On thermodynamically consistent schemes for phase field equations

This paper addresses the issue of thermodynamically consistent derivations of field equations governing nonisothermal processes with diffuse interfaces and their implications for interface conditions in the associated sharp interface theories. We operate within the framework of extended irreversible thermodynamics, allowing gradient terms to be present not only in the free energy and entropy densities but also in the internal energy, for both nonconserved and conserved order parameter theories. These various gradient terms are shown to relate to the splitting of the surface tension into an energetic and an entropic part. It is shown that the principle of nondecreasing local entropy production does not single out uniquely the form of the governing field equations. Instead it leads naturally to a one-parameter family of alternative theories which do not contradict the Curie principle of irreversible thermodynamics. The principal applications are to the solidification of a pure material and of a binary alloy. In the case of a pure substance an asymptotic analysis is developed in the vicinity of the solidification front. The main effect of the alternative theories, as well as of the splitting of the surface tension, on the corresponding free boundary problem is manifested in the first order terms in the nonequilibrium contributions to the interface undercooling. We also consider the possibility of assigning from empirical data some specific values to the phenomenological parameters involved in these models.