area of the sub-manifold of points in the 6N -dimensional phase-space (Γ-space) with prescribed energy E, particle number N , volume V , and some other time invariant constraints which are here suppressed for simplicity. Because it was Planck who coined it in this mathematical form, I will call it the Boltzmann-Planck principle. There are various reviews on the mathematical foundations of statistical mechanics, e.g., the detailed and instructive article by Alfred Wehrl [7]. Wehrl shows how the Boltzmann-Planck formulae, Eqs. (7.3) and (7.5), may be generalized to the famous definition of entropy in quantum mechanics by von Neumann [8] S = −T r[ρ ln(ρ)] ,
(7.6)
addressing general (also non projector like) densities ρ. Wehrl discusses the conventional, canonical, Boltzmann-Gibbs statistics where all constraints on ρ are fixed only to their mean, allowing for free fluctuations. These free, unrestricted
(E − E)2 imply an uncontrolled enerfluctuations of the energy ∆E = gy exchange with the universe, de S in Prigogine’s definition. For the homogeneous phase of a system with short-ranged interactions (∆E/ E) vanishes in
vch 4 Okt 2005 10:43
112
7 Microcanonical Thermostatistics as Foundation of Thermodynamics
the thermodynamic limit. In general this, however, is dangerous; for there are situations where the fluctuations are macroscopic and (∆E/ E) does not vanish in the thermodynamic limit. An example are phase transitions of first oder where ∆E = Elatent , the latent heat of transformation. Another well known is the case of long-range interactions like in gravitating systems. Wehrl points to many serious complications with this definition. The canonical statistics remains up to now the conventional ”microscopic” foundation of thermodynamics (c.f. von Neumann [8], Jaynes [9], Ruelle [10], Balian [11]). Caratheodori [12] gave it an even axiomatic foundation where one intensive variable (mostly the temperature, T ) is a basic control variable. However, in the case of conserved variables, we know more than their mean; we know these quantities sharply provided external perturbations can sufficiently be hold under control. In microcanonical thermodynamics, we do not need von Neumann’s definition Eq. (7.6), and can work on the level of the original, Boltzmann-Planck definition of entropy, Eqs. (7.3) and (7.5). We thus explore statistical mechanics and entropy at their most fundamental level. This approach has the great advantage that the axiomatic level is extremely simple. Because such analysis does not demand scaling or extensivity, it can further be applied to the much wider group of non-extensive systems from nuclei to galaxies and address the original object for which thermodynamics was enunciated some 150 years ago: phase separations. Thus the ubiquitous appearance of water-surfaces which demonstrate phase-separation, controlled by the limited energy resources on earth, get their statistical explanation. The Boltzmann-Planck formula has a simple but deep physical interpretation: W or S are the measure of our ignorance about the complete set of initial values for all 6N microscopic degrees of freedom which are needed to specify the N -body system unambiguously [13]. To have complete knowledge of the system we would need to know (within its semiclassical approximation Eq. (7.4)) the initial position and velocity of all N particles in the system, which means we would need to know a total of 6N values. Then W would be equal to one and the entropy, S, would be zero. However, we usually only know the value of a few parameters that change slowly with time, such as the energy, number of particles, volume and so on. We generally know very little about the positions and velocities of the particles. The manifold of all these points in the 6N -dimensional phase space is the microcanonical ensemble, which has a well-defined geometrical size W and, by Eq. (7.3), a non-vanishing entropy, S(E, N, V, . . .). The dependence of S(E, N, V, . . .) on its arguments determines completely thermostatics and equilibrium thermodynamics.
vch 4 Okt 2005 10:43
7.3 The Zeroth Law in Conventional Extensive Thermodynamics 113 Clearly, Hamiltonian (Liouvillean) dynamics of the system cannot create the missing information about the initial values, i.e., the entropy, S(E, N, V, . . .) cannot decrease. As has been further worked out [14] and more recently in Ref. [15] the inherent finite resolution of the macroscopic description implies an increase of W or S with time when an external constraint is relaxed. Such is a statement of the second law of thermodynamics, which requires that the internal production of entropy be positive for every spontaneous process. Analysis of the consequences of the second law by the microcanonical ensemble is appropriate because, in an isolated system (which is the one relevant for the microcanonical ensemble), the changes in total entropy must represent the internal production of entropy, and there are no additional uncontrolled fluctuating energy exchanges with the environment.
7.3
The Zeroth Law in Conventional Extensive Thermodynamics
This section and the following discuss mainly systems that have no other macroscopic (extensive) control parameter besides energy; the particle density is not changed, and there are no chemical reactions. In conventional (extensive) thermodynamics thermal equilibrium of two systems (1&2) is established by bringing them into thermal contact which allows free energy exchange. Equilibrium is established when the total entropy Stotal (E, E1 ) = S1 (E1 ) + S2 (E − E1 )
(7.7)
is maximal dStotal (E, E1 )|E = dS1 (E1 ) + dS2 (E − E1 ) = 0 .
(7.8)
Under an energy flux ∆E2→1 from 2 → 1 the total entropy changes to lowest order in ∆E by ∆Stotal |E = (β1 − β2 )∆E2→1 ,
β=
1 dS = . dE T
(7.9)
Consequently, a maximum of Stotal (E = E1 + E2 , E1 )|E will be approached when sign(∆Stotal ) = sign(T2 − T1 )sign(∆E2→1 ) > 0 .
(7.10)
vch 4 Okt 2005 10:43
114
7 Microcanonical Thermostatistics as Foundation of Thermodynamics
From here Clausius’ first formulation of the second law follows: Heat always flows from hot to cold. Essential for this conclusion is the additivity of S under the split (Eq. (7.7)). There are no correlations, which are destroyed when an extensive system is split. Temperature is an appropriate control parameter for extensive systems.
7.4
No Phase Separation without a Convex, Non-Extensive S(E)-Dependence
The weight exp (S(E) − E/T ) of the configurations with energy E in the definition of the canonical partition sum ∞ eS(E)−E/T dE
Z(T ) =
(7.11)
0
becomes here bimodal, at the transition temperature it has two peaks, the liquid and the gas configurations which are separated in energy by the latent heat. Consequently S(E) must be convex and the weight in Eq. (7.11) has a minimum between the two pure phases. Of course, the minimum can only be seen in the microcanonical ensemble where the energy is controlled and its fluctuations forbidden. Otherwise, the system would fluctuate between the two pure phases by an, for macroscopic systems even macroscopic, energy ∆E ∼ Elat of the order of the latent heat. I.e. the convexity of S(E) is the generic signal of a phase transition of first order and of phase-separation [16]. Such macroscopic energy fluctuations and the resulting negative specific heat are already early discussed in high-energy physics by Carlitz [17]. The ferromagnetic Potts-model illuminates in a most simple example the occurrence of a convex intruder in S(E) which induces a backbending caloric curve T (E) = (∂S/∂E)−1 with a decrease of the temperature T (E) with rising energy [18]. A typical plot of s(e, N ) = S(E = N e)/N in the region of phase separation is shown in Fig. 7.1. Section 7.5 discusses the general microscopic reasons for the convexity (Moretto et al. [19] have previously put forward errors connected with the use of periodic boundary conditions; these assertions been rebutted [20, 21]). These results have far reaching consequences which are crucial for the fundamental understanding of thermo-statistics and thermodynamics: Let us split the
vch 4 Okt 2005 10:43
7.4 Phase Separation and S(E)-Dependence
115
s - (a + be)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 0.008
e b e3
0.006 s
0.004
0.002
e1
ea
an = sc
2Dssurf
e2
1.48
bmicro
1.46 1.44
ba
2Dssurf
bb
btr
1.42 1.40
latent heat
c(e)
1.38 1.36 80 60 40 20 0 -20 -40 -60 -80 -100 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
e = E/N Fig. 7.1 Ferromagnetic Potts model (q = 10) on a 50 · 50-lattice with periodic boundary conditions in the region of phase separation. At the energy e1 per lattice point the system is in the pure ordered phase, at e3 in the pure disordered phase. At ea little above e1 the temperature Ta = 1/β is higher than T2 and even more than Tb at eb a little below e3 . At ea the system separates into a few bubbles of disordered phase embedded in the ordered phase or at eb into a few droplets of ordered phase within the disordered one. If we combine two equal systems: one with the energy per lattice site ea = e1 + ∆e and at the temperature Ta , the other with the energy eb = e3 − ∆e and at the temperature Tb < Ta , and allowing for free energy exchange, then the systems will equilibrize at energy e2 with a rise of its entropy. The temperature Ta drops (cooling) and energy (heat) flows (on average) from b → a. I.e.: Heat flows from cold to hot! Thus, the Clausius formulation of the second law is violated. This is well-known for self-gravitating systems. However, this is not a peculiarity of only gravitating systems! It is the generic situation at phase separations within classical thermodynamics even for systems with short-range coupling and has nothing to do with long-range interactions
vch 4 Okt 2005 10:43
116
7 Microcanonical Thermostatistics as Foundation of Thermodynamics
system of Fig. (7.1) into two pieces a&b by a dividing surface, with half the number of particles each. The dividing surface is purely geometrical. It exists only as long as the two pieces can be distinguished by their different energy/particle values ea and eb . Constraining the energy-difference eb − ea = ∆e between the two, reduces the number of free, unconstrained degrees of freedom and reduces the entropy by −2∆Ssurf −corr.. Moreover, if the effect of the new surface would also be to cut some bonds: before the split there were configurations with attractive interactions across the surface which are interrupted by the division, their energy shifts upwards outside the permitted band-width 0 , and thrown out of the partition sum Eq. (7.5). I.e. the entropy will be further reduced by the split. If the constraint on the difference eb − ea is fully relaxed and eb − ea can fluctuate freely at fixed e2 = (ea + eb )/2, the dividing surface is assumed to have no further physical effect on the system. For an extensive system [S(E, N ) = N s(e = E/N ) = 2S(E/2, N/2)]. One would argue as follows: The combination of two pieces of N/2 particles each, one at ea = e2 − ∆e/2 and a second at eb = e2 + ∆e/2, must lead to S(E2 , N ) ≥ S(Ea /2, N/2) + S(Eb /2, N/2), the simple algebraic sum of the individual entropies because by combining the two pieces one normally looses information. This, however, is equal to [S(Ea , N ) + S(Eb , N )]/2, thus S(E2 , N ) ≥ [S(Ea , N ) + S(Eb , N )]/2. I.e. the entropy S(E, N ) of an extensive system is necessarily concave (c.f. Fig. (7.2)). For a non-extensive system we have in general S(E, N ) ≥ 2S(E/2, N/2) because again two separated, closed pieces have more information than their unification. Now, if E2 is the point of maximum positive curvature of S(E, N ) (convexity = upwards concave like y = x2 ) we have S(E2 , N ) ≤ [S(Ea , N ) + S(Eb , N )]/2 like in Fig. 7.1. However, the r.h.s. is larger than S(Ea /2, N/2) + S(Eb /2, N/2). I.e. even though S(E, N ) is convex at constant N , the unification of the pieces with Ea /2, N/2 and Eb /2, N/2 can still lead to a larger entropy S(E2 , N ). The difference between [S(Ea , N )+S(Eb , N )]/2 and S(Ea /2, N/2)+S(Eb /2, N/2) we call henceforth ∆Ssurf −corr . The correct entropy balance, before and after establishing the energetic split eb > ea of the system, is Saf ter − Sbef ore =
Sa + Sb − ∆Ssurf −corr. − S2 ≤ 0 2
(7.12)
even though the difference of the first and the last term is positive. In the inverse direction: By relaxing the constraint and allowing, on average, for an energy-flux (∆Eb→a > 0) opposite to Ta − Tb > 0, against the temperaturegradient (slope), but in the direction of the energy-slope, the entropy Stotal → S2
vch 4 Okt 2005 10:43
7.5 The Origin of the Convexities of S(E) and of Phase-Separation
117
Sb
Entropy
S2 (Sa+ S b)/2 Sa Energy
Fig. 7.2 Extensive (concave) S(E) dependence
increases. This is consistent with the naive picture of an energy equilibration. Thus Clausius’ ”energy flows always from hot to cold”, i.e. the dominant control-role of the temperature in thermo-statistics [22] is violated. Of course this shows again that unlike to extensive thermodynamics the temperature is not the appropriate control parameter in non-extensive systems. In the thermodynamic limit N → ∞ of a system with short-range coupling ∆Ssurf −corr. ∼ N 2/3 ,
∆Ssurf −corr. = ∆ssurf −corr. ∝ N −1/3 N
(7.13)
must tend to zero due to van Hove’s theorem.
7.5
The Origin of the Convexities of S(E) and of Phase-Separation
Many applications of microcanonical thermodynamics to realistic examples of hot nuclei, atomic clusters, and rotating astrophysical systems have been presented
vch 4 Okt 2005 10:43
118
7 Microcanonical Thermostatistics as Foundation of Thermodynamics
during the past twenty years which demonstrate convex intruders in the microcanonical entropy and, consequently, negative heat capacities. Such are reviewed in the publication list on the web site http://www.hmi.de/people/gross/ and elsewhere [23, 24, 25]. Here we shall illuminate the general microscopic mechanism leading to the appearance of a convex intruder in S(E, V, N, . . .) as far as possible by rigorous and analytical methods. This is the generic signal of phase transitions of first order and of phase-separation within the microcanonical ensemble. Assume the system is classical and obeys the Hamiltonian
H=
N p2i → + Φint [{ r }] , 2m
→
Φint [{ r }] :=
i
→
→
φ( r i − r j ) .
(7.14)
i 243 K are data with uncertainties reported by Butkovitch (1957), magnified in the lower panel, which were used for the regression
vch 4 Okt 2005 10:43
138
8 A Comprehensive Gibbs Potential of Ice Ih
a) Isentropic Compressibility at 101325 Pa
130
L
-1
125
Compressibility, 10 k/MPa
L D
6
D
D
L
D
L L
D
L
D D
G
G
G
M M G MG G G
D
D P P P P P
0
50
D
D
115 110 105
D D
120
DB
L
D
100
P
95
100
150
200
90
250
Temperature, T/K b) Isentropic Compressibility at -35.5°C
130
Compressibility, 10 k/MPa
-1
125 120
6
115 110 G G
105 G
100
G G
95 0
25
50
75
100
125
150
175
200
90
Pressure, P/MPa
Fig. 8.4 Isentropic compressibilities at normal pressure, upper panel, and at -35.5o C, lower panel, as computed by Eq. (8.9), are shown as curve. D: data computed from the correlation functions for elastic moduli of Dantl (1967, 1968, 1969) with about 3% error shown as lines above and below, P: correspondingly computed data of Proctor (1966) with about 1% error bounds, L: data of Leadbetter (1965), not used, B: Brockamp and R¨ uter (1969), M: Gammon et al. (1980, 1983), G: Gagnon et al. (1988)
vch 4 Okt 2005 10:43
8.4 Comparison with Experiments
139
Tab. 8.4 Selected values for isothermal compressibility K at the normal pressure melting point after different authors. By *) the value is given estimated from the curvature of the melting curve
Source Bridgman (1912a) Richards and Speyers (1914) Franks (1972) Hobbs (1974) Wexler (1977) Yen (1981), Yen et al. (1991) Henderson and Speedy (1987) *) Wagner et al. (1994) *) Tillner-Roth (1998) Marion and Jakubowski (2004) Feistel and Wagner (2005)
K, (TPa)−1 360 120 123 104 134 232 98 190 112 140 118
As shown in Table 8.4, experimental data for K at 0o C and normal pressure vary between 360 TPa−1 (Bridgman (1912a)) and 120 TPa−1 (Richards and Speyers (1914)), and this significant uncertainty remains present in more recent reviews of ice properties, too (Dorsey (1968), Yen et al. (1991)). The former Gibbs potential of Feistel and Hagen (1995) adopted the value 232 TPa−1 from Yen (1981), that of Tillner-Roth (1998), however, used the value 112 TPa−1 . More reliable values are available for isentropic compressibility, 1 κ=− v
∂v ∂P
=K− s
α2 T v , cP
(8.9)
which can be computed from the elastic moduli of the ice lattice (see Feistel and Wagner (2005) for details). The latter ones are determined acoustically or optically with high accuracy. Data at normal pressure computed from elastic constants of Dantl (1967) with error 3%, Proctor (1966) with 1%, Brockamp and R¨ uter (1969) with 8%, and of Gammon et al. (1980) and Gagnon et al. (1988) with errors below 1% are reproduced by the current formulation within their bounds over the temperature interval 60 - 273 K, as are high-pressure data of Gagnon et al. (1988) at -35o C between 0.1 and 200 MPa (Fig. 8.4).
vch 4 Okt 2005 10:43
140
8 A Comprehensive Gibbs Potential of Ice Ih
Heat Capacity cP in J/(kg.K)
Specific Heat at Normal Pressure
+ ++ ++ + + +++ ++++ + + + + + ++++++++++++
0
++ +++++ ++ + + + ++++ +++
50
++
100
+
+ ++
++
++
150
+ ++
++
+
+ ++
200
++
+
+
+ ++
250
2200 2000 1800 1600 1400 1200 1000 800 600 400 200 300
Temperature [K] Specific Heat Deviation at Normal Pressure F F
TR98
F F
8 6
F03
Residual [%]
F
S S S S
G
4
S S S SSS S x F SS S S S S F FF F x FF G S S S S S G G S x x F x S G S FFFFF S S Gx x S S GS S S xxGxxGG GG F FFFFFF G GxG S GS xxxxxxxx Gxx S S xGG G GGG xxxxx S G G F FF F x G G G S G S x G G x G x x S xxx GS G GG x xxxS xx x FFF F xxxxx xx GG xxxx G xxx x S SG G S S G G G G G G Gxxxxx xG x xxSS xxxx G G x xxS xFFFFG Sxxxxxxxx xx xxxxxxxxxx xxxG Gxx xx G xxx xxx SS SSxxxG x x xx xxx FFF FF xG xxxxx G Sxx xS xxx S xxx FFxxx x G x xxxxx xx F SS F S F xxxx G x S FF F S S FF S x SS F FF F S
2 0 -2 -4 -6 -8
0
50
100
150
200
250
300
Temperature [K]
Fig. 8.5 Isobaric specific heat capacity at normal pressure, upper panel, and relative deviation of measurements from Eq. (8.11), lower panel. G, +: by Giauque and Stout (1936), F, +: by Flubacher et al. (1960), S: by Sugisaki et al. (1968), x: Haida et al. (1974). For comparison, the heat capacities are shown as curves derived from the Gibbs functions TR98 (Tillner-Roth (1998)) and F03 (Feistel (2003)). The estimated experimental error of 2% is marked by solid lines
vch 4 Okt 2005 10:43
8.4 Comparison with Experiments
8.4.4
141
Heat Capacity
Compared to many other solids, heat capacity of ice Ih behaves anomalously. It follows Debye’s cubic law in the zero temperature limit, but at higher temperatures it violates the Gr¨ uneisen law which states that the ratio of isobaric heat capacity and isobaric thermal expansion is independent of temperature. Towards the melting temperature most crystalline solids possess a constant heat capacity, but this rule does not apply to ice. Isobaric heat capacities have been measured at normal pressures by several authors (Giauque and Stout (1936), Flubacher et al. (1960), Sugisaki et al. (1968), Haida et al. (1974)); all their results agree very well within their typical experimental uncertainties of about 2% (Fig. 8.5). In order to fit the desired potential function smoothly to this complex shape, a Pade approximation (Luke (1969)) was used with cubic asymptotic behaviour in the low-temperature limit, and being linear towards high temperatures. The resulting rational function was then decomposed into complex partial fractions to support easier analytical treatment (Feistel and Wagner (2005)). The second temperature derivative of the Gibbs potential Eq. (8.1) provides the formula for the isobaric specific heat capacity 2 ∂ g (8.10) cP = −T ∂T 2 P =P0 as 2 1 2 1 cP = t · Re rk − + . cU t − tk t + tk tk
(8.11)
k=1
The unit heat capacity is cU = 1 J kg−1 K−1 . At very low temperatures, Eq. (8.11) converges towards Debye’s cubic law as cP = 0.0091 J kg−1 K−4 , T →0 T 3 lim
(8.12)
which is in good agreement (2%) with the corresponding limiting law coefficient cP = 0.0093 J kg−1 K−4 T →0 T 3 lim
(8.13)
derived by Flubacher et al. (1960) from their experiment. Formula (8.11) properly describes the experimental data within their error range over the entire temperature interval (Fig. 8.5). With formula (8.11), heat capacities can be computed for arbitrary pressures, which were not available before.
vch 4 Okt 2005 10:43
142
8 A Comprehensive Gibbs Potential of Ice Ih
8.4.5
Entropy
Classical thermodynamics defines entropy by heat exchange processes. This way, only entropy differences can be measured for a given substance, thus leaving absolute entropy undefined and requiring an additional reference value like the Third Law. For this reason, the IAPWS-95 formulation of liquid water specifies entropy to vanish at the triple point. Statistical thermodynamics, however, defines entropy theoretically and permits its absolute determination. For water vapour this was done by Gordon (1934) from spectroscopic data at 298.1 K and normal pressure, resulting in the specific entropy of vapour sV = 45.101 cal deg−1 mol−1 = 10474.6 J kg−1 K−1 . The latest such value, reported by Cox et al. (1989), is sL = 69.95 ± 0.03 J K−1 mol−1 for the absolute entropy of liquid water at 298.15 K and 0.1 MPa, which coincides very well with sL = 70.07 ± 0.22 J K−1 mol−1 , as computed using the formulation of this paper. For the ice Ih crystal a theoretical residual entropy s(0, P ) = 189.13 ± 0.05 J kg−1 K−1 was calculated by Pauling (1935) and Nagle (1966) from the remaining randomness of hydrogen bonds at 0 K. This value is very well consistent with Gordon’s (1934) vapour entropy, as Haida et al. (1974) confirmed experimentally with s(0, P ) = 189.3 ± 10.6 J kg−1 K−1 (Petrenko and Whitworth (1999)). The theoretical residual ice entropy leads to a nonzero physical entropy of liquid water at the triple point as sL (Tt , Pt ) = 3522 ± 12 J kg−1 K−1 while the IAPWS95 entropy definition for liquid water requires the residual entropy of ice to be s(0, P ) = −3333± 12 J kg−1 K−1 . Both versions are equally correct, but the latter value has to be used instead of the absolute one if phase equilibria between ice and fluid water are studied in conjunction with the IAPWS-95 formulation. Entropy is computed as the temperature derivative of free enthalpy, Eq. (8.1), s=−
∂g ∂T
,
(8.14)
P
resulting in 2 t tk − t s +2 = s0 + Re rk ln . sU tk + t tk
(8.15)
k=1
The unit specific entropy is here sU = 1 J kg−1 K−1 . Note that specific entropy at 0 K is a pressure-independent constant, in accordance with theory.
vch 4 Okt 2005 10:43
8.4 Comparison with Experiments
143
At the normal melting temperature Tm , entropy of ice, s, can be computed from formulation, and the experthe entropy of water, sL , given by the !IAPWS-95 " L imental melting enthalpies L = Tm · s − s of Giauque and Stout (1936), L = 333.37 ± 0.21 kJ kg−1 , and Osborne (1939), L = 333.5 ± 0.2 kJ kg−1 . The melting enthalpy resulting from Eq. (8.15) is L = 333.376 kJ kg−1 and agrees very well with those data.
8.4.6
Sublimation Curve
The correct analytical solution of the Clausius-Clapeyron differential equation Eq. (8.16) for the sublimation pressure Psubl (T ) s − sV dPsubl = dT 1/ρ − 1/ρV
(8.16)
is given by equal chemical potentials of the solid and the gas phases, g (T, Psubl ) = gV (T, Psubl )
(8.17)
using Eq. (8.1) for ice and the IAPWS-95 formulation for vapour. From Eq. (8.17) the sublimation pressure can numerically be obtained, e.g. by Newton iteration. Vapour pressure measurements, available between 130 and 273.16 K, corresponding to 9 orders of magnitude in pressure from 200 nPa to 611 Pa, are described by the current formulation well within their experimental uncertainties (Fig. 8.6). The Clausius-Clapeyron differential equation Eq. (8.16) can be integrated in lowest order approximation, starting from the triple point (T"t , Pt ), under the assump! tions of constant sublimation enthalpy, Lsubl = T sV − s ≈ Lt = 2834.3 kJ kg−1 , the triple point value of this formulation, and negligible specific volume of ice compared to that of the ideal gas. The result is usually called the Clausius-Clapeyron sublimation law (R is the gas constant): CC Psubl
(T ) = Pt exp
Lt R
1 1 − Tt T
.
(8.18)
The deviation between the very simple law, Eq. (8.18), and the correct sublimation pressure, Eq. (8.17), of this formulation is often smaller than the scatter of experimental vapour pressure data (Fig. 8.16). Other, more complex sublimation formulae are in even much better agreement with the current one, like those of Jancso et al. (1970) for T > 130 K, of Wagner et al. (1994) for T > 150 K, or of
vch 4 Okt 2005 10:43
8 A Comprehensive Gibbs Potential of Ice Ih
b) Sublimation Pressure Deviation
Pressure, Psubl
a) Sublimation Pressure
B B B B
B
120
B B B
M MM J M J M MM M M M M M B MM M M B M MM M M M M M M B M M B M M M M M M KK B K K K K K K KK KK KK K
MM
M MJ
M
M
JM
M
JJJM
DM
D M
D JM
J JJD DJJD J JJJJJ D JJJJJ JD
0.1 kPa B
1 Pa 0.1 Pa 10 mPa 1 mPa 0.1 mPa 1 µPa
200
220
240
60
B B B B
40
B
B B BB BBB B
CC
10 nPa
260
80
B
0.1 µPa 180
100
B
10 µPa
160
120
B
10 Pa
B B BBB B BB B B B BB
140
1 kPa
Residual, 100DP/P
144
120
M J J M K M J J KKKK M M J J MM B KMB M K M M MMM B M MMM M K MM B MM M M M B K M MMB M MM M M J B B K KK MM M JM M MM JM DM JD M JJJJJDD M M M JJJJJD JJJJJD J M MB M M JJ K KK MMM M M M M D JD K K M M K K M M K K K B B K M M M MM M M M M KK KKKM M KK KKKM B K B KK KKK K K B B K KK K K K
140
160
180
Temperature, T/K
200
220
240
260
20 0 -20 -40
Temperature, T/K
c) Sublimation Pressure Deviation
0.4 0.3 D
Residual, 100DP/P
CC
0.2
D J J
J J
J J J J
J
J D J J J J
J
0.1
J J J D
J D J J J
J J J J
J J J
0
J JD J J J J J J J J J
J J
J J
-0.1 -0.2 -0.3
252 254 256 258 260 262 264 266 268 270 272
-0.4
Temperature, T/K
Fig. 8.6 Sublimation curve, left upper panel, and relative vapour pressure deviations, right upper panel, from the solution of Eq. (8.17). Data points are B: Bryson et al. (1974), D: Douslin and Osborn (1965), J: Jancso et al. (1970), K: Mauersberger and Krankowsky (2003), M: Marti and Mauersberger (1993). For the fit only data were used for T > 253 K (P > 100 Pa) with uncertainties about 0.1 - 0.2 %, magnified in the lower panel as IAPWS-95 chemical potential deviation relative to that of ice. CC: Clausius-Clapeyron simplified sublimation law (Eq. (8.18)
Murphy and Koop (2004) for T > 130 K, which remain below 0.01% deviation in sublimation pressure in those temperature regions. Thus, present experimental vapour pressure data hardly provide a suitable means for assessing the accuracy of those formulae. Sublimation enthalpy Lsubl as derived from IAPWS-95 and the current potential function is almost constant over a wide range of pressures and
vch 4 Okt 2005 10:43
8.4 Comparison with Experiments
145
temperatures, it increases to a maximum of Lsubl = 2838.7 kJ kg−1 at 239 K and decreases again to Lsubl = 2811.2 kJ kg−1 at 151 K, thus justifying the success of the simple equation Eq. (8.18).
8.4.7
Melting Curve
The melting pressure equation of Wagner et al. (1994) describes the entire phase boundary between liquid water and ice Ih with an uncertainty of 3% in melting pressure. On the other hand, the freezing temperature of water and seawater derived by Feistel (2003) is more accurate at low pressures but not valid at very high pressures. The formulation given in the current paper takes the benefits of both formulae, i.e. it provides the most accurate melting temperature at normal pressure and reproduces the measurements of Henderson and Speedy (1987) with 50 mK mean deviation up to 150 MPa pressure (Fig. 8.7). Melting temperature TM of ice at given pressure P is given by equal chemical potentials of the solid and the liquid phase, g (TM , P ) = gL (TM , P )
(8.19)
using Eq. (8.1) for ice and the IAPWS-95 formulation for water. From Eq. (8.19) the melting temperature can numerically be obtained, e.g. by Newton iteration. Ginnings and Corruccini (1947) measured the volume change of a water-ice mixture when heating it electrically. They determined their Bunsen calorimeter calibration factor KGC47 to be KGC47 =
Lmelt = 270 370 ± 60 J kg−1 (1/ρ − 1/ρL ) ρHg
(8.20)
and used it for accurate ice density determination by means of melting enthalpy Lmelt , liquid water density ρL , and mercury density ρHg . This way the accuracy of ice density is limited by the accuracy of Lmelt , namely 0.06%, while the better accuracy of the calibration factor itself is 0.02%. The calibration factor is proportional to the Clausius-Clapeyron slope of the melting curve at normal pressure, 1/ρ − 1/ρL Tm dTM = =− = −74.311±0.015 mK MPa −1 .(8.21) dP s − sL ρHg KGC47
vch 4 Okt 2005 10:43
146
8 A Comprehensive Gibbs Potential of Ice Ih
M78
b) Melting Temperature Deviation FH95 B B
B
B
500
B
400 B
WSP94 TR98 B
Residual, DT/mK
B
H H
HB
H
B H
200 100 0
H
-100
B
B
300
B
B
HS87
-200 -300 -400 -500
F03 0
25
50
75
100
125
150
175
200
225
Pressure, P/MPa c) Melting Temperature Deviation FH95
WSP94
10
HS87
8 6
Residual, DT/mK
F03
4
TR98
2 0
GC47
-2
H
-4
M78
-6 -8 0
5
10
15
20
25
30
Pressure, P/MPa
35
40
45
50
-10
Fig. 8.7 Melting temperatures as functions of pressure in comparison to Eq. (8.19) of this paper. The low-pressure range is magnified in the lower panel. Data points are: B: Bridgman (1912a), H: Henderson and Speedy (1987). Melting curves are labelled by M78: Millero (1978), FH95: Feistel and Hagen (1995), WSP94: Wagner et al. (1994), TR98: TillnerRoth (1998), HS87: Henderson and Speedy (1987), F03: Feistel (2003). The cone labelled GC47 indicates the 0.02% uncertainty of the Clausius-Clapeyron slope at normal pressure after Ginnings and Corruccini (1947). The intercept of M78 and FH95 at zero pressure is due to the freezing temperature of air-saturated water
vch 4 Okt 2005 10:43
8.5 Conclusion
147
The Gibbs function of this paper provides for this melting point lowering the Clausius-Clapeyron coefficient χ=−
dTM = 74.305 mK MPa−1 , dP
(8.22)
which fits well into the 0.02% uncertainty interval of Eq. (8.21). Other standard formulae like that of Bridgman (1935), χ = 73.21 mK MPa−1 , of Millero (1978), χ = 75.3 mK MPa−1 , or of Wagner et al. (1994), χ = 72.62 mK MPa−1 , are significantly beyond this error limit (Fig. 8.19). At normal pressure, Eq. (8.19) provides a value of the melting temperature equal 0 = 273.152 518 K. Making use of the fact that triple point temperature and to TM normal pressure are exact by definition, and taking into account the small errors of the triple point pressure (Table 8.1) and of the Clausius-Clapeyron coefficient Eq. (8.21), the possible error of this normal melting temperature is estimated as only 2 µK (Feistel and Wagner (2005)).
8.5
Conclusion
A new, compact analytical formulation for the Gibbs thermodynamic potential of ice Ih is presented. It is supposed to be valid in temperature between 0 and 273.16 K and in pressure between 0 and 210 MPa, thus covering the entire region of stable existence in the (P T )-diagram. Well-known uncertainties as for compressibilities and thermal expansion coefficients could be significantly reduced by combining the various properties into a single, consistent formula. Combined with the IAPWS-95 formulation of fluid water, accurate values for melting and sublimation points can be derived in a consistent manner, replacing former separate correlation functions. This method can directly be extended to other aqueous systems like seawater. Thus, a Gibbs function of sea ice and the freezing points of seawater are made available up to 100 MPa (Feistel and Wagner (2005)). 339 data points of 26 different groups of measurements are reproduced by the new formulation within their experimental uncertainty. The formulation obeys Debye’s theoretical cubic law at low temperatures, and pressure-independent residual entropy as required by the Third Law. The uncertainty in isothermal compressibility of previous formulae is reduced by about 100 times, its new value at normal pressure is 118 ± 1TPa−1 . The uncertainty in the Clausius-Clapeyron slope χ at normal pressure of previous formulae is reduced by 100 times; for
vch 4 Okt 2005 10:43
148
8 A Comprehensive Gibbs Potential of Ice Ih
melting point lowering at normal pressure the Gibbs function of this paper provides the coefficient χ = 74.305 mK MPa−1 with 0.02% uncertainty. The absolute entropy of liquid water at the triple point is found to be 3522 ± 12 J kg−1 K−1 . The corresponding figure of absolute entropy of liquid water at 298.15 K and 0.1 MPa is 3889 ± 12 J kg−1 K−1 ; it agrees well with the latest CODATA key value, 3882.8 ± 1.7 J kg−1 K−1 (Cox et al. (1989)). The melting temperature at normal pressure is found to be 273.152518 ± 0.000002 K. Uncertainties of melting points at high pressures can be estimated by 50 mK. The density of ice at the normal pressure melting point is 916.72 kg m−3 with an estimated uncertainty of 0.01%, in excellent agreement with the value computed by Ginnings and Corruccini (1947). Density measurements of different authors deviate up to 0.3% in an apparently systematic manner. The question if and how newly formed ice crystals expand by about the same amount due to aging is not yet finally answered, neither by experiments nor by theory. The hypothetical shallow density maximum at about 70 K is not reflected in this formulation, further investigation of this point seems necessary in order for its decisive clarification, possibly in conjunction with an improved knowledge about the supposed phase transition to ice XI. The deviations in measured heat capacity at the ferroelectric transition point at about 100 K appear to be systematic but do not rise above the average experimental uncertainty threshold. One order of magnitude improvement in determination is apparently required to resolve those for being included into the theoretical formulation, as well as to meet the accuracy of the CODATA absolute entropy value for liquid water. Deviations of heat capacity due to pressure as determined in this formulation do hardly exceed the 2% range and must therefore be considered uncertain. An extension of the sublimation curve to lower temperatures and pressures is currently not possible because of missing data of water vapour heat capacities below 130 K. The molar cP value at 130 K is about 4R and must decrease exponentially to 1.5R at 0 K due to successively vanishing contributions from vibrational and rotational excitation states of the water molecules (Landau and Lifschitz (1966)). The exact shape of this curve is yet unknown but is required for the computation of the chemical potential of water vapour. Experimental data for ice Ih at high pressures and low temperatures are completely missing. Phase transition curves in this region are only very vaguely known. Verifying the current quantitative knowledge in those ”white areas” of the (P T )diagram remains a future task.
vch 4 Okt 2005 10:43
8.6 References
149
Acknowledgements The authors thank D. Murphy and V. E. Tchijov for hints on additional relevant literature. They are grateful to J¨ urn W. P. Schmelzer for offering the opportunity to present this study on the Nucleation Workshop 2004 in Dubna, and his assistance in preparing a printable version. They further thank C. Guder and U. Overhoff for extended numerical auxiliary works, and A. Schr¨ oder and B. Sievert for accessing special articles. Source code examples in Visual Basic 6, Fortran, and C++ of the Gibbs potential of ice Ih as presented in this paper are freely available on the internet as numerical supplement of the paper of Feistel et al. (2005).
8.6
References
1. S. E. Babb: Some Notes concerning Bridgman’s Manganin Pressure Scale. In: High Pressure Measurements, A. A. Giardanini and E. C. Lloyd (Eds.) (Butterworth, 1963, pgs. 115-123). 2. S. T. Bramwell, Nature 397, 212 (1999). 3. P. W. Bridgman, Proc. Am. Acad. Arts Sci. 47, 441 (1912a). 4. P. W. Bridgman, Proc. Am. Acad. Arts Sci. 48, 309 (1912b). 5. P. W. Bridgman, J. Chem. Phys. 3, 597 (1935). 6. P. W. Bridgman, J. Chem. Phys. 5, 964 (1937). 7. R. Brill and A. Tippe, Acta Cryst. 23, 343 (1967). 8. B. Brockamp and H. R¨ uter, Z. Geophys. 35, 277 (1969). 9. C. E. Bryson III, V. Cazcarra, and L. L. Levenson, J. Chem. Eng. Data 19, 107 (1974). 10. T. R. Butkovich, J. Glaciol. 2, 553 (1955). 11. T. R. Butkovich, SIPRE Res. Rep. 40, 1 (1957). 12. J. D. Cox, D. D. Wagman, and V. A. Medvedev, CODATA Key Values for Thermodynamics (Hemisphere Publishing Corp., 1989). 13. G. Dantl, Z. Phys. 166, 115 (1962); Berichtigung Z. Phys. 169, 466 (1962). 14. G. Dantl, Elastische Moduln und mechanische D¨ ampfung in Eis-Einkristallen (Dissertation, TH Stuttgart, 1967). 15. G. Dantl, Phys. kondens. Materie 7, 390 (1968).
vch 4 Okt 2005 10:43
150
8 A Comprehensive Gibbs Potential of Ice Ih
16. G. Dantl: Elastic Moduli of Ice. In: Physics of Ice, N. Riehl, B. Bullemer, H. Engelhardt (Eds.) (Plenum Press, 1969, pgs. 223-230). 17. G. Dantl and I. Gregora, Naturwissenschaften 55, 176 (1968). 18. N. E. Dorsey, Properties of Ordinary Water-Substance (Hafner Publishing Company, 1968). 19. D. R. Douslin and A. Osborn, J. Sci. Instrum. 42, 369 (1965). 20. R. Feistel, Progr. Oceanogr. 58, 43 (2003). 21. R. Feistel and E. Hagen, Progr. Oceanogr. 36, 249 (1995). 22. R. Feistel and W. Wagner, J. Mar. Res. 63, 95 (2005). 23. R. Feistel, W. Wagner, V. Tchijov, and C. Guder, Ocean Sci. Disc. 2, 37 (2005); http://www.copernicus.org/EGU/osd/2/37; Ocean Sci. 1, 29 (2005); http://www.copernicus.org/EGU/os/1/29. 24. N. H. Fletcher, The Chemical Physics of Ice (Cambridge University Press, 1970) 25. P. Flubacher, A. J. Leadbetter, and J. A. Morrison, J. Chem. Phys. 33, 1751 (1960). 26. F. Franks, F.: The Properties of Ice. In: F. Franks (Ed)., Water - A Comprehensive Treatise (Plenum Press, 1972, vol. 1, pgs. 115-149). 27. S. Fukusako, Int. J. Thermophys. 11, 353 (1990). 28. R. E. Gagnon, H. Kiefte, M. J. Clouter, and E. Whalley, J. Chem. Phys. 89, 4522 (1988). 29. R. E. Gagnon, H. Kiefte, M. J. Clouter, and E. Whalley, J. Chem. Phys. 92, 1909 (1990). 30. P. H. Gammon, H. Kiefte, and M. J. Clouter, J. Glaciol. 25, 159 (1990). 31. P. H. Gammon, H. Kiefte, M. J. Clouter, and W. W. Denner, J. Glaciol. 29, 433 (1983). 32. W. F. Giauque and J. W. Stout, J. Amer. Chem. Soc. 58, 1144 (1936). 33. D. C. Ginnings and R. J. Corruccini, J. Res. Natl. Bur. Stand. 38, 583 (1947). 34. A. R. Gordon, J. Chem. Phys. 2, 65 (1934). 35. L. A. Guildner, D. P. Johnson, and F. E. Jones, J. Res. Natl. Bur. Stand. 80A, 505 (1976). 36. O. Haida, T. Matsuo, H. Suga, and S. Seki, J. Chem. Thermodyn. 6, 815 (1974). 37. S. J. Henderson and R. J. Speedy, J. Phys. Chem. 91, 3096 (1987). 38. P. V. Hobbs, Ice Physics (Clarendon Press, 1974) 39. R. Howe and R. W. Whitworth, J. Chem. Phys. 90, 4450 (1989).
vch 4 Okt 2005 10:43
8.6 References
151
40. R. W. Hyland and A. Wexler, Trans. Am. Soc. Heat. Refrig. Air Cond. Eng. 89, 500 (1983). 41. ISO-1993. ISO STANDARDS HANDBOOK. International Organization of Standardization, 117 pgs. 42. M. Jakob and S. Erk, Mitteilungen Phys.-Techn. Reichsanst. 35, 302 (1929). 43. G. Jancso, J. Pupezin, and W. A. V. Hook, J. Phys. Chem. 74, 2984 (1970). 44. S. Kawada, J. Phys. Soc. Japan 32, 1442 (1972). 45. J.-L. Kuo, J. V. Coe, S. J. Singer, Y. B. Band, and L. Ojam¨ ae, J. Chem. Phys. 114, 2527 (2001). 46. J.-L. Kuo, M. L. Klein, S. J. Singer, and L. Ojam¨ ae: Ice Ih - XI-Phase Transition: A Quantum Mechanical Study. In: Abstracts of Papers, 227(th) ACS National Meeting, Anaheim, CA, USA, March 28-April 1, 2004, PHYS-463; American Chemical Society. 47. L. D. Landau and E. M. Lifschitz, Statistische Physik (Akademie-Verlag, Berlin, 1966). 48. S. LaPlaca and B. Post, Acta Cryst. 13, 503 (1960). 49. A. J. Leadbetter, Proc. Roy. Soc. Lond. 287, 403 (1965). 50. C. Lobban, J. L. Finney, and W. F. Kuhs, Nature 391, 268 (1998). 51. C. Lobban, J. L. Finney, and W. F. Kuhs, J. Chem. Phys. 112, 7169 (1998). 52. D. K. Lonsdale, Proc. Roy. Soc. Lond. 247, 424 (1958). 53. Y. L. Luke, The Special Functions and Their Approximations (Academic Press, New York, London, 1969). 54. G. M. Marion and S. D. Jakubowski, Cold Regions Sci Techn. 38, 211 (2004). 55. J. Marti and K. Mauersberger, Geophys. Res. Lett. 20, 363 (1993). 56. K. Mauersberger and D. Krankowsky, Geophys. Res. Lett. 30, 1121 (2003). 57. H. D. Megaw, Nature 134, 900 (1934). 58. D. M. Murphy and T. Koop, Review of the vapour pressure of ice and supercooled water for atmospheric applications. Submitted to Q. J. Royal Met. Society. 59. J. F. Nagle, J. Math. Phys. 7, 1484 (1966). 60. O. V. Nagornov and V. E. Chizhov, J. Appl. Mech. Techn. Phys. 31, 343 (1990). 61. N. S. Osborne, Research paper 260, J. Res. Nat. Bur. Stand. 23, 643 (1939). 62. V. F. Petrenko, CRREL Report 93-25, 1-21 (1993). 63. V. F. Petrenko and R. W. Whitworth, Physics of Ice (Oxford University Press, 1999).
vch 4 Okt 2005 10:43
152
8 A Comprehensive Gibbs Potential of Ice Ih
64. K. S. Pitzer and J. Polissar, J. Phys. Chem. 60, 1140 (1956). 65. L. Pauling, J. Amer. Chem. Soc. 57, 2680 (1935). 66. R. W. Powell, Proc. Roy. Soc. Lond. 247, 464 (1935). 67. E. R. Pounder, Physics of Ice (Pergamon Press, 1965). 68. H. Preston-Thomas, Metrologia 27, 3 (1990). 69. T. M. Proctor Jr. J. Acoust. Soc. Amer. 39, 972 (1966). 70. T. W. Richards and C. L. Speyers, J. Amer. Chem. Soc. 36, 491 (1914). 71. K. R¨ ottger, A. Endriss, J. Ihringer, S. Doyle, and W. F. Kuhs, Acta Crystall. B 50, 644 (1994). 72. M. Sugisaki, H. Suga, and S. Seki, Bull. Chem. Soc. Japan 41, 2591 (1968). 73. Y. Tajima, T. Matsuo, and H. Suga, Nature 299, 810 (1982). 74. H. Tanaka, J. Chem. Phys. 108 4887 (1998). 75. R. Tillner-Roth, Fundamental Equations of State (Shaker Verlag, 1998). 76. F. K. Truby, Science 121, 404 (1955). 77. W. Wagner and A. Pruß, J. Phys. Chem. Ref. Data 31, 387 (1995). 78. W. Wagner, A. Saul, and A. Pruß, J. Phys. Chem. Ref. Data 23, 515 (1994). 79. A. Wexler, J. Res. Nat. Bur. Stand. 81 A, 5 (1977). 80. Y.-C. Yen, CCREL Report 81-10, 1-27 (1981). 81. Y.-C. Yen, K. C. Cheng, and S. Fukusako: Review on Intrinsic Thermophysical Properties of Snow, Ice, Sea Ice, and Frost. In: Proceedings 3rd International Symposium on Cold Regions Heat Transfer, J. P. Zarling and S. L. Faussett (Eds.), Fairbanks, AL, 187-218.
vch 4 Okt 2005 10:43
9
An Empirical Approach in the Thermodynamics of Amorphous Solids and Disordered Crystals: Glasses, Vitrified Liquid Crystals and Crystals with Orientational Disorder Ivan S. Gutzow(1) , Boris P. Petroff(2) , Snejana V. urn W. P. Schmelzer(4) Todorova(3), and J¨ (1)
Institute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria (2) Institute of Solid State Physics, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria (3) Geophysical Institute, Bulgarian Academy of Sciences, Sofia 1113, Bulgaria (4)
Institut f¨ ur Physik, Universit¨ at Rostock, 18051
Rostock, Germany A word may be said about the inclusion of so many references to older work. In collecting his material an author is under some obligation to the originators of a subject, and there is also a question of integrity involved which does not seem to be as fully appreciated as it should be. The question to whom a scientific idea or experimental method is due was formerly considered important although many now think it is not . . .
154
9 Thermodynamics of Amorphous Solids and Disordered Crystals
The method of presenting material without any reference to the nature of its development, although still favoured in some quarters, has ceased to be acceptable almost overnight. A distinguished and original scientist (Eyring) has recently said that the treatment of a subject is more attractive and informative when it elucidates how particular concepts arose. . . James R. Partington
Abstract
A survey is given on the experimental evidence and the basic ideas concerning the theoretical interpretation of the thermodynamic properties of solids with frozen-in disorder. As the main and best known representatives of this class of solids, common glass-forming systems are analysed, in which an amorphous state of increased disorder is frozen-in in the glass-transition region. The process of vitrification of undercooled liquids is illustrated on the example of several typical cases. The basic thermodynamic and kinetic characteristics of the glass transition are outlined in terms of existing semi-empirical theoretical concepts and approximations. In a second step of the analysis particular attention is given to the process of freezing-in of structural and orientational disorder in nematic or in smectic liquid crystals and in plastic crystals. It is shown that in both cases this process has the main characteristics of glass transition in typical undercooled melt: however, with the limitations imposed by the oriented state in liquid crystals and the particular rheology and the crystalline structure of plastic crystals like cyclohexanol. In both cases solids with frozen-in thermodynamic properties are obtained. They can be called glassy liquid crystals and glassy plastic crystals. In the same connection is also considered the process of formation of amorphous or crystalline magnetic solids with frozen-in orientational spin structure - the so-called spin glasses. The analysis of the thermodynamic and structural evidence of the so-called molecular glassy crystals like CO, N2 O2 etc. shows that the kinetics of their formation is quite different from glass-transition and that it is connected with their crystallization process. In these substances the crystallisation leads to two or more frozen-in energetically nearly equal structural configurations. The methods of thermodynamic characterisation of solids with frozen-in defect structures are described and compared in their significance.
vch 4 Okt 2005 10:43
9.1 Introduction
9.1
155
Introduction
In the present article we consider experimental evidence on the thermodynamic properties of solids with frozen-in configurational disorder: topological, orientational and conformational. Such a categorization of disorder into three main types corresponds both to Ziman’s models [1] and to the hierarchy of disordered structures the present authors have introduced in previous publications [2, 3]. In so-called structural (real or ordinary or typical) glasses which are molecular and polymeric representatives of organic substances or of oxide, silicate, halide or chalcogenide compounds stemming from different fields of inorganic chemistry (including the so-called metallic glasses, i.e. vitrified metallic alloy systems) all three types of disorder are frozen in. The terms ”real” or ”structural” glasses, attributed to both technical and laboratory model glass-forming systems (i.e. to solids with frozen-in amorphous structure) are usually employed to distinguish them from both computer-modelled states of frozen-in disorder or from spin glasses. In the latter mentioned systems (having either an amorphous or a crystalline topological structure) a magnetic, orientational type of disorder is frozen-in [4]. Although considered in many aspects from the standpoint of statistical physics models (see [4, 5]) relatively little experimental evidence is accumulated on the caloric properties of spin glasses. Besides spin glasses there exist also many other crystalline substances, in which different types of topological or orientational disorder is frozen in. Simple molecular disordered crystals (sometimes somewhat exaggeratedly and as shown here: improperly called ”glassy crystals”) like CO, N2 O2 or ice were investigated beginning from 1929. In the early 1980s it was found that several plastic crystals have also thermodynamic properties similar to those of glasses and can be with more justification called glassy crystals. Most results on the thermodynamics of states with frozen-in disorder, as it is also seen from the present analysis, come from hundreds of investigations, calorimetric measurements, performed in the 80 years passed since 1925 with the already (more or less loosely) defined real glasses. These caloric investigations gave in fact the first possibility for a detailed consideration of the thermodynamics of solids with frozen-in disorder and the general aspects of the kinetics of vitrification. In this sense undercooled glass-forming liquids and the ”real” glasses they form gave an instructive and general example of the freezing-in process in metastable systems, both amorphous and crystalline. Many different cases have to be considered in this respect.
vch 4 Okt 2005 10:43
156
9 Thermodynamics of Amorphous Solids and Disordered Crystals
A particularly interesting representative of vitrified melts, as it turns out from our analysis, are glass-forming liquid crystals: these systems, which are in fact partly oriented liquids with complicated structures (somewhat inappropriately called crystals) can be frozen-in to highly ordered glasses with peculiar thermodynamic properties. Thus glassy liquid crystal systems give an example of frozen-in, partially ordered amorphous solids. All frozen-in systems are non-equilibrium, non-ergodic, even non-thermodynamic bodies. Thus they do not obey the Third Principle of thermodynamics in its classical formulation. As seen in the next paragraphs, the first caloric experiments with ”real” glassforming systems gave the first experimental clues how to treat a very general problem. The first result that turned out after a thorough caloric analysis of the initially examined cases was that in real glasses a non-vanishing zero-point entropy, ∆Sg0 > 0, is frozen-in. Nearly of the same significance was also the evidence obtained with the mentioned molecular crystals in which a comparison of caloric and tensimetric measurements with the statistically calculated entropy (the so called ”spectroscopic” entropy of the respective gas because the statistical calculations are based on spectroscopic evidence) gave the proof that here also an orientational disorder and values ∆S 0 > 0 exist in solids with seemingly perfect crystalline structure. Similar results were also obtained with the plastic crystals, in which by thorough caloric measurements it is verified that the orientational disorder is frozen-in in the crystal at the melting temperature. In the present analysis we are trying to find the most significant common features and the differences, characterizing both real glass-forming systems and the other mentioned forms of disordered solids. This can be easily done when glasses and other forms of frozen-in disorder are described in terms of simple thermodynamic invariants, e.g. in connecting the properties of the system as a liquid at melting temperature, T = Tm , and at (or below) the freezing-in temperature (e.g. the temperature of vitrification, T = Tg , for real glass formers). We have organized the present contribution in the following way: first existing experimental evidence on the process of vitrification in the case of real glasses is illustrated by several examples and critically summarized. Then in the following sections evidence on spin glasses and on the vitrification of liquid crystals is compared with the vitrification of ”real” glasses. After that results on the thermodynamic properties of crystals with frozen-in orientational disorder are discussed, summarized and compared with vitrification characteristics of real glasses and liquid crystals. At the end an attempt is made to formulate several consequences showing the common features and the specific differences for various systems with frozen-in disorder.
vch 4 Okt 2005 10:43
9.2 Thermal, Mechanical and Electrical Properties of Disordered Solids
9.2
157
Experimental Evidence on Specific Heats and Change of Mechanical and Electrical Properties in Gaseous and in Disordered Solids. Simon’s Approximation
The first indications on the peculiar state of typical real glasses as kinetically frozen-in, thermodynamically non-stable systems were obtained in the early 1920s as a result of thorough investigations of the temperature dependence of the specific heats, Cp (T ), of two typical glass-forming substances, of SiO2 and of glycerol as crystals, undercooled liquids and glasses. The subsequent measurements were initiated independently in two of the best calorimetric laboratories of those times: in Nernst’s institute in Berlin (see [6-9]) and in Giauque’s laboratory in the United States [10]. They were performed under the guidance of the mentioned scientists, both Nobel prize laureates, who have to be cited among the most famous thermodynamicians of the 20th century. These investigations were initially implemented in order to give an experimental proof of the applicability of the Third Principle of thermodynamics (in the form of Nernst’s heat theorem [2, 3, 6, 11, 12, 13]) also to non-crystalline solids. The results obtained indicated, however, that in contradiction to Nernst’s expectation (see [6, 9]) for both these glass-forming systems (and soon also for ethanol glass [14]) investigated, a non-vanishing, non-zero value of the zero-point entropy of the glass ∆S (T ) |T →0 = ∆Sg (0) = const = ∆Sg (Tg ) > 0 ,
(9.1)
remained as temperatures approach absolute zero. These results are presently confirmed for more than 120 glass-forming substances experimentally investigated (see [2, 3, 13-17]) with compositions ranging from organic high polymers [15] through network glasses like SiO2 , GeO2 and BeF2 [2, 16], with phosphates, borates etc. to the metallic alloy glass-forming systems summarized in Refs. 2, 18 and 19. The first recognition of the exceptional significance of the non-vanishing zeropoint entropy of glasses for the whole development of thermodynamics was given by another well-known 20th century thermodynamician - by F. Simon [8, 12, 13, 20] being at that time one of Walter Nernst’s closest collaborators (see [2, 6] for the detailed historic background). The initial results in this development were first summarized by Kauzmann [21] and by Eitel [22]. In our monograph [2] (as well as in Refs. 3, 14-17, 23) the ∆Sg -values of glasses are compiled for a
vch 4 Okt 2005 10:43
158
9 Thermodynamics of Amorphous Solids and Disordered Crystals
great variety of substances and especially for those with particular structural or methodical significance. An extension to newer problems (spin glasses etc.) may be found in Refs. 24, 25, 26 and 27. In the last above mentioned two papers as well as in Refs. 17, 28 and 29, the first data on Cp (T )-measurements and on zero-point entropies in vitrified liquid crystals are given. The first ∆S (T )-determinations were performed by Sorai et al. [29, 30] in 1973 and 1983. In Refs. 2 and 3, we have given a graphical analysis of the results obtained for ∆Sg with typical glass-formers. A historical survey of the theoretical approaches and ideas connected with the development of the problems of the zero-point entropy of glasses may be also found in Refs. 2 and 16. The essential point to be mentioned in this respect is that Albert Einstein [31] had predicted from the very beginning of the discussions on the statistical foundations of the Third Principle (as early as 1913) that for glasses and for solid solutions as states of constant (i.e. frozen-in) statistical disorder, as a rule zero-point entropies (∆S (T = 0) = const > 0) have to be expected. A further development of these ideas and of the possible natural limits of the values of ∆Sg for glasses was given later on by Pauling and Tolman [32] using general thermodynamic considerations and then by Gutzow [33], who employed a combination of structural models. The first determinations of zero-point entropies ∆S 0 > 0 for disordered molecular crystals were performed by Clusius [34] and again in Giauque’s laboratory (see [35, 36]). It was Giauque [35] and another Nobel laureat, L. Pauling [36], who gave at the time 1930-1935 the first structural models of orientational disorder in molecular crystals and especially in H2 O (ice) and in the structurally corresponding crystalline deuterium oxide, D2 O. Kaischew [37] demonstrated experimentally in 1938 that in CO-crystals two structural arrangements with very narrow potential energy have to exist. According to his measurements even lowest possible crystal growth rates at CO-crystal synthesis or prolonged annealing (as demonstrated by Eucken, see the respective summary in Ref. 34) could not remove or at least decrease the zero-point entropy, ∆S 0 , in disordered molecular crystals. The results of Eucken and Kaischew thus turned out to be of great significance in treating the thermodynamics of disordered crystals. The Cp (T )-measurements performed with glass-forming melts in the early days of the mid-twenties of the last century as well as further investigations of later times, summarized in Refs. 2, 3, 16, 17, and 23, showed that upon vitrification the specific heats Cp (T ) of the undercooled melts of typical glass-formers drop in
vch 4 Okt 2005 10:43
9.2 Thermal, Mechanical and Electrical Properties of Disordered Solids
0.5
Tg 1
0.4
Cp , cal/K.g
159
2 0.3
3
0.2
373
473 T, K
573
Fig. 9.1 Experimental heat capacity curves of B2 O3 melts upon vitrification (after Thomas and Parks ([46], see also [47]): (1) cooling run curve; (2) heating curve, after slow cooling run; (3) heating curve after rapid cooling run
a relatively narrow interval (the glass transformation range, in Tammann’s somewhat non-adequate terminology (see [2, 38])) to low values, alike to those of the corresponding crystal (see Figs. 9.1 and 9.2). A similar Cp (T )-jump was observed also in other thermodynamic properties, which are to be considered as thermodynamic coefficients of the vitrifying system, i.e. as second-order derivatives of the corresponding thermodynamic potential (at a pressure p =const: of the Gibbs free energy difference, ∆G). Typical examples in this respect are besides the already mentioned specific heats Cp (T ) 2 d ∆G (T ) , (9.2) ∆Cp (T ) = dT 2 p the thermal expansion coefficient, ∂G 1 ∂ 1 ∂ 2 ∆G 1 ∂V , = = α= V ∂T p V ∂T ∂p T p V ∂T ∂p
(9.3)
the coefficient of compressibility, ∂G 1 ∂ 1 ∂ 2 ∆G 1 ∂V , =− =− κ=− V ∂p T V ∂p ∂p T T V ∂2p
(9.4)
vch 4 Okt 2005 10:43
160
9 Thermodynamics of Amorphous Solids and Disordered Crystals
a
Cp , cal/K.mol
60
Tm
Tg
Tm
b
60 40
1
40
1, 3
20
20
0
100
200
T, K
1.30
0 113
300
Tm
c
T, K
273
100
353
Tm
d
Tg
5
a .10 , K
Tg
193
140 -1
1.32
r, g/cm3
Tg
80
2
1.34
60 20
1.36 -160
-80
0
80
T, oC
-160
-80
0
80
T, oC
Fig. 9.2 Temperature dependence of thermodynamic properties of glycerol: (a) Specific heats of liquid (◦), crystalline (), and vitreous (•) glycerol; (b) Dielectric constant of liquid (◦), crystalline (), and vitreous (•) glycerol; (c) Density of glycerol. Open and black points of the upper curve refer to liquid and vitreous glycerol, respectively, while the closed circles of the lower curve are experimental data for crystalline glycerol; (d) Thermal expansion coefficient, α, for glycerol. The bold line corresponds to the liquid and the vitreous state while the dashed line refers to the crystal (literature is cited in the text)
the modulus of elasticity, E(T ), E(T ) = −V
∂p ∂V
,
(9.5)
T
which is in fact the reciprocal of the coefficient of compressibility (see [11]), the dielectric constant, which can be represented as ε=
∂ 2 ∆G ∂2E
vch 4 Okt 2005 10:43
, T,p
(9.6)
9.2 Thermal, Mechanical and Electrical Properties of Disordered Solids
161
7
E, Kg/mm2
6 5 4 3 2 1 0
0
100
200
300 400 Temperature,oC
500
600
700
Fig. 9.3 Temperature dependence of the modulus of elasticity, E(T ), of three soda-lime silicate glasses in the transformation region according to measurements by McGraw [45]. Note that the temperature course of E(T ) upon vitrification is reciprocal to that of the Cp (T )- or of the α(T )-curves from Fig. 9.2
where E is the electric field strength (see [2, 39]). The change of Cp (T ), α (T ) and ε (T ) for glycerol upon vitrification is shown in Fig. 9.2. The change of the modulus of elasticity E(T ) of silicate glasses is evident from Fig. 9.3. It is interesting to note that as a reciprocal of the corresponding thermodynamic coefficient (κ) the temperature course of E(T ) shows also a dependence reciprocal to the temperature course of α (T ), Cp (T ) etc. It also turned out that the thermodynamic functions, which have to be considered as the first-order derivatives of the thermodynamic potential (e.g. mo
∂∆G ), molar enthalpy (∆H = − T1 d∆G lar volume (∆V = ∂p dT ), molar entropy T ! ∂∆G " (∆S = − ∂T p )) with respect to temperature T and other external parameters of state (pressure, p, electric, E, or magnetic, M , field strength etc.) display a break point in their temperature course when vitrification takes place in cooling run experimentation. Volumes (and densities, ρ, as given on Fig. 9.2) can be directly measured: enthalpies and entropies have to be calculated from Cp (T )measurements (see, however, also Section 9.3 for alternative ways of direct caloric determination of ∆Hg (T = 0)).
vch 4 Okt 2005 10:43
9 Thermodynamics of Amorphous Solids and Disordered Crystals
5 l.10 , cal/cm.s.K
162
a
120 100 80 60 40
Tm
Tg
20
b
1.50
n(T )
1.49 1.48 1.47 1.46
Tg
153
193
233
273
313
353
T, K Fig. 9.4 Temperature dependence of two of the physical properties of glycerol, connected with either thermodynamic coefficients or thermodynamic functions: (a) The change of the coefficient of thermal conductivity, λ, with temperature for the undercooled melt and the glass (bold lines and black points) and as a crystal (broken line with black points) (see [40]). λ is determined to a large extent by the change of ∆Cp (T ) upon vitrification (note the respective jump in the λ(T )-curve at T = Tg ); (b) The change of the coefficient n(T ) of optical refraction for liquid (high temperatures) and for glassy (low temperatures) glycerol. At Tg a break point from undercooled melt to glass is seen in the n(T )-curve determined by the similar change in the molar volume of glycerol (data according to Ubbelohde [40])
vch 4 Okt 2005 10:43
9.2 Thermal, Mechanical and Electrical Properties of Disordered Solids
163
Out of existing experimental evidence for the mentioned more than one hundred glass-forming substances, critically summarized in previous publications [2, 3, 14, 15, 16, 17, 21, 23], we would like to discuss here in more details the results, obtained for a very typical real glass former - glycerol and only to mention several other also typical or more exotic laboratory models of glass-forming substances.
