An extended Cahn-Hilliard model for interfaces with cubic anisotropy

Abstract

Abstract For studying systems with a cubic anisotropy in interfacial energy σ, we extend the Cahn–Hilliard model by including in it a fourth-rank term, namely, γ ijlm [∂2 c/(∂x i ∂x j )][∂2 c/(∂x l ∂x m )]. This term leads to an additional linear term in the evolution equation for the composition parameter field. It also leads to an orientation-dependent effective fourth-rank coefficient γ (hkl) in the governing equation for the one-dimensional composition profile across a planar interface. The main effect of a non-negative γ (hkl) is to increase both σ and interfacial width w, each of which, upon suitable scaling, is related to γ (hkl) through a universal scaling function. In this model, σ is a differentiable function of interface orientation [ncirc], and does not exhibit cusps; therefore, the equilibrium particle shapes (Wulff shapes) do not contain planar facets. However, the anisotropy in the interfacial energy can be large enough to give rise to corners in the Wulff shapes in two dimensions. In particles of finite sizes, the corners become rounded, and their shapes tend towards the Wulff shape with increasing particle size.