Cahn–Hilliard equation

The Cahn–Hilliard equation (after John W. Cahn and John E. Hilliard)[1] 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.

indicating domains, then the equation is written as where

gives the length of the transition regions between the domains.

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.

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 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.

[2] 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.

Evolution of random initial data under the Cahn–Hilliard equation with and (equal amounts of each phase), demonstrating phase separation.
Evolution of random initial data with and (60/40 mix of the blue and red phases, respectively), demonstrating Ostwald ripening