1
Y (T )
2
3
0
4 Tg
T
Fig. 9.5 Tammann’s definition of the caloric glass transition temperature Tg and of the glass transition region. Curve 1: change of thermodynamic functions with temperature. Curve 2: change of the first derivatives of the thermodynamic functions (i.e. the thermodynamic coefficients e.g. Cp (T )). Curve 3: change of the second-order derivatives of the thermodynamic functions. According to Tammann [38] Tg is defined by the maximum in the (dCp (T )/dT )-curve. A similar but more distinct definition of Tg can be given connecting it with the nullification of the second-order derivative of the specific heats (i.e. at (d2 ∆Cp (T )/dT 2 ) = 0) and the corresponding inflection point is given with curve 4, as proposed by Gutzow et al. [43]
Glycerol is probably the thermodynamically most thoroughly investigated glassforming substance. With this model, measurements have been performed on the temperature dependence of the specific heats, Cp (T ) [8, 9, 10], of the dielectric constant, ε [40], and of the density, ρ [41, 42] of the crystalline, liquid and vitreous substance as they are summarized on Fig. 9.2. It is seen that in the glass tran-
vch 4 Okt 2005 10:43
164
9 Thermodynamics of Amorphous Solids and Disordered Crystals
sition region in the vicinity of the glass transition temperature, Tg is manifested at cooling as a jump-like change described by a sigmoidal curve of the Cp (T ) and ε(T )-dependences and also of the temperature functions of the coefficient of thermal conductivity, λ(T ) (see also Figs. 9.1, 9.3a and 9.4a). The turning point of the S-shaped Cp (T )-curve has been proposed by Tammann [38] as the caloric definition of the glass transition temperature Tg (see Fig. 9.5) where according to this author the changes of a thermodynamic function Y [e.g. ∆H (T ), of its first-order derivative (i.e. Cp (T )) as well as of its second-order derivative (∂ 2 ∆H (T ) /dT 2 )] upon vitrification are given. In a recent series of publications Gutzow et al. [43, 44], following Tammann’s proposal, defined Tg more exactly by the second-order derivative of Cp (T ) (i.e. as a third-order derivative of ∆H (T )). In this way, the circumstance was used point, "which that at T = Tg the sigmoidal shaped Cp (T )-curve has an inflection ! 2 is guaranteed mathematically, when (dCp (T ) /dT ) > 0 and d Cp (T ) /dT 2 = 0. In the temperature dependence of the density, ρ(T ), of the same substance as presented on Fig. 9.2c at T → Tg a typical break point is observed. This break point gives, as shown on Fig. 9.5, another possibility to define the glass transition temperature, Tg . The differentiation of the corresponding ρ(T )-curve gives the temperature dependence of the thermal expansion coefficient, α(T ), for vitrifying glycerol (Fig. 9.2d). Thus similar to the Cp (T ) and ε(T ) dependences a jump also appears on the α(T )-curve for vitreous glycerol, practically at a temperature close to the caloric glass transition temperature, Tg , defined by the above discussed Cp (T )-course. A similar sigmoidal temperature course in the α(T ) or Cp (T )-curves is also observed in heating run experiments, when the already vitrified glass is reheated and thus again transformed into the undercooled liquid (see the two heating Cp (T )curves for B2 O3 -glass on Fig. 9.1, given according to Ref. 46 and data reported in Ref. 47). Here, however, a typical overshot ”nose” maximum in the Cp (T ) heating curves is found (see Figs. 9.1 and 9.6). This ”nose” part is increased with increasing heating rate (see Fig. 9.2 and further experimental evidence in this respect, collected in Refs. 2, 43 and 44). Glycerol gives also an interesting example on the way the physical properties of the system are changed upon glass transition when determined either by a thermodynamic function, or by a thermodynamic coefficient. Thus the coefficient of optical refraction n(T ) (which according to a well-known dependence is determined by the molar volume, V , of the vitrifying melt) displays a break-point, analogous to the one, observed in the ρ(T )-curve (compare Figs. 9.2 and 9.4). On
vch 4 Okt 2005 10:43
9.2 Thermal, Mechanical and Electrical Properties of Disordered Solids
100
T/K 300
200
400
500
165
700
40
Cp (T), J/mol.K
1 DSm 30 2 20
3
Tg
T0
Tm
0 2.0
2.2
2.4 lg T, (K)
2.6
2.8
Fig. 9.6 Temperature course of the specific heats Cp (T ) of the glass to melt transition in As2 S3 . Heating curves 2,1 with a typical ”nose” have to be compared with the curve 3 of the crystal. As a square is given the value ∆S(Tm ) of this substance in Cp (T ) vs log T coordinates. Below T0 in the undercooled fictive melt ∆Cp (T ) = 0 (dotted line) and ∆S(T ) = 0 is to be expected (according to data by Blachnik and Hoppe [51])
the other hand the coefficient λ of heat transfer, which is determined significantly by the change of specific heats, Cp (T ), of the vitrifying system (see [2]) displays at T ≈ Tg , the typical jump, observed for thermodynamic coefficients (see again Figs. 9.2 and 9.4). The viscosity, η, of vitrifying undercooled liquids dramatically changes in the glass transition range. In the vicinity of the calorically defined Tg -value, a breakpoint of the η(T )-dependence is observed (see Figs. 9.7 and 9.8) illustrating this property for both an organic glass (see Fig. 9.7 and Ref. 48) and for a typical silicate glass-forming melt [49]. It is to be noted that the course of the logη(T )
vch 4 Okt 2005 10:43
166
9 Thermodynamics of Amorphous Solids and Disordered Crystals
lg h (cPoise)
15
10
Tg
5
0 3
3.5
4
4.5
5
3
10 /T Fig. 9.7 Viscosity η(T ) vs. temperature curve for a typical organic glass-forming model system (salol) in the temperature range of vitrification. Note the break point of the η(T )-curve in the vicinity of Tg , indicated (according to caloric measurements) by an arrow (see [48]). With a dashed line is given the expected increase in η(T ) of the fictive undercooled melt
vs. T curve near the break-point is reciprocal to the course of the ∆S (T )-curve. In Ref. 50 and in Section 9.9, this peculiar behavior of the logη(T )-curve finds its explanation in terms of an interesting theoretical viscosity model and in the change of the entropy temperature functions upon vitrification. In the transformation range, the viscosity of undercooled melts usually changes from 1010 to 1014 dPa·s and at temperatures approaching Tg this kinetic parameter has values of about 1013 dPa·s. The break point in the η(T )-curve for T → Tg is a very significant finding, as it demonstrates the change of a kinetic property upon vitrification. As far as viscosity η is directly connected with the time of molecular relaxation, τ (T ) (see [2]), of the melt or with the self-diffusion coefficient, D(T ), of the building units of the system, a similar behavior is also to be expected (and observed) for these kinetic characteristics of the vitrifying system. According to classical free volume models the temperature course and the break-point in the logη(T )-curve is caused by the change of the relative free vol-
vch 4 Okt 2005 10:43
9.2 Thermal, Mechanical and Electrical Properties of Disordered Solids
lg h (dPa.sec)
20
167
Tg
15 10 5
0
373
773
1273
1573
T, K
Fig. 9.8 Temperature dependence of the viscosity of a classical glass-forming melt: the change of the slope of the logη(T )-curve upon vitrification for a soda-lime silicate glass in a very broad viscosity interval (after Winter’s analysis [49], based on experimental data of several authors)
ume V (T )f − V (T )c /V (T )c of the melt, where the indices f and c here and below denote the molar volumes of the liquid and the crystal. With the index g in following discussions are indicated values and functions, referring to the glass. Knowing the Cp (T )-dependence for liquid and crystalline glycerol and its enthalpy of melting, ∆Hm (from direct determinations at the temperature of melting, Tm ; thus the entropy of melting, ∆Sm = ∆Hm/Tm , is also known) the temperature course of the difference of the thermodynamic functions of liquid and crystalline glycerol can be established by using a known classical thermodynamic formalism (see [2]). The result is given on Fig. 9.9 using Simon’s calculations. In order to construct the entropy difference, ∆S (T ) = Sf (T ) − Sc (T ), we have to write according to classical thermodynamics (see e.g. [2, 11]) in the range of its expected applicability, i.e. for equilibrium Tm ∆S (T ) = ∆Sm −
∆Cp dT , T
(9.7)
T
vch 4 Okt 2005 10:43
9 Thermodynamics of Amorphous Solids and Disordered Crystals
100 200 300
0
b
Tg
2000
Tm
DG(T )
1000 0
100 200 300
T, K
Tg
Tm
100 200 300 T, K
T, K
5 4 3 2 1 0
Tm
DH(T )
d
DV(T )
cm3/mol
0
Tg
100 200 300
T, K
5 4 3 2 1
c
Tm
cal/mol
20 16 12 8 4
DS(T )
Tm
cal/mol
0
a T0 Tg
cal/mol.K
20 16 12 8 4
DCp (T )/T
cal/mol.K2
168
e
Tg
100 200 300 T, K
Fig. 9.9 Temperature dependence of the thermodynamic functions of vitrifying glycerol as constructed by Simon ([20]; see also [2]). The respective calculations are based on the measurements of the specific heats of the liquid, vitreous and crystalline states of this substance (Fig. 9.2a): (A) Reduced specific heat difference (∆Cp (T )/T ) (the shaded area from Tm to T0 gives the entropy of melting of glycerol if no vitrification has occurred); (B) Entropy difference ∆S(T ); (C) Gibbs free energy difference ∆G(T ); (D) Molar free volume ∆V (T ): the curve is based on density measurements (Fig. 9.2c); (E) Enthalpy difference ∆H(T ). By a dashed line the extrapolated thermodynamic functions of the fictive undercooled melt below Tg are given. The temperature curves for the real undercooled melt (above Tg ) and for the glass (below Tg ) are specified by full lines. All graphs on this figures indicate the difference of liquid or glass with respect to the crystalline state. Note that here typically ∆Sg ≈ (1/3)∆Sm
where ∆Sm = Sf (Tm )−Sc (Tm ) is the entropy of melting (the enthalpy of melting being ∆Hm = ∆Sm Tm ). Assuming as indicated on Figs. 9.2, 9.6, and 9.10 that at T < Tg for various glass-forming systems (including metallic alloy glasses, see
vch 4 Okt 2005 10:43
9.2 Thermal, Mechanical and Electrical Properties of Disordered Solids
DCp (T), cal/K.mol
44
a
T0
36 28 Tm
Tg
20 350
400
450 T, K
500
550 Tm
b DS(T), cal/K.mol
169
50 30 T0
Tg
10 100
200
300
400
500
T, K
Fig. 9.10 Heat capacity Cp (T ) (upper picture) and the entropy (lower picture) of lithium acetate as a liquid (black points above Tg ) and as a glass (below Tg ) according to Ref. 52. Open circles refer to crystalline samples. This is the Cp (T )−∆S(T )-diagram for a nontypical glass former (with relatively low frozen-in ∆Sg -value i.e. ∆Sg ≈ (1/5)∆Sm ). Note that here as in Fig.9.6 a logarithmically scaled abscissa is used for temperature T and that T0 indicates again the temperature at which for the fictive undercooled melt ∆Cp (T ) = 0 and thus ∆S(T ) = 0. On the bottom figure is given the temperature dependence of the entropy difference, ∆S(T ), of the undercooled melt and the glass with respect to the crystal
vch 4 Okt 2005 10:43
170
9 Thermodynamics of Amorphous Solids and Disordered Crystals
Fig. 9.10), we can write ∆Cp (T ) = Cp,g (T ) − Cp,c (T ) = 0, it follows that the entropy value ∆Sg = ∆S (T ) T Tm down to T = Tg in order to calculate ∆Sg . This brought with it Simon’s third necessary assumption: vitrification is a process in which entropy is only frozen-in, no entropy production was considered by him. Simon’s assumptions turned out to be of exceptional significance in creating a general concept of the glass transition and for more than 70 years they determined the development of ideas in glass vitrification. Simon’s first two assumptions gave, moreover, the impetus to use thermodynamics of irreversible processes as the phenomenological basis of glass science. This is clearly seen in the subsequent treatment of the glass transition in the framework of irreversible thermodynamics by Prigogine and Defay [55] and by Davies and Jones [56]. These authors used in fact Simon’s two approximations, also neglecting with him the possibility of entropy production.
vch 4 Okt 2005 10:43
172
9 Thermodynamics of Amorphous Solids and Disordered Crystals
Using more or less complicated structural molecular models, the values of ∆Sg of various glasses can be calculated accepting Simon’s first assumption. In this respect besides the already mentioned paper by Gutzow [33] in which a generalized lattice-hole model was used to estimate ∆Sg of various glasses, particular attention deserve models for calculating ∆Sg in silicate glasses and in vitreous SiO2 by using the arrangement principles of silicate bonding. These models are discussed in more details in Section 9.9. Following the same line of argumentation as in deriving Eq. (9.8) and again accounting for ∆Cp (T ) = 0 at T < Tg it turns out that the enthalpy difference
∆Hg = ∆H (T )T ∆H0 , where ∆H0 is the zero-point enthalpy, corresponding to the equilibrium system. This is in fact another contradiction to the Third Principle. Introducing the thus obtained constant values of ∆Hg and ∆Sg into a well-known thermodynamic dependence ∆G = ∆H − T ∆S
(9.10)
another violation of the Third Law of thermodynamics becomes evident. For systems in equilibrium, obeying classical thermodynamics, the dependence
d∆G (T ) dT
=0
(9.11)
T →0
has to be fulfilled as predicted by Nernst’s theorem. However, with above results we have to write Eq. (9.10) for T < Tg as (see also Section 9.5) ∆Gg (T ) = ∆Hg − T ∆Sg .
(9.12)
Thus it becomes evident (see Figs. 9.9 and 12) that the thermodynamic potential of the glass approaches zero-point temperatures not as predicted by Eq. (9.11) but with a constant non-zero slope
d∆Gg dT
= −∆Sg . T Tr .
DH DHm
b
DG(T ), DH(T )
DG(T ), DH(T )
a
DH DHm
DHg
DH0
DG 0
Tg
Tm T
DG
DH0 0
Tg
Tm T
Fig. 9.12 Nernst’s diagram and Simon’s (∆G(T ), ∆H(T ))-construction [2]: (a) Temperature dependence of the enthalpy, ∆H(T ), and of the Gibbs free energy difference, ∆G(T ), melt-crystal for an equilibrium system, obeying the third law of thermodynamics (Nernst’s (∆H, ∆G)-diagram); (b) Temperature dependence of the enthalpy and the Gibbs free energy for a vitrifying system (full curves) according to Simon’s approximation [13, 20]. The temperature course of ∆G(T ) expected from the third law of thermodynamics is indicated in the construction by a dashed curve. ∆H0 is the zeropoint enthalpy difference of the two different equilibrium states of the same substance (fictive melt, undercooled liquid and crystal), ∆Hg is the zero-point enthalpy difference between the glass and the crystal. Note also that for equilibrium systems (d∆H/dT ) = (d∆G/dT ) = 0 is observed at T → 0, while for a glass at T → 0 the relation (d∆H/dT ) = 0 still holds, however, (d∆G/dT ) < 0
The Third Principle requires also (again in the form of Nernst’s heat theorem) that approaching the absolute zero temperature besides Eq. (9.11) also the relation
∆H (T ) |T →0 = ∆G (T ) |T →0 = ∆H (0)
(9.14)
vch 4 Okt 2005 10:43
174
9 Thermodynamics of Amorphous Solids and Disordered Crystals
has to be fulfilled (see Eq. (9.10)). For the frozen-in system Eq. (9.12) (written again for T → 0) gives in analogy to Eq. (9.14) ∆Hg |T →0 = ∆Gg |T →0 .
(9.15)
However, here ∆Hg > ∆H0 , where ∆H0 corresponds to the fictive equilibrium undercooled liquid at T → T0 . These two deviations from the Third Principle of classical thermodynamics are best illustrated by constructing Nernst’s ∆H (T ) /∆G (T )-diagram and comparing it with the diagram as it corresponds to Eqs. (9.14) and (9.15) as this is done on Fig. 9.12. The above discussed values of ∆Sg and ∆Hg are obtained via Cp (T )-measurements according to Eq. (9.8) and (9.9). This is a tedious experiment, requiring expensive calorimetric apparatus, covering (as done in Simon’s classical investigations [8, 9]) the whole temperature interval from the respective temperature Tm down to temperatures, approaching the zero-point of temperature (Witzel [7] with SiO2 and Gibson and Giauque [10] with glycerol experimented only down to hydrogen boiling temperatures, i.e. T ∼ 20 K). Simon brought the experiments to helium temperatures i.e. to T ≈ 4 K. Below the mentioned low temperatures even significant changes in the Cp (T )-course cannot affect the values of the thermodynamic functions (see [2, 7] and the discussion given in Ref. 2 on contemporary Cp (T )-measurements with glasses at ultra-low temperatures i.e. for T < 1 K). In present day experiments, the convenient differential scanning microcalorimetry (DSM) is mostly used. However, with this technique measurements cannot be extended lower than to nitrogen boiling temperatures (T ≈ 80 K). In most cases authors, analyzing the kinetics of vitrification by DSM-methods, extend their measurements even only to several degrees below the corresponding Tg temperature, more or less correctly assuming that the foregoing summarized classical results allow an extrapolation with ∆Sg =const down to T → 0 K. The application of DSM-measurements is illustrated on Figs. 9.10 and 9.11, where the Cp (T )-course of a vitrifying metallic alloy is also given. Such cases require fast experimentation, because of the high crystallization tendency of vitrifying metallic alloy melts. There exist also alternative methods of determining the caloric properties of glassforming melts and glasses. Most of them are usually applied to the direct determination of the frozen-in enthalpy, ∆Hg . They are discussed in the following section.
vch 4 Okt 2005 10:43
9.3 Alternative Ways of Determining Caloric Properties of Glasses
9.3
175
Alternative Ways of Determining Caloric Properties of Glasses
One possible method of direct determination of the frozen-in value of the enthalpy difference, ∆Hg , is to determine in a calorimeter the reaction heat for a chemical change in which the substance under consideration is reacting once as a glass and in another reaction as a crystal. Care has to be taken that the same reaction products are formed in both cases. A well-known example in this respect gives the determination of the reaction heats of quartz crystals and of silica glass in hydrofluoric acid according to the equation SiO2 + 4HF → SiF4 + 2H2 O + Q .
(9.16)
∆Hg can be thus determined as ∆Hg = Qglass − Qcrystal ,
(9.17)
where Qglass and Qcrystal are the respective reaction heats measured. Usually the solution of silicate glasses in hydrofluoric acid is carried out in platinum calorimeters (see e.g. [22]). Similar experiments can be also performed with water soluble glasses or with vitrified metallic alloys, dissolving them and the corresponding crystals in acid solution or in appropriate solvents. An interesting variation of this method of determination of ∆Hg by dissolution experiments was developed by Jenckel and Gorke ([57], see also the comments on this method in Ref. 2) in application to organic polymer glasses, dissolving them in organic solvents. According to the proposal of these authors instead of the heats of dissolution, ∆lg , of the glassy and of the fully crystalline samples of the polymer, ∆lc (which is difficult to prepare for most polymers), the heats of dissolution of the glassy polymer, ∆lg , and of the respective undercooled melt, ∆lf , are compared for the same temperature slightly above Tg . As far as it can be assumed that between the melt and its solution there is practically no energetic difference (i.e. when ∆lf ≈ 0), ∆Hg ∼ = ∆lg holds. Another method of ∆Hg -determination is to measure the heats of combustion, ∆Q∗c and ∆Q∗g , of the crystalline and of the glassy forms of a carbonaceous substance, e.g. of vitreous carbon and of graphite in pure oxygen in a conventional bomb calorimeter. The advantage of this method is its relative simplicity. Its
vch 4 Okt 2005 10:43
176
9 Thermodynamics of Amorphous Solids and Disordered Crystals
disadvantages are connected with the necessity of determining the relatively low value of ∆Hg as the difference of two comparatively large quantities, ∆Q∗c and ∆Q∗g , resulting in relatively large uncertainties in the ∆Hg -value. Nevertheless, in this way the ∆H-differences between the various forms of carbon (graphite, diamond and so-called vitreous carbon) have been determined with sufficient accuracy (see [58, 59] and results summarized in Ref. 58). Two further methods, which can be applied directly for the simultaneous determination of both ∆Hg and ∆Sg are based on the temperature dependence of the solubility, Cg and Cc , or of the vapor pressures, Pg and Pc , of the glass and the crystal formed of the same substance [2, 23, 60]. It can be shown that the solubility and the vapor pressure of a glass are higher than that of the corresponding crystal. From the temperature dependence of the ratio ln(Pg /Pc ), the Gibbs free energy difference, ∆Gg (T ), glass-crystal can be directly calculated as first quantitatively demonstrated in Refs. 23 and 60. From the respective solubility curve of vitreous carbon and of graphite in Ni-alloys, as it was determined in Refs. 58 and 59, the thermodynamics of the glass → diamond transition can be constructed as done in Ref. 58. In a similar way, measurements of the electromotoric force (EMF) of an electric cell where a suitable conducting glass is used as the anode, the cathode being a crystalline sample of the same substance, can be also employed for a direct determination of ∆Gg (T ). This possibility. discussed in more details in Ref. 61, was applied by Das and Huckle [62] to determine the thermodynamic properties of various vitreous carbon samples by EMF-measurements of the galvanic cell carbon glass/graphite [61, 62]. Of particular significance in analyzing the problems of glass solubility turned out to be measurements of the solubility in water of phenolphtaleine as a glass and as a crystal, performed by Grantcharova, Gutzow et al. [23, 63]. In this case the proof was given that the ∆Sg and ∆Hg -values, obtained from solubility measurements, are in good coincidence with those determined via direct Cp (T )-measurements, according to the formalism given with Eqs. (9.8) and (9.9).
9.4
Consequences of Simon’s Classical Approximation: the ∆G (T )-Course
As already mentioned, Simon [9, 20] assumed that vitrification takes place not in a temperature region (seen on Figs. 9.1, 9.3 and 9.8), but at the temperature
vch 4 Okt 2005 10:43
9.4 Consequences of Simon’s Classical Approximation
177
T = Tg . According to this approximation the change in the Cp (T )-course takes place not along the corresponding sigmoidal curve, but as a simple step-like jump. This step-like change of Cp (T ) at T = Tg , as seen from Figs. 9.1 and 9.2, is only a first approximation and a very substantial idealization. Simon’s approximation introduces a vertical straight line, going through the turning point of the real, sigmoidally changing curve. At heating run experimentation, when the reverse transition glass → undercooled melt takes place, the sigmoidal (i.e. S-shaped) Cp (T )-curves appear again, however, with the mentioned peculiar ”nose”, its height increasing with increasing heating rates. A thorough analysis of the real Cp (T )-course upon glass transition is possible, as shown in Refs. 43 and 44, on the basis of the thermodynamics of irreversible processes and of one of its basic dependences: on its phenomenological law and on de Donder’s and Bragg-Williams’ equations derived from it. In Refs. 43, 44, 64 and 65 the history of the corresponding theoretical development is outlined and the necessary literature is cited. However, in the empirical approach discussed in the present paper Simon’s assumption gives not only a safe basis in defining Tg (as the temperature of the Cp (T )-jump) but also of calculating the change of thermodynamic functions and the thermodynamic potential upon vitrification. The same classical approximation, as pointed out by Gutzow [60] in an article, devoted to the vapor pressure of glasses (see also [23] and the paper of Gupta and Moynihan [66] and the comments in Ref. 67 and 68) defines in a similar way the temperature dependence of the thermodynamic potential of the glass as a tangent to the ∆G (T )-curve of the metastable undercooled liquid. This becomes evident, recalling the definition of a straight line tangenting at the point (x0 ; y0 ) the plane curve f (x) df (x) + y0 (9.18) y = (x − x0 ) dx x→x0 and the thermodynamic significance of the respective derivative: here df (x) /dx = [∂ (∆G (T ) /∂T )]p = −∆S (T ). For the point (Tg ; ∆G (Tg )) we have thus
∂∆G (T ) ∂T
T →Tg
= −∆S (T ) T →Tg = −∆Sg .
(9.19)
Writing Eq. (9.10) for the metastable undercooled liquid (i.e. for an equilibrium system) for T = Tg as ∆G (Tg ) = ∆Hg − Tg ∆Sg ,
(9.20)
vch 4 Okt 2005 10:43
178
9 Thermodynamics of Amorphous Solids and Disordered Crystals
it follows with Eq. (9.9) that Eq. (9.20) corresponds in fact also to the same point (Tg ; ∆G (Tg )) of the straight line
∂∆G (T ) ∆Gg (T ) = (T − Tg ) ∂T
T →Tg
+ ∆G (Tg ) .
(9.21)
This straight line, describing the temperature dependence of the thermodynamic potential of the glass, is tangenting the ∆G (T )-line of the undercooled liquid at T → Tg (see Fig. 9.12). A more general formulation of above considerations in terms of two tangenting (G, T, p)-planes (i.e. in (G, p, T )-coordinates) was given by Gupta and Moynihan ([66], see also [68]).
9.5
Change of Kinetic Properties at Tg and Vitrification Kinetics
The kinetic properties of liquids - their kinetic coefficients (viscosity, η(T ), time of molecular relaxation, τ (T ), coefficient of diffusion, D(T ), etc.) - are at equilibrium unambiguous functions of the state parameters in the same way, as this applies for thermodynamic functions of equilibrium systems. Moreover, as far as the kinetic properties of melts and liquids are determined by the thermodynamic functions of the system (see [2]) the break-point in the temperature dependence of the thermodynamic functions is reflected also in a break-point of the η(T ), τ (T ) and D(T ) curves at T = Tg as seen from Figs. 9.7 and 9.8. In this way the equilibrium ↔ non-equilibrium transition upon vitrification is also manifested by a break-point in the temperature dependence of the kinetic coefficients of the system. In other cases (as with λ (T )), when the respective kinetic coefficient depends significantly on a thermodynamic coefficient (e.g. on Cp (T )) a step-like change is observed, as seen on Fig. 9.4a. Another crucial experimental finding, concerning the kinetic characteristics of vitrification, is that the value of the glass transition temperature, Tg , depends on the cooling rate, q = (−dT /dt). It follows a course established for the first time by Bartenev [69, 70] and Ritland [71] 1 = C1 − C2 log q0 , Tg
(9.22)
where C1 and C2 are constants and q0 is a constant heating rate. Upon heating run experimentation starting with already formed glasses a similar dependence
vch 4 Okt 2005 10:43
9.5 Change of Kinetic Properties at Tg and Vitrification Kinetics
179
2.50
r, g/cm 3
2.51 3 2.52
Tg3
2
Tg2 Tg1
1
4
2.53 400
500
600
o
T, C Fig. 9.13 Temperature dependence of the density, ρ, of a borosilicate glass, measured at three different cooling rates, q0 (after Ritland [71]). Curve 1: with 1 K/min; curve 2: with 2 K/min; curve 3: with 10 K/min; curve 4: annealing curve (i.e. nearly equilibrium ρ(T )-course). Note that Tg increases with increasing cooling rates
is observed determining Tg in the glass to undercooled melt transition (see [70]). The kinetic overshot at Tg leads to somewhat higher Tg -values and to the formation of the already discussed ”nose” in the Cp (T )-curve and in the temperature dependence of any other thermodynamic or kinetic coefficient. At cooling run experiments this ”nose” has never been observed. An explanation of these findings can be given in terms of the already mentioned theoretical approach, based on de Donder’s kinetic equation, as discussed in Refs. 43, 44, and 65. Thus, the temperature Tg has not the character of a thermodynamic equilibrium point like Tm or T2 in first and second-order thermodynamic phase transitions, but is a kinetically determined temperature. This fact is underlying again the non-thermodynamic, kinetic character of the glass transitions.
The Bartenev-Ritland formula (Eq. (9.22)) can be easily derived, employing a dependence introduced years ago in glass science as a postulate by Bartenev [69, 70] attributed by him to Ya. Frenkel and P. Kobeko. It connects the time of
vch 4 Okt 2005 10:43
180
9 Thermodynamics of Amorphous Solids and Disordered Crystals
10 3/Tg , K -1
1.30
1.25
1.20
-2
-1
0 lg q (K/sec)
+1
Fig. 9.14 First proof of the kinetic dependence of Tg on cooling or heating rate q0 (for an alkalisilicate glass, after Bartenev [69, 70]): Experimental proof of Eq. (9.22), giving for this case an activation energy U (Tg ) ≈ 267 J/mol
molecular relaxation, τ (T ), at T → Tg and the (constant) cooling rate, q0 , in the form τ (T ) q0 T →Tg ∼ = const.
(9.23)
The Bartenev-Ritland formula, Eq. (9.22), follows directly from Eq. (9.23), assuming that the τ (T ) or η(T ) dependences can be written as an exponential function of the activation energy, U (T ) τ (T ) = τ0 exp
U (T ) RT
(9.24)
with the approximation that U (T ) = U0 = const. A statement equivalent to Eq. (9.23) has been proposed (in fact: postulated) by Reiner [72, 73] in a somewhat peculiar way: via the introduction of the number Dh, bearing the name of a Biblical character (of the prophetess Deborah). It is the ratio of the time of molecular relaxation, τR (T ), of the system to the time, ∆tg ,
vch 4 Okt 2005 10:43
9.6 Liquid Crystals and Frozen-in Orientational Modes in Crystals
181
of observation (or the stay time of the system at T ≈ Tg in the transformation range) Dh T →Tg =
τR (T ) ∆tg
=1.
(9.25)
T →Tg
For undercooled liquids near the melting point Dh 1 holds, for glasses Dh → ∞ and for T → Tg , according to above proposal, Dh ≈ 1. It is obvious that Reiner’s stay time, ∆tg , is non-sufficiently defined and that Eq. (9.25) is introduced as a ”Deus ex machina”. Only recently Gutzow et al. [43, 44] could show that both Eqs. (9.23) and (9.25) can be in fact derived from general dependences in the framework of the thermodynamics of irreversible processes. In this way, in Refs. 43 and 44, both the kinetics and the thermodynamics of the glass transition become, as was to be expected, a consequence from the same general premise: the phenomenological law of irreversible thermodynamics in a more general, nonisothermal formulation, first employed by Bragg and Williams (see [64, 65] and the literature cited there) and later by Vol’kenstein and Ptizyn [74]. On Fig. 9.13, a typical example is given of the change of the break point of volume (on the figure: the density ρ(T )) of an undercooled silicate melt at different cooling rates, q0 , according to classical experiments performed by Ritland [71]. An experimental proof of Eq. (9.22) is seen on Fig. 9.14 according to Bartenev’s own original measurements [69, 70]. On Fig. 9.15 another significant empirical result due also to Bartenev is given: that vitrification takes place at nearly constant values of the ratio U (Tg )/RTg (see [75]) of the order U (Tg ) ≈ 30.5 2.3RTg
or
U (Tg ) ≈ 70 . RTg
(9.26)
These findings were confirmed for a great variety of glass-forming melts.
9.6
Glass-Transitions in Liquid Crystals and Frozen-in Orientational Modes in Crystals
As liquid crystals are usually denoted liquids (mostly of complicated chain-like organic structures) in which temperature change (thermotropic liquid crystals) or decrease of the percentage of solvent (liotropic liquid crystals) cause different forms of orientation (nematic, smectic or combined) in the essentially liquid
vch 4 Okt 2005 10:43
182
9 Thermodynamics of Amorphous Solids and Disordered Crystals
U(Tg ), Kcal/mol
160 120 80 40
0
200
400
600 Tg , K
800
1000
Fig. 9.15 Value of Tg -reduced activation energy ratio (U (Tg )/(2.3RTg )) = 30.5 upon vitrification: First (1955) experimental proof of this dependence (Eq. (9.26)) according to Bartenev and Lukyanov [75] for several organic and inorganic glass-forming melts as one of the kinetic invariants of glass transition
system - either a melt, or a solution [76]. The oriented liquid state is easily recognized optically under polarized light by the typical birefringens, characteristically changing for different types of orientational order, induced (e.g. by applied electric fields) or existing in the liquid in a given temperature interval. In thermotropic systems the liquid can exist as a smectic or a nematic oriented liquid body in one or more temperature intervals. With appropriate cooling procedures both smectic and nematic liquids can be frozen-in into the respective solid glass, in which the optical birefringence of the initial liquid is preserved [28]. The process of liquid crystal vitrification as far as it is investigated up to now (see Refs. 28, 29, 30, 76, and 77) shows all the typical features of the glass-transition in normal molecular glass-forming liquids already discussed in the foregoing paragraphs (S-shaped decrease of Cp (T ) in the glass transition region, a salient point in the S(T ) and the H(T ) curves at T = Tg ). However, the glass-transition in liquid-crystal systems is preceded by the typical changes in the caloric properties of the liquid, when it undergoes the isotropic
vch 4 Okt 2005 10:43
9.6 Liquid Crystals and Frozen-in Orientational Modes in Crystals
183
1000
a Cp(T), J .K-1mol-1
800 600 400
Tc Tm
200 Tg
0 0
50
100
150
200
250
300
350
400
T, K 600 1
b
S(T), J .K-1mol-1
500 2
400
Tm
300
Tg
3
200
T2
100
0
T0
4
0
50
100
150
200
250
300
350
T, K
Fig. 9.16 Temperature course of the specific heats, Cp (T ), a) and of the entropy, S(T ), b) of a typical liquid crystal (OHMBBA) according to Sorai and Seki [29]. Open circles in both figures a) and b) represent the isotropic, nematic and crystalline phases of the same substance. Black circles in both figures a) and b) represent the glassy liquid crystal and the supercooled liquid crystal of the nematic phase. Tc indicated the so-called ”clearing point” at which a second-order phase transition takes place between the isotropic and nematic liquid. Tm indicates the melting point between the molecular crystal and the nematic liquid, Tg indicates the glass-transition point and T0 the respective Kauzmann temperature, where S(T0 )nematic liquid −S(T0 )crystal = 0. Note that at T0 the frozen-in nematic liquid has a zero point entropy, S0 > 0, corresponding to a glass
state ↔ oriented state (smectic or nematic) transition, which usually shows all the features of a second-order phase transformation (or as stated in Ref. 79) in some cases: the so-called ”weak” first-order phase transitions). This is clearly seen in
vch 4 Okt 2005 10:43
184
9 Thermodynamics of Amorphous Solids and Disordered Crystals
250
a Cp(T), J .K-1mol-1
200 150 Tfus
100
Ttrs
50
Tg
0 0
50
100
150
200
250
300
T, K
S, J .K-1mol-1
200
b
150 Tfus
100 Ttrs
50 0 0
Tg
50
100
150
200
250
300
T, K
Fig. 9.17 Temperature course (a) of the specifc heats, Cp (T ), and (b) of the molar entropy, S(T ), of a typical plastic crystal, cyclohexanol [78]. With Tm is indicated the melting point of the high temperature crystalline modification of this substance and by Ttrs , the transition point between the two crystalline modifications. With Tg is denoted the temperature at which the entropy difference ∆S0 at T → 0 between the two crystalline modifications is frozen-in: again a violation of the Third Law!
vch 4 Okt 2005 10:43
9.6 Liquid Crystals and Frozen-in Orientational Modes in Crystals
185
the ∆Cp (T ) and ∆S (T )-curves of a typical liquid crystal substance (OHMBBA, Fig. 9.16) as they are constructed first by Sorai et al. [29]. However, in going into the quantitative analysis of glass-transitions in liquid crystals peculiar details can be observed, when (as done in the following sections) the main vitrification characteristics in liquid crystals are compared with those of normal glass formers, vitrifying without previous orientational transitions. Some of the dependences outlined in the previous pages are observed not only in the glassy state of typical glass-forming melts and in glass-transition in the oriented liquids somewhat irrelevantly called liquid crystals, but also in crystalline solids with different degrees of orientational structural disorder. A typical example in this respect is given in Fig. 9.17 where as a result of Cp (T )-measurements and calculations similar to the above discussed (see Eqs. (9.7) and (9.8)) the entropy of molten and of crystalline samples of cyclohexanol in its crystalline modifications is constructed [78]. This substance does not give an undercooled melt able to vitrify. However, a break-point in the ∆S(T )-curve analogous to that at Tg upon typical melt vitrification is clearly observable in the temperature dependence of the entropy of the high-temperature crystalline modification of this substance. The same can be also seen in the Cp (T )-curves of cyclohexanol shown on Fig. 9.17a, where the λ-type transitions between the crystalline phases is visible together with the step-like change of ∆Cp (T ) at T = Tg , the freezing-in temperature of the disordered crystalline phase. Cyclohexanol is only one of the representatives of a whole class of crystalline solids, sometimes somewhat misleading called glassy crystals (see [2, 16, 17, 78, 80]) in which different forms of orientational disorder can be frozen in. Because of its particular mechanical properties more exactly cyclohexanol, cyclohexene, cycloheptanol and similar crystals are called plastic crystals. Further examples and experimental evidence on ∆Sg0 -values in this respect may be found in Refs. 16, 17 and 78. It is of significance to note that the freezing-in temperature in plastic crystals depends on cooling rate in a way similar to Bartenev’s dependence Eq. (9.22). This was manifested in an exceptionally instructive way by Suga [78] by analyzing the Tg -dependent change of the dielectric constant ε(T ) in cyclohexanol ”crystalline glasses” at different frequencies of temperature alteration. Very particular, in some respects similar, but also very different with respect to the caloric effects observed upon the vitrification of typical glass-forming melts are the thermodynamic properties of another class of crystals with frozen-in structural disorder. These are the classical ”glassy” crystals [34, 35, 36, 80, 81]. In
vch 4 Okt 2005 10:43
186
9 Thermodynamics of Amorphous Solids and Disordered Crystals
Vapour pressure, P, internal cm
100
a 80 60 40 20 0
0
70
10
20
30
40 50 T, K
60
70
80
b
Cp (T), J .K-1mol-1
60 50 40 T2
30
Tm
20 10 0
0
10
20
30
40
50
60
70
80
T, K
Fig. 9.18 Thermodynamics of a typical disordered molecular crystal, CO (according to [34, 35]): a) the temperature course of the vapor pressure, P (T ), of the second, low-temperature phase (open squares), of the high-temperature CO-crystal (black squares) and (open circles) of the liquid CO; b) temperature dependence of the specific heats, Cp (T ), of above indicated modifications of CO. The same symbols are used as in the P (T )representation
them, by an as yet not clarified mechanism, disorder is frozen-in at the process of crystallization itself as observed by Kaischew [37]. No vitrification step or breakpoint is observed in the temperature course of their Cp (T ) or ∆S(T )-functions
vch 4 Okt 2005 10:43
9.7 ”Spectroscopic” Determination of Zero-point Entropies
187
Cp (T), J .K-1mol-1
20
15
10
5
0
Tm
0
50
100
150
200
250
300
T, K
Fig. 9.19 Specific heats, Cp (T ), of Ice (I)-crystals according to Ref. 90. Note that no singular point is to be seen on the Cp (T )-curve from nitrogen temperatures down to the melting point, Tm
(see Figs. 9.18 and 9.19). Nevertheless the S(T )-curves of all these disordered molecular crystals show a definite zero-point entropy ∆S 0 > 0. The thermodynamic proof of this statement was given by a method, different from the approach, indicated with Eqs. (9.7) and (9.8) used in glass transition.
9.7
”Spectroscopic” Determination of Zero-point Entropies in Molecular Disordered Crystals
In determining the zero-point entropies, ∆S 0 > 0, of above mentioned simple molecular crystals (CO, H2 O, N2 O2 etc.) the entropy of the vapor phase in equilibrium with the respective liquid state (at a temperature T > Tm ) is calculated once by using the vapor pressure formulas, derived from the third thermodynamic principle (see [6, 11]) and secondly - from the statistical partition function of the same vapors. The two values are then compared. According to the Third Principle of thermodynamics the vapor pressure, P , of a
vch 4 Okt 2005 10:43
188
9 Thermodynamics of Amorphous Solids and Disordered Crystals
solid or liquid is given by 1 L (T ) + log P (T ) = − 2.3RT 2.3R
T
dT T2
0
T
Cp (T )vap − Cp (T )cond dT
(9.27)
0
+ I0 (P ) . Here L(T ) is the enthalpy of evaporation (sublimation), Cp (T )vap and Cp (T )cond are the specific heats of the gaseous and the respective condensed phase (liquid, crystalline) in the whole temperature range beginning above Tm and ending as close as possible to the absolute zero of temperatures [6, 11]. The measurements of both P (T ) and Cp (T ) in Clusius and Giauque’s laboratories (see [34, 35]) were possible down to 10 - 20 K. The necessary caloric analysis had to include besides vapor pressure measurements also exact determinations of the enthalpies of melting, ∆Hm, of sublimation, L(T ), and evaporation, ∆Hb (at the melting, Tm , and boiling points, Tb ), and the temperature dependence of L(T ), H(T ) and of both Cp (T )vap and Cp (T )cond . With I0 (T ) in Eq. (9.27) is denoted the so-called chemical constant of the respective gas. For relatively simple molecules like those of the discussed molecular substances, the value and the temperature dependence of Cp (T )vap can be calculated, using the respective formulas of statistical thermodynamics, giving the connection with the thermodynamic functions. These calculations have to be supported by spectroscopic measurements (hence the name of the method of calculation) in order to determine from the band spectra of the vapors the exact values of the molecular constants, determining the partition function and thus Cp (T )vap , Cp (T = 0)vap and I0 (P ). The chemical constant I0 (P ) is determined in the framework of Nernst’s theorem as ! " I0 (P ) = ∆S 0 − Cp0 (T )vap
1 , 2.3R
(9.28)
where Cp0 (T )vap is the specific heat of the gaseous phase at the lowest temperatures, where only the known structure-dependent rotational contribution to Cp (T ) have to be accounted for. In determining via Eq. (9.27) by combined tensimetric and calorimetric data the caloric value of I0 (T ) and comparing it with the statistically determined value of the chemical constant from Eq. (9.28) (when ∆S 0 ≡ 0 is assumed) the true value of the zero-point entropy ∆S 0 for the given crystal is determined.
vch 4 Okt 2005 10:43
9.8 Entropy of Mixing
189
In the subsequent literature [34, 35, 80, 81], usually the values of S(T )vap at the ”normal” thermodynamic reference temperature (T = 298 K) are given as determined from caloric measurements (with the real value of ∆S 0 ). They are compared with the (naturally higher) ∆S(T )vap -value, determined ”spectroscopically”. The briefly summarized procedure, which gave strong evidence for the existence of frozen-in disordered crystals and for another exception from Nernst’s theorem, is thus based on two calculations of the entropy: once on caloric measurements, and in a second approach on its direct determination as S = kB ln Z + T
∂ ln Z ∂T
(9.29) V
from the partition function Z of the respective gas. In order to calculate Z(T ) the mentioned spectroscopic data have to be known. In calculating zero-point entropies of glasses according to Eq. (9.8) the respective crystal is considered as the reference state to both the undercooled liquid and the glass. Moreover it is known that at Tg the specific heats of the liquid decrease to their value in the crystal: thus ∆Sg can be calculated via Eq. (9.8) with a sufficient accuracy. In the discussed molecular crystals, as demonstrated by Figs. 9.18 and 9.19, no change, indicating a freezing-in process, is observed: disorder is frozen-in upon the formation of the defect crystal, giving as it seems, two (for H2 O, D2 O) or three (for CO, N2 O2 , N2 O) energetically nearly equivalent positions, as discussed in details in Refs. 36, 35, 37, 80, and 81 by a number of well-known scientists.
9.8
Entropy of Mixing in Disordered Crystals, in Spin Glasses and in Simple Oxide Glasses
In assuming that the zero-point entropy of disordered crystals can be considered as an ideal entropy of mixing of two structural units arranged in orientationally distinct ways (see [11]) we have to write ∆S 0 ≈ −R [x ln x + (1 − x) ln (1 − x)] .
(9.30)
Here x and (1−x) denote the molar fractions of molecules in the two orientational positions. As far as these two positions are energetically nearly equal (as assumed by Pauling [36, 82], Giauque [35] and Kaischew [37]) we have also to assume
vch 4 Okt 2005 10:43
190
9 Thermodynamics of Amorphous Solids and Disordered Crystals
x ≈ 0.5 and thus ∆S 0 ≈ R ln 2 is to be expected, as it is in fact observed in CO, N2 O2 , N2 O and cyclohexanol. The structure of ice (and of crystalline D2 O) according to Pauling [36] indicates the possible existence of three energetically nearly equal structural positions of structural bonds and thus, via Eq. (9.30) an analogous entropy expression for x1 ≈ x2 ≈ x3 as ∆S 0 ≈ R ln(3/2) is obtained and in fact also verified experimentally in the framework of the formalism derived in the previous section.
TN
a
TK
10
M(T)
8
M(T)
2
1
6 4
qr = -1
qr = 0
2 0
b
0
1
T
2
0
0
1
T
2
Fig. 9.20 Thermodynamics of an antiferromagnetic crystal (a) and of a spin glass (b) in terms of the dependence of magnetization, M (T ), on temperature, T (given in relative units with respect to Neel’s critical temperature, TN ). At T = TN the disordered paramagnet crystal changes into an antiferromagnet (a). In part (b) the magnetic spin disorder is frozen-in at T = TK (full line) or relaxes to M = 0 along the dashed line. Note that here M (T ) corresponds in its temperature behavior, determined by the magnetic susceptibility and the applied low magnetic field strength, to a thermodynamic function. In this sense the process in figure (b) corresponds roughly to the freezing-in of either enthalpy or entropy in common glasses (schematic representation according to Ref. 4)
Very particular, and also in many respects similar to the caloric effects observed upon vitrification of typical glass-forming melts, are the changes in magnetic properties and on freezing-in of magnetic orientational disorder in crystalline and amorphous alloys by changing the magnetic field strength, χ (see [4] and Fig. 9.20). In these cases (at T =const.) changing magnetic field strength yields,
vch 4 Okt 2005 10:43
9.8 Entropy of Mixing
191
as was demonstrated by Dobreva and Gutzow [83], a dependence similar to Eq. (9.23) τ (T, χ) ω0 ∼ = const .
(9.31)
Here ω0 = (dχ/dt) is the rate of change of magnetic field strength and τ (T, χ) is the characteristic time of molecular relaxation, determining the process of magnetization. A detailed description of the properties of magnetic spin glasses may be found in Ref. 4 and in the additional literature given in the introduction to the present chapter. Here of essence is the analogy of the ”magnetic” freezing-in process with the freezing-in in molecular processes, as this is indicated by Eqs. (9.31) and (9.23). In assuming that, in spin glasses, orientations (e.g. parallel and antiparallel) are frozen-in, in terms of Eq. (9.30) again the ∆S 0 -value can be estimated and a result nearly equal to R ln 2 should be expected (see J¨ ackle [25, 84]). However, no direct caloric confirmation of this expectation is given in the cited literature, nor it is known to the present authors. In 1962, Gutzow [33] proposed to consider and calculate the frozen-in topological disorder in glasses in terms of an entropy of mixing: either of free and occupied volume or later-on of differently coordinated structural units in accordance with Bernal’s [85] and Scott’s [86] models of random packing of equal spheres. A detailed description of these calculations is given in Ref. 2. It was proposed many years ago (see [2] and the literature cited therein) to consider glasses as systems with frozen-in free volume (the so-called free volume concept of vitrification). In fact it was found that for most organic polymer and molecular glasses vitrification takes place at a relative free volume v = 0.1, where v is defined as v=
ρglass − ρcryst ρcryst
(9.32)
by the respective densities ρ (or molecular volumes) of glass and crystal. For simple network glasses (like SiO2 , BeF2 , GeO2 etc., see [2]) the thus defined free volume (referred to the respective most stable crystalline modification, e.g. quartz for SiO2 ) is considerably higher, i.e. v ≈ 0.2. By employing Eq. (9.30) the relative contribution of topological disorder to the frozen-in zero-point entropy of glasses turns out to be ∆Sg ≈ R (when v = 0.2 as in the network glasses) or ∆Sg ≈ (1/2)R as in organic molecular substances vitrified at v = 0.1 (see [2]).
vch 4 Okt 2005 10:43
192
9 Thermodynamics of Amorphous Solids and Disordered Crystals
A thorough discussion on this subject was given by Petroff et al. [54], where an estimate of the contribution of various forms of disorder to the total entropy of undercooled liquids is given in the framework of appropriate lattice-hole models of simple and polymer glasses. Thus it follows that the frozen-in topological disorder may be, similar to the already discussed orientational disorder, nearly a constant: at least for simple iso-structural glasses (like SiO2 , BeF2 , GeO2 ), where it gives the main contribution. An additional indication in this respect give also computer results with different models of disorder, frozen-in liquids, constituted of equal spheres. In such calculations [87] also ∆S 0 ≈ R is obtained as in Gutzow’s first estimates [33] and in the already mentioned more exact models of silicate structures given by Beal and Dean [88].
9.9
Generalized Experimental Evidence on the Caloric Properties of Typical Glass-forming Systems
In the preceding sections, results are summarized on the caloric behavior of typical glass-forming systems with very different structures: organic, inorganic, polymeric, metal-like. The obvious question to be answered in considering these and similar results of more than one hundred glassy systems so far investigated is: is there sufficient evidence in the behavior of glass-forming substances permitting, or even requiring, a generalized phenomenological treatment? The positive answer to this question is to some extend guarantied by the existence of several general empirical rules, connecting the caloric and kinetic properties of glasses. The first attempt for a generalized description of the thermodynamic properties of glass-forming melts was given by Kauzmann in 1948 [21]. Summarizing the experimental evidence, available then concerning the temperature dependence of thermodynamic functions (of about 15 typical glass-formers), he has found that: i.) The temperature dependence of entropy ∆S(T ), enthalpy ∆H(T ), molar volumes, ∆V (T ), of undercooled liquids in reduced coordinates (e.g. ∆S(T )/∆Sm vs. reduced temperature T /Tm ) is similar, although not equal, for all substances analyzed. The absolute value of the melting temperature Tm turned out to be a sufficiently reproducible correlating factor for all substances analyzed (see Fig. 9.21). The last circumstance is not surprising, accounting for the well-known fact that
vch 4 Okt 2005 10:43
9.9 Caloric Properties of Typical Glass-forming Systems
193
B2O3
0.8
D S/D Sm
0.6 Ethanol
Propanol
0.4 Glycerol 0.2
0
Glucose NaPO3 Lactic Acid
0.2
0.4 T/Tm
0.6
0.8
Fig. 9.21 The first effort (1948) to summarize experimental evidence on the thermodynamics of the glass-transition: The temperature dependence of the configurational entropy, ∆S(T ), in Kauzmann’s reduced coordinates [21]. Note the fan-like divergence in the temperature dependencies, the break point at T = Tg and the circumstance that the linear extrapolation of ∆S(T ) below Tg would lead to negative values of the zeropoint entropy, ∆S 0 . The composition of the respective glasses is indicated on the drawing itself. On the picture additionally to Kauzmann’s data is given also ∆S(T ) for the NaPO3 melt, as a typical representative of thermodynamically fragile inorganic glasses
at least for simple, non-associating liquids, Tm is directly connected with the critical temperature, Tcrit , via linear dependences, e.g. Tm ∼ = (3/5)Tcrit . ii.) The glass transition temperature obtained at normal cooling rates (e.g. at
vch 4 Okt 2005 10:43
194
9 Thermodynamics of Amorphous Solids and Disordered Crystals
q = 10 to 102 K/s, which were employed for the glass as analyzed by Kauzmann) is for typical glass formers given by 2 (9.33) Tg = Tm . 3 For multi-component glass-forming melts instead of Tm the corresponding liquidus temperature Tl can be used [21]. The expression given with Eq. (9.31) is known as the (2/3)-Kauzmann-Beaman rule (see [2]). However, for the later-on examined metallic alloy glasses (where the cooling rate of vitrification, q0 , is typically of the order 104 to 106 K/s), the relation 1 (9.34) Tg ≈ Tl 2 holds, where Tl is the corresponding liquidus temperature of the metallic alloy (see [2, 3] and literature cited there). iii.) The linear extrapolation of the (∆S(T )/∆Sm )-dependence below Tg indicates that the configurational entropy of the fictive undercooled melt in internal equilibrium would approach zero values (i.e. ∆S(T )/∆Sm → 0) as required by Nernst’s theorem at temperatures T = T0 , which are considerably above the absolute zero, approximately at 1 Tm . (9.35) 2 Thus the linear extrapolation of the (∆S(T )/∆Sm )-dependence below T0 would lead to negative ∆S(T )-values (the so-called Kauzmann paradox, see Fig. 9.22). For metallic alloy glass formers, however, considerably lower T0 -values are found (e.g. T0 = (1/3 to 1/5)Tm or even T0 ≈ (1/5)Tl (see [2])). The thermodynamic impossibility of the linear approximation, leading to negative entropy values ∆S(T = 0) of a fictive state in internal equilibrium (the metastable undercooled liquid) was evident to Kauzmann [21] and he discussed this extrapolation only as a hypothetical procedure. The only permissible extrapolation for equilibrium states has to be on the contrary non-linear and in accordance with the Third Principle as this is also indicated on Fig. 9.22 and argumented in more details in Ref. 2. T0 ≈
The subsequent analysis of experimental data of about 40 mostly organic polymer glass-forming substances performed by Wunderlich [89] has shown further-on (see also [2, 3]) that in general ∆Cp (Tg ) 3 ≈ ∆Sm 2
vch 4 Okt 2005 10:43
(9.36)
9.9 Caloric Properties of Typical Glass-forming Systems
Tm
1.0
DS(T)/DSm
195
0.8 0.6
Tg
DSm
0.4 0.2
DSg
0 0.2 -0.2 -0.4
0.6
1.0
T/Tm
T0 DS0
-0.6 -0.8
Fig. 9.22 Kauzmann’s paradox (schematically): the solid line represents the undercooled metastable liquid (at Tg < T < Tm ) and the vitreous state (for T < Tg ) with a frozen-in entropy, ∆Sg . The dashed curve is the inadmissible linear extrapolation of the ∆S(T )-dependence for the melt down to absolute zero, proposed by Kauzmann [21]. Such an extrapolation would yield negative values of the entropy, ∆S0 , at T → 0. Thermodynamically forbidden consequences and paradoxes do not occur for other, more realistic thermodynamically self-consistent extrapolations of the properties of the fictive undercooled melt as shown, e.g., by the other dashed line. The temperature is given in relative units, T /Tm . With T0 is indicated the temperature, at which ∆S(T ) becomes practically equal to zero
vch 4 Okt 2005 10:43
196
9 Thermodynamics of Amorphous Solids and Disordered Crystals
30
a
10
b
8
16
c
12
Frequency
20 6 8 4
10
4 2 0 0
0.2
0.4
0.6
0.8
0
0.5
Tg / Tm
1.0
1.5
2.0
DCp / D Sm
2.5 3.0
0.2
0.4
0.6
0.8
DSg / D Sm
Fig. 9.23 Thermodynamic invariants of the glass transition after Refs.2 and 3: frequency distribution curves of the major thermodynamic characteristics of typical glass-forming melts for which the respective thermodynamic measurements have been reported till 1992. (a) Tg vs Tm for approximately 110 substances, confirming the KauzmannBeaman formula (Eq. (23)); (b) experimental data of the ratio (∆Cp (Tg )/Sm ) for approximately 80 typical glass-forming melts confirming Eq. (9.37), proposed by Wunderlich; (c) experimental (∆Sg /∆Sm )-values also for 80 substances according to Gutzow’s dependence, Eq. (9.35). Experimental evidence on which these figures are based are summarized in Refs. 2, 3, 15, 23 (see also text)
holds. Two other simple dependences ∆Sg 1 ≈ , ∆Sm 3
∆Hg 1 ≈ ∆Hm 2
(9.37)
have been established by Gutzow also years ago (in 1983 [2, 3]). The degree of fulfillment of all above dependences is to a great extent evident from Figs. 9.23 where the results of a survey of caloric results of about 110 glass-forming substances (organic-molecular and polymer, inorganic, etc., see the figure captions) was summarized (see [3]). At present it is not possible to say if the deviation from above mean values (or more characteristically: from the respective median values) is systematic or accidental (see the discussion in Refs. 2 and 3). It is also essential to note that of above dependences only Eqs. (9.37) are also applicable to metallic glasses (see [2, 3, 19]). However, it may be of particular significance that in all cases of vitrifying liquid crystals analyzed, it turned out that, although the dependences given with Eqs. (9.33), (9.36), and (9.37) are fulfilled for them
vch 4 Okt 2005 10:43
9.9 Caloric Properties of Typical Glass-forming Systems
197
(as far as the only two or three existing measurements are characteristic data), Eq. (9.37) is not obeyed for vitrified liquid crystals (Fig. 9.24; data are taken from Refs. 2, 16, 17, 29, 30, 77 for Fig. 9.24a; from Refs. 16, 34, 35, 36, 77, 80, and 81 for Fig. 9.24b). Recalling Boltzmann’s formula, giving the entropy of a system in dependence on possible microscopic configurations, ∆S = kB ln Ω ,
(9.38)
it follows from Eq. (9.37) that the number of thermodynamically distinguishable configurations at T = Tm and T = Tg for typical glass formers are related approximately by ln Ωm ∆Sm = . ∆Sg ln Ωg
(9.39)
With Eq. (9.37) it follows that ln Ωg =
1 ln Ωm 3
(9.40)
or Ω (Tg ) =
3
Ω (Tm ) .
(9.41)
Thus for typical glass-forming substances, including metallic alloy systems, vitrification takes place when the degeneracy of possible thermodynamic configurations has reached the value indicated by Eq. (9.41). This, may be, explains the generality of the above written simple empirical rules. In vitrifying liquid crystals, where already in the corresponding metastable liquid an increased order (nematic, smectic etc.) is established, the difference to one of the crystalline fully ordered states is smaller and thus a lower coefficient of proportionality between ∆Sg and ∆Sm is to be expected and found. Above simple empirical rules give a simple way to estimate the possible deviations from equilibrium of a glass frozen-in at mentioned standard cooling rates. Considering Eqs. (9.12) and (9.37), we obtain as an approximation for the temperature course of the thermodynamic potential of the glass formed at standard cooling rates 1 1 T ∆Gg (T ) − = (9.42) ∆Sm Tm T →0 2 3 Tm
vch 4 Okt 2005 10:43
198
9 Thermodynamics of Amorphous Solids and Disordered Crystals
40
DSg , J .K-1mol-1
a
DSg = 0.33DSm
35 30 25 20 15 10
LC
D Sg = 0.11 D Sm
5
DS0 , J .K-1mol-1
0
0
20
40
60 80 DSm , J .K-1mol-1
100
120
140
b 10
R ln 2
5 0
R ln 1.5 0
20
40
60
DSm , J .K-1mol-1
Fig. 9.24 Melt-dependent and melt-independent thermodynamic invariants for disordered solids with frozen-in zero point entropy (∆Sg , ∆Sglc , ∆S0 ) vs melting entropy ∆Sm : a) for 39 typical glass-formers (black circles) approximately confirming Eq. (9.35) as ∆Sg ∼ (1/3)∆Sm . The substances here in the order of increasing ∆Sm -values are: (poly)ethylene, SiO2 , Se, GeO2 , rubber, BeF2 , methanol, (poly)dimethylsiloxane, Au0.077 Ge0.136 Si0.094 , H2 O, poly(propylene), NaPO3 , transpolypentanamere, lactic acid, poly(oxacyclobutane), B2 O3 , ethanol, polytetrahydrofurane, n-propanol, isopropylbenzene, butene-1, isopentane, polyglycolide, poly(ethylene-therephtalate), poly-ε-caprolactane, As2 S3 , orthoterphenyle, 2-methylpentane, betol, benzophenone, glycerol, As2 Sl3 , diethylphtalate, phenolphtaleine, H2 SO4 H2 O, H2 O4B2 O3 , Na2 O4B2 O3 . For polymeric substances the gram formula value is taken. For the only three liquid crystal substances, PBBA, OHMBBA, HBBA with known ∆Sglc -values (open circles) giving ∆Sglc ∼ 0.11∆Sm as the zero-point vs ∆Sm dependence in this case. They are also ordered according to increasing melting point entropy values, ∆Sm ; b) the zero-point entropy of the known disordered molecular crystals (black squares), in order of increasing ∆Sm -values (CO, H2 O, D2 O, N2 O, N2 O2 ) and of two plastic crystals (ethanol and cyclohexanol: open squares), all of them are located in between ∆S0 = R ln 2 and ∆S0 = R ln 1.5 (see text)
vch 4 Okt 2005 10:43
9.9 Caloric Properties of Typical Glass-forming Systems
199
as earlier indicated in Refs. 2 and 3. This formula gives the expected deviation from equilibrium of a ”normal” or ”standard” glass. At T → 0, the maximal value thus attainable is ∆Gg (T )|T →0 = (1/2) ∆Hm . Above summarized thermodynamic invariants, determining the thermodynamic behavior of typical glass-forming melts, are also reflected in their kinetic behavior. Thus, we have already mentioned that even in the first applications of Bartenev’s formula (Eq. (9.22)) a relatively stable constancy of the activation energy for viscous flow, U (Tg ), was observed [70, 75] in the sense indicated with Eq. (9.30). This equation is proven by experiment and can be considered as one of the main kinetic invariants of glass transition. The essential point to be noted is that, in assuming any sufficiently correct thermodynamic dependence for the temperature dependence of the thermodynamic functions of the metastable undercooled liquid (i.e. for the equilibrium system) and an appropriate equation for its kinetic properties (i.e. for the viscosity of the undercooled melt), the already mentioned thermodynamic (Eqs. (9.8) to (9.36), (9.38) and (9.43)) and kinetic (Eqs. (9.26), (9.23), (9.33)) invariants can be connected and derived by the combination of only one thermodynamic and one kinetic invariant. Most easily this can be done by accepting the Kauzmann-Beaman ratio (Eq. (9.33)) as the main kinetic invariant and assuming Eq. (9.36) with the median value of (∆Cp /∆Sm ) from Fig. 9.23 to be of universal significance for all glass-forming melts. Now we have to introduce an appropriate thermodynamic relation, connecting the caloric properties of the undercooled liquid. The simplest sufficiently correct thermodynamic equation directly following for ∆S(T ) from the general thermodynamic dependence Eq. (9.7) at the assumption ∆Cp /∆Sm =const=a0 for Tm > T ≥ T0 is ∆S (T ) = ∆Sm (1 + a0 ln x) .
(9.43)
With the same approximation for Cp (T ) it follows from Eq. (9.9) that ∆H (T ) = ∆Hm [1 − a0 (1 − x)] .
(9.44)
The ∆G (T )-course of the metastable melt is then determined by Eqs. (9.43) and (9.44) via Eq. (9.10). In a thorough analysis of Eq. (9.43), Gutzow [2, 61] introduced the ratio a0 =
∆Cp (Tg ) ∆Sm
or
a0 =
∆Cp (Tm ) ∆Sm
(9.45)
vch 4 Okt 2005 10:43
200
9 Thermodynamics of Amorphous Solids and Disordered Crystals
as a generalized thermodynamic structural coefficient. This ratio is in fact the main characteristics of the temperature fragility of the thermodynamic properties of the metastable liquid. In the ratio a0 = (∆Cp (T ) /∆S (T )), two very essential thermodynamic properties of the undercoold liquid are connected. First this is Cp (T ), the susceptibility of the melt with respect to temperature changes. It determines the way temperature is increased when an amount of heat is transferred to the melt. In a0 this quantity is divided by the structural change (measured by the value of ∆S (T ) at T = Tm ) caused in the liquid by the corresponding temperature rise. In Gutzow’s a0 -value the ratio a0 is taken at T = Tm , where most experimental data on both Cp (T ) and ∆S (T ) are tabulated in reference literature. In Eq. (9.36) the value of a0 at T = Tg is taken, as for this temperature most results connected with glass-forming systems are given in the reference literature. If a0 is introduced as a constant, invariant value in Eqs. (9.43) and (9.44) at vitrification, using Kauzmann’s kinetic invariant, Eq. (9.33) for (∆Hg /∆Hm ) and (∆Sg /∆Sm )-values are obtained very close to both thermodynamic invariants empirically proposed by Gutzow (Eqs. (9.37)). From Eq. (9.41) it also follows that, at ∆S0 , we have to write T0 1 = exp − . x0 = Tm a0
(9.46)
Thus values for T0 are obtained as given by Eq. (9.35). In order to derive the kinetic invariants e.g. the one, given with Eq. (9.26), in a third step we have to introduce an appropriate viscosity equation. One of the most universally accepted η(T )-dependences, based on a sound statistical foundation, is given by Adams and Gibbs [90] in the form
log η (T ) = A0 +
B . 2.3T ∆S (T )
(9.47)
Here A0 is a constant (see [2]) and B, according to the original derivation of this viscosity equation, is dependent both on its complexity, expressed by ∆Sm , and its bonding strength, determined by Tm , B ≈ b∗0 ∆Sm Tm .
vch 4 Okt 2005 10:43
(9.48)
9.10 General Conclusions
201
Introducing again the reduced temperature, x = T /Tm , we can write Eq. (9.47) in the form log η (T ) ∼ = A0 +
b∗0 . 2.3x (1 + a0 ln x)
(9.49)
The activation energy for viscous flow corresponding to Eq. (9.43) is now U (T ) =
Rb∗ 1 + a0 ln x
(9.50)
and at T = Tg the ratio (U (Tg )/(2.3RTg )) is to be in accordance with Eq. (9.26) an invariant, as far as xg and a0 in above equations are invariants. Thus invariants are to be also the ratio (U (Tg )/(2.3RTg2 )), which according to Eq. (9.22) determines substantially Bartenev’s equation via the values of C1 , C2 in it and U (T ) d , giving the slope of the activation energy also the expression for dT RT T =Tg
at T → Tg . This slope and the value of the constant in Eq. (9.23) are directly given by this ratio according to theoretical concepts, developed by Gutzow et al. [43, 44], and thus it determines to a great extent the vitrification possibilities of liquids.
