Rouse model monomer displacement

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

Figure 19 Mean square monomer displacements using the CRC model of PB at three temperatures compared with the monomer displacement in an FRC version of the polymer model. Also indicated is the Rouse-like regime with the subdiffusive t0 61 power law entered after the caging regime (CRC at low T) or after the short time dynamics (FRC and CRC at 353 K).

Early-time motion, for segments s such that UgM(s)activated exploration of the original tube by the free end. In the absence of topological constraints along the contour, the end monomer moves by the classical non-Fickian diffusion of a Rouse chain, with spatial displacement f, but confined to the single dimension of the chain contour variable s. We therefore expect the early-time result for r(s) to scale as s. When all prefactors are calculated from the Rouse model [2] for Gaussian chains with local friction we find the form... 

As in the Rouse model, the mean-square displacement of monomer j during time Xp is of the order of the mean-square size of the section containing Njp monomers involved in a coherent motion at this time ... 

Consistent with the fact that the longest relaxation time of the Zimm model is shorter than the Rouse model, the subdiffusive monomer motion of the Zimm model [(Eq. (8.70)] is always faster than in the Rouse model [Eq. (8.58)] with the same monomer relaxation time tq. This is demonstrated in Fig. 8.8, where the mean-square monomer displacements predicted by the Rouse and Zimm models are compared. Each model exhibits subdiffusive motion on length scales smaller than the size of the chain, but motion becomes diffusive on larger scales, corresponding to times longer than the longest relaxation time. ... 

Time dependence of the mean-square monomer displacements predicted by the Rouse and Zimm models on logarithmic scales. 

Tube length fluctuations modify the rheological response of entangled polymers. Reptation dynamics adds a regime to the mean-square monomer displacement that was not present in the free Rouse model. This extra regime is a characteristic signature of Rouse motion of a chain confined to a tube. 

Figure 7 summarizes the data for two key dynamic observables the mean-square monomer displacement gi,mid(0 and i,end(0 and the stress relaxation fimction. Hgures 7(a) and 7(b) show the data as a limction of time in units where f= l,feBT= 1 and unit length is b in Rouse and semiflexible models, Tq in freely jointed and freely rotating models, and a in the LJ+FENE-based models. Since these units are pretty arbitrary and the local details are very different (including presence of inertia), the results are scattered and it is not easy to see similarities and differences between the models. 

Figure 26 Main results for the Rouse model with constraints. Symbols correspond to grid parameter j=1 with chain iengths /V=16,32,64,128, whereas p=2 results are shown by lines for N=32,64,128,256. The four panels show monomer mean-square displacement (a) and (c), end-to-end relaxation (b), and stress relaxation (d).

Our review starts with the general formulation of the GGS model in Sect. 2. In the framework of the GGS approach many dynamical quantities of experimental relevance can be expressed through analytical equations. Because of this, in Sect. 3 we outline the derivation of such expressions for the dynamical shear modulus and the viscosity, for the relaxation modulus, for the dielectric susceptibility, and for the displacement of monomers under external forces. Section 4 provides a historical retrospective of the classical Rouse model, while emphasizing its successes and limitations. The next three sections are devoted to the dynamical properties of several classes of polymer networks, ranging from regular and fractal networks to network models which take into account structural heterogeneities encountered in real systems. Sections 8 and 9 discuss dendrimers, dendritic polymers, and hyperbranched polymers. 

Thus the monomer displacement is faster and the longest relaxation time is shorter in the Zimm model than in the Rouse model. 

These models are designed to reproduce the random movement of flexible polymer chains in a solvent or melt in a more or less realistic way. Simulational results which reproduce in simple cases the so-called Rouse [49] or Zimm [50] dynamics, depending on whether hydrodynamic interactions in the system are neglected or not, appear appropriate for studying diffusion, relaxation, and transport properties in general. In all dynamic models the monomers perform small displacements per unit time while the connectivity of the chains is preserved during the simulation. 

Fig. 5.3. Log-log plot of the self-diffusion constant D of polymer melts vs. chain length N. D is normalized by the diffusion constant of the Rouse limit, DRoUse> which is reached for short chain lengths. N is normalized by Ne, which is estimated from the kink in the log-log plot of the mean-square displacement of inner monomers vs. time [gi (t) vs. t]. Molecular dynamics results [177] and experimental data on PE [178] are compared with the MC results [40] for the athermal bond fluctuation model. From [40]...

In the time scale of t > Ij, the effect of finding the new direction by the chain ends becomes dominant, and the mean square displacement of monomers will become equal to that of the center of mass. In the time scale of f< the motion of the monomers is complicated. At sufficiently short times (t < r ), the monomers will make a diffusional motion without feeling the presence of other monomers, as we have seen for both the Rouse chain and the Zimm model. We can at least say that the dependence of ([r (f) - r (0)] ) on t is sharper at t <... 

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