Approximate solution techniques for the sorption of a finite amount of swelling solvent in a glassy polymer

This study examines a one-dimensional Stefan problem describing the sorption of a finite amount of swelling solvent in a glassy polymer. The polymer is initially in a dry non-swollen state, where the polymer network is dense. The polymer is then injected with a critical concentration of a swelling solvent, causing polymer chain relaxation to occur. A moving boundary separating the swollen rubbery polymer from the dry glassy polymer is created, whose speed is defined by a kinetic law. The form of the kinetic law is typically assumed to be linear, but this is a nonphysical restriction, and thus we consider the case for a general exponent. We present formal asymptotic expansions, for both small and large times as well as for small and large values of the control parameter, as well as considering the heat balance integral method. These approximations are compared with a numerical scheme, which uses a boundary immobilisation technique and correctly identifies the appropriate starting solution.