Binary Alloy Spinodal Decomposition

Cahn-Hilliard Equation: ∂c/∂t = M∇²[∂f/∂c − κ∇²c], where f(c,T) = c ln(c) + (1−c)ln(1−c) + χ(T)c(1−c) is the regular solution free energy (Flory-Huggins). The Flory parameter χ = 1/(2T/Tc) creates a double-well below Tc. Inside the spinodal (∂²f/∂c² < 0), the homogeneous mixture is unstable to all perturbations — no nucleation barrier. The dominant unstable wavevector q* ~ 1/√(2κ) sets the initial length scale. Coarsening follows Lifshitz-Slyozov scaling: domain size L(t) ~ t^(1/3) (diffusive) or t^1 (viscous flow). Colors: orange-rich = A-rich phase, blue-rich = B-rich phase.