Density functional theory of a curved liquid-vapour interface: evaluation of the rigidity constants

It is argued that to arrive at a quantitative description of the surface tension of a liquid drop as a function of its inverse radius, it is necessary to include the bending rigidity k and Gaussian rigidity (K) over bar in its description. New formulae for k and (K) over bar in the context of density functional theory with a non-local, integral expression for the interaction between molecules are presented. These expressions are used to investigate the influence of the choice of Gibbs dividing surface, and it is shown that for a one-component system, the equimolar surface has a special status in the sense that both k and (K) over bar are then the least sensitive to a change in the location of the dividing surface. Furthermore, the equimolar value for k corresponds to its maximum value and the equimolar value for (K) over bar corresponds to its minimum value. An explicit evaluation using a short-ranged interaction potential between molecules shows that k is negative with a value around minus 0.5-1.0 k(B)T and that (K) over bar is positive with a value that is a bit more than half the magnitude of k. Finally, for dispersion forces between molecules, we show that a term proportional to log(R)/R-2 replaces the rigidity constants and we determine the (universal) proportionality constants.