9.10 General Conclusions Considering the available information, summarized in the preceding sections, the following conclusions of more general nature can be made: 1. The experimental evidence on the thermodynamic properties and the kinetics of vitrification of typical glass-forming systems, on the glass-transition in liquid crystals, on the freezing-in of ”glassy” plastic crystals (like cyclohexanol) and most probably also of spin glasses shows that the respective non-equilibrium frozen-in glasses or glass-like states are formed in a very similar process. In any of these cases, dependences equivalent to Eqs. (9.23) and (9.31) are obtained, which can be written as τ (T, P, X) w|Tg ,Pg ,Xg ≈ const .
(9.51)
They determine at a given rate of change ”w” of some external parameter (T, P or X) a critical point (e.g. T = Tg at P =const., X = 0) or more precisely a T , P or X-range at which vitrification or a vitrification-like process takes place.
vch 4 Okt 2005 10:43
202
9 Thermodynamics of Amorphous Solids and Disordered Crystals
At such conditions the characteristic relaxation time τ (T, P, X) at a given rate w of change of the external parameters, the relaxation time of the system, becomes nearly equal to the stay time of the system in the critical region. In any of such processes a dependence, similar to Bartenev’s formula Eq. (9.22), determines the kinetics of relaxation. In liquid systems (glass-forming melts, liquid crystals) the time of molecular relaxation τ (T, P, X) is determined directly by the viscosity of the corresponding fluid. In ”glassy” plastic crystals τ (T, P, X) has obviously to be determined by the relaxation time of the structural units in the respective ”plastic” crystalline phase. In both crystalline and in amorphous spin glass systems, more particular considerations have to be added to determine the true nature of the time τ (T, P, X) (see [5]). However, in all these cases vitrification can and has to be considered as a process taking place at a steadily increasing relaxation time; and when τ (T, P, X) has reached the w-determined stay time of the system, it is frozen-in to a glass-like structure. In Refs. 43 and 44, the phenomenology of such a non-isothermal process of relaxation leading to glass transition is quantitatively established, when w is given by the cooling rate q0 at P, X=const. Similar dependences have been established also for increasing pressure (when w = dP/dt) or at changing magnetic field strength, X, etc. 2. The thermodynamic properties of glasses (and especially of ∆Sg0 ) formed in melt cooling (or quenching) processes are determined by the properties of the corresponding undercooled liquids, and as far as the melt at T = Tm is taken as the reference state, the value of ∆Sg (and of any other thermodynamic function (enthalpy ∆Hg , free volume ∆Vg )) is for all substances a nearly invariant part of the value of the corresponding property at melting temperature, Tm . This result ∆C is a consequence of the fact that the thermodynamic structural factor, a0 = ∆Smp , varies in relatively close limits between 1 and 2 (see Fig. 9.23 and the analysis, given in Ref. 2). The highest values of a0 are observed according to Refs. 20 and 61 for organic molecular or polymeric glass-formers while for simple oxide glasses (SiO2 , BeF2 ) and for metallic glass-forming alloys [19] the relation a0 ∼ = 1 holds. Thus both the relative invariancy of the several ratios (Eqs. (9.37)), mentioned in the preceding two sections, finds its explanation, together with the fan-like spread of the temperature course of the respective ratios (∆S(T )/∆Sm , see Fig. 9.21) as first observed by Kauzmann [21]. In this way the ratio a0 determines via the discussed appropriate equations both the individuality of the (∆S(T )/∆Sm )course of differently structured liquids (”fragile” at a0 = 2, ”strong” at a0 ∼ = 1) for different substances and also its relative invariance at T = Tg . This is the reason ∆Cp (Tg ) the significance of a thermodynamic structural why to the ratio a0 = ∆S m factor has to be attributed.
vch 4 Okt 2005 10:43
9.10 General Conclusions
203
40 35 DHg , KJ .mol-1
30 25 20 15 10 5 0 0
10
20
30
40 50 1 . DHm , KJ mol
60
70
Fig. 9.25 Frozen-in enthalpy ∆Hg of 17 typical glass-formers (black circles) and of the three vitrifying liquid crystal substances shown also in Fig. 9.24 (open circles). Note that here the ∆Hg vs ∆Hm -correlation gives in both cases the same linear dependence Eq. (9.36) with ∆Hg ∼ (1/2)∆Hm . The substances in these figures (also in the order of increasing ∆Hm -values) are: Typical glass-formers (data from Ref. 2): ethanol, n-propanol, Se, 2-methylpentane, ZnCl2 , Au0.77 Ge0.136 Si0.094 , SiO2 , benzophenone, glycerol, betol, Ca(NO3 )2 4H2 O, H2 SO4 3H2 O, B2 O3 , poly(ethyleneterephtalate), LiO2 2B2 O3 , H2 O4B2 O3 Na2 O4B2 O3 ; Liquid crystals: PBBA, OHMBBA, HBBA (data from Refs. 29, 30, and 77)
3. However, when kinetic properties like the course of the viscosity, η(T ), for different glass-forming melts are compared, this individuality is brought into the exponent e.g. of the Adams-Gibbs equation (see Eq. (9.47)) and the fan-like divergence into ”strong” and ”fragile” temperature courses of viscosity becomes more pronounced. A remarkable confirmation in this respect gives Angell’s η(T ) vs (Tg /T ) construction [91], shown on Fig. 9.26. It is to be mentioned (see [50]) that those glass-formers, which are thermodynamically ”fragile” or ”strong” are also more ”fragile” or more ”strong” in their viscosity behavior. This is to be expected by comparing Eqs. (9.43) and (9.47). 4. Individual structural properties of the system (e.g. additional orientational or conformational effects in liquids) are directly reflected in their entropy behavior.
vch 4 Okt 2005 10:43
204
9 Thermodynamics of Amorphous Solids and Disordered Crystals
14 12 12 10
GeO2 .3 Si O Ca ,N 2 O
6
iO 2
. 2S O
SiO2 4
Na 2
2
Cl 2
4
K,
6
2O 3
3
Str
Ca O. Al
lg h, poise
8
g on
8
2
Zn
0
lene l o-xy p. eni com terph o-
nol
etha -2
c organi 0
0.2
0.4
0.6
lg h, Pa.s
10
0.8
0
e
il rag
-2
F
-4 1.0
Tg /T
Fig. 9.26 Kinetic invariants and kinetic diversity in glass transition. Viscosity vs. temperature curves of several glass forming substances in Angell’s coordinates: log η(T ) vs (Tg /T ) [52, 92]. Note the ”strong” type rheological dependence of SiO2 , GeO2 and the ”fragile” type of η(T )-change of systems with temperature-dependent structures (o-therphenyl, ethanolo-xylene, (K,Cu)2 NO3 etc.) and the intermediate position of inorganic chain polymers (Na2 O·2SiO2 , ZnCl2 )
This is demonstrated by the comparison of Figs. 9.24 and 9.25, showing the particular nature of the structural properties of liquid crystals and their reflection in the energetic and relative disorder. 5. No distinct mechanism can be proposed for the formation of defect structures in molecular crystals, like CO, H2 O, etc. Here the common point with typical glass-formers seems to be only the possibility to describe the zero-point entropy in terms of both simple and more complicated forms and expressions of entropies of mixing. In Ref. 54 we have in fact shown that all three forms of configurational disorder existing in liquids - topological, orientational and configurational - can be described as different, generalized forms of entropy of mixing. In defect molec-
vch 4 Okt 2005 10:43
9.11 References
205
ular crystals (such as CO) only orientationally determined structural disorder is expressed in the form of the entropy of mixing, discussed in one of the preceding sections. No process analogous to vitrification has been observed in these cases.
9.11 References 1. J. M. Ziman, Models of Disorder (Cambridge University Press, London, New York, 1979). 2. I. Gutzow and J. Schmelzer, The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization (Springer, Berlin, 1995, pgs. 52-56 and chapters 3 and 5). 3. I. Gutzow and A. Dobreva, J. Non-Cryst. Solids 129, 266 (1991). 4. K. Moorgani and J. M. D. Coey, Magnetic Glasses (Elsevier Publ., Amsterdam, New York, 1984). 5. K. Binder and A. P. Young, Rev. Mod. Phys. 58, 801 (1986). 6. W. Nernst, Die Theoretischen und Experimentallen Grundlagen des Neuen W¨ armesatzes (W. Knapp, Halle, 1918). 7. R. Witzel, Z. Anorg. Allg. Chem. 64, 71 (1921). 8. F. Simon and F. Lange, Z. Physik 38, 227 (1926). 9. F. Simon, Z. Anorg. Allg. Chem. 203, 219 (1931). 10. G. E. Gibson and W. F. Giauque, J. Amer. Chem. Soc. 45, 93 (1923). 11. I. P. Bazarov, Thermodynamics (McMillan, New York, 1964). 12. F. Simon, Physica 4, 1089 (1937). 13. F. Simon, In: Handbuch der Physik, Eds. H. Geiger and H. Scheel, vol. 10 (Springer, Berlin, 1926, p. 350 ff). 14. S. V. Nemilov, Fizika i Khimia Stekla 2, 97 (1976) (in Russian). 15. V. P. Privalko, J. Phys. Chem. 84, 3307 (1980). 16. S. V. Nemilov, Thermodynamic and Kinetic Aspects of the Vitreous State (CRC Press, Boca Raton, London, 1995, pgs. 72-92). 17. E. Donth, The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials (Springer, Berlin, 2001, pgs. 46-49). 18. H. S. Chen and D. Turnbull, J. Chem. Phys. 48, 2560 (1968). 19. L. Battezatti and E. Garone, Z. Metallkunde 75, 305 (1984). 20. F. Simon: 40-th Guthrie Lecture, Year Book Physical Society (London), 1 (1956).
vch 4 Okt 2005 10:43
206
9 Thermodynamics of Amorphous Solids and Disordered Crystals
21. W. Kauzmann, Chem. Rev. 43, 219 (1948). 22. W. Eitel, Thermochemical Methods in Silicate Investigation (Rutgers University Press, New Brunswick, New York, 1952). 23. E. Grantcharova and I. Gutzow, J. Non-Cryst. Solids 81, 99 (1986). 24. G. P. Johari, Phil. Mag. B 41, 41 (11980). 25. J. J¨ ackle, Phil. Mag. B 44, 533 (1981). 26. G. P. Johari, J. Chem. Phys. 112, 7518 (2000). 27. G. P. Johari, J. Chem. Phys. 112, 8958 (2000). 28. J. Grebovicz and B. Wunderlich, Mol. Cryst. Liq. Cryst. 76, 287 (1981). 29. M. Sorai and S. Seki, Mol. Cryst. Liq. Cryst. 23, 299 (1973). 30. M. Sorai, K. Tani, H. Suga, and H., Yoshioka, Mol. Cryst. Liq. Cryst. 97, 365 (1983); 95, 11 (1983). 31. A. Einstein, Verhandlungen der Deutschen Physikalischen Gesellschaft 16, 820 (1914). 32. L. Pauling and R. C. Tolman, J. Amer. Chem. Soc. 47, 2148 (1925). 33. I. Gutzow, Z. Phys. Chem. (Leipzig) 221, 153 (1964). 34. K. Clusius and W. Teske, Z. Phys. Chem. 6, 135 (1938). 35. J. O. Clayton and W. F. Giauque, J. Amer. Chem. Soc. 54, 2610 (1932). 36. L. Pauling, J. Amer. Chem. Soc. 57, 2680 (1935). 37. R. Kaischew, Z. Phys. Chem. B 40, 273 (1938). 38. G. Tammann, Der Glaszustand (Leopold Voss, Leipzig, 1933). 39. L. D. Landau and E. M. Lifshitz, Theoretical Physics, vol. 5, Statistical Physics, Part 1 (Nauka, Moscow, 1955 (in Russian)). 40. A. R. Ubbelohde, Melting and Crystal Structure (Clarendon Press, Oxford, 1965). 41. A. K. Schulz, J. Chem. Phys. Biol. 51, 324 (1954). 42. A. K. Schulz, Kolloid Z. 138, 75 (1954). 43. I. Gutzow, D. Ilieva et al., J. Chem. Phys. 112, 10941 (2000). 44. I. Gutzow, V. Yamakov et al., Glass Phys. & Chem. 27, 148 (2001). 45. D. A. McGraw, J. Amer. Ceram. Soc. 35, 22 (1952). 46. S. B. Thomas and G. S. Parks, J. Chem. Phys. 35, 2091 (1931). 47. G. W. Morey, The Properties of Glass (Reinhold Publ., New York, 1954, p. 212).
vch 4 Okt 2005 10:43
9.11 References
207
48. P. G. Debenedetti, Metastable Liquids (Princeton Univ. Press, 1966, p. 292). 49. A. Winter, Verreset Refract. 7, 217 (1953). 50. I. Gutzow and A. Dobreva, Kinetics of Glass Formation. In: Amorphous Insulators and Semiconductors, Ed. M. F. Thorpe and M. I. Mitkova (NATO ASI Series, Kluwer Academic Publishers, 1997, p. 21). 51. R. Blachnick and A. Hoppe, J. Non-Cryst. Solids 34, 191 (1979). 52. J. Wong and A. Angell, Glass Structure by Spectroscopy (Marcel Decker, New York, 1976, p. 23). 53. D. N. Zubarev, V. G. Morozov, and G. R¨ opke, Statistical Mechanics of Nonequilibrium Processes (Fizmatlit, Moscow, 2002, vol. 1, pgs. 66-68 (in Russian)). 54. B. Petroff, A. Milchev, and I. Gutzow, J. Macromol. Sci.-Phys. B 35, 763 (1996). 55. I. Prigogine and R. Defay, Chemical Thermodynamics (Longmans Green and Co, London, New York, Toronto, 1954). 56. R. O. Davies and G. O. Jones, Proc. Royal Soc. London A 217, 26 (1953). 57. E. Jenckel and K. Gorke, Z. Naturforschung 7a, 630 (1952). 58. I. Gutzow, S. Todorova et al.: Diamonds by Transport Reactions with Vitreous Carbon and from the Plasma Torch. In: Nucleation Theory and Applications, Ed. J. W. P. Schmelzer (Wiley-VCH, Berlin-Weinheim, 2005, pgs. 256-308). 59. W. Weisweiler and V. Mahadevan, High. Temp. High Pressure 3, 111 (1971). 60. I. Gutzow, Z. Phys. Chem. (N. F.) 81, 195 (1972). 61. I. Gutzow, J. Non-Cryst. Solids 45, 301 (1981). 62. S. K. Das and E. E. Huckle, Carbon 13, 33 (1975). 63. E. Grantcharova, I. Avramov, and I. Gutzow, Naturwissenschaften 73b, 95 (1986). 64. I. Gutzow: The Generic Phenomenology of Glass Formation. In: Insulating and Semiconducting Glasses, Ed. P. Boolchand (World Scientific, Singapore, London, 2000, pgs. 65-93). 65. I. Gutzow et al.: Glass Transition: An Analysis in Terms of a Differential Geometry Approach. In: Nucleation Theory and Applications, Eds. J. W. P. Schmelzer, G. R¨ opke, and V. B. Priezzhev (Joint Institute for Nuclear Research Publishing Department, Dubna, Russia, 1999, pgs. 368-418). 66. P. K. Gupta and C. T. Moynihan, J. Chem. Phys. 65, 4136 (1976). 67. S. V. Nemilov: Thermodynamic Significance of the Prigogine-Defay Ratio and the Structural Difference Between Glasses and Liquids. In: The Vitreous State, Ed. E. A. Porai-Koshitz (Nauka, Leningrad, 1988, p.15 (in Russian)).
vch 4 Okt 2005 10:43
208
9 Thermodynamics of Amorphous Solids and Disordered Crystals
68. A. K. Varshneya, Fundamentals of Inorganic Glasses (Academic Press, Boston, New York, 1993, p.273/275, 292/295). 69. G. M. Bartenev, Doklady Akad. Nauk SSSR 76, 227 (1951). 70. G. M. Bartenev, Structure and Mechanical Properties of Inorganic Glasses (Building Materials Press, Moscow, 1966, p.20/21, 26/31 (in Russian)). 71. H. N. Ritland, J. Amer. Ceram. Soc. 37, 370 (1954). 72. M. Reiner, Phys. Today 17, 62 (1964). 73. M. Stevels, J. Non-Cryst. Solids 6, 307 (1971). 74. M. V. Vol’kenshtein and O. B. Ptizyn, J. Exp. Theor. Phys. (JETF) USSR 26, 2204 (1956) (in Russian). 75. G. M. Bartenev and I. A. Lukyanov, J. Phys. Chem. (USSR) 29, 1486 (1955) (in Russian). 76. H. Kelker and R. Hatz, Handbook of Liquid Crystals (Verl. Chemie, Weinheim, Deerfield Beach, Florida, Basel, 1980, p. 340). 77. H. Yoshioka, M. Sorai, and H. Suga, Mol. Cryst. Liq. Cryst. 95, 11 (1983). 78. H. Suga: Thermodynamic Aspects of Glassy Crystals. In: Dynamic Aspects of Structural Changes in Liquids and Glasses, Eds. C. A. Angell and M. Goldstein Ann. New York Acad. Sci. 484, 249 (1986). 79. P.-G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974). 80. R. Haase, Thermodynamik der Mischphasen (Springer, Berlin, 1956). 81. J. Wilks, The Third Law of Thermodynamics (Oxford Univ. Press, 1961, Chapter 5.4). 82. D. Eisenberg and W. Kauzmann, Structure and Properties of Water (1986; p.77/80, 93/99 of the Russian translation: Gidrometeoizdat, Leningrad, 1993). 83. A. Dobreva and I. Gutzow, J. Non-Cryst. Solids 220, 235 (1997). 84. J. J¨ ackle, Physica B 127, 79 (1984). 85. J. D. Bernal, Scientific American 8, 908 (1961). 86. G. D. Scott, Nature (London) 188, 908 (1960). 87. L. V. Woodcock, J. Chem. Soc. & Farad. Trans. II 72, 1661 (1976). 88. R. J. Beal and R. P. Dean, Phys. Chem. Glasses 9, 125 (1968). 89. B. Wunderlich, J. Phys. Chem. 69, 1052 (1960). 90. W. F. Giauque and J. W. Stout, J. Amer. Chem. Soc. 58, 1144 (1936). 91. L. H. Adams and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965). 92. C. A. Angell, J. Non-Cryst. Solids 102, 205 (1988).
vch 4 Okt 2005 10:43
10
Modifying Fundamental Equations to fulfill the Third Law of Thermodynamics J¨org M¨ oller scitecon, Bayreuther Str. 13, 01187 Dresden, Germany Jede wissenschaftliche Entdeckung ist entweder schon gemacht oder fehlerhaft oder beides. J¨ org M¨ oller
Abstract The laws of classical thermodynamics constrain the form the equations of state may take. These constraints are not met by most equations of state in use, especially with regard to the third law of thermodynamics. A method is proposed by which existing equations of state can systematically be modified to fulfill Nernst’s theorem.
10.1 Introduction In order to describe properties of a substance in the framework of classical thermodynamics, equations of state are needed for that particular substance. The most general kind of such equations is called a fundamental equation. A minimal requirement for such an equation is that it has to fulfill the laws of thermodynamics. Of many fundamental equations in use, few fulfill the third law of thermodynamics, also known as Nernst’s theorem. On the other hand side, these equations have some useful properties to those who invented them or apply them in practice, at
210
10 Fundamental Equations and Third Law of Thermodynamics
higher temperatures. We will address the violation of the third law in particular, and look for systematic amendments to existing fundamental equations retaining all its properties at higher temperatures. This contribution is structured as follows. First we give a comprehensive review of what may appropriately be called the thermodynamic method; dealing with the problem of how to obtain a fundamental equation when the thermal and caloric equations of state are given. This will also lead us to the formulation of the compatibility condition between these two equations, which is a consequence of the second law, resulting in the existence of entropy. This procedure also gives us a powerful and systematic way of dealing with the central problem to be addressed here, the third law of thermodynamics. We review the content of the third law, after which we subject a test sample of equations to the thermodynamic method as introduced. We then modify existing equations of state to fulfill the third law, taking the well-known van der Waals equation as an example. The paper is completed by a summary and outlook.
10.2 The Thermodynamic Method One might view the term ,thermodynamic method’ a bit broader, but we will simply use it for the following problem. From measuring temperature T , pressure p, and volume V , we find that for a given one-component substance, if two of these quantities are given, the third follows. This relation, p = p (T, V ) ,
(10.1)
is called a thermal equation of state. We may also have carried out some measurements concerning heat and work, yielding U = U (T, V ) ,
(10.2)
which is called the caloric equation of state of the particular system. We want to have a fundamental equation, for example, F = F (T, V ) ,
(10.3)
of the system. As F is defined as F = U − TS ,
vch 4 Okt 2005 10:43
(10.4)
10.2 The Thermodynamic Method
211
this task is equivalent to introducing the entropy S as a property of state, as we will see next. We may choose to express the physical quantities F , U , T , and S as functions of whatever variables we like. We choose F (T, V ) = U (T, V ) − T S(T, V ) .
(10.5)
The reason for this choice is that F (T, V ) is a fundamental equation, U (T, V ) is not, but is given as the caloric equation of state. So if we can obtain S(T, V ) from somewhere, our problem would be solved. Once we have F (T, V ), or if we suppose we have it, we find S := −
∂F (T, V ) , ∂T
p := −
∂F (T, V ) , ∂V
(10.6)
which can be taken as the definitions of S and p.
10.2.1 General Solution Integration of the second of Eqs. (10.6) leads to −
p(T, V )dV = F + X(T ) ,
F =−
p(T, V )dV − X(T ) , (10.7)
with X(T ) being some initially arbitrary function of temperature occuring in the integration with respect to V . For this integration, p needs to be represented as a function of T and V . If the expression for F in the first of Eqs. (10.6) is replaced by the second of Eqs. (10.7), one obtains S=
∂
$
p(T, V )dV dX(T ) + = S(T, V ) . ∂T dT
(10.8)
Replacing S in Eq. (10.5) by this expression yields $ ∂ p(T, V )dV dX(T ) + . F (T, V ) = U (T, V ) − T ∂T dT
(10.9)
Comparison of F of Eq. (10.7) and Eq. (10.9) yields −
$ ∂ p(T, V )dV dX(T ) + . (10.10) p(T, V )dV − X(T ) = U (T, V ) − T ∂T dT
vch 4 Okt 2005 10:43
212
10 Fundamental Equations and Third Law of Thermodynamics
After rearranging some terms we obtain dX(T ) − X(T ) = U (T, V ) + T dT
p(T, V )dV − T
∂
$
p(T, V )dV , ∂T
(10.11)
and, after some simplification, X(T ) d ∂ T = U (T, V ) − T 2 T2 dT
$
p(T, V )dV T . ∂T
(10.12)
Solving for X(T ) yields X(T ) = T
U (T, V ) dT − T2
p(T, V )dV + K(V )T ,
(10.13)
with K(V ) being the integrational constant in the integration with respect to T . When determining K(V ), we first notice that the left hand side of Eq. (10.13) does not depend on V . Consequently, the right hand side must not depend on V either. Consequently ∂K(V ) =− ∂V
∂
U (T, V ) dT p(T, V ) T2 + ∂V T
(10.14)
and hence ∂ K(V ) = −
U (T, V ) dT p(T, V ) T2 dV + dV + S0 . ∂V T
(10.15)
Substitution of K(V ) into Eq. (10.13) yields ⎞ ⎛ U (T, V ) ∂ dT ⎟ ⎜ U (T, V ) T2 dV + S0 ⎟ dT − X(T ) = T ⎜ 2 ⎠ . ⎝ T ∂V
(10.16)
The connection between thermal and caloric equation of state can now be obtained in the following way. Eq. (10.13) is divided by T , after which the derivative
vch 4 Okt 2005 10:43
10.3 Procedures 213 of the result with respect to T is computed. In this way, K(V ) disappears. Solving the result for U (T, V ) yields T2
U (T, V ) =
∂ ∂T
p(T, V ) T
dV + T 2
d dT
X(T ) T
(10.17)
or
∂ T ∂T 2
U (T, V ) =
p(T, V ) T
dV + Y (T ) ,
(10.18)
with Y (T ) being a function of temperature only. This is the compatibility requirement between thermal and caloric equations of state in this case. As a consequence we find that the volume dependence of U (T, V ) is completely determined by the thermal equation of state. By comparing the last two equations we find that of X(T ) and Y (T ), one can be obtained if the other is given. They are connected by the relations
Y (T ) = T
2
∂
X (T ) T ∂T
⇐⇒
X (T ) = T
Y (T ) dT + S0 T2
We then find that the entropy can be expressed in the form $ ∂ p (T, V ) dV ∂X (T ) + . S (T, V ) = ∂T ∂T
.
(10.19)
(10.20)
The fundamental equation can then be obtained by solving Eq. (10.17) for F , F (T, V ) = U (T, V ) − T S (T, V ) .
(10.21)
10.3 Procedures The second law puts mutual constraints on the thermal and caloric equations of state. This leads to two separate tasks that may have to be solved. Either the thermal equation of state is known and the caloric equation of state is needed, or, vice versa, the caloric equation of state is known and the thermal equation of state is required. We now list the respective equations in the correct order, like ready-to-use recipes. In brackets we indicate the connections between them.
vch 4 Okt 2005 10:43
214
10 Fundamental Equations and Third Law of Thermodynamics
10.3.1 A: Thermal Equation of State Given If p = p(T, V ) is known, p(T, V ) 2 ∂ dV + Y (T ) U (T, V ) = T ∂T T
(10.22)
follows, with a still undetermined or unknown function Y (T ). This equation confines the choice for a possible, or the form of the real, caloric equation of state. At this point, a caloric equation of state needs to be chosen that fits this constraining equation. This caloric equation of state is inserted into p(T, V ) ∂ dV (10.23) Y (T ) = U (T, V ) − T 2 ∂T T (obtained by means of comparing with, or solving of, Eq. (10.22)). Hence we use Y (T ) dT + S0 , (10.24) X(T ) = T T2 (found by solving the definition Y (T ) for X(T ) to find X(T ).
10.3.2 B: Caloric Equation of State Given If U = U (T, V ) is known instead, we use ⎞ ⎛ U (T, V ) ∂ dT ⎟ ⎜ U (T, V ) T2 dV + S0 ⎟ dT − X(T ) = T ⎜ 2 ⎠ ⎝ T ∂V
(10.25)
(as Eq. (10.16) yields). Now we obtain (as follows from Eq. (10.14), by rearranging ) terms and using the abreviation K2 (V ) = ∂K(V ∂V ) ∂ p(T, V ) = T
U (T, V ) dT T2 + T K2 (V ) ∂V
(10.26)
with an unknown function K2 (V ). This equation limits the choice of the form a thermal equation of state may take. A thermal equation needs to be chosen that fits this constraint.
vch 4 Okt 2005 10:43
10.4 Demonstration of the Method
215
10.3.3 Completion of the Method With both thermal and caloric equations now at hand, we may proceed with completely solving the problem of finding a fundamental equation. The entropy is given by S(T, V ) =
∂
$
p(T, V )dV dX(T ) + ∂T dT
(10.27)
(as Eq. (10.8) yields). Finally our fundamental equation reads F (T, V ) = U (T, V ) − T S (T, V )
(10.28)
(which is identical to Eq. (10.21)).
10.4 Demonstration of the Method 10.4.1 A: Thermal Equation of State Given As a first demonstration of the method, we choose the ideal gas. Its thermal equation of state reads p (T, V ) =
RT . V
(10.29)
By means of Eq. (10.18) we obtain U (T, V ) =
⎛
RT V
⎞
⎜∂ T ⎟ ⎟ T2 ⎜ ⎝ ∂T ⎠ dV + Y (T ) .
(10.30)
Its evaluation yields U (T, V ) = Y (T ) .
(10.31)
So the requirement of compatibility between thermal and caloric equations of state for the ideal gas is that U (T, V ) is a pure function of temperature. This
vch 4 Okt 2005 10:43
216
10 Fundamental Equations and Third Law of Thermodynamics
statement on the caloric equation follows directly from the thermal equation of state. One possible caloric equation of state for the ideal gas would be U (T, V ) = cRT .
(10.32)
Whether this equation is part of the model of an ideal gas or not is regarded differently by different authors. Eq. (10.19) then yields X (T ) = cRT ln (T ) + S0 T .
(10.33)
The entropy then follows from Eq. (10.20), S (T, V ) = cR ln (T ) + R ln (V ) + cR + S0 .
(10.34)
So, starting from the thermal equation of state, we obtain the compatibility condition with the caloric equation of state and an expression for the entropy after four lines of equations. For the fundamental equation we obtain F (T, V ) = −cRT ln (T ) − RT ln (V ) − S0 T .
(10.35)
The only requirement on the caloric equation of state when the thermal equation of state is taken to be the one of the ideal gas is that it must depend on temperature only. So a different example for U (T, V ) would be U (T, V ) =
cRT 2 . T0
(10.36)
We then obtain, respectively, Y (T ) dT + S0 , X (T ) = T T2 X (T ) =
cRT 2 + S0 T , T0
S (T, V ) = R ln (V ) +
(10.37)
(10.38)
2cRT + S0 , T0
(10.39)
and F (T, V ) = −RT ln (V ) −
vch 4 Okt 2005 10:43
cRT 2 − S0 T . T0
(10.40)
10.5 The Third Law of Thermodynamics 217
10.4.2 B: Caloric Equation of State Given Let the caloric equation of state be U (T, V ) = cRT .
(10.41)
Then the compatibility equation yields cRT dT ∂ T2 + T K2 (V ) p(T, V ) = T ∂V
(10.42)
and hence p(T, V ) = T K2 (V ) .
(10.43)
So, with the caloric equation of state given in the form above, all we can say on the thermal equation of state is that it must be a product of T with some arbitrary function of V . The assumption of a dependence of the form R , V
K2 (V ) =
p(T, V ) =
RT V
(10.44)
would be one special choice, which we already considered. Another example would be p(T, V ) =
RT V . V02
(10.45)
10.5 The Third Law of Thermodynamics 10.5.1 Basic Statement The third law may be stated like this: If A and B are two equilibrium states of a closed system, and P describes an arbitrary reversible process, then lim B T →0 dS = 0 ∀ A, B, P (10.46) lim A T →0 P
vch 4 Okt 2005 10:43
218
10 Fundamental Equations and Third Law of Thermodynamics
(cf. [1]). So the entropies of A and B are the same if both states approach T = 0. Sometimes, a weaker form of Nernst’s theorem is given in the form lim S(T, A\T ) − S(T, B\T ) = 0 ,
T →0
(10.47)
where A and B are formally decomposed into temperature T and the remaining description of the state, such that A and B are identical to {T, A\T } and {T, B\T }, respectively. This weaker form, however, does not ensure that the entropy converges at all when T approaches zero. There is a still weaker claim, that is S(T = 0, A\T ) − S(T = 0, B\T ) = 0 .
(10.48)
Such a formulation makes little sense because it is shown in the literature in various ways that T = 0 cannot be reached. We will not discuss this last formulation any further. If, on the other hand side, we stipulate that limT →0 S(T ) = 0, [2], then the third law applies to open systems as well. Even more generally, it then applies to variable amounts of substance. We will not expound on this item any further either, but simply remember that for the thermodynamic analysis it is sufficient if carried out on closed systems, and on one work coordinate, and confine our analysis to such cases. Furthermore we note that all thermodynamic information on a substance is contained in its fundamental equation, also called its thermodynamic potential. For example, if F is given as a function of temperature and volume, F (T, V ), it is a fundamental equation. The same goes for U (S, V ), that is the internal energy as a function of entropy and volume.
10.5.2 Differential Form of the Third Law Because of the limit transition contained in the weak form of Nernst’s theorem, Eq. (10.47) above, there is a differential form for the weak form of the third law, which is equivalent if the functions by which state properties can be represented of each other are continuous and differentiable in the considered range of these properties. As this is usually taken for granted in any thermodynamic consideration, it is fair to consider the differential form completely equivalent to the weak form of Nernst’s theorem. For a differential form of the third law we repeat the two equations ∂F (T, V ) = −S (T, V ) , ∂T
vch 4 Okt 2005 10:43
(10.49)
10.6 State of Fundamental Equations in Use
219
and ∂F (T, V ) = −p (T, V ) . ∂V
(10.50)
They lead to one of Maxwell’s relations, namely ∂ 2 F (T, V ) ∂ 2 F (T, V ) = , ∂T ∂V ∂V ∂T
(10.51)
resulting in ∂p (T, V ) ∂S (T, V ) = . ∂V ∂T
(10.52)
The third law states that upon approaching T = 0, S must not depend on volume (or any other work variables or their conjugates). So the differential form of the third law reads ∂S (T, V ) =0. T →0 ∂V
(10.53)
lim
Using the Maxwell relation just obtained consequently leads to ∂p (T, V ) =0. T →0 ∂T
(10.54)
lim
The important consequence is that the thermal equation of state determines whether the third law is fulfilled or not.
10.6 State of Fundamental Equations in Use We now list a number of different thermal equations of state to find out whether the third law is fulfilled for them. We will just indicate the results without going into details of computation, and leave it to the interested reader to verify the results. A (+) indicates that the third law is fulfilled, a (−) indicates that it is not. Berthelot: (+)
a RT exp − RT V . p (T, V ) = V −b
(10.55)
vch 4 Okt 2005 10:43
220
10 Fundamental Equations and Third Law of Thermodynamics
Dieterici: (−)
p (T, V ) =
a (b − V ) + RT 2 V 2 . T V 2 (V − b)
(10.56)
a (b − V ) + RT 1.5 V (b + V ) √ . T V (b + V ) (V − b)
(10.57)
Redlich: (−)
p (T, V ) =
Wohl (explicite form fairly complex): (−) p=
RT a c − + 2 3 . V − p T V (V − b) T V
(10.58)
Beatti-Bridgeman: (−)
c F 1 − a b V T3 V . − V +e 1− 2 V V V2
RT 1 − p (T, V ) =
(10.59)
Clausius-Callendar: (−)
p (T, V ) =
(b − V ) Ψ (T ) + RT 2 (c + V )2 T (V − b) (c + V )2
Ψ (T ) = 0.075
273 T
.
(10.60)
10/3 .
(10.61)
van der Waals: (−)
p (T, V ) =
3pc Vc2 8pc T Vc − . Tc (3V − Vc ) V2
(10.62)
Ideal gas, which is a limit case of the van der Waals’ equation of state: (−)
vch 4 Okt 2005 10:43
10.7 Modifying Thermal Equations of State to Fulfill the Third Law
221
10.7 Modifying Thermal Equations of State to Fulfill the Third Law We will now see what basic modification of the thermal equations of state is necessary to make it fulfill the third law. We chose as an example van der Waals’ equation, Eq. (10.62), ⎞ ⎛ ⎜p 3 ⎟ T V ⎟ ⎜ −1 =8 , (10.63) ⎜ + 2 ⎟ 3 Vc Tc ⎠ ⎝ pc V Vc which is equivalent to Eq. (10.62). This equation is very nice for discussing basic features of first-order and second-order phase transitions in text books. First let us demonstrate the thermodynamic method as stated above. For the compatibility condition we immediately obtain U (T, V ) = Y (T ) −
3pc Vc2 . V
(10.64)
As the ideal gas is a limiting case of van der Waals’ equation of state, and the caloric equation for the ideal gas discussed first shall now be assumed to be part of the model, our assumption for van der Waals shall be that limV →∞ U (T, V ) shall coincide with this caloric equation of state for the ideal gas, whence we obtain Y (T ) = cRT .
(10.65)
Consequently X (T ) = T
Y (T ) dT + S0 T2
(10.66)
yields X (T ) = cRT ln (T ) + S0 T .
(10.67)
Substituting the respective expression for p(T, V ) into Eq. (10.64) yields U (T, V ) = cRT −
3pc Vc2 , V
(10.68)
vch 4 Okt 2005 10:43
222
10 Fundamental Equations and Third Law of Thermodynamics
and for S we obtain $ ∂ p (T, V ) dV ∂X (T ) + , S (T, V ) = ∂T ∂T
(10.69)
or simply S (T, V ) = cR ln (T ) +
8pc Vc ln (3V − Vc ) + cR + S0 , 3Tc
(10.70)
the fundamental equation turns out to be F (T, V ) = −T (cR ln (T ) + S0 ) −
3pc Vc2 8pc T Vc ln (3V − Vc ) − . V 3Tc
(10.71)
We now turn again to the thermal equation of state, Eq. (10.62). The basic idea for a modification of this equation is this: Every monotonous function of a temperature scale is itself a temperature scale. Hence, we may assume that the temperature scale in the respective given thermal equation of state is not the absolute temperature, and replace it by a function whose argument is, instead, stipulated as being the absolute temperature. One difficulty remains, that is finding an appropriate temperature function. Our first attempt for van der Waals’ equation shall be the following replacement: T →
T Tc
0.5
2 + h2
−h.
(10.72)
For the thermal equation of state we obtain ⎞ ⎛ ⎛ ⎞ 0.5 2 T ⎜8 ⎝ ⎟ + h2 − h⎠ Vc ⎜ ⎟ T c 2 ⎜ 3Vc ⎟ ⎜ − 2 ⎟ . p (T, V ) = pc ⎜ Tc (3V − Vc ) V ⎟ ⎜ ⎟ ⎝ ⎠
(10.73)
First we verify that it fulfills the third law, ∂p (T, V ) =0. T →0 ∂T lim
vch 4 Okt 2005 10:43
(10.74)
10.7 Modifying Thermal Equations of State to Fulfill the Third Law
223
It contains an additional parameter h. This parameter can be regarded as a temperature shift. In this regard it is interesting to consider the so-called critical ratio s=
RTc pc Vc
(10.75)
with Tc being the critical temperature, pc being the critical pressure, Vc being the critical volume, and R = 8.314 J/K being the ideal gas constant. For van der Waals’ equation of state one finds Tc = 8/(27R)(a/b)
pc = 1/27(a/b2 )
Vc = 3b ,
(10.76)
a and b being some material parameters, and consequently s=
8 . 3
(10.77)
Substance van der Waals He H2 O2 CO2 H2 O
Tc
pc
5.1 K 33 K 154 K 304 K 647 K
2.26 atm 12.8 atm 49.7 atm 73 atm 217.7 atm
Vc 58 65 74 96 55
cm3 cm3 cm3 cm3 cm3
s 2.6666 3.192 3.25449 3.4358 3.5594 4.433
h 0K 0.9227 K 6.617 K 39.445 K 89.006 K 343.311 K
Tab. 10.1 One illustration of the method
The observed critical ratios are, however, greater for real gases. With the replacement for the temperature, Eq. (10.76) takes a different form, Tc =
(1/2) 2 8a 2 +h −h , 27Rb
(10.78)
and so does Eq. (10.77) 8 s= 3
' 1+
27bhR . 4a
(10.79)
vch 4 Okt 2005 10:43
224
10 Fundamental Equations and Third Law of Thermodynamics
Solving for h yields h=
a(9s2 − 64) . 432bR
(10.80)
So, we get s = 8/3 for the van der Waals equation of state, while for real gases s is greater. The additional parameter h can be determined from the observed critical ratios. Table 10.1 gives a respective overview over some important substances. These are the preliminary consequences: • In order to modify a fundamental equation of state so that it fulfills the third law, it is sufficient to modify the thermal equation of state. • The modification consists in a replacement of the temperature scale.
10.8 Further Aspects The correction introduced, completely solves the problem of fulfilling the third law, as we have shown. However, the correction introduced is not satisfactory, because it also modifies properties of the equations of state which we might like to retain. For example, in order to guarantee that p(Tc , Vc ) = pc , we need to introduce an additional parameter, q. For the thermal equation of state we now obtain ⎞ ⎛ ⎛ ⎞ 0.5 2 T ⎜8 ⎝ ⎟ + h2 − h⎠ Vc ⎜ ⎟ T c 2 ⎜ 3Vc ⎟ ⎜ − 2 ⎟ (10.81) p (T, V ) = pc q ⎜ Tc (3V − Vc ) V ⎟ ⎜ ⎟ ⎝ ⎠
with q=
Tc
0.5
4 (h2 + 1)0.5 − 4h − 3Tc
.
(10.82)
This equation fulfills the third law, i.e. ∂p (T, V ) =0. T →0 ∂T lim
vch 4 Okt 2005 10:43
(10.83)
10.8 Further Aspects
225
However, U (T, V ) =
⎞ p (T, V ) ∂ ⎟ ⎜ T T2 ⎝ ⎠ dV + Y (T ) ∂T ⎛
(10.84)
leads to an equation such that lim U (T, V ) = ∞ .
(10.85)
V →∞
A coincidence with the caloric equation of state of an ideal gas with V → ∞, which is considered an important property of van der Waals’ equation, cannot be obtained. We can correct this by a further modification, i.e., the extension h = h(V ) in such a way that limV →∞ p(T, V ) = 0. We then have to modify the thermal equation of state once more ⎛ ⎞ 2 2 0.5 T hVc hVc ⎠ 8⎝ + − Tc V V 3Vc p (T, V ) = − 2 , (10.86) Vc pc q Tc (3V − Vc ) V q=
Tc 4 (h2
+ 1)
0.5
− 4h − 3Tc
.
(10.87)
Now we encounter a more formal problem: Eq. (10.18) cannot be evaluated analytically. For formal manipulations this is not very satisfactory. So we need to look for an alternative for Eq. (10.72). One is this: T →
T3 . hTc2 + T 2
(10.88)
As the thermal equation of state we now obtain p (T, V ) =
3qpc Vc2 8qpc T 3 Vc − 2 2 Tc (3V − Vc ) (hTc + T ) V2
(10.89)
with q=
h+1 . 1 − 3h
(10.90)
vch 4 Okt 2005 10:43
226
10 Fundamental Equations and Third Law of Thermodynamics
The third law is fulfilled: ∂p (T, V ) =0. lim T →0 ∂T
(10.91)
Again we need to make sure that h disappears with V → ∞. We choose the same kind of modification here: p (T, V ) =
q=
3qpc Vc2 8qpc T 3 Vc − hVc 2 V2 Tc + T 2 Tc (3V − Vc ) V
h+1 . 1 − 3h
(10.92)
(10.93)
U (T, V ) can be obtained in a closed (though lengthy) form; we also obtain its limit with V → ∞, which we assume must coincide with the ideal gas: 16hqpc T 3 Tc Vc (ln (3) − 2 ln (T )) (3hTc2 + T 2 )2
+ Y (T ) = cRT ,
(10.94)
which, in turn, leads to Y (T ) and consequently to ! " 8qpc Vc ln 3hTc2 + T 2 X (T ) = cR ln (T ) + S0 − T 3Tc
(10.95)
" ! 8qpc Vc 2T 2 ln (T ) + 3hTc2 ln (3) . + 3Tc (3hTc2 + T 2 ) The expressions for S(T, V ) and for F (T, V ) can also be obtained via S(T, V ) =
∂
$
p(T, V )dV dX(T ) + ∂T dT
(10.96)
and F (T, V ) = U (T, V ) − T S (T, V ) .
(10.97)
They can be obtained in a closed form, which is straightforward. However, there is no point in writing down these very lengthy expressions.
vch 4 Okt 2005 10:43
10.9 Discussion of Literature
227
Two more points shall still be mentioned. One is the critical point. It is given by those values of p, T , and V , at which the first and second derivatives of the thermal equation of state in the form p(T, V ) are both zero. We did not carry out this analysis, so the respective abbreviations in the modified equations above are no critical values with respect to these equations. Obtaining them was not the focus of this paper. Furthermore, it would lead to extremely lengthy expressions, if closed expressions can be obtained at all. Another point is the connection to statistical physics. One may deduce a special form for the replacement of the temperature from the Einstein or the Debye model for heat capacity. We did not proceed with this here, as van der Waals’ equation contains the phase transition between liquid and gas, but does not describe solids. The cited models for heat capacity, however, are supposed to work for solids. For different equations of state this is certainly a fruitful consideration, as the basic physical reason is the same in both cases, it is the circumstance that a system in complete internal equilibrium upon approaching absolute zero can be said to occupy exactly one state.
10.9 Discussion of Literature As is true of virtually all scientific papers, many individual aspects of this contribution can be found elsewhere. So, the compatibility aspect between thermal and caloric equations of state can be found in various forms, a concise one being ([3]; page 87). That the weak form of Nernst’s theorem can be expressed in a differential form only using the thermal equation of state can be found, e.g., in [4]. It has been known, of course, that there are other aspects in the realm of statistics and quantum mechanics as to why Nernst’s theorem is valid. And, as I was to find out lately, there is an approach in literature virtually identical to the one presented here, published by K. Popoff [5]. However, Popoff does not carry his basic analysis through to the end, as he fails to conserve certain properties at higher temperatures. Essentially, the equation given by him, namely pv = RT α , does not converge to the ideal gas law at higher temperatures. To ensure such a convergence, a parameter h needs to be introduced as in Eq. (10.72). Without the introduction of h which would have been necessary in his equation as well, the internal energy diverges as V tends to infinity, requiring further modification as done in this publication, from Eq. (10.85) on. So he does not reach the point at which he could recognize that the modification of the temperature definition
vch 4 Okt 2005 10:43
228
10 Fundamental Equations and Third Law of Thermodynamics
leads to increased critical ratios as well, which even can be made to coincide with experimental values, as was shown here.
10.10 Summary • A set of equations is proposed that allow us to obtain explicite compatibility requirements between thermal and caloric equations of state, as well as a fundamental equation, in a very efficient way. • In order to modify a fundamental equation of state so that it fulfills the weak form of the third law, it is sufficient to modify the thermal equation of state. • To modify existing equations of state such that the weak form of the third law is fulfilled can be done by modifying the temperature scale (c.f. [5]). • Other properties may also get lost, which have to be taken care of by further modifications of the thermal equation of state. These modifications can be carried out in such a way that the third law keeps being fulfilled. • It is best to start with fundamental equations already fulfilling the third law.
Acknowledgements This talk was given at the VII. Research Workshop on Nucleation Theory and Applications in Dubna, in April 2003. I am grateful to the organizers in Rostock and Dubna for preparing, and inviting me to, such an interesting and nice workshop. I am also grateful to the Humboldt Foundation, which allowed me a stay at the Bulgarian Academy of Sciences in 2005, resulting in a significant addition to this contribution as well, and to Prof. I. Gutzow for many fruitful discussions.
10.11 References 1. W. Nernst, The New Heat Theorem (Dover Publ. Inc., New York, 1969). 2. M. Planck, Vorlesungen u ¨ber Thermodynamik (Walther de Gruyter, Berlin, 1964, 11. edition). 3. F. Henning, Handbuch der Physik, Band IX: Theorien der W¨ arme (Julius Springer, Berlin, 1926).
vch 4 Okt 2005 10:43
10.11 References
229
4. L. D. Landau and E. M. Lifschitz, Lehrbuch der theoretischen Physik, vol. V, Statistische Physik (Akademie-Verlag, Berlin, 1983). 5. K. Popoff: Nernstsches Theorem und Theorie realer Gase. Annuaire de l’Universit´e de Sofia, pgs.137-153, 1948-1949.
vch 4 Okt 2005 10:43
11
Nucleation and Growth of Gas-Filled Bubbles in Low-Viscosity Liquids Vitali V. Slezov(1), Alexander S. Abyzov(1) and Zhanna V. Slezova(2) (1)
National Science Center ”Kharkov Institute of Physics and Technology”, ul. Akademicheskaya 1, Kharkov, 61108 Ukraine (2)
Scientific and Technological Enterprise ”Institute
for Single Crystals”, pr. Lenina 60, Kharkov, 61001 Ukraine Ich bin aus vielf¨ altiger Erfahrung u ¨ berzeugt, daß die wichtigsten und schwersten Gesch¨ afte in der Welt, die der Gesellschaft den meisten Vorteil bringen, durch die sie lebt und sich erh¨ alt, von Leuten getan werden, die zwischen 300 und 800 oder 1000 Taler Besoldung geniessen, zu den meisten Stellen, mit denen 20, 30, 50, 100 Taler oder 2000, 3000, 4000, 5000 Taler verbunden sind, k¨ onnte man nach halbj¨ ahrigem Unterricht jeden Gassenjungen t¨ uchtig machen, und sollte der Versuch nicht gelingen, so suche man die Schuld nicht im Mangel an Kenntnissen, sondern in der Ungeschicklichkeit, diesen Mangel mit dem geh¨ origen Gesicht zu verbergen. Georg Ch. Lichtenberg
11.1 Introduction
231
Abstract Bubble nucleation and growth in low-viscosity liquids supersaturated with gas is theoretically studied. It is shown that, in a certain parameter range, the bubble size is adjusted to the amount of gas in a bubble, and the state of the bubble can be described using one variable: the bubble size or the number of gas atoms in a bubble. Expressions for the nucleation time, bubble size distribution function, flux of nuclei in bubble size space, and the maximum number of bubbles formed in the system are determined. After the nucleation period an intermediate stage of the process starts, when the number of bubbles per unit volume remains virtually constant, whereas the amount of the gas dissolved per unit volume of the solution significantly decreases, almost attaining the equilibrium value. At the late stage (coalescence), small subcritical bubbles disappear due to gas transfer to large supercritical bubbles; as a result, the number of bubbles in the liquid decreases. For all these stages, the kinetics of evolution of the bubble size distribution function and the amount of gas per unit volume of the solution is determined.
11.1
Introduction
The process of formation of bubbles in a solution supersaturated with a gas is an example of a multi-component phase transition of large practical importance, for example, it determines major parts of the technology of formation of polymeric foams. For the case when the heat conductivity of the solution is sufficiently high, the temperature of a bubble does not differ from that of the solution and the evaporation of a liquid into the bubble can be neglected. The process can be characterized then by the bubble radius and the number of gas atoms in the bubble and a particular consideration of the evolution of the temperature of the bubbles is not required. Processes of phase transitions can be subdivided in a variety of different applications into three stages. At the nucleation stage, the bubbles with sizes larger than the critical size are formed. In view of a strong dependence of the nucleation rate on the degree of supersaturation, this stage is characterized by almost time-independent conditions and is terminated by the formation of the spectrum of the finite-size particles (bubbles) of a new phase. At the second stage, new bubbles are virtually not formed, formed earlier supercritical bubbles grow, and the supersaturation decreases substantially. At the third (final) stage, when the supersaturation of the solution almost vanishes, the
vch 4 Okt 2005 10:43
232
11 Gas-Filled Bubbles in Low-Viscosity Liquids
part of bubbles with pre-critical sizes dissolves and supplies the substance (gas) for the growth of supercritical bubbles. In this case, the total amount of gas in the bubbles is conserved, their mean size increases, but the number of bubbles decreases. The kinetics of multicomponent nucleation was first theoretically treated by Reiss [1] who assumed that, in composition space, the flux of the clusters to the new phase passes the saddle point of the characteristic thermodynamic potential. Later, this idea was elaborated in a number of works (e.g., see [2, 3]). Thus, the problem becomes efficiently one-dimensional, because the particle size is completely determined by the position of the saddle point. Such scenario can be considered as the rule, but exceptions from this general rule are possible as well (see e.g. [4-7]). Such exceptions correspond to situations when the main flux to the new phase passes not the saddle but some ridge point of the thermodynamic potential. General expectations and detailed analyses [8, 9] allow us to conclude that ridge crossing will be the preferred channel of the transformation in the case of significant excess of one of the components in the solution and when the thermodynamic barrier is relatively low. In Refs. 10-12, the nucleation of gas-filled bubbles in a viscous liquid was studied, and the diffusion regime of their growth was considered. For the case of boilingup of gas dissolved in low-viscosity liquids, Kuni and Zhuvikina [13] derived the kinetic equations for homogeneous nucleation and discussed the general regime of bubble growth. This regime includes both the diffusion regime treated in Refs. 11 and 12 and free molecular regime, both being considered as limiting cases. The bottleneck of the molecular regime is the boundary kinetics of the addition of gas molecules to the bubble. Subsequent growth of supercritical bubbles was discussed with the account of solvent fugacity in Ref. 14. The first two stages of the process (nucleation and the growth of supercritical bubbles) were treated [15] for the case of instantaneous creation of the initial metastability in high-viscosity liquids, when the amount of gas in the bubble was adjusted to its volume and when the state of the bubble is unambiguously determined by its size. This work is devoted to the study of all three successive stages of the process of the nucleation [16] and growth [17] of gas-filled bubbles in low-viscosity liquids. Note that metal melts can also be considered as low-viscosity liquids within a rather wide temperature range; hence, such an approximation has fairly wide applications. The method employed in this work makes it possible to determine the main parameters (the rate of bubble growth, characteristic times, and the total volume of supercritical bubbles [14]), as well as the bubble size distribution function at all stages of the process.
vch 4 Okt 2005 10:43
11.2 Nucleation (tlag < t < tN )
233
Nucleation (tlag < t < tN )
11.2
11.2.1 Reduced Equations describing the Process of Bubble Nucleation in a Low-Viscosity Liquid As a rule, both the liquid and the gas in a bubble in the process of bubble nucleation in a liquid can be considered as being in a local thermodynamic equilibrium, whereas the system as a whole is in an nonequilibrium state. Therefore, to describe the nucleation process, it is convenient to introduce the free energy that determines the corresponding fluctuations under the conditions of thermodynamic equilibrium. Upon the nucleation of a bubble in the metastable medium, its variation ∆F (V, N ) is defined by the expression [15-19] ∆F (V, N ) = V (pL − pV ) + N (µV − µL ) + 4πR2 σ .
(11.1)
Here, V , R and N are the volume, radius of a bubble, and the number of gas molecules in the bubble, respectively; pV and pL are the pressures in the bubble L and! the liquid " afterV its transition to the metastable state, respectively; µ = L L µ p(n ), T and µ are the chemical potentials of the gas atom in a liquid and in a bubble, respectively; p(nL ) is the external saturating gas pressure, i.e., the pressure at which the gas in the bubble with a size of R is in equilibrium with the gas dissolved in a liquid with density nL ; and σ is the surface tension of the liquid. Variables describing the bubble are regarded as continuous so that (1/N ) 1. At the range where this ratio is of the order of unity, the calculation accuracy for the averages will be quite adequate, provided that the range of their variation is rather wide (for the case in question, it is sufficient that the amount of gas in a critical bubble is large, Nc 1). Differentiating ∆F with respect to N at constant volume and taking into account that N ∂µV /∂pV T =const = N ω = V (ω is the volume per gas molecule in the bubble), we obtain ∂∆F = µV − µL . ∂N V =const
(11.2)
Similarly 2σ ∂∆F . = pL − pV + ∂V N =const R
(11.3)
vch 4 Okt 2005 10:43
234
11 Gas-Filled Bubbles in Low-Viscosity Liquids
The parameters of the equilibrium bubble (Rc , Nc ) are found from the extremum of the ∆F function. From Eq. (11.3), we arrive at pV = pL +
2σ . Rc
(11.4)
This expression determines the pressure in a bubble with a size of R, which is in mechanical equilibrium with the liquid at pressure pL . From Eq. (11.2), it follows " ! µV pV , T = µL (p, T ) .
(11.5)
! " Here, p = p nL is the external (saturating) pressure of the gas that is in equilibrium with the gas dissolved in a liquid with density nL at a given moment of time. This means that the chemical potential of the gas in the liquid can be replaced in the right-hand side of Eq. (11.5)! by "the equal chemical potential of the saturating gas. Assuming that pV and p nL values are rather small (≤ 107 Pa) and using the equation of state for the ideal gas with constant heat capacity (µ = kB T ln p + χ (T ), kB is Boltzmann’s constant), we obtain from Eq. (11.5) ! "
ln pV /p nL = 0 ,
! " pV = p n L .
(11.6)
Thus, critical size Rc of the bubble containing Nc gas atoms is determined from Eqs. (11.4) and (11.6) V
p = Nc kB T
4π 3 R 3 c
−1
= p(nL ) .
(11.7)
As! a "rule, the critical size at the nucleation stage is fairly small; then, pL p nL ≈ 2σ/Rc and Rc =
2σ 2σ 2σ , = ≈ pV − pL p (nL ) − pL p (nL )
pV =
! " Nc kB T = p nL , Vc
Nc =
pL +
vch 4 Okt 2005 10:43
2σ Rc
(11.8)
(11.9) Rc2
2σ Vc 8πσ Vc ≈ = . kB T Rc kB T 3 kB T
11.2 Nucleation (tlag < t < tN )
235
Hereafter, we assume that, for the gas dissolved in a liquid, Henry’s law is fulfilled, which is applicable within a wide range of parameters nL (t) =
δ kB T
p(nL (t)) = δnG ,
L nL e ≡ n (∞) =
δ kB T
(11.10)
" ! pL nL (∞) = δnG e ,
(11.11)
where δ is the solubility parameter, nL = nL (t), nL (0) is the initial gas density, nG is the saturating gas density equal to the density of the gas in the ambient space (we suppose that the gas in a liquid is distributed homogeneously, it is true in the case when the metastable state is created quickly as compared to the time interval before the start of nucleation [18]) and nG e corresponds to the gas L (nL ) ≡ pL in a liquid. at the pressure p equilibrium density nL e e As was shown in Ref. 15, depending on the value of parameters, one can consider two limiting cases: a high-viscosity liquid where the amount of gas in a bubble is adjusted to its volume and a low-viscosity liquid where the bubble volume is determined by the amount of gas in a bubble. The first case is realized upon the fulfillment of inequality τg τR [15], where τg = 2lR (3δD)−1 is the char! "−1 is the acteristic time of filling the bubble with the gas, τR = 4η 3pL + 8σ/R characteristic time of bubble size variations, D is the diffusion coefficient for gas atoms in the liquid, l is the length of an elementary displacement (of the order of the distance between atoms in a liquid), and η is the liquid viscosity. In the case of a low-viscosity liquid, the inverse inequality is valid τg τR .
(11.12)
Upon the adjustment of the bubble size R = R (N ) to the amount of gas in the bubble, we arrive at the mechanical equilibrium for the virtually entire spectrum of sizes where the condition pV = pL +
2σ R
(11.13)
is fulfilled.
vch 4 Okt 2005 10:43
236
11 Gas-Filled Bubbles in Low-Viscosity Liquids
According to Eq. (11.6), N = 8πσR2 (N ) /3 (for Nc , Eq. (11.10) is valid). Substituting the condition Eq. (11.4) into Eq. (11.1), we find 2σ 4 V L L (11.14) , T ) − µ + πR2 σ . ∆F (V, N ) = N µ (p + R 3 After the adjustment, Eq. (11.3) is retained with the account of ∂pV /∂R = −2σ/R2 and ∂µV /∂pV = ω. As a result, we get ∆F (V ) = ∆F (V (N ) , N ) = N kB T ln
1 pV + N kB T L p (n ) 2 (11.15)
Nc 1 N kB T ln + N kB T . = 2 N 2 The bubble volume V (N ) = 4πR3 /3 is unambiguously determined by the number of gas atoms in the bubble using the equation of state of the gas and Eq. (11.13). Thus, for the ideal gas, we obtain V (N ) = N kB T
2σ p + R L
−1 .
(11.16)
The bubble size distribution function in a low-viscosity liquid can be represented as Eq. (11.16) f (N, V, t) = ψ (N, t) δ (V − V (N )) .
(11.17)
After integration of the general equation with respect to the adjusting variable V , the reduced distribution function ψ (N, t) satisfies the equation ∂I ∂ψ =− ∂t ∂N
(11.18)
with the boundary conditions ψ(N, t)|N =1 = nL ,
ψ(N, 0)|N >1 = 0 ,
(11.19)
where the flux I in the space of gas amount in the bubble is determined by Eq. (11.20) ∂ψ 1 ∂∆F ψ+ . (11.20) I = −D(N ) kB T ∂N ∂N
vch 4 Okt 2005 10:43
11.2 Nucleation (tlag < t < tN )
237
Here, D(N ) = WN,N +1 is the diffusion coefficient of bubbles in the space of the amount of gas in the bubble or the probability of absorption of one atom by the bubble from the solution per unit time [18]. The first and the second terms in the brackets of Eq. (11.20) are correspondingly the thermodynamic and the diffusion flow rates in the space of numbers of gas atoms in the bubbles, N . At the initial stage of bubble nucleation with the gas when there are still no diffusion clouds around the bubbles [18], the rate of filling the bubble with the gas has the form 1 ∂∆F dN = −D(N ) , dt kB T ∂N
D(N ) = α
D 2l
4πR2 nL ,
(11.21)
where α is the coefficient accounting for the additional barrier, which can exist for the last jump of gas atoms into the bubble, 0 < α ≤ 1. From Eq. (11.16), we also find 1 ∆F 1 Nc = ln . kB T ∂N 2 N
(11.22)
Using Eq. (11.17), we arrive at the law of conservation of the total amount of gas in the system [18] ⎡ n0 − n (t) = ⎣1 −
∞
L
0
⎤−1 V (N )ψ(N, t)dN ⎦
∞ N ψ(N, t)dN .
(11.23)
0
Here, n0 ≡ nL (0) is the initial density of the gas dissolved in the liquid; the expression in square brackets on the right-hand side of Eq. (11.23) accounts for an increase in the total volume of a system due to bubble growth. Eq. (11.18), together with the law of conservation of the total amount of gas in a system (Eq. (11.23)), represent the total set of equations describing the nucleation of bubbles with gas in a low-viscosity liquid.
11.2.2 Time of Establishment of Steady-state Conditions of Nucleation At the instantaneous transition of a system into the metastable state, the flow of critically sized nuclei emerges and the quasi-steady-state regime is established after a certain lag-time tlag . Let us estimate this time.
vch 4 Okt 2005 10:43
238
11 Gas-Filled Bubbles in Low-Viscosity Liquids
By the conditions determining the parameters of the critical bubbles, the first derivatives of the thermodynamic potential with respect to N and R are equal to zero. This makes it possible to find the bubble size Rc and the amount of gas in the bubble, Nc . From Eq. (11.5), we obtain pV R=Rc = p(nL (t)); then Rc =
2σ , − pL
(11.24)
p(nL )
where pL is the pressure in a liquid corresponding to the equilibrium density of L the dissolved gas nL e = δp /kB T . For the transition stage, where the metastability of the system decreases substantially but is still rather high, we have Rc =
2σδ 1 1 2σ 2σδ = = Rc0 , = L L p kB T n kB T n (0) 1 − Z(t) 1 − Z(t) (11.25)
nL (0) − nL (0) + nL = 1 − Z(t) , nL (0)
where Z (t) = nL (0) − nL (t) /nL (0) is the relative (per unit volume) number of gas atoms in the bubbles. Taking into account that, as a rule, for pre-critical bubbles, the case pL 2σ/R is realized, from the equation of state, we obtain 8π 8π σ 1 L 2 p V + σR ≈ R2 , (11.26) N= kB T 3 3 kB T nL
nL (0)
=
1 8π 8π σ L 2 p Vc + σRc ≈ R2 , Nc = kB T 3 3 kB T c
(11.27)
R2 N = 2 , Nc Rc
(11.28)
2
R =
2 Rc0
N , Nc0
N R2 = 2 , Nc0 Rc0 Nc0
! " p nL (0) Vc0 . = kB T
(11.29)
The use of Eq. (11.26) yields Nc = (1 − Z (t))−2 . Nc0
vch 4 Okt 2005 10:43
(11.30)
11.2 Nucleation (tlag < t < tN )
239
3 /3 is the volume of a critical bubble where the pressure is equal Here, Vc0 = 4πRc0 " ! to the saturation pressure of the dissolved gas at the initial moment, p nL (0) and Nc0 is the amount of gas in the critical bubble at the initial moment. Then, we have
∞ L
n (0)Z (t) =
N ψ(N, t)dN .
(11.31)
0
Let us write the expression for the diffusion coefficient D(N ) (Eq. (11.21)) as D(N ) = α
N 1 3D 4π 3 L R n (0) (1 − Z (t)) = N (1 − Z (t)) , (11.32) 2lRc0 3 c0 Nc0 t0
where t0 =
2lRc0 . 3αDδ
(11.33)
When deriving this relation, we used the Henry law Eq. (11.10) and the equality p(nL (0))Vc0 4π 3 L Rc0 n (0) = δ = Nc0 δ . 3 kB T
(11.34)
Using Eqs. (11.22), (11.28), and (11.32), we can write Eq. (11.21) in the following form 1 N dN = N (1 − Z) ln , dt 2t0 Nc
(11.35)
or, with account of Eq. (11.30), as N N 1 dN = N (1 − Z (τ )) ln , = N (1 − Z(τ )) ln + 2 ln dτ Nc Nc 1 − Z(τ )
(11.36)
where τ=
t . 2t0
With the notation y = N/Nc , Eq. (11.18) gets the form ∂ 2 ∂ψ ∂ψ = y −ψ ln y + . ∂τ ∂y Nc ∂y
(11.37)
(11.38)
vch 4 Okt 2005 10:43
240
11 Gas-Filled Bubbles in Low-Viscosity Liquids
For the range of pre-critical nuclei, 2 ≤ N < Nc , it is convenient to introduce the variables ψ˜ = N ψ = ex ψ .
x = ln N ,
(11.39)
Then ln y = ln(
N ) = x − xc , Nc
(11.40)
and Eq. (11.37) is transformed into ∂ 2 ψ˜ ∂ ψ˜ ∂ ψ˜ = b 2 + a ψ˜ − dψ˜ , ∂τ ∂x ∂x
ψ˜
N =1
= nL ,
(11.41)
1 d = (1 − 2e−x ) > 0 . 2
(11.42)
where b = e−x ,
a=
1 (xc − x) − 4e−x > 0 , 2
At variable coefficients a, b and d, Eq. (11.41) cannot be solved exactly. However, taking into account the constant signs of these coefficients, let us estimate the upper limit of τlag . Assuming that the coefficients are constant, we arrive at the exact solution of this equation. Using the substitution a 2 a τ exp − x ψ˜ = p(x, τ ) exp − d + 4b 2b
(11.43)
we obtain ∂ 2 p(x, τ ) ∂p(x, τ ) =b , ∂τ ∂x2
x N ψ = ψ˜ = n √ πb
τ
L
×
dτ (τ − τ )3/2
vch 4 Okt 2005 10:43
.
