Curvature-dependence of the liquid-vapor surface tension beyond the Tolman approximation

Surface tension is a macroscopic manifestation of the cohesion of matter, and its value $\sigma_\infty$ is readily measured for a flat liquid-vapor interface. For interfaces with a small radius of curvature $R$, the surface tension might differ from $\sigma_\infty$. The Tolman equation, $\sigma(R) = \sigma_\infty / (1 + 2 \delta/R)$, with $\delta$ a constant length, is commonly used to describe nanoscale phenomena such as nucleation. Here we report experiments on nucleation of bubbles in ethanol and n-heptane, and their analysis in combination with their counterparts for the nucleation of droplets in supersaturated vapors, and with water data. We show that neither a constant surface tension nor the Tolman equation can consistently describe the data. We also investigate a model including $1/R$ and $1/R^2$ terms in $\sigma(R)$. We describe a general procedure to obtain the coefficients of these terms from detailed nucleation experiments. This work explains the conflicting values obtained for the Tolman length in previous analyses, and suggests directions for future work.