Extended Donnan model for ion partitioning in charged nanopores: Application to ion-exchange membranes

Membranes consist of pores and the walls of these pores are often charged. In contact with an aqueous solution, the distribution of ions between external and pore solution at chemical equilibrium is governed by Donnan theory, or Donnan equation. Including a partitioning coefficient that does not depend on salt concentration results in an extended Donnan equation 'of the first kind'. Recently, an electrostatic model was proposed for ion partitioning in ion exchange membranes based on the arrangement of ions around strands of polymer charge. The model also includes ion activity coefficients in solution, which leads to an extended Donnan equation 'of the second kind' by having extra factors depending on ion concentrations in the pores and salt concentration in external solution. In the present work, we set up another Donnan model of the second kind by evaluating the Coulombic interactions of ions in a cylindrical pore, including the interaction of ions with the charged pore walls and between the ions. We derive an analytical expression for the chemical potential of ions in such a pore. This expression depends on coion concentration, pore size, and other geometrical factors, but there is no additional dependence on counterion concentration and charge density. The model predicts the Coulombic contribution to the chemical potential in the pore to be small, much smaller than that predicted by the electrostatic model from literature. Instead, we predict that up to around 1 M salt concentration, activity effects of ions in external solution are more important. We demonstrate the good agreement between the extended Donnan model and experimental ion partitioning data of both single and mixed salts in ion exchange membranes.