Combining Manning's theory and the ionic conductivity experimental approach to characterize selectivity of cation exchange membranes

Abstract The transport selectivity of different cations through cation exchange membranes (CEMs) could be estimated with the partitioning selectivity factor ( K j i ) and the cation mobility ratio in the membrane ( u m i / u m j ), which in turn can be related to corresponding membrane conductivity and dimensional swelling ratio data [Journal of Membrane Science, 2020, 597, 117645]. This method has been validated in two hydrocarbon-based CEMs, and the obtained K+/Na+ selectivity equals to the one obtained with conventional electrodialysis (ED) method. However, the K+/Na+ selectivity of perfluorosulfonic acid (PFSA) membranes, and the bi-/monovalent cation (Mg2+/Na+) selectivity of all three types of CEMs estimated with this ionic conductivity experimental approach deviate noticeably from corresponding values obtained with ED. In this work, it is proved that this deviation is mostly due to the simplification of cation activity coefficients in the membrane. Here, the cation activity coefficients in three types of CEMs are calculated according to Manning's counter-ion condensation model. In this model, the Manning parameter ( ξ ) characterizing the dimensionless linear charge density is determined by the average distance between two adjacent fixed sulfonate groups ( b ) and the permittivity of hydrated membranes ( e ). In hydrocarbon-based CEMs, the average distance between fixed sulfonate groups can be estimated by assuming homogeneous distribution of the fixed groups, while in PFSA membranes three representative structure models are employed to estimate this average distance. After accounting for the cation activity coefficients in the membrane, the cation transport selectivity obtained with the ionic conductivity experimental approach agrees well with the selectivity obtained with the ED method. This work shows the importance of cation activity coefficients in the membrane phase in interpreting the membrane transport properties, and complements the proposed conductivity approach to characterize the counter-ion selectivity of ion exchange membranes.