0
p(x, τ )|x=0
x2 exp − 4b(τ − τ )
a2 τ , = n exp − d + 4b L
a2 exp − d + (τ − τ ) × 4b
(11.44)
(11.45)
11.2 Nucleation (tlag < t < tN )
241
−1
instead of τ ; then we get Let us introduce variable the ξ = x 2 b(τ − τ ) 2 ψ˜ = nL √ π
∞ √ x(2 bτ )−1
α 2 exp − − ξ dξ , ξ
(11.46)
where α=
a2 d+ 4b
x2 . 4b
(11.47)
The integrand in Eq. (11.46) has a sharp maximum at ξmax = α1/4 . If this maximum fits the integration domain, ψ˜ is virtually independent of τ that corresponds to the quasi-stationary state. Then, we derive the equation for the time of establishment of such state, τlag , equating Eq. (11.47) to the lower integration limit in Eq. (11.46) at τ = τlag x x2 = α1/2 = √ 4bτlag 4b
' d+
a2 , 4b
(11.48)
thus arriving at τlag = √
x . 4db + a2
(11.49)
The parameters a, b, and d are functions of N and Nc . Let us estimate the maximal time τlag ; to do so, we substitute the minimal values amin = 0, bmin = e−zc = Nc−1 , d = 1/2, and x = xc and get τlag ≤
Nc0 Nc0 ln √ , 2
(11.50)
and for the dimensional time tlag ≤ t0
2 lRc Nc0 Nc0 Nc0 ln √ = Nc0 ln √ . 3α Dδ 2 2
(11.51)
Thus, the quasi-stationary distribution of pre-critical bubbles with a distribution function ψ(N, t) = ψ(N, Nc )
(11.52)
vch 4 Okt 2005 10:43
242
11 Gas-Filled Bubbles in Low-Viscosity Liquids
and a flux I(N, t) = I(N, Nc ) is established after the time t > tlag . The number of gas molecules in a critical nucleus, Nc , depends on time due to the depletion of the solution; however (at the stage of intensive nucleation), this dependence, as will be shown below, turned out to be rather weak. Therefore, ψ(N, Nc ) and I(N, Nc ) can be regarded as slowly varying functions of the parameter Nc .
11.2.3 Quasi-Stationary Distribution of Sub-critical Bubbles As was shown in Refs. 21 and 22, it is convenient to describe the nucleation process after the establishment of the quasi-stationary state (t > tlag ) in terms of the flux I(N, t). For this purpose, let us express the distribution function via I(N, t), using Eq. (11.20)
∆F ψ(N, t) = exp − kB T
Nmax
dN I(N , t) exp D(N )
∆F (N ) kB T
,
(11.53)
N
where Nmax is the upper bound of the region of nonzero values of flux and distribution function in space N . Since exp (∆F (N )/kB T ) has a sharp maximum in the point N = Nc , the integral can be taken using the saddle point approximation and we get Nmax ∆F (N ) ∆F (N ) I(Nc ) exp − . dN exp ψ(N, t) = D(Nc ) kB T kB T
(11.54)
N
Expanding the function ∆F into a series in the vicinity of the saddle point, we obtain ∆F (Nc ) 1 ∆F (N ) = + N (11.55) kB T kB T kB T ∂ 2 ∆F 1 ∂∆F N = (N − N ) + (N − Nc )2 , = c ∂N Nc 2kB T ∂N 2 Nc 1 Nc 1 ∂∆F = ln , kB T ∂N 2 N
vch 4 Okt 2005 10:43
1 1 ∂ 2 ∆F 1 . ≡ = 2 2 (∆N ) 2kB T ∂N 4N
(11.56)
11.2 Nucleation (tlag < t < tN )
243
Taking into account that, at the stage of intensive nucleation, Nmax ≥ Nc + ∆N holds and, at homogeneous nucleation, each gas atom is a potential nucleus of a bubble, i.e., ψ(N, t)|N →1 = nL (0) ,
(11.57)
we arrive at √ nL (0) Nc Nc √ exp − , I(Nc ) = t0 2 π 2
N ψ(N, t) = n (0) exp − 2 L
(11.58)
Nc 1 N − Nc ln +1 1 − erf , N 2 ∆Nc
(11.59)
where ∆Nc =
−1/2 1 ∂ 2 ∆F − ≈ 2 Nc . 2kB T ∂N 2 N =Nc
(11.60)
Expression similar to Eq. (11.59) have been first derived in Ref. 23. It demonstrates that the behavior of the distribution function ψ(N ) should be significantly changed at N ≈ Nc .
11.2.4 Distribution Function of Bubbles in the Range ˜ Nc < N < N Let us calculate the distribution function for bubbles within the range Nc < ˜ , where N ˜ is determined from the condition ∆F (N ˜ ) = 0. In this range, N N becomes gently sloping and the growth in size space can be considered as purely ˜ = eNc . In variables x = ln N hydrodynamic. From Eq. (11.16), we find that N and τ = t/2t0 , system acquires the form [18-20] ! " ∂I ∂2I ∂I = − x − xc + 2e−x + 2e−x 2 , ∂τ ∂x ∂x
(11.61)
I(x, τ )|x=xc = I (Nc (τ )) ,
(11.62)
I(x, τ )|τ =0, x>xc = 0 .
vch 4 Okt 2005 10:43
244
11 Gas-Filled Bubbles in Low-Viscosity Liquids
In order to estimate the time of the establishment of the quasi-stationary state, one can omit, with a good accuracy (it is true already at Nc > 10), in the right-hand side of Eq. (11.62) the term 2e−x xc in the coefficient at the first derivative and substitute e−xc for e−x in the coefficient at the second derivative. Then ∂I ∂I ∂2I ˜ c) . = − (x − xc ) + 2e−xc 2 , I(x, τ )|x=xc = I (Nc (τ )) = I(x ∂τ ∂x ∂x ! " After the substitution I(x, τ ) = I φ(x, τ ), t˜(τ ) , we have ∂I ∂I ∂φ ∂I ∂ t˜ = + ∂τ ∂φ ∂τ ∂ t˜ ∂τ ∂I ∂φ 2 ∂I ∂ 2 φ 2 ∂ 2 I ∂φ 2 + . + = − (x − xc ) ∂φ ∂x Nc ∂φ2 ∂x Nc ∂φ ∂x2
(11.63)
(11.64)
The function φ(x, τ ) is chosen as φ(x, τ ) = (x − xc ) e−τ . Then ∂2φ =0, ∂x2
∂φ = e−τ , ∂x
∂φ = − (x − xc ) e−τ . ∂τ
(11.65)
1 − e−2τ t˜(τ ) = , Nc
(11.66)
Selecting the function t˜(τ ) such that 2 ∂ t˜ = ∂τ Nc
∂φ ∂x
2 =
2e−2τ , Nc
we arrive at equations ∂2I ∂I = , ∂ t˜ ∂φ2
I(x)|x=xc = I(Nc ) ,
φ(x, τ )|x=xc = 0 ,
(11.67)
which can be solved exactly. As the result, we obtain
I = I(Nc0 ) 1 − erf
φ √ = I(Nc0 ) [1 − erf (p(τ )(x − xc ))] , 2 t˜
(11.68)
where p(τ ) = e−τ
vch 4 Okt 2005 10:43
Nc0 / (1 − e−2τ ) .
(11.69)
11.2 Nucleation (tlag < t < tN )
245
Hence, we find the relaxation time (induction time) as trel ≤ t0 ln Nc0 .
(11.70)
The total relaxation time needed to establish a quasi-stationary state in the range N ≤ eNc0 is determined from Eqs. (11.51) and (11.70) as tf ull = t0
Nc0 Nc0 ln √ + t0 ln Nc0 ≈ t0 ln Nc0 ≈ tlag . 2
(11.71)
Expanding the integrand in Eq. (11.53) near the lower bound at t > trel , we find for the distribution function the result I(Nc ) ψ(N, t) = D(Nc )
N 1 1 (N − N )2 dN exp − ln (N − N ) − 2 Nc 4N
N max
N
I(Nc ) 2 = πNc √ D(Nc ) π nL (0) = 2
'
A(N )
2
e−ξ dξ
C(N )
Nc [1 − erf(C (N ))] exp N
(11.72)
N 2 N Nc ln − 4 Nc 2
,
where C (N ) =
N 1√ N ln , 2 Nc
A (N ) =
Nmax − N √ + C (N ) 1 . 2 N
(11.73)
The definitions D(Nc ) (Eq. (11.72)) and I(Nc ) (Eq. (11.58)) are taken into account in this equation. Thus, within the size range 2 ≤ N ≤ eNc0 , the distribution function of bubbles depends on time only via the critical size ψ(N, t) = ψ(N, Nc (t)). Note that, with an accuracy of small terms in the first derivative (of the order of 1/Nc 1), the derived distribution function is joined at N = Nc with the function represented by Eq. (11.59). In view of quasi-stationary state, ∂ψ/∂τ = ∂I/∂N = 0 holds and I(N, τ )|N τlag = I(eNc (τ )) = I(Nc (τ )) ,
(11.74)
i.e., the flux is almost steady at t > tlag (Eq.(11.71)) at the 2 ≤ N ≤ eNc0 range.
vch 4 Okt 2005 10:43
246
11 Gas-Filled Bubbles in Low-Viscosity Liquids
11.2.5 Distribution Function of Bubbles within the Range ˜ N >N ˜ = eNc . We omit the Let us now consider the case of large bubbles when N > N −x term 2e in Eq. (11.62) (as will be shown below, it is proportional to the number of gas atoms in the critical bubble at the time of intensive nucleation τN ) and the second derivative because of the weak dependence I(N ) and the smallness of the coefficient in front of this dependence within the studied range x ≥ xc + 1 (N ≥ eNc ). As a result, we obtain the equations ∂I ∂I = −(x − xc ) , ∂τ ∂x
I(x)|xc +1 = I(Nc (τ )) ,
I(N, τ ) = I(Nc (τch )) ,
(11.75)
where τch = τ − ln(x − xc ) = τ − ln ln (N/Nc (τ ))
(11.76)
is the characteristic of the differential equation Eq. (11.75). Thus, we have N = Nc (τ ) exp(exp(τ − τch )) .
(11.77)
At τch = 0, we obtain that the maximum number of atoms in the bubble is equal to Nmax (τ ) = Nc exp (exp (τ )) = Nc0 exp(exp τ ) .
(11.78)
Here, we neglected the small variation in Nc (τ ) in the process of bubble nucleation (the validity of such approximation is shown below, see Eq. (11.104)). This result implies that all bubbles emerge at different times, however, with the identical initial amount of gas, Nc (τ ) = Nc0 . ˜ , the distribution function also changes slowly; hence, the In the range N > N diffusion term in Eq. (11.20) is small. Ignoring the second-order derivative of ψ(N, t) with respect to N and taking into account Eqs. (11.58) and (11.33), we obtain that the bubble size distribution function within the eNc < N ≤ Nmax range has the following form ' Nc 1 dN −1 Nc L exp − × = n (0) ψ(N, t) = I(Nc ) dt π 2 N ln(N/Nc ) N − Nmax (t) 1 1 − erf . (11.79) × 2 ∆Nc
vch 4 Okt 2005 10:43
11.2 Nucleation (tlag < t < tN )
247
Here we have introduced (similar to Eq.(11.59)) the error function for the smooth correction of the distribution function at N > Nmax (t) (this procedure is proven in Ref. 24). To determine the time of intensive nucleation, τN , one needs to employ the conservation law for the amount of gas atoms Eq.(11.23), which can be conveniently written as nL (0) − nL (τ ) =
N$ max
N ψ(N, τ )dN,
0
ψ(N, τ )|N =Nmax (τ ) = 0 .
(11.80)
From Eq. (11.80), we derive the following expression for the variation of the relative amount of gas in the bubbles with time, Z = (nL (0) − nL (τ ))/nL (0) , 2t0 dZ = L dτ n (0)
∞
2t0 ∂ψ N dN = L ∂τ n (0)
0
(11.81) ∞
∂ψ N dN . ∂τ
(11.82)
eNc (τ )
It is accounted for in Eq. (11.82) that, at t > trel , the relations ∂I ∂ψ =0, =0 ∂t N ≤eNc ∂N N ≤eNc
(11.83)
hold. As is seen from Eq. (11.58), the determining role in the variation of the flux with time is played by its exponential dependence on ∆F (Nc (τ ))/kB T = Nc (τ )/2. Even a small change of this quantity strongly affects the length of the time interval where the particles of the new phase are intensively formed. Indeed Nc 1 pVc 1 4π (2σ)3 ∆F (Nc ) = = ≈ kB T 2 2 kB T 2 3 p2 −2 nL (0) − nL (τ ) Nc0 Nc0 1− (1 + 2Z(τ )) . ≈ = 2 nL (0) 2
(11.84)
When deriving Eq. (11.84), we accounted for the relation 4π 3 4π R = Vc = 3 c 3
2σ p
3 (11.85)
vch 4 Okt 2005 10:43
248
11 Gas-Filled Bubbles in Low-Viscosity Liquids
and the Henry law Eq. (11.10). From Eqs. (11.58) and (11.84), we obtain that the flux sharply decreases at 1 nL (0) − nL (τ ) = Z(τN ) = 1, nL (0) Nc0
(11.86)
where τN is the time period of intensive nucleation. At the same time, we obtain that, at Nc0 1, the relative change in the amount of gas in the critical nucleus is rather small and almost everywhere [except for the multiplier with exponential dependence on Nc (τ )] one can assume that Nc (τ ) ∼ = Nc0 ≡ Nc (τ )|τ =0 1 .
(11.87)
Thus, Eq. (11.86) is responsible for the time of intensive nucleation τN . For the dependence of the flux on time at eNc (τ ) (Eq. (11.74)), we have the same dependence as in the point Nc0 I(Nc (τ )) = I(Nc0 ) exp (−Nc0 Z(τ )) ,
Z(0) = 0 .
(11.88)
With the account of the characteristic Eq. (11.76), the general solution as a function of Z can be written as I(Nc (τ ))|τ >τlag = I(Nc0 ) exp (−Nc0 Z(τ0 )) .
(11.89)
Expressing the term ∂ψ/∂τ via the continuity equation Eq. (11.18) and substituting Eq. (11.89), we arrive at the equation for the definition of Z(τ ) ⎛ 2t0 ⎜ dZ = L ⎝eNc I(Nc ) + dτ n (0)
Nmax (τ )
⎞
⎟ I(Nc0 )e−Nc0 Z(τ0 ) dN ⎠ .
(11.90)
eNc (τ )
Here, we took into account that the flux is equal to zero in the N ≥ Nmax range.
vch 4 Okt 2005 10:43
11.2 Nucleation (tlag < t < tN )
249
Let us substitute Eq. (11.89) into (11.90) and go over to the integration variable τ0 = τ0 (N, τ ), integrate by part, and take into account that, according to Eq. (11.76), τ0 (τ, eNc ) = τ and τ0 (0, Nmax ) = 0. Then we have ⎛
⎞
0
dN 2t0 ⎝ dZ dτ0 ⎠ = L eNc I(Nc ) + I(Nc (τ )) (11.91) dτ n (0) dτ0 τ ⎛ ⎞ τ dZ 2t0 ⎝ I(Nc0 )Nmax (τ ) + I(Nc0 )e−Nc0 Z(τ0 ) N (τ − τ0 , Nc )dτ0 ⎠ . = L n (0) dτ0 0
This equation can be solved by the method of successive approximations. Since the integral term is of second-order of smallness with respect to I(Nc0 ) 1, in the first approximation, we have 2t0 dZ = L I(Nc0 )Nmax (τ ) = A exp(exp τ ) , dτ n (0)
Z(0) = 0 ,
(11.92)
where (Nc0 )3/2 Nc0 2t0 I(Nc0 )Nc0 = √ exp − . A= nL (0) π 2
(11.93)
From Eq. (11.92), we obtain τ Z(τ ) = A
exp(exp τ )dτ ≈ Ae−τ [exp(exp τ ) − e]
(11.94)
0
≈ Ae−τ [exp(exp τ )] . In view of the very fast increase, the integrand gives the main contribution to the integral at the upper limit. The time period of intensive nucleation can be determined, using Eq. (11.86) and substituting τ = τN into Eq. (11.95), as Nc0 Ae−τN exp (exp τN ) = 1 .
(11.95)
vch 4 Okt 2005 10:43
250
11 Gas-Filled Bubbles in Low-Viscosity Liquids
Taking the logarithm of Eq. (11.95), we find the equation , √ π 1 Nc0 τN , = ln exp e − τN = ln 5/2 ANc0 2 N
(11.96)
c0
which is solved by the method of successive approximations (see Appendix 1). At ln (1/ANc0 ) 1 1 1 1 τN + ln ln + ln ln + ln(. . .) (11.97) e ≈ ln ANc0 ANc0 ANc0
−1 or, with an accuracy of the terms of the order of ln ln AN1 c0 ln AN1 c0 eτN ≈ ln
1 1 + ln ln . ANc0 ANc0
(11.98)
Eq. (11.95) is satisfied with the same accuracy. Retaining principal terms in Eq. (11.95), we obtain 5/2 2 N (N Nc0 ) c0 c0 1− , τN = ln >1, (11.99) ln √ eτN = 2 Nc0 π 2 tN = 2t0 ln
4lRc Nc0 Nc0 = ln . 2 3αDδ 2
(11.100)
Combining Eqs. (11.95) and (11.95), we find that, at τN > 1, we have Z (τ ) = Nc0 Z (τ ) = eτN −τ exp [− (eτN − eτ )] . Z (τN )
(11.101)
Differentiating with respect to τ , we arrive at Nc0
dZ (τ ) = −eτN −τ exp [− (eτN − eτ )] + eτ eτN −τ exp [− (eτN − eτ )] . dτ
(11.102)
Taking into account Eqs. (11.84), (11.86) and (11.102 ), Eq. (11.76) yields 1 1 dNc 1 dZ ∂τ0 =1+ =1+ 2 (11.103) ∂τ ln (N/Nc0 (τ )) Nc dτ ln (N/Nc (τ )) dτ
! " 2 1 (eτ − 1) eτN −τ exp −eτN 1 − eτ −τN ≈ 1 . =1+ ln (N/Nc (τ )) Nc0
vch 4 Okt 2005 10:43
11.2 Nucleation (tlag < t < tN )
251
It follows from Eq. (11.103) that in the range where the inequality exp (e−τN ) 1 holds true, i.e. almost within the entire range of values 0 ≤ τ ≤ τN − e−τN ≈ τN ,
(11.104)
we have dτ0 (N, τ ) /dτ = 1, and only in a very narrow range (with a width of ≈ (ln τN ) e−τN ) in the vicinity of τN , this value increases up to 2. Hence, at the stage of intensive nucleation, the variations in Nc (τ ) with time can be ignored, as it was the case in Eq. (11.78). The number of bubbles in unit volume, Nb , is determined by t Nb =
I(Nc )dt ∼ = I(Nc0 )t = 2t0 I(Nc0 )τ .
(11.105)
0
At τ = τN , the number of bubbles formed at the nucleation stage reaches its maximum value Nbmax = 2t0 I(Nc0 )τN ,
(11.106)
where τN is determined by Eq. (11.99). The same expression can be derived, using the definition of Nmax (Eq. (11.78)) N max
N max
N
eNc
ψ(N, t)dN ∼ = 2t0 I(Nc )
Nb =
dN N ln(N/Nc ) (11.107)
Nmax (t) ∼ = 2t0 I(Nc0 )τ . = 2t0 I(Nc ) ln ln Nc It can be easily seen that this equation yields the expression for the maximum number of bubbles that is analogous to Eq. (11.106). The passage to the size distribution function ϕ(R, t) is made in accordance with ϕ(R, t) = ψ(N, t)dN/dR, where the dependence of N on R is determined by Eq. (11.16).
vch 4 Okt 2005 10:43
252
11 Gas-Filled Bubbles in Low-Viscosity Liquids
11.3
The Intermediate Stage (tN < t < tf )
After the end of nucleation the intermediate stage starts, when the amount of excess gas in the solution is sufficiently large for the growth of the formed bubbles but too small for intensive nucleation of new bubbles (since the nucleation rate depends on the excess amount of the gas to a very high extent). Therefore, the number of bubbles per unit volume remains virtually constant during this stage. The amount of excess gas at the intermediate stage decreases, tending to the equilibrium value, so Eq. (11.36) can be used to determine the amount of gas in a bubble. At N ≥ eNc , for the main part of the spectra in the N -space one can neglect the diffusion term in the definition of the flux. Then, we obtain a complete set of equations, which includes the continuity equation for the distribution function ψ(N, t) with the initial condition at τ = τN (which is defined by the distribution function formed at the end of the nucleation stage, Eqs. (11.59) and (11.79)) and the law describing the change in the number of gas atoms ∂ψ ∂IN + =0, ∂τ ∂N
IN =
dN ψ, dτ
ψ(N, τ )|τ =τN = ψ0 (N, τN ) .
(11.108)
The distribution function formed after the completion of the intensive nucleation stage at N ≥ eNc0 determines the growing bubbles at τ > τN , because bubbles with N ≤ eNc0 rapidly disappear and make a minor contribution to the law of conservation, owing to a small amount of gas. That is the reason why the distribution function ψ0 (N, τN ) (which is determined by Eq. (11.79)) can be used with sufficient accuracy. Let us write the law of conservation with respect to the number of gas atoms in the form of Eq. (11.23) as
L # )] = nL n0 − nL (τ ) [1 − ϕ(τ 0 Z(τ ) =
N max
N ψ(N, τ )dN
(11.109)
0 N max
≈
N ψ(N, τ )dN = Q , eNc0
where ϕ(τ # )=
$∞
V (N )ψ(N, τ )dN is the relative volume of bubbles and Q is the
0
total number of gas atoms in bubbles that are contained in a unit volume of the
vch 4 Okt 2005 10:43
11.3 The Intermediate Stage (tN < t < tf )
253
liquid. If ϕ # is not very small, ϕ # ≤ 1, one should consider the relative increase in the volume of the solution. With Eq. (11.10), we obtain N=
nL (τ ) pV V = nV V = V . kB T δ
(11.110)
With Eq. (11.110), the law of conservation Eq. (11.110) takes the form
L nL (τ ) n0 − nL (τ ) [1 − ϕ(τ ϕ(τ # ), # )] = δ
(11.111)
which yields L
n (τ ) =
nL 0
# ) 1 ϕ(τ 1+ δ 1 − ϕ(τ # )
−1 .
(11.112)
Thus, we obtain the relative swelling of the liquid V (τ ) /V (0) as 1 δnL δ V (τ ) = =1+ L 0 −δ ≈1+ . V (0) 1 − ϕ(τ ˜ ) n (τ ) 1 − Z(τ )
(11.113)
When deriving this relationship, we take into account Eq. (11.26) and the fact that the solubility coefficient δ is usually much smaller than unity. Further, we will L ˜ ) ≈ 1 it is necessary to assume ϕ(τ # ) 1 and, accordingly, δnL 0 /n (τ ) 1 (if ϕ(τ take into account the increase in the total volume of a solution [24]). The continuity equation in the space of N Eq. (11.108) in the time interval tf of the existence of the intermediate stage has the following form with allowance for Eq. (11.36) ∂ ∂ψ + (1 − Z(τ )) ∂τ ∂N
N ψ ln Nc
=0,
(11.114)
and the initial condition is defined by Eq.(11.79) at τ = τN : 2t0 I(Nc0 ) 1 ψ(N, τN ) = N ln(N/Nc0 ) 2
N − Nmax (τN ) 1 − erf ∆Nc
θ(N − eNc0 ) .
(11.115)
Here we have introduced the θ-function to take into account the fact that bubbles with N ≤ eNc0 rapidly dissolve, making a minor contribution to the balance of
vch 4 Okt 2005 10:43
254
11 Gas-Filled Bubbles in Low-Viscosity Liquids
gas atoms. The time is calculated referred to the moment τN when the stage of nucleation is completed. As it is known, the solution of Eq. (11.114) can be expressed via the characteristic of this equation and the distribution function at the initial time moment as N0 − Nmax (τN ) 2t0 I(Nc0 ) 1 1 − erf × ψ(N, τ ) = N0 ln(N0 /Nc0 ) 2 ∆Nc (11.116) ∂N0 . ×θ(N0 − eNc0 ) ∂N Here N0 = N0 (N, τ ) is the characteristic of Eq. (11.114), which is determined by Eq. (11.36) with the initial condition N |τ =0 = N0 , where N0 is arbitrary point of the initial distribution function within the range of values from eNc0 to Nmax (τN ). Let us introduce the variables 2 N0 Nc (τ ) Ze N , x0 = ln , xc = ln = ln (11.117) x = ln Nc0 Nc0 Nc0 Ze − Z(τ ) L with allowance for the fact that Ze ≈ 1 at nL e /n (0) 1 and rewrite Eq. (11.36) as
dx = (1 − Z(τ )) (x − xc ) , dτ
x|τ =0 = x0 = ln
N0 . Nc0
(11.118)
The general solution of Eq. (11.118) has the form x = β(τ ) (x0 − f (τ )) , τ ϕ(τ ) =
!
β(τ ) = exp ϕ(τ ) ,
(11.119)
" 1 − Z(τ ) dτ ,
(11.120)
0
τ f (τ ) =
!
" 1 − Z(τ )
1 xc (τ )dτ β(τ )
0
(11.121) τ = 0
vch 4 Okt 2005 10:43
e−ϕ(τ
)
!
" 1 − Z(τ ) ln
Ze Ze − Z(τ )
2
dτ .
11.3 The Intermediate Stage (tN < t < tf )
255
From these formulas, we derive the relationships N0 N = β(τ ) ln − f (τ ) , ln Nc0 Nc0
ln
N N0 1 ln = + f (τ ) , Nc0 β(τ ) Nc0
(11.122)
Nmax (τN ) − f (τ ) , Nmax (τ ) = Nc0 exp β(τ ) ln Nc0
(11.123)
Nc0 1 1 N0 ∂N0 N0 = exp ln . + f (τ ) = ∂N N β(τ ) β(τ ) Nc0 β(τ )N
(11.124)
Using Eq. (11.122), let us write Eq. (11.117) in the form 2t0 I(Nc0 ) 1 N0 − Nmax (τN ) ψ(N, τ ) = 1 − erf β(τ )N ln(N0 /Nc0 ) 2 ∆Nc (11.125) ×θ(N0 − Nmin (τN )) , where Nmin (τ ) is determined by Eq. (11.122) at N0 = eNc0 and represents the lower bound for the bubbles that give the main contribution to the conservation law for the number of gas atoms (smallest bubbles have been dissolved). The number of bubbles per atom of the liquid Nb remains virtually invariable during the time of the intermediate stage tf : indeed, Nmax (τ )
Nb =
2t0 I(Nc0 ) ∂N0 dN = N0 ln(N0 /Nc0 ) ∂N
Nmin (τ )
≈ 2t0 I(Nc0 ) ln ln
Nmax (τN )
2t0 I(Nc0 ) dN0 N0 ln(N0 /Nc0 )
eNc0
Nmax (τN ) = 2t0 I(Nc0 )τN , eNc0
(11.126)
which coincides with the maximum number of bubbles (Eq. (11.106)) formed at the stage of nucleation. Note that, when the initial lower bound of the spectrum eNc0 reaches the lowest value Nmin (τ ) ≈ 1 during the dissolution of small bubbles, the size spectrum will be formed from the initial distribution of bubbles with sizes from eNc0 to Nmax (τN ). The moment τ ∗ of reaching N ≈ Nmin (τ ) is determined by Eq. (11.122).
vch 4 Okt 2005 10:43
256
11 Gas-Filled Bubbles in Low-Viscosity Liquids
Using the law of conservation for the number of gas atoms, we obtain an equation for the relative number of gas atoms Z (τ ) from Eq. (11.26) 1 nL (τ ) ≡ Z (τ ) = L 1− L n0 n0 2t0 I(Nc0 ) = L n0 β (τ ) =
Nmax (τ )
N ψ(N, τ )dN
(11.127)
Nmin (τ )
Nmax (τ )
Nmin (τ )
ln
dN N0 (N,τ ) Nc0
≈
2t0 I(Nc0 )Nmax (τ )
Nmax (τ ) nL β(τ ) ln 0 Nc0
2t0 I(Nc0 )Nc0 1 N (τ )
max . L N (τ ) Nc0 n0 β(τ ) ln max Nc0
When writing Eq. (11.128), we used Eq. (11.123) and replaced the slowly changing logarithm ln (N0 (N, τ ) /Nc0 ) by its value ln (N0 (Nmax (τ ), τ )/Nc0 ) (here we take into account that N0 (Nmax (τ ), τ ) is the maximum value at the upper limit of the size spectrum at the nucleation stage, i.e. at the point mainly responsible for the value of the integral) and also considered that Nmin (τ ) Nmax (τ ). Then Eq. (11.78) yields N0 (Nmax (τ ), τ )/Nc0 = exp(exp(τN ))
(11.128)
and Eq. (11.128) takes the form Z(τ ) =
1 1 exp [β (eτN − f (τ ))] . exp(−eτN ) Nc0 β(τ )
(11.129)
During the lifetime of the intermediate stage 0 ≤ τ ≤ τf (remember that the time is taken with reference to the completion of the nucleation stage τN ), the relative amount of gas in the solution changes within the limits Z(τ )|τ =0 =
1 n∗ n∗ ≤ Z(τ ) ≤ Z(τf ) = 1 − L ≤ 1 − . Nc0 n0 − nL nL e 0
(11.130)
The excess amount n∗ of gas in the solution at the end of the intermediate stage is determined with an accuracy at which the excess of the substance is regarded as negligible (usually n∗ /nL 0 ≤ 1/e). The relative decrease in the amount of excess gas in the solution with respect to its initial amount changes within the range L ∗ L ∗ nL e /n0 n /n0 ≤ 1. When τ → ∞, n tends to the equilibrium amount of
vch 4 Okt 2005 10:43
11.3 The Intermediate Stage (tN < t < tf )
257
gas per unit volume of the liquid nL e . The condition Eq. (11.130) determines the lifetime of the intermediate regime τf . Using Eqs. (11.117) and (11.120), let us estimate f (τ ) from Eq. (11.121) in the range 0 ≤ τ ≤ τf τ f (τ ) =
!
" 1 − Z(τ ) e−ϕ(τ ) xc (τ )dτ =
!
⎧ ⎡ τ ⎤⎫ ⎨ ⎬ " " ! 1 exp ⎣− 1 − Z(τ ) 1 − Z(τ ) dτ ⎦ ln dτ . ⎩ ⎭ (1 − Z(τ ))2
0
τ = 0
(11.131)
0
Considering that e−ϕ(τ ) < 1, we arrive at τ f (τ )
tf )
¯ = n|r→∞ = n
δ
L
2σ p + Rc L
kB T
265
.
(11.175)
On this basis, we obtain −j|r→R = 4πR2
" D! L n ¯ −n ˜L . R
(11.176)
From the condition of continuity of fluxes Eqs. (11.169) and (11.174) in the point r = R, we determine the gas density near the bubble boundary n ˜ L as n ˜L =
n ¯L −
αR L n 2l e
1+
αR 2l
−1 .
For the flux of gas atoms onto the bubble surface, we get " ! L αR/2l 2D n ¯ − nL −j|r→R = 4πR R . R 1 + αR/2l
(11.177)
(11.178)
Apparently, Eq. (11.178) describes both limiting situations: at αR/2l 1, we arrive at the ”boundary” growth kinetics (the free-molecular regime Eq. (11.10)); at αR/2l 1, we have the growth kinetics determined by the diffusion supply of gas atoms to the bubble surface. If α is not very small and R l, the case of αR/2l 1 is implemented at the late stage (the intermediate growth regime was taken into account in Eq. (11.13)). Then, substituting n ¯ L and nL Rc from Eqs. (11.170) and (11.171) into Eq. (11.178) and using Eq. (11.167), we obtain 2σ 1 dR = Dδ L dt p R
1 1 − Rc R
.
(11.179)
Note that the set of equations (11.179), (11.163), and (11.166) fully coincides with the complete set of equations derived in Refs. 25-27 but one parameter α = 2σωc∞ /kB T (c∞ is the equilibrium concentration of the admixture near the boundary and ω is the volume per admixture atom in the solution) with the dimension of length is substituted by the parameter ξ ≡ 2σδ/pL having the same dimension. Thus, both the method and the results of Refs. 25-27 are applicable to the solution of this set of equations. As it is shown in mentioned papers, this set of equations at n ¯ L (t) − nL e → 0 starts to ”forget” about its initial conditions in time and acquires an increasingly universal nature, asymptotically tending to a self-similar form, which is independent
vch 4 Okt 2005 10:43
266
11 Gas-Filled Bubbles in Low-Viscosity Liquids
of the initial conditions. The only parameter dependent on the initial distribution function is the time moment at which the bubble distribution becomes fairly well describable by the universal distribution function. Naturally, the actual pattern of the time-asymptotic distribution function depends on the mass transfer mechanism. The process of the transformation of the distribution function from an arbitrary to the asymptotically universal one is thoroughly considered in Refs.25 and 27 for the case where the initial distribution function has an infinitely long tail at R → ∞. In our case, after the intermediate stage, there is a finite tail at R > Rc different from the fluctuational one, and the process of the transformation to the asymptotically universal function occurs sufficiently rapidly [25]. Let us introduce, just as in Ref. 27, the reduced variable u = R/Rc and rewrite Eq. (11.179) in the form 2σ δ u − 1 u dRc du =D L 3 2 − , dt p Rc u Rc dt
(11.180)
where Rc is determined by Eq. (11.172). Since dRc/dt > 0, it is convenient to 3 instead of time, t. The we introduce the new time variable τ = ln Rc3 (t)/Rc0 can rewrite Eq. (11.180) in the canonical form [27] as du3 = γ (u − 1) − u3 , dτ
(11.181)
"−1 ! is a dimensionless parameter. With ϕ (u, τ ) du = where γ = 3Dξ dRc3 /dt ϕ (R, τ ) dR, the complete set of equations (11.163) and (11.166) gets the form ∂ ∂u ∂ϕ (u, τ ) + ϕ (u, τ ) = 0 , ∂τ ∂u ∂τ ϕ (u, τ )|τ =0 = ϕ0 (u) = ϕ0
3 1 4π pL (nL e )Rc0 τ e Z(τ ) = L kB T n0 3
vch 4 Okt 2005 10:43
R Rc0
(11.182)
,
(11.183)
∞ ϕ (u, τ ) u3 du . 0
(11.184)
11.4 The Late Stage (t > tf )
267
At the late stage, nL (τ )/nL 0 1; therefore, let us write Eq. (11.184) in a simpler form ∞ τ
ϕ (u, τ ) u3 du ,
1 = κe
κ=
0
3 1 4π pL (nL e )Rc0 . kB T nL 0 3
(11.185)
As is shown in Refs. (25)-(27), the resulting set of equations Eqs. (11.181)-(11.185) has a stable solution at τ → ∞ if γ|τ →∞ → γ0 = const. This condition is fulfilled if du/dτ < 0 for all sizes except for u = u0 , for which, in the zeroth approximation, ∂ du du =0, =0. (11.186) dτ u=u0 , γ=γ0 ∂u dτ u=u0 , γ=γ0 On this basis, we find u0 =
3 , 2
γ = γ0 =
27 . 4
(11.187)
Naturally, these values are determined by the form of Eq. (11.180), i.e., by the mechanism of mass transfer or even by several simultaneously acting mechanisms [25]. At the same time, ϕ (u, τ ) in our approximation τ → ∞ tends to the universal function, virtually forgetting about the initial distribution and becoming nullified at u ≥ u0 together with all its derivatives with respect to u. This is what determines the only stable (with respect to fluctuations) solution [25]. Thus, in this approximation and at with γ = 27/4, Eq. (11.181) yields 1 3 du = −g(u) = − 2 2 u − (u + 3) , (11.188) dτ 3u 2 and the solution of Eq. (11.183) has the form ⎧ 1 ⎪ ⎨ χ(τ + φ) g(u) , ϕ (u, τ ) =
⎪ ⎩
u < u0 = 23 , (11.189)
0,
u ≥ u0 = 23 ,
where u φ (u) = 0
4 5 du = ln (u + 3) + ln g (u) 3 3
3 1 33 e −u + − ln 5/3 . 2 1 − 2u/3 2
(11.190)
vch 4 Okt 2005 10:43
268
11 Gas-Filled Bubbles in Low-Viscosity Liquids
tlag
tlag, tN, tf, tlate
10 tN tlate
1
tf
0.1
20
30
40
50
Nc0
Fig. 11.1 Dependencies of τlag , τN , τf and τlate on the initial critical bubble size
Substituting Eq. (11.189) into Eq. (11.200), we find that χ (τ + φ) = A exp [− (τ + φ)]
(11.191)
and, using Eqs. (11.189), obtain ⎧ −(τ +φ(u)) 1 , ⎪ ⎨ Ae g(u) ϕ (u, τ ) =
⎪ ⎩
u < 23 , (11.192)
0,
u ≥ 32 ,
where ⎞−1 3/2 3 u ⎟ ⎜ du⎠ . e−ϕ(u) A = ⎝κ g(u) ⎛
0
vch 4 Okt 2005 10:43
(11.193)
11.4 The Late Stage (t > tf )
269
The condition γ|τ →∞ → γ0 determines Rc3 (t) from Eq. (11.181). We get 2σ 4 3 . Rc3 (t) = δD L t + Rc0 9 p
(11.194)
With this relation, Eq. (11.172) yields 2σ n ¯ L (t) − nL e =δ L nL p e
2σ 4 3 δD L t + Rc0 9 p
−1/3 .
(11.195)
The time t is referred here to the end tf of the intermediate stage. 18
18
Distribution function, j (R,t)
10
a
10
b
17
10
17
10
c
6
12
10
16
10
7
15
16
10
10
11 5.10
14
8
10 15
10
9
13
1
2 3
14
10 0
1
3 5 6 2 4 Reduced radius, R/Rc0
10
12
10
3 4
5
0 20 40 60 80 100 120 140 Reduced radius, R/Rc0
0
0
100 200 300 400 Reduced radius, R/Rc0
500
Fig. 11.2 Distribution functions of bubbles for nL (0) = 3 · 1019 , Nc0 = 17 at different times: a) nucleation, (1) t = 0.2tN , (2) t = 0.75tN , (3) t = tN ; b) intermediate stage, (4) t = tN + 0.5tf , (5) t = tN + tf ; c) late stage, (6) t = tlate , (7) t = 5t0 , (8) t = 6t0 , (9) t = 8t0
The number of bubbles per unit volume is diminished with time according to the law
Nb (t) = Ae−τ
3/2 3/2 Rc0 3 du −φ −τ −φ −τ = Ae e e dφ = Ae = A . g(u) Rc (t) 0
(11.196)
0
The bubble size distribution function for the variable u has the form ϕ (u, τ ) = Nb (t)P (u) ,
(11.197)
vch 4 Okt 2005 10:43
270
11 Gas-Filled Bubbles in Low-Viscosity Liquids
where ⎧ 1 ⎪ 2 ⎪ u exp − ⎪ ⎪ ⎪ 1 − 2u/3 ⎨ 34 e , 5/3 7/3 2 P (u) = (u + 3) (3/2 − u)11/3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩0 ,
u < 23 ,
(11.198)
u ≥ 32 .
Using Eq. (11.181) at γ = 27/4, we obtain 3/2 3/2 du e−φ (u − 1) e−φ [u (φ) − 1] dφ = g (u) 0
(11.199)
0
3/2 u3 −
=
du3 dφ
e−φ
2γ dφ = 0 . u
0
Hence, u = 1, Rc (t) = R (t); i.e., the average bubble size at τ → ∞ approaches Rc (t). To!achieve a"higher accuracy, one should use a dependence of the form γ(τ ) = γ0 1 − ε2 (τ ) , which was used in Refs. 26-28. As it is shown in those papers, ε2 (τ ) > 0 and τ → ∞ at ε2 (τ ) → 0; also, the behavior of ε2 (τ ) depends on the form of the tail of ϕ0 (u) at u u0 . The distribution function ϕ(u, τ ) at u > u0 is determined by the initial distribution function. At du/dτ < 0, bubbles with u > u0 move in size space in such a way that, passing over the region of the blocking point ε2 (τ ), they consume the whole gas from the dissolving bubbles, increasing their own dimensions and reducing the total number of bubbles. At the same time, naturally, the total amount of gas (dissolved and evolved into the bubbles) in the system remains constant. A nonzero ε2 (τ ) is necessary for bubbles with u > u0 to pass into the u ≤ u0 region without sticking in the u0 point, which would lead to a violation of the conservation law Eq. (11.184) [11.24]. Note that, if we consider fluctuations in the nucleation of bubbles, when bubbles formed in the direct vicinity of each other merge, then ε2 (τ ) → ε2 (∞) =const. These collisions determine the ε2 (∞) value and the distribution of bubbles beyond the blocking point [25] (this conclusion has been experimentally confirmed for solid solutions [29]). The ε2 (τ ) → ε2 (∞) value, which is determined by the decreasing tail of the initial distribution function near the blocking point and
vch 4 Okt 2005 10:43
11.4 The Late Stage (t > tf )
150
100
Rmax
Intermediate stage
Nucleation
Rmax/Rc0, Rc/Rc0
200
Rc
50
Late stage
0 1.5
Nb
271
tN
tN+tf
tN+tlate
1.0 0.5 0
0
1
2
3
4
5
6
7
8
Time, (t - tlag )/t0
Fig. 11.3 Time evolution of Rmax (t), Rc (t) (dashed curve (Eqs.(11.144), (11.194)), solid curve (Eq. (11.201)) and Nb (t) for Nc0 = 17
beyond it, decreases with time; therefore, the zero approximation asymptotically becomes exact with time. Since it is very difficult to determine the time of beginning of the late stage, tlate , and, accordingly, the critical size of bubbles in the system after the intermediate stage, one should require good joining of Rc and Rmax corresponding to the end of the intermediate and beginning of late stages. Taking into account that on the late stage Rmax = (3/2)Rc (see Eq. (11.187)), we obtain the equation for the definition of tlate in the form 3 Rc (tlate ) , Rmax (tlate ) = 2
(11.200)
vch 4 Okt 2005 10:43
272
11 Gas-Filled Bubbles in Low-Viscosity Liquids
18
18
Distribution function, j (R,t)
10
10
a
17
16
13 4.10
15
13 3.10
10
16
10
10
15
10
14
10
14
10
13 2.10
13
c
6
7
10
13
10
12
11
0
5 10 Reduced radius, R/Rc0
15
10
0
8
13
10
12
10 10
13 5.10
b
17
10
10
20 40 60 Reduced radius, R/Rc0
80
0
0
9
50 100 Reduced radius, R/Rc0
150
Fig. 11.4 Distribution functions of bubbles for nL (0) = 3 · 1019 , Nc0 = 24 at different times: a) nucleation, (1) t = 0.2tN , (2) t = 0.75tN , (3) t = tN ; b) intermediate stage, (4) t = tN + 0.5tf , (5) t = tN + tf ; c) late stage, (6) t = tlate , (7)-t = 4t0 , (8) t = 6t0 , (9) t = 8t0
where Rmax (t) and Rc (t) are defined by Eqs. (11.143) and (11.144), correspondingly. Thus, we prolong a little the intermediate stage until the time tlate , when Eq. (11.200) will be fulfilled. Let us note that such definition does not allow a smooth connection for the both functions, Rmax (t) and Rc (t), simultaneously. Therefore we shall require a smooth connection only for the maximum radius Rmax , thus the critical radius Rc will have a kink at t = tlate (see Figs. 11.3 and 11.5 below). It is consequence of the fact that the definitions of Rmax (t) and Rc (t) (Eqs. (11.143) and (11.144)) are asymptotical ones, and are not valid in the beginning of a late stage. Therefore it is necessary to use a more rigorous analysis describing the transformation of the distribution function which was formed at the end of an intermediate stage, Eq. (11.153), into the asymptotical one, Eq. (11.197), as it has been made in Ref. 25 (see p.134 ff), this topic is beyond the given work. Taking into account that the value of the critical radius, defined by Eq. (11.144), very sharply grows at t > tlate , for the definition of the critical radius it is possible to use the simple smoothing procedure Rc (t) =
1 Rcint (t)
+
1 Rclate (t)
−1 ,
where Rcint (t) is defined by Eq. (11.144), and Rclate (t) by Eq. (11.194).
vch 4 Okt 2005 10:43
(11.201)
11.5 Results of Numerical Computations
11.5
273
Results of Numerical Computations
Fig. 11.1 shows the results of calculation of the lag-time (Eq.(11.50)), τlag , the nucleation time (Eq. (11.99)), τN , the time of the intermediate stage (Eq. (11.147)), τf , and the time of the beginning of the late stage (Eq. (11.200)), tlate , in dependence on the initial critical bubble size. One can see that the time-lag and the nucleation time increase with increasing critical bubble size, and the time of the intermediate stage and time of the beginning of the late stage decrease with increasing critical bubble size.
50
Nucleation
Rmax/Rc0, Rc/Rc0
40
Intermediate stage
Rmax
30 20
Rc
Late stage
10 0
tN
tN+tf
tN+tlate
1
2
3 4 5 Time, (t - tlag )/t0
Nb
1.0
0.5
0
0
6
7
8
Fig. 11.5 Time evolution of Rmax (t), Rc (t) (dashed curve (Eqs.(11.144), (11.194)), solid curve (Eq. (11.201)) and Nb (t) for Nc0 = 24
In Figs. 11.2 and 11.4 the distribution functions of bubbles are shown for a) nucleation, b) intermediate and c) late stages for Nc0 = 17 and Nc0 = 24, correspondingly. In Figs. 11.3 and 11.5 the time evolution of critical size Rc (t) (dashed
vch 4 Okt 2005 10:43
274
11 Gas-Filled Bubbles in Low-Viscosity Liquids
curve (Eqs.(11.144), (11.194), solid curve (Eq. (11.201)), maximal size Rmax (t) (Eqs. (11.78), (11.143) and (11.187)), and number of bubbles per unit volume, Nb (t) (Eqs. (11.106) and (11.196)) are shown for Nc0 = 17 and Nc0 = 24, correspondingly.
11.6
Conclusions
For the case of bubble nucleation in a low-viscosity liquid supersaturated with the gas, we derived expressions for the nucleation time Eq. (11.186), bubble size distribution function Eqs. (11.134), (11.156), (11.167), the flux of nuclei in the space of bubble sizes Eqs. (11.133) and (11.158), and the maximum number of formed bubbles Eq. (11.196) with the assumption of a small volume fraction occupied by bubbles. The derived size distribution is the initial condition for the next (transient) stage, when still there is a sufficient amount of excess gas, but the number of bubbles does not vary practically remaining at the level reached at the end of the nucleation stage. For such stage the bubble distribution functions Eqs. (11.126) and (11.135), the flux of nuclei in the size space Eq. (11.23), and the maximum number of the formed bubbles Eq. (11.126) are obtained. For the late stage, the universal bubble size distribution function Eq. (11.197) is derived; it does not depend on the initial distribution in zero approximation and becomes asymptotically exact in the course of time. The number of bubbles per unit volume Eq. (11.196) is also obtained for this stage. Thus, we describe all stages of the process, starting from the initial stage of the nucleation of gas-filled bubbles and ending with coalescence (the late stage). Our approximation of a small volume fraction occupied by bubbles does not allow us to describe an even later stage, where the volume of bubbles exceeds that of the liquid (foam). This stage requires a separate consideration.
11.7
Appendices
11.7.1 Some Mathematical Transformations To solve the equation exp(exp τ ) = Beτ
vch 4 Okt 2005 10:43
(11.202)
11.7 Appendices 275 at B = 1/A + e ≥ e and ln ln B/ ln B < 1 (at B = e, the solution to this equation is τ = 0), let us use the method of successive approximations eτ = ln B + τ ,
eτ0 = 0 ,
eτ1 = ln B ,
eτ2 = ln B + ln ln B
(11.203)
etc., resulting in
τe
e
1 1 = ln B + ln ln B 1 + ln ln B 1 + [ln [ln B [1 + . . .]]] ln B ln B 1 1 ln ln B 1 + [ln [ln B [1 + . . .]]] . = ln B + ln ln B + ln 1 + ln B ln B
Expanding this expression for ln ln B/ ln B < 1, we arrive at ln lnB ln ln B ln ln B + + + ... 2 ln B (ln B) (ln B)3 ∞ ln ln B 1 . = ln B + ln ln B n = ln B + (ln B) 1 − ln1B
eτe = ln B + ln ln B +
(11.204)
n=0
Substituting the resulting solution into Eq. (11.202), we have τe
exp e
= exp ln B + ln ln B + ln ln B
∞ n=1
, (11.205)
, 1 ≈ B ln B 1 + ln ln B (ln B)n n=1 , , ∞ ∞ 1 1 . = B ln B + ln ln B n−1 = B ln B + ln ln B (ln B)n n=1 (ln B) n=0 ∞
1 = B ln B exp ln ln B (ln B)n n=1
,
1 (ln B)n
∞
11.7.2 Estimation of the Conditions when Merging of Colliding Bubbles can be Neglected Gas-filled bubbles in a liquid are affected by the buoyancy force; as a result, they float up at a rate depending on their radius. Therefore, collision and merging of bubbles are possible. Let us estimate the conditions when merging can be neglected.
vch 4 Okt 2005 10:43
276
11 Gas-Filled Bubbles in Low-Viscosity Liquids
The characteristic time of the diffusion-caused change in the content of gas in a bubble is given by tdif ≈ t0 =
2 lRc . 3δα D
(11.206)
The change in the content of gas due to merging of bubbles can be approximately described by the equation dV = 4πR2 Nb V ∆vA . (11.207) dt coll Here Nb is the density of bubbles and ∆vA is the spread in the rate of the steadystate motion of bubbles with respect to its average value, which is due to the balance between the Archimedean buoyancy force and the Stokes force of viscous friction
2 − R2 gρ R 2 ∆vA = , (11.208) 9 η where g is the gravitational acceleration, ρ is the density of the liquid, and η is its viscosity. Using Eq. (11.207), we can find the characteristic time of the change in the content of gas due to merging of bubbles tmerg =
R 1 = . V Nb ∆vA 4R2 Nb ∆vA
The effect of merging is not manifested at 1 4R2 Nb ∆vA
(11.209) ! dV " dt
coll
2 lRc 3δα D
! dV " dt
dif f
, which leads to (11.210)
or −1 2 2 3 2 2 αδηD 4Rc R Nb l gρ R − R 1. 2 9
(11.211)
Estimates show that latter inequality is fulfilled for water if R < 10−3 cm and Nb > 1011 cm−3 , i.e., for the initial and intermediate stages. At the late stage, this condition becomes less strict because of a decrease in Nb and, accordingly, an increase in the interbubble distance, as well as because of a decrease in the bubble size spread, which leads to a decrease in the rate ∆vA of the relative motion of bubbles according to Eq. (11.208).
vch 4 Okt 2005 10:43
11.8 References
277
Acknowledgments This work was partially supported by the CRDF Foundation, grant UE1-2523CK-03.
11.8
References
1. H. Reiss, J. Chem. Phys. 18 (1950) 840. 2. D. J. Stauffer, J. Aerosol Sci. 7 (1976) 319. 3. K. J. Kremer, J. Aerosol Sci. 9 (1977) 243. 4. H. Trinkaus, Phys. Rev. B 27 (1983) 7372. 5. Jin-Song Li, I. L. Maksimov, and G. Wilemski, Phys. Rev. E 61 (2000) R4710 and references cited therein. 6. Z. Kozisek and P. Demo, J. Cryst. Growth 132 (1993) 491. 7. B. E. Wyslouzil and G. Wilemski, J. Chem. Phys. 103 (1995) 1137 and references cited therein. 8. M. Sanada, I. L. Maksimov, and K. Nishioka, J. Crystal Growth 199 (1999) 67. 9. I. L. Maksimov and K. Nishioka, Phys. Lett. A 264 (1999) 51. 10. F. M. Kuni, V. M. Ognenko, L. N. Ganyuk, and L. G. Grechko, Kolloidn. Zh. 55, (1993) 22 (in Russian). 11. F. M. Kuni, V. M. Ognenko, L. N. Ganyuk, and L. G. Grechko, Kolloidn. Zh. 55 (1993) 28 (in Russian). 12. F. M. Melikhov, F. M. Trofimov, and F. M. Kuni, Kolloidn. Zh. 56 (1994) 201 (in Russian). 13. F. M. Kuni and I. A. Zhuvikina, Colloid Journal 64 (2002) 166. 14. F. M. Kuni, I. A. Zhuvikina, and A. P. Grinin, Colloid Journal 65 (2003) 201. 15. V. V. Slezov, J. Colloid Interface Sci. 255 (2002) 274. 16. V. V. Slezov, A. S. Abyzov, and Zh. V. Slezova, Colloid Journal 66 (2004) 575. 17. V. V. Slezov, A. S. Abyzov, and Zh. V. Slezova, Colloid Journal 67 (2005) 94. 18. V. V. Slezov, Fiz. Tverd. Tela (St. Petersburg) 42 (2000) 733. 19. V. V. Slezov and J. Schmelzer, Phys. Rev. E 65 (2002) 0215. 20. V. V. Slezov, J. Schmelzer, and Ya. Y. Tkach, J. Phys. Chem. Solids 57 (1996) 8340. 21. V. V. Slezov, Phys. Solid State 45 (2003) 335.
vch 4 Okt 2005 10:43
278
11 Gas-Filled Bubbles in Low-Viscosity Liquids
22. V. V. Slezov and J. Schmelzer, Fiz. Tverd. Tela (St. Petersburg) 39 (1997) 2210. 23. H. Trinkaus and H. Yoo, Philos. Mag. A 55 (1987) 269. 24. V. V. Slezov and J. Schmelzer, J. Colloid Interface Sci. 255 (2002) 274. 25. V. V. Slezov, Sov. Sci. Rev., Sect. A, vol. 17, Part 3, 1995. 26. I. M. Lifshitz and V. V. Slezov, J. Phys. Chem. Solids 19 (1961) 35. 27. I. M. Lifshitz and V. V. Slezov, Zh. Eksp. Teor. Fiz. 35 (1958) 479 (in Russian). 28. I. M. Lifshitz and V. V. Slezov, Fiz. Tverd. Tela (St. Petersburg) 1 (1959) 1401 (in Russian). 29. C. K. L. Davies, P. Nash, and R. N. Stevens, Acta Met. 28 (1980) 179.
vch 4 Okt 2005 10:43
12
Thermodynamics and Kinetics of Deliquescence of Small Soluble Particles Alexander K. Shchekin and Ilya V. Shabaev Department of Statistical Physics, Research Institute of Physics, St. Petersburg State University, St Petersburg, Ulyanovskaya 1, Petrodvoretz, 198504 Russia All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident. Arthur Schopenhauer
Abstract The thermodynamic theory of deliquescence of small soluble solid particles in an undersaturated vapor is presented. The work of droplet formation as a function of droplet size and size of the solid residue of the particle in the droplet is analyzed. Two-dimensional and one-dimensional approaches are considered. The first refers to the case when droplet size and size of the soluble solid core are considered as independent. The second assumes that the soluble solid core is in chemical equilibrium with the solution in the droplet, and its size depends on droplet size. The chemical potential of the condensing matter serving as a solvent and the chemical potential of a dissolved particle matter within the solution in the droplet are derived. The limits of chemical equilibrium of the residue of the particle with a solution in the droplet, as well as the limits of one-dimensional thermodynamic
280
12 Deliquescence of Small Soluble Particles
approach and its relation to the two-dimensional approach are considered. The quasiequilibrium distributions of droplets with partially and completely dissolved solid cores and kinetics of establishing the final aggregative equilibrium between these droplets in undersaturated vapor are described within the one-dimensional approach. The specific kinetic times of establishing these distributions are found.
12.1
Introduction
The stage of nucleation by soluble particles in the atmosphere of solvent vapor, when arising droplets consist of a liquid solution film around incompletely dissolved particles, is called the deliquescence stage. In a supersaturated vapor, this stage is the initial one in the whole condensation process, and the particles inside the droplets will completely dissolve in the growing droplets with time. In an undersaturated vapor, this stage finishes by establishing the aggregative equilibrium between droplets of different sizes, with residues of the particles (solid cores within the droplets) varying in size down to complete dissolution. Experimenters are able now to monitor the equilibrium droplet size as a function of the undersaturated vapor concentration [1-3]. By increasing the vapor concentration, they observe thickening of a liquid film around the soluble nucleus with gradual dissolution of the nucleus until it completely disappears. Subsequent decreasing the vapor concentration leads to evaporation of the droplet until the solid nucleus crystallizes inside the droplet. The experiment clearly demonstrates a hysteresis effect at increasing and decreasing vapor concentration, and locations of the corners in the hysteresis loop can provide us with important information on solubility and effective surface tension of small particles serving as nuclei of condensation and the parameters of the solution films around the particles. A thermodynamic theory of the deliquescence has been considered within two approaches: the one-dimensional approach [4-6] and the two-dimensional approach [7-11]. The one-dimensional approach assumes that the chemical equilibrium between the soluble solid core and the liquid film establishes fast at every droplet size. As a result, the solution concentration within a droplet equals the solubility of the core matter at the core size. Thus the droplet size can be considered as the only independent variable of the droplet state. The two-dimensional approach is more sophisticated and assumes that the internal chemical equilibrium in the droplet between the residue of the particle and the solution may not be achieved. Thus two independent variables are considered, for instance, the solution concentration and the number of condensate molecules in the droplet, or the sizes of the
vch 4 Okt 2005 10:43
12.2 Work of Droplet Formation on Condensation Nuclei
281
soluble core and the droplet. Which physical situation is realized in experiment depends upon the relations between the parameters of the problem. In fact, as will be shown in this paper, in many cases these relations allow us to use the one-dimensional approach. To understand some peculiarities of the deliquescence and its experimental observation, the thermodynamics of deliquescence should be supplemented with kinetics. Activation energy for transition between the metastable droplet with partially dissolved solid core and the critical droplet, which corresponds to unstable equilibrium with the vapor, determines the deliquescence barrier for transformation into metastable droplet with completely dissolved core at the same vapor chemical potential. Intensive fluctuating overcoming of the deliquescence barrier by growing droplets starts at vapor concentrations below the maximum in the curve of the undersaturated vapor concentration as a function of the droplet size. This maximum realizes for condensation nuclei, with not small solubility in the condensing solvent, in the region of undersaturated vapor concentrations. Thus the kinetics of the transitions between states with incompletely and completely dissolved particles can be observable in an undersaturated vapor [1-3]. The quasi-steady kinetic solution of the problem has been recently given within the framework of the two-dimensional approach by Djikaev [9]. Nevertheless, the one-dimensional kinetics of deliquescence may be of interest in many cases, and can provide us with rather simple theoretical and numerical estimates. Therefore we will consider the one-dimensional approach to the kinetics of deliquescence in this paper. Even though our approach will have here a phenomenological form, by contrast to the classical approach it takes into account the effects of overlapping of the surface layers from the solid and the vapor sides of a liquid film around the residue of the particle. In this way, the theory considered here will be able to give both a qualitative and quantitative thermodynamic and kinetic description of the deliquescence of small soluble particles.
12.2
The Work of Droplet Formation on Partially or Completely Dissolved Nuclei of Condensation
Let us denote by W the work of droplet formation. We suppose that, in the initial state, the system under consideration consists of a solid nucleus and vapor within a fixed volume V at absolute temperature T . In the final state, the system includes a droplet, with partially or completely dissolved condensation nucleus, and vapor. If volume V , temperature T and numbers of molecules of
vch 4 Okt 2005 10:43
282
12 Deliquescence of Small Soluble Particles
every component stay fixed in both states of the system, then the work W can be determined as a difference of the free energies Φ2 and Φ1 in the final and initial states, correspondingly. Thus we have W ≡ Φ2 − Φ1 .
(12.1)
The free energy Φ1 can be written as Φ 1 = µ β N + µ n νn + Ω 1 ,
(12.2)
where µβ is the vapor chemical potential, N is the number of vapor molecules, µn is the chemical potential of the solid particle matter in the initial state when the number of molecules within the particle equals νn and there is no droplet, Ω1 is the grand potential of the system in the initial state. Analogously, the free energy Φ2 of the system in the final state has the form ! " Φ2 = µβ (N − ν) + µν ν + µn νn + µαn νn − νn + Ω2 ,
(12.3)
where ν is the number of molecules condensed in the droplet from vapor (number of solvent molecules), µν is the chemical potential of solvent molecules in the droplet, µn and νn are the chemical potential and the number of molecules of the soluble particle matter within the residue of the particle in the droplet in the final state of the system, µαn is the chemical potential of solute in the droplet coming into solution film from the particle, Ω2 is the grand potential of the system in the final state. Though the matter of soluble condensation nucleus is typically an electrolyte and dissociates into ions under dissolution in a polar condensate, we will consider below a simpler situation of the solid particle matter with molecular dissolution in the condensing solvent without dissociation. The grand potentials Ω1 and Ω2 of the system in the initial and final states, can be represented under assumption that the particle and its residue have a spherical form as
(12.4) Ω1 = −P1γ (µn ) VRn − P β µβ (V − VRn ) + 4πRn2 σ γβ ,
! ! " " Ω2 = −P2γ µn VRn − P β µβ (V − VR ) − P α (µν , µαn ) VR − VRn + 4πRn σ αγ + 4πR2 σ αβ + ΩD , 2
vch 4 Okt 2005 10:43
(12.5)
12.2 Work of Droplet Formation on Condensation Nuclei
283
where VRn , VRn , and VR are the volumes of the initial particle, the residue of the particle and the droplet, respectively. Indices α, β, γ mark the quantities referred to the liquid film, vapor and solid particle, correspondingly, while the double indices γβ, γα, and αβ mark the quantities referred to the interfaces between the solid particle and vapor, solid particle and liquid film, the film and the vapor, respectively. ! " We assume here, that the vapor pressure P β µβ remains the same in the initial and final states of the system; P1γ (µn ) and P2γ (µn ) are the scalar pressures in the initial particle and in its residue (for simplicity, we consider a solid particle as a structureless body characterized by a scalar pressure); σ γβ , σ γα , and σ αβ are the surface tensions referred to indicated interfaces; 4πRn 2 , 4πRn 2 , and 4πR2 are the corresponding spherical surface areas with radii Rn , Rn , and R. P α (µν , µαn ) is the pressure in the bulk liquid solution with the same chemical potentials µν and µαn of the solvent and solute as they are in the solution film around the residue of the particle [12]. It does not coincide with a pressure in that film if the film is thin and the surface layers from opposite film sides overlap [12-14]. The effect of interface overlapping is described by the last term in Eq. (12.5). Let ! "0 x = νn − νn ν
(12.6)
be the relative solute concentration in the film. The partial molar volumes v α and vnα for solvent and solute, respectively, can be defined as (∂µν /∂P α ) ≡ v α ,
(∂µαn /∂P α ) ≡ vnα .
(12.7)
Assuming the solution concentration x to be not large, we will neglect the dependencies of v α and vnα on x. The number of condensate molecules in the droplet with partially dissolved nucleus may be related to the droplet volume VR and the nucleus residue volume VRn by the relation ! "0 α
(12.8) v . ν = VR − VRn − vnα νn − νn We ignore in Eq. (12.8) the contributions to ν from adsorptions on the opposite sides of the liquid film and assume that the surface tensions σ γα and σ αβ are independent of solution concentration. Because of the spherical shape of the droplet, the soluble particle and its residue, we have VR =
4π 3 R , 3
VRn =
4π 3 Rn , 3
VRn =
4π 3 R , 3 n
(12.9)
vch 4 Okt 2005 10:43
284
12 Deliquescence of Small Soluble Particles 0 νn = 4πRn 3 3vn ,
νn = 4πRn
3
1 3vn .
