Several problems related to hydrodynamic interaction in flexible chain polymers without excluded volume are investigated. First, a correction to the Oseen hydrodynamic interaction tensor for a system of many spheres is derived, taking into account the finite volume of the spheres. The new hydrodynamic interaction tensor gives the positive definite diffusion tensor identical with that of Rotne and Prager. Second, the possible effects of the correction term in the hydrodynamic interaction tensor on the translational diffusion coefficient and intrinsic viscosity are examined when preaveraging the hydrodynamic interaction tensor is avoided following the procedure of Pyun and Fixman. If the Rouse free-draining normal coordinates together with the diagonal approximation are used, the effects are shown to be negligibly small in the case of flexible chains. Third, however, the intrinsic viscosity and diffusion coefficient are re-evaluated, taking the lower limit of | i − j | as unity in the evaluation of sums over the segment indices i and j. The results are similar to those derived by Hearst and Stockmayer for wormlike chains. That is, the present theory also predicts that the draining effect vanishes under certain conditions. It is emphasized that the Oseen tensor has still some practical value in the case of flexible chains when it is used carefully.
The mean-square end-to-end distance and the second virial coefficient are evaluated for stiff chains with excluded volume. The calculation is based on a wire-bead model whose backbone in the unperturbed state obeys wormlike-chain statistics. In order to establish the necessary distribution functions, results derived in the second Daniels approximation in the previous paper are used, and suitable extrapolations from the coil region to the high-stiffness (low temperature) limit are made. The ring closure probability in this limit is also derived by applying a variation principle to a functional-integral representation of the partition function in terms of the bending elastic energy of the chain. The result is used for the first-order perturbation calculation of the mean-square end-to-end distance. Evaluation of the second virial coefficient is carried out in the double-contact approximation. The interpenetration function ψ is also calculated in an approximate fashion, and is shown to be appreciably smaller than for the coil limit over the ordinary range of interest for typical stiff chains. This result is consistent with experimental data for cellulose nitrates.
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