A new approach to multi-phase formulation for the solidification of alloys

This paper demonstrates that the standard approach to the modelling of multi-phase field dynamics for the solidification of alloys has three major defects and offers an alternative approach. The phase field formulation of solidification for alloys with multiple solid phases is formed by relating time derivatives of each variable of the system (e.g. phases and alloy concentration), to the variational derivative of free energy with respect to that variable, in such a way as to ensure positive local entropy production. Contributions to the free energy include the free energy density, which drives the system, and a penalty term for the phase field gradients, which ensures continuity in the variables. The phase field equations are supplemented by a constraint guaranteeing that at any point in space and time the phases sum to unity. How this constraint enters the formulation is the subject of this paper, which postulates and justifies an alternative to current methods.