Rouse model diffusion constant

Although this athermal bond fluctuation model is clearly not yet a model for any specific polymeric material, it is nevertheless a useful starting point from which a more detailed chemical description can be built. This fact already becomes apparent, when we study suitably rescaled quantities, such that, on this level, a comparison with experiment is already possible. As an example, we can consider the crossover of the self-diffusion constant from Rouse-like behavior for short chains to entangled behavior for longer chains. 

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]...

Within the Rouse model for polymer dynamics the viscosity of a melt can be calculated from the diffusion constant of the chains using the relation [22,29,30] ... 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

Now in the Rouse model the diffusion constant is given by the Einstein relation... 

As for the multidimensional freely jointed chain, it is possible to relate a to the parameters which describe a Rouse chain by evaluating the translational diffusion constant D for the center of mass. In the stochastic model, we determine the square of the displacement per unit time of a single bead averaged over an equilibrium ensemble. For bead j,... 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

The primitive chain reptates along itself with a diffusion constant that can be identified as the diffusion coefficient of the Rouse model. Under the action of a force /, the velocity of the polymer in the tube is v =f /, where is the overall friction coefficient of the chain. It is expected that C is related to the friction coefficient of the individual segments, Q, by the expression... 

Figure 5.14 shows high-pressure isobars for the self-diffusion data of the n-alkanes in a plot of log D against log where is the molecular weight. These are the first data obtained with the titanium autoclave described in Section 1.4.2. Such results are commonly described by the Rouse model or by the reptation model, which both predict a linear correlation in this type of plot at constant pressure and temperature this linear correlation is clearly established in Fig. 5.14. Judging from the chain length of the polymethylenes the Rouse model should apply. This model predicts, that D should be proportional to M while the experiments give a D correlation, which is...

In summary, we have used the Rouse chain model to obtain the diffusion constant of the center of mass and the time-correlation function of the end-to-end vector, which reflects the rotational motion of the whole polymer molecule. Since N is proportional to the molecular weight M, and K is independent of molecular weight, Eqs. (3.41) and (3.62) indicate that Dq and Tr depend on the molecular weight, respectively, as... 

As indicated by Eqs. (3.41) and (3.55), the molecular translational motion and the internal modes of motion of a Rouse chain ultimately depend on the diffusion constant of each individual Rouse bead, D = kT/ The diffusion of a Brownian particle (Eq. (3.3)) can be simulated by the random walk model as shown in Appendix 3.D, which in turn can be used to introduce the diffusion process into different discrete-time models of polymer dynamics (Chapters 8 and 16-18). 

This result shows that the diffusion constant of a long polymer chain in a concentrated system, because of the constraint effect of entanglement, is inversely proportional to the square of the molecular weight. This molecular-weight dependence is distinctively different from the result, Dg oc M, given by the Rouse model (Chapter 3) and its observation is often regarded as the indication of the reptational motion. As shown in... 

The reptation model, like the Rouse model, supposes that the friction involved in dragging the chain through its tube is proportional to the chain length, 5 = N i, Equation (33.33). The diffusion constant Dtubo for the chain moving through the tube is given by the Einstein-Smoluchowski relation, Equa-... 

Experiments give D oc N - , which differs somewhat from this reptation model prediction of D oc N (see Figure 33.12). In contrast, the self-diffusion constant for a Rouse chain, which is based on t oc (Equation (33.34)), is... 

Rouse-like behavior is not in fact observed in dilute solutions, for which it is necessary to take into account the influence of the chain on the motion of the solvent, and deviations from Gaussian statistics arising from polymer-solvent interactions [17, 18]. These factors are incorporated in the Zimm model, which predicts the diffusion constant to be proportional to N, for example, where v depends on the solvent quality, in better agreement with experimental data [4,14]. Indeed, although it was first proposed for isolated chains, the Rouse model turns out to be more appropriate to polymer melts, where flexible linear chain conformations are approximately Gaussian and hydrodynamic interactions are relatively unimportant [4, 14-16]. 

Within the Rouse model g3(t) t for all times. The diffusion constant D N) - limi->cx>U3 t)/ t is expected to reach the asymptotic value... 

The next relaxation time is for the entire object to rotate or to move a distance comparable to its own diameter, since the same time is needed for the star to move its own distance or to make a complete rotation. For an assembly of Nf objects subjected to independent random forces as in the Rouse model, the diffusion constant is given by D r j Nf)- Within the diffusion time td, the system moves a distance of about its own diameter,... 

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