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Accounting for concentration-polarization (CP) is critical for modeling solute transport in membrane separation processes. In a mixed-electrolyte solution, ions CP is affected not only by diffusion and advection but also by electromigration. Yet, the classic film model, lacking an electromigration term, is frequently used for modeling ion CP. Often, ion CP is altogether neglected to reduce the computational load. Here, we study the CP of trace ions in a dominant salt solution, a case relevant for many reverse-osmosis and nanofiltration processes. First, we revisit the solution-diffusion-electromigration-film theory to obtain an analytical solution for the CP and membrane-transport of trace-ions in a dominant salt solution. Secondly, we consider limiting conditions relevant to reverse-osmosis and nanofiltration, from which we derive two compact equations that emerge as a seamless extension to the classic film theory. These equations can be used to account for the effect of electromigration on CP with minimal effort. Thirdly, we use our theory to quantify the effect of electromigration on ion CP in different dominant salt solutions. Finally, by analyzing two environmental membrane processes, we demonstrate how our theory deviates from the conventional one and quantify the implications on membrane scaling potential and the transport of ionic contaminants., Comment: 15 pages, 3 figures and a graphical abstract

Narrow carbon nanotubes (CNTs) desalinate water, mimicking water channels of biological membranes, yet the physics behind selectivity, especially, the effect of the membrane embedding CNTs on water and ion transfer, is still unclear. Here, we report $ab$ $initio$ analysis of the energies involved in transfer of water and K$^+$ and Cl$^-$ ions from solution to empty and water-filled 0.68 nm CNTs, for different dielectric constants $\epsilon$ of the surrounding matrix. The transfer energies computed for $1 \leq \epsilon < \infty$ permit a transparent breakdown of the transfer energy to three main contributions: binding to CNT, intra-CNT hydration, and dielectric polarization of the matrix. The latter scales inversely with $\epsilon$ and is of the order $10^2$/$\epsilon$ kJ/mol for both ions, which may change ion transfer from favorable to unfavorable, depending on ion, $\epsilon$, and CNT diameter. This may have broad implications for designing and tuning selectivity of nanochannel-based devices., Comment: 17 pages, 3 figures (main text), 8 pages, 7 figures (ancillary material)