(12.10)
Thus Eqs. (12.6)-(12.10) determine the relations between x, ν, νn , and νn and the radii R, Rn , and Rn . Below we will consider for simplicity the case of diluted solutions. For diluted solutions, the chemical potentials µν and µαn of solvent and solute in the liquid film can be represented as functions of pressure P α and concentration x in the form [15] µν (P α , x) = µ∞ − kB T x + v α (P α − P∞ ) , µαn (P α , x) = µn∞ + kB T ln
x + vnα (P α − P∞ ) , x∞
(12.11) (12.12)
where µn∞ and x∞ are the chemical potential and the solubility of the nucleus matter at equilibrium with a flat interface between solid phase of the nucleus substance and the solution, kB is the Boltzmann constant. Analogously, the chemical potentials µn and µn of the substance in the condensation nucleus may be represented as functions of pressure in the phase γ µn = µn∞ + vn (P1γ − P∞ ) ,
µn = µn∞ + vn (P2γ − P∞ ) .
(12.13)
All Eqs. (12.11)-(12.13) assume incompressibility of phases α and γ. A substitution of Eqs. (12.11)-(12.13) into Eqs. (12.2)-(12.5) yields, taking into consideration Eqs. (12.1), (12.9), and (12.10), the following expression for the work of droplet formation on partially dissolved condensation nucleus
2 (12.14) W = 4πRn σ αγ + 4πR2 σ αβ − 4πRn 2 σ γβ − µβ − µ∞ ν
! " x + ΩD + P β − P∞ (VR − VRn ) . − kB T νx + kB T νn − νn ln x∞ The term ΩD in Eq. (12.14) can be determined with the help of the disjoining pressure of a thin liquid film. In accordance with Refs. 12 and 16, we define the disjoining pressure ΠD as " ! ΠD ≡ PN Rn , h − P α ,
(12.15)
where PN is the normal component of the pressure tensor inside the spherical liquid film with thickness h ≡ R − Rn , taken at the surface of the spherical
vch 4 Okt 2005 10:43
12.2 Work of Droplet Formation on Condensation Nuclei
285
solid residue of the nucleus. By analogy with the case of a pure solvent film on a wettable substrate [13], the term ΩD has the form
ΩD =
∞
2 4πRn
ΠD dh ,
(12.16)
R−Rn
where we have taken into account that the surface of the substrate is spherical with a radius Rn and the outer surface of the film is also spherical with a radius R. Using Eq. (12.16) in Eq. (12.14) and neglecting the difference P β − P∞ gives
2 W = 4πRn σ αγ + 4πR2 σ αβ − 4πRn 2 σ γβ − µβ − µ∞ ν (12.17) ! " x 2 + 4πRn − kB T νx + kB T νn − νn ln x∞
∞ ΠD dh .
R−Rn
In the case of a completely dissolved nucleus we !have Rn = " 0, νn = 0 and, in β view of Eq. (12.16), ΩD = 0. Neglecting the term P − P∞ VR , one can reduce Eq. (12.14) in this case to the form
W = 4πR2 σ αβ − 4πRn 2 σ γβ − µβ − µ∞ ν − kB T νx + kB T νn ln
x x∞
(12.18)
.
Another limiting situation realizes in the case of an insoluble nucleus with νn = νn , Rn = Rn and, in view of Eq. (12.6), with x = 0. As follows from Eqs. (12.17), we have in this case [13, 14]
W = 4πR2 σ αβ + 4πRn 2 σ αγ − σ γβ − µβ − µ∞ ν (12.19) ∞ + 4πRn 2
ΠD (h)dh .
R−Rn
vch 4 Okt 2005 10:43
286
12 Deliquescence of Small Soluble Particles
12.3
The Generating Properties of the Work of Droplet Formation
According to the definition of the work of droplet formation, the partial derivatives of the work W with respect to the number ν of solvent molecules and to the number νn of molecules in the residue of the condensation nucleus determines the chemical potentials µν and µαn ∂W = µν − µβ , ∂ν
∂W = µn − µαn . ∂νn
(12.20)
Before we start to analyze Eq. (12.20), let us simplify the problem. As was said in Section 12.2, surface tensions σ αγ and σ αβ are considered to be independent of the solution concentration. Moreover, the dependence of the disjoining pressure ΠD on concentration x can be also omitted, since the solution concentration in thin films is almost constant and equals approximately the solubility x∞ . Using Eq. (12.17) in the first equation from Eqs. (12.20) yields µν − µβ = µ∞ − µβ − kB T x +
2σ αβ v α Rn 2 − 2 ΠD v α , R R
(12.21)
where we took into account Eqs. (12.6), (12.8)-(12.10). Eq. (12.21) represents indeed the expression for the chemical potential of the solvent in the droplet. It can also be derived from Eq. (12.11) if one neglects the difference P β − P∞ and recognizes that the condition of mechanical equilibrium of the spherical liquid film with inner radius Rn and outer radius R has the form [16] Pα = Pβ +
R 2 2σ αβ − n2 ΠD . R R
(12.22)
Using Eq. (12.17) in the second equation from Eq. (12.20) yields 2σ γα vn 2σ αβ (vnα − vn ) x − kB T ln − Rn R x∞ , ∞ Rn 2 2vn α ΠD dh + ΠD vn − 2 (vn − vn ) , + Rn R
µn − µαn =
R−Rn
vch 4 Okt 2005 10:43
(12.23)
12.3 The Generating Properties of the Work of Droplet Formation
287
where we took into account Eqs. (12.6), (12.8)-(12.10). In order to prove that Eq. (12.23) represents indeed the expression for the difference in the chemical potential of molecules in the condensation nucleus and the chemical potential of solute in the droplet, one can use Eqs. (12.12) and (12.13). It follows from Eqs. (12.12) and (12.13) that µn − µαn = −kB T ln
x + vn (P2γ − P∞ ) − vnα (P α − P∞ ) . x∞
(12.24)
The condition of mechanical equilibrium for a spherical residue of the condensation nucleus within the droplet can be written as ! " 2˜ σ γα , P2γ = PN Rn + Rn
(12.25)
where σ ˜ αγ is the surface tension of the inner side of the thin film that does not generally coincide with the macroscopic value σ αγ . Thermodynamics of thin flat films [17] provides us with the relationship ˜ αγ ) ∂(˜ σ αβ + σ = −ΠD ∂h
(12.26)
with σ ˜ αβ being the surface tension of the outer side of the thin film. According to the definitions Eqs. (12.15) and (12.16) (i.e., according to the fact that the disjoining pressure is defined by the normal component of the pressure at the surface of the nucleus residue, and ΩD is proportional to the residue area), we can replace σ ˜ αβ for a spherical film by the macroscopic value σ αβ and reduce Eq. (12.26) to the form ∂σ ˜ αγ = −ΠD . ∂h
(12.27)
Integrating Eq. (12.27) and substituting the result, as well as Eqs. (12.15) and (12.22), into Eq. (12.25) yields ⎛ P2γ = P β +
2σ αβ R
−
Rn 2 ΠD R2
+ ΠD +
2 ⎜ γα ⎝σ + Rn
∞
⎞ ⎟ ΠD dh⎠ .
(12.28)
R−Rn
vch 4 Okt 2005 10:43
288
12 Deliquescence of Small Soluble Particles
In the limit of a flat film as Rn → ∞ and R → ∞, P2γ = P β . Substituting Eqs. (12.28) and (12.22) into Eq. (12.24) and neglecting the difference P β − P∞ leads to Eq. (12.23). If the chemical equilibrium with respect to both components in the droplet, the soluble component in the nucleus residue, and the condensing component in vapor is reached, then µν = µβ and µn = µαn . In view of Eq. (12.20), we have at complete equilibrium ∂W =0, ∂ν
∂W =0. ∂νn
(12.29)
As follows from Eqs. (12.21), (12.23), (12.6), (12.8)-(12.10), equations
µ∞ − µβ − kB T x +
⎛ 2vn ⎜ γα ⎝σ + Rn
∞ R−Rn
2σ αβ v α Rn 2 − 2 ΠD v α = 0 , R R
(12.30)
⎞ ⎟ 2σ αβ (vnα − vn ) ΠD dh⎠ − R
(12.31)
, x Rn 2 α + ΠD vn − 2 (vn − vn ) = 0 − kB T ln x∞ R
determine the values ν and νn corresponding to the minimum and the saddle points of the work W which are important for barrier kinetics. Eq. (12.30) is a generalization of the Gibbs-Kelvin-K¨ ohler equation of the theory of nucleation on soluble particles. Eq. (12.31) is a generalization of the OstwaldFreundlich equation of the theory of solutions. It gives the dependence of nucleus residue solubility on the residue size and size of the droplet. In view of Eq. (12.6) for the concentration x as a function of ν and νn , Eq. (12.31) can be regarded as a relation between ν and νn (or between R and Rn ) at the chemical equilibrium in droplet between the solute component and the nucleus residue.
vch 4 Okt 2005 10:43
12.4 Two- and One-Dimensional Theories of Deliquescence
12.4
289
Two- and One-Dimensional Theories of Deliquescence
Let us now consider the relation between two- and one-dimensional theories of deliquescence stage of nucleation on soluble particles in undersaturated vapor. The two-dimensional theory starts with the investigation of the two-dimensional plot of the work of droplet formation as a function of the number ν of condensate molecules in the droplet and number νn of molecules in the solid nucleus residue. To deal with undersaturated vapor, we will consider the vapor chemical potential µβ fixed somewhere below µ∞ , µβ − µ∞ < 0. As a good approximation for the disjoining pressure ΠD , we will employ the approximation [12-14]
R − Rn ΠD = K exp − l
(12.32)
with l being the correlation length for the condensation film around the nucleus and the factor K being related to surface tensions σ βγ , σ αγ , σ αβ and length l by the formula [13, 14] 1
l . K = σ βγ − σ aγ − σ aβ
(12.33)
As we noted in Section 12.3, we do not consider the dependence of the quantities l and K on the solution concentration and take them at a concentration x∞ . Using Eq. (12.32), we can rewrite the work W given by Eq. (12.17) in the form
2 (12.34) W = 4πRn σ αγ + 4πR2 σ αβ − 4πRn 2 σ γβ − µβ − µ∞ ν ! " R − Rn x 2 . − 4πRn lK exp − − kB T νx + kB T νn − νn ln x∞ l We define the numerical values for the parameters entering Eq. (12.34) as Rn = 15 · 10−7 cm ,
vn = 2 · 10−23 cm3 ,
v α = 3 · 10−23 cm3 ,
vnα = 2.2 · 10−23 cm3 , (12.35)
T = 298 K , x∞ = 0.2 ,
σ αγ = 200 dyn/cm , l = 2 · 10−7 cm ,
σ αβ = 72 dyn/cm ,
0 K = 3 · 109 dyn cm2 .
vch 4 Okt 2005 10:43
290
12 Deliquescence of Small Soluble Particles
Fig. 12.1 The two-dimensional work F of droplet formation on a soluble nucleus in undersaturated vapor at the vapor chemical potential b = −0.2
These values of the parameters are close to those for real condensation nuclei and water as a condensate. It is convenient to deal with dimensionless work of droplet formation, F ≡ ! "0 W /kB T , dimensionless chemical potentials of vapor, b ≡ µβ − µ∞ kB T , and solvent, bν ≡ (µν − µ∞ )/kB T , all of them expressed in terms of thermal units kB T . The plot of F as a function of ν and νn is shown in Figs. 12.1 and 12.2 for two values of the vapor chemical potential: b = −0.2 and b = −0.25. The relief of the ”waterfall path” in Figs. 12.1 and 12.2 shows the trajectory for a droplet transition from the state with partially dissolved condensation nucleus to the state with completely dissolved nucleus. Both considered cases demonstrate the existence of the activation barrier for such a transition, at this Fig. 12.1 illustrates the situation when such a transition occurs with a high probability, and
vch 4 Okt 2005 10:43
12.4 Two- and One-Dimensional Theories of Deliquescence
291
Fig. 12.2 The two-dimensional work F of droplet formation on soluble nucleus in undersaturated vapor at vapor chemical potential b = −0.25
the state with completely dissolved nucleus is more stable than the state with partially dissolved nucleus. Fig. 12.2 refers to the opposite case when a droplet state with partially dissolved nucleus is more stable. Let us now try to describe the situation, illustrated in Figs. 12.1 and 12.2, by using a one-dimensional approach. This approach assumes that chemical equilibrium between solute and the soluble core establishes rather quickly, and Eq. (12.31) holds for every value of the number ν of solvent molecules in a droplet. It is convenient to rewrite Eq. (12.31) in view of Eqs. (12.6) and (12.32) in the form kB T ln
νn − νn (ν) ν
2vn 2σ αβ (vnα − vn ) R − Rn γα σ − + lK exp − Rn (ν) l R , R 2 (ν) R − Rn vn − n 2 (vn − vnα ) . (12.36) + K exp − l R =
vch 4 Okt 2005 10:43
292
12 Deliquescence of Small Soluble Particles
7
6
nn' .10-5
5
4
3
2
1
0
2
4
6
8
10 12 14 16 18 20 22
ni
26 28
n .10-5
Fig. 12.3 Plot of νn as a function of ν according to Eq. (12.36)
The number of molecules νn (ν) in the equilibrium residue of the condensation nucleus, found as a function of the number of solvent molecules ν by solving Eq. (12.36) with account of Eqs. (12.35), (12.8)-(12.10), is shown in Fig. 12.3. The plot demonstrates that there is no equilibrium solution for νn above a certain value ν = νi (νi 2.414 · 106 in Fig. 12.3). This means that rather small soluble solid cores (νn < 62300 in Fig. 12.3) become unstable and inevitably dissolve if the limiting value ν = νi is reached. The whole picture becomes clearer if one considers the one-dimensional work F˜ = F (νn = νn (ν)) with νn being determined as a function of ν in accordance with Eq. (12.36) and the curve depicted in Fig. 12.3. The plots of F˜ are presented in Figs. 12.4 and 12.5 for two values of the vapor chemical potential: b = −0.2 and b = −0.25. Both plots have two branches corresponding to the regions of partial and complete dissolution of condensation nucleus along the ν-axis. In Fig. 12.4, the branch for the partial dissolution of the nucleus has minimum and maximum points located before the turning point ν = νi to solution instability (νi 2.414 · 106 , the same as in Fig. 12.3) and the point of intersection with the branch for complete dissolution. This means that the one-dimensional theory is applicable in the situation in Fig. 12.4 for the kinetic analysis of droplet transition between states with partial and complete dissolution of the nucleus.
vch 4 Okt 2005 10:43
12.4 Two- and One-Dimensional Theories of Deliquescence
n .10-5 -320
4
8
12
16
20
ni
28
32
36
293
40
-328 -336 -344
partial dissolution of the condensation nucleus
~ -3 F .10
-352
complete dissolution of the condensation nucleus
-360 -368 -376 -384 -392 -400
Fig. 12.4 Plot of the one-dimensional work F˜ of droplet formation at vapor chemical potential b = −0.2
n .10-5 -200
4
8
12
16
20
ni
28
32
36
40
-225
~ -3 F .10
-250
-275
partial dissolution of the condensation nucleus
complete dissolution of the condensation nucleus
-300
-325
Fig. 12.5 Plot of one-dimensional work F˜ of droplet formation at vapor chemical potential b = −0.25
vch 4 Okt 2005 10:43
294
12 Deliquescence of Small Soluble Particles
Another situation is shown in Fig. 12.5. Here the maximum point appears somewhere beyond the turning point and a direct application of the one-dimensional theory is questionable. But we will see in Sections 12.5 and 12.6 that the situation shown in Fig. 12.5 is not interesting from the point of view of the kinetics.
n .10-5 4
6
8
10
nth14
16 18 20 22
bth
ni
26 28 30 32
b
-0.2 -0.225
~ bn
-0.25 -0.275
partial dissolution of the condensation nucleus complete dissolution of the condensation nucleus
-0.3 -0.325 -0.35
Fig. 12.6 Plot of one-dimensional chemical potential ˜bν of the solvent in the droplet
The relation between the two- and one-dimensional approaches can be finally clarified with the help of Eq. (12.21). Substituting Eqs. (12.6) and (12.32) into Eq. (12.21), we obtain 2σ αβ v α Rn 2 v α R − Rn νn − νn + − 2 K exp − . (12.37) bν = − ν RkB T R kB T l Using Eqs. (12.35), (12.8) - (12.10) and the dependence νn (ν) given by Eq. (12.36) (as shown in Fig. 12.3) to calculate the one-dimensional chemical potential of the solvent ˜bν ≡ bν (νn = νn (ν)) as a function of ν leads to the plot presented in Fig. 12.6. As well as the plots in Figs. 12.4 and 12.5, the plot of ˜bν has also two branches corresponding to the regions of partial and complete dissolution of condensation nucleus along the ν-axis.
vch 4 Okt 2005 10:43
12.4 Two- and One-Dimensional Theories of Deliquescence
295
From a thermodynamic point of view, the phenomenon of deliquescence is connected with the existence of the maximum in the branch of the chemical potential of the solvent in the region of partial dissolution of the nucleus. This maximum is clearly seen in Fig. 12.6 at ν = νth . The stage of deliquescence is long-living when the value of the vapor chemical potential b appears to be below from that maximum (as shown in Fig. 12.6). A prompt transition from a droplet state with partially dissolved nucleus to a state with complete dissolution of the nucleus occurs if the vapor chemical potential reaches the close vicinity of the maximum from below or exceeds the maximum. In this sense, the maximal value ˜ bth = bν of the solvent chemical potential determines the deliquescence ν=νth
threshold, i.e. a certain value of the vapor chemical potential, above which all the droplets barrierlessly transform into a state with completely dissolved condensation nuclei. Below the threshold, the transition between the states requires an activation energy which may be called the deliquescence barrier. As follows from Eq. (12.37), bth = −
νn − νn (νth ) νth
vα + kB T
2σ αβ −K Rth
(12.38)
Rn (νth ) Rth
2
exp −
Rth − Rn (νth ) l
where νth is the root of the equation ∂bν dνn (ν) ∂bν d˜bν + = =0 dν ∂ν ∂νn dν =ν (ν ) ν=νth ,νn n th
, ,
(12.39)
ν=νth
with νn (ν) and dνn (ν)/dν found from Eq. (12.36), Rth ≡ R|ν=νth ,νn =νn (νth ) and Rn (νth ) ≡ Rn |νn =νn (νth ) determined with the help of Eqs. (12.6), (12.8)-(12.10). As can be seen from Fig. 12.6, if the vapor chemical potential is close enough to the deliquescence threshold from below, the one-dimensional approach is applicable of as far as the vapor chemical potential will be above the value bi ≡ ˜bν ν=νi
the chemical potential of the solvent at the turning point corresponding to the beginning of the instability (the same turning point ν = νi as in Figs. 12.312.5). It is the case for the situation shown in Fig. 12.1. It is not the case for the situation shown in Fig. 12.2. The part of the ˜bν -curve, located in Fig. 12.6 beyond the turning point on the branch with partial dissolution of a nucleus, has no physical meaning.
vch 4 Okt 2005 10:43
296
12 Deliquescence of Small Soluble Particles
12.5
Kinetics of Droplet Growth over the Deliquescence Barrier
Let us consider, now, the kinetics of droplet growth over the deliquescence barrier in the situation when the one-dimensional approach is applicable, i.e. in the (1) (2) situation illustrated by Fig. 12.4. We denote by νs , νc , and νs the positions of the extrema of the one-dimensional work of droplet formation. They are roots of the equation b = ˜bν
(1)
(12.40)
(2)
ν=νs ,ν=νc ,ν=νs
and correspond to the intersection points of the b-line with the ˜bν -curve in Fig. 12.6 (note that the vapor is undersaturated in the situations illustrated (1) by Figs. 12.4 and 12.6). The root ν = νs corresponds to the stable equilibrium droplet with νn > 0 (partly dissolved nucleus) at the bottom of the first potential well of the one-dimensional work F˜ in the undersaturated vapor at a given vapor chemical potential b. The root ν = νc corresponds to the critical droplet with νn > 0 (the residue of the nucleus still present in the droplet), which is in unstable equilibrium with the vapor at the same b. This root determines the location of the maximum in the curve of the one-dimensional work F˜ of droplet formation. (2) The root ν = νs refers to the droplet with completely dissolved nucleus, which is in a stable equilibrium with the vapor at the same b at the bottom of the second potential well of the one-dimensional work F˜ . (1) (2) ˜ ˜ , and Fs ≡ F (νn = 0)|ν=ν (2) , let us Introducing Fs ≡ F (1) , Fc ≡ F ν=νs
ν=νc
define the activation barriers ∆F (1) ≡ Fc − Fs(1) ,
∆F (2) ≡ Fc − Fs(2)
s
(12.41)
for the direct (denoted with index (1)) and reversal (denoted with index (2)) transitions of droplets over the deliquescence barrier. As follows from general formulas of heterogeneous nucleation ([14], Eq. (2.11)) and can be seen from Figs. 12.4 and 12.5, the following chain of inequalities, (2)
∂Fs ∂b
(1)
(2 − 3)νc . These droplets grow regularly with the mentioned law of growth. (C) The process of intensive nucleation takes place at Φ ≥ ζ ≥ Φ − 1/p. (D) Later new periods of nucleation (may be not so intensive but rather extended) will take place. The statements (A) and (B) are necessary for construction of the kinetic equations. The statement (C) allows us to use the approximation Eq. (13.78). The statement (D) is required to continue the consideration of the nucleation process which will be done below. The last step to do is to give an approximation for the number of droplets. After the evident rescaling the kinetic equation can be presented as z (z − x)3 exp(−G(x))dx .
G=4
(13.98)
0
vch 4 Okt 2005 10:43
330
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
The number of droplets appearing until z can be calculated as z exp(−x4 )dx .
N (z) =
(13.99)
0
We shall use the following approximation Nappr (z) = Θ(0.9 − z)z + Θ(z − 0.9)0.9 .
(13.100)
The relative error |Nappr − N |/N is small for all x. 13.3.1.2 The Case ∆1 z zlim This case is much more simple than the previous one. Here the kinetic equation can be written as z−z lim
exp(p(ζ(x) − Φ))dx
Φ = ζ(z) + Aνlim
(13.101)
0
or z−z lim
exp(−pG(x))dx
G = Aνlim
(13.102)
0
for z−z lim
exp(p(ζ(x) − Φ))dx .
G = Aνlim
(13.103)
0
It can be solved very easily. Namely, after the differentiation we come to dG = Aνlim exp(−pG(z)) , dx
(13.104)
which can be easily integrated 1 (exp(pG(z)) − 1) = Aνlim z . p
vch 4 Okt 2005 10:43
(13.105)
13.3 Kinetics of Nucleation at Growth Termination
331
The spectrum has the form f ∼ A exp(−pG(z)) .
(13.106)
One can see that here the total number of droplets ∞ f (x)dx
N (∞) =
(13.107)
0
is infinite. It is certainly a wrong result. The reason is the inapplicability of the approximation Eq. (13.78). The head of the spectrum is described quite satisfactorily, but the tail is not described well. Really at n = n∞ the nucleation according to this approximation does not stop. To describe the tail we have to use the precise equation for the stationary nucleation rate Is = Z exp(−Fc ) ,
(13.108)
where Z is the Zeldovich factor [9], Fc is the free energy of the critical embryo, taken in some approach but without the approximate expansion Eq. (13.78). For example, in the capillary approximation, we have 1 Fc = aνc2/3 , 3
νc1/3 =
2a , 3 ln(ζ + 1)
(13.109)
where a is the renormalized surface tension. Then t
νlim Is (ζ(t ))dt = Φ − ζ(t)
(13.110)
0
holds or νlim Is (ζ(t)) = −
dζ(t) . dt
(13.111)
The last equation can be easily integrated resulting in ζ
dζ =t νlim Is (ζ(t))
(13.112)
Φ
vch 4 Okt 2005 10:43
332
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
or ζ
exp
4a3 27 ln2 (ζ + 1) νlim Z(ζ)
dζ =t.
(13.113)
Φ
One can get an analytical solution for the tail of the spectrum having noticed that at small ζ one can write ln(ζ +1) ≈ ζ. The values of ζ cannot be too small, at least νc (ζ) < νlim and the behavior of Z is not singular. So we can approximately reduce the last equation to B2
exp(B3 ζ −2 )dζ = t + B1
(13.114)
with known constants Bi , i = 1, 2, 3. The integral can be reduced to the error function. This solves the problem of an explicit solution in terms of standard special functions. Principally the main result here and in all other situations is evident - the total number of droplets is
Ntotal =
n − ninf ty . ρ3lim
(13.115)
The problem is the shape of the spectrum. Here one has no need to obey the property (B) because this property begins to 1/3 be violated only when νlim strongly exceeds ∆z1 and then practically all droplets attain the final values. As for the quasi-stationarity (property (A)) we see that |dζ/dt| decreases in time and if (A) is obeyed in the initial moment of time it will be obeyed also later. The absence of the property (C) was taken into account just above. Instead of (D) one can state that this stage will be the last stage of the whole nucleation process in the system.
vch 4 Okt 2005 10:43
13.3 Kinetics of Nucleation at Growth Termination
333
13.3.1.3 The Case ∆1 z ≈ zlim The intermediate case is the most complex one and can be investigated employing already presented solutions. Let the kinetic equation be rescaled to have G = Θ(z − zlim )
⎧ ⎨ ⎩
z (z − x)3 exp(−G(x))dx
4 z−zlim
z−z lim
exp(−G(x))dx
+ 4νlim 0
(13.116)
⎫ ⎬ ⎭
z
+ Θ(zlim − z)4
(z − x)3 exp(−G(x))dx . 0
Certainly, here zlim will be also rescaled. Now we shall give the approximate method to solve this equation. The value G can be presented as G = g+ + g− ,
(13.117)
where g+ is the rescaled number of molecules in the droplets which have already attained zlim , g− is the rescaled number of molecules in all other droplets. For further constructions we have to write approximations for g− and g+ . The function g− can be easily found on the basis of an iteration procedure presented for the case δ1 z zlim . Then ∞ g− = Θ(z − ∆1 z)4A (z − x)3 exp(−g− − g+ )dx
(13.118)
0
z + Θ(∆1 z − z)4A
(z − x)3 exp(−g− − g+ )dx . 0
Here A is the amplitude which can differ from 1 due to the possible large value of g+ . At the current moment this remark is not too clear, but in any case we can say that for further purposes it is convenient to conserve A here. Now we shall explain latter relation.
vch 4 Okt 2005 10:43
334
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
At z < zlim , we have g+ = 0 and z (z − x)3 exp(−g− (x))dx .
g− (z) = 4A
(13.119)
0
It is important that it is the close one and it is identical to the case of unlimited growth. So, it can be successfully solved by iterations. Particularly, we have g−(0) = 0, g−(1) = Az 4 , etc. At z ≥ zlim , we have z (z − x)3 exp(−g− (x) − g+ (x))dx .
g− = 4A
(13.120)
z−zlim
Now we give a qualitative picture of the phenomena observed. At the first moments of time, the spectrum in the region ρ < ρlim is formed and g+ = 0 but g− is growing rather rapidly ∼ z 4 . Later the substance is going to be accumulated in droplets with ρ = ρlim . Then, a growth of g+ occurs. If the size of the cut-off zlim is essentially smaller than ∆1 z ≡ A−1/4 then g− 4 accumulates not so many molecules of substance g− = g−(in) ≡ Azlim 1 and later due to the decrease of the nucleation rate the value of g− will also decrease, i.e. g− < g−(in) In this case, g+ grows as z−z lim
g+ ≈
exp(−g+ (x) − g− (x))dx .
3 zlim 4A
(13.121)
0
It will grow slowlier than 3 4A(z − zlim ) . g+(in) = zlim
(13.122)
One can state that dg+ 3 < 4Azlim . dz
(13.123)
Then the variation δg+ in g+ during ”the time” of establishing of the quasistationary state in the region [0, zlim ] at intensive formation of embryos can be estimated as δg+
zlim the last inequality allows us to take exp(−g+ ) out of the integral. Then approximately z (z − x)3 exp(−g− )dx
g− (z) = 4A exp(−g+ (z))
(13.125)
z−zlim
holds and on the basis of the initial iteration approximation 4 g− (z) = A exp(−g+ (z))zlim
(13.126)
is obtained. For g+ , we get the following approximate equation z−z lim
g+ =
exp(−g− (x) − g+ (x))dx .
3 4Azlim
(13.127)
0
Here z is the current ”moment of time”, zlim is the ”moment” when the spectrum attains zlim . Here we are interested in the other characteristic scales of time which are greater than 1/(4Azlim ). At these times, we can suppose that exp(−g− ) is approximately constant and take it out of the integral. Then approximately z−z lim
g+ =
3 4Azlim
exp(−g− )
exp(−g+ (x))dx
(13.128)
0
holds. When g− is essential (i.e. exp(−g− ) is small) then g+ is not necessary and we can take out exp(−g− ). But when g− is small, then exp(−g− ) = 1 and we can also take it out. The substance consumption in g− occurs in avalanche manner. We can get the solution of this equation quite analogously to the case ∆1 z zlim , but here the initial conditions will be different i.e. g+ |z=zlim = 0 .
(13.129)
The solution will be dg+ 3 = 4Azlim exp(−g− ) exp(−g+ ) dz
(13.130)
vch 4 Okt 2005 10:43
336
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
and 3 exp(−g− )(z − zlim ) . exp(g+ ) − 1 = 4Azlim
(13.131)
The spectrum will be f = 4A
exp(−g− ) exp(g+ )
(13.132)
or f = 4A
(1 +
exp(−g− ) 3 4Azlim exp(−g− )(z
− zlim ))
.
(13.133)
In the initial approximation we consider A = 1 and get f = f0 ≡ 4
exp(−g− ) . 3 exp(−g )(z − z (1 + 4zlim − lim ))
(13.134)
4 and get Then we take g− = zlim
f = f0 ≡ 4
4 ) exp(−zlim . 3 exp(−z 4 )(z − z (1 + 4zlim lim )) lim
(13.135)
It is necessary to refine the expression for the spectrum obtained in zero-order approximation. We have to take into account that g+ is formed under the influence of g− . But the spectrum in the last expression is written through g− (but not through g+ ). Here is the difficulty. We shall overcome it via the special approximate approach. At first we assume that g− (but not g+ ) is formed with the 4 ) instead of exp(−z 4 ). It is evident renormalized intensity. We take exp(−αzlim lim that we do this only in the first factor which is responsible for the influence of g− . So, we get f =4
4 ) exp(−αzlim . 3 exp(−z 4 )(z − z (1 + 4zlim lim )) lim
(13.136)
For α we have to take the renormalized amplitude. The natural candidate for this amplitude is f0 . As the result we come to the following expression for the spectrum f1 = F (f0 ) ≡ f0 K ,
vch 4 Okt 2005 10:43
(13.137)
13.3 Kinetics of Nucleation at Growth Termination
337
where K=
4 ) exp(−f0 zlim 4 ) , exp(−zlim
f0 =
4 ) exp(−zlim . (13.138) 3 exp(−z 4 )(z − z 1 + 4zlim lim ) lim
Now let us realize what we have done. In reality we had to take into account the inverse influence, i.e. to take into account that g+ is formed under the influence of g− , but not the influence of g+ on g− . To take into account the inverse influence we shall use the inverse transformation F −1 . So, approximately f = F −1 (f0 ) .
(13.139)
We can calculate this transformation rather easily as f = F −1 (f0 ) =
f0 . K
(13.140)
This will solve the problem of the adequate construction of the approximate spectrum. One can see that the relative error in the droplets number is small for reasonable zf in ∼ 10. Statements (A), (B), (C) are obeyed here. Statement (D) is not necessary. 13.3.1.4 Further Nucleation The case ∆1 z zlim requires the analysis of the further nucleation. This analysis is not too complex. On the basis of ζf in,1 we can calculate the length ∆2 z as ∆2 z = ∆1 z(ζf in,1 )
(13.141)
i.e. we use the same formula but with ζ1,f in instead of Φ. Since in Eq. (13.84) there is an amplitude of the spectrum A which is a very sharp function of supersaturation we see that ∆1 z ∆2 z
(13.142)
holds. This strong inequality allows us to neglect the time interval [0, ∆1 zτ /Φ] in the nucleation at ζ2 . Then the situation is reduced to the nucleation with an initial supersaturation ζ2 . We have to repeat the steps of the three previous sections. Again we can come to the possibilities (A), (B) and (C). So, we see that the structure of the spectrum is not too complex. It includes the plateau, the region of partial collapse of surplus substance and the tail. Now the analysis of nucleation after instantaneous creation of a metastable state is completed.
vch 4 Okt 2005 10:43
338
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
13.3.2
Nucleation at Smooth Behavior of External Conditions
The straightforward generalization of the kinetic equation to the case of a smooth change of the external conditions leads to the following kinetic equation z (z − x)3 exp(p(ζ(x) − Φ∗ ))dx
Φ(z) = ζ(z) + A
(13.143)
z−zlim z−z lim
exp(p(ζ(x) − Φ∗ ))dx .
+ Aνlim −∞
Here Φ(z) is the ideal supersaturation, i.e. the supersaturation which should be found in the system without any processes of vapor consumption and heat release effects. The value of Φ at some characteristic moment t∗ will be denoted as Φ∗ . We shall choose t∗ later. The number of droplets (in renormalized units) will be calculated as z exp(p(ζ(x) − Φ))dx .
N (z) =
(13.144)
−∞
Under the conditions of unlimited growth of droplets the period of nucleation is well localized in time. Moreover it is rather short in time which allows to linearize the ideal supersaturation during this period. Here the process of nucleation cannot be localized in time, it is seen simply by the fact that the number of droplets in a liquid phase in limited from above by N νlim . Then to compensate the ”action” of external conditions it is necessary to have nucleation again and again. To present concrete calculations we suppose that the ideal supersaturation can be linearized dΦ(z) z, Φ(z) = Φ∗ + dz z=0
(13.145)
but now it is no more than a model. Later we shall see how to construct the theory for a rather arbitrary behavior Φ(z). But even with this linearization the situation here cannot be solved by a simple generalization of an ordinary iteration procedure because here the process is not limited in time.
vch 4 Okt 2005 10:43
13.3 Kinetics of Nucleation at Growth Termination
339
13.3.2.1 Rescaling The kinetic equation Eq. (13.143) can be written in the following form z−x lim
x3lim f (x)dx
g= −∞
z (z − x)3 f (x)dx
+
(13.146)
z−xlim
with xlim = zlim and a spectrum f (x) that is given by f (x) = a exp(bx − g(x)) .
(13.147)
It is essential that after the evident rescaling one can put a = 1 and b = 1. The third parameter xlim remains and this leads to the absence of universality. The restrictions a = 1 and b = 1 do not result in a special meaning of the point z = 0 (in homogeneous condensation z = 0 was the point of the maximum of supersaturation, in heterogeneous condensation z = 0 was the point where half of droplets have been appeared). One has simply to require that Φ(z = 0) is not too far from the maximum of supersaturation. This fact can be also analytically proven here. When xlim = ∞ then one can get the half-width of the spectrum by the condition ∆z = 1 .
(13.148)
13.3.2.2 Pulse Regime When xlim 1, the process of nucleation can be described rather simple. To describe the first peak of nucleation intensity we have to solve the equation without limitations on growth. This case is well known and can be solved both by iterations [4] and by employing the universal solution [7]. Thus, we get N1 , the number of droplets formed in the first peak. One can see that N1 ∼ 1 holds. Then the supersaturation ζ continues to fall until z ∼ xlim . The inequality xlim 1 allows here to speak about the well formed peak of nucleation intensity and to use the standard procedure of unlimited growth. At z ∼ xlim , the supersaturation begins to grow. At z ∼ xlim , we have f ∼ exp(xlim − N1 x3lim ) 1. At z ∼ N1 x3lim , the second peak of nucleation begins. The second peak of nucleation can be described absolutely analogously to the first one. So, we have a set of identical peaks. Consequently, nucleation occurs in the pulse regime.
vch 4 Okt 2005 10:43
340
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
13.3.2.3 Smooth Regime The opposite situation when zlim 1 can be also solved analytically. Here the kinetic equation can be written as z−z lim
exp(p(ζ(x) − Φ∗ ))dx
Φ(z) = ζ(z) + Aνlim
(13.149)
−∞
and z Φ(z) = ζ(z) + Aνlim
exp(p(ζ(x) − Φ∗ ))dx .
(13.150)
−∞
After differentiation with account of linearization we have dg = Aνlim exp(bz − pg(z)) . dz
(13.151)
In our units (one can prove that it is possible to choose units in such a way), p = 1, Aνlim = 1, b = 1. Integration with initial conditions g(z = −∞) = 0
(13.152)
exp(g(z)) = Aνlim exp(z) + 1 .
(13.153)
gives
This value will be denoted as gst . Then the spectrum f will be f = fst ≡ exp(x − ln(exp(x) + 1)) .
(13.154)
At x → ∞, we see fst → fst,lim ≡ 1. These limiting situations will form the basis for the description of nucleation in all situations. But here we have to stress two features: • The limiting solutions have to be radically modified. The power of the already presented methods for these solutions is not sufficient to give a complete description. • Description of the basis of the iteration method can not be suitable because the process is not restricted in time. At first we shall present the general structure of methods and then we shall show how to determine the elements included in these approaches.
vch 4 Okt 2005 10:43
13.3 Kinetics of Nucleation at Growth Termination
341
13.3.2.4 Advanced Pulse Regime To describe situations with xlim ≥ 2 and |xlim −2| 2 one has to use the rescaled peaks approach. This approach is based on the following simple considerations. Due to the previous analysis we shall approximate the spectrum by several peaks. Every peak is formed by some external effective source (initiated both by external conditions and by the action of the tail of the previous peak). The distance between peaks of nucleation is approximately ∆f x = N x3lim ,
(13.155)
where N is the number of droplets formed in the previous peak. Certainly, between different peaks the distance can be different because N will be different, but the suitable approximation is to consider N to be the ideal value, i.e. at the first peak. The last formula will be rather evident if we notice that the action of external conditions has simply to compensate the loss of substance which is equal to N x3lim . Here we shall present a recurrent procedure to calculate peaks of nucleation. Suppose that we have described the current peak and know the coordinate of maximum xm . Now we have to rescale the amplitude of the next peak. We neglect the non-linear behavior of the effective source. Then the next peak will be similar after rescaling to the previous one. But without rescaling in the same units the next peak will have a new (ordinary smaller) height and another width (ordinary wider). So, now we have to determine the new intensity b of the effective source (the initial intensity was b0 = 1). We know the expression for the intensity of vapor consumption by droplets dg = 3A dz
z (z − x)2 exp(x − g(x))dx .
(13.156)
z−xlim
The subintegral function has the form (z − x)2 exp(x − g(x))
(13.157)
and approximately has the maximum at (z − x) = ρa = 2 ≡ xa for xlim > 2 and (z − x) = ρa = xlim for xlim < 2 (here we can put z = 0 at maximum because this choice will affect only the parameters but not the spectrum form).
vch 4 Okt 2005 10:43
342
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
The current peak has a coordinate xm , the next peak will have the coordinate xm + ∆f x. The maximum of the subintegral expression for dρ/dz for the next peak will be attained at xb = xm + ∆f x − xa . The main effect is connected with the property that at x = xb the essential quantity of droplets from the previous peak has attained the value xlim . But not all droplets have attained this value. So, the supersaturation ζ lies higher than the supposed value b0 z − N x3lim ,
(13.158)
which would ensure the similarity of the next peak to the previous one. The supersaturation at the corresponding coordinate xc = xm − xa for the previous peak was lower than at the current peak. We suppose that at x = xb + xa = xm + ∆f x approximately corresponding to the coordinate of the next peak all droplets of the previous peak will attain the limit value N x3lim and there will be no difference between the value of the effective ideal supersaturation (the supersaturation imaginary formed without droplets of the current peak taken into account) Ω at xm and at xm + ∆f x Ω(xm ) = Ω(xm + ∆f x) .
(13.159)
The effective intensity b of the external source can be approximated as b ∼
Ω(xm + ∆f x) − Ω(xm + ∆f x − xa ) xa
(13.160)
and has to be compared with b0 ∼
Ω(xm ) − Ω(xm − xa ) . xa
(13.161)
Then taking into account |Ω(xm ) − Ω(xm + ∆f x)| 1
(13.162)
|Ω(xm ) − Ω(xm − xa )| ∼ xa ≥ 1
(13.163)
and
one can come to b = b0 −
vch 4 Okt 2005 10:43
∆ζ xa
(13.164)
13.3 Kinetics of Nucleation at Growth Termination
343
where ∆ζ = Ω(xm + ∆f x − xa ) − Ω(xm − xa )
(13.165)
holds. This completes the procedure. The first peak calculations have to be performed in explicit manner. One can see that the calculated peaks are very near to the real solution. Even such simple rescaling brings a rather accurate result for xlim = 2. When xlim > 2 the accuracy will be better. The simple account of variations of ∆f x will give a better accuracy also. All other sources of error can be taken into account by the standard perturbation technique.
13.3.2.5 Advanced Smooth Regime We see that already at xlim = 2 the first peak is located not so far from the stationary solution Eq. (13.154). Later all peaks will be even smaller than the first one. So, we can consider (f − fst )/fst as the small parameter and linearize the kinetic equation with respect to this parameter. One can also use instead of fst the value fst,lim . Then this solution can be solved analytically and can give only oscillations relaxing to fst,lim. We can avoid these long analytical formulas and simply draw the relaxing oscillations fosc = fst,lim + k1 exp
(x − xm ) k2
cos
(x − xm ) ∆f x
(13.166)
with two parameters k1 and k2 . The arguments for the derivation of ∆f x apply here also. The parameters k1 and k2 can be determined by a coincidence of approximate and real solution at the maxima of the two first peaks. We choose the maxima for coincidence of real solution and approximate solution because namely here the intensity of nucleation has maxima. The first local minimum demonstrates at xlim = 2 the deviation of the approximate solution from the real one but here the intensity of nucleation is small. One can see that the precise solution is very close to the approximate one.
vch 4 Okt 2005 10:43
344
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
13.3.2.6 Iterations for Parameter Determination Now we know the methods to describe nucleation adequately, but the parameters in these approximations are unknown and our next task will be to determine these parameters. To do this we have to know the course of the evolution during several first peaks of nucleation. In the advanced pulse method we have to know the evolution during the first peak including the back side of the peak and in the advanced relaxation method we have to know the positions and values of the two first maxima. These characteristics will be determined on the basis of iteration methods. But here the iteration methods cannot be obtained by direct generalization of already presented procedures in the case of condensation under the pure free molecular regime of vapor consumption [4, 7]. Really, even in the pure free molecular regime it is difficult to describe even the back side of the spectrum on the basis of standard iterations and here we need to describe the second peak. The limitation of growth will also diminish the converging power of iterations. So we need to reexamine the iteration procedure. The initial standard iteration procedure can be determined as z
z−z lim
(z − x) exp(x − gi (x))dx + νlim
exp(x − gi (x))dx .
3
gi+1 = z−zlim
(13.167)
0
It will converge at every initial approximation. As suitable initial approximations one can propose g0 = 0
(13.168)
g0 = gst ,
(13.169)
or
gst = 0
at
x < ln(fst
lim )
,
gst = x
at
x ≥ ln(fst
lim )
.
(13.170)
Since here it will be impossible to take only two first iterations there is no special significance what approximation we shall use. We prefer to use the first one because here the chain of inequalities g0 < g2 < . . . < g2i < . . . < g < . . . < g2i+1 < . . . < g3 < g1 for every fixed x is observed.
vch 4 Okt 2005 10:43
(13.171)
13.3 Kinetics of Nucleation at Growth Termination
345
Expansions in Iterations The analytical calculation of iterations will be terminated at the second step. For the unlimited free molecular regime this number of iterations is sufficient, but here we have to calculate the next iterations. It is possible to do it analytically if we choose the expansion of the exponent in the subintegral function. The following difficulty appears. The value of g goes to infinity and it is not possible to expand exp(−g(x)) over g(x). The other possibility is to expand exp(x − g(x)) over x − g near some characteristic value of the exponent argument. But at x → −∞, the value of g goes to zero and the spectrum looks like exp(−|x|). Then it is impossible to use the second expansion at initial moments of time (i.e. at negative x). But this very period is governing the evolution during the first peak of nucleation and, thus, plays the main role in the description of nucleation. So, the pure second approach cannot lead to a suitable result. So, we need to employ here a more sophisticated approach. We shall proceed in the following manner. Already the first iteration gives us the approximate position of the first maximum xmax,1 and the value of this maximum fmax,1 . Then we propose the following procedure: When x < xmax,1 then exp(−g) ≈
4 (−g)i i=0
i!
.
(13.172)
Here the first four terms are taken into account. It is sufficient to ensure the high relative accuracy. When x ≥ xmax,1 then exp(x − g) ≈ exp(ζbase )
4 (x − g − ζbase )i
i!
i=0
,
(13.173)
where ζbase = ln
fmax
1
+ fst 2
lim
(13.174)
is the basis of the expansions. Here the expansion is limited also by the first four terms. Then the iterations can be calculated analytically at every step and they give the adequate approximation for the solution. They converge to the solution of the
vch 4 Okt 2005 10:43
346
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
kinetic equation where instead of exp one should write the presented approximation. Already the four first terms of expansion ensure the high accuracy. Instead of the first four terms one can use an arbitrary number of terms. One can see that already the several first iterations give practically the precise positions of the first maximum and the first minimum fmin,1 at xmin,1 . Later iterations converge but every new iteration gives the ”step” of coincidence with the real solution which becomes smaller and smaller. The reason of this failure is the following: really, the precise solution does not go far from fst,lim and the expansion works, but every iteration inevitably goes to zero or to infinity and on every iteration the expansion does not work. This error leads to the crisis in the convergence of iterations. So, we need to modify the iterations. Cut-off of Iterations Now we know fmax,1 and fmin.1 and can use these characteristics. One can analytically prove that for every x > xmax,1 fmin,1 < f < fmax,1 .
(13.175)
Then we can require that before we use in the expansion the following cut-off: • When exp(x−gi (x)) > fmax,1 , we take instead of exp(x−gi (x)) the value of the constant fmax,1 and perform no further expansions (one can prove analytically that in iterations, if fi > fmax,1 at some x = xcrit , then it will be fi > fmax,1 at all x > xcrit . So, the iterations cannot come back to the real solution. They continue to escape from the real solution). • When exp(x−gi (x)) < fmin,1 , we take instead of exp(x−gi (x)) the value of the constant fmin,1 and perform no further expansions (one can prove analytically that in iterations, if fi < fmin,1 at some x = xcrit , then it will be fi < fmin,1 at all x > xcrit ). • When fmin,1 < exp(x − gi (x)) < fmax 1 no special actions are required and we have to make the mentioned expansions. The convergence of new iterations is higher than the convergence of previous iterations. It evidently converges to the real solution (this does not mean the convergence in a mathematical sense but the fact that the step of adequate approximation for the real solution produced by every new iteration does not go to zero as in the previous case but remains a finite positive value). Also it is
vch 4 Okt 2005 10:43
13.3 Kinetics of Nucleation at Growth Termination
347
very easy to estimate rigorously the error of expansion because now we have an estimate fmin,1 ≤ fi ≤ fmax,1 for every x. The same estimates allow us to prove analytically the high rate of convergence for the new iterations. To complete the description of iterations one can prove the following important statements • The cut-off iterations without expansions converge to the real solution of the kinetic equation and they converge faster than the first-type iterations. The chains of inequalities Eq. (13.171) remain valid for the cut-off iterations without expansions. • The iterations with expansions converge to the solution of the kinetic equation where the function exp is treated according to the above mentioned expansions. The chains of inequalities Eq. (13.171) remain valid (here g is the solution of the kinetic equation where the function exp is treated according to the mentioned above expansions; to show this we have to see that the expansion of exp conserves the monotonuous properties. This fact can be proven if we differentiate the approximation and note that the derivative of the approximation is the approximation for the derivative of the exponent, i.e. for the same exponent. Then with the necessary terms taken into account and with fmin,1 ≤ fi ≤ fmax,1 this derivative can be made positive. Then the monotonuous properties remain even after expansions). For practical needs one can show that for xlim ≤ 2 already the first six iterations ensure the correct values of fmax,1 , xmax,2 and fmax2 , xmax,2 (this is the value and coordinate of the second peak; when xlim decreases from xlim = 2 then the iterations will approximate these values even better. It can be proved analytically). This is all we need to construct the relaxation oscillations. Here one can see the high rate of convergence. When xlim > 2, we can use the advanced pulse method and here we need to describe only the first peak. It can be done by the first four iterations (Really, if it would be necessary to get the coordinate of the second peak in frames of the iteration method we might have difficulties because of the very small value of fmin.1). This procedure solves the problem of obtaining the parameters of approximation and completes the description of the nucleation under the break of the blow-up growth. Here we do not consider the effect of the blurring of diffusional profiles of density around embryos. It will be the matter of a separate publication.
vch 4 Okt 2005 10:43
348
13 Kinetics of Nucleation with Specific Regimes of Droplet Growth
13.4
References
1. R. Becker and W. D¨ oring, Annalen der Physik 24, 749 (1935); J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, Oxford, 1946). 2. A. A. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Handbook on Integrals and Series. Elementary Functions (Moscow, 1981 (in Russian)). 3. V. B. Kurasov, Physica A 226, 117 (1996). 4. F. M. Kuni, A. P. Grinin, and V. B. Kurasov, Heterogeneous Nucleation in Vapor Flow. In: Mechanics of Inhomogeneous Systems, Ed. G. Gadiyak (Novosibirsk, 1985, p. 86 (in Russian)). 5. V. B. Kurasov, Theoretical and Mathematical Physics 131, 503 (2002) (in Russian). 6. V. B. Kurasov, Universality in the Kinetics of First-order Phase Transitions (St. Petersburg, 1997, 400p). 7. V. B. Kurasov, Phys. Rev. 49, 3948 (1994). 8. V. B. Kurasov, Physica A 207, 541 (1994). 9. Ya. B. Zeldovich, JETF 24, 749 (1942) (in Russian). 10. F. M. Kuni and A. P. Grinin, Colloid Journ. 46, 28 (1984) (in Russian).
vch 4 Okt 2005 10:43
14
Simulation of Performance of a Laminar Flow Diffusion Chamber in Nucleation Experiments Sergey P. Fisenko and Anton A. Brin A. V. Luikov Heat Mass Transfer Institute, National Academy of Sciences of Belarus, P. Brovka Str. 15, Minsk 220072, Belarus
Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius - and a lot of courage - to move in the opposite direction. Albert Einstein Abstract A new mathematical model of laminar flow diffusion chamber (LFDC) performance is proposed, which includes effects of homogeneous (heterogeneous) nucleation and droplet growth. A qualitative investigation of this model is made. It was shown that for the simulation it is very important to take into account correct temperature and composition dependences of transfer coefficients of vapor-gas mixture. The limits of applicability of an earlier developed mathematical model of the laminar flow diffusion chamber are established. The numerical simulation of the new mathematical model of the performance of the is presented. It is discovered that the nature of the carrier gas substantially affects the volume of the nucleation zone in the laminar flow diffusion chamber. In particular, for argon as the carrier gas the volume of the nucleation zone is about an order higher then for helium as the carrier gas. Effects of local structure of the supersaturation
350
14 Laminar Flow Diffusion Chamber in Nucleation
field near the growing droplets are discussed. Their application for the interpretation of experimental data is proposed.
14.1
Introduction
For experimental studies of nucleation in vapor-gas mixtures there are several types of devices. Until recent time experimentalists used the Wilson chamber, including its modifications, and diffusion cloud chambers. It is worth to note that the range of nucleation rates, studied with every of these devices do not overlap. The diffusion cloud chamber is used for measurements of relatively small nucleation rates (about 106 droplets/(m3 s)) in steady-state regime. Usually the Wilson chamber, which uses adiabatic expansion of a vapor-gas mixture in lowpressure volume, is aimed for measurements of much higher nucleation rates. It is important that the Wilson chamber and its modifications work at an impulse regime and usually at lower temperatures in comparison with diffusion cloud chambers. For the steady-state regime, the range of nucleation rates, which can be obtained in a laminar flow diffusion chamber (LFDC), is found usually in between these ones [1-7]. A sketch of LFDC is shown in Fig. 14.1. The idea of LFDC performance is that the hot vapor-gas mixture (after saturator) enters into a vertical cylindrical chamber (condenser) with a cold wall. There is a preliminary formed thin film of the condensate on the wall. Vapor diffusion and heat conductivity start to change temperature, composition and average velocity of the mixture. As a rule, the composition of the mixture is chosen in such manner that the Lewis number Le is greater than one. The density of the saturated vapor exponentially depends on its temperature, therefore even a small cooling of the mixture leads to the appearance of a supersaturated medium in the central part of the channel. If the supersaturation is relatively high, homogeneous nucleation and subsequent growth of new phase droplets take place. The optical count of the number of droplets permits to determine the dependence of the nucleation rate on many parameters of LFDC. For Le < 1, a supersaturated state will be found only near the surface of the cold film. For a mixture of air-water vapor in LFDC, when Le < 1, detailed calculations of field temperature and supersaturation are presented in Ref. [8]. It was established by experimentalists, that the nature of the carrier gas affects the LFDC performance; in particular, the effect of a number of carrier-gases on nucleation-growth processes in LFDC was investigated in Ref. [2]. Among them
vch 4 Okt 2005 10:43
14.1 Introduction
351
Helium
Argon
1
2
3
Fig. 14.1 Scheme of the laminar flow diffusion chamber
were hydrogen, helium, argon and nitrogen, as the condensing vapor 1-propanol (C3 H8 O) was employed. For the same supersaturation, the authors of Ref. [3] discovered the interesting fact that in LFDC the nucleation rate of propanol with argon as the carrier gas is higher than the one with helium taken as the carriergas. In other words, it was experimentally discovered that a carrier gas affects the kinetics of nucleation. This conclusion seriously contradicts the classical theory of the nucleation kinetics. For mixtures of higher alcohols with helium the measurements of homogeneous nucleation rates in LFDC have been made and reported in Ref. [4]. The discrepancy between theoretical results and the best experimental data is about three orders of magnitude with respect to the nucleation rate. An interesting fact
vch 4 Okt 2005 10:43
352
14 Laminar Flow Diffusion Chamber in Nucleation
was noted here that at the measurement zone almost 90% of the nuclei had radii have sizes of about 1-5 micron [5]. One of the basic aims of the present chapter consists in the search of the solution of abovementioned contradiction and the further experimental results. In order to proceed in this direction, we consider not only heat and mass transfer processes in the gas phase but also processes related with droplet growth and motion in the nucleation zone. As shown, these processes really depend on the nature of the carrier gas and the total pressure in LFDC [7]. Their appropriate account allows us in this way to arrive at a solution of abovementioned problems. It is worth to underline that for the interpretation of the results of nucleation studies calculated fields of temperature and supersaturations play a very substantial role. Below for calculations of LFDC performance we use a new mathematical model, initially presented in Ref. [9]. No doubts that applications of LFDC to different nanotechnologies will be expanded in future. In particular we would like to mention here covering of the surface of nanoparticles by different substances [8, 10]. The further development of the theoretical understanding of the respective processes is, this way, not only of academic but also of direct technological significance.
14.2
Mathematical Model of Laminar Flow Diffusion Chamber Performance
For a correct performance of a laminar flow diffusion chamber the flow of the mixture consisting of vapor and carrier-gas is organized in such manner that it is a laminar one. The Reynolds number is smaller that one hundred. The inlet profile of the velocity u(r) of the gaseous mixture is u(r) = 2u0 1 −
r 2 R
,
(14.1)
where u0 is the velocity averaged over the cross-section, R is the radius of the condenser. The convective heat diffusivity equation with heat source describes the field of temperatures of the vapor-gas mixture in the condenser. It is given by 1 ∂T 1 ∂ ∂T (r, z) = λ(r, z)r + It [n(r, z) − ns (T (r, z))] , (14.2) u(r, z) ∂z ρm cm r ∂r ∂r
vch 4 Okt 2005 10:43
14.2 Mathematical Model of LFDC Performance
353
where λ(r,z), ρm , and cm are, correspondingly, the coefficient of heat conductivity, the density and heat capacity of the gaseous mixture, r and z are, correspondingly, the radial and axial coordinates. It [n(r, z) − ns (T (r, z))] is the expression for the heat source, generated by the release of the latent heat of condensation during droplet growth, ns (T ) is the numerical density of saturated vapor at temperature T. The convective diffusion equation with a source describes the field of the vapor density, n(r, z), in the condenser via 1 ∂ ∂n(r, z) = u(r) ∂z r ∂r
∂n D(T (r, z))r ∂r
+ In (r, z) [n(r, z) − ns (T (r, z))] ,
(14.3)
where D(T (r, z)) is the binary diffusion coefficient of the vapor, depending on temperature, In (r, z) [n(r, z) − ns (T (r, z))] is the sink, describing the condensation of the supersaturated vapor on newly formed droplets. For modern laminar flow diffusion chambers, our estimations show that the contribution of thermodiffusion is negligible. Therefore we neglect contributions of the conjugate processes in Eqs. (14.2) and (14.3). There is a permanent condensation of the vapor onto the cold wall of the condenser; as a consequence, there is a thin film flow of condensed vapor. The boundary conditions for vapor are given then as T (R, z) = Tw ,
n(R, z) = ns (Tw ) ,
(14.4)
where Tw is the temperature of the cold wall. We have two standard conditions in the center of the flow, ∂n(0, z) ∂T (0, z) = =0. ∂r ∂r
(14.5)
The integral term In in Eq. (14.2) is z N (z0 , r)Rd2 (z, z0 , r)L(Rd (z, z0 ))dz0 ,
In (r, z) = 4π
(14.6)
0
where Rd (z, z0 ) and N (z0 , r) are, correspondingly, radius and the number of droplets, formed near the point with the coordinates (z0 , r).
vch 4 Okt 2005 10:43
354
14 Laminar Flow Diffusion Chamber in Nucleation
In Eq. (14.5) the function L(Rd ) depends on the Knudsen number Kn= λ/Rd , where λ is the mean free path of vapor molecules. This function is useful for the description of the isothermal mass transfer of droplet and vapor for different regimes of droplet growth [11] Dm 1 , (14.7) L(Rd ) = ρl Rd 1 + (D/Rd ) 2πm/kB T where m is the mass of a vapor molecule, kB is the Boltzmann constant, ρl is the density of liquid propanol. We take into account the temperature dependence of the diffusion coefficient D. For the number density of droplets per unit volume, N (z, r), we have at steadystate the continuity equation with the source, which describes the nucleation contribution J(z, r) ∂N (z, r) = , ∂z u(z, r)
(14.8)
where J(z, r) is the local nucleation rate. For the description of the growth of the droplets in the supersaturated vapor we use the equation n(r, z) − ns (T (r, z)) ∂Rd (z, z0 ) = L(Rd (z, z0 )) , ∂z u(z, r)
(14.9)
obtained in Ref. [11]. In Eq. (14.2), the integral term It is It =
U In , N 4πRd3 ρcd ρm cm + 3
(14.10)
where U is the latent heat of the phase transition per vapor molecule, cd is the specific heat capacity of the droplet (cluster). We assume that the inlet flow has a uniform distribution of vapor density and temperature T (0, r) = Tc ,
n(0, r) = ns (Ts ) ,
(14.11)
where Ts is the temperature of the saturator of the laminar flow diffusion chamber and Tc is the inlet temperature of the vapor-gas mixture before it comes into contact with the condenser of the laminar flow diffusion chamber [2]. Before reporting numerical results of the solution of above given set of equations, we present in the next section some qualitative estimates [12].
vch 4 Okt 2005 10:43
14.3 Qualitative Estimates
14.3
355
Qualitative Estimates
For obtaining qualitative estimates for the fields of temperature and vapor density in the condenser, we approximately integrate Eqs. (14.2)-(14.3) using the Galerkin method [13]. For a cylindrical condenser as the trial function we choose the Bessel function of zeroth order. Below we use only the two first terms of the expansion in the expression for the fields of temperature and density in the series of Bessel functions. Approximately the temperature field in the condenser is given then by
r + Tw , (14.12) T (r, z) = At(z)J0 b1 R where At(z) is the unknown function, which will be determined later. Correspondingly, the expression for the field of the vapor density is described by a similar expression with the unknown function An(z)
r + ns (Tw ) , (14.13) n(r, z) = An(z)J0 b1 R where b1 is the least positive root of the equation J0 (b1 ) = 0 .
(14.14)
Equations (14.12)-(14.13) satisfy the boundary conditions Eqs. (14.4)-(14.5) exactly. Substituting Eqs. (14.12)-(14.13) into Eqs. (14.3)-(14.4) and after simple transformations we have two ordinary differential equations for the determination of the functions An(z) and At(z) 2u0 k1 ∂z An(z) = −kr An(z) ,
(14.15)
2u0 k1 ∂z At(z) = −ktr An(z) .
(14.16)
The solution of the first differential equation is z kr , An(z) = An0 exp − 2u0 k1
(14.17)
where An0 = ns (Ts ) − ns (Tw ) holds and the following notations are employed R
r 2 r dr = 0.107R2 , J02 b1 k1 = r 1 − R R
(14.18)
0
vch 4 Okt 2005 10:43
356
14 Laminar Flow Diffusion Chamber in Nucleation
b2 kr = 12 R
R
r dr . rD(T (r))J12 b1 R
(14.19)
9
Correspondingly, the expression for At(z) is
z ktr At(z) = At0 exp − u0 2k1
,
(14.20)
where At0 = Tc − Tw holds and the following notation is used
ktr =
b21 R2
R λ(T (r))J 2 b1 r 1 R dr . r ρm (T (r))cm (T (r))
(14.21)
9
It is helpful here to introduce the characteristic length of decay of the temperature field lt , which is directly proportional to the square of the chamber radius and the averaged velocity of flow and inversely proportional to the heat conductivity of the mixture lt =
2u0 k1 2u0 R2 ρm cm = 0.8 , kr λ(Tw )b21
(14.22)
and the characteristic length of the decay of the field of vapor density lv lv =
2u0 k1 2u0 R2 = 0.8 . ktr D(Tw )b21
(14.23)
It is easy to show that for mixtures with helium, the strong inequality lv lt holds due to high heat conductivity of helium. Calculation gives us that for a mixture of argon-propanol lv > lt is also fulfilled. Note that due to the continuity equation the product (u0 R2 ρc) remains constant along the condenser. This circumstance significantly improves the efficiency of the Galerkin method. In particular, for conditions of the run 1 from Ref. [2] (mixture helium-propanol) we have (lv /R) = 4 and (lt /R) = 0.13. At the same conditions, for a mixture argon-propanol we have (lv /R) = 14.8 and (lt /R) = 10.9.
vch 4 Okt 2005 10:43
14.3 Qualitative Estimates
357
Using obtained above formulas, we have got an approximate expression for the supersaturation field S(r, z) = n(r, z)/ns (r, z) of the form
r z (ns (Ts ) − ns (Tw )) exp − + ns (Tw ) J0 b1 l R v
. S(r, z) = z r ns At0 exp − J0 b1 + Tw lt R
(14.24)
If lv > lt and far from the inlet of the condenser, where the temperature field already equalized, we have the more simple expression S(r, z) =
z r ns (Ts ) − 1 exp − +1 . J0 b1 ns (Tw ) lv R
(14.25)
Equation (14.24) gives a qualitatively correct description of the behavior of the field of supersaturation in the laminar flow diffusion chamber. A similar behavior of the supersaturation field in LFDC was numerically obtained in Refs. [2-7]. The supersaturation field in the laminar flow diffusion chamber, calculated via Eq. (14.24), is displayed in Fig. 14.2.
