Explaining Uphill Ion Transport in Multicomponent Solutions Across Membranes in Reverse Electrodialysis

Diffusion against a concentration gradient, also called uphill transport, has been reported for magnesium and sulfate ions in solutions of NaCl and MgSO4 in reverse electrodialysis (RED). Here we derive transport equations for such systems and explain the observed uphill transport using non-equilibrium thermodynamics (NET). A set of Nernst-Planck equations was reformulated into a set of flux equations with neutral salt driving forces. By applying the condition of entropy production invariance to the sets of variables, we show how an ideal ion selectivity model provides Onsager coupling coefficients for the ion transport. These coupling coefficients can predict and explain the observed uphill transport of magnesium and sulfate. A linear fitting scheme is developed to fit the transport coefficients to experimental data for the ideal ion selectivity model and for a more general model. The results show that even the simplest ideal ion selectivity model captures the uphill transport, and the phenomenon occurs due to diffusional ion exchange, as in Donnan dialysis. Under open-circuit conditions, deviations are large, and the simplest model is no longer sufficient; co-ion leakage corrections are essential. The results provide a basis for future modelling of RED processes; in particular, one can determine and optimise the process efficiency, as the model gives direct access to the process's local entropy production. Osmosis is identified as a potential source of error.