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 {ε}. In the limit as {ε} 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 -{σ}({α}v+{κ}) =[s]E{T-TB-[(TA-TB)/2M]ln[(1-c⁺)/(1-c^-)]}, where {σ} is surface tension, v is (normal) velocity of the interface, {κ} is the curvature, [s]E is the jump in entropy density between phases, TA and TB are the melting temperatures of the two materials, M is related to the phase diagram, and {α} is a dynamical constant.