Nonlinear swelling of polymer gels

Abstract

We present a continuum theory for the nonlinear swelling of polymer gels. The approach is to solve the continuity equations for the network and liquid under the constraint that the instantaneous free energy be minimal. Constitutive relations are needed for the free energy density of the gel, W, and for the liquid diffusion current, J. The James–Guth phantom network model and Bastide’s scaling model are used for W; Fick’s law is used for J which is tantamount to neglecting the gel’s shear modulus relative relative to its osmotic compressive modulus. The theory is applied to the free swelling of networks in good solvents to a semidilute gel for three geometries: sphere, long cylinder, and thin slab. We find a geometry-dependent, nonlocal effect influencing the measurables (gel shape and liquid uptake). This arises from the dependence of the network’s deformation at the gel/liquid interface on the whole deformation field for the sample. The theory gives reasonable agreement with experiment; discrepancies are likely due to nonlocal effects in J not accounted for by Fick’s law.