# Like-charge polymer-membrane complexation mediated by multivalent cations: one-loop-dressed strong coupling theory

## TL;DR

This paper develops a theoretical model explaining how multivalent counterions induce like-charge polymer adsorption onto membranes, highlighting the roles of counterion valency, membrane charge, and salt effects in complexation.

## Contribution

It introduces a one-loop-dressed strong coupling theory combining monovalent salt and multivalent counterions to explain polymer-membrane complexation mechanisms.

## Key findings

- Multivalent counterions induce polymer adsorption via strong condensation at the membrane.
- Higher counterion valency lowers the critical concentration needed for adsorption.
- Adding monovalent salt suppresses counterion condensation and causes polymer desorption.

## Abstract

We probe the electrostatic mechanism driving adsorption of polyelectrolytes onto like-charged membranes upon the addition of tri- and tetravalent counterions to a bathing monovalent salt solution. We develop a one-loop-dressed strong coupling theory that treats the monovalent salt at the electrostatic one-loop level and the multivalent counterions within a strong-coupling approach. It is shown that the adhesive force of the multivalent counterions mediating the like-charge adsorption arises from their strong condensation at the charged membrane. The resulting interfacial counterion excess locally maximizes the screening ability of the electrolyte and minimizes the electrostatic polymer grand potential. This translates into an attractive force that pulls the polymer to the similarly charged membrane. We show that the high counterion valency enables this adsorption transition even at weakly charged membranes. Additionally, strongly charged membranes give rise to salt-induced correlations and intensify the interfacial multivalent counterion condensation, strenghtening the complexation of the polymer with the like-charged membrane, as well as triggering the orientational transition of the molecule prior to its adsorption. Finally, our theory provides two additional key features as evidenced by previous adsorption experiments: first, the critical counterion concentration for polymer adsorption decreases with the rise of the counterion valency, and second, the addition of monovalent salt enhances the screening of the membrane charges and suppresses salt correlations. This weakens the interfacial multivalent counterion condensation and results in the desorption of the polymer from the substrate.