Understanding long-range opposite charge repulsion in multivalent salt solutions

The electrostatic correlations between ions profoundly influence the structure and forces within electrical double layers. Here, we apply the modified Gaussian renormalized fluctuation theory to investigate the counter-intuitive phenomenon of repulsion between two oppositely charged surfaces and discuss its relationship with overcharging. By accurately accounting for the effect of spatially varying ion-ion correlations, we capture these repulsive forces for divalent, trivalent, as well as tetravalent ions, in quantitative agreement with reported simulation results. We show that the opposite-charge repulsion is long-ranged with an effective length scale of a few nanometers. The strength of opposite-charge repulsion increases monotonically with the multivalent salt concentration, in stark contrast with the non-monotonic salt concentration dependence of other ion correlation-driven phenomena, such as overcharging and like-charge attraction. We also elucidate that the origin of the opposite-charge repulsion is the large number of ions attracted to the double layer as a result of ion-ion correlations, leading to higher osmotic pressure and stronger screening of the electrostatic attraction, which results in an overall repulsive force between two oppositely charged surfaces. Furthermore, we demonstrate that there is no causal relationship between opposite-charge repulsion and the overcharging of the surface. Opposite-charge repulsion is accompanied by overcharging at large separation distances but can also occur in normal double layers without overcharging at intermediate separation distances.