  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.
