Charge regulation of a surface immersed in an electrolyte solution

In this paper, we investigate theoretically a model of charge regulation of a single charged planar surface immersed in an aqueous electrolyte solution. Assuming that the adsorbed ions are mobile in the charged plane, we formulate a field theory of charge regulation where the numbers of adsorbed ions can be determined consistently by equating the chemical potentials of the adsorbed ions to that of the ions in the bulk. We analyze the mean-field treatment of the model for electrolyte of arbitrary valences, and then beyond, where correlation effects are systematically taken into account in a loop expansion. In particular, we compute exactly various one-loop quantities, including electrostatic potentials, ion distributions, and chemical potentials, not only for symmetric (1, 1) electrolyte but also for asymmetric (2, 1) electrolyte, and make use of these quantities to address charge regulation at the one-loop level. We find that correlation effects give rise to various phase transitions in the adsorption of ions, and present phase diagrams for (1, 1) and (2, 1) electrolytes, whose distinct behaviors suggest that charge regulation, at the one-loop level, is no longer universal but depends crucially on the valency of the ions.