Introduction

The electrostatic interaction between charge-stabilized colloidal particles is mediated and modified by simple ions dissolved in the supporting electrolyte. For like-charged particles, the resulting effective interaction, described by the Poisson-Boltzmann mean field theory (1), is predicted to be always purely repulsive (2,4,5,3) independent of the strength of the electrostatic coupling, the valency of the suspended ions or the state of confinement of the suspension.

This reasonable prediction is known to fail under some conditions. For example, both simulations and non-mean-field theoretical studies predict the possibility of an attraction in bulk suspensions of like-charged colloidal particles if multivalent counterions are present (see (6,7) and references therein). This attraction is mediated by strong ion-ion correlations induced by coupling to the charged colloidal particles. Experimental observations have confirmed similar predictions for highly charged parallel plates (8) and cylinders (9).

These departures from mean-field behavior appear even in the so-called Primitive Model (PM), which treats both the colloidal particles and the dissolved ions as charged hard spheres and describes the suspending fluid as a uniform dielectric continuum. Similar effects are also predicted in more general models, when image charges (10), salt-specific dispersion forces (11), or the sizes of simple ions (12) become important.

In all such cases, mean-field theory fails because the relevant electrostatic interactions exceed the thermal energy scale over length scales of interest. Colloidal spheres in symmetric monovalent electrolytes, by contrast, are expected to satisfy the conditions of the mean-field approximation. Indeed, PM simulations of charged colloidal spheres in symmetric monovalent electrolytes are found to agree well with mean-field predictions (13,14,15), a class of systems for which Poisson-Boltzmann theory is expected to be accurate. Experimental results, however, have been more mixed. Direct measurements of colloidal pair interactions (17,18,16) in bulk dispersions qualitatively agree with mean-field predictions. When applied to dispersions confined to thin layers by charged glass surfaces, however, these same methods have repeatedly revealed long-ranged attractions (22,20,21,19) that are too strong to be accounted for by van der Waals interactions (22,23), and are inconsistent with Poisson-Boltzmann theory (4,2,3). Measurements on colloidal spheres confined by just a single glass wall, by contrast, have not revealed anomalous attractions (26,24,18,25). These measurements were performed at much lower ionic strengths than those in thin samples, however. One explanation for this seeming inconsistency is that the appearance of confinement-induced like-charge attractions might depend strongly on the ionic concentration. Perhaps, then, even a single bounding surface could induce anomalous attractions if the ionic strength were in the appropriate range.

Figure 1: (Color online) Measured pair potentials $u(r)$ for $\sigma = 1.58~\ensuremath{\unit{\mu m}}\xspace$ diameter silica spheres near a single wall ( $H \simeq 200~\ensuremath{\unit{\mu m}}\xspace$ ). Dashed curves are uncorrected results, $\tilde u(r)$ . Discrete points have been corrected for imaging artifacts. Inset: schematic of the geometry. (a) Glass surface with $\kappa^{-1} = 60 \pm 10~\unit{nm}$ (circles), induces a long-ranged attraction. (b) Increasing the screening length to $\kappa^{-1} \approx 180~\unit{nm}$ (squares) yields monotonic repulsion. The solid curve is a fit to Eq. 10 with $Z = 6500 \pm 1000$ . (c) Coating the confining surface with gold eliminates attractions at $\kappa^{-1} \approx 100~\unit{nm}$ : Single surface (diamonds). (d) Parallel gold-coated surfaces separated by $H = 15~\ensuremath{\unit{\mu m}}\xspace$ (triangles).

Another possible explanation is that the measurements reporting anomalous attractions among like-charged colloidal spheres were in error. Several experimental artifacts have been proposed that might mimic the reported observations. These include a geometric bias introduced by projecting a colloidal sphere's three-dimensional position onto the two-dimensional focal plane (27), a statistical bias resulting from the experiments' limited sample size (24), nonequilibrium hydrodynamic interactions induced by small in-plane drifts (28), and uncorrected many-body contributions to the apparent pair potential (25). Thermodynamic self-consistency checks demonstrate that none of these can account for the observed like-charge attractions in more recent experiments (21,29,30).

Recently, a subtle imaging artifact in widely used particle tracking algorithms (17) has been demonstrated to mimic long-ranged attractions in systems with purely repulsive pair interactions (31,32). Its influence on the long ranged repulsive pair potentials observed in low ionic strength bulk and surface experiments should be minimal, but the effect could play a major role at the smaller separations relevant for observations of like-charge attractions (32). Could the appearance of confinement-induced like-charge be due entirely to experimental artifacts?

The experiments and simulations reported in this Article confirm the appearance of confinement-induced like-charge attractions, even when all known experimental artifacts are taken into account. They furthermore contribute three new insights into the effect's phenomenology: (1) Confinement by a single charged glass surface can induce anomalous attractions among charge-stabilized colloidal silica spheres. (2) Confinement-induced attractions may be masked by electrostatic repulsion at very low ionic strengths. (3) Coating the confining surfaces with conducting gold layers eliminates the attraction, even under conditions of ionic strength that foster attractions in glass-bounded samples. These results are summarized in the data plotted in Fig. 1.

Section 2 describes the experimental and analytical methods used to obtain these results. Of particular importance are the methods introduced in Secs. 2.2 and 2.3 to measure and correct for imaging artifacts. The additional insights these experimental results provide into the nature of like-charge colloidal attractions suggest possible mechanisms for the effect. On this basis, we present an idealized model for confinement-induced like-charge attractions in Sec. 3. This interpretation of our experimental results relies the accuracy and efficacy of our analytical techniques, which we demonstrate through Monte Carlo simulations in Sec. 4. Section 5 summarizes our results and conclusions while placing them in the context of recent advances in the theory of macroionic interactions.