Zimm model diffusion constant

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. 

To show the usefulness of the scaling argument, let us consider the diffusion constant Dq- As indicated by the Zimm model, the parameters appearing in the problem are N, b, ksT and ij. From the dimensional analysis. Da is written as... 

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 predictions of the Zimm model for a Gaussian chain have a very simple interpretation the Zimm limit is a non-draining limit, the solvent flow does not penetrate into the polymer chain and, as far as hydrodynamic properties are concerned, a polymer chain can be considered as a hard sphere. The diffusion constant is then given by Stokes law, and the intrinsic viscosity by the Einstein equation for dilute solutions of hard spheres, the only difference being in the numerical factors. 

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