Coion exclusion properties of cation exchange membranes in 2:1 divalent salt solutions

The selectivity of ion exchange membranes hinges on their capacity to exclude coions. Yet, understanding how they reject coions in multivalent salt solutions remains challenging. This study presents a new molecular theory to understand coion exclusion. It examines counterion and coion condensation, including electrosteric effects. Supported by experimental data [Gokturk et al., Nat. Commun., 13 (2022)], we analyze cation exchange membrane potential in a 2:1 salt solution, revealing insights into coion selectivity and exclusion mechanisms. Notably, our theory deviates from earlier research by accommodating condensed divalent counterions that bind to one or two fixed-charged monomers. Additionally, we account for many-body electrostatic interactions, allowing the former to bind to sorbed coions. Our findings highlight the importance of coion condensation, especially in low-concentration salt conditions where coions are mainly condensed. This effect becomes less significant as the external solution concentration rises. These findings clarify the suitability of the ideal Donnan model in high-concentration salts, but its accuracy decreases in low-concentration situations. To summarize, our study emphasizes the need to account for counterion our study emphasizes the need to account for counterion and coion condensation to understand membrane selectivity in low-concentration multivalent salt solutions.