Reformulating the permselectivity-conductivity tradeoff relation in ion-exchange membranes

Polymer membranes used in separation applications exhibit a tradeoff between permeability and selectivity. That is, membranes that are highly permeable tend to have low selectivity and vice versa. For ion-exchange membranes used in applications such as electrodialysis and reverse electrodialysis, this tradeoff is expressed in terms of membrane permselectivity (i.e., ability to selectively permeate counter-ions over co-ions) and ionic conductivity (i.e., ability to transport ions in the presence of an electric field). The use of membrane permselectivity and ionic conductivity to illustrate a tradeoff between counter-ion throughput and counter-ion/co-ion selectivity in ion-exchange membranes complicates the analysis since permselectivity depends on the properties of the external solution and ionic conductivity depends on the transport of all mobile ions within a membrane. Furthermore, the use of these parameters restricts the analysis to ion-exchange membranes used in applications in which counter-ion/co-ion selectivity is required. In this study, the permselectivity-conductivity tradeoff relation for ion-exchange membranes is reformulated in terms of ion concentrations and diffusion coefficients in the membrane. The reformulated framework enables a direct comparison between counter-ion throughput and counter-ion/co-ion selectivity and is general. The generalizability of the reformulated tradeoff relation is demonstrated for cation-exchange membranes used in vanadium redox flow batteries.