The Stefan Problem with Nonlinear Kinetic Undercooling

Abstract

The behaviour of the one-phase Stefan problem with nonlinear kinetic undercooling is studied. This system is physically relevant in a number of contexts, in particular as the sharp-interface (fast-reaction) limit of a variety of reaction-diffusion systems. The similarities and differences with the linear kinetic condition (studied by Evans and King) are highlighted for both one- and two-dimensional problems. Asymptotic results (both in time and in the Stefan number) are obtained for the power-law form of the kinetic condition. Significantly, the one-dimensional growth behaviour of the moving boundary is seen to be relatively insensitive to the precise form of the nonlinear kinetic condition, and this in effect has hindered its experimental determination in applications such as silicon oxidation. By contrast, the two-dimensional development of the moving boundary around a mask edge depends strongly on the form of the kinetic condition and consequently a method, similar to the Boltzmann-Matano method for determining nonlinear diffusivities, is described to determine the kinetic undercooling relation from experiment