Multiple similarity solutions for solidification and melting

Abstract

We explore multiple similarity solutions in one spatial dimension during the solidification or melting of a binary alloy. The configuration is analogous to that of a diffusion couple, for which similarity multiple solutions in isothermal ternary alloys were discovered by Coates and Kirkaldy and explored further by Maugis et al. We proceed by finding analytical solutions to the transport equations for heat and solute. Under the assumption of local equilibrium at the interface, a transcendental equation for the parabolic growth rate constant is obtained, which we solve numerically. We use a lead–tin alloy as an example, and examine the situation in which each member of the diffusion couple is a stable phase, one solid and one liquid. The “diffusion path” begins at the composition and temperature of one such phase, crosses the two-phase region at a tie line, and terminates at the composition and temperature of the other phase. For the cases examined here in which there are multiple solutions, such diffusion paths enter the two-phase region before crossing the tie line. The concomitant temperature and concentration profiles are also explored and, in some cases, are quite counterintuitive. For a pure material, we have only found multiple solutions when the solid is superheated, which may be difficult experimentally. Experiments to explore the binary alloy case should, however, be possible. We speculate about the physical manifestation of these multiple solutions by considerations including adjustment to initial conditions, interface kinetics, non-similarity solutions, and instabilities.