We follow the conventional DLVO theory  in modeling
the long-range electrostatic
interaction between colloidal spheres as a screened-Coulomb repulsion,
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 . 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
Eq. (24) alone is not sufficient to estimate both
Z and .
accurately in the confined sample volume is
difficult, particularly at the low ionic strengths of the present
A second condition relating Z to
recently was deduced
from interaction measurements on isolated pairs of spheres .
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  charge number, Z,
then is related to
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 .
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.