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Next: Conclusions Up: Interactions in Colloidal Suspensions: Previous: Electrohydrodynamic Coupling

Electrostatics Near One Wall

Figure 9: The measured potential of mean force for three different particle concentrations, obtained from the inset radial distribution functions. Curves are offset by 0.25 for clarity. Inset: Corresponding pair correlations functions. Peaks near $ r = \sigma $ result from roughly one aggregated dimer in the field of view at any time and were suppressed in calculating $ U(r)$.
Figure 10: The pair interaction energy calculated in the HNC (open circles) and Percus-Yevick (full dots) approximations together with a fit of the HNC result to Eq. (9). Inset: Logarithmic representation of HNC results and best fits for data obtained at different areal densities.

We studied suspensions of silica spheres with diameter $ \sigma = 2a = 1.58~\ensuremath{\mathrm{\mu m}}\xspace $ (Catalog # 8150, Duke Scientific, Palo Alto, CA) sedimented in deionized water into monolayers above the bottom wall of a 200  $ \mathrm{\mu m}$-thick glass sample cell. The balance of electrostatic and gravitational forces maintained the spheres in a two-dimensional layer at $ h = 0.9 \pm 0.1~\ensuremath{\mathrm{\mu m}}\xspace $. As in previous studies [23,24], we extracted the sphere's equilibrium pair potential from their pair correlation function $ g(r)$. Unfortunately, $ g(r)$ cannot be related directly to $ U(r)$ using Eq. (10) because we cannot be sure a priori that the monolayer is sufficiently dilute. Finite concentration introduces many-body correlations into $ g(r)$ and, in general,

$\displaystyle W(r) = -k_BT \ln g(r)$ (48)

is the density-dependent potential of mean force, rather than the pair potential. To make matters worse, no exact relationship is known between $ W(r)$ and $ U(r)$, although approximations are available provided that errors in $ g(r)$ are sufficiently small. Achieving the necessary accuracy requires avoiding several subtle sources of error.

The average number of spheres separated from a given sphere by $ r \pm \delta/2$ is $ 2 \pi r \delta \, \bar \rho \, g(r)$, where $ \bar \rho$ is the two-dimensional particle concentration. Given $ A \bar \rho$ spheres in field of view $ A$, the total number of such $ r$-pairs should yield the pair correlation function through $ N(r) = N_0(r) \, g(r)$, with $ N_0(r) = \pi \delta A \, \bar \rho^2 \, r$. In a limited field of view, however, spheres near the edge have fewer neighbors to sample at range $ r$ than those near the middle. Applying periodic boundary conditions would introduce spurious correlations, while limiting the calculation to a central region of the field of view would drastically degrade statistics. Instead, we count the $ \vec r$-pairs in a recorded image, normalize by the number of particles further than $ \vec r$ from any edge, and sum over angles to calculate $ g(r)$.

The limited field of view contains very few particles when $ \bar \rho$ is small, so that $ N_0(\sigma)$ typically is not much larger than 1 for a resolution $ \delta \approx 0.1\sigma$. Limiting relative errors in $ g(r)$ to $ \varepsilon$ would require $ N_0(r = \sigma) > 1/\varepsilon$. Measuring $ g(r)$ to this accuracy therefore requires averaging $ M > 1/(\pi \delta A \bar \rho^2 \sigma \varepsilon)$ uncorrelated snapshots of the suspension's structure. The time required for correlations to decay is roughly ten percent of the interval $ \tau = (4D \bar \rho)^{-1}$ a particle needs to diffuse the mean inter-particle separation. Thus the time $ T = M \tau$ required to adequately sample $ g(r)$ is $ T > 0.1/(4 \pi \delta A D \sigma \varepsilon \,\bar \rho^3)$.

On this basis, we suspect that some previously reported equilibrium measurements [23] of like-charge attractions may have been undersampled by as much as a factor of 20 in time. Excess correlations observed under these conditions might be interpreted, therefore, as transients, rather than as evidence for equilibrium attractions. This raises concern because the attractions reported in these studies are comparable to the quoted energy resolutions and thus could be particularly sensitive to incomplete averaging.

