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.
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The Cahn--Hilliard equation transports a conserved composition field down gradients of its chemical potential: with . Advection adds to the left-hand side.