Weak long-ranged Casimir attraction in colloidal crystals

– We investigate the influence of geometric confinement on the free energy of an idealized model for charge-stabilized colloidal suspensions. The mean-field Poisson-Boltzmann formulation for this system predicts pure repulsion among macroionic colloidal spheres. Fluctuations in the simple ions’ distribution provide a mechanism for the macroions to attract each other at large separations. Although this Casimir interaction is long-ranged, it is too weak to influence colloidal crystals’ properties. Experimental evidence collected over 20 years [1] suggests that similarly charged colloidal spheres dispersed in water need not simply repel each other. Under some circumstances they instead experience an unexpected long-ranged attraction. For example, like-charge attractions are implicated in the cohesion of metastable superheated colloidal crystals [2, 3] even though isolated pairs of the constituent spheres are observed to repel each other [4, 5]. Comparable attractions have been measured for pairs of spheres confined by two [5, 6] charged planar walls. Recent calculations [7, 8] reveal that such confinement-induced attractions cannot be accounted for by local density theory nor by electrohydrodynamic coupling [9, 10]. Such anomalous effects in charge-stabilized colloid therefore challenge our general understanding of interactions and dynamics in macroionic systems. This letter addresses fluctuations’ contribution to the free energy of highly charged colloidal spheres surrounded by a neutralizing cloud of small singly charged counterions. Highly symmetric monopolar fluctuations in the counterion distribution increase the system’s free energy. We demonstrate that their suppression by boundary conditions at the spheres’ surfaces introduces a long-range attraction into the crystal’s free energy analogous to the Casimir force in quantum electrodynamics, but that it is too weak to account for anomalous behavior in charge-stabilized suspensions. Our treatment is based on the Wigner-Seitz cell model introduced by Wennerström, Jönsson and Linse [11] which has been studied extensively [12] both theoretically and through Monte Carlo simulation. It consists of a single spherical macroion of radius a carrying a uniformly distributed surface charge −Ze and surrounded by a thermal cloud of Z point-like counterions at temperature T , each carrying a single charge e. The macroion and counterions