The same spheres which repel each other in isolation and form metastable crystals when compressed also can develop a strong and long-ranged pair attraction when confined by one or two glass walls. Such attractions were first observed by Kepler and Fraden  who measured by tracking particles in dilute monolayers sandwiched between parallel glass plates, and supplemented Eq. (10) with molecular dynamics simulations to obtain . Carbajal-Tinoco, Castro-Román and Arauz-Lara reproduced this result, further applying liquid structure theory to demonstrate that many-body correlations could not account for the apparent pair attraction . This interpretation was supported by Rao and Rajagopalan's analysis of the experimental data with a predictor-corrector algorithm .
Optical tweezer measurements  such as the middle two examples in Fig. 2.4 confirm the walls' role by demonstrating their influence on two otherwise isolated spheres. Spheres between widely separated walls (Fig. 2.4 top) move freely in three dimensions; their pair potential is described accurately by the DLVO theory, in this case with and .
The sample container's glass walls also carry a negative surface charge due to the dissociation of silanol groups. As they move together, they tend to confine the negatively charged spheres to their midplane. This confinement appears to induce a long-ranged attraction which, together with the screened electrostatic repulsion, yields a minimum in the pair potential roughly 0.5 deep at a separation of about 3 diameters.
Neu  and Sader and Chan  (NSC) recently proved that these observations cannot be explained by mean-field theory. The NSC proof holds for constant potential boundary conditions, at least some variants of constant charge boundary conditions, and for confining pores of arbitrary cross-section. Trizac and Raimbault extended the proof to include steric effects based on the the small ions' finite size . These proofs appear to contradict  recent perturbative many-body calculations  which find wall-induced attractions for constant charge boundary conditions.
As a further complication, still closer confinement appears to induce a long-ranged repulsion in the pair potential, as in the bottom trace of Fig. 2.4. Stillinger  and Hurd  analyzed the related problem of charged colloid interacting near an air-water interface. The resulting unscreened dipole repulsion appears to be too weak  to explain the measurement, however.