The gas–liquid surface of the penetrable sphere model

Abstract

A symmetry inherent in the penetrable sphere model of gas-liquid equilibrium allows the energy density $\phi (Z)$ to be expressed in terms of the number density of the particles, $\rho (Z)$, where $Z$ is measured in a direction normal to the planar gas-liquid surface. This expression leads, in a meanfield approximation, to a non-linear integral equation for $\rho (Z)$ which can, in some circumstances, be solved analytically. The potential distribution theorem for the determination of the local activity $\lambda (Z)$ is shown to be consistent with the constancy of $\lambda (Z)$ at all $Z$. It is shown that $\phi (Z)$ and $\lambda (Z)$ are functions not only of $\rho (Z)$ but of two further effective densities, $\sigma (Z)$ and $\tau (Z)$ respectively, the first of which is an average of $\rho (Z)$ over the distance of 2$a$, from ($Z+a$) to ($Z-a$), and the second a different average from ($Z+2a$) to ($Z-2a$), where 2$a$ is the range of the intermolecular potential. These results are used to calculate the excess surface energy and the surface tension, and to examine their behaviour at low temperatures and in the critical region. This examination shows both the value and the limitations of the concept of a free energy density in the surface zone.