Statistical Mechanics of Surface Tension

Abstract

It is shown that we can derive by a comparatively simple procedure the formula of surface tension in terms of distribution functions from purely statistical considerations. Surface tension is derived as the increase of free energy which accompanies the increase of unit area of the interface between the liquid and the vapor phases.
The expression obtained is the same as MacLellan’s:
γ=\frac12∫∫∫∫\fracdφ₁₂dR₁₂\fracx₁₂²−z₁₂²R₁₂ρ_s⁽²⁾(z,R₁₂)dz₁dv₁₂,
where γ is the surface tension, φ₁₂ is the intermolecular potential and ρ_s⁽²⁾(z,R₁₂) is the excess pair density reckoned relative to an arbitrary Gibbs dividing surface. The above expression can be transformed into the form given by Bakker,
γ=∫(p_N−p_T)dz₁,
where
p_N(z₁)=kTρ⁽¹⁾(z₁)−\frac12∫∫∫dv₁₂∫_z1−z12^z₁\fracdφ₁₂dR₁₂\fracz₁₂R₁₂ρ⁽²⁾(ζ,R₁₂)dζ,
p_T(z₁)=kTρ⁽¹⁾(z₁)−\frac12∫∫∫dv₁₂\fracdφ₁₂dR₁₂\fracx₁₂²R₁₂ρ⁽²⁾(z₁,R₁₂).
This result coincides with that of Kirkwood and Buff which was derived by calculating stressed directly. The assumption of density discontinuity, which is often made in the theory of surface tension, is examined closely.