Self-consistent theory to describe charge inversion and like-charge attraction in multivalent electrolytes*

The standard mean-field Poisson-Boltzmann (PB) theory for electrical double layers does not account for three crucial factors: inhomogeneous ion correlations, spatially varying dielectric permittivity, and finite-size effects of ions and solvent molecules. This does not allow PB to even qualitatively explain the phenomena of charge inversion and like-charge attraction in multivalent electrolytes. Here we present the first rigorous and self-consistent theory to explain them. We capture the surface charge-induced continuous transition from a normal double layer to an overcharged double layer. The strength of charge inversion is decided by ion correlation and excluded volume effects with only a minor contribution from long-range image forces. An increase in the counterion valency and surface charge leads to enhanced charge inversion and even ionic layering and oscillations in ion density profiles near the surface. In quantitative agreement with experiments, the inverted zeta potential behaves non-monotonically with respect to bulk multivalent salt concentration. For two approaching like-charged plates, an increase in surface charge leads to a greater accumulation of multivalent counterions in the double layer and hence a stronger ion correlation effect. The gain in ion correlations overcomes the entropic repulsion and an attractive force is obtained. In agreement with simulations, the strength of the attractive force is again found to depend non-monotonically on bulk salt concentration, highlighting the origins of reentrant condensation in charged colloids. The addition of monovalent salt to a multivalent salt solution cancels charge inversion and reduces the strength of like-charge attraction.