10
Axial position, z/R
8 6 4
1.500 3.000
2
3.500 0
0
0.2
2.000 2.500
0.4 0.6 0.8 Radial position, r/R
1.0
Fig. 14.2 Contour plot of supersaturation field (conditions of run 1 from Ref. [2])
vch 4 Okt 2005 10:43
358
14 Laminar Flow Diffusion Chamber in Nucleation
3.2
50.00
3.0
45.00
Axial position, z/R
2.8
40.00
2.6 2.4
35.00
2.2 30.00
2.0 1.8 25.00
1.6 1.4 1.2
0
0.2 0.4 Radial position, r/R
Fig. 14.3 Contour plot of free energy of critical cluster formation (conditions of run 1 from Ref. [2])
The two-dimensional field of the dimensionless free energy for propanol critical clusters is demonstrated in Fig. 14.3. We used the same conditions as in Fig. 14.2 also for Fig. 14.3. As seen in Fig. 14.3, the global minimum of the free energy is located along the axes of symmetry of the condenser; it is equal to (∆Φ∗ /kB T (r, z)) ≈ 23. The axial width of the nucleation zone, determined by the condition that the free energy at the border is equal to (∆Φ∗ /kB T (r, z)) + 2, is about 0.6R, and at radial direction the width is smaller than 0.2R. We determine the actual volume of the nucleation zone, in which the vast majority of new droplets should be formed. As we shall see later, two terms in the expansion Eq. (14.12)-(14.13) determine the value of the global minimum not accurately enough. Nevertheless, Fig. 14.3 gives a qualitatively correct impression about the sizes of the nucleation zone. Following the estimation, obtained in Ref. [9], we can neglect the depletion of vapor due to condensation on clusters and droplets in laminar flow diffusion chambers if for the number of droplets per volume Nd , the following inequality is valid Nd R−2 Rd−1 .
vch 4 Okt 2005 10:43
(14.26)
14.3 Qualitative Estimates
359
In other words, we can use mathematical models of the performance of LDFC described in Refs. [2-5]. For LDFC, described in Ref. [2], the inequality (14.26) means that Nd has to be much smaller than 109 droplets/m3 . Otherwise droplets with a radius about one micron will substantially change the state of the vaporgas mixture (temperature and vapor density). Using Eq. (14.2), we can make an additional estimation. In order to neglect the release of the latent heat of phase transition in the laminar flow diffusion chamber, the number of droplets per unit volume has to satisfy the following inequality Nd Le
U kB Tw
−1
R−2 Rd−1 ,
(14.27)
where the Lewis number Le is defined as Le =
λ . ρcD
(14.28)
Gaseous mixtures, used at the laminar flow diffusion chamber, have a Lewis number which is greater than one, and (U/kB Tw ) ∝ 10. For a mixture of heliumpropanol the Lewis number Le = 31, for a mixture argon-propanol Le = 1.3. Note the important results of Ref. [14], where analytical calculations have been made of some parameters of the laminar flow diffusion chamber versus variations of the inlet profile of velocity. There is a connection between Nd and the total nucleation rate J. After approximate integration of Eq. (14.8) we have the following expression J≈
vf N , d
(14.29)
where d is the width of the nucleation zone in the laminar flow diffusion chamber, vf is the averaged velocity of flow after its cooling. Let us estimate the way of growth of newly formed droplets, using Eqs. (14.9) and (14.24). The droplet growth stops in the laminar flow diffusion chamber if the supersaturation dropped to the value one. It follows from Eq. (14.24) that the supersaturation practically dropped to one if (z/lv = 3). Therefore the path of droplets growth is equal to three characteristic lengths of the decay of vapor density lv .
vch 4 Okt 2005 10:43
360
14 Laminar Flow Diffusion Chamber in Nucleation
Approximately integrating Eq. (14.9), we have the dependence of droplet radius versus the path z along the axes of the chamber as 4 5 z 5 D(Tw )mlv 1 − exp − 5 6 ns (Ts ) lv −1 , (14.30) Rd (z) ∝ ns (Tw ) 2u0 ρ or substituting the expression for lv into Eq. (14.14), we have 4 5 z 5 m 1 − exp − 5 R 6 ns (Ts ) lv −1 . Rd (z) ∝ b1 ns (Tw ) ρ
(14.31)
Thus the highest possible droplet radius Rm in the laminar flow diffusion chamber is proportional to 7 R m ns (Ts ) Rm (z) ∝ −1 . (14.32) b1 ns (Tw ) ρ We see that the saturator temperature, Ts , substantially affects the maximum radius of the droplets formed in the LFDC. Interestingly, Rm is proportional to the radius of the condenser.
14.4
Methods of Numerical Analysis
For an approximate calculation of the velocity profile of the mixture, which changes during the cooling process, we use the following equations: The continuity equation for vapor-gas mixture ρ(Tc )u0 = ρ(Tw )uf ,
(14.33)
where Tw is the temperature of the cold wall of LFDC, ρ(Tw ) and ρ(Tc ) are the densities of the mixture, correspondingly, at temperatures Tw and Tc and atmospheric pressure, uf is the final velocity of the mixture; in addition, we use the equation of state of the ideal gas. The expression for the final velocity uf is uf = u0
Tw . Tc
vch 4 Okt 2005 10:43
(14.34)
14.5 Simulation Results
361
For the transient part of the flow in the condenser, where cooling takes place, we use the simplest linear approximation for the averaged velocity ⎧ (u − u0 ) z ⎪ ⎨ u0 + f , L u(z) = ⎪ ⎩ z≥L uf ,
z 0.2, ξ < 10%, and (T − T0 )/T0 ≈ 10%. In this case, the difference of the values of cv and T from those of c0v and T0 does not exceed 10%. With the same accuracy, one can replace A in Eq. (15.11) by A0 =
c0v L2 µv (RCp T02 )
(15.16)
and integrate Eq. (15.11) with allowance for relations Eqs. (15.15) within the interval from τi to τc resulting in 1=
πab2 (1 − a3 )VT r˙02 A0 I0 τc4 . 3
(15.17)
When deriving Eq. (15.17), it was taken into account that, by definition, during the time of condensation relaxation, the term ln s does not vary and remains of the order of unity; within the time interval from 0 to τi , temperature and the supersaturation ratio remain virtually unchanged.
15.3.2
Scaling Relations
The derived expression Eq. (15.17) turned out to give the basis for the analytical consideration of the process of condensation relaxation. First, relations connecting the time of condensation relaxation and number droplet density (i.e., the resultant
vch 4 Okt 2005 10:43
15.3 Results of Simulation and Their Analysis
(a)
1.9
383
650
2 630
1
lns
1.5
T, K
610
1.1
0.7
ti
1
0
590
tc 2 t, ms
3
4
570
3
4
1'
1
2
rd , mkm
nd , 1010m
-3
3
2
2
1 1
2'
(b) 0
0
1
2 t, ms
3
4
0
Fig. 15.1 (a) Dependences of temperature (1) and supersaturation ratio (2) on time; (b) Dependences of droplet number density (1, 1 ) and average droplet radius (2, 2 ) on time; (1 , 2 ) give the results of the computations based on Eqs. (15.15).
vch 4 Okt 2005 10:43
384
15 Condensation-Relaxation of Supersaturated Vapor
process characteristics) with nucleation rate at the initial moment follow directly from Eq. (15.17) like −1/4
τc = Pτ I0
,
3/4
nd = Pn I0
.
(15.18)
If we take into account that r˙0 and A0 ∝ c0v , we can also determine the dependence of the factors Pτ and Pn on vapor concentration in a mixture at the initial moment of the relaxation process Pτ ∝ (c0v )−3/4 ,
Pn ∝ (c0v )−3/4 .
(15.19)
3/4
Note that the relation nd ∝ I0 was obtained for the first time in Ref. [10] as an asymptotic relation (at t → ∞) for isothermal condensation. More genweral results (concerning also nucleation time value) have been obtained recently in slightly different approximation in Ref. [11]. Coincidence (in this point) of the results for non-isothermal (in this paper) and isothermal [10, 11] cases follows from the fact noted above: the formation of droplets and their size distribution function proceed at practically unchanged values of temperature and supersaturation ratio. Note also that relations similar to Eq. (15.18) were derived when considering the kinetics of first-order phase transition in solid solutions and glasses [12, 13]. Second, the use of the classical theory of nucleation allows us to derive, in accordance with Eq. (15.17), explicit dependences of τc and nd on the initial values of the supersaturation ratio, temperature, and vapor concentration in a mixture τc = Qτ exp
C [(ln s0 )2 T03 ]
τc ∝ Qτ ∝ (c0v )−5/4 ,
,
nd = Qn exp −
3C [(ln s0 )2 T03 ]
nd ∝ Qn ∝ (c0v )3/4 .
,
(15.20)
(15.21)
Here, C is a constant, which is a combination of universal constants and individual characteristics of matter (molar mass, surface tension, and liquid density). In was taken into account in Eq. (15.21) that, in classical theory of nucleation, I ∝ (cv )2 holds. The obtained dependences Eqs. (15.20)-(15.21) correctly describe the limiting transition: as approaching the region of stability (s0 → 1) and lowering the vapor concentration in a mixture (cv → 0), the time of condensation relaxation tends to infinity and the number droplet density, to zero.
vch 4 Okt 2005 10:43
15.3 Results of Simulation and Their Analysis
385
Tab. 15.1 Comparison of the results of numerical simulation and analytical dependences
Parameter τc nd Pτ Pn Qτ Qn
Power of exponent Analytical Approximation Value of numerical results -0.25 -0.256 0.75 0.758 -0.75 -0.68 -0.75 -0.70 -1.25 -1.19 0.75 0.80
Data of numerical simulation were approximated by Eqs. (15.18)-(15.21) that allowed us to determine the corresponding powers of the exponents. Results of the approximation are shown in the Table 15.1. The comparison of analytical results and data of numerical simulation demonstrates their satisfactory agreement (see Table 15.1). In addition, we could, based on the data of approximation, establish that 1 nd = I0 τc . 4
(15.22)
I.e., the part of the induction period a of the total time of condensation relaxation (see Eq. (15.15)) is equal to 0.25.
15.3.3
On the Possibility of Experimental Determination of the Nucleation Rate
Equation (15.17) with allowance for Eq. (15.15) can be presented in the form I0 = Bτi−4 , B=
3a3 , [πb2 r˙02 VT A0 (1 − a3 )]
(15.23) (15.24)
that allows us to use it for measuring the nucleation rate. Indeed, the duration of the ”induction period” τi can be determined by the time dependence of temperature in the volume of condensing vapor, the a value is known, and the remaining
vch 4 Okt 2005 10:43
386
15 Condensation-Relaxation of Supersaturated Vapor
parameters in Eq. (15.24) are characteristics of the studied substance or they are given by the initial experimental conditions. The attractiveness of a similar experiment is determined by the following circumstance. At the moment of nucleation, the droplets are about 1 nm in size and they cannot be detected by optical diagnostics. In this connection, it seems important to simultaneously use methods of experimental determination of nucleation rate unrelated to the optical registration of droplets. There is the possibility to estimate the accuracy of the experimental data received by the suggested method as follows. From general arguments, we can write (see Eq. (15.23)) Accuracy(I0 ) = Accuracy(B) + 4 × Accuracy(τi ) . For construction of Eq. (15.17) from which Eq. (15.23) follows, the quantity A has been replaced by A0 within the accuracy of 10 %. Hence, the factor B brings an error not less than 10 %. The error in the definition of the duration of the ”induction period”, τi , is connected with the accuracy of experimental definition of temperature depending on time in the concrete experimental situation. As a result we have Accuracy(I0 ) ≥ 4 × Accuracy(T (t)) + 10% .
15.4
Condensation-Relaxation of Supersaturated Vapor under Static and Dynamic Conditions
Using Eq. (15.23) as a theoretical basis for experiments requires to analyze the assumptions under which this expression was derived, in particular, the assumption that one can ignore drop nucleation and growth at the step of rapid attainment of a supersaturated state, which precedes relaxation. For this purpose, in this paper, we consider two condensation relaxation modes without making this assumption. The results to be obtained will allow one, in particular, to determine the conditions for the applicability of Eq. (15.23). The equation describing the change in the degree of supersaturation during expansion has the form d ln V d ln s = A1 − πrd2 nd VT A2 , dt dt
vch 4 Okt 2005 10:43
(15.25)
15.4 Static and Dynamic Conditions
387
where A1 = (γ − 1)
A2 = 1 + cv
γ Lµv − RT γ−1
L Cp T
Lµv −1 RT
,
(15.26)
.
(15.27)
The first term in Eq. (15.25) describes the increase in the degree of supersaturation during adiabatic expansion; and the second, the decrease in the degree of supersaturation during condensation relaxation, which is caused both by the depletion of the vapor phase (the first term in the expression for A2 ) and by the increase in the temperature owing to the release of the latent heat of phase transition (the second term in the expression for A2 ). The quantities nd and rd can be found from the set of moment equations Eq. (15.5). Above this process was studied under the assumption that, at the rapid expansion step, the second term in Eq. (15.25) can be neglected, and at the relaxation step, because of the constancy of the volume, the first term in Eq. (15.5) was zero. In this paper, as noted above, two condensation relaxation modes are considered: • Eq. (15.1): Relaxation during continuous vapor expansion, when only the expansion rate is specified. We call such a relaxation mode the dynamic mode. • Eq. (15.2): Relaxation after attaining a given degree of expansion at finite expansion rate. In this case, at the expansion step, both terms in Eq. (15.25) are taken into account, and after attaining a given expansion ratio, the first term in Eq. (15.25) becomes zero. We called such a relaxation mode the static mode. Let us begin our consideration of the dynamic mode of condensation relaxation with analyzing Eq. (15.25). This equation implies that the supersaturation ratio during condensation relaxation attains a maximum under the condition A1
d ln V = πrd2 nd VT A2 . dt
(15.28)
The left-hand side of Eq. (15.28) is an increasing function of the expansion rate, and the right-hand side of Eq. (15.28) is an increasing function of the supersaturation ratio. Thus, the higher the expansion rate, the higher the supersaturation ratio attained during condensation relaxation. The calculation results presented in Fig. 15.2 confirm this conclusion.
vch 4 Okt 2005 10:43
388
15 Condensation-Relaxation of Supersaturated Vapor
3.0 2.5
3
lns
2.0 1.5
2 1.0
1
0.5 0 1.0
1.1
1.2
1.3
1.4
1.5
V/V0
Fig. 15.2 Supersaturation ratio versus expansion ratio in the dynamic mode of condensation relaxation at various expansion rates: V˙ 2 = 10V˙1 , V˙ 3 = 100V˙1
Following our previous paper [6], let us introduce, for dynamic relaxation, a time interval τcd in which the degree of supersaturation decreases from the value τcd by a factor of e ((ln s) decreases to the value ln smax − 1). By analyzing the results of numerical modelling, approximate expressions are obtained for the characteristics of condensation relaxation (relaxation time, attained supersaturation ratio) as functions of the expansion rate. The obtained expressions can be used to solve two problems: • Eq. (15.1): from a given expansion time, it is necessary to determine the expansion ratio at which the supersaturation ratio during dynamic relaxation attains a maximum; • Eq. (15.2): for the given expansion ratio, it is necessary to determine the expansion rate at which the maximum of the supersaturation ratio during dynamic relaxation is attained at the given expansion ratio. The second problem is necessary to solve, in particular, for analyzing condensation relaxation in the static mode. Let us choose a certain value (V /V0 )1 of the
vch 4 Okt 2005 10:43
15.4 Static and Dynamic Conditions
389
expansion ratio and, by the above method, determine the corresponding expansion rate V˙ 1 . Here, V and V0 are the volumes of the mixture at the current and initial moments of time, respectively. It is clear that, for the found combination of the expansion ratio and the expansion rate, consideration of condensation relaxation in both the dynamic and the static modes gives the same results (cf. Fig. 15.3, solid and dashed curves 1). The difference between the relaxation modes is exhibited with an increase in the expansion rate (Fig. 15.3, solid and dashed curves 2 and 3). In the dynamic mode, as already noted, an increase in the expansion rate leads to attaining a higher maximal supersaturation ratio, which, according to Eq. (15.2), corresponds to higher expansion ratios. However, the qualitative shape of the relaxation curves with an increase in the expansion rate in the dynamic mode remains unchanged (Fig. 15.3a, solid curves 2 and 3). Conversely, an increase in the expansion rate at constant expansion ratio (the static mode) changes the qualitative shape of the relaxation curves: in them, a characteristic plateau emerges, which corresponds to the induction period in condensation relaxation (Fig. 15.3a, dashed curves 2 and 3). The relaxation portions of the curves, which describe the behavior of the supersaturation ratio, temperature, the number density of drops, and the nucleation rate in the static mode, are presented in Figs. 15.4 and 15.5 as functions of the relative time t − tm . Here, tm is the moment of time (depending on the expansion rate) at which the maximal supersaturation ratio is attained. It is seen that, with an increase in the expansion rate from the value corresponding to the dynamic mode, the induction period increases from zero to the limiting value, which is characteristic of a given expansion rate. It is due to this value of the induction period that the results obtained above correspond. Thus, one can state that the increase in the induction period with an increase in the expansion rate at a given expansion ratio is indicative of a transition from the dynamic to the static mode of condensation relaxation. The minimal expansion rate at which the relaxation mode can be considered static was determined as follows. A variety of calculations at different expansion ratios gave the expansion rates at which, in the expansion time by the beginning of the relaxation step (0 < t < tm ), the number density of drops had reached no more than 1% of the final value of this quantity. It turned out that, throughout the considered ranges of the parameters being varied (which for the expansion
vch 4 Okt 2005 10:43
390
15 Condensation-Relaxation of Supersaturated Vapor
3.0
(a) 2.5
3
lns
2.0
2
1
1.5 1.0 0.5 0 0.01
0.1
1
10
t, ms 1.5
(b) 1.4
3
2
1
V/V0
1.3 1.2 1.1 1.0 0.01
0.1
1
10
t, ms
Fig. 15.3 (a) Supersaturation ratio and (b) expansion ratio versus time in the dynamic (solid lines) and static (dashed lines) modes of condensation relaxation at various expansion rates: V˙ 2 = 10V˙ 1 , V˙ 3 = 100V˙1
vch 4 Okt 2005 10:43
15.4 Static and Dynamic Conditions
2.0
391
620
1.8
610
1.6
2
3
T, K
lns
600
1
1.4
590 1.2 580
1.0 0.8
0.1
0
0.2 0.3 t - tm , ms
0.4
570
1.2
6
1.0
5
2
3
0.8
4
-12
0.6 3
1 0.4
2
0.2
1 0
-3
7
10 nd , m
-15
10 I, m s
-3 -1
Fig. 15.4 Supersaturation ratio (solid lines) and temperature (dashed lines) versus time in the static mode of condensation relaxation at various expansion rates: V˙ 2 = 10V˙ 1 , V˙ 3 = 100V˙ 1
0
0.1
0.2 0.3 t - tm , ms
0.4
0
Fig. 15.5 Number density of drops (solid lines) and nucleation rate (dashed lines) versus time in the static mode of condensation relaxation at various expansion rates: V˙ 2 = 10V˙ 1 , V˙ 3 = 100V˙ 1
vch 4 Okt 2005 10:43
392
15 Condensation-Relaxation of Supersaturated Vapor
rate was as wide as six orders of magnitude), the condition imposed is satisfied by the expansion rates obeying the relation V˙ st ≥ 50V˙d ,
(15.29)
where V˙ d is the expansion rate found by the above method (problem 2) in the dynamic mode for a chosen degree of expansion and V˙ st is the expansion rate in the static mode for the same expansion rate. Thus, Eq. (15.11) solves the problem stated in the beginning of this paper, namely, it indicates a method for choosing the expansion rate at which Eq. (15.2) can be used to find the nucleation rate from experimental data.
15.5
Concluding Remarks
The analysis of the results of numerical simulation of the process of condensation relaxation revealed the possibility of an approximate analytical description of the process. As a result, scaling relations for resultant values (the time of condensation relaxation and number density of forming droplets) as a function of the nucleation rate at the initial moment (i.e., at the known values of temperature, supersaturation ratio, and vapor concentration in the vapor-gas mixture) were established. The result obtained for droplet number density agrees with the published data on the condensation of supersaturated vapor under isothermal conditions. Agreement (in this point) between the results for non-isothermal (treated in this paper) and isothermal cases follows from the fact mentioned in this work, namely, the formation of droplets and their size distribution function proceed at virtually constant values of temperature and supersaturation ratio even if the heat release is taken into account. Scaling relations similar to those indicated above were obtained also when considering the kinetics of first-order phase transition in solid solutions and glasses. The results obtained are indicative of the essential possibility of experimental determination of nucleation rates by measuring the time dependence of the temperature of condensing vapor, because the analytical consideration of condensation relaxation was carried out without recourse to the specific expression for the nucleation rate. The relation between the time of condensation relaxation and number density of forming droplets, on one hand, and the temperature, supersaturation ratio, and vapor concentration at the initial moment, on the other hand, can be established using the specific expression for the nucleation rate. Interrelated numerical
vch 4 Okt 2005 10:43
15.6 References
393
and analytical power dependences of the resultant values of condensation relaxation on the initial vapor concentration were obtained within the framework of the classical theory of nucleation used in numerical simulation.
Acknowledgments The authors would like to thank A. L. Tseskis for help and discussions. The Russian Foundation for Basic Research supports the given work (Contract No 03-02-16646).
15.6
References
1. D. Kashchiev, Nucleation. Basic Theory with Applications (Butterworth-Heinemann, Oxford, 2000). 2. A. L. Itkin, J. Chem. Phys. 108, 3360 (1998). 3. V. G. Baidakov, Peregrev Kriogennykh Zhidkostei (Overheating of Cryogenic Liquids) (Ural Branch Russian Academy of Sciences Publishers, Ekaterinburg, 1995). 4. J. W¨ olk, R. Strey and B. E. Wyslouzil: Homogenous Nucleation Rates of Water: State of the Art. In: M. Kasahara and M. Kulmala (Eds.), Nucleation and Atmospheric Aerosols 2004: 16th Int. Conf. 5. N. M. Kortsenstein and E. V. Samuilov, Doklady Physics 46, 864 (2001). Translated from Doklady Akademii Nauk 381, 777 (2001). 6. L. E. Sternin, Osnovy Gazodinamiki Dvukhfaznykh Techenii v Soplakh (The Fundamentals of Gas Dynamics of Two-Phase Flows in Nozzles) (Mashinostroenie, Moscow, 1974). 7. M. Frenclach and S. J. Harris, J. Coll. Interface Science 118, 252 (1987). 8. D. A. Frank-Kamenetskii, In: R. I. Soloukhin (Ed.), Diffuziya i Teploperedacha v Khimicheskoi Kinetike (Diffusion and Heat Transfer in Chemical Kinetics) (Nauka, Moscow, 1987). 9. N. A. Fuks, In: Isparenie i Rost Kapel’ v Gazoobraznoi Srede (Evaporation and Growth of Drops in a Gaseous Medium) (Akad. Nauk SSSR, Moscow, 1958). 10. N. N. Tunitskij, Zh. Fiz. Khim. 15, 1061 (1941), in Russian. 11. F. M. Kuni, A. P. Grinin, et al., Colloid Journal (in Russian) 62, N1, 39 (2000).
vch 4 Okt 2005 10:43
394
15 Condensation-Relaxation of Supersaturated Vapor
12. V. V. Slezov and J. W. P. Schmelzer: Kinetics of Nucleation-Growth Processes: The First Stages. In: J. W. P. Schmelzer, G. R¨ opke and V. B. Priezzhev (Eds.) Nucleation Theory and Applications (Joint Institute for Nuclear Research Publishing House, Dubna, Russia, 1999). 13. V. V. Slezov and J. W. P. Schmelzer, Phys. Rev. E 65, 031506-1 (2002).
vch 4 Okt 2005 10:43
16
Critical Opalescence: Application of the Diffusion Equation to Multiple Light Scattering Dmitri Yu. Ivanov Saint-Petersburg State University of Refrigeration and Food Engineering, Lomonossov str. 9, Saint-Petersburg, 191002, Russia Nature is doing nothing in vain but for the best of purposes. Aristotle
Abstract Critical opalescence is one of the fundamental forms of appearance of critical behaviour. The nature of critical opalescence consists in multiple acts of scattering of transmitted light on growing critical fluctuations. Movement of a photon through the ”critical” medium is here the basic diffusion process similar to the Brownian motion of an ordinary particle. In the present chapter, two theoretically and practically important examples of the application of the diffusion equation approach to dynamical multiple light scattering are considered. The first problem is the determination of the half-width of multiple light scattering spectra when a laser light propagates through a dispersed non-absorbing turbid medium. The second one is the evaluation of substance heating near its critical point by the laser beam. Based on the diffusion equation the solutions of both problems are found.
396
16 Diffusion Equation and Multiple Light Scattering
16.1
Introduction
In the two last decades considerable attention has been focused on dynamical light scattering in high extinction media [1-14]. The single scattering correlation spectroscopy has long been in use as a routine procedure of determination of particle dimensions and their kinetics for dispersed systems of the different physicochemical nature [15, 16]. It has been extended then to account for multiple scattering employing the diffusion equation approach. It has been the stochastic nature of the multiple scattering which has dictated the diffusion approach to the problem [3, 4, 18]. Once the multiple scattering theory has been developed [3-7], the practical application of the method is expanded to a wide range of strongly turbid systems such as milk-like media, blood, systems near the critical point [8-10, 14, 15]. In the present work, two theoretically and practically important examples of the diffusion equation application to the dynamical multiple light scattering will be considered. The first problem is the multiple light scattering spectra half-width determination when a laser light propagates through a dispersed nonabsorbing turbid medium. The second one consists in the estimatetion of the substance heating near its critical point by the laser beam. Based on the diffusion equation approach, the solution of both problems has been found and will be presented in the present chapter.
16.2
Dispersed Nonabsorbing Turbid Media
The mathematical model employed for this analysis is the well-known diffusion equation [19] D∇2 ρ (r, t) =
∂ρ (r, t) , ∂t
(16.1)
where D is the diffusion coefficient, ρ (r, t) is the density of diffusing particles at a point, r, at time, t. Generally, such equations are denoted as ”thermal conductivity equations” and their properties are well presented in the corresponding literature (see e.g. [20]). In particular, the well-known solution of this equation for a pulse source located in an unbounded medium is given by 9 8 |∆r|2 = 6Dt ,
vch 4 Okt 2005 10:43
(16.2)
16.2 Dispersed Nonabsorbing Turbid Media
397
where ∆r is the displacement of a diffusing particle from its original position. The distribution of the scattered radiation per scattering multiplicity can be also defined by the diffusion equation. With this aim in view one can imagine that there is an input of the initial pulse of radiation into the scattering medium. It is well known that such non-stationary problems are described by an equation like Eq. (16.1). It is evident that in this case the scattering multiplicity N varies in direct proportion to time t. This means that the form of the equation remains valid despite of the change of the variable t by N . The only difference is connected with the coefficient in the left side of Eq. (16.1) which must be modified. It can be shown [6, 23] that this equation in such a case will look like ∂I (r, N ) l2 ∇2 I (r, N ) = , 3(1 − µ) ∂N
(16.3)
where I (r, N ) is the N -fold scattered intensity, l is the mean free path of a photon in the medium, µ is the value of cosθ averaged over a single scattering indicatrix, θ is the scattering angle. It should be noted that the function I (r, N ) is a continuous one with respect to N , inasmuch as the function I (r, t) is continuous with respect to t. The mathematical model of the multiple scattering process under consideration is based on Eq. (16.3). But the complete formulation of this mathematical problem demands in addition the determination of the initial and boundary conditions. Upon integrating Eq. (16.3) with respect to N we obtain an equation which must coincide with the well-known diffusion one for the total intensity of the scattered radiation [18, 21]. The initial condition being conform to this consideration looks like I (r, 0) = (1 − µ)−1 I0 (r) ,
(16.4)
where I0 (r) has the meaning of the coherent radiation intensity penetrated into the scattering medium via its boundary. The boundary conditions for Eq. (16.3) may be presented as Eq. (16.5). It is convenient to introduce an effective boundary r + z0e instead r [18] I (r + z0e, N ) = 0 ,
(16.5)
vch 4 Okt 2005 10:43
398
16 Diffusion Equation and Multiple Light Scattering
where r describes any point at the boundary with no external source of radiation, e is the outward normal to the boundary in this point, z0 is the so-called ”extrapolated length”, it is determined as follows (see e.g. [22]) z0 =
0.7l 2l ≈ . 3 (1 − µ) 1−µ
(16.6)
The mean multiplicity $∞
N =
N I (r, N )dN
0 $∞
(16.7) I (r, N ) dN
0
may be found after the solution of Eqs. (16.3)-(16.5). Naturally, the use of the model is limited by conditions of the applicability of the diffusion approximation: the absorption of the radiation in a turbid medium must be negligible, and the characteristic dimension L of a sample must be in accordance with the inequality L z0 . Now general characteristics of the solution of Eqs. (16.3)-(16.5), which are the consequence of the symmetry with respect to a similarity transformation, may be considered. The symmetry of Eq. (16.3) allows us to express the solution of the initial problem through a geometrically similar one [23] I (r, N ) =
1 I1 L2
r N , L L2
,
(16.8)
where I1 (r, N ) is the solution for the geometrically similar problem with L = 1. More accurately, L + 2z0 would have to be taken as a coefficient of similarity, however, because of the inequality L z0 a difference in the first-order approximation is negligible. Substituting Eq. (16.8) into Eq. (16.7) one can arrive at the following conclusion: N is proportional to the square of the ratio (L/l). As a result we have 1−µ
N d = 2
2 L F , l
(16.9)
where the subscript d denotes the diffusion model. In comparison with the simplest diffusion model discussed earlier [14] this formula contains an additional
vch 4 Okt 2005 10:43
16.2 Dispersed Nonabsorbing Turbid Media
399
factor F , which depends on the form of the scattering medium. As it is expected, an analytical calculation is possible only for some high-order symmetric problems, but this is not unfortunately a case for the cylindrical geometry [23] and the following cases: • Scattering matter fills a sphere with a coherent radiation source at the center. In this case for the radiation which pass through the spherical boundary, the condition F = 1 holds; • Scattering medium fills a hemisphere, a narrow pencil of the coherent radiation is directed to the plane boundary center. In this case at the spherical boundary, we have F = 0.6; • At the planar boundary the value of the coefficient F changes from 0.6 near the outer edge to 3.0 in the center. All these examples analyzed by Pavlov [23] show that the scattering average multiplicity fundamentally depends on the form of the scattering medium. Consequently, any model not taking into account the sample shape will be unsatisfactory for practical applications. Unfortunately, in the general case (for an arbitrary form of the scattering volume) an analytical solution of such a problem is impossible and only the separate examples with the high-order symmetry allow us to calculate the factor F analytically. In all other cases the coefficient F must be determined by means of calibration measurements on the systems with well-known values of Γ, µ and l (Γ is the single scattering spectrum halfwidth averaged over a single scattering indicatrix). As an alternative, the Monte Carlo method may be proposed (see, e.g., [24]). The basic idea of the diffusion model of any level of complexity, it will be recalled, may be expressed as [1-3] Γm = N Γ ,
(16.10)
where N is a mean multiplicity as a result of the averaging over different trajectories of the photons. Thus in order to finish the mathematical model construction it is necessary to substitute Eq. (16.9) into Eq. (16.10). However, it should be remembered that the parameter N d is only an approximate estimate for the scattering average multiplicity, N . It is believed that the difference between these values tends to the constant N0 for high values of the multiplicity. As a result [15] one can arrive at a closer approximation of Γm like 2 1 L F Γ + N0 Γ . (16.11) Γm = (1 − µ) 2 l
vch 4 Okt 2005 10:43
400
16 Diffusion Equation and Multiple Light Scattering
Latter formula permits to repeat once more the general conclusion of the theoretical and experimental investigations [3, 6, 14, 15, 23]: in strongly turbid media at geometrically similar conditions the measured scattering spectrum half-width, Γm , is a linear function of the square of the scattering sample optical thickness. Moreover, this formula predicts the existence of an additional term, N0 Γ, independent of optical thickness, which has been really found experimentally [6] (for an extended discussion see our previous article [14] and also the book [15]).
16.3
”Critical” Medium Heating by Probing Radiation
The problem under consideration is of particular significance and interest in the studies of the critical opalescence. Despite the fact that in any optical study light may be commonly considered as a noninvasive probing instrument, for critical opalescence this is not the case. When one deals with the strong dependence of the properties of matter on temperature a success of the study is not possible without a correct account of medium heating by probing radiation. Evaluation data show a considerable (for critical conditions) temperature increase ∼ 1 mK/mW even for low-turbid binary mixtures (e.g. nitroethane-isooctan [25, 26]). In Ref. [27] the problem of heating of the liquid crystal isotropic phase by probing radiation near the nematic transition point has been solved by a single scattering approximation with respect to the absorption. The problem is, how can multiple scattering be compared with the single one in the context of such a problem? The scattering medium heating in the case of developed multiple scattering in a low-absorbing but very turbid medium (e.g. binary mixture aniline-cyclohexane) is an extremely important problem for the critical opalescence experimental investigations. Let a narrow laser light beam fall into a cuvette filled with a strongly scattering medium [28]. When the scattering multiplicity tends to infinity the light intensity of the transmitted beam is negligible, and one can take into account only the scattering energy. It is required to find the temperature distribution in a cuvette at the additional condition that the boundary of the cuvette is kept at a constant temperature, T0 . Let us write the equation for the diffusion of radiation ∆I = −
3 P0 δ (r − r0 ) , 4πtr
vch 4 Okt 2005 10:43
(16.12)
16.3 ”Critical” Medium Heating by Probing Radiation
401
where I is, as usual, the intensity of the light scattering, r0 is the equivalent source location, and P0 is the power of the laser. Following the sense and the determination of tr a direct laser light penetrates only to a depth of the order of magnitude of tr . After that the knowledge concerning the initial direction of propagation is lost. Therefore it is quite reasonable to choose the equivalent source position (r0 ) at a distance tr counted off the boundary point down the beam direction. The boundary condition for Eq. (16.12) I (r)|r∈(∂S+dif f ) = 0 ,
(16.13)
where ∂S +dif f is a surface located at a distance dif f from the sample boundary in the direction of the outward surface normal. If α is a radiation absorption coefficient, then αIdω is a power of the heat sources in the volume unit produced by photons with velocity directions distributed in solid angle dω. Then 4παI is the full volume density of a energy source. In order to find the temperature volume distribution it is necessary to solve the thermal conductivity equation with a 4παI energy source ∆Θ =
4π αI , cp DT
(16.14)
where Θ is the temperature difference between the cuvette (T ) and the thermostat (T0 ), cp is the heat capacity of the volume unit. The boundary condition for Eq. (16.14) is Θ|r∈∂S = 0 .
(16.15)
Applying the ∆-operation to both parts of Eq. (16.14) and taking into account Eq. (16.12), we get ∆2 Θ =
3αP0 δ (r − r0 ) ≡ Aδ (r − r0 ) , cp DT tr
(16.16)
where A=
3αP0 . cp DT tr
(16.17)
The boundary conditions Eqs. (16.13) and (16.15) have to be consistent. In order to obtain the solution of Eqs. (16.12) and (16.14), which would be the upper
vch 4 Okt 2005 10:43
402
16 Diffusion Equation and Multiple Light Scattering
estimate for them, it is necessary to employ the boundary condition Eq. (16.18) instead of Eq. (16.15) Θ|r∈(∂S+dif f ) = 0 .
(16.18)
Let us consider the Green function for the Helmholtz equation ∆I (r, r0 ) − k2 I (r, r0 ) = δ (r − r0 ) ,
(16.19)
and let us write a formal expression for I as I (r, r0 ) =
! " 1 δ (r − r0 ) = ∆−1 δ (r − r0 ) + k2 ∆−2 δ (r − r0 ) + O k4 . (16.20) ∆−k
Now it becomes evident that the multiplier at k2 is an expression which coincides up to a constant factor A with the solution of Eq. (16.16). In other words, in order to find a solution of Eq. (16.16) it is sufficient to obtain the Green function for the Helmholtz equation up to k2 -order terms. For a sphere with the zeroth boundary conditions on its surface this problem may be solved analytically for an arbitrary position of the radiation source [28]. Here we will consider and analyse only the final result [15] "1/2 ! 2 "1/2 AR1 ! 1 + 2ρρ0 cosθ + ρ2 ρ20 − ρ + 2ρρ0 cosθ + ρ20 8π "! " (ρρ0 )1/2 ! 1 − ρ2 1 − ρ20 y2 dy , + 3/2 1 + 2y 2 cosθ + y 4 ρ3/2 ρ
Θ(r, ϕ) =
0
(16.21)
0
where ρ=
r , R1
ρ0 =
r0 , R1
R1 = R + dif f .
(16.22)
Here r0 and r are distances between the center of the sphere and the source position and the point of observation, correspondingly; θ is a scattering angle. Let us derive an upper estimate for Eq. (16.21). We get Θ(r, ϕ) ≤
"1/2 ! 2 "1/2 AR1 ! 1 + 2ρρ0 cosθ + ρ2 ρ20 − ρ + 2ρρ0 cosθ + ρ20 . 8π
vch 4 Okt 2005 10:43
(16.23)
16.3 ”Critical” Medium Heating by Probing Radiation
403
Let us denote the expression in square brackets in Eq. (16.23) as H, then "! " ! 1 − ρ2 1 − ρ20 ∂H = sin(θ)ρ0 ρ , ∂θ F (ρ, ρ0 , θ)
(16.24)
where ! "1/2 ! 2 "1/2 (16.25) ρ + 2ρρ0 cosθ + ρ20 F (ρ, ρ0 , θ) = 1 + 2ρρ0 cosθ + ρ2 ρ20 ! ! " " 1/2 1/2 × 1 + 2ρρ0 cosθ + ρ2 ρ20 + ρ2 + 2ρρ0 cosθ + ρ20 ≥0. It is clear that H(ρ, ρ0 , θ) reaches maximum at ρ0 = ρ and θ = π. Then the both estimates Eqs. (16.26) and (16.27) will be true ! " H ≤ 1 − ρ20 ρ0 →0 ∼ = 2 (1 − ρ0 ) ,
(16.26)
r0 AR1 A AR1 2 (1 − ρ0 ) = 1− (R1 − r0 ) = Θ (r, θ) ≤ 8π 4π R1 4π 5 Θ∗ 3αP0 5 A (tr + dif f ) = = · , = 4π 4πcp DT 3 2 2π
(16.27)
where Θ∗ =
αP0 cp DT
(16.28)
is a quantity with the dimension of temperature. For reasonable parameters of the medium and radiation source (such as α = 0.1 m−1 , P0 = 2 mW, cp = 1.8J/(m3 K), DT = 1.5 · 10−7 m2 /s) one can get Θ∗ ∼ = 7 · 10−4 K. It should be noted that Θ∗ does not depend on tr . This feature may be qualitatively explained: the smaller is tr , the longer are the photon’s trajectories and as a consequence the absorption increases. On the other hand, the smaller is tr , the smaller is the depth of the equivalent source position. These two tendencies are mutually exclusive. As a result, Θ∗ does not depend on tr . Near the point of falling of a beam, in domain with a radius of the order of tr , the diffusion theory for the radiation intensity will fail. Thus, this domain requires special consideration [28]. Let a photon fall on the surface of a cuvette and then it goes to the limit of the sphere with radius of tr measured from the input point.
vch 4 Okt 2005 10:43
404
16 Diffusion Equation and Multiple Light Scattering
Then the heat power which evolves in this volume approximately is αtr P0 . If we substitute this distributed source by the point one which should be offset by tr from the sphere boundary, we can obtain the set of equations ∆Θ = −Θ∗ tr δ (r − r0 ) ,
Θ||r|=R = 0 ,
(16.29)
where R − r0 = tr . The solution of Eqs. (16.29) is Θ∗ tr Θ (r) = 4π
1
R
! "1/2 − ! 2 "1/2 r 2 + 2rRcosθ + r02 r0 r 2 + 2rr0 R2 cosθ + R4 ! " 2 R2 − r 2 ∗ Θ tr ∼ · . = 4π (r 2 + 2rRcosθ + R2 )3/2
, r0 →R
(16.30)
Then the upper estimate of Eq. (16.30) will be [28] ! " 2tr R2 − r 2 Θ∗ Θ∗ 2tr (R − r) Θ(r) = · · ≤ , 3 4π (r 2 + 2rRcosθ + R2 )3/2 2π |r − R|
(16.31)
= (0, 0, −R). If we denote β = π − θ and use the obvious inequality where R R − rcosβ ≥ R − r, we get
2tr r − R
Θ∗ Θ∗ 2tr (R − rcosβ) z · · = 3 3 2π 2π |r − R| |r − R|
∗ 2 ∗ tr Θ r − R = − Θ 2tr fz r − R . · cos R, =− 2 2π |r − R| 2π
Θ (r) ≤
(16.32)
It should be noted that
= fz r − R
1 2 |r − R|
r − R cos R,
(16.33)
may be considered as the z-component of a certain Coulomb force which acts on the unit charge located at the point with coordinates (0, 0, −R) from the side of the volume element of a charged sphere with unit charge density. It is well known that under these conditions the full force acting from the side of a sphere is not changed if this sphere will be replaced by the point charge with
vch 4 Okt 2005 10:43
16.4 References
405
the magnitude (4πR3 )/3 and located in the center of the sphere. After volume averaging Eq. (16.32) can be finally written as [28] 1 1 Θ∗ 2
dr tr fz r − R Θ= 4 3 Θ (r) dr ≤ 4 3 2π 3 πR 3 πR =
Θ∗ 2π
| r|≤R
tr R
| r|≤R
2 .
(16.34)
In the case of the multiple scattering the value of the ratio tr /R 1. Then comparing Eqs (16.34) and (16.27) one can see that the volume averaged contribution from this domain into the heating of scattering medium is negligible in comparison with the contribution from diffusively scattered light. From the result obtained, it can be unexpectedly concluded that the heating of matter by multiply scattered light when tr /R does not depend on the scattering multiplicity. This feature of the multiple scattering gives the unique possibility to investigate the substance characteristics near the critical point without any necessity of continuous power corrections as the medium temperature approaches the critical one. Furthermore, the calculation made here has confirmed the evaluations of other authors [25, 26] by which the heating of the weakly absorbing media by a probing radiation near the critical point does not exceed ∼ 1 mK per 1 mW of the output power of the laser beam penetrated in the medium investigated.
Acknowledgements I gratefully acknowledge my colleagues A. F. Kostko, V. A. Pavlov and A. V. Soloviev for fruitful efforts turned to solving the above problems.
16.4
References
1. C. M. Sorensen, R. C. Mockler, and W. J. O’Sullivan, Phys. Rev. A 17, 2030 (1978). 2. A. Bøe and O. Lohne, Phys. Rev. A 17, 2023 (1978). 3. D. Yu. Ivanov and A. F. Kostko, Opt. Spectrosk. 55, 950 (1983) [Opt. Spectrosc. (USSR) 55, 573 (1983)]. 4. G. Maret and P.-E. Wolf, Z. Phys. B 65, 409 (1987). 5. D. Y. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzhemer, Phys. Rev. Lett. 60, 1134 (1988).
vch 4 Okt 2005 10:43
406
16 Diffusion Equation and Multiple Light Scattering
6. D. Yu. Ivanov, A. F. Kostko, and V. A. Pavlov, Phys. Lett. A 138, 339 (1989). 7. D. A. Weitz, D. Y. Pine, P. N. Pusey, and R. J. A. Tough, Phys. Rev. Lett. 63, 1747 (1989). 8. S. Fraden and G. Maret, Phys. Rev. Lett. 65, 512 (1990). 9. D. A. Weitz, J. X. Zhu, D. J. Durian, Hu Gang, and D. Y. Pine, Physica Scripta 49, 610 (1993). 10. J. C. Earnshaw and A. H. Jaafar, Phys. Rev. E 49, 5408 (1994). 11. A. F. Kostko and V. A. Pavlov, Applied Optics 36, 7577 (1997). 12. P.-A. Lemieux, M. U. Vera, and D. J. Durian, Phys. Rev. E 57, 4853 (1998). 13. J. A. Lock, Appl. Optics 40, 4187 (2001). 14. D. Yu. Ivanov: Critical Opalescence: Models-Experiment. In: Mathematical Modelling: Problems, Methods, Applications, L. A. Uvarova and A. V. Latyshev (Eds.) (Kluwer Academic/Plenum Publishers, New York, 2001, pgs. 37-51). 15. D. Yu. Ivanov: Critical Behavior of Nonideal Systems (Fizmatlit, Moscow, 2003, in Russian). 16. Photon Correlation and Light Beating Spectroscopy. H. Z. Cummins and E. R. Pike (Eds.) (Plenum Press, New York, 1974). 17. B. J. Bern and R. Pecora: Dynamic Light Scattering (Krieger, Malabar, Fla., 1990). 18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978). 19. Ph. M. Morse and H. Feshbach: Methods of Theoretical Physics (McGraw-Hill Book Co. Inc., New York, Toronto, London, 1953). 20. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford, 1959). 21. L. A. Apresyan and Yu. A. Kravtsov: Radiation-Transport Theory: Statistical and Wave Aspects (Nauka, Moscow, 1983, in Russian). 22. K. M. Case and P. F. Zweifel, Linear transport theory (Addison-Wesley, Reading, Mass., 1967). 23. V. A. Pavlov, Opt. Spectrosk. 64, 828 (1988) [Opt. Spectrosc. (USSR) 64, N 4 (1988)]. 24. Monte Carlo Methods in Statistical Physics, K. Binder (Ed.) (Springer-Verlag, Berlin, 1979). 25. D. Beysens: Status of the Experimental Situation in Critical Binary Fluids. In: Phase Transitions, Carg`ese 1980, M. L´evy, S. C. Le Guillou, and J. Zinn-Justin (Eds.) (Plenum Press, New York, London, 1982, pgs. 25-62).
vch 4 Okt 2005 10:43
16.4 References
407
26. D. Beysens, A. Bourgou, and G. Paladin, Phys. Rev. A 30, 2686 (1984). 27. L. A. Zubkov, V. P. Romanov, and T. Kh. Salikhov, Opt. Spectrosk. 68, (1990) [Opt. Spectrosc. (USSR) 68, 110 (1990)]. 28. D. Yu. Ivanov and A. V. Soloviev, VINITI, 24.02.92, N 1357-B92, 12 pages (in Russian).
vch 4 Okt 2005 10:43
17
Iron Matrix Effects on Cluster Evolution in Neutron Irradiated Reactor Steels Alexander R. Gokhman(1), Frank Bergner(2) , and Andreas Ulbricht(2) (1)
South Ukrainian Pedagogical University, Department of Physics, 65020 Odessa, Ukraine
(2)
Forschungszentrum Rossendorf e.V., Institut f¨ ur
Sicherheitsforschung, PF 510 119, 01314, Dresden, Germany Computers are useless. They can only give you answers. Pablo Picasso
Abstract The results of the determination of the binding energy by molecular dynamics are used to describe the interaction between clusters in the iron matrix. It is found that the change in Gibbs’s energy in cluster formation can be described as the sum of two terms proportional to n2/3 and n, respectively. Here n is the number of particles in a cluster. Comparison of the results of cluster dynamics modelling of cluster vacancy evolution in neutron irradiated steels with experimental data indicates that the iron-vacancy system can be considered as a real solution.
17.1 Introduction
17.1
409
Introduction
The interaction between cluster and matrix is included into the rate theory and in particular into the cluster dynamics (CD) modelling. The interaction term is accounted for in the determination of the emission rate from the absorption rate in accordance with the principle of detailed balance. The latter one of these coefficients can be presented in terms of the binding energy, E B , or Gibbs’s free energy change, ∆G E B (n) , αn = βn−1 exp − kB T
∆G(n − 1) − ∆G(n) αn = βn−1 exp − kB T
(17.1) .
(17.2)
Here βn and αn are the absorption and emission rates, respectively, for clusters containing n particles; kB is the Boltzmann constant; T is the temperature in Kelvin; ∆G(n) is the change in free energy when a cluster containing n monomers are formed; E B (n) is the binding energy of one particle with a cluster containing n monomers. Usually, ∆G(n) is given by ∆G (n) = −n∆µ + 4πRn2 γ ,
(17.3)
where ∆µ is the change in chemical potential and γ is the specific surface energy. In Ref. [1] an additional term, ∆(ε) G(n), is considered for diffusion limited type of cluster growth, resulting from the evolution of elastic stresses. This term is found proportional to (V − Vo )2 /Vo , where V is the actual cluster volume and Vo is the cluster volume at the transition from the kinetic limited type of cluster growth to the diffusion limited type affected by stress. In Ref. [2], the approach developed in Ref. [1], has been applied successfully to phase formation in porous systems. These approaches [1, 2] are used in Ref. [3] to account for the interaction between vacancy clusters (VC) and iron matrix. Because Vo is considered as the fitting parameter in Ref. [3] and the total number of vacancy monomers is not constant in irradiated steels the problem of iron matrix effects on the evolution of VC is not solved completely yet. Moreover, till now the interaction between iron matrix and Cu precipitates (CRP) is considered only on Eq. (17.3) in CD modelling. In continuum mechanics, the strain energy, Ws (n), in absence of external stresses is
vch 4 Okt 2005 10:43
410
17 Cluster Evolution in Neutron Irradiated Reactor Steels
determined for a spherical inclusion with bulk modulus K ∗ and specific volume of atom v ∗ using Eq. (17.4) from Ref. [4], i.e., Ws (n) = 2n
4 3K ∗
(1 + ν) +2 E
−1
(v ∗ −v)2 , 3v ∗
(17.4)
where E, ν and v are Young modulus, Poisson ratio and specific volume of the atoms in the matrix, respectively. Since for VC the relation K ∗ = 0 holds, the additional term in Eq. (17.4) is absent in Eq. (17.3) for VC. For CRP the additional term in Eq. (17.4) is present, proportional to n and determined by misfit of Cu-atoms in the iron matrix. The next problem is the determination of ∆µ for the process under consideration. As a rule, for VC in iron matrix the model of an ideal solution is used ∆µi = kB T ln
Cv CRn
,
(17.5)
where the Thomson-Freundlich term, CRn , is given by Ef 2γ Ωa + , CRn = exp 3 − kB T kB T Rn
(17.6)
and Cv , Ef and Ωa are the concentration of free vacancies, energy of formation of a vacancy, and volume of one vacancy, respectively; Rn is the radius of VC consisting of n vacancies. For CRP in an iron matrix ∆µ is calculated according to [5] ! 2 " ω 1 − xα + xα − x2 ∆µ = −kB T (1 − xα ) ln 1−x kB T
ω xα 2 2 + (1 − xα ) − (1 − x) . + xα ln x kB T
(17.7)
Here xα and x are the molar fraction of Cu in an Cu-Fe cluster and the matrix respectively; ω is the mixture energy of a Cu-Fe pair. The present status of accounting for the interaction between clusters and iron matrix as described in Eqs. (17.5)-(17.7) is not satisfactory. First, these equations contain a set of parameters that cannot be determined with good accuracy. Second, already in Ref. [6] the negative mixing energy of the vacancy-iron system
vch 4 Okt 2005 10:43
17.2 Method
411
(for ideal solution model the mixing energy is zero) is used in a Monte Carlo simulation. That is why the additional investigations on the interaction problem of clusters and iron matrix in molecular dynamics modelling [7, 8] are very helpful. In Refs. [7, 8], the binding energy E B (n) is calculated as the difference E1 − E2 , where E1 is the sum of potential energy of N atoms of matrix (Fe) + cluster of n particles + 1 particle far from the cluster; E2 is the sum of potential energy of N atoms of matrix + cluster of n particles + 1 particle in the surface of the cluster. Because the discussion on the validity of the results of Ref. [8] on the values of E B (n) for CRP is not completed (cf. [9]), in our present investigation a consideration of the interaction between VC and iron matrix in CD modelling has been done. Tab. 17.1 Material parameters. Data taken from Ref. [11]
Vacancy migration energy Vacancy pre-exponential Interstitial migration energy Interstitial pre-exponential Vacancy formation energy, Ef Recombination radius, r Surface energy, γ Dislocation density, ρ Lattice parameter, a
17.2
Method
17.2.1
Assumptions
1.3 eV 0.5 · 10−4 m2 /s 0.4 eV 5 · 10−6 m2 /s 1.64 eV 0.574 nm 1-2 J/m2 5 · 1014 m−2 0.2866 nm
The following assumptions are made in our analysis: • Single vacancies, VC’s and self interstitial atoms (SIAs) are continuously generated during irradiation as a result of the cascade stage of neutron-lattice interaction. Generation rates are adopted from molecular dynamics (MD) calculations by Jumel cited by Christien & Barbu [9]. • VC’s are assumed to be spherical. A sensitivity study by Gokhman et al. [10] along with measured SANS curves supports the applicability of this approximation and renders the applicability of a model based on planar VC’s questionable.
vch 4 Okt 2005 10:43
412
17 Cluster Evolution in Neutron Irradiated Reactor Steels
Tab. 17.2 Direct generation rate, 10−9 Gn,v in dpa/s, of cascade VC’s containing n vacancies. A0: Data taken from Fig. 8 of Christien & Barbu [9], A1 and A2: Values calculated for alloy A employing the condition A1 and A2, respectively
n 1 2 3 4 5 6 7 8 >9
A0 17.5 0.26 0.25 0.24 0.23 0 2.5 1.75 0
A1 0.0732 0.0011 0.0010 0.0010 0.0010 0 0.0105 0.0073 0
A2 0.549 0.0082 0.0078 0.0075 0.0072 0 0.0784 0.0549 0
• Both agglomeration and direct generation of clusters of SIA’s are neglected as done also by Odette [11]. However, planar clusters of SIA’s (dislocation loops) are considered in current work. • Single vacancies can be absorbed by or emitted from VC’s. SIA’s can be absorbed by VC’s but not emitted. VC’s can neither be absorbed by other VC’s nor be emitted from them (that means VC’s are immobile).
17.2.2
System of Equations
The rate of change of the number density of VC’s is specified by the following master equation dCn = βn−1,v Cn−1 + (βn+1,i + αn+1,v ) Cn+1 dt − (βn,i + βn,v + αn,v ) Cn + Gn,v
(17.8)
with Cn : concentration of VC’s consisting of n vacancies (n > 1); αn,v : rate of emission of single vacancies by VC’s containing n vacancies; βn,v(i) : rate of absorption of single vacancies (SIA’s) by VC’s containing n vacancies; Gn,v : rate
vch 4 Okt 2005 10:43
17.2 Method
413
of direct generation of VC’s containing n vacancies. In the diffusion-limited regime the rate, at which vacancies (SIA’s) are absorbed for spherical VC’s, is given by
βn,v(i) =
4πRn Dv(i) Cv(i) , Ωa
(17.9)
where Dv(i) is the diffusivity of vacancies (SIA’s), and Cv(i) is the number density of free vacancies (SIA’s). The evolution of Cv(i) with time is given by the integration of the kinetic equations [11] dCv(i) 4πr (Dv + Di ) Cv Ci − Dv(i) Cv(i) Sv(i) . = Gv(i) − dt Ωa
(17.10)
Here r is the recombination or trap radius, Gv(i) is the production rate of free vacancies (SIA’s), and Sv(i) is the sink strength for vacancies (SIA’s). The system of rate equations, Eqs. (17.8) for n > 1, is completed by the separate equation for n = 1, Eq. (17.10), to obtain a full set of equations. Tab. 17.3 Irradiation conditions and results from SANS experiments for the low copper alloy A [13]
Irradiation condition Neutron dose rate, 10−9 dpa/s Mean radius of VC’s, nm Volume fraction of VC’s, %
A1 0.4 1.0 ± 0.1 0.02 ± 0.005
A2 3 1.0 ± 0.1 0.10 ± 0.01
In Eq. (17.10), it is assumed that the rate of emission of vacancies from VC’s is much less than the production rate. Sv(i) is taken to be proportional to the dislocation density, ρ, [11] with a factor of proportionality 1.2 for vacancies (SIA’s) [9]. Gv(i) is taken to be proportional to the neutron dose rate expressed in units of dpa/s (dpa = displacements per atom) with a factor of proportionality of 0.183 for both vacancies and SIA’s. Taking into account that the recombination term in Eq. (17.10) for SIA’s is much less than the sink term, Eq. (17.10) can be solved analytically [3]. Odette [11] used the stationary values of Cv and Ci obtained from dCv(i) /dt = 0 instead of the time-dependent solution of Eq. (17.10).
vch 4 Okt 2005 10:43
414
17 Cluster Evolution in Neutron Irradiated Reactor Steels
17.2.3
Computations
The maximum considered number of vacancies in a cluster corresponds to the number of equations in the system, Eq. (17.8). We have chosen nmax = 9000 in agreement with the maximum VC radius of about 3 nm observed in SANS experiments on the low-copper Fe-based model alloy A [12]. The values of the material parameters used in the calculations are listed in Table 17.1. These values have been adopted from Odette [11]. The direct generation rates of VC’s, listed in Table 17.2, have been adopted from MD calculations performed by Jumel as cited by Christien & Barbu [9]. The program D02EJF (NAG Fortran Library Manual) has been applied to solve the master equations. It integrates a stiff system of first-order ordinary differential equations using a variable order, variable step method implementing the backward differential formulae. We have observed that the mean radius of the calculated VC distribution strongly depends on the dislocation density, ρ. Furthermore, the total volume fraction of VC’s turned out to depend strongly on the surface energy, γ. Therefore, these quantities have been taken as fit parameters in order to reproduce the mean radii and volume fractions observed by means of SANS experiments for the irradiation conditions of the low-copper Fe-based model alloy A (Table 17.3) [13].
17.2.4
Cluster-Matrix Interactions
Cluster-matrix interaction is considered by introducing an additional term, W (n), into Eq. (17.3). ∆G (n) = −n(∆µ)i + 4πRn2 γ + W (n) .
(17.11)
The meaning of the terms on the right-hand side of Eq. (17.11) is schematically depicted in Fig. 17.1. The first, second and third terms correspond to missing atoms (number of vacancies in a VC), missing bonds of the atoms at the cluster matrix interface, and relaxation of the adjacent lattice, respectively. Using this expression, the binding energy, EbV (n) = Ef + ∆G (n − 1) − ∆G (n)
(17.12)
gets the general form EbV
(n) = Ef − x n
vch 4 Okt 2005 10:43
2/3
− (n − 1)
2/3
+y ,
x = 2.5
8π 3
1/3 a20 γ ,
(17.13)
17.2 Method
a)
b)
c)
d)
415
Fig. 17.1 The vacancy formation energy consists of three parts: that of the missing atom (a), that of the missing bonds of the four (eight in the bcc lattice) nearest neighbours (b), and that due to relaxation of the adjacent lattice (c). The metric for clustering is the energy difference between the formation energy of supercells containing non-interacting single vacancies (d) and a supercell of equal total number of atoms containing a cluster
vch 4 Okt 2005 10:43
416
17 Cluster Evolution in Neutron Irradiated Reactor Steels
where a0 is the lattice parameter of iron and y = [ln (Cv ) − 3] kB T − [W (n) − W (n − 1)] .
365
0.0012
Size Distribution Function, nm-1
(17.14)
Rate theory model: Time in days
0.001
300 240
0.0008
SANS measurement 289 days
0.0006 180 0.0004
120 60
0.0002 1 0 0
0.5
1
1.5
2
2.5
3
Cluster Radius R, nm
Fig. 17.2 Calculated (with interaction term neglected) evolution of cluster size distribution for a dose rate of 4 · 10−10 dpa/s and comparison with SANS-based experimental results
Fitting results of MD calculations by Soneda & Diaz de la Rubia [7] with Eqs. (17.13) and (17.14), we obtained x = 2.79 eV and y = 0.11 eV. This fit is as least as good as the one-parameter fit (Eq. (17.13) with y set equal to 0) as done in Ref. [7], which gave x = 2.59 eV. The term W (n) − W (n − 1) to be introduced into Eq. (17.1) was calculated from Eqs. (17.13) and (17.14) using the value obtained for y. The remaining procedure corresponds to the one described above. It is easy to find that the implementation of the term W (n) into Eq. (17.11) corresponds to the determination of ∆µ or the real solution model (∆µr ) ∆µr = ∆µI + kB T (3 − ln(Cv )) ,
vch 4 Okt 2005 10:43
∆µr = Ef −
a20 γ
3 8πn
1/3 .
(17.15)
17.3 Results
17.3
Results
17.3.1
The Ideal Solution Model (W (n) = 0)
417
In a sensitivity study the effect of several material parameters on the size distribution function (SDF) of VC’s has been investigated. Depending on the value of the dimensionless quantity c, c = Ci Di /Cv Dv , the results are essentially different. If c > 1, the calculated SDF has only one peak at a radius of about 0.3 nm. This peak corresponds to the direct generation of VC’s as a result of the cascade stage (Table 17.2). If c < 1, a second peak appears at a radius greater than 0.3 nm. The radius of the second peak is an increasing function of irradiation time. The SDF calculated for the same set of material parameters but three different irradiation times is shown in Fig. 17.2. After one year of irradiation, the volume fraction that corresponds to the second peak is much larger than the volume fraction that corresponds to the first peak. We have also observed that, after a certain irradiation time, the mean radius of VC’s is mainly determined by the dislocation density, whereas the volume fraction of VC’s is mainly influenced by the surface energy. The experimental values of the mean radius (Table 17.3) are reproduced best, if a dislocation density, ρ = 1.84 · 1014 m−2 , is assumed. For irradiation conditions A1 and A2 (Table 17.3), best fit with respect to the volume fraction is obtained for values of the surface energy, γ = 0.82 J/m2 and γ = 0.67 J/m2 , respectively.
17.3.2
Account of Cluster-Matrix Interaction, W (n)
The behaviour of the model with cluster-matrix interaction taken into account is similar as described above (Fig. 17.2). In this case, however, best fit with respect to the volume fraction is obtained for values of the surface energy, γ = 1.94 J/m2 (condition A1) and γ = 2.02 J/m2 (condition A2).
17.4
Discussion
The first peak of the calculated SDF of VC’s (at about 0.3 nm) cannot be resolved by means of SANS, because measuring errors strongly increase for radii below 0.5 nm. Experimental evidence is, however, available from positron annihilation measurements reported by Cumblidge et al. [14] for the same alloy. The second peak of the calculated SDF corresponds to the experimental results obtained from
vch 4 Okt 2005 10:43
418
17 Cluster Evolution in Neutron Irradiated Reactor Steels
SANS measurements. As a variation of the irradiation time in model calculations shows, the mean radius of VC’s increases according to a t1/5 -law, which is slower than the prediction by the Lifshitz & Slyozov theory [15] (resulting in ∼t1/3 ). On one hand, the view that irradiation-induced clusters grow only slowly or not at all after long-term neutron-irradiation under in-service conditions of an RPV is generally accepted. On the other hand, due to the restrictions to vary irradiation time in real irradiation experiments it is rather impossible to establish a quantitative growth law by experimental means. Clearly, this is an argument in support of modelling (but not against experiment). The values of the dislocation density and the surface energy obtained by calibration of the model with experimental data are reasonable estimates (compare Table 17.1). In principle, the difference in the surface energy obtained for the two irradiation conditions, A1 and A2, may have two explanations. Firstly, there may be a discrepancy between model and experiment. For instance, the true nature of the irradiation-induced defect clusters might be slightly different for the irradiation conditions considered. Secondly, the model itself may be incomplete. In this respect, the closer agreement of the surface energies obtained in the second set of calculations indicates both that consideration of cluster-matrix interaction is important and that the method is appropriate.
17.5
Conclusions
The present model of vacancy cluster evolution in bcc iron under neutron irradiation, which is based on rate theory according to Ref. [11], can be adjusted with respect to volume fraction and mean radius of vacancy clusters obtained by means of SANS measurements on a low-copper iron alloy. The values of the specific surface energy of clusters and the dislocation density taken to adjust the model are both reasonable. The expectation that the surface energies for the two dose rates considered should agree is best fulfilled, if the real solution model for vacancies in iron is taken into account.
