The experiments described below were carried out on suspensions of polystyrene sulfate spheres of diameter dispersed in deionized water (Duke Scientific catalog number 5065A). These spheres are synthesized with a large number of ionizable sulfate salt groups chemically bonded to their surfaces. These groups dissociate in water, each leaving a single negative charge bound to its sphere's surface and a compensating positively charged counterion in solution. One sulfate group subtends roughly 10 nm so that a micrometer-scale sphere has a titratable charge on the order of electron equivalents. Not all surface groups dissociate at once, though, so the effective charge number is typically much smaller. While these specially prepared spheres have consistent and highly uniform surface charges, many naturally occurring and industrially prepared colloids also acquire static surface charges . Results discussed in this article should pertain to them as well.
The interaction between an isolated pair of similarly charged spheres is purely repulsive and has been shown both theoretically [14, 15] and experimentally [16, 17, 18] to adopt the screened-Coulomb form
where r is the spheres' center-to-center separation. This expression for the interaction energy was originally derived by Derjaguin, Landau, Verwey, and Overbeek (DLVO) [14, 15] and accounts approximately for the counterions' exclusion from the spheres' interiors. The range of the interactions is limited by the concentration n of simple ions in the water and is described by the Debye-Hückel screening length , where is the characteristic separation between unit charges at temperature T in a fluid of dielectric constant .
The complete DLVO theory includes a term accounting for van der Waals attractions. This term is neglected in the following discussion because it contributes less than to the interaction potential for polystyrene microspheres immersed in water and separated by more than 100 nm .
The suspension at an initial volume fraction around is contained in the space between two parallel glass walls separated by . The edges of this sample volume are hermetically against contamination by airborne CO and makes diffusive contact with reservoirs of ion exchange resin. We estimate the steady state concentration of residual stray ions in this system to be less than M. Such low ionic strengths provide for long-ranged electrostatic interactions with Debye-Hückel screening lengths extending to nm.
If we assume that the interactions among a collection of spheres can be built up by superposing pair-wise repulsions of the form described by Eq. (1), then the phase diagram for the ensemble of spheres is well understood [7, 8, 9, 10, 11]. In particular, a fluid interacting through screened-Coulomb repulsions can be frozen to a crystal of face-centered cubic (fcc) or body-centered cubic (bcc) symmetry by increasing the volume fraction of spheres in the suspension while leaving other parameters fixed. Transformations qualitatively consistent with this picture have been observed experimentally [12, 13] in charge-stabilized colloidal suspensions.
Conventional crystals are rigid enough to support themselves against gravity and so determine their own density. Colloidal crystals exist only within containers. Were they not confined, we might expect mutually repelling spheres to drift apart until the forces causing them to order were overwhelmed by randomizing thermal agitation. At this point, the crystal would melt. We have found, however, that some colloidal crystals made of like-charged spheres maintain their integrity even without confining walls. The structure and dynamics of these metastable crystals provide strong evidence for the influence of long-ranged attractions not accounted for in the DLVO theory.