Thermodynamically consistent models of phase-field type for the kinetic of phase transitions

A general framework is given for the phenomenological kinetics of phase transitions in which not only the order parameter but also the temperature may vary in time and space. Instead of a Ginzburg-Landau free energy functional, as used in formulating the Cahn-Hilliard equation, we use the analogous entropy functional. Model entropy functionals, and the kinetic equations resulting from them, are constructed for various cases: phase transitions with and without a critical point, and (in the former case) with or without a latent heat. The class considered is general enough to include the entropy functionals for the Ising model in mean-field approximation, the van der Waals fluid, and a simplified version of the density-functional theory of freezing. A case without critical point, for which the energy is conserved but the order parameter is not, provides a thermodynamically consistent derivation of the phase-field equations studied by Caginalp, Fix and others, and also leads in a natural way to the Lyapunov functional given by Langer for these equations; but the treatment also suggests that a modified version of the phase-field equations might provide a more realistic model of freezing. © 1990.