ISSN 0965545X, Polymer Science, Ser. A, 2012, Vol. 54, No. 6, pp. 505–511. © Pleiades Publishing, Ltd., 2012. Original Russian Text © T.M. Shakirov, N.F. Fatkullin, P.G. Khalatur, S. Stapf, R. Kimmich, 2012, published in Vysokomolekulyarnye Soedineniya, Ser. A, 2012, Vol. 54, No. 6, pp. 907–914.
MODELING
ComputerAided Simulation of the Influence of Collective Effects on PolymerMelt Dynamics in a Straight Cylindrical Tube: Observation of the Onset Stage of the Corset Effect1 T. M. Shakirova,*, N. F. Fatkullina, P. G. Khalaturb, S. Stapfc, and R. Kimmichd a
b c
Institute of Physics, Kazan Federal University, Kremlevskaya ul. 18, Kazan, 420008 Tatarstan, Russia Nesmeyanov Institute of Organoelement Compounds, Russian Academy of Sciences, ul. Vavilova 28, Moscow, 119991 Russia Fachgebiet Technische Physik II/Polymerphysik,Technische Universität Ilmenau, PO Box 100565, 98693 Ilmenau, Germany d Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany *email:
[email protected] Received November 1, 2011; Revised Manuscript Received January 10, 2012
Abstract—The results of the computeraided simulation of the dynamics of a polymer melt consisting of Fraenkel chains in straight cylindrical tubes and in bulk are discussed. Two different models are studied. In the first model, the dynamics of the polymer melt is simulated via the molecular dynamics simulation. The interaction of unbound polymer segments is described by the LennardJones potential, which excludes any chain crossing of macromolecules and generates collective acoustic waves. In the second model, which serves as a reference, the system is studied via the Brownian dynamics method, in which intermolecular interactions are allowed for phenomenologically via friction and stochastic Langevin forces. In this case, cooperative effects are absent and the effect of spatial confinements makes itself evident only in a narrow nearwall layer. For the two models under consideration, there is a significant difference in the decay of dynamic correlation functions C αβ (t ) = bα (t ) bβ (t ) bα ( 0) bβ ( 0) bα2bβ2
−1
, where averaging is performed over all macromolecular
segments and bα (t ) is the α component of the endtoendsegment vector (α ≠ β = x, y , and the cylindrical axis of the tube is directed along the z axis). For the first model allowing for collective effects, the dynamics of decay of C αβ (t ) functions is much slower than that for the melt in bulk, and for the second model, in which the presence of the tube leads only to spatial confinements for the polymer segments in the direct vicinity of walls. This difference indicates the fundamental significance of the collective effects in the dynamics of poly mer melts confined in porous media. This phenomenon is the first computersimulated evidence of the onset stage of the socalled corset effect, which was first observed experimentally with the use of NMR relaxometry. DOI: 10.1134/S0965545X12050100
1 The dynamics of polymer melts confined in porous nano and mesoscale media has been intensively stud ied in recent years [1–29]. These studies were to a large extent stimulated by the unusual phenomena detected at frequencies of 0.1–100 MHz that were first observed in the experiments devoted to the frequency dependence of the nuclear spin–lattice relaxation rate of polymer melts in porous systems [5, 7, 16, 24]. These phenomena are known as the corset effect. In general, the basis of the corset effect is as follows. The dynamic properties of polymer melts in porous sys tems may considerably differ from the corresponding properties of polymer melts without any spatial con straints at high frequencies ω (small times t ∝ 1 ω), when the rms displacements of polymer segments 1 This
work was supported by the Russian Foundation for Basic Research (project no. 100300739a) and the DAAD program Mikhail Lomonosov II
r 2(t ) are much smaller than the squared characteris 2 tic radius of a tube, rtube . Note that the specified differ ences were observed even for tubes whose radii were several times larger than the characteristic lengths of polymer chains. In other words, a mutual noncross ability of polymer chains and a low compressibility of the polymer melt lead to the specific collective effect: Polymer segments span the medium via rapidly prop agating acoustic waves and respond to the presence of impermeable tube walls by changing the character of thermal fluctuations relative to those of the common melt much earlier than the direct physical contact with the tube wall ensues.
