Allen-Cahn model — dendrite growth and undercooling
0.45
0.04
2
0.02
Allen-Cahn equation: ∂φ/∂t = M[∇·(W²∇φ) + φ(1−φ)(φ−½+mΔT)]
where φ∈[0,1] is the solid fraction. Anisotropy W(θ)=W₀(1+ε cos 4θ) creates fourfold symmetry, generating dendritic patterns characteristic of snowflakes and metal casting.