The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard) is an equation of mathematical physics which describes the process of phase separation, spinodal decomposition, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. If c is the concentration of the fluid, with c=\pm1 indicating domains, then the equation is written as where D is a diffusion coefficient with units of and gives the length of the transition regions between the domains. Here is the partial time derivative and \nabla^2 is the Laplacian in n dimensions. Additionally, the quantity is identified as a chemical potential. Related to it is the Allen–Cahn equation, as well as the stochastic Allen–Cahn and the stochastic Cahn–Hilliard equations.

Features and applications

Of interest to mathematicians is the existence of a unique solution of the Cahn–Hilliard equation, given by smooth initial data. The proof relies essentially on the existence of a Lyapunov functional. Specifically, if we identify as a free energy functional, then so that the free energy does not grow in time. This also indicates segregation into domains is the asymptotic outcome of the evolution of this equation. In real experiments, the segregation of an initially mixed binary fluid into domains is observed. The segregation is characterized by the following facts. The Cahn–Hilliard equation finds applications in diverse fields: in complex fluids and soft matter (interfacial fluid flow, polymer science and in industrial applications). The solution of the Cahn–Hilliard equation for a binary mixture demonstrated to coincide well with the solution of a Stefan problem and the model of Thomas and Windle. Of interest to researchers at present is the coupling of the phase separation of the Cahn–Hilliard equation to the Navier–Stokes equations of fluid flow.