Direct Evidence for Curvature-Dependent Surface Tension in Capillary Condensation: Kelvin Equation at Molecular Scale

Capillary condensation is the first-order vapor-to-liquid phase transition taking place in confined geometries. Such heterogeneous nucleation has been well described by thermodynamic laws such as the Kelvin equation, but the equation's applicability at the nanoscale is still unresolved. Here, we show that the Kelvin equation is valid down to approximately 0.5 nm radius of curvature when the curvature dependence of surface tension is taken into account. By the shear-mode atomic force microscopy, we have measured directly and accurately the critical tip-surface distance (dc) at which the water meniscus is capillary condensed in ambient condition; e.g., dc≈1.2 nm at 10% relative humidity. In particular, we can determine the Tolman length, the unique characteristic of the curvature-dependent surface tension, as the single fitting parameter (δ=0.21±0.05 nm). Our results that unify the validity of the Kelvin equation at molecular scale and the characterization of the curvature effect of surface tension may provide a better understanding of general nucleation phenomena in nature, including the role of nanometric aerosols in cloud formation.