On the accuracy of the Derjaguin approximation for depletion potentials

The accuracy of the Derjaguin expressions for the depletion potential of large spheres due to small spheres, discs, and rods is analysed. To that end, we subdivide the generalised Gibbs adsorption equation into three contributions. We determine these contributions both in first order of curvature of the large spheres as well as exact for small spheres, discs, and rods up to first order in the number density of the depletion agents. The gain of volume in the bulk when the depletion zones of the large spheres overlap is the same for spheres, discs, and rods of equally characteristic sizes. The Derjaguin approximation underestimates the exact solution because of the neglected curvature. The number of particles that fit in the gap between the two large spheres as well as the amount that can enter a single depletion zone, are both underestimated by the Derjaguin approximation for discs and rods to relatively the same extent. Due to the intermediate excluded volume of the discs, the correction of the last two contributions just cancels the error in the first one. Small spheres only exhibit the first contribution, whereas for rods the other parts overcompensate the first. We can thus explain why the depletion potential of large spheres due to small spheres is underestimated by the Derjaguin approximation, is surprisingly accurate for discs, and is overestimated for rod-like depletion agents.