Spatiotemporal pH dynamics in concentration polarization near ion-selective membranes

We present a detailed analysis of the transient pH dynamics for a weak, buffered electrolyte subject to voltage-driven transport through an ion-selective membrane. We show that pH fronts emanate from the concentration polarization zone next to the membrane and that these propagating fronts change the pH in the system several units from its equilibrium value. The analysis is based on a 1D model using the unsteady Poisson-Nernst-Planck equations with nonequilibrium chemistry and without assumptions of electroneutrality or asymptotically thin electric double layers. Nonequilibrium chemical effects, especially for water splitting, are shown to be important for the dynamical and spatiotemporal evolution of the pH fronts. Nonetheless, the model also shows that at steady state the assumption of chemical equilibrium can still lead to good approximations of the global pH distribution. Moreover, our model shows that the transport of the hydronium ion in the extended space charge region is governed by a balance between electromigration and water self-ionization. On the basis of this observation, we present a simple model showing that the net flux of the hydronium ion is proportional to the length of the extended space charge region and the water self-ionization rate. To demonstrate these effects in practice, we have adopted the experiment of Mai et al. (Mai, J.; Miller, H.; Hatch, A. V. Spatiotemporal Mapping of Concentration Polarization Induced pH Changes at Nanoconstrictions. ACS Nano 2012, 6, 10206) as a model problem, and by including the full chemistry and transport, we show that the present model can capture the experimentally observed pH fronts. Our model can, among other things, be used to predict and engineer pH dynamics, which can be essential to the performance of membrane-based systems for biochemical separation and analysis.