Metallic supercooled-liquid to solid (glass or crystal) transitions were studied previ- ously by ... i.e., n. = 500, 1000 and 2000 Na atoms. The values of the mean atomic volume, Q, and the ... is an extension of the pure supercooled-liquid's Q(T).
CHINESE JOURNAL OF PHYSICS
FEBRUARY 1995
VOL. 33, NO. 1
A Molecular-Dynamics Study of Liquid-Solid Transitions II: System-Size aud Pressure Effects Shiow-Fon Tsay, C. F. Liu, and S. Wang’ Department of Physics, National Sun Yai-Sen University, Kaohsiung, Taiwan 804, R. 0. C. (Received October l?, 1994)
A molecular-dynamics (hlD) calculation of the transitions between Na-liquid and Na-solid (crystal or glass) is performed for (a) different system sizes, (b) different initial conditions of the prepared Na liquids, and (c) low and high pressures. In addition, the system-size and pressure effects on the transitions of interest are demonstrated. It appears that (i) the system-size effect is much more significant for the transitions between metallic liquid and crystal than for the metallic glass transitions, lending support to the validity of the frozen model of glasses, (ii) the system-size effect decreases rapidly as the number of atoms, R, increases from 1500 and becomes negligible as n approaches 3500 in a realistic constant-pressure MD calculation of the metallic liquid-crystal transition, (iii) high pressure significantly affects the time of formation of critically sized nuclei
and crystal growth through the change of volume in the crystallization of a liquid and (iv) for a system,to be a good glass-former, the atoms in this system must have a large probability of forming non-crystalline-like local ordered units. PACS. 61.20. Ja - Computer simulation. PACS. 61.20. Lc - Time-dependent properties; glass transit.ions. PACS. 61.25. hlv - Liquid metals and alloys.
_I
I. INTRODUCTION Metallic supercooled-liquid to solid (glass or crystal) transitions were studied previously by making use of molecular-dynamics (AID) simulations on a small metallic system [l-4]. The obtained results have only been used to show (i) the qualitative feature of homogeneous nucleation and growth of crystalline phases and (ii) the temperature effects on the microstructure of rapidly quenched metals. This limitation occurs because (a) the variation of system volume with both the temperature and coolin,m rate, occurring in the esperimental cooling process, was not included adequately in these previous hlD studies and (b) the size of the system in those studies is small compared to real metals. Employing highly reliable first-principles atomic potentials, such as that proposed in [s], the inclusion of point (a) in 75
@ 199.5 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA
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a MD calculation, using the Verlet algorithm [G] with the integration time step equal to 1.5 x lo-r5 set, for some metallic systems is possible by explicitly introducing the variation of volume with temperature at constant pressure in the calculation. This calculation was performed in Ref. [7] (to b e referred to as Paper I hereafter) in a similar manner as in the well known constant-volume calculation [8] for given constant pressure at each temperature. The obtained thermodynamic and structural results appear to be reliable as can be seen from the comparison of theoretical and experimental results in Figs. 2 and 8 in Paper I and those in Ref. [9]. The effects of point (b) have been studied for va.rious model systems [lo-191. Some investigators have concluded that the periodic boundary conditions used in the MD simulations interfere with the process of nuclea.tion and crystal growth for systems of up to 500 atoms [lO,ll]. From the simulations of a system of 4000 soft spheres, others [13,14] have concluded that a 4000-atom system is la.rge enough to observe the supercooled-liquid to solid (glass or crystal) transition free from the effects of periodic boundaries. However, the results of a recent constant-volume simulation of homogeneous nucleation in a large Lenneard-Jones system (lo6 atoms) [18] lea.ds to the conclusion that the system-size effects become unimportant only for those model systems each containing more than lo4 atoms. If this should be the case for the MD study of all the physical properties involved in the liquid to solid transition, the reliability of most earlier MD studies of this transition should be in doubt, because the number of atoms used in these studies is small compared to lo4 atoms. In view of this and of the fact that the liquid-crystal transition depends upon the interatomic pair potential used [l], in this work we first examine if the above noted conclusions apply to a realistic constant-pressure MD simulation of metallic solidification. Following this, in order to gain insight in the basic nature of the supercooled-liquid to solid transition, we check the pressure effects on the liquid to solid transitions. All of this-is presented in Sec. II. Some conclusions are dralrn in Sec. III. II. CALCULATIONS AND RESULTS II- 1. System-size effects The h!ID simulation on the Na-liquid t.o crystal transition of n Na atoms, as described in Paper I, is re-performed with a cooling ra.te R, = 1.5 x 10 l2 K/s for different system-sizes, i.e., n. = 500, 1000 and 2000 Na atoms. The values of the mean atomic volume, Q, and the pair distribution functions (PDF) for the crystals (bee [4]). obtained from this simulation, are displayed in Figs. 1 and 2, respectively. The latter figure shows that the increase of n from 500 in the present h1D simulation has little effect on the basic structure of the crystal produced in the process of the liquid to crystal transition. Now, we come to clarify the n-dependence of crystallization, as shown in Fig. 1. For this, we apply the pair-analysis
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SHIOW-FONTSAY,C.F.LIU,AND
S.WANG
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275 -
270 -
/
250' 60
fl,’ 160
I
260
360
-UK) FIG. 1. The calculated values of Q(r) In tt le case of the Sa-liquid to crystal transition at constant pressure (5 bar) for n = 500, 1000 and 2000 atoms (see test for detail). The dashed line is an extension of the pure supercooled-liquid’s Q(T). Each horizontal line with arrows
represents the standard deviaton of T< arising from different initial conditions.
