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Multiple Similarity Solutions for Solidification and Melting



Sam R. Coriell, Geoffrey B. McFadden, R F. Sekerka, William J. Boettinger


We explore multiple one-dimensional similarity solutions during the solidification of a binary alloy and a pure material. The configuration is analogous that of a diffusion couple, for which multiple solutions in ternary alloys were discovered by Coates & Kirkaldy and explored further by Maugis et al. We proceed by finding analytical solutions to the relevant transport equations, which results in a transcendental equation for the parabolic growth rate constant, which we solve numerically. In our case, for a lead-tin alloy each member of the diffusion couple in a stable phase, one solid and on liquid. The diffusion path begins at the composition and temperature of one each phase, crosses the two-phase region at a tie line and terminates at the composition and temperature of the other phase. For cases in which there are multiple solutions such diffusion paths often enter the metastable two-phase region before crossing the tie line. The concomitant temperature and concentration profiles are also explored and, in some cases, are quite counter-intuitive. For the pure material, the only multiple solutions we have found occur when the solid is superheated, which is probably unrealistic experimentally. Experiments to explore the binary alloy case should, however, be possible. We speculate about the origin of these multiple solutions by considerations including adjustment to initial conditions, interface kinetics, non-similarity solutions, and instabilities.
Journal of Crystal Growth
No. 3


alloy solidification, similarity solutions


Coriell, S. , McFadden, G. , Sekerka, R. and Boettinger, W. (1998), Multiple Similarity Solutions for Solidification and Melting, Journal of Crystal Growth (Accessed April 20, 2024)
Created July 1, 1998, Updated June 2, 2021