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,

$$\beta U(r) = Z^2 \lambda_B \left( \frac{e^{\kappa \sigma}}{1 + \kappa \sigma} \right)^2 \, \frac{e^{-\kappa r}}{r},$$ (21)

parameterized by the spheres' effective charge number Z and by the electrolyte's screening length $\kappa^{-1}$. The term in parentheses accounts for the exclusion of simple ions from the interiors of spheres of radius $\sigma$under the linear superposition approximation. The Bjerrum length, $\lambda_B = e^2/(\epsilon k_B T)$, is the thermal separation for unit charges dispersed in a medium of dielectric coefficient $\epsilon$ at temperature T. For water at the experimental temperature $T = (24.0 \pm 0.2)^\circ$C, $\lambda_B = 0.714$ nm. Van der Waals attraction is negligibly weak for polystyrene spheres separated by more than a hundred nanometers [26] and will not be considered in the following discussion. Eq. (21) has been found to accurately describe interactions between isolated pairs of the spheres in this study under comparable conditions of ionic strength [4,6,12,28]. It therefore seems the natural candidate to describe interactions within a crystal of these spheres.

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

$$\beta W(r) = c \, U(a) \, \left( \frac{\sinh \kappa r}{\kappa r} - 1 \right)$$ (22)

where c = 12 is the coordination number for an FCC crystal. Taylor expanding Eq. (22) to lowest non-trivial order yields

$$W(r) \approx 2 \, U(a) \, (\kappa r)^2.$$ (23)

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:

$$\frac{1}{2} \beta k a^2 \approx 2 (\kappa a)^2 \beta U(a).$$ (24)

Eq. (24) alone is not sufficient to estimate both Z and $\kappa$. Measuring $\kappa$ accurately in the confined sample volume is difficult, particularly at the low ionic strengths of the present experiments. A second condition relating Z to $\kappa$ recently was deduced from interaction measurements on isolated pairs of spheres [12]. The two parameters are coupled because the effective surface potential, $\zeta$, for a highly ionizable sphere must saturate at a thermally regulated value, $\zeta_0$. The corresponding renormalized [30] charge number, Z, then is related to $\kappa$ and $\zeta_0$ through [12]

$$Z = \beta e \zeta_0 \frac{\sigma}{\lambda_B} (1 + \kappa \sigma).$$ (25)

The dimensionless surface potential, $\Phi_0 = \beta e \zeta_0$, indeed has been found to saturate in the limit of low ionic strength at $\Phi_0 = 5.7$for isolated pairs of the spheres used in the present study [12]. Combining Eqs. (24) and (25) yields an expression,

$$(\kappa a)^2 \, e^{-\kappa a \left(1 - \frac{2 \sigma}{a} \... ...2} \beta k a^2}{2 {\Phi_0}^2} \, \frac{\lambda_B a}{\sigma^2},$$ (26)

which can be solved for the screening length, $\kappa^{-1}$.

Substituting values obtained for the dilute crystal into Eq. (26) yields $\kappa a = 10.6 \pm 0.2$, which is equivalent to a screening length of $\kappa^{-1} = 333 \pm 5$ nm and corresponds to a renormalized charge number of $Z = 5200 \pm 100$. The corresponding values for the dense crystal are $\kappa a = 10.3 \pm 0.2$, $\kappa^{-1} = 243 \pm 5$ nm and $Z = 6100 \pm 100$. 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 $\kappa$ can account for both the spheres' dynamics and their crystals' elasticity.