Generalized Cahn-hilliard Equations Based on a Microforce Balance

The Cahn-Hilliard equation is very important in materials science. It is a conservation law which describes important qualitative behaviors of two-phase systems, namely, the phase separation process in a binary alloy. This phenomenon can be observed, for instance, when the alloy is cooled down sufficiently. One then observes a partial nucleation (i.e., the apparition of nucleides in the material) or a total nucleation (the socalled spinodal decomposition: the mixture quickly becomes inhomogeneous, forming a fine-grained structure). We refer the reader to [15] for more details; see also [54, 55] for a qualitative study of the spinodal decomposition. The starting point of the Cahn-Hilliard theory is a free energy ψ = ψ(ρ,∇ρ) of the form