Charged Plate in Asymmetric Electrolytes: One-loop Renormalization of Surface Charge Density and Debye Length due to Ionic Correlations

Abstract

The self-consistent field theory (SCFT) is used to study the mean potential near a charged plate inside a $m:-n$ electrolyte. A perturbation series is developed in terms of $g = 4 \pi b/\ell_{\rm {\scriptscriptstyle DB}}$, where $b, \ell_{\rm{\scriptscriptstyle DB}}$ are Bjerrum length and {\em bare} Debye length respectively. To the zeroth order, we obtain nonlinear Poisson-Boltzmann theory. For asymmetric electrolytes ($m \neq n$), the first order (one-loop) correction to mean potential contains a {\em secular term}, which indicates the breakdown of regular perturbation method. Using a renormalizaton group transformation (RG), we remove the secular term and obtain a globally well-behaved one-loop approximation with {\em a renormalized Debye length} and {\em a renormalized surface charge density}. Furthermore, we find that if the counter-ions are multivalent, the surface charge density is renormalized substantially {\em downwards}, and may undergo a change of sign, if the bare surface charge density is sufficiently large.