Depletion forces in fluids

We investigate the entropic depletion force that arises between two big hard spheres of radius $R_b,$ mimicking colloidal particles, immersed in a fluid of small hard spheres of radius $R_s.$ Within the framework of the Derjaguin approximation, which becomes exact as ${s=R}_s{/R}_b\to 0$, we examine an exact expression for the depletion force and the corresponding potential for the range $0<h<\mathrm{}{2R}_s,$ where $h$ is the separation between the big spheres. These expressions, which depend only on the bulk pressure and the corresponding planar wall-fluid interfacial tension, are valid for all fluid number densities ${\rho }_s.$ In the limit ${\rho }_s\to 0$ we recover the results of earlier low density theories. Comparison with recent computer simulations shows that the Derjaguin approximation is not reliable for $s=0.1\mathrm{}$ and packing fractions ${\eta }_s=4\mathrm{}\pi {\rho }_s{R}_s^3/3\mathrm{}\gtrsim 0.3$. We propose two new approximations, one based on treating the fluid as if it were confined to a wedge and the other based on the limit ${s=R}_s{/R}_b\to 1$. Both improve upon the Derjaguin approximation for $s=0.1\mathrm{}$ and high packing fractions. We discuss the extent to which our results remain valid for more general fluids, e.g., nonadsorbing polymers near colloidal particles, and their implications for fluid-fluid phase separation in a binary hard-sphere mixture.