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

Abstract

We show that the standard approach to the modeling of multi-phase field dynamics for the solidification of alloys has three major defects and offer an alternative more successful formulation. The model contains an action made up of a free energy functional of the temperature, concentration and the phase variables. In addition there is a penalty functional for the gradients of the phase variables, which keeps all field variables continuous – the phase field method. A variation of this action is related to time derivatives of the variables in the field. At any physical point there exists up to N phases, ϕi, each of which represent the proportion of each phase at that point, thus implying the constraint, ΣNi = 1 ϕi = 1. The standard Lagrange multiplier treatment for imposing this constraint has three major defects: non-reduction to standard single phase formulation; a dependence on the value of N; generation of unphysical additional phases. We demonstrate a multi-phase formulation that avoids these two defects and, partly as a consequence, does not generate spurious additional phases. Moreover, this aim is achieved without losing the active part that three non-zero phases should play if present at any point.