Extension of scaled particle theory to inhomogeneous hard particle fluids. III. Entropic force exerted on a cavity that intersects a hard wall

We present a further development of an inhomogeneous scaled particle theory (I-SPT) for hard particle fluids confined by hard walls, such that the reversible work of cavity insertion can now be determined for all cavities that intersect one of the walls. Building upon a previous version of I-SPT [D. W. Siderius and D. S. Corti, Phys. Rev. E, 71, 036141 (2005)], a new function, F, is introduced, which is proportional to the net force on the surface of the cavity in the direction normal to the wall. The reversible work of cavity insertion is then determined by an integral over the force required to "push" the cavity of fixed size into the fluid starting from a position behind the wall. An exact relation for F at certain cavity locations and radii is derived and an accurate interpolation scheme is proposed for the computation of F beyond these exact limits. The chosen interpolation incorporates a large number of exact and nearly exact conditions, several of which follow from the surface thermodynamics of macroscopic cavities. Work predictions using F are highly accurate as compared to simulation results at low to moderate fluid densities. Good agreement still persists at densities near the hard-sphere freezing transition. The interpolation of F is also used to estimate the depletion force between a hard sphere solute and the wall. The I-SPT entropic force predictions are in good agreement with simulation results presented in the literature. Due to its reliance upon physical and geometric arguments, I-SPT provides important insights into the origin of various depletion effects such as how the interplay between geometry and the varying local density at the cavity surface gives rise to the appearance of multiple attractive regions at intermediate solute sizes and a universal repulsive region, both within solute to wall separations that are less than the diameter of a solvent particle. Finally, all of the scaled particle theory-based methods presented here can, in principle, be extended to describe hard particle fluids confined by nonplanar surfaces, thereby providing estimates of the depletion force between a solute and a variety of surfaces of interest.