Perturbation Methods for Solid Diffusion in a Stefan Problem

Consider a one-dimensional, two-phase Stefan problem where one phase is a semi-infinite solid and the second, a semi-infinite liquid, and where the dependent variable represents a diffusive impurity concentration. Assume that the diffusion coefficient for the solid phase is much less than that for the liquid and that temperature is constant in space. In this paper, singular perturbation techniques are used to study this problem when the movement of the solid-liquid interface is governed by a thermodynamic perturbation in time which is large compared to the solid diffusion coefficient. Asymptotic expansions for the solid impurity concentration are given for solids that decay, grow, or have both periods of growth and decay. It is shown that when the thermodynamic perturbations lead to decay, the boundary layer in the solid impurity concentration is substantially narrower than in the absence of the thermodynamic perturbations. The significance of this narrowing is illustrated using the liquid-phase epitaxial decay of semiconductor crystals.