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Next: Effective Pair Interactions Up: The Potential of Mean Previous: The Harmonic Approximation

Diffusion in the Potential of Mean Force

Time-resolved particle trajectories yield complementary insights into the potential of mean force. The conditional probability density for a sphere initially at location r to diffuse through the potential of mean force a distance $\Delta {\mathbf r}$in time $\tau$ is
$\displaystyle P(\Delta {\mathbf r}, \tau \vert {\mathbf r} )$ = $\displaystyle P_0 \, e^{- \frac{\Delta r^2}{4D \tau}} \,
e^{- \beta [W({\mathbf r} + \Delta {\mathbf r}) - W({\mathbf r})]}$ (9)
  = $\displaystyle P_0 \, e^{- \frac{\Delta r^2}{4D \tau}} \,
e^{- \frac{1}{2} \beta k
(\Delta r^2 + 2 {\mathbf r} \cdot \Delta {\mathbf r})},$ (10)

where D is the free-particle self-diffusion coefficient. The normalization constant, P0, is set by the condition

\begin{displaymath}\int_{-\infty}^{\infty} P( \Delta {\mathbf r}, \tau \vert {\mathbf r} ) \,
d\Delta {\mathbf r} = 1.
\end{displaymath} (11)

Straightforward integration gives

\begin{displaymath}P( \Delta {\mathbf r}, \tau \vert {\mathbf r}) =
\left( \fra...
...\cdot \Delta {\mathbf r}} \,
e^{-(\gamma + \ell) \Delta r^2},
\end{displaymath} (12)

where $\gamma = \frac{1}{2} \beta k$ and $\ell(\tau) = (4D \tau)^{-1}$.

If we now average over all possible starting positions r, the probability for a particle to move by $\Delta {\mathbf r}$ in time $\tau$is

$\displaystyle P(\Delta {\mathbf r}, \tau)$ = $\displaystyle \int_{-\infty}^{\infty}
P(\Delta {\mathbf r}, \tau \vert {\mathbf r}) \, P({\mathbf r}) \,
d {\mathbf r}$ (13)
  = $\displaystyle \frac{s^3}{\pi^\frac{3}{2}} \, e^{-s^2 \Delta r^2},$ (14)


\begin{displaymath}s(\tau) = \frac{\gamma + \ell(\tau)}{[2\gamma + \ell(\tau)]^\frac{1}{2}}.
\end{displaymath} (15)

Since this result is separable, the probability distribution for one-dimensional displacements in the three-dimensional effective potential well is

\begin{displaymath}P(\Delta x, \tau) = \frac{s}{\pi^\frac{1}{2}} \,e^{-s^2 \Delta x^2}.
\end{displaymath} (16)

The mean-squared one-dimensional displacement is then
$\displaystyle \langle \Delta x^2 \rangle$ = $\displaystyle \frac{s}{\pi^\frac{1}{2}} \,
\int_{-\infty}^{\infty} \Delta x^2 \, e^{-s^2 \Delta x^2} \, d\Delta x$ (17)
  = $\displaystyle \frac{1}{2 s^2}$ (18)
  = $\displaystyle \frac{1}{2} \frac{\beta k + \frac{1}{4D \tau}}{
\left( \frac{1}{2} \beta k + \frac{1}{4D \tau} \right)^2}.$ (19)

The time evolution of $\langle \Delta x \rangle^2$ provides an independent estimate of the potential well's curvature, $\beta k$, as well as a measure of the free-particle self-diffusion coefficient, D. However, it does not account for the motion of lattice defects such as edge dislocations. We account for this with an additional diffusive contribution to the mean-squared in-plane displacement:

 \begin{displaymath}\langle \Delta \rho^2 \rangle =
\langle \Delta x^2 \rangle +...
...rac{1}{2} \beta k + \frac{1}{4D\tau} \right)^2}
+ 4 D_d \tau,
\end{displaymath} (20)

where Dd is the effective single-particle diffusion coefficient due to defect diffusion. In general, we expect $D_d \ll D$ even quite near melting.

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, $\frac{1}{2} \beta k a^2 = 59 \pm 12$ and $199 \pm 60$for the dilute and dense crystals respectively. When combined with the result from Section 3.1, this provides an overall estimates of $\frac{1}{2} \beta k a^2 =63 \pm 7$and $221 \pm 32$which we will use in following Sections.

The extracted self-diffusion coefficients for the dilute and dense crystals, $D = (2.4 \pm 0.3) \times 10^{-2}~a^2/\mathrm{sec} =
0.29 \pm 0.03~\mu\mathrm{m}^2/\mathrm{sec}$and $D = (4.2 \pm 0.3) \times 10^{-2}~a^2/\mathrm{sec} =
0.26 \pm 0.03~\mu\mathrm{m}^2/\mathrm{sec}$respectively, agree very well with each other, but are only half the value for a freely diffusing sphere, $D_0 = k_B T / (6 \pi \eta \sigma) = 0.67~\mu\mathrm{m}^2/\mathrm{sec}$. 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, $D_d = 4.3 \times 10^{-5}~a^2/\mathrm{sec}$, for the dense crystal is a factor of 3000 smaller than D, as expected. Even in the softer crystal, Dd is less than 2 percent of D.

Figure 6: Time evolution of the mean-squared displacement from arbitrary starting positions for spheres in the two colloidal crystals. The solid lines are fits to Eq. (20). (a) $\phi = 0.011$. (b) $\phi = 0.026$.
\begin{figure}\centering\includegraphics[width=3in]{figures/r2bar.epsf} \end{figure}

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.

next up previous
Next: Effective Pair Interactions Up: The Potential of Mean Previous: The Harmonic Approximation
David G. Grier