Phase-field and sharp-interface alloy models

We consider a phase-field model of a binary mixture or alloy which has a phase boundary. The model identifies all macroscopic parameters and the interface thickness. In the limit as approaches zero, an alternative two-phase alloy solidification model (with a sharp interface) is obtained. For small concentrations, we recover the classical sharp-interface problems, the theory of which is reviewed. We obtain, in the simplest phase-field system, a new (nonlinear) interface relation for concentration c which is discontinuous across the interface and subject to ln[c/ (1-c) ]-^+=-2M, coupled with - (+) =s₄T-{T₁- (T₀-T₁) /2Mln (1-c^+) / (1-c^-) }, where is surface tension, v is (normal) velocity of the interface, is the curvature, s₄ is the jump in entropy density between phases, T₀ and T₁ are the melting temperatures of the two materials, M is related to the phase diagram, and is a dynamical constant.