Selective divalent/monovalent ion partitioning in cation exchange membranes

The selective partitioning of divalent and monovalent counterions through cation exchange membranes immersed in 2:1 and 1:1 binary salt solutions remains unclear, especially at the molecular level. We present a theoretical framework that quantifies these membranes’ sorption of divalent and monovalent ions, incorporating condensed and free variants to reconcile deviations from the ideal Donnan picture observed in published experimental data. This framework explores the membranes’ global and local selective ion sorption characteristics by assessing the effects of competitive ion condensations, where these counterions compete to bind to the fixed charge groups. First, we calculate the thermodynamic property—the global partition ratio—of the membranes immersed in these binary salt solutions, quantifying the partitioning between the binary salt solutions and the membranes. Our theoretical results successfully capture the published experimental partitioning data for membranes immersed in equimolar and non-equimolar 2:1 and 1:1 binary salt solutions. Furthermore, the Donnan potential of the membranes reciprocally changes with the total concentration of 2:1 and 1:1 external salt solutions. However, for the non-equimolar systems at a constant low total concentration, varying the binary solution concentration ratio of 2:1 to 1:1 salts maintains a strong electric potential intensity due to high ionic imbalance. Second, we quantify the local partitioning of the sorbed counterions that are condensed on the crosslinked polymeric chains and those that are mobile in the interstitial solution of the membrane, a variable that is difficult to access experimentally. Accordingly, we can fractionate the sorbed ions into condensed and free states, which mutually govern the membrane’s kinetic and thermodynamic (global and local partitioning) properties.