Adsorption of hard spheres: structure and effective density according to   the potential distribution theorem
  We propose a new type of effective densities via the potential distribution theorem. These densities are for the sake of enabling the mapping of the free energy of a uniform fluid onto that of a nonuniform fluid. The potential distribution theorem gives the work required to insert a test particle into the bath molecules under the action of the external (wall) potential. This insertion work W_ins can be obtained from Monte Carlo (MC) simulation (e.g. from Widom's test particle technique) or from an analytical theory. The pseudo-densities are constructed thusly so that when their values are substituted into a uniform-fluid equation of state (e.g. the Carnahan-Starling equation for the hard-sphere chemical potentials), the MC nonuniform insertion work is reproduced. We characterize the pseudo-density behavior for the hard spheres/hard wall system at moderate to high densities (from \rho^*= 0.5745 to 0.9135). We adopt the MC data of Groot et al. for this purpose. The pseudo-densities show oscillatory behavior out of phase (opposite) to that of the singlet densities. We also construct a new closure-based density functional theory (the star-function based density functional theory) that can give accurate description of the MC density profiles and insertion works. A viable theory is established for several cases in hard sphere adsorption. 
L. L. Lee and G. Pellicane
cond-mat.soft