Y. Mao 
, M. E. Cates 
and H. N. W. Lekkerkerker 
Cavendish Laboratory, Madingley Road,
Cambridge, CB3 OHE, UK.
Department of Physics and Astronomy,
University of Edinburgh, JCMB King's Buildings,
Mayfield Road, Edinburgh EH9 3JZ, UK.
Van't Hoff Laboratory, University of Utrecht,
Padualaan 8, 3584 Utrecht, The Netherlands.
The entropic depletion force, in colloids,
arises when large particles are placed in a solution of smaller
ones, and sterically constrained to avoid them. In this paper,
we consider a system of two parallel plates
suspended in a semidilute solution of long
thin rods of length L and diameter D. By numerically solving an integral
equation, which is exact in the ``Onsager limit" ( 
),
we obtain the depletion force between the plates.
The second integral of this determines (via the Derjaguin approximation)
the depletion potential between two large hard spheres of radius R,
immersed in a solution of hard rods (satisfying 
).
The results for this potential are compared with our
previous second order perturbation treatment (Y. Mao, M. E. Cates
and H. N. W. Lekkerkerker, Phys. Rev. Lett. 75, 4548 (1995)), as well
as with newly computed third order perturbation results. There is good
agreement at low and intermediate densities (which validates our numerical
procedures for the
integral equation) but the gradual failure of the perturbative
treatments is revealed as the density increases. From the numerical results,
we conclude that for typical colloidal sphere/rod mixtures, the repulsive
barrier in the depletion interaction is likely to be less than the thermal
energy 
throughout the semidilute concentration range of the rods. This
reverses our previous conclusion, based on second order perturbation theory,
that this barrier could easily lead to kinetic stabilization of the mixture.
However, it is possible that terms of order D/L (neglected here) could
further modify the results. Within the present theory, the main modification
to the well-known (attractive) first order interaction, as the density is
raised, is by changing the depth and range of the primary attractive well.