Physical, mathematical, and numerical derivations of the Cahn–Hilliard equation

Abstract

We review physical, mathematical, and numerical derivations of the binary Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard). The phase separation is described by the equation whereby a binary mixture spontaneously separates into two domains rich in individual components. First, we describe the physical derivation from the basic thermodynamics. The free energy of the volume Ω of an isotropic system is given by NV∫Ω[F(c)+0.5∊2|∇c|2]dx, where NV, c, F(c), ∊, and ∇c represent the number of molecules per unit volume, composition, free energy per molecule of a homogenous system, gradient energy coefficient related to the interfacial energy, and composition gradient, respectively. We define the chemical potential as the variational derivative of the total energy, and its flux as the minus gradient of the potential. Using the usual continuity equation, we obtain the Cahn–Hilliard equation. Second, we outline the mathematical derivation of the Cahn–Hilliard equation. The approach originates from the free energy functional and its justification of the functional in the Hilbert space. After calculating the gradient, we obtain the Cahn–Hilliard equation as a gradient flow. Third, various aspects are introduced using numerical methods such as the finite difference, finite element, and spectral methods. We also provide a short MATLAB program code for the Cahn–Hilliard equation using a pseudospectral method.