Interactions in charge-stabilized colloid

Considerable attention has been focused in recent years on colloidal interactions, particularly in light of experimental observations that challenge theoretical predictions. For example, charged colloidal spheres of diameter $\sigma$ and charge number $Z$ dispersed in an electrolyte interact with each other directly through their Coulomb repulsion and also indirectly through their influence on the surrounding distribution of simple, atomic-scale ions. Poisson-Boltzmann mean-field theory predicts an overall screened-Coulomb repulsion (8) of the form

$$\beta u(r) = Z^2 \lambda_B \, \frac{\exp(\kappa \sigma)}{\left(1 + \frac{\kappa \sigma}{2}\right)^2} \, \frac{\exp(-\kappa r)}{r},$$ (23)

where $\beta^{-1} = k_BT$ is the thermal energy scale at absolute temperature $T$, $r$ is the spheres' center-to-center separation, and $\lambda_B = e^2/(\epsilon k_B T)$ is the Bjerrum length in a medium of dielectric constant $\epsilon$. If the electrolyte has a total concentration $c$ of monovalent ions, then the Debye-Hückel screening length, $\kappa^{-1} = 1/\sqrt{4 \pi \lambda_B c}$, sets the range of the effective electrostatic interaction in the mean-field approximation. This is an effective interaction because it results from an average over the simple ions' degrees of freedom. When viewed in this light, it is not surprising that measurements might differ from predictions based on Eq. (23). More surprising is that like-charged colloidal spheres appear to attract each other under some circumstances, in qualitative disagreement with Eq. (23). One of the goals of the present study is to apply the configurational temperature formalism to resolve some of the outstanding questions regarding these anomalous like-charge attractions.

Unlike atoms, which travel ballistically within the potential energy landscape established by inter-atomic interactions, colloidal particles are immersed in a viscous fluid that randomizes their trajectories over intervals longer than $\tau = \beta m D$, the momentum relaxation time. Given a typical colloidal diffusion coefficient $D \approx 1~\unit{\ensuremath{\unit{\mu m}}\xspace ^2/sec}$, $\tau \approx 1~\unit{\mu sec}$. Consequently colloidal suspensions' microscopic temperatures are not easily monitored with the usual kinetic definition of the temperature, Eq. (2). The configurational temperature, by contrast, can be measured from snapshots and so provides an ideal alternative.

The fluid also acts as an intimately coupled heat bath whose heat capacity vastly exceeds the colloidal particles'. Consequently, the dispersion's thermodynamic temperature is all but guaranteed to be the fluid's, which is readily monitored with standard techniques. It is natural, therefore, to compare estimates of the configurational temperature based on microscopic dynamical measurements with this bulk thermodynamic temperature.

Our samples consist of negatively charged silica spheres $\sigma = 1.58~\ensuremath{\unit{\mu m}}\xspace$ in diameter (Duke Scientific Lot. 24169) dispersed in water and confined within a slit pore of height $H$ formed between a glass microscope slide and a cover slip. The glass surfaces also develop large negative charge densities in contact with water (9), which repel the spheres and prevent them from sticking under the influence of van der Waals attraction. Silica spheres are roughly twice as dense as water and sediment to a height of roughly 300 nm above the lower wall in a matter of seconds (10). The low-concentration samples used in this study thus form a dilute monolayer once they reach equilibrium. Reservoirs of mixed-bed ion exchange resin help to maintain a total ionic strength around $c = 5 \times 10^{-6}~\unit{M}$ in the $1 \times 4~\unit{cm^2}$ visible sample area.

The hermetically sealed sample is allowed to equilibrate at ambient temperatures on the stage of a microscope, with the bulk temperature being gauged with a standard mercury thermometer to an accuracy of about 0.5$^\circ$C. The particles' motions are imaged with a CCD (charge-coupled device) camera and video taped at 30 frames/sec before being digitized. Standard methods of digital video analysis (11) identify the particles in each video frame and report their locations in the plane with a resolution of 30 nm. The resulting distribution,

$$\rho(\ensuremath{{\mathbf r}}\xspace , t) = \sum_{j=1}^{N(t)} \delta(\ensuremath{{\mathbf r}}\xspace - \vecr_j(t)),$$ (24)

of $N(t)$ particles in the field of view at time $t$ provides detailed information regarding the particles' dynamics under the combined influences of random thermal forces and their mutual interactions. Distilling this information into an easily interpreted form requires further analysis.

One of the most commonly used tools for analyzing colloidal microscopy data is the radial distribution function, $g(r)$, which is computed as

$$g(r) = \frac{1}{n^2} \, \left< \rho(\ensuremath{{\mathbf r}}\xspa... ...ath{{\mathbf r}}\xspace , t) \rho(\ensuremath{{\mathbf r}}\xspace , t) \right>,$$ (25)

where $n = \left< \rho \right> = N/A$ is the areal density of particles in a field of view containing $N = \left< N(t) \right>$ particles. Angle brackets in Eq. (25) denote averages over both angles and time. A typical example appears in the inset to Fig. 1.