Experimental Results: Glass Walls

The a posteriori artifact corrections described in Secs. 2.2 and 2.3 were applied to the two data presented in Fig. 1. Pair potentials were tested for thermodynamic self-consistency using previously reported methods (29,30).

The data in Figs. 1(a) and (b) were obtained for colloidal silica spheres hovering above clean glass walls and demonstrate two of our principal points: Even a single glass wall can induce attractive interactions among like-charged colloidal spheres, and this attraction is not apparent at low ionic strengths.

The discrete points in Fig. 1 reflect estimates for the pair potential obtained with Eqs. (1) through (7), each of which was corrected for imaging artifacts with an individually measured calibration curve, $\Delta (r)$ . The dashed curves are results obtained without correcting for imaging artifacts. Under the conditions in Figs. 1(a) and (b), correcting for imaging artifacts leads to only small quantitative changes in the estimated pair potential, rather than qualitative changes as have been reported elsewhere (31). More specifically, the minimum in the measured pair potential in Fig. 1(a) is not eliminated by correcting for imaging artifacts. A single clean glass wall, therefore, can induce like-charge attractions between nearby pairs of charge-stabilized spheres. All previous reports of like-charge attractions in equilibrium involved particles confined by pairs of parallel glass walls (21,22,29,17,19).

This result should not be taken to contradict previous reports of colloidal interactions near single glass surfaces (40,41,25,18,42,33), all of which described monotonically repulsive interactions. Unlike these previous reports, the data in Fig. 1(a) were obtained in deionized water that had been allowed to equilibrate with air, thereby increasing the concentration of dissolved monovalent ions to roughly $50~\unit{\mu M}$ . Equilibrating the sample against mixed-bed ion exchange resin, as in previous reports, reduces the ionic strength, increases the range of the particles' electrostatic interaction, and yields the monotonically repulsive pair potential plotted in Fig. 1(b).

The purely repulsive pair potential may be compared with the screened-Coulomb form predicted by mean-field theory (1),

$$\beta u(r) = {Z^\ast}^2\, \lambda_B \, \frac{e^{-\kappa r}}{r}.$$ (10)

Here, $Z^\ast = Z \, \exp(-\kappa \sigma/2) / (1 + \kappa \sigma/2)$ is the effective charge of a sphere of bare charge $Z$ in an electrolyte with Debye-Hückel screening length $\kappa^{-1}$ . This is related to the ionic concentration, $n_c$ , through $\kappa^2 = 8 \pi n_c \lambda_B$ , where $\lambda_B = 0.717~\unit{nm}$ is the Bjerrum length in water at room temperature. Fitting Eq. (10) to the data in Fig. 1(b) yields $Z = 6500 \pm 1000$ , which agrees quantitatively with previous reports (24). The fit screening length, $\kappa^{-1} = 180 \pm 10~\unit{nm}$ , corresponds to a concentration $n_c = 2.8~\unit{\mu M}$ of monovalent ions. The result is plotted as a solid curve in Fig. 1(b).

The screening length obtained from Fig. 1(b) also is comparable to values obtained from previous measurements of colloidal interactions near single walls (26,24,18,25). The attraction's dependence on ionic strength suggested by Figs. 1(a) and (b) recalls a similar trend from the earliest report of attractions induced by two walls (22). It seems plausible to suggest, therefore, that the attraction evident from the minimum in Fig. 1(a) either is eliminated by reducing the ionic concentration, or else becomes masked by the stronger and longer-ranged screened-Coulomb repulsion. This also would explain why simulations of colloidal dispersions under salt-free conditions have found no evidence of confinement-induced attractions.