Frictional Coefficient of Polymer Molecules in Solution

Abstract

The coefficient ks is calculated in the expression for the frictional coefficient f=f0(1+ksc+···), where f0 is the frictional coefficient at infinite dilution, and c is the concentration. Results are given for hard spheres and for interpenetrable spheres of uniform segment density. Difficulties caused by the long-rangedness of hydrodynamic interactions are removed by taking the translational velocity of the spheres relative to the mean solvent velocity; f is computed after transformation to a laboratory frame. All orders of ``reflections'' are correctly summed within the approximation of slow spatial variation of the velocity perturbation arising from a sphere in the vicinity of another. The derived velocity agrees well with that obtained exactly by others for the special case of the line of centers of the two interacting spheres in the direction of the external force. For hard spheres, a value of 7.157 for ks is found when c is the volume fraction of the spheres in the solution. For soft spheres where ks is smaller, the dependence of ks on the segment—segment interaction constant is obtained with the approximation that the binary clusters of overlapping spheres are treated as prolate ellipsoids of varying axial ratio. A substantial drop in ks, but not quite to zero, at the theta temperature is predicted by this model. Also the dependence of ks at the theta point on the polymer molecular weight is briefly discussed.