Bracket formalism applied to phase field models of alloy solidification

We present a method for coupling current phase field models of alloy solidification into general continuum modelling. The advantages of this approach are to provide a generic framework for phase field modelling, give a natural and thermodynamically consistent extension to non-isothermal modelling, and to see phase field models in a wider context. The bracket approach, introduced by Beris and Edwards, is an extension of the Poisson bracket of Hamiltonian mechanics to include dissipative phenomena. This paper demonstrates the working of this formalism for a variety of alloy solidification models including multi phase, multi species with thermal and density dependency. We present new models by deriving temperature equations for single and more general phase field models, and postulate a density dependent formulation which couples phase field to flow.