We first discuss, within perturbation theory, the scalings of
the three contributions to the depletion interaction
in Eq.(32). The first order term 
influences the depth of the attractive (primary) minimum, which is of
order 
, and this is the same scaling as for depletion by small
spheres whose diameter 
is equal to the length of the
rods L, Ref.[5].
For a given 
, however, much larger attractions are
possible with rods since the maximum 
is of
order 
(corresponding to 
)
which far exceeds that attainable for spheres, of order
.
The second order contribution in (32), which
provides a repulsive barrier,
arises directly from the excluded volume interaction among rods, and
accordingly is smaller than the first by one power of 
.
The resulting barrier height scales as 
in the perturbative
regime. Finally, the third order correction 
multiplies
a term reduced by one further power of , on which basis we expect
the previous estimate of the barrier height to become unreliable
for of order unity. In practice, the form of 
and are such
that the range of validity of the second/third order treatment
is roughly 
, depending on the separation h as can be
seen in Figures 3 and 4.
These show the full numerical results compared with the perturbation
treatment, for the cases 
.
At low concentrations, the numerical results and perturbation treatment
agree well, which is expected since the expansion
leading to Eq.(16) is then justified. Put differently,
the second and especially the third order results
provide very powerful checks of our numerical solution; for intermediate
concentrations ( 
) the effect of the third order correction
is quantitatively
reproduced by the full results. This makes us confident that
our computer programme indeed provides a correct solution of
the full self-consistent integral equation. Another check
is provided by the limit 
for which
the Onsager pressure is recovered, as mentioned above.
At concentrations above 
the full results show the second order
perturbation theory to somewhat overestimate the main depletion attraction.
A more surprising and important feature of the full calculation is the relative
smallness of the repulsive barrier now found in the depletion potential for
of order unity. For example, in a
system of size ratios R/L=10,L/D=20
(reasonable parameters for colloidal sphere-rod mixtures), it
was argued in Ref.[7] that a barrier
of 
could be expected at 
and one of 
for 
.
In contrast to these predictions, which were based on second order
perturbation theory,
the full numerical solution
presented here shows that the maximum repulsive barrier height in the
depletion potential is of order 
, attained
at a reduced density
of 
, giving for the aforementioned parameters a maximum
barrier height of 
. This will now have very little
effect on either phase equilibria or kinetic stabilization, although
the repulsive features of the potential might still be measurable between
parallel plates in a force machine.
As Fig.2 shows, the third order correction has an
attractive part largely coinciding with the repulsive part of
. This overlap seems to be mainly responsible for the reduction (and
perhaps ultimately the disappearance) of the repulsive depletion potential at
high densities.
Although in our earlier predictions we were aware of the dangers of
extrapolating the second order
perturbation results beyond their valid density range, it seems
very surprising that the discrepancy should already be so large for
of order unity.
We can find no deep-seated reason for this except
perhaps that in going from the force between plates to the interaction
energy between particles there are two integrations. This means that
the magnitude (once) and the range (twice) of the repulsive force
enter multiplicatively; at each factor is perturbatively
overestimated by roughly a factor two which leads to an order of magnitude
reduction
in barrier height. Even this does not explain the
unusual smallness of the dimensionless prefactor
in the expression for the maximum barrier height
( ).
We now consider an important limitation to our treatment, arising from the
finite aspect ratio of the rods. For practical situations the aspect
ratio L/D is never more than 100, so the Onsager limit ( 
),
on which Eq.(14) is based, may not be fully satisfied.
A conservative estimate of this error can be obtained from examining
the virial series for rods in bulk. Onsager showed that in the bulk the
virial expansion of pressure
would terminate at the second
order in for . For finite aspect ratio rods, the virial
coefficients 
have been computed numerically by Frenkel
[23], and
we find that at and L/D=100 this pressure deviation from the
Onsager limit is of the same
order of magnitude as the maximum repulsive depletion force (which arises
for 
).
This deviation increases with increasing concentration and with decreasing
aspect ratio. Accordingly we cannot rule out the possibility that for
aspect ratio of (say) 20, as used in the numerical examples discussed
above, a significant repulsive barrier could once again be restored by
these higher order virial effects. This issue is probably best addressed
by computer simulation.
A second limitation is our assumption that the density distribution
function possess the uniaxial symmetry imparted by the parallel plate
geometry (or the geometry of two spheres in solution).
This would of course be untrue if the rods spontaneously broke the uniaxial
symmetry to form a biaxial system. This excluded-volume driven transition
could occur near that occurring in bulk at the I-N (Isotropic-Nematic)
transition ( 
); however the role of the plates should help
to suppress such a transition, because the restricted geometry partially
aligns the rods thereby alleviating the excluded-volume interaction.
Accordingly for hard plates (or spheres) without specific surface treatments
we do not expect the uniaxial symmetry to be broken way before the bulk I-N
transition is reached. Indeed the stability analysis in Ref.[20]
suggest that the uniaxial symmetry should be preserved for 
.
There should be some adjustment, however, for rods whose aspect ratio
is not large. In the case of uniaxial symmetry being broken, a grid
of higher dimension will be needed in the numerical solution, demanding
significantly more computing power in solving the Eq.(14).