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In the literature, two quite different phase-field formulations for theproblem of alloy solidification can be found. In the first, the material in thediffuse interfaces is assumed to be in an intermediate state between solid andliquid, with a unique local composition. In the second, the interface is seenas a mixture of two phases that each retain their macroscopic properties, and aseparate concentration field for each phase is introduced. It is shown herethat both types of models can be obtained by the standard variational procedureif a grand-potential functional is used as a starting point instead of afree-energy functional. The dynamical variable is then the chemical potentialinstead of the composition. In this framework, a complete analogy withphase-field models for the solidification of a pure substance can beestablished. This analogy is then exploited to formulate quantitativephase-field models for alloys with arbitrary phase diagrams. The precision ofthe method is illustrated by numerical simulations with varying interfacethickness.

Unified derivation of phase-field models for alloy solidification from a grand-potential functional