Crystal growth rarely begins in a perfectly uniform environment. In most real systems, crystallization starts through heterogeneous nucleation: the formation of a new crystalline phase on a pre-existing surface such as dust particles, container walls, impurities, grain boundaries, or catalytic substrates. Unlike homogeneous nucleation, where molecules must spontaneously organize within the bulk phase, heterogeneous nucleation lowers the Gibbs free energy barrier required for stable nucleus formation. The foreign surface reduces the interfacial energy penalty between phases, making nucleation thermodynamically favorable at lower supersaturation or undercooling conditions. The critical free energy barrier for nucleation is strongly dependent on surface geometry and interfacial tensions: ΔGheterogeneous = f(θ)ΔGhomogeneous where f(θ) is a wetting-dependent geometric factor determined by the contact angle between the nucleus and substrate. Because 0 < f(θ) < 1, heterogeneous nucleation...