Reptation simulations

For such systems Fixman s model however would be more of a molecular level model. 

From the discussion in Section 4.3, it is clear that for long-time, large-scale phenomena the term molecular level still means coarse-grained models. As shown before it is impossible to perform simulations for melt dynamics for the case of a very detailed, chemically accurate model. In this subsection we report on recent MD and MC results as well as one hybrid method simulation which incorporates both MD and MC to study the dynamics of a melt of linear chains. In all cases the moves are employed in a way that the appearance of a slowed down motion is a direct consequence of the noncrossability of the chains and not of the model. 

The MD simulations were performed at a density of p = 0.85ct and the bond fluctuation MC simulations at two different volume fractions = 0.4,0.5. For the hybrid method of Schulz et al., we used iVe = 40 as 

From these results it is clear that the simulations are in a position to analyze the crossover towards the reptation regime in some detail, however 

A key signature of the reptation is given by the mean-square displacements of the monomers and the center of mass of the whole chain. Since all present computer simulations still only can deal with rather short chains, it is necessary to confine the analysis to the innermost monomers in order to identify the long chain behavior. The ends show an enhanced mobility, which dominates the data. Thus we define 

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Many simulations attempt to determine what motion of the polymer is possible. This can be done by modeling displacements of sections of the chain, Monte Carlo simulations, or reptation (a snakelike motion of the polymer chain as it threads past other chains). These motion studies ultimately attempt to determine a correlation between the molecular motion possible and the macroscopic flexibility, hardness, and so on. 

MC simulations and semianalytical theories for diffusion of flexible polymers in random porous media, which have been summarized [35], indicate that the diffusion coefficient in random three-dimensional media follows the Rouse behavior (D N dependence) at short times, and approaches the reptation limit (D dependence) for long times. By contrast, the diffusion coefficient follows the reptation limit for a highly ordered media made from infinitely long rectangular rods connected at right angles in three-dimensional space (Uke a 3D grid). 

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

While the simulation times for N=350 were long enough to reach the diffusive regime, the data for N=700 and 10,000 just reach far into the predicted reptation regime. After an initial Rouse-like motion for inner chain... 

On the other hand, the group of Briels [54] recently proposed a coarse graining in terms of blobs with coarse grained properties determined beforehand by atomistic simulations. The mean-square displacement of these blobs displays only a very narrow Rouse regime and then is slowed down further, but never reaches the power law of reptation. The authors relate this observation to the shortness of the chains ( 8 entanglement lengths) and their averaging over all blobs (not only the innermost ). 

It is not clear how improvements can be made without real progress on the difficult fundamental problems of diffusion in media with obstacles and cooperation of large-scale motions between interpenetrating chains which do not violate chain connectivity. The DeGennes reptation model (225) makes a significant contribution to the first problem, although in an admittedly simplified system. Rigorous calculations or computer simulations on well-defined models which relate to the second problem would be extremely valuable, even if the models themselves were not completely faithful representations of the assumed physical situation. It is not obvious how even to pose solvable problems, simplified or not, which relate to interchain cooperation. 

Barkema, G. T., Marko.J. F., and Widom, B. (1994). Electrophoresis of charged polymers Simulation and scaling in a lattice model of reptation. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 49(6), 5303—5309. 

The results of estimation of coefficient of self-diffusion due to simulation for macromolecules with different lengths are shown in Fig. 12. The introduction of local anisotropy practically does not affect the coefficient of diffusion below the transition point M, the position of which depends on the coefficient of local anisotropy. For strongly entangled systems (M > M ), the value of the index —2 in the reptation law is connected only with the fact of confinement of macromolecule, and does not depend on the value of the coefficient of local anisotropy. At the particular value ae = 0.3, the simulation reproduces the results of the conventional reptation-tube model (see equation (5.21)) and corresponds to the typical empirical situation (M = 10Me). 

Kholodenko AL (1996) Reptation theory geometrical and topological aspects. Macromol Theory Simul 5 1031-1064... 

The reptation theory has been controversial. In large part, this is because experimental data and computer simulations usually show some deviations from the behavior expected for pure... 

We sume that disengagement by pure reptation is negligible for star molecules with diffidently long arms in an entangled medium. (For a contrary opmion, however, based on computer simulation of star molecule motions, see Ref. 33). Relaxation for an f-arm star in a topologically invariant medium is then equivalent to the relaxation of f tethered chains, where is the tethered chain relaxation time (Eq.66) for individual aruK (Rg.l2). 

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