Finite-size scaling Casimir force function: Exact spherical-model results.

Abstract

The excess free energy due to the finite-size contributions to the free energy of the system in a film geometry characterizes a fluctuation-mediated interaction that is termed the Casimir force, or, in the case of a fluid confined between two parallel walls, the solvation force (or the disjoining pressure). The analog of these forces within the three-dimensional mean-spherical model with periodic boundary conditions and geometry L\ifmmode\times\else\texttimes\fi{}${\mathrm{\ensuremath{\infty}}}^{2}$ is investigated in the presence of an external magnetic field. The corresponding analytical expressions for the finite-size scaling functions of the excess free energy and the Casimir (solvation) force and their asymptotic behavior in the vicinity, below and above the critical temperature ${\mathit{T}}_{\mathit{c}}$, are derived and evaluated numerically. In contrast to the Ising-like case the scaling functions of the excess free energy and of the Casimir force below ${\mathit{T}}_{\mathit{c}}$ in zero magnetic field do not tend exponentially fast with L to zero, but, tend to some universal constant. The last is supposed to be true for all O(n), n\ensuremath{\ge}2 models. \textcopyright{} 1996 The American Physical Society.