Free energy vs. Grand Potential Energy formulations in phase field modelling of alloy solidification

We review and compare the [Wheeler-Boettinger-McFadden Phys. Rev A 45, 1992] (WBM), free energy based, phase field formulation of alloy solidification, with the grand potential energy formulation (GPF) of [Plapp Phys. Rev E 84 2011] and so, by association, the two phase approach of [Kim-Kim-Suzuki Phys. Rev. E 60 1999]. We ask what the effective differences are between these approaches: are they equivalent? We then advocate an approach that lies within the WBM scheme, yet remains consistent with the GPF. This has the flexibility to apply, with some modification, to arbitrary bulk free energies, including CALPHAD type descriptions such as Redlich-Kister relations for solution phases and sublattice models for non-stoichiometric intermetallics. The proposed model avoids some inherent complications implicit in the grand potential formulation, e.g. inverting the relation between chemical potential and solute concentration.