ΔG(r) — Free Energy vs Nucleus Radius
Nucleation Rate J vs Supercooling
r* = 2γ / (ΔGᵥ) | ΔG* = 16πγ³ / (3ΔGᵥ²)
Classical Nucleation Theory (Volmer & Weber 1926, Becker & Döring 1935):
A nucleus of radius r has free energy ΔG(r) = −(4π/3)r³·|ΔGᵥ| + 4πr²·γ,
balancing bulk driving force (negative, ∝ supercooling) against surface penalty.
The maximum at r* = 2γ/|ΔGᵥ| is the critical nucleus; smaller clusters dissolve,
larger ones grow. Nucleation rate J ∝ exp(−ΔG*/kT) is strongly nonlinear in supercooling.
The Zeldovich factor and attachment kinetics modify prefactors but the barrier controls the rate.