 Cahn–Hilliard equation

The Cahn--Hilliard equation transports a conserved composition field down gradients of its chemical potential: ${\partial }_t\varphi =\nabla \cdot (M\nabla \mu )$ with $\mu =\delta F/\delta \varphi$. Advection adds $\mathbf{v}\cdot \nabla \varphi$ to the left-hand side.

The Cahn–Hilliard equation is a partial differential equation that describes the phase separation and motion of interfaces in a binary mixture or alloy. It is particularly important in materials science, as it models the process by which two phases of a material (such as solid and liquid) separate from each other, leading to the formation of distinct microstructures over time. The equation was introduced by John W. Cahn and John E.