Phase-Field Crystal Solidification

The Phase-Field Crystal (PFC) model describes solidification at atomic resolution using an order parameter ψ(r,t) representing atomic number density fluctuations. The free energy F[ψ] = ∫[ψ(ε + (1+∇²)²)ψ/2 + ψ⁴/4] dr drives dynamics ∂ψ/∂t = ∇²(δF/δψ). Crystalline phases appear as periodic density waves; the solid-liquid interface is resolved over lattice spacings. Dendritic growth emerges from anisotropic surface energy and diffusion-limited solidification. The undercooling parameter ε controls whether liquid (ε > 0) or crystal (ε < 0) is preferred.