Divalent ion partitioning through dense ion exchange membranes

Understanding the mechanism behind the permeation of multivalent ions through ion exchange membranes remains an ongoing challenge. Here we address this knowledge gap by presenting a molecular theory to elucidate the influence of counterion condensation and electrosteric effects on the (multivalent) divalent counterion permeation through such membranes. Building upon published experimental data (Galizia et al., 2017), we analyze the energetic tendencies of divalent counterions to associate with the fixed charge groups and their entropic preferences to disassociate to gain translational freedom, allowing us to quantitatively assess the partitioning of divalent ions at two distinct levels. (i) The global partitioning between the external 2:1 salt solution and the membrane, and (ii) the local partitioning within the interstitial solution and the crosslinked polymeric chains within the membrane. Notably, our observations reveal that the global partitioning of divalent counterions does not differ when the membranes operate separately in magnesium chloride (MgCl2) and calcium chloride (CaCl2) solutions. However, at the local partitioning level, we identify a higher preference for calcium counterions to associate with the fixed charge groups than magnesium ions. This preference arises from calcium ions' more favorable charge density (physical size), facilitating their association/pairing with the fixed charge groups. The key takeaway message from this work is that when dealing with multivalent ions, we must downlevel our investigation, looking at how the counterions relax between the interstitial solution and the crosslinked polymeric chains and, in return, comprehending how it ultimately controls the global transport property of these membranes (such as their Donnan potential and water uptake).