Validity of the Boltzmann equation to describe Donnan equilibrium at the membrane–solution interface

To describe Donnan equilibrium at the membrane–solution interface, the simplest approach uses the classical Boltzmann equation, based on a mean-field description of ions as ideal point charges, in combination with the assumption of fully overlapped electrical double layers in the membrane pores. We test the Boltzmann equation by measurement of the equilibrium counterion and co-ion concentration in densely charged membranes equilibrated with various NaCl solutions (0.01–3 M). To obtain a good fit of data it was found necessary to express the membrane charge and ion concentration per volume of aqueous solution phase in the membrane, and to include a volume-exclusion term in the Boltzmann relation. A discrepancy between theory and experiment data is found at low external NaCl concentrations. Similar deviations from the Donnan model have been reported for over half a century, but do not yet have a convincing explanation. Agreement between experiment data and theory at low external NaCl concentrations is obtained when we model the desorption experiment taking into account the role of H+ and OH− ions in closing the charge balance, and postulating the presence in the membrane of a tiny amount of fixed groups with a charge opposite to overall fixed membrane charge.