= | (9) | ||
= | (10) |
(11) |
(12) |
If we now average over all possible starting positions
r,
the probability for a particle to move by
in time is
= | (13) | ||
= | (14) |
(15) |
(16) |
The degree to which Eq. (20) describes single-particle dynamics in colloidal crystals can be gauged from Fig. 6. Nonlinear least-squares fits to Eq. (20) yield the dimensionless curvatures of the potentials of mean force, and for the dilute and dense crystals respectively. When combined with the result from Section 3.1, this provides an overall estimates of and which we will use in following Sections.
The extracted self-diffusion coefficients for the dilute and dense crystals, and respectively, agree very well with each other, but are only half the value for a freely diffusing sphere, . The two crystals should yield comparable self-diffusion coefficients since hydrodynamic coupling to neighboring spheres is quite weak for the large separations in the present systems [10,21]. The overall suppression of the self-diffusion coefficients can be ascribed to a combination of hydrodynamic coupling to the bounding glass wall [22,23,24] and ionic drag [22].
The estimated defect diffusivity, , for the dense crystal is a factor of 3000 smaller than D, as expected. Even in the softer crystal, D_{d} is less than 2 percent of D.
Extracting reasonable values for the diffusion coefficients serves as an additional consistency check for the estimated curvatures of the potential of mean force. The potential of mean force then can be related to parameters describing the colloidal pair potential. The pair potential, in turn, provides insight into the crystal's anticipated elastic properties. We therefore seek a model for the pair interaction which accounts not only for the potential of mean force, but also for the crystals' elastic moduli. In particular, we will assess whether or not the interaction parameters for isolated pairs of spheres can be used to describe the many-body behavior of strongly interacting colloidal crystals.