At present, controversial opinions about the possi bility of the corset effect can be found in the literature. In studies denying the existence of the corset effect, it is assumed that the influence of confinements due to tube walls on the dynamics of polymer chains can be
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reduced only to the spatial confinement of segments that directly contact the walls [8, 9]. This study concerns the computeraided simula tion of dynamic properties of polymer melts placed in cylindrical tubes with impermeable side walls, whose effect on the polymer segments is described by the repulsive potential. The results are compared with similar properties predicted for an individual chain in a tube and for a melt consisting of chains of the same molecular mass in bulk. Main attention is given to the comparative analysis of the time dependence of dynamic correlation functions: b (t ) bβ (t ) bα ( 0) bβ ( 0) Cαβ (t ) = α , 2 2 (1) bαbβ (α ≠ β = x, y, z), where bα (t ) is the α component of the endtoend seg ment vector and averaging was performed over all macromolecular segments. The Fourier transform of this correlation function determines the frequency dependence of the intramolecular contribution to the nuclear spin–lattice relaxation rate [30, 31]. Experiments of this study are based on comparison of (i) time dependences of autocorrelation functions of the two components of the segment vector that are perpendicular to the tube axis, C xy (t ), and (ii) the autocorrelation functions of the components parallel to the tube axis and perpendicular to the tube axis, C z ⊥(t) (⊥ = x, y ). The dynamics of polymer melts was simulated in bulk and in straight cylindrical tubes with a segmentlength diameter of 6–12. In order to distin guish between the collective effects of confinements imposed on the dynamic properties of melts (i.e., the corset effect) and the trivial effect of the nearwall layer, changes in the dynamics of two model systems were compared. The systems consisting of the Fraenkel chains, the simplest generalized variant of the freely jointed Kuhn chain, are considered. The bond energy of segments is described by the function N
U ({ri }) = kBT
∑ κ2 ( r
n+1
− rn − b)
n =1 N
= kBT
(2)
∑ κ2 (b
n
n =1
2
− b) , 2
where {ri} is the combination of radius vectors that describe the spatial position of N + 1 particles com prising the polymer chain, b is the specified average segment length, bn = rn+1 − rn is the length of the nth segment of the chain, and κ is the bondrigidity parameter. In the base model (hereinafter, model 1), the poly mer melt was simulated via the molecular dynamics method. The interaction between directly unbound polymer segments was described via the Lennard
Jones potential; its parameters were selected to exclude chain crossing. It is natural to expect manifes tation of collective effects related to the collective acoustic waves and the corset effect if the latter effect is substantial or exists for the studied time range. The melts of polymer chains with the lengths N = 100 and 400 segments and the rigidity parameter κ = 40 were considered. In the simulation, the reduced tempera ture and the segment length were assumed to be unity: kBT = 1 and b = 1. Without any spatial constraints, the polymer melt was a translationally periodic system of identical chains placed in a cubic cell. In the reference model (hereinafter, model 2), the system was simulated via the Brownian dynamics method. In fact, an individual macromolecule with a Langevin thermostat in a tube was considered. Hence, intermacromolecular interactions and collective effects were absent in this model. It is evident that the influence of spatial confinements in this type of system can manifest itself only in a narrow nearwall layer, where polymer segments are in direct contact with tube walls. The interactions between the segments and the medium were taken into account phenomenologi cally with the use of friction forces and the Langevin stochastic forces related to them via the dissipation– fluctuation theorem. In this case, the computeraided simulation was performed in the inertialess approxi mation, identically to the previous calculation [32]. The interaction with the walls was simulated with the use of the force acting from tube walls on chain seg ments:
