Quantifying the coupled monovalent and divalent ions sorption in dense ion-exchange membranes

As a universal phenomenon in solution-membrane systems, the coupled competitive monovalent and divalent ions sorption significantly impacts the transport process in ion-exchange membranes. Often, the quantitative and qualitative relationships between them are unclear. Here, we present a lattice-based molecular model to capture the interplay effect of ion-pairing and electrostatic interaction on the competitive partitioning process. In particular, we utilize statistical mechanics to formulate how the monovalent and divalent counterions competitively pair with the fixed charge group in the membranes. To this end, the model can quantify (a) the counterions sorption and (b) the coion repulsion within the membranes, which, collectively, allow us to quantify the ion partitioning and the pairing site saturation. To showcase the capability of our model, we compare our results with published experiment data, where we achieve a relatively good agreement between them. We demonstrate that the competitive ion partitioning depends highly upon (i) the entropy change generated by the counterions pairing with the fixed charge group, (ii) the binding energy, and (iii) the osmotic pressure, and (iv) the electrostatic interaction. These different factors can explain the favorable sorption of divalent counterions in the cation-exchange membranes but monovalent counterions in the anion ones, as previously reported. Altogether, our theoretical approach provides an insight into the fundamental understanding of coupled competitive monovalent and divalent ions sorption within the dense ion-exchange membranes, further enhancing the knowledge of the competitive ion partitioning process.