Model for Counterion-Membrane-Fixed Ion Pairing and Donnan Equilibrium in Charged Membranes

We have studied theoretically the effects of counterion-membrane-fixed ion pairing on the Donnan equilibrium in charged membranes with low dielectric constant regions. First, the formal relationship between the ionic activity coefficients in the membrane and the membrane effective charge concentration is established by comparing the results obtained from a model based on the ideal Donnan equilibrium extended to account for the ion pairing effect with the results obtained from a thermodynamic formalism including ionic activities instead of concentrations. Second, we calculate the membrane effective charge concentration and the ionic activity coefficients by using a simple model of ion pairing based on the Fuoss approach for contact ion pairs in electrolyte solutions with a correction term for the entropy change of the counterion undergoing ion pairing. Finally, we study the activity coefficients effects on the co-ion exclusion in the charged membrane as a function of the external salt concentration. Though it seems established that membrane inhomogeneity is the mainly responsible for the deviations from the ideal Donnan equilibrium observed experimentally in charged membranes, the theoretical approach presented here can be useful for those cases where conclusive experimental evidence for ion pairing exists.