Nucleation & Crystal Growth

Classical Nucleation Theory: Forming a new phase (e.g., ice crystal) requires overcoming a Gibbs free energy barrier ΔG(r) = 4πr²γ − (4/3)πr³·|ΔGv|, where γ is surface energy and ΔGv is the volumetric free energy gain due to supercooling. The critical radius r* = 2γ/|ΔGv| maximizes ΔG — nuclei smaller than r* dissolve, larger ones grow. Increasing supercooling ΔT lowers r* and the barrier ΔG*, exponentially enhancing nucleation. The crystal grows with 6-fold symmetry (snowflake: n=6) due to hexagonal lattice anisotropy.