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Erratum: ``Interactions, Dynamics, and Elasticity in Charge-Stabilized Colloidal Crystals'' [J. Chem. Phys. 109, 8659 (1998)]

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 [1]. 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

\frac{d{\vec r}}{dt} = - \frac{k}{\gamma} \, {\vec r} -
\frac{1}{\gamma} \, {\vec \eta},
\end{displaymath} (1)

where $\vec r(t)$ is the sphere's position at time t, k is the curvature of a spherically-symmetric harmonic potential of mean force centered at r = 0, and $\gamma = 6 \pi \nu \sigma$ is the drag coefficient for sphere of radius $\sigma$ moving through a fluid of viscosity $\nu$. $\vec \eta (\vec r, t)$ is a stochastic force with ensemble average behavior $\langle \vec \eta \rangle = 0$ and $\langle \vec \eta(t) \cdot \vec \eta(t') \rangle =
6k_BT\gamma \, \delta(t-t')$, where T is the temperature and kB is Boltzmann's constant.

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

\langle \Delta \rho^2(t) \rangle =
\frac{4}{\beta k} \, \left(
1 - e^{-\beta k D t} \right) + 4D_d t,
\end{displaymath} (2)

where $\beta = 1/(k_B T)$ and $D = k_B T / \gamma$ is the free-particle self-diffusion coefficient. The last term in Eq. (2) accounts for slow diffusion of the the harmonic well itself due to defect motion. This is the correct form for Eq. (20) in Ref. [1].

Fig. 1 shows nonlinear least-squares fits of trajectory data from Ref. [1] to Eq. (2) for crystals of volume fractions $\phi = 0.011$ and $\phi = 0.026$. The revised values for the fitting parameters are summarized in Table 1. They agree quantitatively with those in Ref. [1] because the correct and incorrect forms for $\langle \Delta \rho^2(t) \rangle$ have the same limiting behavior at short and long times. Consequently, the conclusions drawn in Ref. [1] are not affected.

We are grateful to Jean-Pierre Hansen for independently bringing this error to our attention.

Figure 1: Mean-square in-plane displacements for spheres in two colloidal crystals. (a) $\phi = 0.011$. (b) $\phi = 0.026$. Solid lines are nonlinear least-squares fits to Eq. (2). Dashed lines indicate the expected error in $\Delta \rho ^2(t)$ given the estimated error in locating sphere centers.
\includegraphics[width=3in]{fig1.eps} \vspace{1ex}

Table 1: Interaction and diffusion coefficients for spheres in colloidal crystals with volume fractions $\phi = 0.011$ and $\phi = 0.026$. Where indicated, lengths are expressed in units of the crystals' lattice constants, a, for comparison with Ref. [1].
$\phi$ $\frac{1}{2} \, \beta k a^2$ $D~(\mu \mathrm{m}^2/s)$ Dd (a2/s)
0.011 $73 \pm 1$ $0.37 \pm 0.02$ $(6.0 \pm 0.2) \times 10^{-4}$
0.026 $212 \pm 2$ $0.23 \pm 0.02$ $(2.8 \pm 0.4) \times 10^{-5}$

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David G. Grier