Electrostatistics of counter-ions at and between planar charged walls: From Poisson-Boltzmann to the strong-coupling theory

The Poisson-Boltzmann (PB) approach gives asymptotically exact counter-ion density profiles around macroscopic charged objects and forces between macr oscopic charged objects in the weak-coupling limit of low counter-ion valen cy, low surface-charge density, and high temperature. In this paper we deri ve, using field-theoretic methods, a theory which becomes exact in the oppo site limit of strong coupling (SC). Formally, it corresponds to a standard virial expansion. Long-range divergences render the virial expansion intrac table for homogeneous bulk systems, giving rise to non-analyticities in the low-density expansion of the free-energy density of electrolyte solutions. We demonstrate that for the case of inhomogeneous density distribution fun ctions at macroscopic charged bodies these divergences are renormalizable b y a systematic expansion in powers of the fugacity. For a single planar cha rged wall, we obtain the counter-ion density profile in the SC limit, which decays exponentially, in contrast to the PB result, which predicts algebra ic decay, and in agreement with previously published numerical results. Sim ilarly and highly charged plates in the presence of multivalent counter-ion s attract each other in the SC limit and form electrostatically bound state s, in contrast to the PB limit, where the interaction is always repulsive. By considering next-leading corrections to both the PB and SC theories, we estimate the range of validity for both theories.