A simple model for the response of an anion-exchange membrane to variation in concentration and pH of bathing solution

Swelling is an important property of charged gels and membranes. The size of pores and, therefore, membrane properties such as conductivity, diffusion and hydraulic permeability, permselectivity depend on water content and degree of swelling. In this paper we propose a simple model for equilibrium swelling of an ion-exchange membrane, allowing calculation of water content and membrane thickness as functions of the concentration and pH of the bathing solution. The model parameters include the equivalent volume of dry polyelectrolyte gel, the volume fraction of macropores and others. Three types of ion-exchange functional groups, namely, the secondary, tertiary and quaternary amino groups in an anion-exchange membrane are taken into account. The osmotic pressure exerted by micro- and mesopores, appearing in the gel when swelling, is expressed using the Gregor equation, which employs the mole fractions of free and bound water. The equilibria between protonated and deprotonated amino groups are assumed as well as the Donnan and ion-exchange equilibria between the membrane and bathing solution. The results of calculations are compared with experimental data on the membrane thickness and effective exchange capacity obtained for two heterogeneous anion-exchange membranes MA-40 and MA-41 (Shchekinoazot) differed by the composition of functional groups. The procedure of determining membrane structural and thermodynamic parameters is described. A good quantitative agreement between the theory and experiment is found for both membranes using the same set of ion hydration numbers and chemical equilibrium constants for secondary and tertiary amino groups. In particular, it is shown that with increasing pH of the bathing solution, the membrane thickness (and, hence, water content) pass through a local maximum and a local minimum. The simplicity of the model, which does not reduce adequacy, would enable it to be included later in other models describing the transport of ions and water to account for the swelling and change in the membrane structure with a change in pH and concentration of the bathing solution.