Analyzing ion uptake in ion-exchange membranes using ion association model

Predicting ion uptake and selectivity in ion-exchange membranes is desired for many applications, yet a suitable physical description defining the most appropriate ion-specific parameters is still challenging. Here, we systematically develop an ion-association-based approach to modeling ion uptake in ion-exchange membranes from solutions of symmetric and non-symmetric salts. The model treats association in an ion-specific manner, self-consistently accounting for equilibria between free ions in solution and within the membrane phase (salt injection) and between free and associated species within the membrane (association equilibria), subjects to overall membrane electroneutrality. The resulting models, including different possible association equilibria, were employed to fit the reported data for Nafion 117 and CR61 cation-exchange membranes in equilibrium with NaCl, MgCl₂, CaCl₂, and Na₂SO₄ single-salt solutions. The results are compared with the previously reported fits to the Manning condensation model, which shows that both models produce similarly good fits for NaCl, MgCl₂, and CaCl₂ solutions in the 0.01 to 1 M range. However, the greater flexibility and specificity of the association model allow addressing deviations observed for Na₂SO₄ solutions and for CaCl₂ above 1 M as free-ion paring and possible formation of charged NaSO₄^- and CaCl⁺ pairs, respectively. The results demonstrate the present model may be a sound non-mean-field alternative to the Manning condensation model, capable of addressing ion-specificity and multiple modes of association.