(
F tube
)
2 ⎧0, x 2 + y 2 < rtube ⎪ ⎪ ⎡ ⎧x 2 + y 2 ⎫ ⎤ = ⎨ A ⎢exp ⎨ 2 − 1⎬ − 1⎥ , ⎩ rtube ⎭ ⎦ ⎪ ⎣ ⎪ x2 + y2 ≥ r 2 tube ⎩
(
(3)
)
where x and y are particle coordinates, rtube is the effec tive radius of a cylindrical tube, and A is the parameter determining the intensity of interaction between the particles and tube walls. In the calculations that will be discussed below, parameter А was assumed to be 100. Note that force F tube increases exponentially when any particle leaves the tube. Note also that, if the tube walls are replaced with periodic boundary conditions, model 2 satisfactorily describes the dynamic proper ties of unentangled melts of macromolecules (i.e., the melts of chains with molecular masses lower than the critical molecular mass) in bulk [32–35]. The calculations were performed with the LAM MPS package [36] supporting the multiprocessor cal culations with the MPI interface and paralleling based on spatial decomposition. In order to increase the role of collective effects in model 1, the concentration of the Kuhn segments, ρ, and the Kuhn segment length, b , were selected so that the packaging length—a value linearly related to the correlation length of the melt— POLYMER SCIENCE
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pl ≡ ρ −1b −2 was approximately 0.18b. Note that the correlation length of the melt is the measure of the dis tance between the nearest segments of neighboring macromolecules. The interaction between polymer units was described with the use of the standard LennardJones potential with parameters ε = 0.1kBT and σ = 0.5b . The radius of action of the potential was rc = 21/6 σ . The unit mass of the particles forming polymer chains, m, was 1. The step of integration of the motion equa tions was Δ t = 0.001τ nat , where τ nat is the characteris tic time unit τ nat = m ( kBT )σ . The numberaverage density of the simulated system was ϕ = 5.4b −3 . A tube was generated by the standard command from the LAMMPS package simulating cylindrically symmet ric walls; the interaction with them is specified by the Lennard–Jones potential, where the distance between the cylinder of a given radius and the polymer chain particle serves as a distance parameter. The wall poten tial was clipped at the minimum point. The periodic (toroidal) boundary conditions were imposed along the tube axis (direction z); thus making it possible to simulate a tube of a formally infinite length. The cal culations were performed in the NVE ensemble. Prior to calculations, the relaxation simulation was per formed with ~107 steps of integration of the motion equations. The correlation functions and the time dependences of various static parameters of a chain, such as the Flory radius and the radius of gyration of macromolecules, were calculated. All the results given below were obtained through averaging over a total of 20 000 chains. The time of segmental relaxation, τ s , determined from the relaxation rate of the Rouse mode with the maximum
N
number,
X N (t )
=
∑
N +1 n =1
rn (t ) ×
πN ( n − 1/2) , was selected as the time unit. The N +1 accurate 1 τ s value was calculated as the rate of exponen X (t)X (0) tial decay of the correlation function N 2 N = XN cos
⎧ ⎫ exp ⎨− t ⎬ ; an adjustment was made over the two ⎩ τs ⎭ order decay in the magnitude of the function. For all systems within model 1 both in tubes and in bulk, the value of τ s = 0.56τ nat . In model 2 τ s corresponded to ~ 6600 steps. To make the transition from dimensionless units to conventional units, the size of the Kuhn segment in real melts of flexiblechain macromolecules lie within the range b = 6–20 Å, whereas the characteristic time of segmental relaxation τs is approximately 10–11– 10 ⎯9 s at temperatures above the glasstransition tem perature. POLYMER SCIENCE
