Fluctuation and correlation effects in electrostatics of highly-charged surfaces

This work explores the statistical mechanics of counterions associated with their oppositely charged surfaces, which is relevant to many systems in soft condensed matter physics: charged colloids, membranes, and polyelectrolytes immersed in solutions containing mobile neutralizing counterions. The mean-field treatment for these systems is the Poisson-Boltzmann (PB) theory, or its linearized version, the Debye-Hückel theory. Among other results, the mean-field theory predicts repulsion between two like-charged plates. However, recent experimental and numerical works suggest that two like-charged objects may attract! After reviewing the main features of PB theory, we describe an approach which takes charge fluctuations explicitly into account to improve the mean-field picture and demonstrate that charge-fluctuations can induce a long-ranged attraction, similar to the van der Waals interaction. We also analyze the effects of charge fluctuations on the bending properties of a charged membrane. Furthermore, we argue that fluctuations may induce a novel condensation phenomenon in an overall neutral system, consisting a single charged plate and its oppositely charged counterions. Finally, we study the interactions between two 2-dimensional Wigner crystals, which may be the ground state of the counterions condensed onto charged surfaces. In particular, we show that at low temperatures, quantum zero-point fluctuations of the plasmon modes (charge-fluctuations) of two mutually coupled 2D Wigner crystals give rise to a novel long-range attractive force.