Anisotropic Dendritic Growth — Pattern Selection

Symmetry-broken crystal growth via diffusion-limited solidification

ε = 0.15
n = 6 (hexagonal)
Δ = 0.50
η = 0.05

Solvability Theory of Dendritic Growth

Dendrites form when surface tension anisotropy breaks the circular Mullins-Sekerka symmetry. The phase-field model with anisotropic surface energy:

τ ∂φ/∂t = W²∇²φ + φ(1−φ²) + λu(1−φ²)² , a(n̂) = 1 + ε·cos(n·θ)

The n-fold anisotropy selects specific tip orientations. Solvability theory (Ivantsov 1947, corrected): the selected tip velocity V* and radius ρ* satisfy σ* = 2Dα/V*ρ*² = fixed constant dependent on anisotropy. Higher ε → faster tips, thinner stalks, richer sidebranching.