Free energy barrier ΔG* = 16πγ³/3Δg² and critical nucleus r* = 2γ/Δg
Classical Nucleation Theory (CNT) competes surface energy cost 4πγr² against bulk free energy gain −(4π/3)r³Δg. The total ΔG(r) = 4πγr² − (4π/3)r³Δg peaks at the critical radius r* = 2γ/Δg with barrier ΔG* = 16πγ³/(3Δg²). The nucleation rate J ∝ exp(−ΔG*/kT) is exponentially sensitive to supersaturation — a key mechanism in cloud formation, crystallization, and first-order phase transitions.