Molecular Models for Polymer Dynamics

Rouse solved the m — 1 simultaneous equations [Eqs. (45)] by transforming them into a set of uncoupled equations for the normal modes of motion of the chain. 

Because the length of chain corresponding to each Gaussian spring is the shortest unit that can relax in the model, these expressions are only physically meaningful for m 1. In this limit, Eq. (48) may be replaced by Eq. (49), where Co = ClN is a monomeric friction coefficient and N = mN is the total number of statistical segments per chain. 

the longest Rouse relaxation time, xr, is proportional to N. For t xr, the motion of the chain becomes essentially diffusive, and the diffusion constant of the center of gravity of the chains is given by the Einstein relation, Eq. (50). 

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]. 

The Rouse model is nevertheless inadequate to describe the high-frequency response associated with bond rotations and local cooperative motions, important for the glassy state (see, for example. Ref. 4). Moreover, as the chain length increases (or the concentration increases in a solution of long chains), the fact that 

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The starting point for molecular models for polymer dynamics based on the ideas introduced in Section 14.2.3 is the Rouse model for an isolated chain in a viscous medium, in which the chain is taken to behave as a sequence of m beads linked by Gaussian springs [Figure 14.9(a)] [13-16]. The chain interacts with the solvent via the beads, and the solvent is assumed to drain freely as the chain moves. Hence, Eq. (22) leads to Eqs. (45), where N is the number of links between adjacent beads, C is a friction coefficient per bead and r is the position of the ith bead. 

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