A kinetic theory for polymer melts. I. The equation for the single-link orientational distribution function

Abstract

In this series of papers we show how a kinetic theory of undiluted polymers can be developed using the Curtiss–Bird–Hassager phase‐space formulation. The polymer molecule is modeled as a Kramers freely jointed bead–rod chain. The objective is to obtain a molecular‐theory expression for the stress tensor from which the rheological properties of polymer melts can be obtained. This development is put forth as an alternative to the Doi–Edwards theory; using a very different approach, we have rederived some of their results and generalized or extended others. In this first paper we develop the partial differential equation for the chain configurational distribution function, and then proceed to get the equation for the orientational distribution function for a single link in the chain. A modification of Stokes’ law is introduced that includes a tensor drag coefficient, characterized by two scalar parameters ζ (the friction coefficient) and ε (the link tension coefficient). In addition, to describe the increase of the drag force on a bead with chain length, at constant bead density, a ’’chain constraint exponent’’ β is used, which can vary from zero (the Doi–Edwards limit) to about 0.5. Solutions to the partial differential equation for the single‐link distribution function are given in several forms, including an explicit series solution to terms of third order in the velocity gradients.