Because of difficulties in controlling concentration, temperature, and ionic strength constant over long periods, statistical accuracy strongly favors larger particle concentrations. On the other hand, many-body contributions obscure pair interactions for larger concentrations. The areal densities between $ \bar \rho \sigma^2 = 0.05$ and 0.1 chosen for the present study represent a compromise between statistical and interpretive accuracy. Adequate sampling for our lowest density monolayer required half and hour of sampling.

Even if $ g(r)$ is calculated appropriately, spurious correlations could be introduced by lateral variations in the glass wall's interaction with the spheres. Preferred regions would collect spheres, mimicking an inter-sphere attraction. We compared two-dimensional histograms of recorded particle positions with analogous histograms for uniformly distributed random data sets with the same number of particles. Differences in the first two moments of these histograms vanish with increasing delay time between consecutive snapshots, suggesting that each particle's position becomes uncorrelated over time as expected. Thus the substrate potential appears to be featureless on the length- and timescales of our experiment, to within our resolution. Previous studies [23,24] do not appear to have addressed this possible source of error.

Our experimental results for $ g(r)$ and $ W(r)$ at different concentrations appear in Fig. 9. The curves indicate a repulsive core interaction causing particle depletion extending to about $ 2\sigma$. Beyond this, they reveal a preferred nearest-neighbor separation between two and three diameters, as well as the onset of the oscillatory correlations typical of a structured fluid. The depth of the minimum in the potential of mean force depends on $ \bar \rho$ and so reflects at least some many-body contributions.

Provided that $ g(r)$ is free from experimental artifacts, reliable approximations for $ U(r)$ can be obtained from the Ornstein-Zernike integral equation with appropriate closure relations [65]. Good results for ``soft'' potentials typically are achieved with the hypernetted chain (HNC) approximation, whereas the Percus-Yevick (PY) approximation is known to be a better choice for hard spheres. The pair interaction potential can be evaluated numerically as

$\displaystyle \beta U(r) = -\ln[g(r)] + \begin{cases}\bar \rho \, \ensuremath{{...
...\\  \ln [1 + \bar \rho \, \ensuremath{{\cal I}}(r)] & \mathrm{(PY)} \end{cases}$ (49)

where $ I(r)$ is the convolution integral

$\displaystyle \ensuremath{{\cal I}}(r) = \int \left[ g(r') - 1 - \bar \rho \, \ensuremath{{\cal I}}(r) \right] \left[ g(\vert{\bf r'-r}\vert)-1 \right] d^2r'$ (50)

to be solved iteratively with $ \ensuremath{{\cal I}}(r) = 0$ as initial value [66]. Evaluating $ \ensuremath{{\cal I}}(r)$ directly rather than with numerical Fourier transforms avoids introducing misleading features into $ U(r)$ due to experimental noise.

Figure 10 shows the pair potential obtained for $ \bar \rho \, \sigma^2 = 0.051$ together with a fit of the HNC approximation to Eq. (9) for $ Z = 5000 \pm 500$, $ \kappa = 0.32 \pm 0.05~\ensuremath{\mathrm{\mu m}}\xspace $, and an arbitrary additive offset. Having obtained comparable results in both the HNC and PY approximations lends confidence in the accuracy of $ U(r)$. Corresponding pair potentials extracted from data at $ \bar \rho \, \sigma^2 = 0.079$ and 0.083 are essentially indistinguishable on this scale, and the extracted coupling constants are consistent with those obtained at lower areal density. The absence of minima in the pair potential confirms that oscillations in the potential of mean force (Fig. 9) resulted from crowding, while the underlying interaction is purely repulsive. The similarity of the results at different areal densities suggests that the interaction responsible for the monolayer's liquid structure indeed can be resolved into an additive pair potential. Values extracted for the charge number and screening length are consistent with surface charging due to silanol dissociation in a solution with an ionic strength around $ 10^{-6}$ M.

Monte Carlo simulations of particles interacting with comparable pair potentials confirm our energy resolution to be about $ 0.1~k_BT$. This uncertainty is too large to resolve van der Waals attraction or image charge repulsions [49,51]. Long-range attractions of the previously reported strength [2,23,24,27], however, would have been resolved.

next up previous
Next: Conclusions Up: Interactions in Colloidal Suspensions: Previous: Electrohydrodynamic Coupling
David G. Grier 2001-01-16