Models of phase segregation and diffusion of atomic species on a lattice

TLDR: In this paper, two mathematical models for phase segregation and diffusion of an order parameter are derived, within one and the same continuum mechanical framework, respectively of the Allen-Cahn type and of the Cahn-Hilliard type.

Abstract: Two mathematical models for phase segregation and diffusion of an order parameter are derived, within one and the same continuum mechanical framework. These models are, respectively, of the Allen-Cahn type and of the Cahn-Hilliard type. Our framework is similar to that used in [1], in that a postulated balance of microforces plays a central role in both deductive paths, but differs from it, mainly in three ways: imbalance of entropy replaces for a dissipation inequality, whose form depends on the case, restricting the growth of free energy; balance of energy replaces for the mass balance introduced in [1] to arrive at (a generalization of) the C-H equation; and chemical potential is given the same role played by coldness in the deduction of the heat equation. When appropriate constitutive prescriptions are made, different in the cases of segregation and diffusion but consistent with the entropy imbalance, it is found that standard A-C and C-H processes are solutions of constant chemical potential of the corresponding generalized equations; in particular, the stationary solutions are the same. Keywords: Phase segregation, Diffusion, Allen-Cahn equation, Cahn-Hilliard equation, Phase-field methods Mathematics Subject Classification (2000): 74N25, 74A50, 35K60