General formulae for the curvature dependence of droplets and bubbles

Abstract

General equations have been developed that describe the curvature dependence of the surface tension of droplets in the gas phase and of bubbles in a liquid. These are based on thermodynamic arguments and a special assumption concerning the Tolman parameter, δ. The equations derived by Gibbs, Tolman and Rasmussen are obtained as special cases for particular values of a parameter dependent both on the system considered and the thermodynamic constraints. Although the behaviour of the Tolman coefficient δ as a function of the radius, R_s, of the surface of tension can be quite different, the resulting equations describing the curvature dependence of surface tension are similar, thus confirming Tolman's equation as a first, but accurate approximation.

It is further shown that physically meaningful results for the surface tension of bubbles can be obtained only if the Tolman coefficient changes its sign for a specific value R_s=|b| of the radius of the bubble. Therefore in this case the surface tension has a maximum for a finite value of the radius of the surface of tension and there are two regions in which the surface tension either decreases (R_s < |b|) or increases (R_s > |b|) with decreasing size of the bubble.