Cahn–Hilliard equation

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 ${\displaystyle c}$ is the concentration of the fluid, with ${\displaystyle c=\pm 1}$ indicating both domains, then the equation is written as$${\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}\left(c^{3}-c-\gamma \nabla ^{2}c\right),}$$where ${\displaystyle D}$ is a diffusion coefficient with units of ${\displaystyle {\rm {{L}^{2}/{\rm {T}}}}}$, ${\displaystyle {\sqrt {\gamma }}}$ gives the length of the transition regions between the domains, ${\displaystyle \partial /{\partial t}}$ is the partial time derivative and ${\displaystyle \nabla ^{2}}$ is the Laplacian in ${\displaystyle n}$ dimensions (typically three). Additionally, the quantity ${\displaystyle \mu \equiv c^{3}-c-\gamma \nabla ^{2}c}$ is sometimes identified as a chemical potential.

It is related to the Allen–Cahn equation, as well as the stochastic Allen–Cahn and the stochastic Cahn–Hilliard equations.