We have presented an integral equation, exact in the limit of long thin rods,
whose numerical solution we used to predict the depletion force between
parallel plates,
and that between large
spheres, due to rod-like colloidal particles in semidilute solution.
The results were
compared with our previous perturbation treatment, showing good agreements
at low densities (confirming the numerics) but also the gradual failure of
perturbation theory
as the density goes up. From the numerical results, we concluded that
the depletion barrier is typically less than the thermal energy 
,
and therefore unlikely to significantly alter the phase diagram of colloidal
sphere-rod mixtures, either in equilibrium or kinetically.
Compared to the simplest first order results, the main correction from the
full numerical results is to modify the depth and the range of the
primary depletion well. The same method allows predictions for the surface
tension of rod solutions to be made, along the lines suggested in
Ref.[19]; see Ref.[13].
In principle our treatment could be extended to the nematic phase of rods,
but the reduced symmetry of the integral kernel will make this numerically
unfeasible; indeed even for the isotropic case the maximum density of rods
we could
achieve was only about half that of a nematic solution. All our results are
subject to corrections of order D/L, the
inverse aspect ratio of the rods, which may be significant for realistic
values of this quantity and which remain to be quantified in detail.
Acknowledgements: We thank R. C. Ball, M. E. Fisher, C.-Y. D. Lu, H. P. Marro, J. Melrose, P. van der Schoot, and M. S. Turner for useful discussions. Y.M. is grateful to Trinity College for a Research Studentship. This collaboration was funded in part by the Colloid Technology Programme. MEC is grateful to the Isaac Newton Institute for Mathematical Sciences (Cambridge) where part of this work was completed.