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RESULTS OF THE COMPUTERAIDED SIMULATION The influence of spatial constraints on the dynam ics of polymer chains, which is observed for model 2, which has no collective effects, manifests itself as small acceleration of relaxation of autocorrelation function C xy (t ) relative to functions C z ⊥ (t ) (Fig. 1). The influ ence of the walls in this model is related to direct inter action with chain particles solely; therefore, differ ences between autocorrelation functions rapidly decrease with an increase in the tube diameter. As expected, both correlation functions are identical in the system without constraints (C xy (t ) = C z ⊥ (t ) ). In model 1, where collective effects can be present, fundamentally different distinctions are observed in the behavior of the considered functions. The decay in function C xy (t ) with time with respect to C z ⊥ (t ) becomes slower, as opposed to its acceleration in model 2 (Fig. 2). The dependence of the decay of functions C xy (t ) on tube diameter becomes consider ably weaker. For tubes with diameters in the range d = 2rtube = (6–12)b, the decay of function Cxy(t) is notice ably weaker than that of function C z ⊥ (t ) . Starting with times on the order of several τ s , the decay of function Cxy(t) differs from the analogous dependence in bulk (Fig. 2). Note that, at these times, thermal transla tional displacements of polymer segments are much smaller than the tube diameter; i.e., most segments of the system are not in direct contact with the walls. In particular, for 100unit chains with gyration radius Rg ∼ 4.4b in tubes with the diameter d = 12b , func tions C xy (t ) and C z ⊥ (t ) are almost indistinguishable for model 2 (Fig. 1). There is a simple explanation for this behavior. In the considered system, collective effects are absent and the tube radius is more than twice as large as the radius of gyration of macromolecules; hence, the number of segments in the nearwall layer is small. Therefore, the dynamics of macromolecules in the porous medium differs slightly from their dynamics in bulk. In contrast, if the geometric ratios between the size of macromolecules and the diameter of tube are the same, as assumed in model 1, the col lective effects related to low compressibility of the melt and the mutual noncrossability of chains are responsi ble for a significant difference in the kinetics of the dis cussed correlation functions. This difference becomes more illustrative if the binary correlation functions of polymer segments are examined:
cα (t ) =
bα (t ) bα ( 0) , (α = x, y, z) 2 bα
(4)
The ratio between functions Cαβ (t ) and cα (t ) is sen sitive to the microscopic details of the dynamics of macromolecules [32]. Thus, equality Cαβ (t ) =
508
SHAKIROV et al. Cxy(t) 100
Cz⊥(t) 100
(а)
10−1
(b)
10−1
10−2
10−2
1
1 2
2 10−3 Cxy(t) 100
10−3 Cz⊥(t) 100
(c)
10−1
(d)
10−1
10−2
10−2
1
1 2
2 10−3
10−1
100
101
t/τs
10−3
10−1
100
101
t/τs
Fig. 1. Dynamic correlation functions of segmentvector components in model 2: (1) the correlation function of two different components of the segment vector in bulk, C αβ (t ), and (2) correlation functions (a, c) C xy (t ) and (b, d) C z ⊥ (t ) for the system in a tube. The tube diameters are (a, b) 6 and (c, d) 12 segments. The chain length is N = 100 segments.
cα (t ) cβ (t ) = cα2 (t ) should be valid for the isotropic motion models at t > τ s and α ≠ β. For the reptation model, at times longer than entanglement time τe and shorter than terminal relaxation time τ1 ∝ τ s N 3N e−1 , where N e is the average number of Kuhn segments between entanglements, Cαβ (t ) ∝ cα (t ) . At times t < τ s , when the effects of rigidity of the polymer seg ments are significant, Cαβ (t ) ∝ cα3 (t ) [33]. For model 1 and model melts without constraints, the relationship C z ⊥ (t ) ~ cz (t ) c⊥ (t ) is valid at t ≥ 6τ s (Figs. 3, 4). How ever, the behavior of autocorrelation functions C xy (t ) is substantially different: Even at times ~100τs, C xy (t ) > c x (t ) c y (t ) = c⊥2 (t ) and C xy (t ) ~ Bc⊥ (t ) (Figs. 5, 6). This behavior reflects the anisotropy of motion induced by the collective effects. Note that, in the classical reptation model, C xy (t ) ~ Bc⊥ (t ); however, C xy (t ) = C z ⊥ (t ) . Small differences observed for functions C z ⊥ (t ) for tubes of different diameters (Fig. 2) are explained by the effect of tube walls on singlecomponent functions c⊥ (t ) . The interaction with walls accelerates relaxation
