Classical nucleation theory (CNT): forming a nucleus of radius r costs surface energy 4πr²γ but gains bulk energy (4π/3)r³|ΔG_v|. The free energy barrier ΔG(r) peaks at critical radius r* = 2γ/|ΔG_v|, with barrier ΔG* = 16πγ³/(3ΔG_v²). Nucleation rate J = J₀exp(−ΔG*/kT) — exponentially sensitive to supersaturation. Spinodal decomposition (right panel): no barrier, spontaneous phase separation when inside the spinodal region (∂²G/∂c² < 0). Shown via simulated Cahn-Hilliard-like coarsening.