Viscoelastic Theory of Branched and Cross-Linked Polymers

A model of polymer chain motions is developed which in the limit of long, flexible chains can readily be applied to branched chain and cross-linked systems. The equation governing the motion of a chain takes the form of a differential equation rather than the form of the large set of difference equations that occur in most other treatments. The present model gives essentially the same distribution of relaxation times for linear polymers as previous treatments, and it also gives the same dependence of viscosity upon branching as the theories based on the effective radius of randomly coiled chains. In addition to this information, the theory yields the distribution of relaxation times for branched chain polymers, permitting, for example, the calculation of the energy stored in steady state shear. The relaxation time distribution of several models of a cross-linked polymer is also calculated, and the relationship of this distribution function to that of the uncross-linked materials is discussed.