Cahn-Hilliard equation — spontaneous phase separation
The Cahn-Hilliard equation: ∂c/∂t = M ∇²(∂f/∂c − κ∇²c) governs phase separation. f(c) = c²(1−c)² is the double-well free energy. Below the spinodal (T < Tc), concentration fluctuations grow spontaneously without nucleation. The result is a bicontinuous microstructure with characteristic wavelength λ ∝ √(κ/|f″|). Coarsening follows Lifshitz-Slyozov-Wagner: R̄ ∝ t^(1/3).