POLYMER MATERIAL COORDINATES FOR MUTUAL DIFFUSION IN POLYMER-PENETRANT SYSTEMS

Abstract

This paper presents a general framework, for formulating polymer-penetrant mutual diffusion problems using polymer material (Lagrangian) coordinates. The material approach has two advantages over the spatial (Eulerian) framework: it allows a simple specification of boundary conditions and, for systems with negligible excess volume, it obviates the need to solve two continuity equations. We derive in curvilinear polymer material coordinates the field equations needed for multi-dimensional problems: species continuity; species and mixture equations of motion. These enable the analysis of a broad class of diffusion problems by either (a) simultaneously solving the species continuity equations and the equations of motion for the mixture with constitutive equations for the diffusion current and mixture's stress tensor or (b) using the formalism natural in kinetic theory or in modern continuum theories where the species continuity equations and species equations of motion are solved together with relations specifying the species interaction forces and species stress tensors. We show that for simple problems in Cartesian, cylindrical, and spherical coordinate systems and for three-dimensional isotropic diffusion in a Cartesian system, a kinematic constraint completely specifies the transformation from ordinary, spatial to polymer material coordinates. In these cases, the approach confirms the intuitive transform of Hartley and Crank (1949) for a one-dimensional process in a slab and supplies new transforms for radial diffusion and swelling in a cylinder and sphere. The passive, two-dimensional swelling of a gel slab is studied in detail to demonstrate the use of the material approach for multi-dimensional problems.