Modeling non-linear ion transport phenomena in ion-selective membranes: Three simplified models

In electromembrane processes employing the ion concentration polarization phenomenon, such as electrodialysis (ED), ion concentration polarization (ICP) desalination, and reverse electrodialysis, the ion-selective membrane (ISM) has been modeled with an assumption of an ideal ISM permits only the passage of counter-ions with all the co-ions blocked. However, in the works that study the effects of co-ion flux on the response of system, this assumption has been questioned about the adequate answer to the simulation of the characteristic properties of the membrane. In this work, we evaluate the three models which are used widely in modeling ISM including surface-charged, space-charged and nanochannel array model. The results show that the nanochannel array and space-charged models, the alternative model for the model ISM in the case of the studies which consider the effects of co-ion leakage on the performance of the system, sufficiently mimic the key characteristic of the membrane. Their current–voltage response (I-V) curve conforms to a typical one, yet they differ due to the varying co-ion fluxes. The nanochannel array model, which is the direct model of the actual ISM, closely mimics the seed electroconvection vortices near the membrane, which does not appear in the surface-charged and space-charged model. This model is computationally expensive because of resolving the nonlinearity of the variables near the entrance of nanochannels, so it is not the appropriate model to employ in a study that considers the micrometer or millimeter order scale system. Finally, the space-charged model is more appropriate to mimic the ISM but requires more careful consideration of the simplification condition. By investigating the volume charge density of the membrane in the space-charged model, we suggest a value that brings the most similar I-V curve to that of the nanochannel array model.