A Monte Carlo Test of the Fisher-Nakanishi-Scaling Theory for the Capillary Condensation Critical Point

Abstract

Extending the Swendsen-Wang cluster algorithm to include both bulk (H) and surface fields (H_1) in L x L x D Ising films of thickness D and two free L x L surfaces, a Monte Carlo study of the capillary condensation critical point of the model is presented. Applying a finite-size scaling analysis where the lateral linear dimension L is varied over a wide range, the critical temperature T_c(D) and the associated critical field H_c(D) are estimated for 4 <= D <= 32 lattice spacings, for a choice of the surface field H_1 small enough that the dependence of H_c(D) on H_1 is still linear. It is shown that the results are consistent with the power laws predicted by Fisher and Nakanishi [M.E. Fisher and H. Nakanishi, J. Chem. Phys. 75, 5857 (1981)], namely T_c(\infty)-T_c(D) \propto D^{-1/ν}, H_c(D) \propto D^{-(Δ-Δ_1)/ν}, where νis the bulk correlation length exponent of the three-dimensional Ising model, and Δ, Δ_1 are the corresponding ``gap exponents&#39;&#39; associated with bulk and surface fields, respectively. As expected, the order parameter of the thin film near its critical point exhibits critical behavior compatible with the universality class of the two-dimensional Ising model.