A planar solidification front becomes unstable when diffusion destabilizes faster than surface tension stabilizes — leading to dendritic growth.
Mullins-Sekerka (1963): perturbation of mode k grows at rate sigma(k) = V*k - d0*k^3. Unstable modes: k < sqrt(V/d0). Surface tension provides cutoff at small scales, diffusion field drives instability at large scales.