Title: Reexamination of Tolman's law and the Gibbs adsorption equation for curved interfaces,
Abstract: The influence of the surface curvature on the surface tension of small
droplets in equilibrium with a surrounding vapour, or small bubbles in
equilibrium with a surrounding liquid, can be expanded as $\gamma(R) = \gamma_0
+ c_1\gamma_0/R + O(1/R^2)$, where $R = R_\gamma$ is the radius of the surface
of tension and $\gamma_0$ is the surface tension of the planar interface,
corresponding to zero curvature. According to Tolman's law, the first-order
coefficient in this expansion is assumed to be related to the planar limit
$\delta_0$ of the Tolman length, i.e., the difference $\delta = R_\rho -
R_\gamma$ between the equimolar radius and the radius of the surface of
tension, by $c_1 = -2\delta_0$.
We show here that the deduction of Tolman's law from interfacial
thermodynamics relies on an inaccurate application of the Gibbs adsorption
equation to dispersed phases (droplets or bubbles). A revision of the
underlying theory reveals that the adsorption equation needs to be employed in
an alternative manner to that suggested by Tolman. Accordingly, we develop a
generalized Gibbs adsorption equation which consistently takes the size
dependence of interfacial properties into account, and show that from this
equation, a relation between the Tolman length and the influence of the size of
the dispersed phase on the surface tension cannot be deduced, invalidating the
argument which was put forward by Tolman [J. Chem. Phys. 17 (1949) 333].