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Schwinger-Dyson equations for composite electrolytes governed by mixed electrostatic couplings strengths

The electrostatic Schwinger-Dyson equations are derived and solved for an electrolyte mixture composed of mono- and multivalent ions confined to a negatively charged nanoslit. The closure of these equations is based on an asymmetric treatment of the ionic species with respect to their electrostatic coupling strength; the weakly coupled monovalent ions are treated within a gaussian approximation while the multivalent counterions of high coupling strength are incorporated with a strong-coupling approach. The resulting self-consistent formalism includes explicitly the interactions of the multivalent counterions with the monovalent salt. In highly charged membranes characterized by a pronounced multivalent counterion adsorption, these interactions take over the salt-membrane charge coupling. As a result, the increment of the negative membrane charge brings further salt anions into the pore and excludes salt cations from the pore into the reservoir. The corresponding like-charge attraction and opposite-charge repulsion effect is amplified by the pore confinement but suppressed by salt addition into the reservoir. The effect is particularly pronounced in high dielectric membranes where the attractive polarization forces lead to a dense multivalent cation layer at the membrane walls. These cation layers act as an effective positive surface charge, resulting in a total monovalent cation exclusion and a strong anion excess even in the case of neutral membrane walls.