Phenomenological analysis of transport of mono- and divalent ions in nanofiltration

A phenomenological approach was used to deduce concentration-dependent permeabilities of salts containing divalent ions (CaCl₂, MgCl₂ and Na₂SO₄) for NF200 membrane and compare them with results for monovalent salts and predictions of the steric-dielectric-Donnan (electric) (SDE) model. As a part of this effort the importance of the correction for concentration polarization was demonstrated and the error associated with the use of single average values of mass transfer coefficients and permeate concentration was analyzed. The dependence of permeability on salt concentration was found incompatible with the standard SDE picture for all salts. The discrepancy was more drastic for divalent cations, which showed a permeability decreasing with concentration in the whole analyzed range, in opposite to the standard model. It may be suggested that, a localized binding or association of counter-ions with fixed charges accompanied by charge reversal for divalent cations may be responsible for the observed behavior, provided the bound ions may retain osmotic activity. Such association may be promoted by the low dielectric constant of the membrane hence a large Bjerrum length commensurate with spacing of fixed charges.