Irreversible thermodynamic basis of phase field models

We develop the irreversible thermodynamic basis of the phase field model, which is a mesoscopic diffuse interface model that eliminates interface tracking during phase transformations. The phase field is an auxiliary parameter that identifies the phase; it is continuous but makes a transition over a thin region, the diffuse interface, from its constant value in a growing phase to some other value in the nutrient phase. All phases are treated thermodynamically as viscous liquids, even crystalline solids. Phases are assumed to be isotropic for simplicity with reference to works that include anisotropy. The basis is an entropy functional which is an integral of an entropy density that includes non-classical gradient entropies. Equilibrium is investigated to identify a non-classical temperature and non-classical chemical potentials for a multicomponent system that are uniform at equilibrium in the absence of external forces. Coupled partial differential equations that govern the time evolution of the phase field and accompanying fields (such as temperature and composition) are formulated on the basis of local positive entropy production subject to suitable constraints on energy and chemical species. Fluxes of energy and chemical species, Korteweg stresses due to inhomogeneities and an equation for phase field evolution are obtained from the rate of entropy production by postulating linear constitutive relations. The full phase field equations are discussed and illustrated for the simple case of solidification of a pure material from its melt assuming uniform density.