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Thermodynamically consistent models of phase-field type for the kinetics of phase transitions. (English) Zbl 0709.76001

Summary: 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 e.g., G. Caginalp [Arch. Ration. Mech. Anal. 92, 205-245 (1986; Zbl 0608.35080)], and also leads in a natural way to the Lyapunov functional given by J. S. Langer [Models of pattern formation in first-order phase transitions, in: Directions in Condensed Matter Physics, 165-186 (1986)] 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.

MSC:

76A02 Foundations of fluid mechanics
35Q35 PDEs in connection with fluid mechanics

Citations:

Zbl 0608.35080
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