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

Abstract

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: fW = -2fN + 3fS, where fW = W/W∗CNT, fN = n/n∗CNT, and fS = 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.