Density Functional Theory of a Curved Liquid-Vapour Interface: Evaluation of the rigidity constants

Abstract

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_bar in its description. New formulas for k and k_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_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_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 kT and that k_bar is positive with a value which 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.

28 pages; 5 figures; accepted for publication in J. Phys.: Condens. Matter (2013)

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• Tolman lengths and rigidity constants from free-energy functionals -- General expressions and comparison of theories
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• A Mesoscale Perspective on the Tolman Length
• Virial series for inhomogeneous fluids applied to the Lennard-Jones wall-fluid surface tension at planar and curved walls
• Triezenberg-Zwanzig expression for the surface tension of a liquid drop
• Bending and Gaussian rigidities of confined soft spheres from second-order virial series
• Reexamination of Tolman's law and the Gibbs adsorption equation for curved interfaces