Accounting for frame of reference and thermodynamic non-idealities when calculating salt diffusion coefficients in ion exchange membranes
Abstract Accurate evaluation of salt diffusion coefficients from transport rate data in ion exchange membranes requires accounting for frame of reference and non-ideal thermodynamic effects. Due to a lack of models and experimental data quantifying membrane ion activity coefficients, it has been impossible to evaluate the impact of non-ideal thermodynamic effects on observed salt diffusion coefficients. Here, a framework is presented that includes both frame of reference (i.e., convection) and non-ideal thermodynamic effects in calculating salt diffusion coefficients in ion exchange membranes. Effective concentration averaged NaCl diffusion coefficients were determined as a function of upstream NaCl concentration in commercial ion exchange membranes from NaCl permeability and sorption measurements via the solution-diffusion model. Frame of reference effects were evaluated using a version of Fick's law that accounts for convection. The factors necessary to account for non-ideal thermodynamic effects were developed using Manning's counter-ion condensation theory. At low upstream NaCl concentrations, frame of reference and non-ideal thermodynamic effects on diffusion coefficients were negligible. However, at higher upstream NaCl concentrations (e.g., >0.1 M), both effects contribute measurably to NaCl diffusion coefficients. Correcting for frame of reference effects increased apparent NaCl diffusion coefficients. However, correcting for thermodynamic non-idealities of the ions sorbed into the membranes reduced apparent NaCl diffusion coefficients. Fortuitously, for the materials considered in this study, frame of reference and non-ideal thermodynamic effects nearly cancel each other.