CahnHilliard

The Cahn–Hilliard equation, named after John W. Cahn and John Hilliard, is a mathematical model for phase separation in binary mixtures. It describes how a composition field evolves over time while conserving the total amount of each component, making it a mass-conserving diffuse-interface model. It is widely used in materials science to describe spinodal decomposition in alloys and polymer blends.

Let c(x,t) denote the local concentration (or order parameter). The dynamics derives from a free-energy functional F[c] = ∫ [ f0(c) + (κ/2)|∇c|^2 ] dV, where f0(c) is a double-well potential with minima at the pure phases and κ>0 is the gradient-energy coefficient. The chemical potential is μ = δF/δc = f0'(c) - κ∇^2 c. The evolution equation is ∂c/∂t = ∇·[M ∇μ], where M is the mobility (often taken as a constant). Substituting μ yields the fourth-order PDE ∂c/∂t = ∇·[M ∇(f0'(c) - κ∇^2 c)].

This equation conserves the average concentration and drives phase separation with coarsening, producing diffuse interfaces whose thickness is set by κ. Numerical treatment is challenging due to stiffness; common approaches include semi-implicit or convex-splitting schemes, finite differences, finite elements, and spectral methods. Boundary conditions are typically no-flux, ensuring mass conservation.

Extensions and variants include composition-dependent mobility M(c), coupling to elasticity or fluid flow, and multi-component generalizations. The Cahn–Hilliard framework is used to model microstructure evolution in alloys, polymer blends, porous materials, and battery materials, where accurate capture of phase separation and domain growth is essential.