Analysis of ion transport in nanofiltration using phenomenological coefficients and structural characteristics

The analysis of salt transport in nanofiltration using extended Nernst-Planck equations or similar models often suffers from the difficulties to establish and independently and transparently verify the consistency between the filtration results, assumed mechanism, and fitted values of parameters. As a general alternative, we propose here a procedure that reduces filtration data to two general phenomenological coefficients, concentration-dependent salt permeability omega(s) and Peclet coefficient A, which does not require that a specific exclusion mechanism be assumed and thus allows a transparent test on consistency with commonly used models. This approach was demonstrated using concentration polarization-corrected filtration data for NF-200 membrane and four monovalent salts, NaCl, NaBr, KBr, and KCl. The coefficient A was found to be very small, which points to the negligible contribution of convection to salt transport. The smallness of A was verified through estimates of the effective pore radius of the membrane, found to be between 0.2 and 0.3 nm, and comparing them with similar independent estimates from the hydraulic permeability L(p) using the data on the thickness and swelling of the selective polyamide layer obtained by AFM. The concentration dependence of omega(s) and its variation for different salts suggested that in the concentration range above 0.01 M the salt exclusion may be dominated by a combination of Donnan and dielectric mechanisms. The values of omega(s) obtained for single salts were also consistent with the selectivity observed for equimolar feed mixtures of NaCl and NaBr. However, the observed variation of omega(s) with concentrations of single salts below 0.01 M reveals a new regime that is inconsistent with all commonly used models of NF based on a Donnan mechanism modified with dielectric and steric effects. In particular, omega(s) appeared to approach a constant value at low salt concentrations, whereas the standard mechanisms predict a linear or even steeper decrease as concentration decreases. This puzzling discrepancy could have passed unnoticed in the standard multiparameter fitting extended Nernst-Planck equations and demonstrates the benefits of the present phenomenological analysis.