Interpretation

It is possible that the gold surfaces' conductivity alone could suppress like-charge attractions, for instance through a mechanism based on image charges. This would not explain, however, why glass walls induce attractions in the first place. Non-mean-field treatments of macroionic interactions (43,45,47,46,44), suggest that charged surfaces can induce oscillatory correlations in the distribution of simple ions surrounding charged colloidal spheres. Such local departures from electroneutrality could inject extra counterions between pairs of spheres, thereby inducing an effective attraction.

Modeling the predicted diffuse space charge density as a discrete point charge $q$ , midway between the spheres' centers, yields the phenomenological pair potential (26)

$$\beta u(r) = Z^\ast \, \lambda_B \, \left( Z^\ast \, \frac{e^{-\kappa r}}{r} - 4 q \, \frac{e^{- \kappa r/2}}{r} \right).$$ (11)

Equation (11) agrees well with the data in Fig. 1(a) for $Z = 6500 \pm 1000$ , $\kappa^{-1} = 60 \pm 10~\unit{nm}$ and $q = 10 \pm 5$ . It also agrees with the purely repulsive potential in Fig. 1(b) for $Z = 7000 \pm 1000$ , $\kappa^{-1} = 198 \pm 10~\unit{nm}$ and $q = 7 \pm 3$ . In this interpretation of the data, the wall-induced attraction is indeed masked by the electrostatic repulsion at low enough salt concentrations.

Whereas Fig. 1(a) suggests that the wall's surface charge injects an equivalent of 10 counterions between the spheres, and Fig. 1(b) is consistent with this interpretation, the data in Figs. 1(c) and (d) both are consistent with $q = 0$ . The other fitting parameters are $Z = 8000 \pm 1000$ and $\kappa^{-1} = 95 \pm 10~\unit{nm}$ for Fig. 1(c), and $Z = 7000 \pm 1000$ and $\kappa^{-1} = 105 \pm 10~\unit{nm}$ for Fig. 1(d). One interpretation of this result is that the weakly charged gold walls induce no attraction because they have no counterions to contribute. Even a single glass wall, by contrast, carries enough charge to induce non-monotonic correlations in the distribution of simple ions and thus to qualitatively alter the dynamics of nearby charged colloidal spheres.

Recent results on colloidal polystyrene spheres confined to the air-water interface suggest another possible explanation for our observations (48). These interfacial colloids also displays strong long-ranged attractive interactions that contrast with predictions (49). This was interpreted as resulting from a nonuniform distribution of charged groups on the particles' surfaces. Fluctuations in neighboring particles' dipole moments then could induce long-ranged attractions. The proposed connection between nonuniform surface charge and the measured attraction has proved controversial (50,51). Furthermore, the charged groups in our silica particles result from dissociation of silanol groups, which are uniformly distributed on the spheres' surfaces. It seems unlikely, therefore, that dipolar interactions due to surface charge inhomogeneities are responsible for the confinement-induced like-charge attractions that we observe.