17.6
References
1. H. Ulbricht, J. Schmelzer, and R. Mahnke, F. Schweitzer, Thermodynamics of Finite Systems and the Kinetics of First-Order Phase Transitions (Teubner, Leipzig, 1988). 2. J. Schmelzer, G. R¨ opke, and R. Mahnke, Aggregation Phenomena in Complex Systems (Wiley-VCH, Berlin, 1999, p. 459 ff.).
vch 4 Okt 2005 10:43
17.6 References
419
3. A. Gokhman and J. B¨ ohmert, Rad. Eff. & Def. in Solids 158 499 (2003). 4. L. E. Murr, Interfacial Phenomena in Metals and Alloys (Addison-Wesley, New York, 1975, p. 377). 5. J. W. P. Schmelzer: Comments on Curvature Dependent Surface Tension and Nucleation Theory. In: J. W. P. Schmelzer, G. R¨ opke, V. B. Priezzhev, Nucleation Theory and Applications (Joint Institute for Nuclear Research Publishing House, Dubna, Russia, 1999). 6. F. Soisson, G. Martin, and A. Barbu, Annales de Physique, Colloque C3, 13 (1995). 7. N. Sonneda and T. Diaz de la Rubia, Phil. Magazine A 78, 995 (1998). 8. S. I. Golubov, A. Serra, Yu. N. Osetsky, and A. V. Barashev, Journal of Nuclear Materials 227, 113 (2000). 9. F. Christien and A. Barbu, Journal of Nuclear Materials 324, 90 (2004). 10. A. Gokhman, J. B¨ ohmert, and A. Ulbricht: Contribution to the Determination of Microstrucural Parameters from Small Angle Scattering Experiments at Reactor Pressure Vessel Steels, FZR-288, February 2000, p. 29. 11. G. R. Odette: Modelling Irradiation Embrittlement In Reactor Pressure Vessel Steels. M. Davies (Ed.), Neutron Irradiation Effect in Reactor Pressure Vessel Steels and Weldments (Vienna, 1998, p.438-504). 12. A. Ulbricht and F. Bergner: Detecting Irradiation Induced Damage in RPV Steels by SANS. In: Proceedings of the 30th MPA-Seminar in conjunction with 9th GermanJapanese Seminar ’Safety and Reliability in Energy Technology (MPA Stuttgart, 2004, Vol. 1, pp. 10.1-10.13). 13. J. B¨ohmert, A. Ulbricht, A. Kryukov, Y. Nikolaev, and D. Erak: Composition effects on the radiation embrittlement of iron alloys. In: Effects of Radiation on Materials: 20th International Symposium, ASTM STP 1405, S.T. Rosinski, M. L. Grossbeck, T. R. Allen, and A.S. Kumar (Eds.) (American Society of Testing and Materials, West Conshohocken, PA, pgs. 383-398). 14. S. E. Cumblidge, A. T. Motta, G. L. Catchen, G. Brauer, and J. B¨ ohmert, Journal of Nuclear Materials 320, 245 (2003). 15. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961).
vch 4 Okt 2005 10:43
18
Statistical Theory of Electrolytic Skin Effects Rainer Feistel(1) and Werner Ebeling(2) (1)
Leibniz-Institut f¨ ur Ostseeforschung, Seestr. 15, D-18119 Rostock-Warnem¨ unde, Germany (2)
Institut f¨ ur Physik, Humboldt-Universit¨ at,
Newtonstr. 15, 12489 Berlin, Germany Zwei Dinge sind zu unserer Arbeit n¨ otig: Unerm¨ udliche Ausdauer und die Bereitschaft, etwas, in das man viel Zeit und Arbeit gesteckt hat, wieder wegzuwerfen. Albert Einstein
Abstract Inspired by Ebeling’s Electrolyte Phase Transition, which hypothetically may possess a two-phase region in the (pV )-diagram with different degrees of dissociation of the dissolved salt on both sides of the interface, we derive the statistical expression for the relaxation force at the boundary of a dilute electrolyte. The analogies and differences to the classical theories of Onsager-Samaras and Buff-Stillinger for the surface tension of aqueous electrolytes are discussed. Suitable integral hierarchy equations for reduced distribution functions are briefly derived, and the solution for the pair distribution functions is calculated analytically by means of Fourier, Hilbert, and Hankel integral transforms.
18.1 Introduction
18.1
421
Introduction
The current study on the statistical properties of a dilute electrolyte in the neighbourhood of an interface, which is preventing the ions from diffusion into the pure solvent behind it, was inspired by the Electrolyte Phase Transition. Even though both phenomena are not immediately related, we will briefly introduce the theory of this phase transition in the following, and explain their potential connection. The Electrolyte Phase Transition was discovered theoretically by Ebeling (1971). The equation of state of dilute, associating electrolytes exhibits a critical point, and a region of thermodynamic instability in the (pV )-diagram. Since its theoretical-statistical description is available in analytical form, it may provide new insights into critical behaviour in general. This aspect was emphasised later by Fisher and Levin (1993), and Aqua and Fisher (2004). It may contribute to the ongoing discussion about the effect of gravity on the critical behaviour of fluids (Anisimov (1991), Wagner et al. (1992), Kurzeja et al. (1999), Ivanov (2003), Skripov and Ivanov (2004)), since ionic criticality is unlikely to be influenced by the symmetry-breaking gravity force, which is suspected to be responsible for the apparent contradictions between theory and experimental findings. The remarkable contrast between the universality claim of the renormalisation group theory on the one hand (Sengers and Levelt Sengers (1986), Anisimov (1991)), and the opposite universality claim of the catastrophe theory on the other hand (Thom (1975), Poston and Stuart (1980)), is still a scientific challenge. Since the statistical theory of the electrolytic surface tension is easier treatable than that of the liquid-gas interface, the current study is mainly devoted to further progress into this specific direction. We consider the dissociation equilibrium of a symmetrical electrolyte, between the neutral molecule AC and its anion A− and cation C+ , i.e., AC ←→ A− + C+ .
(18.1)
K(T )
The equilibrium constant is defined by the cut-off condition (Falkenhagen et al. (1971)), d r 2 dr exp (−l/r) ≈ 4πa3
K (T ) = 4π
exp (b) b
for
b 1.
(18.2)
a
vch 4 Okt 2005 10:43
422
18 Statistical Theory of Electrolytic Skin Effects
Here, the Landau length is l = q 2 / (4πεε0 kB T ), a is the ion contact distance, q the ion charge, b = l/a the Bjerrum parameter, and d = l/2 the association distance. Of course, in Eq. (18.2), d > a is supposed, or b > 2. We note that in the strict theory [10] the mass action constant is given by a modified expression which, however, is asymptotically identical with Eq. (18.2). The mass action law for the neutral (nAC ) and ion (n± = n+ = n− ) particle densities K (T ) =
nAC fAC n − n± = 2 2 n+ f+ n− f− n± f±
(18.3)
can be solved for the conserved total particle density, n, n = nAC + n+ = nAC + n− = nAC +
1 (κa)2 , 8πa3 b
(18.4)
using the Debye parameter, κ2 = 8πl · n± , and the activity coefficients (Falkenhagen and Ebeling (1971)), ln f± = −
κl , 2 (1 + κa)
ln fAC = 0 ,
(18.5)
resulting in the total particle density expressed by the Debye and the Bjerrum parameter as 1 (κa)2 (κa)2 b 1 1+ . (18.6) exp n= = v 8πa3 b 2b2 1 + κa This formula is to be used in the expression for the osmotic pressure (Falkenhagen and Ebeling (1971)) 1 κa p = nAC + n+ + n− − − κa 2 ln (1 + κa) − kB T 8πa3 1 + κa , κa (κa)2 1 + 2 ln (1 + κa) − − κa =n+ 8πa3 b 1 + κa 3 3 κ2 2 − κa + (κa) + . . . . =n+ 24πa b 2
(18.7)
With κa as running dummy parameter, the partial volume, ν (κa) (Eq. (18.6)), can be plotted versus the osmotic pressure, p (κa) (Eq. (18.7)), as shown in
vch 4 Okt 2005 10:43
18.1 Introduction
423
p b =16.2 b =16 b =15.8
Fig. 18.1 Critical behaviour at the Electrolyte Phase Transition. Shown are osmotic pressure, p, versus partial volumes, v, for different Bjerrum parameters, b
Fig. 18.1 for the critical region and selected values of b. The critical Bjerrum parameter of Eq. (18.7) is bc = 16, found at the Debye radius (1/κc ) = a, and the corresponding critical temperature is Tc =
q2 . 64πεε0 kB
(18.8)
For T < Tc , we observe van der Waals’s wiggles in the osmotic pressure curves. In this region, the electrolyte divides into two regions with different ionic concentrations. In the limit T Tc , one of the phases is a neutral salt solution and the other one is a fully ionized electrolyte. This situation is the motivation for the studies in the subsequent sections, since only little is known about the behavior of the electrolyte in the instability region, (∂v/∂p)T > 0. Although the behaviour resembles very much the van der Waals equation, note that the instability region appears at supercritical pressures, however. If there is a stable spatial separation of phases, their thermodynamic and statistical properties at the interface are of substantial theoretical interest. In Section 18.2 we briefly introduce the usual approach to the electrolytic surface tension based on the image force method, and why our problem - possessing
vch 4 Okt 2005 10:43
424
18 Statistical Theory of Electrolytic Skin Effects
a homogeneous dielectric solvent background - is distinct. In Section 18.3, we introduce the required integral hierarchy equations, specialized for the case of ions confined to a certain spatial region, as the starting point of the statistical theory. In Section 18.4, the analytic solution for the pair distribution function is derived, using the integral transforms of Fourier, Hilbert and Hankel, as an alternative to the ”classical” mirror image method. That solution is the very aim of this paper.
18.2
Electrolytic Skin Effects
The physical problem we are going to treat in this article is sketched in Fig. 18.2. We imagine a membrane separating a half-space, containing a dilute electrolyte,
Electrolyte
Pure Solvent
Relaxation Force
Debye Cloud
Fig. 18.2 Forces onto an ion near the interface between electrolyte and pure solvent. Due to the cut-off part of the ion cloud, the cloud charge centre is displaced from the central ion position, causing a relaxation force repelling the ion from the surface
from the other half-space, filled with the pure solvent. The ions are confined to the half-space of the electrolyte by, say, an idealised thin non-conducting membrane, not influencing the electrostatic fields of the ions. Ions located farther away from the interface than the radius of the Debye screening cloud do not recognise the existence of the boundary; their distribution is the
vch 4 Okt 2005 10:43
18.2 Electrolytic Skin Effects
425
same as for a homogeneous electrolyte. Ions near the surface, however, extend their unscreened Coulomb field into the other half-space and miss the oppositely charged part of the cloud behind the barrier. This causes the charge centre of the cloud to be displaced off the interface, and a resulting relaxation force pulling the ion away from the surface. Thus, the ion concentration near the surface will be lowered until the diffusion force against the density gradient will balance the electrostatic force.
Water
Air
Image Force
Debye Cloud
Image Cloud
Fig. 18.3 Forces onto an ion near the surface between an aqueous electrolyte and air. Due to the image charge, resulting from the difference between the dielectric constants of water and air, the image force is repelling the ion from the surface
It is helpful to consider here the similar case of electrolytic surface tension at the water-air interface, which is studied extensively in the literature. The physical situation is sketched in Fig. 18.3. Considering the electrostatic problem of a point charge in a discontinuous dielectric, forces onto the charge appear trying to move it into a position with lowest potential energy of the polarisation field. The case of a planar interface is analytically solvable and exactly corresponds to the existence of a virtual image charge behind the surface. This image charge is proportional to the difference between the dielectric constants of both media, q (im) = q
(εH2 O − εair ) . (εH2 O + εair )
(18.9)
vch 4 Okt 2005 10:43
426
18 Statistical Theory of Electrolytic Skin Effects
It becomes zero if both media have the same dielectric properties, and changes its sign if the external medium has a higher dielectric constant than that of the internal solvent. Note that similar closed solutions for this problem are not found for other physically very interesting but non-planar interfaces like bubbles or droplets (Landau and Lifschitz (1967)). An overview over known such analytical solutions can be found in Grinberg (1948) or Smythe (1950). The limiting law of the surface tension of a dilute electrolyte based on the imageforce models was derived by Wagner (1924), later corrected by Onsager and Samaras (1934). Their result was confirmed and generalised by several authors in the following. Buff and Stillinger (1956) re-derived it starting from a strictly statistical approach; we will briefly consider their paper below again because it is methodically closely related to the calculations performed in this paper. Nakamura et al. (1982) improved the Onsager-Samaras theory by a self-consistent approach. Bhuiyan et al. (1991) have extended the Buff-Stillinger method to high concentrations, using a modified Poisson-Boltzmann approximation. For the same purpose, Li et al. (1999) propose a one-dimensional box model of the interface and the application of Pitzer functions for the ion activities. A similar description, but based on a modified mean spherical approximation for the osmotic coefficient, is given by Yu et al. (2000). Levin and Flores-Mena (2001) replace the grandcanonical formulation for the computation of surface tension of Onsager-Samaras by a simpler, canonical one. Hu and Lee (2004), in distinction to Li et al. (1999) or Yu et al. (2000), propose the use of Patwardhan-Kumar expressions for the activity coefficients at higher concentrations. Buff and Stillinger (1956) apply a modified form of the Kirkwood integral equation, leading to an integral equation for the pair distribution function Fab (r1 , r2 ) = 1 + µza zb f (r1 , r2 ) ,
(18.10)
describing the probability density of finding one ion ”a” with valence number za at position r1 and another ”b” at r2 : f (r1 , r2 ) +
1 1 κ2 + (im) + r12 r 4π 12
f (r2 , r3 ) dr3 = 0 r13
(18.11)
whole space
Here, the distance r12 between the particles 1 and 2 enters into the (scaled) Coulomb potential 1/r between the particles. By including their images, denoted
vch 4 Okt 2005 10:43
18.3 BBGKY-Hierarchy Equations
427
by superscript “(im)”, the integral is extended to the whole space and solved by those authors using Fourier transform, resulting in a superposition of two Debye distributions,
f (r1 , r2 ) = −
exp (−κr12 ) − r12
(im) exp −κr12 (im)
.
(18.12)
r12
The Debye parameter κ (the reciprocal radius of the ion cloud) and the plasma parameter µ used here are defined in Section 18.4 in which our corresponding solution will be derived, obtained by a statistical approach similar to that of Buff and Stillinger. We emphasise, however, that the Buff-Stillinger method briefly recalled here uses strictly equal expressions for the direct and the image force, which in the case of water-air surfaces has in fact a ratio of (80 − 1) (εH2 O − εair ) ≈ ≈ 0.98 (εH2 O + εair ) (80 + 1)
(18.13)
but not exactly 1. If the difference between the dielectric constants on both sides of the interface vanishes, however, the image force disappears, and Eqs. (18.11) and (18.12) reduce to the normal Debye-H¨ uckel bulk theory, without any surface effects. Thus, the relaxation effect we are going to consider in the following chapters is completely neglected in the Buff-Stillinger theory.
18.3
BBGKY-Hierarchy Equations
The systematic statistical theory of electrolytes is based on the BogolyubovBorn-Green-Kirkwood-Yvon (BBGKY) hierarchy of coupled integro-differential equations for the molecular unary, binary etc. distribution functions Fa (r1 ), Fab (r1 , r2 ), . . . (see Falkenhagen et al. (1971)), whom we are following in this section. The s-particle function Fs is defined as Vs Fs (r1 ...rs ) = QN
exp (−βUN ) drs+1 . . . drN .
(18.14)
$ Here, QN = exp (−βUN ) dr1 ...drN is the so-called configuration integral, β = 1/kB T , and V is the volume. The N -particle mean interaction potential UN is
vch 4 Okt 2005 10:43
428
18 Statistical Theory of Electrolytic Skin Effects
supposed here to consist of pair and single-particle contributions, the latter in distinction to other derivations found for only homogeneous systems:
βUN =
N i=1
N 1 Ψi (ri ) + ψij (ri , rj ) . 2
(18.15)
i,j=1
Taking the derivative with respect to the first coordinate, we have N ∂Us ∂ ∂ UN = β + ψij (r1 , rj ) β ∂r1 ∂r1 ∂r1
(18.16)
j=s+1
with the s-particle potential, βUs =
s
Ψi (ri ) +
i=1
ψij (ri , rj )
(18.17)
1≤i 0 coincides with f (r), and vanishes for z < 0. After these preliminary remarks, we rewrite now Eq. (18.38) in the form of an integral over the whole space, 1 dx3 [− (1 + sgn (z3 )) h (x3 ) (18.48) (1 + sgn (z1 )) h (x1 ) = 4π |x1 − x3 | ¯ Ω+Ω
+ (1 − sgn (z3 )) g (x3 )] , multiply with exp (ikx1 ), and integrate x1 over the whole space, with subscript ”+” denoting the half-space containing the ions: 1 ˜ dx1 exp (ikx1 ) (1 + sgn (z1 )) h (x1 ) (18.49) h+ (k) ≡ 2 ¯ Ω+Ω 1 dx3 exp (ikx3 ) [− (1 + sgn (z3 )) h (x3 ) + (1 − sgn (z3 )) g (x3 )] = 2 2k ¯ Ω+Ω
1 ˜ (k) + g ˜ (k) . ≡ 2 −h + − k
vch 4 Okt 2005 10:43
18.4 Analytical Solution for the Pair Distribution Function
435
˜ + (k): This way, we can easily solve this equation for h g˜− (k) ˜ . h+ (k) = 1 + k2
(18.50)
Thus, we have to calculate the right-hand side function g˜− (k), and then to per˜ + (k). First, we compute g˜ (k) by Fourier transform the backward transform of h form, exp (ikx2 ) exp (− |x1 − x2 |) = −4π (18.51) g˜ (k) = − dx1 exp (ikx1 ) |x1 − x2 | 1 + k2 and apply to the result the Hilbert transform, i.e. the operator P− , Eq. (18.47): ! " ! " (18.52) g˜− (k) = −2π dk δ kx − kx δ ky − ky × ! " exp (ik x2 ) i × δ kz − kz − π (kz − kz ) 1 + k2 exp (ikx2 ) = −2π + 2i exp (ikx x2 + iky y2 ) × 1 + k2 , z ) exp (ik 2 z ! " . × dkz (kz − kz ) 1 + kx2 + ky2 + kz2 Inserting g˜− (k) into Eq. (18.50) yields exp (ikx2 ) exp (ikx x2 + iky y2 ) ˜ × h+ (k) = −2π 2 + 2i 2 (1 + k2 ) (1 + k ) , exp (ikz z2 ) ! " . × dkz (kz − kz ) 1 + kx2 + ky2 + kz2
(18.53)
The backward Fourier transform leads to the 4-dimensional integral 1 ˜ + (k) (18.54) dk exp (−ikx1 ) h h (x1 , x2 ) = (2π)3 exp (ik (x2 − x1 )) 1 = − 2 dk 4π (1 + k2 )2 , − k )z ) i exp (ik (x2 − x1 )) exp (i (k z 2 ! z " . dkz + 3 dk 4π (1 + k2 ) (kz − kz ) 1 + kx2 + ky2 + kz2
vch 4 Okt 2005 10:43
436
18 Statistical Theory of Electrolytic Skin Effects
The required calculation using known standard techniques and integral tables (Abramowitz and Stegun (1965), Brychkov and Prudnikov (1977), Prudnikov et al. (1981)) is somewhat lengthy and eventually results in exp (− |x2 − x1 |) 1 + h (x1 , x2 ) = − 4 4
∞ [(1 + |z2 − z1 | t) exp (− |z2 − z1 | t) 1
dt
(18.55) − exp (− (z2 + z1 ) t)] J0 ρ t2 − 1 2 . t Here, J0 is the Bessel function, and ρ = (x2 − x1 )2 + (y2 − y1 )2 is the particle distance parallel to the interface, scaled to the Debye radius, in cylinder coordinates. Referring to tables of Hankel transforms (Bateman and Erdelyi (1954), Wheelon (1968), Oberhettinger (1972)), we mention the mathematical identity ∞
1 + β y 2 + α2 exp −β y 2 + α2 J0 (xy)
0
=
ydy (y 2 + α2 )3/2
1 exp −α x2 + β 2 . α
= (18.56)
This very special but physically important Hankel transform was used in similar form already by Sommerfeld (1909); its proof is found e.g. in Watson (1995). Its application to Eq. (18.55) shows that the bulk terms, i.e. those independent of the distance from the surface, cancel each other completely as was expected for physical reasons. Then, the solution simplifies significantly to only a surface correction of the Debye distribution: 1 h (x1 , x2 ) = − 4
∞
dt J0 ρ t2 − 1 exp (− (z2 + z1 ) t) 2 . t
(18.57)
1
Making use of Eq. (18.56) again in a modified form, we can further reformulate Eq. (18.57) into the simpler integral 1 h (x1 , x2 ) = − 4
∞ 1
vch 4 Okt 2005 10:43
2 2 dt 2 exp − ρ + (z2 + z1 ) t . t2
(18.58)
18.4 Analytical Solution for the Pair Distribution Function
437
An explicitly evaluated analytical form of this integral is not known to the authors. For two ions on the surface itself, z1 = z2 = 0, the special case solution is simply h (x1 , x2 )|surface = −
exp (−ρ) . 4
(18.59)
Another integrable, interesting special case is ρ = 0, i.e. two ions located on a line normal to the interface, at different distances.
h (x1 , x2 )|normal
1 =− 4
∞ exp (− (z2 + z1 ) t)
dt t2
(18.60)
1
1 = − [exp (−z2 − z1 ) + (z2 + z1 ) Ei (−z2 − z1 )] . 4 It has a constant limit of −1/4 at the surface with first deviations in the order of (z1 + z2 ) ln (z1 + z2 ), and an exponential decay towards the bulk. To summarise, the generalized Debye law for the binary distribution function of a dilute electrolyte at a planar interface, impenetrable for ions, is, after returning to usual, unscaled length units
exp (−κr12 ) + (18.61) r12 ⎫ ⎬ ∞ ! " dt κ exp −κ ρ212 + (z2 + z1 )2 t2 2 + O µ2 . + 4 t ⎭
fab (r1 , r2 ) = 1 − za zb l
1
This is the final result we have been looking for.
Acknowledgements The authors thank Wolfgang Wagner for providing his experimental results on the critical behaviour of fluids under gravity, and Michail A. Anisimov for helpful discussions about this issue. They are grateful to J¨ urn W. P. Schmelzer for offering the opportunity to present this study on the Nucleation Workshop 2005 in Dubna, and his assistance in preparing a printable version. The authors further thank A. Schr¨ oder and B. Sievert for their permanent help in accessing special literature.
vch 4 Okt 2005 10:43
438
18 Statistical Theory of Electrolytic Skin Effects
18.5
References
1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1965). 2. M. A. Anisimov, Critical Phenomena in Liquids and Liquid Crystals (Gordon and Breach Science Publishers, Philadelphia, Reading, Paris, Montreux, Tokyo, Melbourne, 1991). 3. J.-N. Aqua and M. E. Fisher, Phys. Rev. Lett. 92, 135702-1 (2004). 4. H. Bateman and A. Erdelyi, Tables of Integral Transforms, Vol. II. (New York, Toronto, London, McGraw-Hill Book Company, Inc. 1954). 5. L. B. Bhuiyan, D. Bratko, and C. W. Outhwaite, J. Chem. Phys. 95, 336 (1991). 6. Yu. A. Brychkov and A. P. Prudnikov, Integralnye Preobrazovaniya Obobshchonnykh Funktsiy (Nauka, Moskva, 1977). 7. F. P. Buff and F. H. Stillinger Jr., J. Chem. Phys. 25, 312 (1956). 8. B. Buttkus, Spectral Analysis and Filter Theory in Applied Geophysics (SpringerVerlag, Berlin, Heidelberg, New York, 2000). 9. W. Ebeling, Z. Phys. Chem. 247, 340 (1971); W. Ebeling, W. D. Kraeft, and D., Kremp, Theory of Bound States and Ionization Equilibria (Akademie-Verlag, Berlin, 1976). 10. H. Falkenhagen and W. Ebeling: Equilibrium Properties of Ionized Dilute Electrolytes. In: S. Petrucci, Ionic Interactions (Academic Press, New York, London 1971, pgs. 2-59). 11. H. Falkenhagen, W. Ebeling, and H. G. Hertz, Theorie der Elektrolyte (S. Hirzel, Leipzig, 1971). 12. M. E. Fisher and Y. Levin, Phys. Rev. Lett. 71, 3826 (1993). 13. G. A. Grinberg, Izbrannye Voprosy Matematicheskoy Teorii Elektricheskykh i Magnitnykh Yavleniy (Izdatelstvo Akademii Nauk SSSR Moskva, Leningrad, 1948). 14. Y.-F. Hu and H. Lee, J. Coll. Interf. Sci. 269, 442 (2004). 15. D. Yu. Ivanov, Kriticheskoye Povedeniye Neidealizirovannykh System (Fizmatlit., Moskva, 2003). 16. N. Kurzeja, Th. Thielkes, and W. Wagner, Int. J. Thermophys. 20, 531 (1999). 17. L. D. Landau and E. M. Lifschitz, Elektrodynamik der Kontinua; Lehrbuch der Theoretischen Physik, Bd. VIII (Akademie-Verlag, Berlin, 1967). 18. Y. Levin and J. E. Flores-Mena, Europhys. Lett. 56, 187 (2001). 19. Z.-B. Li, Y.-G. Li, and J.-F. Lu, Ind. Eng. Chem. Res. 38, 1133 (1999).
vch 4 Okt 2005 10:43
18.5 References
439
20. T. Nakamura, T. Tanaka, and Y. Izumitani, J. Phys. Soc. Japan 51, 2271 (1982). 21. A. D. Poularikas, The Transforms and Applications Handbook (CRC Press Inc., Boca Raton, 1996). 22. T. Poston and I. Stuart, Teoriya Katastrof i Yeyo Prilozheniya (Mir, Moskva, 1980). 23. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integraly i Ryady (Nauka, Moskva, 1981). 24. F. Oberhettinger, Tables of Bessel Transforms (Springer-Verlag, Berlin, Heidelberg, New York, 1972). 25. L. Onsager and N. T. Samaras, J. Chem. Phys. 2, 528 (1934). 26. J. V. Sengers and J. M. H. Levelt Sengers, Ann. Rev. Phys. Chem. 37, 189 (1986). 27. V. P. Skripov and D. Yu. Ivanov: On the Second Crossover Near to the Critical Point. Lecture on the 8 Research Workshop on Nucleation Theory and Applications, Dubna, Russia, October 2004. 28. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill Book Company, Inc., New York, Toronto, London, 1950). 29. A. Sommerfeld, Ann. Physik Chemie 28, 682 (1909). 30. R. Thom, Structural Stability and Morphogenesis (W. A. Benjamin, Inc., Reading, Massachusetts, 1975). 31. C. Wagner, Phys. Z. 25, 474 (1924). 32. W. Wagner, N. Kurzeja, and B. Pieperbeck, Fluid Phase Equil. 79, 151 (1992). 33. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1995). 34. A. D. Wheelon, Tables of Summable Series and Integrals involving Bessel Functions (Holden-Day, San Francisco, Cambridge, London, Amsterdam, 1968). 35. Y.-X. Yu, G.-H. Gao, and Y.-G. Li, Fluid Phase Equil. 173, 23 (2000).
vch 4 Okt 2005 10:43
19
On the η 4 -Model in the Displacive Limit: Critical Temperature, Specific Heat Maximum and Critical Indices Alexander L. Tseskis Am Weidenhof 29, 51381 Leverkusen, Germany
Ich denke, daß das Problem der kritischen Exponenten soviel Aufmerksamkeit erregt hat, weil die Physiker glaubten, es werde wahrscheinlich eine sch¨ one L¨ osung haben . . . Auf eine sch¨ one L¨ osung deutet vor allem die Universalit¨ at des Ph¨ anomens hin . . . aber auch die Tatsache, daß Physiker immer wieder entdeckt und sich daran gew¨ ohnt haben, daß die wesentlichsten Eigenschaften physikalischer Ph¨ anomene sich in Gesetzen niederschlagen, die physikalische Gr¨ oßen mit Potenzen anderer Gr¨ oßen verkn¨ upfen . . . Im Unterschied dazu ist die Berechnung der pr¨ azisen Temperaturen von Phasen¨ uberg¨ angen ein vertracktes Problem . . . und deshalb wird es entweder studiert, weil es von praktischer Bedeutung ist, oder weil man nichts Besseres zu tun hat. Steven Weinberg Now, you see that you do not see anything. And why you see nothing, you will see, now. Physicist’s joke
19.1 Introduction
441
Abstract The so-called displacive limit for the continuous η 4 -model is considered. In this approach, the transformation from the initial to the Landau Effective Hamiltonian (LEH) together with the definition of the critical temperature are realized. This result can be obtained via an integration over the Fourier components with large wave numbers so that only components with small wave numbers remain in LEH. The approach also allows us to analyze the behavior of the specific heat and to define critical indices for the case under consideration.
19.1
Introduction
It is usually suggested that an universal description of second order phase transitions can be based on the continuous η 4 -model [1] with the Hamiltonian H=
−aη 2 + bη 4 + g (∇η)2 dV
(19.1)
with a > 0, b > 0, and g > 0 and V being the volume of the system. Inserting Eq. (19.1) into the statistical integral, Z=
H Dη , exp − T
(19.2)
where Dη in this continuous (or path [2]) integral denotes the summation over all the distributions of the (scalar) field, η, in the volume V , one believes to define (at least, qualitatively) the macroscopic properties of the system in the vicinity of the transition temperature, Tc . Such expectations originate from the fact that Eq. (19.1) generalizes the so-called lattice model [3], Hl =
=
2 2 2 2 dηr J/2 [η (r) − η (r + q)] + λ η (r) − η0 . q,r
(19.3)
r
Here r are the coordinates of the cubic lattice and q represents its shortest period, so that only the nearest neighbor interaction is taken into account; J and λ are constants. At λ → ∞ and finite J, the variables η(r) adopt the values ±η0 and Eq. (19.3) becomes isomorphic to the Ising model. Namely this feature was considered for a long time to provide the model with the Hamiltonian defined by
vch 4 Okt 2005 10:43
442
19 On the η 4 -Model in the Displacive Limit
Eq. (19.1) belonging to the same universality class as the Ising model. It turns out, however, that the statement fails at the opposite condition, when J → ∞ at finite λ which presents the so-called displacive limit (see for example [4]). In this limit, the system displays properties close to the ones of a continuous model with Landau Effective Hamiltonian (LEH) [5], Hef f =
α ˜ tη 2 + bη 4 + g (∇η)2 dV ,
t = T − Tc
(19.4)
with critical exponents different from those in the Ising universality class. Moreover, according to various R(enormalization) G(roup) and numerical investigations [6, 7] the model Eq. (19.3) exhibits a wide range of power laws ranging from Ising-like to those with seemingly classical (mean-field) properties as J/λ is varied from zero to infinity. It also follows from the results of Monte-Carlo simulations [7] that the quadratic coefficient α ˜ in Eq. (19.4) becomes dependent on t already at values of the parameter J/λ of the order unity which is a strong evidence that the theory with LEH must not be exploited, generally speaking, outside the displacive limit. One knows however that namely LEH is used in one or the other way in the schemes when this condition is not fulfilled. For instance, it happens when one of the so-called irrelevant fields is identified with t in the RG method [1, 4] leading to Ising-like behavior or after splitting - aη 2 from Eq. (19.1) into ˜ t) η 2 as it was done in Ref. 8. In both these α ˜ tη 2 and the counterterm − (a + α cases, LEH is used as some intermediate subsidiary tool which is not sufficient for the description of critical phenomena without the RG method. On the other hand, it was suggested in Ref. 5 that namely LEH as such being inserted into the statistical integral would give a complete description of the phenomena without any additional hypothesis and we shall see below that the corresponding results are different from those for Ising-like systems. What is also important here is that being a powerful heuristic tool LEH by itself should be obtained from the basic one, Eq. (19.1), so that the transition from Eq. (19.1) to Eq. (19.4) has to lead to the definition of, say, α ˜ and Tc . We focus on this problem in this work and show that such a transformation becomes possible namely in the displacive limit. Another difficulty of RG and numerical method realizations originates from the fact that they represent some type of lattice model rather than the continuous one. Belonging to the Ising universality class, such (lattice) models show the same Ising-like behavior of the thermodynamic functions. In particular, it follows immediately from the equality between free energy F and thermodynamic potential
vch 4 Okt 2005 10:43
19.2 Critical Temperature and Landau Effective Hamiltonian (LEH)
443
Φ in Ising-like systems that the specific heats, cp and cv , coincide. Such a result is in strong contradiction to the inequality cv < cp which is always fulfilled in real situations. Moreover, only such physical systems are implied below for which the whole line of the transitions exists on the (p, T )- or (V, T )-plane, leading one to the well-known conclusion according to which cv must be finite while cp can tend to infinity as T → Tc . In order to verify the validity of this requirement in the η 4 -model the variable V is retained in the explicit form in all the calculations below in contrast to numerous approaches in which V is excluded in one or another way reducing the problem to the solution of the >lattice ? model. In this context, we follow Ref. 8 where some averages, for instance η 2n with natural n, were estimated in the high-temperature phase (it is easy to see that specific heat is represented just in terms of such a form). In contrast to Ref. 8, we work just in space of dimension d = 3 and with the low-temperature phase, t < 0, which enables us to find the specific heat maximum for real physical situations. The chapter is organized as follows. In Section 19.2, the conditions at which LEH can be obtained from the basic Hamiltonian in the continuous η 4 -model ˜ are calculated. In Section 19.3 we are formulated and relevant values of Tc and α discuss the problem concerning specific heat and show that cv is finite, as it is has to be. The possible behavior of cp is also discussed. In Section 19.4, critical indices of the continuous η 4 -model in the displacive limit are calculated. In conclusion we touch upon the corresponding problems for the cases of lower and upper critical space dimensions, d =2 and d = 4, respectively.
19.2
Critical Temperature and Landau Effective Hamiltonian (LEH)
We start in this section with the Hamiltonian given by Eq. (19.1) expressed in terms of Fourier components of the field η, H=V
−a
k
ηk2
+b
p,q,r
ηp ηq ηr η−p−q−r + g
k2 ηk2
,
(19.5)
k
is fulfilled, η−k = ηk∗ . where V = L3 and the known condition for the real field η dηk becomes valid. At this form of the Hamiltonian the replacement Dη ⇒ k
Our goal, now, is to integrate in Z with H from Eq. (19.5) over the components
vch 4 Okt 2005 10:43
444
19 On the η 4 -Model in the Displacive Limit
ηk with large k, and we assume here that such a small but finite wave number p exists that a gp2 ,
b gp2 ,
pK,
(19.6)
where K is the upper bound of the spectrum. It is clear that in our notations the inequalities (19.6) present nothing else but the displacive-limit condition. At the first step those ηk will be integrated for which p ≤ k ≤ K holds. In order to do it let us for some time change the variables in the following way, ηk4 V = ξ 4 .
(19.7)
All the calculations must be carried out at the thermodynamic limit, V → ∞, and it is clear that only the terms of the Hamiltonian (−a + gk2 )ξk2 V 1/2 (19.8) k
include the large multiplier V 1/2 . Because of the inequalities (19.6) all these terms are positive and provide the Gaussian integrals appearance in the following calculations. Now taking into account the well-known representation of the δfunction, ! " s exp −s2 x2 , s→∞ π
(19.9)
δ (x) = lim
one can easily see that all ξk with finite k mustbe considered to be small. Really, πT 1/2 (−a + gk2 )V −1/4 with all after multiplying the statistical integral by k
k ≥ p and dividing the integrand by the same quantity one obtains for such components δ(ξk ) in the integrand. The additional factor . . . gives, after taking k
the logarithm, some non-singular term in the free energy. So all these ξk and corresponding ηk terms are small. Denoting the latter ones as η k and returning to Eq. (19.5) one can omit three- and fourth-order terms in η k by virtue of their smallness. It is easy to see that the terms linear in η k can be also omitted as for them the summation takes place in the k-space in the interval [p; 3p] which is small (due to the inequalities (19.6)) in comparison to the whole interval [p; K]. Doing so and remembering that integrals upon η k are Gaussian one can write (we keep the old notation η for the long-wave, k < p, part of the field) H (η) √ −1 Λ , (19.10) Z = Dη exp − T
vch 4 Okt 2005 10:43
19.2 Critical Temperature and Landau Effective Hamiltonian (LEH)
445
where Λ is the determinant of the matrix Λki kj /π of the coefficients at ηki ηkj in Eq. (19.5). Using the relation Det[Λij ] = exp{Sp ln[Λij ]}
(19.11)
and extracting the diagonal matrix [Λii ] one can easily see that the lowest in η order of the rest of the terms is four. Including the factor b2 /gk2 (k > p), these terms give a small correction to the coefficient b in the Hamiltonian and then can be omitted. Neglecting also, as it is usually done, the terms of the orders higher than four one obtains ⎧ ⎫ ⎨ 1 πT ⎬ H (η) ln exp , (19.12) Z = Dη exp − ⎩2 T Ak ⎭ p≤k
where Ak =
!
" 6bη 2 − a + gk2 dV .
(19.13)
After converting the sum in Eq. (19.12) into an integral and taking into account the fact that integration is to be carried out over half of the k−space (because of the condition η−k = ηk∗ ) it reads Z=
⎧ ⎫ K ⎨ 2 1 2πk dk ⎬ Ak H (η) exp − × ln . Dη exp − ⎩ 2 T πT (2π)3 ⎭
(19.14)
p
Expanding the logarithms in Eq. (19.14) and carrying out the integration one can easily see that the terms independent of$ η give corresponding non-singular terms in F (compare [9]). The first order in η 2 dV terms give the correction to the quadratic term in H, that is 3b (K − p) T η 2 dV. (19.15) −a η 2 dV ⇒ −a η 2 dV + 4π 2 g $ As to the term including ( η 2 dV )2 it is apparent that, by virtue of the CauchyG¨ older inequality 2
2
η dV
≤V
η 4 dV ,
(19.16)
vch 4 Okt 2005 10:43
446
19 On the η 4 -Model in the Displacive Limit
it leads to a small correction to b (the correction is small due to the factor a/gpK in this term). Neglecting the latter and higher order terms one can write down from Eq. (19.15) Tc (p) =
4π 2 ag . 3b(K − p)
(19.17)
It is easy to see however that the procedure of the passage from the basic Hamiltonian to the effective one is yet far from complete. Really, defining Tc (p) we took into account components up to small but finite k ≥ p, while at Tc per definition one has to carry out such an integration procedure as far as it is possible [5]. Due $ to the fact that the quadratic term in H (η) now takes the form α ˜ t η 2 dV where α ˜≡α ˜ (p) =
3b(K − p) 4π 2 g
(19.18)
with arbitrary small t = T − Tc (p), one can continue this integration procedure up to wave numbers of order k ≥ V 1/4 as for them k2 V 1/2 → ∞ holds when V → ∞. Repeating the employed above procedure and denoting the upper limit of integration (in the k−space) by κ one has ˜t η 2 dV + b η 4 dV + g (∇η)2 dV (19.19) Hef f = α k 2 ? η0 V exp − const · tη02 V + bη04 V dη0 "E D ! . (19.46) η0 V = $ exp − const · tη02 V + bη04 V dη0 It follows from Eq. (19.46) that it is to be (compare with Eq. (19.25)) > 2 ? η0 V ∝ t−1
(19.47)
so that again γ = 1. As outside the fluctuation region the η 4 -model with LEH transforms to Landau (mean-field) theory it follows automatically that in the non-symmetric phase where η ∝ |t|β the index β is equal to β = 1/2 so that the known relation α + 2β + γ = 2
(19.48)
: is also fulfilled; the same certainly can be obtained calculating the value | ηi | at t → 0. As it was mentioned above, the field η, in the calculations based on the division of the the volume V into cubes of size rc , in each cube is homogeneous. So in order to obtain the index ζ which describes the decrease of the correlation function with a distance r by means G(r) ∝ r −(d−2+ς)
(19.49)
one should, generally speaking, use LEH in its full form including the gradient term. The latter presents a problem which can hardly be resolved. Instead one
vch 4 Okt 2005 10:43
454
19 On the η 4 -Model in the Displacive Limit
>$ 2 3 ? $ η d x inside the cube can identify the integral G (r) r 2 dr with the average of size rc . This way of treating the problem is equivalent to the usual approach leading to the known relation ν (2 − ς) = γ ,
(19.50)
so it gives ζ = 1/2. The rest of the known equations gives the next values of the ”field” indices: ε = 0, µ = 4/9, and δ = 3. As α, β and γ in this case coincide with those in the mean-field theory it is often said [6, 7] that the displacive limit shows the same features as the mean-field theory. From the results obtained above we can see however that such an affirmation is not exact because ν, µ and ζ differ from those in the mean-field consideration. It is to be added here that γ = 1 at the displacive limit was obtained in the numerical study [11] and confirmed by analysing these numerical results within the framework of the RG-method [6]. There are also some experimental evidence about the crossover from Ising-like to mean-field (concerning indices β and γ) behavior when the parameters of H tends to mentioned here displacive limit (see also for example references in [6, 7]).
19.5
Conclusions
It is seen from the previous sections that the displacive limit in the case of the continuous η 4 -model allows one to obtain both critical temperature (reducing at the same time the basic Hamiltonian to the Landau Effective Hamiltonian) and define all the critical exponents at a dimension d = 3. In conclusion we present here for completeness the corresponding results for space dimensions d = 2 and d = 4 (see [9]). At the upper critical dimension d = 4, one obtains
Tc =
32π 2 ag 3bK 2
(19.51)
and the critical indices coincide with those in the mean-field model; the same follows from RG consideration of Ising-like systems.
vch 4 Okt 2005 10:43
19.6 References
455
At the lower critical dimension d = 2, the transition temperature at L → ∞ tends to zero emphasizing the fact of the two-dimensional η 4 -model anomaly. At large but finite L, it reads Tc =
4πag 3b ln K/κ
(19.52)
with κ ∝ L−1/2 . So at this condition the phase transition occurs at small but finite Tc ; the characteristic width of the transition region is of order κ which is small in comparison to Tc . Critical exponents take here the values α = 0, β = 1/2, γ = 1, ν = 1, ζ = 1, δ = 3, ε = 0, µ = 2/3. We are to emphasize that in this case the index ν also coincides with the one for Ising-like systems; namely this coincidence (just as at d = 3) provides the validity of the superscaling equation Eq. (19.38) and makes the theory with LEH scale-invariant.
19.6
References
1. K. Wilson and J. Kogut, Phys. Rep. C 12, 75 (1974). 2. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (Mc GrawHill, New York, 1965). 3. A. Z. Patashinsky and V. L. Pokrovsky, Fluctuation Theory of Phase Transitions (Pergamon Press. Oxford, 1979). 4. A. D. Bruce, Adv. Phys. 29, 111 (1980). 5. L. D. Landau and E. M. Lifshitz, Statistical Physics, vol. 1, 3rd edition (Pergamon Press, Oxford, 1979). 6. M. A. Anisimov, E. Luijten, V. A. Agayan, J. V. Senders, and K. Binder, Phys. Lett. A 264, 63 (1999). 7. S. Radescu, I. Extebarria, and J. M. Perez-Mato, J. Phys.: Condens. Matter 7, 585 (1995). 8. E. Brezin and J. Zinn-Justin, Nucl. Phys. B 257, 867 (1985). 9. A. Tseskis, Phys. Scripta 55, 403 (1997). 10. N. M. Kortsenstein and A. L. Tseskis, Doklady Physics 46, 773 (2001). 11. E. Luijten and K. Binder, Phys. Rev. E 58, R 4060 (1998).
vch 4 Okt 2005 10:43
20
A Model Explaining Necking in Polymer Spinning J¨org M¨ oller Scitecon, Bayreuther Str. 13, 01187 Dresden, Germany The First Clarke’s Law: If an elderly but distinguished scientist says that something is possible he is almost certainly right, but if he says that it is impossible he is very probably wrong. Arthur C. Clarke
Abstract The extension of a freely joining phantom chain in a volume stable, purely deforming, non-rotating flow field is analyzed. The flow field surrounding the chain is assumed to produce an average drag on the ends of each joint of the chain. The extension of the chain along the direction of maximum extension rate is shown to depend on a combination of temperature, fibre extension rate and several constants. A substantial chain extension sets on at a critical extension rate of the fibre, which scales with the reciprocal of the square root of the chain length. The existence of a critical extension rate leads to an enhanced orientation, especially for long chains. This orientation effect may help explaining the considerable increase of the nucleation rate of oriented crystals and the occurence of necking as it is observed in polymer melt spinning.
20.1 Introduction
20.1
457
Introduction
Melt spinning of polymers is a technologically very important process. One mayor application of melt spun fibers is the textile industry. There are at least two observations in fiber spinning to the explanation of which the work presented here may contribute. The first observation is that under certain circumstances, crystallization in high speed fiber melt spinning may occur. Often an amorphous orientation within the molten, non-crystalline material is experimentally observed (c.f. [1]). On the other side, theories on nucleation in polymers predict a nucleation rate that is very sensitive to amorphous orientation and increases quickly with it (c.f. [2, 3, 4]). The second observation is the so-called necking, a sudden considerable reduction of the cross section of a fiber along a very short distance along the spin line [1]. In Ref. [1], a plot of the temperature along the fiber as it is spun can be found. Along the fiber, the temperature decreases due to cooling. However, the plot shows an increase in the region of the neck, while temperature drops again a short distance down the spinline, while the fiber solidifies. However, solidification and crystallization may take place without the occurence of a neck as well. Several attempts have been made to compute different aspects of melt spinning [1]. Mainly, the description of flow and deformation is dealt with on a theoretically sound basis, using momentum, angular momentum, and energy conservation equations, which, in turn, are based on homogeneity and isotropy of space and the independence of the description of a stationary process on the choice of the zero in time. However, constitutive equations of materials frequently lack such a clear theoretical foundation. They often represent mere fit formulae with many parameters. For example, the formula for the extensional viscosity alone as given in Ref. [1] contains 15 constants and parameters. These equations may fit observations fairly well. However, the explanation of the observed features can be accomplished very easily with much simpler and more general assumptions. The results obtained give a fairly reasonable explanation of the observations cited above. The work presented here consists in the analysis of a simple model describing some basic features of behaviour of a polymer chain in a continuous deforming viscous liquid. The theoretical description presented is based on classical statistical mechanics, and similar to the treatment of rubber elasticity by non-Gaussian chains as it can be found, e.g., in Ref. [5]. The chain is treated in two different ways. The most exact treatment is carried out first. This is done to present the model and its implications in full, and to have
vch 4 Okt 2005 10:43
458
20 A Model Explaining Necking in Polymer Spinning
the exact expressions for the various quantities at hand. The discussion of this model leads to a simplified model which can be fully understood and is sufficient to understand the physical behaviour. The simplified model is easier to handle computationally as well. At the end of the paper, an even more simplified model is presented, which is designed to explain in general terms why the chain behaves the way it does.
20.2
Model Assumptions
A freely joint, freely dredging, phantom chain in an extensional flow field is analyzed. The chain is assumed to consist of joint segments. These joints will be referred to as atoms, although chemically they are, in general, groups of atoms. They are called atoms here because they are not divisible into smaller units in the model. Likewise, the segments connecting these atoms are assumed to be Kuhn elements. The chain is assumed to be surrounded by a polymer melt. This environment of the chain is assumed to be a continuum and not to interact with the chain other than by means of viscous friction. Heat of both mixing and friction is assumed to be negligible, and a swelling due to entropic effects is also assumed to be small. The swelling can be assumed to be small practically in all polymer melts, the exact condition for which can be found in Ref. [6]. The embedding medium is subjected to extensional flow, which is assumed not to be disturbed locally to any considerable degree by the presence of the chain, a condition usually referred to as a freely dredging chain. As the whole chain moves along with the flow, there is a force free reference point on the chain. Relative to this reference point, there is a drag on its different parts due to the friction between them and the gradient of the flow field. The extensional force acting on the different parts of the chain is considered large as compared to accelerational forces acting on the chain as a whole with respect to its flowing environment, so that this acceleration can be neglected. Fig. 20.1 depicts the model situation. The leading term in the Taylor expansion of the extensional velocity field of the flow is the velocity gradient. The x-axis be parallel to the direction of maximum extension rate of the surrounding deforming medium, which is the fiber. Taking into account only the leading Taylor expansion term of the velocity field relative to the reference point, the relative velocity of the surrounding deforming medium
vch 4 Okt 2005 10:43
20.2 Model Assumptions 459
Fig. 20.1 Model chain in a flow field: On the left, a chain in a flow field with increasing velocity can be seen. On the right, only the velocity field relative to the chain is plotted
with respect to the reference point of the chain, vx , is a linear function of the location, x, and the deformation rate, γ, ˙ ˙ = γ(x ˙ − x0 ) . vx = vx0 + γx
(20.1)
We choose the origin of x to coincide with the reference point, such that vx (x0 ) = 0. With respective stipulations in the y- and z-directions the flow field can, in the case of rotational symmetry about the x-axis and essentially volume stable flow, be described by γ˙ γ˙ ˙ − y, − z} . v(x, y, z) = {γx, 2 2
(20.2)
The frictional forces are assumed to act on the atoms of the chain, not on the bonds. In this section, let r be the frictional force acting on one atom. Then, in general, r = r(v, ρ(x)) .
(20.3)
In Eq. (20.3), ρ(x) describes the average number density of chain atoms projected on the x-axis. In general, we may assume that r is a monotonous function of vx .
vch 4 Okt 2005 10:43
460
20 A Model Explaining Necking in Polymer Spinning
In the y- and z-directions, the chain is squeezed, whereas in the x-directions it is pulled. Hence the chain is, on average, elongated in the x-direction. As a first assessment, the forces on the chain atoms are assumed to be proportional to the respective velocity components, whereas the velocity components are, in turn, proportional to the respective components of the difference vector relative to the origin. Consequently, the y- and z-components of the forces acting on the atoms become small, whereas those in x-direction may become large (c.f. Fig. 20.1). This means that the difference vector components and consequently the components of the force acting in the y- and z-directions can be neglected as compared with the effect of the force in x-direction. The treatment is hence continued one-dimensionally, all extensions, locations, forces, and velocities will be those in the x-direction. In general, r may depend on the angles between the relative direction of flow and the two bonds adjacent to the atom under consideration, as well as on the number of atoms of the chain in the vicinity of this atom. Hence, in general, ρ(x) occurs in the argument of r. As mentioned above, we neglect any disturbance of the chain on the flow, and with it the action of such a disturbance on neighboring atoms. The environment of a chain is treated as a viscous homogeneous continuum. This also means that entanglements are neglected as well. Furthermore, to each atom is assigned the same drag coefficient. Under these assumptions, r is independent of ρ(x). Furthermore, the atoms are assumed to be exposed to Stokes viscous friction. This model on the friction is very similar to a consideration first made by Rouse (c.f. [6], p. 167). With cη being the friction coefficient we obtain ˙ η xr , r(v(x), ρ(x)) = r(v(x)) = γc ˙ η x = γac
(20.4)
where a is the bond length between two atoms and xr = x/a is a reduced distance.
20.3
Chain in the Frame of Classical Statistics
20.3.1
Theoretical Background
In classical statistics, which may serve as the basis of description, the density of states for movements in unconstrained directions is given by a continuous phase space density dΓ. This a priori probability density is provided by the homogeneity and isotropy of space, i.e., the basis for classical theoretical mechanics. A simple example to elucidate this point is a model consisting of a freely moving particle in a box. The particle has the same probability to be in any part of the box provided
vch 4 Okt 2005 10:43
20.3 Chain in the Frame of Classical Statistics
461
we compare parts of the box that have the same volume. Our judgement on the equality of volumes, in turn, depends on our concept of homogeneity and isotropy of space as it is introduced by classical mechanics.
20.3.2
Chain Statistics
The treatment presented here essentially follows a statistical analysis of a chain very similar to that classically carried out for polymer chains forming a rubber, explaining rubber elasticity (c.f. [5, 7]). The classical equilibrium occupation density of states that have the same a priori probability is given by the Boltzmann factor. Hence, a very direct way of a statistical treatment would be to generate random chains and calculate the probability of each randomly created chain to exist in a state of thermodynamic equilibrium by means of the Boltzmann factor
E fB = exp − kB T
,
(20.5)
and then calculate conditional probability densities with respect to the quantities needed. The Monte Carlo (MC) method is commonly employed for this purpose (c.f. Ref. [8] for an introduction). ΞN be a randomly chosen configuration for a chain of N atoms. It is completely characterized by the locations, xi , and momenta, pi , of all atoms, ΞN = {x1 , x2 , . . . , xN , p1 , . . . , pN } .
(20.6)
The force on the overall chain in the flow field is F (ΞN )
=
N l=1
rl = cη γ˙
N
xl = cη γN ˙ x
(20.7)
l=1
with x being the mean value of all xl . The configuration needs to be shifted translationally to let x = 0 in order to meet the requirement that the resulting force on the chain be zero, F = 0. The resulting configuration is ΞN = {x1 , x2 , ...xN , pl } = {x1 − x, x2 − x, ...xN − x, pl } ,
(20.8)
vch 4 Okt 2005 10:43
462
20 A Model Explaining Necking in Polymer Spinning
i.e., ΞN represents the chain in the center of mass system. Extensive and intensive quantities, Qext. and Qint. , respectively, of one chain can, in general, be written as Qext. (ΞN ) =
N
q(xl ) , Qint. (ΞN ) =
l=1
N
q(xl )/N = Qext. /N .
(20.9)
i=l
Extensive and intensive quantities in this sense would be, e. g., the end-to-end distance of the chain and Herrman’s orienting function, respectively. The internal energy, E(ΞN ), of the chain, which occurs in the Boltzmann factor, is given by theoretical mechanics as the chain’s Hamiltonian, H(pl , q l ) = Ekin (pl , q l ) + Epot (pl , q l ) .
(20.10)
So, for the computation of it, all locations and momenta need to be known or obtained. Although the force acting on each atom of our model chain is caused by friction, it entirely depends on x (c.f. Eq. (20.4)). Hence, it can be formally treated as a conservative force. As the location of each atom has an expectation value, the equation of motion of the atom yields that the time average of the force on it is zero. If ric is the force on atom number i exerted on it by the chain, then, on average ri + ric = 0 .
(20.11)
Consequently ric = −acη xr,i .
(20.12)
Its potential is (xi ) = −
ric dxi =
γa ˙ 2 cη 2 xr,i . 2
(20.13)
Because the bonds are stiff, this external drag is the only potential that may do work on the chain. Summation over all atoms yields
Epot =
N i=l
vch 4 Okt 2005 10:43
N γa ˙ 2 cη xl 2 l = . 2 a l=1
(20.14)
20.3 Chain in the Frame of Classical Statistics
463
This expression can, in turn, rightfully be called the potential energy of the chain, because its derivative leads to forces that depend on locations only, in such a way that the integral over the force is also path independent. Normalizing Epot with respect to temperature T we obtain N N N γa ˙ 2 cη xl 2 k xl 2 Epot (ΞN ) 1 l = = , = kB T kB T 2kB T a 2 a l=1
l=1
(20.15)
l=1
where we introduced k = γa ˙ 2 cη /(kb T ) as a reduced extension rate. Consequently the Hamiltonian reads N N p2l k xl 2 H = + . kB T 2mkB T 2 a l=1
(20.16)
l=1
For a particular configuration, the Boltzmann factor according to Eq. (20.14) can be written as N N 2 k xl 2 pl (20.17) − fB (ΞN , k) = exp − 2 a 2mkB T l=1
l=1
or
N k xl 2 fB (ΞN , k) = exp − 2 a
exp −
l=1
N l=1
p2l 2mkB T
.
(20.18)
If n chains, each of length N , are generated numerically, then estimates for expectation values for extensive and intensive quantities of one chain would be, in general, n :
< Q >=
i=1
(i) (i) Q ΞN fB k, ΞN n : i=1
(i) fB k, ΞN
,
(20.19)
(i)
with i in ΞN being the sample index of the chain. Arbitrary distribution functions of these quantities can also be calculated. Let Θ2 (a, b, c) := Θ(c− b)+ Θ(b− a)− 1 be an auxiliary function, where Θ(x) is Heaviside’s step function; the ”:=” sign standing for a definition. Θ2 (a, b, c) is unity if b lies in the interval (a, c) and is
vch 4 Okt 2005 10:43
464
20 A Model Explaining Necking in Polymer Spinning
zero anywhere else, a < c tacitly assumed. If Q1 and Q2 are two fixed values of the variable Q then an estimate for the probability of finding a randomly chosen chain with this property in this range is the relative frequency of this event, n :
h(Q1 ≤ Q < Q2 ) =
i=1
(i)
(i)
Θ2 (Q1 , Q(ΞN ), Q2 )fB (k, ΞN ) n : i=1
.
(20.20)
(i)
fB (ΞN )k
The chain is assumed to go through a representative fraction of its phase space at any given temperature. So the temperature is controlled, and supposed to be constant for the statistical treatment. In other words, all chains in one ensemble are supposed to have the same temperature. The temperature represents the mean kinetic energy of the particles in the system. If the temperature is constant, the second factor in Eq. (20.18) is constant. In Eq. (20.19), this factor cancels, hence we may use an abbreviated Boltzmann factor fB (ΞN , k) = exp
N k xl 2 2 a
.
(20.21)
l=1
With the definition fB0 (ΞN ) := fB (ΞN , 1) = exp
1 xl 2 2 a N
,
(20.22)
l=1
Eq. (20.21) can obviously be rewritten in the form fB (ΞN , k) = fB0 (ΞN )k
(20.23)
As k is a reduced expansion rate, Eq. (20.23) leads to a factorization between fB0 and k (taking the logarithm of Eq. (20.23)), and thus to a considerable reduction in computational effort. Furthermore, this simplifies Eqs. (20.19) and (20.25) to n :
< Q >=
i=1
(i)
n : i=1
vch 4 Okt 2005 10:43
(i)
Q(ΞN )fB0 (ΞN )k , (i) fB0 (ΞN )k
(20.24)
20.3 Chain in the Frame of Classical Statistics
465
and n :
h(Q1 ≤ Q < Q2 ) =
i=1
(i)
(i)
Θ2 (Q1 , Q(ΞN ), Q2 )fB0 (ΞN )k n : i=1
,
(20.25)
(i) fB0 (ΞN )k
respectively.
20.3.3
Random Chain Generation: Trivial Approach
The first approach taken here in the creation of random chains is to start with one atom and then generate a random position for the next atom in such a way that all solid angles of the same size around the first atom have the same probability of containing the bond to the next atom. Consequently, the second atom is located somewhere on a spherical shell around the first atom with a probability density which is constant everywhere on the shell. In our model, we are only interested in the projection of such a bond on the x-direction, because only this projection enters the Hamiltonian. As the surface content of a spherical segment can be represented as a function of the radius, r, of the sphere and the height, h, of the segment, A = 2πrh, the probability distribution for the projection on any direction, e. g., the x-direction, is constant within the interval −a and a around the first atom and zero outside this interval. Consequently, only this projection in x-direction needs to be generated for the chain. This represents a mayor simplification for the simulation. The x-component of the position of the next atom is placed randomly in the interval −a and a relative to the first atom, with the same probability for choosing any admissible value. Then the x-component of the position of the third atom is chosen in the same way relative to the second atom, and so forth until a desired number, N , of atoms is in the chain. After that, all locations can be corrected according to Eq. (20.8), causing the resulting force on the chain to be zero. After that the Hamiltonian and all quantities of interest can be evaluated.
20.3.4
Simulation with a Biased Generator
The statistical information we want to extract is a measure for the extension of the chain along the x-axis. One characteristic figure for the chain extension is its end-to-end distance. One problem in the statistical treatment of this kind is
vch 4 Okt 2005 10:43
466
20 A Model Explaining Necking in Polymer Spinning
that for a long chain it is very unlikely that a random chain is generated which is highly extended, but that the Boltzmann factor for such an extended chain is very large (c.f. Eq. (20.17)), making these chains occur relatively frequently in a statistical ensemble but rare events in the simulation with a great Boltzmann factor making up for it. This still means that our estimates for the probability distributions by counting relative abundances of chains are very poor unless we generate an astronomical number of chains. To at least partly overcome this difficulty, use was made of the fact that the chain is statistically symmetric about x = 0, and that we are not interested in information on the variance due to the finiteness of the number of chains selected. Consequently, a biased random generator was introduced, which produced more elements pointing towards one direction than to the other as each segment was generated. Based on the bias, the probability for an element to point in the selected direction in the biased generator, W (ao , b), can be calculated, with b and ao being the bias parameter and the random value for the projection of a bond, respectively. Because all segments of a chain are statistically independent as they are generated, the relative weight of a generated chain in the distribution could be given as
W (ΞN ) =
N =
W (xi ) .
(20.26)
i=2
With the biased generator, Eqs. (20.24) and (20.25) turn into n :
< Q >=
i=1
(i)
(i)
(i)
Q(ΞN )fB0 (ΞN )k /W (ΞN ) n : i=1
(20.27) (i)
(i)
fB0 (ΞN )k /W (ΞN )
and n :
h(Q1 ≤ Q < Q2 ) =
i=1
(i)
n : i=1
vch 4 Okt 2005 10:43
(i)
(i)
Θ2 (Q1 , Q(ΞN ), Q2 )fB0 (ΞN )k /W (ΞN ) . (i) (i) fB0 (ΞN )k /W (ΞN )
(20.28)
20.3 Chain in the Frame of Classical Statistics
467
probability for end-to-end distance of chain
probability
1.10
-2
7.5.10
-3
5.10
-3
2.5.10
-3
kcr
0 -50
-40
-30
-20
-10
0
10
20
30
40
50
location of chain end in units of a Fig. 20.2 The probability density for the end to end distribution of a chain of N = 100. The curve is mirrored around zero for better illustration. The sample size is 108 , the bias b is 1.0. The different curves correspond to values of k increasing in equal steps. kcr according to the simplified model is indicated
An instance of a chain is now not a chain in an ensemble, but ”a chain times its a priori probability”, which now depends on the sample. The biased generator used was
1 ao (b, X) = − ln(1 − X (1 − exp(−2b))) , b
(20.29)
where X is a random variable in the range 0 to 1 and b as the bias. It was
vch 4 Okt 2005 10:43
468
20 A Model Explaining Necking in Polymer Spinning
2.10
-2
probability
1.5.10-2
1.10
-2
5.10
-3
kcr 0 -50
-40
-30
-20
-10
0
10
20
30
40
location of chain end in units of a
Fig. 20.3 Exactly like Fig. 20.2, with an unbiased random generator. With a sample size of 108 , the statistics is still extremely poor
designed to partly anticipate the influence of the Boltzmann factor. The function W belonging to Eq. (20.29) is
W (ao , b) =
2b exp(−b ao ) . exp(b) − exp(−b)
(20.30)
$ ao W (ao , b)dao in the random process of (With Y (ao , b) = P (Ao < ao , b) = −∞ segment creation, the inverse ao (b, X) = Y −1 (b, X) is the respective random generator.) Even with a biased random generator, the computational effort is tremendous, and the resulting curves needed considerable smoothening.
vch 4 Okt 2005 10:43
20.4 Simplified Model for Longer Chains 469
20.3.5
Results
Figs. 20.2 and 20.3 show the result of two simulations with 108 instances of chains (20.2) and with a bias of b = 1 (20.2), with N = 100 in both cases. The relative frequency for the location of the one end towards which the simulation was biased for equidistant values of k is plotted. The results of the simulation were taken from that one side of the curve that was smoother due to the bias, which was then mirrored about x = 0, after which the whole curve was smoothened with a Gauss curve with a variance of 10. The simulation kept a Silicon Graphics-WS Indigo2 busy for one week. The result was surprising: The simulations, albeit containing very much noise, indicate that comparatively little happens to the expectation value of the end to end distance as k increases, until k reaches some critical value, at which the maximum of the end to end distance probability density curve splits up into two, and the chain extends considerably. As a reminder, k = γa ˙ 2 cη /(kb T ). This means that there is a critical extension rate above which a considerable extension of the chain sets on, at a given temperature. Likewise, there is a temperature at a given extension rate, below which such an extension sets on. If, in an experiment, the temperature drops, whereas the extension rate increases, as it is the case in fiber spinning, an even more sudden onset of extension must be expected.
20.4
Simplified Model for Longer Chains
20.4.1
Description of the Model
A simplified description may allow a more thorough analysis of the effect of sudden onset of chain extension. This model especially applies for long chains, for which no reasonable statistics according to MC-calculations can be carried out. For example, if the chain length is N = 1000, and for every bond only two specified orientations are to be considered, this would give a complete sample size of 21000 ≈ 10333 . Although there may be other ways to proceed in MC, this still gives a fairly good idea of the problems encountered in computation. In the simplified model treated next, the actual probability density for the projection of each segment along the x-axis is replaced by its expectation value. A justification for such an approach is given along with the derivation. The expectation value of the projection of a segment of length a along x be a ¯, so that the actual probability a − a0 ). density is replaced by p(a = a0 ) = δ(¯
vch 4 Okt 2005 10:43
470
20 A Model Explaining Necking in Polymer Spinning
2.23
tension on central link
N = 20 1.78
1.34
0.89
0.45
0 0
2.10-2
4.10-2
6.10-2
8.10-2
0.1
k
Fig. 20.4 The tension of the central bond of a chain of N = 20 versus k is plotted. A sudden onset of extension is indicated by an increase of this tension from a critical value, kcr
We refer to Eq. (20.27). The chain is again supposed to be statistically in equilibrium. The temperature of the system, T , be a given quantity. Relative to the position of the predecessor atom, xi−1 , the location of an atom along the xaxis can be anywhere from −a to +a. If the variation of the force acting on the atom is neglected within this interval, the Boltzmann factor in the interval is approximately exp(−f (xi−1 ) · x/(kB T )). The a priori probability is constant, as was mentioned above. Consequently, the probability density is given by the Boltzmann factor alone. In this way we obtain
a ¯ =L a
f (x)a kB T
$1
,
L(x) =
y exp(xy )dy
−1 $1
= exp(xy )dy
−1
L(x) is the Langevin function (c.f. [5]).
vch 4 Okt 2005 10:43
ex + e−x 1 − . ex − e−x x
(20.31)
20.4 Simplified Model for Longer Chains 471
20.4.2
Justification of the Simplified Model
If we consider a long chain at the ends of which a force pulls along the x-direction then every segment of the chain will have a projection the expectation value of which is given by Eq. (20.31), so the expectation value of a piece of the chain ¯. The variance can be estimated by N0 a ¯2 , which can containing N0 bonds is N0 a be found from the central limit theorem of probability calculus, even for less restrictive models (c.f. [9]). Consequently, if the number of bonds in a considered interval is not too small (≈ N√ 0 > 20), about 99.5% √ of all random chains have a 2 ¯ and N a ¯+3 Na ¯2 . That also means that the section length between N a ¯−3 Na number of chain elements that fall into a certain, not too short, interval along the x-axis divided by the length of this interval is approximately given by a ρ(x) ≈ . (20.32) a ¯
7.47
tension on central link
N = 28 5.97
4.48
2.99
1.5
0 0
2.10-2
4.10-2
6.10-2
8.10-2
0.1
k
Fig. 20.5 The same as Fig. 20.4, for N = 28, kcr has decreased. For a given value of k, the tension has increased considerably
For extremely long chains, the discrete chain model may even be replaced by a continuous one, and a frictional force acting per unit length considered. The tension along the chain changes by df = r(v(x), ρ(x))ρ(x)dx →
df = r(v(x), ρ(x))ρ(x) . dx
(20.33)
vch 4 Okt 2005 10:43
472
20 A Model Explaining Necking in Polymer Spinning
At this point, we do not have ρ(x), which would have to constitute a part of the solution.
