In the past few years, suspensions consisting of colloid-polymer and colloid-colloid mixtures have attracted considerable attention as a result of their rich phase behaviour . The phase transitions in these systems arise from the non-additivity of the excluded volumes leading to the so called depletion interaction. This depletion interaction was first recognized and formulated for a number of cases by Asakura and Oosawa [2, 3]. Their results can be viewed as the first order approximation of a virial expansion in depletant concentration. More recently, work has been done [4, 5, 6, 7] to extend these calculations to second and third orders. A notable result is that at second order, the depletion forces can be repulsive, leading to a barrier in the depletion potential. This can be viewed as the precursor, within the virial expansion, of the crowding and layering effects arising at higher density, which have been extensively discussed for binary sphere systems in the context of solvation (rather than depletion) forces [8, 9].
It is clear that both the range and the absolute value of the depletion interaction can increase when rod-like macromolecules [3, 7] are used instead of spherical ones [2, 4, 5, 6] as the depletant. Bolhuis and Frenkel  presented a numerical study of the phase diagram of a mixture of spherical and (infinitely) thin rod-like colloids, using simulations and first order perturbation theory. Their work suggests in addition to the fluid-solid transition the possibility of a fluid-fluid phase separation. Experimentally, however, no phase separation induced by rod-like macromolecules or particles has been observed [11, 12]. In Ref., we reported results of a second order perturbation calculation of the depletion interaction due to rods, which was found to include a repulsive tail. On this basis it was suggested that, for realistic size ratios and concentrations, the repulsion could lead to a barrier in the depletion potential between two spheres in a solution semidilute rods, with a height large compared to . Such a barrier, if indeed present, could have significant consequences for colloid-rod mixtures, not only in terms of the equilibrium phase diagram, but also by the mechanism of kinetic stabilization: the long time scales required to cross the barrier might lead to a very long-lived, metastable homogeneous fluid, even for compositions where the true equilibrium state was a multiphase one.
In this paper we provide a more comprehensive treatment of the depletion interaction mediated by rod-like particles, by again considering the depletion interaction between flat plates, and then between two large spheres (of radius R), caused by mutually avoiding thin hard rods of length L, diameter D ( ) and bulk number density . We derive below (Section 2) a self-consistent integral equation (exact in the limit ), describing the density profile in the confined geometry of two parallel plates. Results for the profiles themselves will be presented elsewhere ; in this paper we focus on the resulting depletion forces. Expanding and truncating this integral equation recovers the previous perturbation treatment , valid for low rod concentrations, of which we present the full details in Section 3, which also contains results from a new third-order calculation. Then in Section 4, by numerically solving the full integral equation between two walls, we obtain (via the usual Derjaguin approximation, valid when ) essentially exact results for the depletion interaction between two spheres suspended in a fluid of long thin rods at various concentrations in the semi-dilute regime. The latter corresponds to where is the reduced density. (At the Onsager transition to a nematic phase occurs in the bulk; our calculations, which assume rotational symmetry about the common axis of the two colloidal spheres, should remain valid up to some transition point, in the vicinity of this threshold, at which the rotational symmetry is broken.) These results, which to within any numerical error are exact as , are compared with the perturbation treatments, and show good agreements at low densities (confirming the numerics) but also the gradual failure of the perturbation theory as the density increases. One unexpected consequence of this is that for of order unity, the repulsive barrier is, for reasonable sphere/rod size ratios, typically much smaller than the thermal energy , limiting severely the scope for kinetic stabilization. These and other results are discussed in Section 5 and Section 6 gives a brief summary.