Surface tension and energy of a classical liquid-vapour interface

Rigorous general expressions for the surface tension σ and the surface energy per unit area e 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, n) we obtain the following results: (i) Both σ and e are proportional to (n 1 - n v)2. (ii) The expressions for σ and e 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, e = B(n 1 - n v)2λ from the microscopic theory (λ is a measure of the surface thickness). The same forms for σ and e follow from (iii), with explicit expressions for A and B. These forms for σ and e 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 crit...