Phase-field and sharp-interface alloy models.

Abstract

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 \ensuremath{\epsilon}. In the limit as \ensuremath{\epsilon} 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)]${]}_{\mathrm{\ensuremath{-}}}^{+}$=-2M, coupled with -\ensuremath{\sigma}(\ensuremath{\alpha}v+\ensuremath{\kappa}) =[s${]}_{\mathit{E}}${T-${\mathit{T}}_{\mathit{B}}$-[(${\mathit{T}}_{\mathit{A}}$-${\mathit{T}}_{\mathit{B}}$)/2M]ln[(1-${\mathit{c}}^{+}$)/(1-${\mathit{c}}^{\mathrm{\ensuremath{-}}}$)]}, where \ensuremath{\sigma} is surface tension, v is (normal) velocity of the interface, \ensuremath{\kappa} is the curvature, [s${]}_{\mathit{E}}$ is the jump in entropy density between phases, ${\mathit{T}}_{\mathit{A}}$ and ${\mathit{T}}_{\mathit{B}}$ are the melting temperatures of the two materials, M is related to the phase diagram, and \ensuremath{\alpha} is a dynamical constant.