c⊥ (t ) relative to that of the same functions in bulk. It is easy to understand that this behavior is caused by direct interaction (contact) between polymer seg ments and walls. Indeed, it follows from calculations that the effect of the walls is reduced with an increase in the tube diameter; at d = 12b , spatial confinements practically cease to play a role. As was mentioned above, in model 2, the effect of the confining walls weakens with an increase in the tube diameter. Thus, the effect described for model 1 cannot be associated with the direct interaction with tube walls. The major differences for the two models under discussion are the low compressibility of the melt, the mutual noncrossability of polymer chains, and the possibility of acousticwave propagation in a dense medium. These factors transmit the influence of walls at distances that are considerably larger than the thin nearwall layer and, thus, entail qualitative changes in the dynamics of the polymer melt, as embodied in the appearance of the collective corset effect. Thus, the computer simulation of the dynamics of polymer melts in porous media demonstrates that, in addition to purely geometric constraints imposed on translational displacements of polymer segments in POLYMER SCIENCE
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509
Cz⊥(t) (a)
100
(b)
100 10−1
10−1 ~t−0.85
1 2
−2
10
~t
10−2
~t−1.2
−1.2
10−3 −3
10 Cxy(t)
Cz⊥(t)
100
(c)
10−1
10−1
10−2
~t−0.85 10−2 ~t−1.2
10−3 10−1
(d)
100
100
101
~t−1.2
10−3 10−1
t/τs
100
101
t/τs
Fig. 2. Dynamic correlation functions of segmentvector components in model 1: (1) the correlation function of two different components of the segment vector in bulk, C αβ (t ), and (2) correlation functions (a, c) C xy (t ) and (b, d) C z ⊥ (t ) for the system in a tube. The tube diameters are 6 (a, b) and 12 (c, d) segments. The chain length is N = 100 segments.
the nearwall layer, a specific collective effect related to the low compressibility of the melt and the mutual noncrossability of polymer chains exists. These factors induce the fluctuation anisotropy of polymer segments that are not located in the immediate proximity of the tube walls. Cz⊥(t), cz(t)c⊥(t)
Cz⊥(t), cz(t)c⊥(t) (a)
100
The correlations of fluctuations of the vector com ponents of the segment that are perpendicular to the tube axis remain for a longer time than the same cor relations in bulk and the correlations of fluctuation of the vector components of the segment that are parallel and perpendicular to the tube axis. The results of this
10−1
10−1
2 1
10−2
(b)
100
2 1
10−2
10−3
10−3 10−1
100
101
t/τs
10−1
100
101
t/τs
Fig. 3. (1) Dynamic correlation functions of segmentvector components that are parallel and perpendicular to the tube axis, C z ⊥ (t ) , and (2) the product of binary correlation functions c z (t ) c⊥ (t ) in model 1 in (a) bulk and in (b) a tube with a diameter of 12 segments. The chain length is N = 100 segments. POLYMER SCIENCE
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Cz⊥(t), cz(t)c⊥(t)
100
100
1
10−1
10−1
2
2
1 10−2
10−2
10−3
10−3 −1
10
10
0
10
1
study are in qualitative agreement with the experimen tal results obtained through NMR relaxometry at fre quencies of 0.1–100 MHz [5, 7, 16, 24] and indicate the onset of the collective corset effect. Note that the accurate reproduction of the experimental results in a computer simulation presents a problem because a rough melt model must be used in calculations to study the dynamics during the time when the corset effect may manifest itself. Moreover, because the cor set effect is closely related to the reflection of acoustic waves from tube walls, the details of the observed dynamic behavior may considerably depend on the Cz⊥(t), Bc⊥(t) 100 1 10−1 2 10−2
100
101
100
101
t/τs
Fig. 4. (1) Dynamic correlation function of segmentvec tor components that are parallel and perpendicular to the tube axis, C z ⊥ (t ) , in the tube with a diameter of 10 seg ments and (2) the product of correlation functions c z (t ) c⊥ (t ) for the same system in model 1. The chain length is N = 400 segments.
