Surface tension and energy of a classical liquid-vapour interface

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Abstract
Rigorous general expressions for the surface tension σ and the surface energy per unit area ε are derived in the form of three-fold integrals. In the approximation n 2(r 1, r 2) = n(z 1)n(z 2)g(r 12, ñ) we obtain the following results: (i) Both σ and ε are proportional to (n 1 - n v)2. (ii) The expressions for σ and ε are formally reduced to a single integral, with integrands determined in terms of the density profile n(z). (iii) Explicit expressions are given for an exponential density variation. (iv) In the limit of a density variation which is slow on the scale of the molecular diameter, we derive the general expressions σ = A(n 1 - n v)2/λ, ε = B(n 1 - n v)2λ from the microscopic theory (λ is a measure of the surface thickness). The same forms for σ and ε follow from (iii), with explicit expressions for A and B. These forms for σ and ε are shown to be very good approximations even well away from the critical point. It is argued that the critical power laws have the same range of validity. (v) The critical exponents of A and B are determined, that for A agreeing with the result of Fisk and Widom. (vi) The surface thickness is determined for Ar, Kr and Xe near their triple points, using our theory for ε and experimental data on the bulk energy of the liquids. The results are in excellent agreement with other estimates. Similar results are obtained with the direct correlation function expression for the surface tension: the general expression is reduced to a three-fold integral, and results analogous to (i) through (iv) are obtained in the approximation c(r 1, r 2) = c(r 12, ñ). The equivalence of the c(r 1, r 2) and g(r 1, r 2) formulations for σ is proved in the low density limit.

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