A unifying identity for the work of cluster formation in heterogeneous and homogeneous nucleation theory

A unifying identity is derived relating the reversible work of cluster formation (W) and its molecular number content (n) and surface work (Φ) components, each ratioed to the corresponding values for a spherical capillary drop of critical size in classical nucleation theory. The result is a relationship that connects these ratios: f_W = -2f_N + 3f_S, where f_W = W/W^*_CNT, f_N = n/n^*_CNT, and f_S = Φ/Φ^*_CNT. Shown to generalize two early thermodynamic relationships of Gibbs, the new result is demonstrated here for Fletcher's model of heterogeneous nucleation, resulting in a unified treatment of condensation on flat and curved substrates and smooth passage to the homogeneous limit. Additional applications are made to clusters of non-critical as well as critical size and to a molecular-based extension of classical nucleation theory based on the Kelvin relation. The new identity serves as a consistency check on complicated theoretical expressions and numerical calculations and can be used to guide the construction of theory and interpretation of measurements.