10−1
10−1
t/τs
Fig. 6. (1) Dynamic correlation function of two segment vector components that are perpendicular to the tube axis, C xy (t ) , in the tube with a diameter of 12 segments and (2) the function proportional to the binary correlation func tion of the segment vector component perpendicular to the tube axis, Bc ⊥ (t ) , in model 1. The chain length is N = 400 segments, and the proportionality factor is B = 0.11.
t/τs
Fig. 5. (1) Dynamic correlation function of two segment vector components that are perpendicular to the tube axis, C xy (t ) , in the tube with a diameter of 12 segments and (2) the function proportional to the binary correlation func tion of the segment vector component perpendicular to the tube axis, Bc ⊥ (t ) , in model 1. The chain length is N = 100 segments, and the proportionality factor is B = 0.14.
geometry of spatial confinements (e.g., on the shape of a tube and the degree of roughness of its wall surface). Recent experimental studies [1] presumably attest to the existence of a strong effect of the confinement geometry on the dynamic properties of the polymer melt. Note also that, in the pioneering studies devoted to investigation of the corset effect, the microscopic interpretation of the dynamics of polymer chains was based on the Doi–Edwards reptation model. Our sim ulation indicates the existence of anisotropy of trans lational displacements of polymer segments that is similar to that in the reptation model at times much shorter than the terminal relaxation time. However, this analogy is incomplete. For example, the decay of correlation functions C z ⊥ (t ) and C xy (t ) in the repta tion model must be identical. In the reptation model, the anisotropy of spatial displacements of polymer segments, which is generated by their correlation with the initial conformation of the macromolecule, results in the fact that C z ⊥ (t ) = C xy (t ) ∝ c⊥ (t ) = c z (t ), rather than C xy (t ) ∝ c⊥2 (t ) , as in isotropic models, which are similar, for example, to the nrenormalized Rouse models. Our simulation shows that, for a polymer melt in tubes, whose behavior in the absence of constraints over the studied time interval is close to the Rouse behavior, correlation functions C xy (t ) ~ Bc⊥ (t ) at t ≥ 6τ s , as in the reptation model, whereas C z ⊥ (t ) ~ cz (t ) c⊥ (t ) , a result that corresponds to predictions of the isotropic models of the dynamics of polymer melts. Investigation of this problem is of considerable interest for polymer science and can stimulate a fur ther advance in the dynamics of polymer melts both in POLYMER SCIENCE
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spatially confined systems and in systems without any confinements.
15. R. Kimmich, R.O. Seitter, U. Beginn, M. Möller, and N. Fatkullin, Chem. Phys. Lett. 307, 147 (1999).
ACKNOWLEDGMENTS
16. C. Mattea, N. Fatkullin, E. Fischer, U. Beginn, E. Anoardo, M. Kroutieva, and R. Kimmich, Appl. Magn. Reson. 27, 371 (2004).
The authors are grateful to Carlos Mattea for his assistance and valuable discussions. Most calculations were performed at the Kazan Branch of the Joint Supercomputer Center of the Russian Academy of Sciences (http://kbjscc.knc.ru).
17. R. Kausik, C. Mattea, R. Kimmich, and N. Fatkullin, Eur. Phys. J., Spec. Top. 141, 235 (2007). 18. N. Fatkullin, R. Kausik, and R. Kimmich, J. Chem. Phys. 126, 0949041 (2007). 19. R. Kausik, N. Fatkullin, N. Hüsing, and R. Kimmich, Magn. Reson. Imaging 25, 489 (2007).
REFERENCES 1. S. Ok, M. Steinhart, A. Serbescu, C. Franz, F. V. Chavez, and K. Saalwächter, Macromolecules 43, 4429 (2010).
20. R. Kausik, C. Mattea, N. Fatkullin, and R. Kimmich, J. Chem. Phys. 124, 1149031 (2006). 21. J. Krzaczkowska, M. Strankowski, S. Jurga, K. Jurga, and A. Pietraszko, J. NonCryst. Solids 356, 945 (2010).
2. M. Hofmann, A. Herrmann, S. Ok, C. Franz, D. Kruk, K. Saalwächter, M. Steinhart, and E. A. Roessler, Mac romolecules 44, 4017 (2011).
