Effective Pair Interactions

We follow the conventional DLVO theory [25] in modeling
the long-range electrostatic
interaction between colloidal spheres as a screened-Coulomb repulsion,

parameterized by the spheres' effective charge number

To do so, we
further assume that the potential of mean force, *W*(*r*),
can be built up by linear superposition of pairwise contributions.
This is a risky assumption given the body of experimental evidence
that confined and crowded spheres can develop long-range attractions
not accounted for by the DLVO theory [5,12,27,28].
It is equivalent, however, to the linear superposition approximation
used in deriving Eq. (21).
Furthermore,
van Roij and Hansen have shown that a many-body cohesion
arising from nonlinear correlations between spheres and simple ions
need not distort the underlying DLVO repulsion [29].
If this is the case, then the DLVO pair potential still may
suffice for describing colloidal crystals' elastic properties,
although perhaps with renormalized interaction parameters.

Considering nearest-neighbor interactions and averaging over angles,
we obtain for the potential of mean force

where

Comparing this to Eq. (6) allows us to relate the parameters describing the pair interaction to the measured curvature of the potential of mean force:

Eq. (24) alone is not sufficient to estimate both
*Z* and .
Measuring
accurately in the confined sample volume is
difficult, particularly at the low ionic strengths of the present
experiments.
A second condition relating *Z* to
recently was deduced
from interaction measurements on isolated pairs of spheres [12].
The two parameters are coupled because
the effective surface potential, ,
for a highly ionizable sphere
must saturate at a thermally regulated value, .
The corresponding renormalized [30] charge number, *Z*,
then is related to
and
through
[12]

The dimensionless surface potential, , indeed has been found to saturate in the limit of low ionic strength at for isolated pairs of the spheres used in the present study [12]. Combining Eqs. (24) and (25) yields an expression,

which can be solved for the screening length, .

Substituting values obtained for the dilute crystal
into Eq. (26) yields
,
which is equivalent to a screening length
of
nm and corresponds to a renormalized charge
number of
.
The corresponding values for the dense crystal are
,
nm and
.
The difference in effective charge number for identical spheres
reflects the difference in screening length for the two suspensions.
This, in turn, could be due to the different sphere densities
since each sphere
contributes *Z* counterions to the local electrolyte
within the crystal [31].

If these values were to account for the elastic constants of
the crystals, then we would have a self-consistent understanding
of how colloidal interactions give rise to colloidal crystals'
macroscopic properties.
However, the elastic constants calculated
from these interaction parameters turn out to be significantly larger
then the measured values.
Indeed, no values of *Z* and
can account for both
the spheres' dynamics and their crystals' elasticity.