Phase-field model for solidification of a eutectic alloy

In this paper we discuss two phase-field models for solidification of a eutectic alloy, a situation in which a liquid may transform into two distinct solid phases. The first is based on a regular solution model for the solid with a chemical miscibility gap. This model suffers from the deficiency that, in the sharp interface limit, it approximates a free-boundary problem in which the surface energy of the solid-solid interface is zero and consequently mechanical equilibrium at a trijunction requires that the solid-solid interface has zero dihedral angle (locally planar). We propose a second model which uses two order parameters to distinguish the liquid phase and the two solid phases. We provide a thermodynamically consistent derivation of this phase-field model which ensures that the local entropy production is positive. We conduct a sharp interface asymptotic analysis of the liquid-solid phase transition and show it is governed by a free-boundary problem in which both surface energy and interface kinetics are present. Finally, we consider a sharp interface asymptotic analysis of a stationary trijunction between the two solid phases and the liquid phase, from which we recover the condition that the interfacial surface tensions are in mechanical equilibrium (Young's equation). This sharp interface analysis compares favourably with numerical solutions of the phase-field model appropriate to a trijunction.