The equilibrium structure of a hard-sphere fluid confined in a small spherical cavity is investigated. In such systems the statistical mechanical ensembles are no longer equivalent and we consider both open (grand canonical) and closed (canonical) cavities in order to analyze the effects of size and packing constraints on the density profile of the confined fluid. For systems in the grand canonical ensemble the profiles are obtained from grand canonical ensemble Monte Carlo simulations and from density functional theory. The profiles of the closed (canonical) systems are obtained by means of canonical ensemble Monte Carlo simulations. A scheme is proposed which expands the canonical ensemble density profiles in terms of grand canonical averages; this is formally a series in powers of the inverse average number of particles. By comparing canonical ensemble Monte Carlo data with the results of the expansion applied to grand canonical ensemble Monte Carlo data and to the results of density functional theory the series expansion is shown to converge very quickly in most situations, even when the cavity contains only a few particles. However, as a consequence of packing constraints, in certain situations the density profile develops a pronounced peak in the center of the cavity. Then significant differences arise between the canonical and grand canonical profiles and the convergence of the series is much slower in the central zone where the peak develops. Describing accurately the various terms in the expansion and, hence, the detailed shapes of the profiles provides a searching test of density functional approximations. We find that recent modifications of Rosenfeld’s fundamental measure theory, which are designed to describe situations of low effective dimensionality, perform better than his original theory and yield accurate results for all cases except those near maximum packing.

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We present a density-functional approach for dealing with inhomogeneous fluids in the canonical ensemble. A general relation is proposed between the free-energy functionals in the canonical and the grand canonical ensembles. The minimization of the canonical-ensemble free-energy functional gives rise to Euler-Lagrange equations which involve averaged Ornstein-Zernike equations of second and third order. The theory is especially appropriate for systems with a small, fixed number of particles. As an example of application we obtain accurate results for the density profile of a hard-sphere fluid in a closed spherical cavity that contains only a few particles.

When thermodynamic properties of a pure substance are transformed to reduced form by using both critical- and triple-point values, the corresponding experimental data along the whole liquid-vapor coexistence curve can be correlated with a very simple analytical expression that interpolates between the behavior near the triple and the critical points. The leading terms of this expression contain only two parameters: the critical exponent and the slope at the triple point. For a given thermodynamic property, the critical exponent has a universal character but the slope at the triple point can vary significantly from one substance to another. However, for certain thermodynamic properties including the difference of coexisting densities, the enthalpy of vaporization, and the surface tension of the saturated liquid, one finds that the slope at the triple point also has a nearly universal value for a wide class of fluids. These thermodynamic properties thus show a corresponding apparently universal behavior along the whole coexistence curve.

Explicit size corrections in the calculation of the fluctuations in the number of particles in a finite subvolume of a hard-disk fluid composed of a fixed number of particles are considered. The size corrections are obtained on the basis of a Taylor series expansion of the pair distribution function of the N-particle system in powers of 1/N. Analytical density dependent expressions are obtained at low density. These expressions show that not only explicit size effects (due to consideration of a fixed number of particles) but also edge effects that result from considering a finite subvolume must be taken into account. A general density dependence study is also reported by relating the relative fluctuation in the number of particles to the equation of state. Numerical results for the Henderson equation of state are obtained. These theoretical results are compared with Monte Carlo computer simulation results.