Depletion force in colloidal systems

The entropic depletion force, in colloids, arises when large particles are placed in a solution of smaller ones, and sterically constrained to avoid them. We calculate the interaction between large spheres (of radius R) in a dilute solution of mutually avoiding small spheres (of diameter σ R and volume fraction φ), to third order in φ. In addition to the well-known attractive force for 0 < h < σ, we find a repulsive barrier at larger separations, and beyond that a secondary minimum. Except for unusually large size ratios (perhaps abetted by relatively high density φ), these features of the interaction potential are too small, compared to kBT, for kinetic stabilization (arising from the barrier) or flocculation into the secondary minimum, to be widespread, although such effects are possible in principle. For feasible size ratios, the same features should have observable consequences for the radial distribution function of the large spheres. Such effects can be viewed as precursors, at low density, of liquidlike structuring (solvation forces) expected at higher φ. Our third order calculation gives satisfactory agreement with a recent computer simulation at moderate density and size ratio (2R/σ = 10; φ = π/15).