28.56
tension on central link
N = 50 22.85
17.14
11.43
5.72
0 0
2.10-2
4.10-2
6.10-2
8.10-2
0.1
k
Fig. 20.6 Same as Figs. 20.4 and 20.5, for N = 50, kcr has decreased further
It is hard to solve this differential equation for a given chain length N . The discrete model can be solved numerically: One can take a ratio a ¯/a as being given at the center of the chain, and integrate to that value x0 at which the force drops to zero, i.e. changes signs. Then x0 N =a
ρ(x)dx
(20.34)
−x0
yields the respective chain length.
20.4.3
Results
The evaluation of this concept leads to Figs. 20.4-20.7. Figs. 20.4-20.6 show, how the chain extension sets in at a very definite deformation rate k, which is clearly a critical value, kcr . For longer chains, kcr decreases. A guess on the functional
vch 4 Okt 2005 10:43
20.5 Understanding the Physics
473
relationship, gained by sharply looking at the Hamiltonian, was confirmed: kcr ∝ 1/N 2 . A least mean square fit of 1/kcr versus N 2 as given in Fig. 20.7, yielded 1 ≈ 0.0334792N 2 − 3.95996 . kcr
(20.35)
For example, for N = 100 we obtain kcr ≈ 3.0 · 10−3 . The respective curve corresponding to this value is indicated in the MC-results, Figs. 20.2 and 20.3. Figs. 20.4-20.6 clearly show an onset of the extension which resembles Hopf bifurcations, second-order phase transitions, or buckling phenomena: The curves are continuous, there is a critical value of a parameter, and the curve has a vertical tangent at kcr + dk. It may be also interesting to note that for Gaussian chains, this critical onset cannot be observed because N = ∞ for Gaussian chains.
20.5
Understanding the Physics
We already realized that a complete solution of the mechanical equations is not possible. A strongly simplified mechanical model of a polymer chain which does not rest on statistical physics may still serve to elucidate the underlying physical processes. It should not be taken too seriously in scientific terms, but it may illustrate the pulling of a polymer chain. The model of the freely joining chain may be further modelled as consisting of a massless cord, representing two bonds, and a mass in the middle that can swivel around, which stands for the atom. The distance of the two ends of the cord from each other is equated to the relative extension of the polymer chain. Consequently, a fully stretched polymer chain corresponds to a cord pulled straight. The temperature is identified with the kinetic energy of the chain simply by 2kB T =
m 2 v , 2
(20.36)
due to two degrees of freedom for the mass. So, the temperature with mean external force f¯ = 0 is given by T0 =
m v02 . 4 kB
(20.37)
Let the radius of the circular orbit of the mass be r, then ¯2 + r 2 , a2 = a
(20.38)
vch 4 Okt 2005 10:43
474
20 A Model Explaining Necking in Polymer Spinning
1.10
4
1/kcr
7.5.10
3
5.10
3
2.5.10
3
0 0
5.104
1.105 1.5.105 2.105 2.5.105 2
N
Fig. 20.7 1/kcr is plotted versus N 2 . Slope and correlation coefficient are indicated
where a is half of the length of the cord. When the two ends of the cord are pulled from each other, the angular momentum of the movement of the mass is conserved, i.e., av0 = rv → v =
av0 . r
(20.39)
This model is analoguous to an adiabatic expansion. Let x := a ¯/a. Then we can write for the kinetic energy ' T 1 a ¯ T0 m 2 m 2 1 v = v0 . (20.40) → = ↔ x= = 1− 2 2 2 2 1−x T0 1−x a T In isothermal conditions T = T0 , i.e., the speed of the mass is kept constant. If the cord is held on its two ends, the radial force is fr =
mv 2 . r
(20.41)
Furthermore, the force along the connecting line between the two ends is connected to the radial force f¯ a ¯ fr a ¯v 2 m 2kB T a x 2kB T ¯ fr = ↔ f¯ = = . = 2 = 2 2 2r a ¯ 2r 2r a −a ¯ 1 − x2 a
vch 4 Okt 2005 10:43
(20.42)
20.6 Dumpbell Model
475
We obtain f¯T =
x 2kB T0 1 − x2 a
f¯ad =
x (1
− x2 )2
2kB T0 a
(20.43)
for the isothermal and adiabatic case, respectively. The interesting feature to be observed is that the force f¯ is a strictly monotonous function of x = a ¯/a for small values of x, and diverges for x → 1. So the behaviour of either function is identical to that of the inverse of Langevin’s function.
20.6
Dumpbell Model
Fig. 20.8 illustrates a particular result of an MC-calculation, which can be deduced from the simplified model or the analytical differential equation in the form of Eq. (20.33). It shows the tension on the bonds over the linear distance from the chain center. Because the ends of the chain pull the central part to stretch it, the chain is highly extended in the middle. Furthermore, the drag on the individual atoms is greater the further away these atoms are from the center, because of the greater relative velocity these atoms experience. This means that there is a competition of a contractive force acting essentially from the center on the ends which is due to the rubber elastic, or entropic force of the central part, and the frictional forces acting on the two outer areas of the chain, which are less extended, i.e. more compact, than the central part. In order to understand the critical behaviour of the chain, it can be modelled as two compact areas on which only frictional forces act and pull them apart, and a central rubber band pulling these two compact areas together. This model would look like a dumpbell. The attractive force in dependence on the extension can now be taken from rubber elasticity theory as the inverse of the Langevin function, L−1 (x) (c.f. Eq. (20.31)). This function carries in itself the explanation for the critical behaviour observed. • It is always attractive, • it has a non-vanishing slope at zero extension, namely 3, • it grows faster than linear, diverging at full chain extension. The explanation for the critical behaviour is now this: Suppose k = 0. Both compact areas would be at x = 0. If the system is variationally disturbed and the two centers are pulled away from each other, the rubber band between them will pull them back to the origin.
vch 4 Okt 2005 10:43
476
20 A Model Explaining Necking in Polymer Spinning
-2
1.6.10
-2
f
1.2.10
8.10
-3
4.10
-3 -5
1.11 .10
1
267.4
533.8
800.2
1060
1330
l
Fig. 20.8 The tension in the bonds is plotted versus the location of the atoms. The central part is highly stretched, the tension is high. In the outer regions, the tension is less, resulting, at the same time, in a higher density of atoms in the outer areas, which is ρ(x).
Now suppose k = 0, but very small. To be definite, k = 0.1. If the system is now disturbed, the frictional forces acting on the compact areas will pull them away from each other. This force acting on them will grow linearly with the distance. However, the force acting on them exerted by the rubber band connecting them also increases, at least linearly, with a definite slope of 3. So the sum of these forces will pull the ends back to the origin. This will always be the case for k < 3. If k > 3, a small disturbance will first start to grow. The frictional force will always grow linearly with the distance, but the attractive force will grow faster. Eventually, it will equal the frictional force, which can be regarded as a repulsive force between the two compact centers. So the non-vanishing slope of the Langevin function of 3 at its zero establishes the existence of a critical value for k, which is, in this case, kcr = 3. The sum of the attractive and repulsive forces for different values of k are plotted in Fig. 20.9. One remark which may be important in terms of the generality of the considerations is that the existence of a critical flow rate is ensured for all friction laws for the individual chain for which df /dk has a positive upper bound. Besides this, the existence of a critical value for k does not depend on any specific features
vch 4 Okt 2005 10:43
20.7 Discussion
477
0.6 0.4
f
0.2 0 0.2
0.4
0.6
0.8 x
-0.2 -0.4
Fig. 20.9 The sum of the frictional and restoring forces is plotted versus the normalized chain extension. Forces < 0 pull towards the origo, forces > 0 pull away from it. For values of k < 3, the only stable position is at zero extension. For each value for k with k > 3, there is one value for the extension = 0 which is stable. For extension = 0, the force is = 0 in this range for k as well, but this value for the extension is not stable against a disturbance
of the frictional force law of a chain in its environment. This astonishing fact results from the divergence of L−1 (x) at a definite chain extension, namely its full extension, and the fact that for k = 0 the frictional force is necessarily zero, too. So, if the frictional force increases with the deformation rate with a upper bound of this increase, it will cross over with L−1 (x) at least once.
20.7
Discussion
In melt spinning especially at high speeds, two observations are that necking may occur, and that solidification/crystallization may start after necking. A considerable extension of the polymer chains sets on fairly suddenly, at some critical deformation rate. This extension along the direction of extension, i.e., in the direction of the fibre, will lead to a contraction perpendicular to this direction, which could explain the necking as it is observed. The sudden extension will also
vch 4 Okt 2005 10:43
478
20 A Model Explaining Necking in Polymer Spinning
lead to a reduction in configurational entropy, which facilitates the phase transition, reducing the term T ∆S for the transition. The sudden reduction in entropy would also lead to a sudden increase of temperature as it is observed as well.
Acknowledgement The author is grateful to the National Textile Center, project # M96-G19, Deutsche Forschungsgemeinschaft, as well as Humboldt-Stiftung, for financial support. The author also wishes to thank Prof. Sinha, Argonne National Laboratory, and Prof. Abhiraman of Georgia Institute of Technology for many fruitful discussions, as well as A. Milchev from Sofia for proofreading.
20.8
References
1. K. F. Zieminsky and J. E. Spuriell, J. Applied Polymer Science 35, 2223 (1988). 2. L. Mandelkern, Crystallization of Polymers (McGraw-Hill, 1964). 3. T. Amanato, K. Nakamura, and K. Katayama, J. Polymer Science 17, 1031 (1972). 4. K. Nakamura, T. Watanabe, K. Katayama, and T. Amanato, Journal of Polymer Science 1077 (1972). 5. L. Treloar, The Physics of Rubber Elasticity (Clarendon Press, Oxford, 1958). 6. P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979). 7. M. Toda, R. Kubo, and N. Saito, Statistical Physics, vol. 1 (Springer, Berlin, 1992). 8. K. Binder and D. W. Heermann, Monte Carlo Simulations in Statistical Physics (Springer, Berlin, 1992). 9. W. J. Adams, The Life and Times of the Central Limit Theorem (Kaedmon Pub. Co., New York, 1974).
vch 4 Okt 2005 10:43
21
Appendices Science is a wonderful thing if one does not have to earn one’s living at it. Albert Einstein
21.1
Programs of the Research Workshops 2003, 2004 and 2005
21.1.1
Research Workshop 2003
Sunday, April 13, 9.30 1. Alexander P. Grinin, D. S. Grebenkov (St. Petersburg, Russia): Time Evolution of Ensembles of Molecular Aggregates in a Micellar Solution after an Instantaneous Change of the Thermodynamic Parameters 2. Alexander K. Shchekin, V. V. Borisov (St. Petersburg, Russia): The Role of Ionic Adsorption in Nucleation on Soluble Particles of Electrolytes 3. Sergey A. Kukushkin (St. Petersburg, Russia): Mechanism and Kinetics of GaN Thin Film Growth 4. Rimma S. Bubnova, S. K. Filatov (St. Petersburg, Russia): Isosymmetrical First-Order Phase Transitions and Thermal Expansion of Borates 5. Stanislav K. Filatov, E. N. Kotelnikova (St. Petersburg, Russia): Mechanism of Thermal Polymorphic Transitions in Rotary Crystals on the Example of Normal Paraffins 6. Elena V. Charnaya, B. F. Borisov, A. V. Gartvik, Yu. A. Kumzerov (St. Petersburg, Russia): Crystallization and Melting of Gallium and Mercury in Confined Geometry Monday, April 14, 9.00
480
21 Appendices
1. Dieter H. E. Gross (Berlin, Germany): Geometric Foundation of ThermoStatistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit: I. Basic Postulates 2. Alexander V. Kosarim, E. A. Andreev, B. M. Smirnov (Moscow, Russia): Transition from Mechanics to Statistical Mechanics 3. R. S. Berry (Chicago, USA), Boris M. Smirnov (Moscow, Russia): Heat Capacity of Clusters Near the Phase Transition 4. Victor B. Kurasov (St. Petersburg, Russia): Specific Behavior in the Kinetics of First-order Phase Transitions 5. Andriy M. Gusak (Cherkasy, Ukraine), K. N. Tu (Los Angeles, USA): A Kinetic Theory of Flux-Driven Ripening 6. J¨ org M¨ oller (Dresden, Germany): Modifying the Fundamental Equations to fulfil the Third Law of Thermodynamics (30 min) 7. Vladimir M. Fokin (St. Petersburg, Russia), P. Guanabara Jr., G. T. Niitsu, A. C. Rodrigues, F. M. Spiandorello (S˜ ao Carlos, Brazil), Miguel O. Prado (Bariloche, Argentina): Effect of Devitrification on Electric Conductivity of Na2 O 2CaO 3SiO2 glass (30 min) Special Talks (Monday, 14. 4., 19. 00): 1. Vladimir P. Skripov (Ekaterinburg, Russia): Some Thoughts about Scientific Research Arising from the Own Experience in this Field 2. Discussion: Is there a spinodal in (one-component) melt crystallization? Tuesday, April 15, 9.00 1. Vladimir G. Baidakov (Ekaterinburg, Russia): A New Model in the Theory of Homogeneous Nucleation: Theory and Computer Modelling 2. Aleksey Vishnyakov (Princeton, USA): Calculation of Nucleation Barriers by Monte Carlo Simulations 3. Henry E. Norman, V. V. Stegailov (Moscow, Russia): Homogeneous Nucleation in Superheated Crystals at Constant Temperature and Under Heating Conditions 4. Grey Sh. Boltachev, V. G. Baidakov (Ekaterinburg, Russia): The Size Dependence of the Surface Tension: Connection with the Interaction Potential and the Pair-Distribution Function
vch 4 Okt 2005 10:43
21.1 Programs of the Research Workshops 2003, 2004 and 2005
481
5. Dmitrii I. Zhukhovitskii (Moscow, Russia): Towards the Theory of a Highly Curved Liquid-Vapor Interface 6. Naoum M. Kortsenstein, E. V. Samuilov (Moscow, Russia): The Moment Method in the Theory of Binary Condensation Wednesday, April 16, 9.00 1. Dieter H. E. Gross (Berlin, Germany): Geometric Foundation of ThermoStatistics, Phase Transitions, Second Law of Thermodynamics, but without Thermodynamic Limit: II. Applications 2. J¨ urn W. P. Schmelzer (Rostock, Germany & Dubna, Russia): Is Gibbs’ Thermodynamics Really Applicable to Nucleation 3. Nicolay P. Malomuzh (Odessa, Ukraine): Peculiarities of Nucleation in Supercooled States of Glass-forming Liquids 4. Leonid M. Landa, K. A. Landa (Carlton, USA): Paradoxes and Prejudices in Glass Science 5. Galina G. Boiko, A. Parkachev (St. Petersburg, Russia): Dynamical and Thermodynamical Properties of Supercooled Pyrophosphate Melts 6. German V. Berezhnoy, G. G. Boiko (St. Petersburg, Russia): Defects and Oxygen Diffusion in Alkali Silicate Glass Modelled by Molecular Dynamics Simulation (30 min) Special Talks (Wednesday, 16. 4., 19. 00): 1. Vyatcheslav I. Zhuravlev (Dubna, Russia): The Bogoliubov Laboratory of Theoretical Physics: Past, Present and Future 2. J¨ urn W. P. Schmelzer (Dubna & Rostock): Some Proposals Concerning Future Common Scientific Activities Thursday, April 17, 9.00 1. Vladimir Ya. Shur (Ekaterinburg, Russia): Correlated Nucleation and Kinetics of Ferro-electric Domains unster, Germany), V. 2. Andriy M. Gusak (Cherkasy, Ukraine), G. Schmitz (M¨ Vovk (G¨ ottingen, Germany), M. Pasichny (Cherkasy, Ukraine): Nucleation of the Co2 Al9 -Intermetallic Phase at the Co-Al Interface after ”Concentration Preparation”
vch 4 Okt 2005 10:43
482
21 Appendices
3. Oleg V. Potapov, V. M. Fokin (St. Petersburg, Russia), J. W. P. Schmelzer (Rostock & Dubna): Influence of Elastic Stress Evolution and Relaxation on Crystal Nucleation in Lithium Disilicate Glass ao Carlos, Bra4. Vladimir M. Fokin (St. Petersburg, Russia), E. D. Zanotto (S˜ zil): Effect of Elastic Strains on Nucleation in Li2 O · 2SiO2 -glass 5. Irina G. Polyakova (St. Petersburg, Russia): Crystallization of Glasses in the Vicinity of Stoichiometric and Eutectic Compositions 6. Olga Yu. Golubeva, Yu. K. Startsev (St. Petersburg, Russia): Specific Features of One- and Two-Alkali Borate Glasses Containing Water 7. Aram S. Shirinyan, A. M. Gusak (Cherkasy, Ukraine): ”Critical Supersaturation” of Phase-Separating Nano - Alloys (30 min) Friday, April 18, 9.00 1. Werner Ebeling, U. Erdmann (Berlin, Germany), G. R¨ opke (Rostock & Dubna), S. Trigger (Moscow, Russia): Two-Dimensional Dynamics of Clusters far From Equilibrium 2. Alexander P. Chetverikov (Saratov, Russia), J. Dunkel (Berlin, Germany): States of Aggregation of Ensembles of Particles Interacting via Morse Potentials ohmert, A. Ulbricht (Rossen3. Alexander R. Gokhman (Odessa, Ukraine), J. B¨ dorf, Germany): Experimental and Theoretical Background of Multi-component Cluster Formation in Reactor Pressure Vessel Steels Due to Neutron Irradiation 4. Igor L. Maksimov (Saratov, Russia): Self-organized Critical States in Condensed - Matter Systems 5. Eugene D. Nikitin (Ekaterinburg, Russia): Critical Properties of Substances Consisting of Long Chain Molecules 6. Timur V. Tropin, M. V. Avdeev (Dubna, Russia): Aggregation Processes of Fullerenes: Some New Results Saturday, April 19, 9.30 1. Vitali V. Slezov, A. S. Abyzov (Kharkov, Ukraine): Gas Bubble Formation in Liquids and Melts with Small Viscosities aki, M. Kulmala (Helsinki, Fin2. Madis G. Noppel (Tartu, Estonia), H. Vehkam¨ land): Binary Nucleation of Sulfuric Acid Water Systems: The Effect of Hydration
vch 4 Okt 2005 10:43
21.1 Programs of the Research Workshops 2003, 2004 and 2005
483
3. Anton Yu. Lavrenov, I. A. Lubashevsky, A. A. Katsnelson (Moscow, Russia): Multiple Defect Model for Non-monotonic Relaxation in Binary Alloys like Pd-Er Charged with Hydrogen 4. Boris Z. Pevzner, I. G. Polyakova (St. Petersburg, Russia): BaO·2B2 O3 : Specific Character of Glass Crystallization and Thermal Expansion of the Glass and the Crystal 5. Vitali V. Slezov (Kharkov, Ukraine): Phase Transitions in Condensed Matter under Finite Rate of Formation of Metastable States
21.1.2
Research Workshop 2004
Sunday, October 10: 9. 30 1. V. P. Skripov, Dmitry Yu. Ivanov (Ekaterinburg, St. Petersburg, Russia): On the Second Crossover Near the Critical Point 2. Werner Ebeling (Berlin, Germany): Investigation of Cnoidal Waves as NonEquilibrium Clusters 3. Alexander P. Chetverikov (Saratov, Russia): Thermodynamics of Clustering and Phase Transitions in Driven Morse Chains 4. Mikhail V. Avdeev (Dubna, Russia): Structural Changes of Clusters in Diamond Powders Under High Pressure (45 min) 5. Olga A. Shilova (St. Petersburg, Russia): Fractal Structures in Sol-Gel Derived Nano-Composites (Xerogels and Glassy Films) combined with Valeri V. Shilov, V. I. Padalko, O. A. Shilova (Kiev, Ukraine; St. Petersburg, Russia): Microphase and Fractal Structure of Sol-Gel Nanocomposites filled with Nanodiamonds Monday, October 11: 9. 00 1. Vladimir G. Baidakov (Ekaterinburg): Molecular Dynamics Investigations of the Liquid-Crystal Phase Transition 2. Sergey P. Protsenko, V. G. Baidakov, E. R. Zhdanov (Ekaterinburg, Russia): Computer Modelling of Nucleation in Gas-Supersaturated Solutions 3. Dmitry I. Zhukovitskij, S. V. Stepanov (Moscow, Russia): Microscopic Bubbles in Liquids as Traps for Positronium 4. Sergey N. Burmistrov, Leonid B. Dubovskij (Moscow, Russia): Sub-barrier Nucleation Kinetics in a Metastable Quantum Liquid near the Spinodal
vch 4 Okt 2005 10:43
484
21 Appendices
5. Victor N. Grigor’ev, A. S. Rybalko (Kharkov, Ukraine): Evidence of Homogeneous Nucleation and General Features of the Phase Transition Kinetics in Solid 3 He-4 He Mixtures 6. Alexander K. Shchekin, F. M. Kuni, A. I. Rusanov, A. P. Grinin (St. Petersburg, Russia): Micellization in Solutions with Spherical and Cylindrical Micelles Special Talks (19.00): 1. Vitali V. Slezov (Kharkov, Ukraine): Some Reflections on Science and History 2. Alexander L. Tseskis (Ludwigshafen, Germany): Adventures of a Russian in Germany Tuesday, October 12: 9. 00 1. Vitali V. Slezov (Kharkov, Ukraine): Phase Transitions in Condensed Matter under a Finite Rate of Formation of Metastable States 2. R. S. Berry (Chicago, USA), Boris M. Smirnov (Moscow, Russia): Void Theory for Nucleation in Simple Liquids 3. Alexey Vishnyakov, A. Neimark (Princeton, USA): Vapor-to-Droplet Transition in a Lennard-Jones Fluid: Simulation Study of Nucleation Barriers Using the Ghost Field Method 4. Ivan S. Gutzow (Sofia, Bulgaria): Crystallization and Glass Transition in Biological Fluids and the Problems of Life, Health and Death of Organisms 5. Sergey P. Fisenko (Minsk, Belorussia): Instantaneous and Observable Nucleation Rates: Analysis of Laminar Flow Chamber Experiments 6. Timur V. Tropin, M. V. Avdeev (Dubna, Russia): Aggregation Processes of Fullerenes: Some New Results (45 min) 7. J¨ urn W. P. Schmelzer (Rostock & Dubna): Nucleation Theory and Applications: State of affairs with the VCH-WILEY publication and some first unexpected positive consequences and possible future plans Wednesday, October 13: 9. 00 unde, Germany), W. Wagner (Bochum, Ger1. Rainer Feistel (Rostock-Warnem¨ many): A Comprehensive Gibbs Thermodynamic Potential of Ice 2. J¨ urn W. P. Schmelzer (Rostock, Germany): Is Gibbs’ Theory of Heterogeneous Systems Really Perfect? Some Further Developments
vch 4 Okt 2005 10:43
21.1 Programs of the Research Workshops 2003, 2004 and 2005
485
3. Alexander S. Abyzov (Kharkov, Ukraine), J. W. P. Schmelzer (Rostock & Dubna): Nucleation vs. Spinodal Decomposition in Phase Formation Processes in Multi-component Solutions 4. T. V. Lokotosh, Nikolay P. Malomuzh (Odessa, Ukraine): Clusterization and Nucleation in Water 5. Maksim A. Zakharov, S. A. Kukushkin, A. V. Osipov (St. Petersburg, Russia): Switching Kinetics in Ferroelectrics and Ferroelastics 6. Boris M. Smirnov, S. A. Ivanenko (Moscow, Russia): Nucleation Processes in Cluster Plasmas Special Talks (19:00): 1. Irina G. Polyakova (St. Petersburg, Russia): Crystallization and Recrystallization: A Picture Book 2. Vladimir M. Fokin (Sao Carlos, Brazil & St. Petersburg, Russia): Adventures of a Russian in Brazil Thursday, October 14: 9. 00 1. Vladimir Ya. Shur (Ekaterinburg, Russia): Kinetics of Nano- and Micro-scale Domains 2. Boris A. Klumov (Garching, Germany): Numerical Modelling of Complex Plasmas 3. Alexander L. Tseskis (Leverkusen, Germany): Critical Temperature in Statistical Field Theory in the Displacive Limits 4. Naoum M Kortsenstein, E. V. Samuilov (Moscow, Russia): Features of Phase and Chemical Transformations in C-H-O Mixtures with Metal Additives 5. Alexander O. Bogatyrev, A. M. Gusak, M. O. Pasichnyi (Cherkasy, Ukraine): Kinetics of Nucleation in a Concentration Gradient 6. Aram S. Shirinyan, M. O. Pasichnyi (Cherkasy, Ukraine): Hysteresis in the Process of Phase Separation in an Ensemble of Small Isolated Particles (45 min) Friday, October 15: 9. 00 1. Vladimir M. Fokin, E. N. Soboleva, N. S. Yuritsyn (St. Petersburg, Russia), E. D. Zanotto (S˜ ao Carlos, Brazil): Crystal Nucleation in Meta-silicate Glasses of Compositions between Na2 O·2CaO·3SiO2 and Na2 O·CaO·2SiO2 . The Problem
vch 4 Okt 2005 10:43
486
21 Appendices
of Critical Nuclei Compositions combined with N. S. Yuritsyn, A. G. Cherepova, Vladimir M. Fokin (St. Petersburg, Russia): Comparative Study of Concentration Dependencies of Nucleation and Growth Rates in Lithium Silicate Glasses 2. Irina G. Polyakova (St. Petersburg, Russia): Surface and Volume Nucleation of Low-alkali Sodium Borate Glasses 3. Elena O. Litovtschik, I. G. Polyakova, B. Z. Pevzner (St. Petersburg, Russia): Specific Features of Crystallization and Liquid Phase Separation of Strontium Borate Glasses (30 min) 4. Galina Sycheva (St. Petersburg, Russia): Formation of Bubble Structure and Homogeneous - Heterogeneous Nucleation in Glass 26 Li2 0 - 74 SiO2 (45 min) 5. Galina G. Boiko, G. V. Berezhnoi (St. Petersburg, Russia): Microheterogeneous Structure of Eutectic Melts. Problems of Molecular Dynamics Simulations (45 min) 6. Boris Z. Pevzner (St. Petersburg, Russia): Polycationic Effect in Borate Glasses (45 min) Saturday, October 16: 9. 30 1. Leonid M. Landa, K. A. Landa, L. L. Landa (Carleton, USA): Second-Order Phase Transitions under Negative Pressure and Amorphous Solidification of Liquids 2. Henry E. Norman, V. V. Stegailov (Moscow, Russia): Superheating, Decay and Melting of Crystals 3. A. Yu. Kuksin, H. E. Norman, V. V. Stegailov (Moscow, Russia): Decay of Crystals under Stretching: Spinodal at Negative Pressures 4. Vitali V. Slezov, Oleg A. Osmayev, Zh. V. Slezova (Kharkov, Ukraine): Diffusional Decomposition of Supersaturated Solutions of Admixtures at Grain Boundaries (30 min) 5. Eduard R. Zhdanov (Ufa, Russia): Molecular Theory of Binary Liquid Solutions (30 min) 6. J¨ urn W. P. Schmelzer (Rostock, Germany), V. G. Baidakov (Ekaterinburg, Russia): On Different Possibilities of a Thermodynamically Consistent Determination of the Work of Critical Cluster Formation in Nucleation Theory
21.1.3
Research Workshop 2005
Sunday, June 26: 9. 30
vch 4 Okt 2005 10:43
21.1 Programs of the Research Workshops 2003, 2004 and 2005
487
1. Ivan S. Gutzow, B. Petroff (Sofia, Bulgaria): Glass Transition and Landau’s Phenomenological Theory of 2nd -Order Phase Transitions 2. Leonid M. Landa, K. A. Landa (Carleton, USA): Thermodynamic Nature of the Glass Transition Interval 3. Boris Petroff (Sofia, Bulgaria): Glass Transition and Crystallization in Liquid Crystals 4. Elena N. Soboleva, N. S. Yuritsyn, V. M. Fokin, V. P. Klyuev (St. Petersburg, Russia): Concentration Dependence Analysis of the Crystal Nucleation Rate in Meta-Silicate Glasses of Composition Na2 O2CaO3SiO2 and Na2 OCaO2SiO2 5. Nikolay S. Yuritsyn (St. Petersburg, Russia), J. W. P. Schmelzer (Rostock & Dubna), V. M. Fokin (St. Petersburg, Russia), E. D. Zanotto (Sao Carlos, Brazil): Estimation of the Time-Lag for Crystal Nucleation in Lithium Silicate Glasses from Nucleation and Growth Kinetics Monday, June 27: 9. 00 1. V. P. Skripov, Mars Z. Faizullin (Ekaterinburg, Russia): On the Role of the Internal Pressure in the Phase Transformation Kinetics 2. Rainer Feistel (Rostock-Warnem¨ unde, Germany): Statistical Theory of Electrolytic Skin Effects 3. Dmitry Yu. Ivanov (St. Petersburg, Russia): Critical Opalescence 4. Vladimir G. Baidakov, S. P. Protsenko (Ekaterinburg, Russia): Extended Phase Diagram of Simple Substances (Computer Experiment) 5. Sergey P. Protsenko, V. G. Baidakov (Ekaterinburg, Russia): Thermal and Caloric Equations of State of a Lennard-Jones Fluid 6. Alexander R. Gokhman (Odessa, Ukraine), F. Bergner, A. Ulbricht (Rossendorf, Germany): Iron Matrix Effects on Cluster Evolution in Neutron Irradiated Reactor Steels Special lectures: 18. 00 1. Georgi T. Guria (Moscow, Russia): Regular and Chaotic Procedures in Teaching Processes 2. Henry E. Norman (Moscow, Russia): Irreversibility and Predictability in Physics, Chemistry and Biology Tuesday, June 28: 9. 00
vch 4 Okt 2005 10:43
488
21 Appendices
1. Dieter H. E. Gross (Berlin, Germany): The Microscopic Origin of Phase Separation 2. Alexander L. Tseskis (Leverkusen, Germany): Non-Physical Problems in Statistical Physics (45 min) 3. J¨ urn W. P. Schmelzer (Rostock, Germany & Dubna, Russia): Classical vs. Generalized Gibbs’ Approaches: Experimental Confirmations and Open Problems 4. Sergey P. Fisenko (Minsk, Belorussia): Maximum Information Entropy and the Size-Distribution of Nano-Clusters (45 min.) 5. Vyatcheslav B. Priezzhev (Dubna, Russia): Master Equation for Asymmetric Exclusion Processes 6. Sergey N. Burmistrov, Leonid B. Dubovskij (Moscow, Russia), Y. Okuda (Tokyo, Japan): Quantum Decay of a Metastable Liquid near the Spinodal of the Liquid-Gas Transition urn W. P. Schmelzer (Rostock, 7. Alexander S. Abyzov (Kharkov, Ukraine), J¨ Germany & Dubna, Russia): Thermodynamic Analysis of the Transition from Metastability to Instability in Phase Separating Solutions Wednesday, June 29: 9. 00 1. Victor N. Grigor’ev (Kharkov, Ukraine): Nanoclusters of 4 He Atoms with Delocalized Vacancies in Solid Mixtures of 4 He in 3 He 2. Yevgenii Syrnikov (Kharkov, Ukraine): Giant Asymmetry Between Phase Separation and Homogenization of Solid 3 He-4 He Mixtures at Low Temperatures (30 min) 3. Dmitry I. Zhukovitskij (Moscow, Russia): Surface Vibration Spectra of Argonlike Clusters from Molecular Dynamics Simulations 4. Vladimir V. Stegailov (Moscow, Russia): Theory and Molecular Dynamics Simulations of the Decay of Metastable Solids and Molten Metals (45 min) 5. Alexey Yu. Kuksin (Moscow, Russia): Decay of Metastable Lennard-Jones Solids (20 min) 6. Timur T. Bazhirov (Moscow, Russia): Cavitation in Stretched Molten Lead (20 min) 7. Alexey V. Yanilkin (Moscow, Russia): Decay of Solid Iron under Superheating and Stretching (20 min) 8. Henry E. Norman (Moscow, Russia): On the Thermodynamic Instability of Ultra-Cold Non-Ideal Plasmas (20 min)
vch 4 Okt 2005 10:43
21.1 Programs of the Research Workshops 2003, 2004 and 2005
489
Special lectures: 19. 00 1. Ivan S. Gutzow (Sofia, Bulgaria): Some Personal Reflections on Science and History Thursday, June 30: 9. 00 1. Werner Ebeling (Berlin, Germany): Past, Present and Future of Thermodynamics – 100 Years After Einstein and Nernst 2. Alexander P. Chetverikov (Saratov, Russia): Structure Formation In Ensembles of Active Brownian Particles 3. E. A. Andreev, Alexander V. Kosarim, B. M. Smirnov (Moscow, Russia): Structure Evolution of a System of Hard Spheres in a Box with Oscillating Walls 4. G. T. Guria, Evgenij A. Katrukha (Moscow, Russia): Instabilities and Catastrophies in Tubuline Dynamics 5. Vitali V. Slezov (Kharkov, Ukraine): Diffusional Decomposition of Supercooled Liquids in Heat-Insulated Systems 6. Naoum M. Kortsenstein (Moscow, Russia): Modelling of Heterogeneous Chemical Reactions in the Upper Layers of the Atmosphere 7. Victor B. Kurasov (St. Petersburg, Russia): Kinetics of Nucleation with Specific Regimes of Droplet Growth Friday, July 1: 9. 00 1. Vladimir M. Fokin (St. Petersburg, Russia), J. W. P. Schmelzer (Rostock, Germany), E. D. Zanotto (Sao Carlos, Brazil), and I. Avramov (Sofia, Bulgaria): Stress Development and Stress Relaxation During Crystal Growth in GlassForming Melts 2. J¨ org M¨ oller (Dresden, Germany): Structure Formation in Polymeric Liquids under Flow 3. Sergey P. Fisenko, A. A. Brin (Minsk, Belorussia): Simulation of the Performance of a Laminar Flow Chamber in Nucleation Experiments 4. Alexander K. Shchekin, I. V. Shabaev (St. Petersburg, Russia): Thermodynamics and Kinetics of Deliquescense of Small Soluble Particles Saturday, July 2: 9. 30
vch 4 Okt 2005 10:43
490
21 Appendices
1. Andriy M. Gusak, T. Zaporozhetz (Cherkassy, Ukraine), A. Vairagar, S. Mhaisalkar (Singapore), K. N. Tu (Los Angeles, USA): Nucleation, Migration, Trapping and Coalescence of Nano-Voids at the Interface Copper/Dielectric under a Strong Current 2. Victor M. Lebedev, V. T. Lebedev, S. P. Orlov (St. Petersburg, Russia), I. N. Tolstikhin, B. Z. Pevzner (St. Petersburg, Russia): Formation, Evolution and Structure of Radiation Defects in Natural and Synthetic Quartz Irradiated by Fast Neutrons 3. Irina A. Zhuvikina, A. P. Grinin, G. Yu. Gor (St. Petersburg, Russia): NearestNeighbour Drop Approximation in the Kinetics of Homogeneous Condensation 4. Alexander P. Grinin, I. A. Zhuvikina, G. Yu. Gor (St. Petersburg, Russia): Non-stationary Vapor Concentration and Temperature Fields in the Vicinity of a Growing Drop: The Balance of Condensing Substance and Phase Transition Heat
21.2
Content of the Proceedings Nucleation Theory and Applications for the Periods 1997-1999 and 2000-2002 and of the Monograph Nucleation Theory and Applications with Overview Lectures Published by WILEY-VCH in 2005 Nucleation Theory and Applications
J. W. P. Schmelzer, G. R¨ opke, V. B. Priezzhev (Eds.) Joint Institute for Nuclear Research Publishing Department Dubna, Russia, 1999 1. Introduction
1
2. V. V. Slezov, J. Schmelzer: Kinetics of Nucleation-Growth Processes: The Initial Stages
6
3. J. W. P. Schmelzer, G. R¨ opke, J. Schmelzer, Jr., V. V. Slezov: Shapes of Cluster Size Distributions Evolving in Nucleation-Growth Processes
82
4. W. Kleinig, J. W. P. Schmelzer, G. R¨ opke: Fragmentation in Dissipative Collisions: A Computer Model Study 130 5. F. M. Kuni, A. P. Grinin, V. B. Kurasov: Kinetics of Condensation at a Gradual Creation of the Metastable State
vch 4 Okt 2005 10:43
160
21.2 Content of Proceedings
491
6. N. M. Kortsenstein, E. W. Samuilov: The Effect of Temperature Pulsations on the Rate of Homogeneous Nucleation in Vapor Condensation 194 7. F. M. Kuni, A. K. Shchekin, A. P. Grinin: Kinetics of Condensation on Macroscopic Solid Nuclei at Low Dynamic Vapor Supersaturation
208
8. J. W. P. Schmelzer, J. Schmelzer Jr., I. S. Gutzow: Reconciling Gibbs and van der Waals: A New Approach to Nucleation Theory
237
9. J. W. P. Schmelzer: Comments on Curvature Dependent Surface Tension and Nucleation Theory 268 10. V. B. Priezzhev: The Upper Critical Dimension of the Abelian Sandpile Model 290 11. A. M. Gusak, A. O. Bogatyrev: Thermodynamic and Kinetic Constraints on the Nucleation of Intermediate Phases During Reactive Diffusion 310 12. A. K. Shchekin, D. V. Tatianenko, F. M. Kuni: Towards Thermodynamics of Uniform Film Formation on Solid Insoluble Particles 320 13. S. I. Bastrukov, D. V. Podgainy, I. V. Molodtsova, V. S. Salamatin, O. I. Streltsova: Elastodynamics of Nuclear Fission in the Elastic Globe Model 341 14. B. M. Smirnov: Excitations and Melting of a System of Bound Atoms
355
15. I. Gutzow, D. Ilieva, V. Yamakov, Ph. Babalievski, L. D. Pye: Glass Transition: An Analysis in Terms of a Differential Geometry Approach
368
16. G. A. Sycheva: Nucleation of Sodium Zinc Phosphate: Whether it is Homogeneous or Not?
419
17. J. W. P. Schmelzer, T. N. Vasilevskaya, N. S. Andreev: On the Initial Stages of Spinodal Decomposition
425
18. B. J. Mokross: Nucleation in Non-Homogeneous Enthalpy Systems
445
19. J. M¨ oller, J. W. P. Schmelzer, I. Gutzow: Shapes of Critical Clusters under Stress
464
20. Appendices
492
Nucleation Theory and Applications J. W. P. Schmelzer, G. R¨ opke, V. B. Priezzhev (Eds.) Joint Institute for Nuclear Research Publishing Department Dubna, Russia, 2002
vch 4 Okt 2005 10:43
492
21 Appendices
1. Introduction
1
2. V. P. Skripov, M. Z. Faizullin: Solid-Liquid and Liquid Vapor Phase Transitions 4 3. V. G. Baidakov: Boiling-Up Kinetics of Solutions of Cryogenic Liquids
19
4. J. W. P. Schmelzer, J. Schmelzer Jr.: Kinetics of Bubble Formation and the Tensile Strength of Liquids 88 5. J. W. P. Schmelzer, V. G. Baidakov: Kinetics of Boiling in Binary Liquid-Gas Solutions 120 6. S. P. Fisenko, R. H. Heist: High Pressure Nucleation Experiments in Diffusion Cloud Chambers 146 7. N. M. Kortsenstein, E. V. Samuilov: Effect of Pulsation of Thermodynamic Parameters on Condensation 165 8. V. B. Kurasov: Heterogeneous Decay of Metastable Phases
177
9. S. Todorova, I. Gutzow, J. W. P. Schmelzer: Nucleation at Increasing and Decreasing Supersaturation 215 10. V. Soares, E. Meyer: Non-Gaussian Distributions of Volume and Temperature in Nucleation 235 11. E. M. de Sa, E. Meyer, V. Soares: Adiabatic Nucleation in Strongly Supersaturated Vapors 242 12. V. P. Koverda, V. N. Skokov, A. V. Reshetnikov: (1/f)-Noise at NonEquilibrium Phase Transitions 259 13. S. N. Burmistrov, L. B. Dubovskij, T. Satoh: Quantum-Kinetics of FirstOrder Phase Transitions 273 14. A. R. Gokhman, J. B¨ ohmert, A. Ulbricht: A Kinetic Study of Cluster Ensemble Evolution in VVER-Reactors 313 15. R. S. Berry, B. M. Smirnov: The Void Concept for Phase and Glass Transitions in Clusters 340 16. J. Dunkel, W. Ebeling, J. W. P. Schmelzer, G. R¨ opke: Dissipative Collisions of One-Dimensional Morse Clusters 367 17. L. Landa, K.Landa, S. Thomsen: Crystalloids, Polymorphism, and Configurational Pressure in Glasses 395
vch 4 Okt 2005 10:43
21.2 Content of Proceedings
493
18. N. M. Kortsenstein, A. L. Tseskis: Finite-Size Effects in Ginzburg-Landau Model of Second-Order Phase Transitions 416 19. I. Gutzow, Ts. Grigorova, J. W. P. Schmelzer: Irreversible Thermodynamics, Reaction Kinetics, and Relaxation 428 20. Micro- and Nanostructures: A Little Picture Book
469
21. Final Remarks
489
22. Appendices
490
Copies of the proceedings 1999, 2002, 2006 can be ordered via: Dr. J¨ urn W. P. Schmelzer Institute of Physics University of Rostock 18051 Rostock, Germany Email:
[email protected] [email protected]
J. W. P. Schmelzer (Editor): Nucleation Theory and Applications Wiley-VCH Publishers, Berlin-Weinheim, 2005 1. J. W. P. Schmelzer: Introductory Remarks
1
2. V. P. Skripov, M. Z. Faizullin: Solid-Liquid and Liquid-Vapor Phase Transitions: Similarities and Differences 4 3. V. V. Slezov, J. W. P. Schmelzer, A. S. Abyzov: A New Method of Determination of the Coefficients of Emission in Nucleation Theory 39 4. V. M. Fokin, N. S. Yuritsyn, E. D. Zanotto: Nucleation and Crystallization Kinetics in Silicate Glasses: Experiment and Theory 74 5. V. G. Baidakov: Boiling-Up Kinetics of Solutions of Cryogenic Liquids
126
6. V. Ya. Shur: Correlated Nucleation and Self-Organized Kinetics of Ferroelectric Domains 178 7. S. A. Kukushkin, A. V. Osipov: Nucleation and Growth Kinetics of Nanofilms 215
vch 4 Okt 2005 10:43
494
21 Appendices
8. I. Gutzow, S. Todorova, L. Kostadinov, E. Stoyanov, V. Guencheva, G. V¨olksch, C. Dunken, Ch. R¨ ussel: Diamonds by Transport Reactions with Vitreous Carbon and from the Plasma Torch: New and Old Methods of Metastable Diamond Synthesis and Growth 256 9. A. K. Shchekin, A. P. Grinin, F. M. Kuni, A. I. Rusanov: Nucleation in Micellization Processes 312 10. A. M. Gusak, F. Hodaj: Nucleation in a Concentration Gradient
375
11. J. W. P. Schmelzer, G. Sh. Boltachev, V. G. Baidakov: Is Gibbs’ Thermodynamic Theory Really Perfect? 418 12. J. W. P. Schmelzer: Summary and Outlook
447
13. Index
453
21.3
List of Participants
1. Dr. Alexander Sergeevich Abyzov Kharkov Institute of Physics and Technology, Academician Street 1, 310 108 Kharkov, Ukraine Phone: (+38 0572) 98 68 85; Email:
[email protected] 2. Prof. Victor L. Aksenov Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia Phone: (+7 09621) 65796, Fax: (+7 09621) 65484; Email:
[email protected] 3. Dr. Mikhail V. Avdeev Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia Phone: (+7 09621) 65796, Fax: (+7 09621) 65484; Email:
[email protected] 4. Prof. Vladimir Georgievich Baidakov Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsena 106, 620016 Ekaterinenburg, Russia Phone: (+7 3432) 678806 (office), (+7 3433) 312129 (home), Fax: (+7 3432) 678800; Email:
[email protected] 5. Timur Tynlybekovich Bazhirov Institute for High Energy Densities, 125412 Moscow, Izhorskaya 13-19, Russian Academy of Sciences Phone: (+7 095) 4842456, Fax: (+7 095) 485 7990; Email: timur
[email protected]
vch 4 Okt 2005 10:43
21.3 List of Participants
495
6. Dr. German Vitalievich Berezhnoy Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, 199155 St. Petersburg, Russia Phone: (+7 812) 351 08 29, Fax: (+7 812) 3510811; Email:
[email protected] 7. Dr. Alexander Olegovich Bogatyrev Cherkasy State University, Department of Applied Mathematics, 81 Shevchenko blvd., Cherkasy, Ukraine, 18017 Phone: (+38 0472) 361355, Fax: (+38 0472) 372142; Email:
[email protected] 8. Dr. Galina Grigorievna Boiko Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, 199155 St. Petersburg, Russia Phone: (+7 812) 351 40 07, Fax: (+7 812) 351 08 05; Email:
[email protected] 9. Dr. Grey Shamilovich Boltachev Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsena 106, 620016 Ekaterinenburg, Russia Phone: (+7 3432) 678802 (office), Fax: (+7 3432) 745450; Email:
[email protected] 10. Dr. Boris Fedorovich Borisov Institute of Physics, St. Petersburg State University, Petrodvoretz, 198 504 Russia Phone: (+7 812) 42 84 330, Fax: (+7 812) 42 87 240; Email:
[email protected] 11. Anton Anatolievich Brin A. V. Luikov Heat and Mass Transfer Institute, Academy of Sciences of Belarus, P. Brovka Str. 15, Minsk 220072, Belarus Phone: (+37 5295) 549170; Email:
[email protected] 12. Dr. Rimma Sergeevna Bubnova Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, 199155 St. Petersburg, Russia Phone: (+7 812) 328 85 91; Email: rimma
[email protected] 13. Dr. Sergey Nikolaevich Burmistrov RRC Kurchatov Institute, Moscow Phone: (+7 095) 1969384, Fax: (+7 095) 9430074; Email:
[email protected] 14. Dr. Elena Vladimirovna Charnaya Institute of Physics, St. Petersburg State University, Petrodvoretz, 198 504 Russia Phone: (+7 812) 42 84 330, Fax: (+7 812) 42 87 240; Email:
[email protected] 15. Prof. Alexander Petrovich Chetverikov Saratov State University, Department of Nonlinear Processes, Astrakhanskaya ul. 83, 410012 Saratov, Russia Phone: (+7 8452) 370 630 (home), (+7 8452) 273385 (office), Fax: (+7 8452) 278529; Email:
[email protected]
vch 4 Okt 2005 10:43
496
21 Appendices
16. Dr. Leonid Nikolaevich Davydov Kharkov Institute of Physics and Technology, Academician Street 1, 61108 Kharkov, Ukraine Phone: (+38 0572) 45 00 26 (home), (+38 0572) 35 65 70 (office), Fax: (+38 0572) 352683; Email:
[email protected] 17. Dr. Karlygash Nurmanova Dshumagulova Al Farabi Kazakh National University, IETP, Almaty, 480012, Tole bi 96, Kazakhstan Phone: (+7 3272) 416494; Email:
[email protected];
[email protected] 18. Dr. Leonid Borisovich Dubovskij RRC Kurchatov Institute, Moscow Phone: (+7 095) 1969384, Fax: (+7 095) 9430074; Email:
[email protected] 19. J¨ orn Dunkel Institut f¨ ur Physik, Humboldt-Universit¨ at Berlin, Invalidenstr. 110, 10115 Berlin, Germany Fax: (+40-030) 2093 7638; Email:
[email protected];
[email protected] − berlin.de 20. Prof. Werner Ebeling Institut f¨ ur Physik, Humboldt-Universit¨ at Berlin, Newtonstr. 15, 12489 Berlin, Germany Phone: (+49 30) 2093 7637 (office), Fax: (+49 30) 2093 7638; Email:
[email protected];
[email protected]; werner
[email protected] 21. Dr. Mars Zakievich Faizullin Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsena 106, 620016 Ekaterinenburg, Russia Phone: (+7 3432) 679586 (office), Fax: (+7 3432) 678800; Email:
[email protected] 22. Dr. Rainer Erich Feistel Institut f¨ ur Ostseeforschung Rostock - Warnem¨ unde, Seestr. 15, 18119 Warnem¨ unde Phone: (+49 381) 5197 152, Fax: (+49 381) 5197 4818; Email: rainer.feistel@io − warnemuende.de 23. Prof. Stanislav Konstantinovich Filatov St. Petersburg State University, Dept. of Crystallography, 199034 St. Petersburg, Russia Phone: (+7 812) 328 96 47; Email: fi
[email protected] 24. Prof. Sergey Pavlovich Fisenko A. V. Luikov Heat and Mass Transfer Institute, Academy of Sciences of Belarus, P. Brovka Str. 15, Minsk 220072, Belarus Phone: (+37 5172) 842 222, Fax: (+37 5172) 322 513; Email:
[email protected] 25. Dr. Vladimir Mikhailovich Fokin S. I. Vavilov State Optical Institute, Russian Academy of Sciences, ul. Babushkina 36-1, 193171 St. Petersburg, Russia Phone: (+7 812) 355 30 38; Email:
[email protected]
vch 4 Okt 2005 10:43
21.3 List of Participants
497
26. Prof. Alexander Rafailovich Gokhman South Ukrainian Pedagogical University, Department of Physics, Staroportofrankovskaya 26, 65020 Odessa, Ukraine Phone: (+38 048) 732 5107, Fax: (+38 048) 732 5103; Email:
[email protected] 27. Dr. Olga Yurievna Golubeva Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, 199155 St. Petersburg, Russia Phone: (+7 812) 328-85-89, Fax: (+7 812) 3510811; Email: olga
[email protected] 28. Prof. Victor Nikitich Grigoriev Institute for Low Temperature Physics and Engineering, Academy of Sciences of Ukraine, 61103 Kharkov, Ukraine, prospect Lenina 47 Phone: (+38 0572) 308581 (office); (+38 0572) 7040514 (home), Fax: (+38 0572) 335593; Email:
[email protected] 29. Prof. Alexander Pavlovich Grinin Department of Statistical Physics, St. Petersburg State University, Research Institute of Physics, Ulyanovskaya Str. 1, Petrodvoretz, St. Petersburg 198904 Phone: (+7 812) 5346415 (home), Fax: (+7 812) 428 72 40 30. Prof. Dieter E. H. Gross Hahn-Meitner Institut Berlin, Abt. Theoretische Physik, Glienicker Str. 100, 14109 Berlin, Germany Phone: (+49 30) 8062 2677 (office), Fax: (+40-030) 2093 7638; Email:
[email protected] 31. Evelyn Gross 32. Prof. Georgi Theodorovich Guria National Science Center Hematology, Russian Academy of Sciences, Moscow 125167, Russia Phone: (+7 095) 214 9948, Fax: (+7 095) 212 4252; Email:
[email protected] 33. Prof. Andriy Michailovich Gusak Cherkasy State University, Chair of Theoretical Physics, Shevchenko Str. 81, 18017 Cherkasy, Ukraine Phone: (+38 0472) 471220 (office), (+38 0472) 541655 (home), Fax: (+38 0472) 472233; Email:
[email protected];
[email protected] 34. Prof. Ivan Stoyanov Gutzow Full Member of the Bulgarian Academy of Sciences, Institute of Physical Chemistry, Bulgarian Academy of Sciences, Academician Bonchev Street 11, Sofia 1113, Bulgaria Phone: (+359 2) 797 2552 (office), (+359 2) 72 49 93 (home), Fax: (+359 2) 971 26 88; Email:
[email protected]
vch 4 Okt 2005 10:43
498
21 Appendices
35. Dr. Dmitri Yurievich Ivanov St. Petersburg State University of Refrigeration and Food Engineering, Lomonossov str. 9, St. Petersburg, 191002 Russia Phone: (+7 812) 164 2198 (office); (+7 812) 373 0587 (home); Email:
[email protected] 36. Yevgen Alexandrovich Katrukha National Science Center Hematology, Russian Academy of Sciences, Moscow 125167, Russia Phone: (+7 095) 257 3903, Fax: (+7 095) 212 4252; Email:
[email protected] 37. Tatyana Sergeevna Ketova Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsena 106, 620016 Ekaterinenburg, Russia Phone: (+7 3432) 678801 (office), Fax: (+7-3432) 678800; Email:
[email protected] 38. Dr. Boris Alexandrovich Klumov Max-Planck-Institut f¨ ur Extraterrestrische Physik, Center for Interdisciplinary Plasma Physics, Giessenbachstr., 85740 Garching Phone: (+49 89) 3 0000 3396, Fax: (+49 89) 3 0000 3950; Email:
[email protected] 39. Alexey Yurievich Kuksin Institute for High Energy Densities, Izhorskaya 13/19, 125412 Moscow, Russia Phone: (+7 095) 484 2456; Email:
[email protected] 40. Prof. Sergey Arsenevich Kukushkin Institute of Mechanical Engineering, Russian Academy of Sciences, Bolshoy Prosp. 61, Vasilievski Ostrov, St. Petersburg, 199178, Russia Phone: (+7-812) 217 31 13 (office), Fax: (+7-812) 217 86 14; Email:
[email protected] 41. Alexander Victorovich Kosarim Moscow Energetic Institute Phone: (+7 095) 5290080; 8-903-1191715 (mobile); Email:
[email protected] 42. Dr. Naoum M. Kortsenstein Krzhizhanovsky Power Engineering Institute, Leninskij Prosp. 19, 117927 Moscow, Russia Phone: 095 362 7841, Fax: 095 954 4250; Email: kor@ihpc − i.mcsa.ru 43. Dr. Victor Borisovich Kurasov Department of Statistical Physics, St. Petersburg State University, Research Institute of Physics, Ulyanovskaya Str. 1, Petrodvoretz, St. Petersburg 198904 Fax: (+7 812) 428 72 40; Email: Victor
[email protected] 44. Prof. Leonid Mendel Landa Guardian Industries Corporation, 14511 Romine Road, Carleton, MI 48117, USA Phone: (+1 734) 654 4715, Fax: (+1 734) 654 4750; Email:
[email protected]
vch 4 Okt 2005 10:43
21.3 List of Participants
499
45. Dr. Anton Yurievich Lavrenov Department of Solid State Physics, Physics Department, Moscow State University, 119899 Moscow Phone: (+7 095) 939 4610; Email:
[email protected] 46. Dr. Victor Mikhailovich Lebedev St. Petersburg Nuclear Physics Institute of Russian Academy of Sciences Phone: (+7 81371) 46684; Email:
[email protected] 47. Elena Olegovna Litovtschik Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia Phone: (+7 812) 4284515 (office), (+7 812) 2327459 (home), Fax: (+7 812) 3510811 48. Prof. Igor Lavrentjevich Maksimov Nizhny Novgorod University, 23 Gagarin Avenue, 603000 Nizhny Novgorod, Russia Phone: (+7 831) 265 6255, Fax: (+7 831) 265 8592; Email:
[email protected] 49. Prof. Nikolay Petrovich Malomuzh Department of Theoretical Physics, Odessa National University, Dvoryanskaya str. 2, 65026 Odessa, Ukraine Phone: (+38 0482) 230418, Fax: (+38 0482) 238936; Email:
[email protected] 50. Dr. J¨ org M¨ oller Scitecon, Bayreuther Str. 13, 01187 Dresden, Germany Phone: 0173 4176567 (mobile); Email:
[email protected] 51. Dr. Evgenij Dmitrievich Nikitin Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsena 106, 620016 Ekaterinenburg, Russia Phone: (+7 3432) 678811 (office), 312129 (home), Fax: (+7 3432) 678800; Email:
[email protected] 52. Dr. Madis Grigori Noppel ¨ Institute of Environmental Physics, University of Tartu, Ulikooli 18, 50090 Tartu, Estonia Email:
[email protected] 53. Prof. Henry E. Norman Institute for High Energy Densities associated with the Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13-19, Moscow 125 412, Russia Phone: (+7 095) 4842456 (office), (+7 095) 2901018 (home), Fax: (+7 095) 4857990; Email: henry
[email protected] 54. Dr. Andrey V. Osipov Institute of Mechanical Engineering, Russian Academy of Sciences, Bolshoy Prosp. 61, Vasilievski Ostrov, St. Petersburg, 199178, Russia Phone: (+7 812) 217 31 13 (office), Fax: (+7 812) 217 86 14; Email:
[email protected]
vch 4 Okt 2005 10:43
500
21 Appendices
55. Dr. Oleg Adanevich Osmayev 56. Dr. Boris Zelmanovich Pevzner Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, 199155 St. Petersburg, Russia Email:
[email protected] 57. Boris Petrov Petrov Institute of Solid State Physics, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, Sofia 1784, Bulgaria Phone: (+359 2) 7144 285 (office), (+359 2) 852 4914 (home); Email: petroff@issp.phys.bas.bg 58. Dr. Irina Georgievna Polyakova Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia Phone: (+7 812) 3510829, Fax: (+7 812) 3510811; Email: ira
[email protected] 59. Dr. Oleg Vasilevich Potapov Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia Email: oleg
[email protected] 60. Prof. Vyatcheslav B. Priezzhev Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia Phone: (+7 09621) 65333, Fax: (+ 7 09621) 65084; Email:
[email protected] 61. Dr. Sergey Pavlovich Protsenko Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsena 106, 620016 Ekaterinenburg, Russia Phone: (+7 3432) 678806 (office), (+7 3433) 312129 (home), Fax: (+7 3432) 678800; Email:
[email protected] 62. Elena Nikolaevna Rusakovich Joint Institute for Nuclear Research, 141 980 Dubna, Russia Phone: (+7 09621) 63890; Email:
[email protected] 63. Galina Gavrilovna Sandukovskaya Bogoiliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia Phone: (+7 09621) ; Email:
[email protected] 64. Dr. J¨ urn W. P. Schmelzer Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia & Institut f¨ ur Physik, Universit¨ at Rostock, 18051 Rostock, Germany Phone: (+7 09621) 63703 (office), (+ 7 09621) 47008 (home), Fax: (+ 7 09621) 65084; Email:
[email protected],
[email protected]
vch 4 Okt 2005 10:43
21.3 List of Participants
501
65. Prof. Alexander Kimovich Shchekin Department of Statistical Physics, St. Petersburg State University, Research Institute of Physics, Ulyanovskaya Str. 1, Petrodvoretz, St. Petersburg 198904 Phone: (+7 812) 2327459 (home), (+7 812) 428 7459 (office), Fax: (+7 812) 428 72 40; Email:
[email protected] 66. Dr. Olga Alexeevna Shilova Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia Phone: (+7 812) 328596 (office), (+7 812) 18505667 (home), Fax: (+7 812) 3285401; Email:
[email protected] 67. Dr. Aram Sergeevich Shirinyan Cherkasy State University, Chair of Theoretical Physics, Shevchenko Str. 81, 18031 Cherkasy, Ukraine Phone: (+38 0472) 471220 (office), Fax: (+38 0472) 354463; Email:
[email protected] 68. Prof. Vladimir Yakovlevich Shur Ural State University, Lenin Avenue 51, 620083 Ekaterinburg, Russia Phone: (+7 3432) 617436, Fax: (+7 3432) 615 978; Email:
[email protected] 69. Prof. Vladimir Pavlovich Skripov Full Member of the Russian Academy of Sciences, Institute of Thermal Physics, Ural Branch of the Russian Academy of Sciences, ul. Amundsena 106, 620016 Ekaterinenburg, Russia Phone: (+7 3432) 493578 (office), Fax: (+7 3432) 745450; Email:
[email protected] 70. Prof. Vitali Valentinovich Slezov Corresponding Member of the Ukrainian Academy of Sciences, National Science Center, Kharkov Institute of Physics and Technology, Institute for Theoretical Physics, Academician Street 1, 310108 Kharkov, Ukraine Phone: (+38 0572) 40 42 54 (office), (+38 0572) 35 15 20 (home), Fax: (+38 0572) 35 17 38; Email:
[email protected] 71. Prof. Boris Michailovich Smirnov Institute of High Temperatures, Russian Academy of Sciences, Izhorskaya Str. 13/19, Moscow, 127 412 Phone: (+7 095) 190 42 44 (office), (+7 095) 196 83 10 (home), Fax: (+7 095) 4859922; Email:
[email protected] 72. Elena Nikolaevna Soboleva Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia Phone: (+7 812) 3510829, Fax: (+7 812) 3510811
vch 4 Okt 2005 10:43
502
21 Appendices
73. Dr. Vladimir Vladimirovich Stegailov Institute of High Temperatures, Russian Academy of Sciences, Izhorskaya Str. 13/19, Moscow, 127 412 Email:
[email protected] 74. Dr. Galina Alexandrovna Sycheva Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, St. Petersburg, 199155 Russia Phone: (+7 812) 3510829, Fax: (+7 812) 3510811; 75. Evgenij Vladimirovich Syrnikov Institute for Low Temperature Physics and Engineering, Academy of Sciences of Ukraine, 61103 Kharkov, Ukraine, prospect Lenina 47 Phone: (+38 0572) 308581 (office), (+38 057) 7040514 (home), Fax: (+38 0572) 335593; Email:
[email protected] 76. Timur V. Tropin Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia Phone: (+7 09621) 65796, Fax: (+7 09621) 65484; Email:
[email protected] 77. Dr. Alexander Lvovich Tseskis Am Weidenhof 29, 51381 Leverkusen, Germany Email:
[email protected] 78. Dr. Aleksey Michailovich Vishnyakov TRI Princeton, 601 Prospekt Avenue, Princeton NJ 08542, USA Phone: (+1 609) 430 4825, Fax: (+1 609) 683 7149; Email:
[email protected] 79. Alexei Vitalievich Yanilkin Institute for High Energy Densities, Russian Academy of Sciences, 125412 Moscow, Izhorskaya 13-19 Phone: (+7 095) 4842300, Fax: (+7 095) 485 7990; Email:
[email protected] 80. Dr. Nikolai Sergeevich Yuritsyn Institute of Silicate Chemistry, Russian Academy of Sciences, ul. Odoevskogo 24/2, 199155 St. Petersburg, Russia Phone: (+7 812) 351 08 29, Fax: (+7 812) 3510811); Email:
[email protected] 81. Prof. Maksim Anatolievich Zakharov Novgorod State University, Department Solid State Physics and Mechanics, 173003 Novgorod, Peterburgskaya 41 Phone: (+7 8162) 11 68 91 (office), Fax: (+7 8162) 24110; Email:
[email protected] 82. Eduard Rifonovich Zhdanov
vch 4 Okt 2005 10:43
21.3 List of Participants
503
83. Dr. Dmitry Igorievich Zhukovitskii Institute of High Temperatures, Russian Academy of Sciences, Izhorskaya Str. 13/19, Moscow, 127 412 Phone: (+7 095) 362 5310 (office), (+7 095) 196 83 10 (home), Fax: (+7 095) 485 9922; Email:
[email protected] 84. Dr. Vyatcheslav Ivanovich Zhuravlev Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141 980 Dubna, Russia Phone: (+7 09621) 65277 (office), Fax: (+ 7 09621) 65084; Email:
[email protected] 85. Dr. Irina Alexeevna Zhuvikina St. Petersburg State University, Rectorate Email:
[email protected]
vch 4 Okt 2005 10:43