22. A. Schonhals, H. Goering, and Ch. Schick, J. Non Cryst. Solids 305, 140 (2002).
3. A. Sanz, A. Nogales, and T. A. Ezquerra, Soft Matter 7, 6477 (2011).
23. C.Y. Liu, B. Zhang, J. He, R. Keunings, and C. Bailly, Macromolecules 42, 7982 (2009).
4. D. Kruk, A. Herrmann, and E. A. Rössler, Progress in Nuclear Magnetic Resonance Spectroscopy, Accepted Manuscript (2011).
24. N. Fatkullin, R. Kimmich, E. Fischer, C. Mattea, U. Beginn, and M. Kroutieva, New J. Phys. 6, 461 (2004).
5. U. Beginn, E. Fischer, T. Pieper, F. Mellinger, R. Kim mich, and M. Möller, J. Polym. Sci., Part A: Polym. Chem. 38, 2041 (2000).
25. S. Stapf and R. Kimmich, Macromolecules 29, 1638 (1996).
6. E. Fischer, U. Beginn, N. Fatkullin, and R. Kimmich, Macromolecules 37, 3277 (2004). 7. E. Fischer, R. Kimmich, U. Beginn, M. Möller, and N. Fatkullin, Phys. Rev. E: Stat. Phys., Plasmas, Flu ids, Relat. Interdiscip. Top. 59, 4079 (1999). 8. E. Fischer, U. Beginn, N. Fatkullin, and R. Kimmich, Magn. Reson. Imaging 23, 379 (2005). 9. R. Kimmich and N. Fatkullin, Macromolecules 43, 9821 (2010). 10. R. Kimmich and N. Fatkullin, J. Chem. Phys. 134, 057101 (2011). 11. G. D. Smith, D. Y. Yoon, R. L. Jaffe, R. H. Colby, R. Krishnamoorti, and L. J. Fetters, Macromolecules 29, 3462 (1996). 12. M. Kroutyeva, J. Martin, A. Arbe, J. Colmenero, C. Mijangos, G. J. Schneider, T. Unruh, Y. Su, and D. Richter, J. Chem. Phys. 131, 1749011 (2009). 13. J. Martin, M. Krutyeva, M. Monkenbusch, A. Arbe, J. Allgaier, A. Radulescu, P. Falus, J. Maiz, C. Mijan gos, J. Colmenero, and D. Richter, Phys. Rev. Lett. 104, 1978011 (2010). 14. K. Lagrené, J.M. Zanotti, M. Daoud, B. Farago, and P. Judeinstein, Phys. Rev. E: Stat. Phys., Plasmas, Flu ids, Relat. Interdiscip. Top. 81, 0608011 (2010). POLYMER SCIENCE
Series A
Vol. 54
No. 6
2012
26. M. Kehr, N. Fatkullin, and R. Kimmich, J. Chem. Phys. 126, 0949031 (2007). 27. M. Kehr, N. Fatkullin, and R. Kimmich, J. Chem. Phys. 127, 0849111 (2007). 28. A. Milchev, J. Phys.: Condens. Matter 23, 103101 (2011). 29. G. J. Schneider, K. Nusser, L. Willner, P. Falus, and D. Richter, Macromolecules 44, 5857 (2011). 30. A. Gubaidullin, T. Shakirov, N. Fatkullin, and R. Kim mich, Solid State Nucl. Magn. Reson. 35, 147 (2009). 31. N. Fatkullin, A. Gubaidullin, and S. Stapf, J. Chem. Phys. 132, 094903 (2010). 32. N. F. Fatkullin, T. M. Shakirov, and N. A. Balakirev, Polymer Science, Ser. A 52, 72 (2010) [Vysokomol. Soedin., Ser. A 52, 67 (2010)]. 33. Yu. Ya. Gotlib, A. A. Darinskii, and Yu. E. Svetlov, Physical Kinetics of Macromolecules (Khimiya, Lenin grad, 1986) [in Russian]. 34. K. Binder and W. Paul, J. Polym. Sci., Part B: Polym. Phys. 35, 1 (1997). 35. E. M. Pestryaev, J. Phys., Conf. Ser. 324, 012031 (2011). 36. http://lammps.sandia.gov/
