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The Harmonic Approximation


  
Figure 5: Probability distribution P(x) for in-plane displacements about spheres' mean positions for single-particle trajectories averaged over 1024 video frames (33 seconds). Statistics were acquired along an arbitrary direction, taking advantage of the separability of Eq. (7). The solid lines are fits to Eq. (8). (a) $\phi = 0.011$. (b) $\phi = 0.026$.
\begin{figure}\centering\includegraphics[width=3in]{figures/disp.epsf} \vspace{2ex}
\end{figure}

Each sphere in a colloidal crystal diffuses in a potential well created by interactions with its neighbors. Regardless of the form of the underlying pair potential, the potential of mean force for a stable solid may be approximated by a spherically symmetric harmonic well,

 \begin{displaymath}
W(r) = \frac{1}{2} k r^2,
\end{displaymath} (6)

centered about each sphere's equilibrium position. The adequacy of this model can be judged by comparing the measured probability distribution for sphere displacements from their mean positions in d dimensions with

 \begin{displaymath}P({\mathbf r}) = \left(\frac{\beta k}{2 \pi}\right)^{\frac{d}{2}} \,
e^{-\frac{1}{2} \beta k r^2},
\end{displaymath} (7)

where $\beta^{-1} = k_B T$ is the thermal energy scale at temperature T. Figure 5 shows $\ln P(x)$ for the crystals in this study binned along one Cartesian coordinate for a period of 33 seconds. The choice of projection is arbitrary and quantitatively equivalent results are obtained with other projections. The solid lines in Fig. 5 are two-parameter fits to the form

 \begin{displaymath}\ln P(x) = - \frac{1}{2} \beta k x^2 + \alpha,
\end{displaymath} (8)

where $\alpha$ is an arbitrary additive offset. We expect Eq. (7) to fail for displacements larger than the Lindemann limit, $x \approx \Delta$. Consequently, the data in Fig. 5 are fit over the range $x \in [-\Delta,\Delta]$ but plotted over a wider range to illustrate limitations of the harmonic approximation for large displacements. The resulting estimates for the dimensionless curvature of the potential of mean force, $\frac{1}{2} \beta k a^2 = 66 \pm 6$ and $243 \pm 20$ for the dilute and dense crystals respectively, will be used in Sections 4 and 5 to estimate the crystals' elastic parameters.


next up previous
Next: Diffusion in the Potential Up: The Potential of Mean Previous: The Potential of Mean
David G. Grier
1998-06-08