Statistical Mechanics of Interfaces

Abstract

The mean-field approximation for interfacial structure may be derived from a form of the potential-distribution theory appropriate to inhomogeneous fluids:

$$\rho (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{r} ) = \lambda < e^{ - \psi /kT} > _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{r} }$$

(1)

where ψ is the potential measured by a test particle at ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{r}},\rho ({\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{r}})$ is the mean density there, and λ is the uniform activity. Applied to a fluid of attracting hard spheres, with the hardsphere repulsions treated exactly and the attractions by the mean-field approximation, (1) yields a functional equation for the density profile ρ(z) (density as a function of height z through the liquid-vapor interface):

$$\mu = M[\rho (z)] + \int\limits_{{r > b}} {\phi (r)[\rho (z') - \rho (z)]d\tau }$$

((2))

with dτ an element of volume at the variable height z′, with r the distance of that volume element from a fixed point at the height z, with b the sphere diameter and φ (r) the potential energy of intermolecular attraction, with M(ρ) the chemical potential of the bulk fluid as a function of density in meanfield approximation (with its van der Waals loops), and with μ the uniform chemical potential of the fluid, obtained from M(ρ) by the equal-areas (Maxwell) construction. For small gradients, (2) reduces to a form analogous to the laws of motion for a particle moving on a line subject to a prescribed potential, and so may be analyzed and solved by the methods of particle dynamics.