Eric R. Dufresne and David G. Grier
February 8, 1999
We recently reported measurements of single-sphere trajectories in colloidal crystals near their melting point and related the individual spheres' dynamics to their crystals' bulk elastic properties . By doing so, we demonstrated that the potential of mean force in cannot be constructed by linear superposition of pairwise interactions of the standard DLVO form. The DLVO theory's failure to consistently describe both single-particle dynamics and many-particle elasticity suggests that the linear superposition approximation breaks down in strongly interacting charge-stabilized colloidal suspensions.
As part of this study, we estimated the effective harmonic potential maintaining each colloidal particle in its equilibrium position using two independent approaches: (1) by looking at the equilibrium probability distribution for particles' positions in their wells, and (2) by tracking spheres' mean square fluctuations away from arbitrary starting positions. The second of these methods was implemented incorrectly. Fortunately, the correct formulation yields results in quantitative agreement with the published conclusions, as we now show.
The Langevin equation for an overdamped Brownian sphere diffusing in a
harmonic potential well is
The mean-square in-plane displacement for an
ensemble of particles
starting from random positions in the potential
well and moving according to
Eq. (1) is
Fig. 1 shows nonlinear least-squares fits of trajectory data from Ref.  to Eq. (2) for crystals of volume fractions and . The revised values for the fitting parameters are summarized in Table 1. They agree quantitatively with those in Ref.  because the correct and incorrect forms for have the same limiting behavior at short and long times. Consequently, the conclusions drawn in Ref.  are not affected.
We are grateful to Jean-Pierre Hansen for independently bringing this error to our attention.