technique (see Ref. [20] for details) to the determination of the change in the number of the crystalline bee units involved in the transition from the supercooled liquid to bee-like crystal noted here. Of the obtained results, those for (a) n. = 500 and (b) n = 1000 are summarized in Table I along with the relative numbers of the associated atomic bonded pairs, obtained using the pair-analysis technique and denoted by Nijkl herein. Nrcsr and Nr44r denote, respectively, the relative numbers of lGG1 and 1441 a.tomic bonded pairs. These pairs are characteristic of bee-crystalline units, each unit consisting of eight lGG1 pairs and six 1441 pairs. Nrssi denotes the relative number of 1551 atomic bonded pairs in the system considered, these pairs being characteristic of non-crystalline (icosahedral) structure (201. N1541 and Nidsr represents the relative number of 1541 and 1431 atomic bonded pairs, which are also characteristic of non-crystalline structure. From Table I, the number of the crystalline bee units in the rapidly quenched Na liquid under study increases faster for 7~ = 500 than for n 1 1000 in going down from 300 I< to a low T. That is, in the
I..“_
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A MOLECULAR-DYNAMICS STUDY OF LIQUID-SOLID .
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TABLE I. The number of bee-type polyhedra per 500 Na atoms, A4rt,oc, and the relative numbers of the associated a.tomic bonded pairs in the supercooled-liquid to crystal transition for (a) n = 500 and (b) n = 1000 at selected T ’ s .
T(K) 300
Nbcc
N1661
A’1 441
N1551
hT1541
N1431
a b
0 0
0.09 0.12
0.97 0.09
0.23 0.27
0.18 0.15
0.18 0.14
;
1 0
0.15 0.15
0.11 0.11
0.28 0.26
0.15 0.16
0.12 0.13
b”
129 10
0.38 0.17
0.30 0.13
0.08 0.26
0.08 0.16
0.05 0.13
250 220
4
3 ,z zl 2
1 I t t
-
loo0
--- So0
i i
1
i : /
1
FIG. 2. The calculated g(r)‘s at T = 100 K (2 x) I. n the case of the Na-liquid to crystal transition.
case presently considered the time of formation of potential nuclei for the start of crystallization is smaller for n = 500 than for n 2 1000, this being consistent with the literature [12,15]. Consequently, crystallization occurs faster (or say, at higher T) for R = 500 than
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SHIOW-FON TSAY, C. F. LIU, AND S. WANG
for n 2 1000, as shown in Fig. 1. To proceed further, the above-mentioned MD simulation is also re-performed for five different initial conditions of the prepared liquid state (which can be obtained by producing ion configurations separated by several thousand steps on the trajectory of n atoms under the condition of the same total energy [2]). Tlle obtained results show that the end point of crystallization, Ti (that is, the T at which the liquid to crystal transition tends to complete, see Fig. l), depends on the initial conditions (Table II), this being consistent with the literature [2]. However, this dependence decreases as the number of atoms involved in the simulation increases, as indicated by the presently determined standard deviation of each Ti in Fig. 1. Accordingly, the end point of crystallization for the 2000-Na system is only somewhat above To noted in Fig. 1. This To is the crossing point of the hypercooled-liquid and crystal states in Fig. 1 and can be considered as the lower limit of Ti in the present case. It thus appears that Ti is insensitive to the increase of n above n = 2000 atoms. Thereby, the relation between the (7’;)‘s and n noted in Fig. 1 leads to a smooth T;-n curve (Fig. 3). This curve is obtained actually from the least-square regression of the best curve using the values of (Ti) for rz = 500, 1000 and 2000 atoms (Fig. 1) and the lower limit of Ti (TO = SO I
