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 [1]. 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 [10]
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.[7], 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 [13]; in this paper
we focus on the resulting depletion forces. Expanding and truncating this
integral equation recovers the previous perturbation treatment [7],
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.