Ja. Wesselingh et al., EXPLORING THE MAXWELL-STEFAN DESCRIPTION OF ION-EXCHANGE, Chemical engineering journal and the biochemical engineering journal, 57(2), 1995, pp. 75-89
In ion exchange, water and several ions diffuse simultaneously, with d
ifferent velocities. They are driven by activity, electrical and press
ure gradients. We describe these complicated processes with the Maxwel
l-Stefan equation. This equation for multicomponent diffusion requires
one diffusivity or friction coefficient for each pair of components i
n the mixture. In this article, we explore the behaviour of these coef
ficients in the liquid and solid phases of ion exchange. Friction coef
ficients between ions and liquid are similar to those between uncharge
d species. They increase with the size of the ions, and depend only we
akly on composition. Friction coefficients between positive and negati
ve ions are much larger, especially between ions with multiple charges
. They decrease with increasing concentrations. Ions of like charge us
ually have negative friction coefficients; their behaviour mirrors tha
t of ions with unlike charges. The many coefficients in the solid can
only be obtained by combining data on ion exchange and electrodialysis
membranes. In the matrix, we can estimate friction coefficients with
water from free solution values and a 'tortuosity' correction. Frictio
n between counter-ions and the charged matrix is important. It is much
larger than the (free solution with tortuosity) prediction. This is e
specially so for ions with multiple charges. This type of friction als
o seems to depend on the polymer morphology. At the high concentration
s in the matrix, friction coefficients between different counter-ions
with a like charge are usually positive. The friction can be appreciab
le. Our knowledge of friction coefficients within the solid is still i
ncomplete; we end with a few remarks on this.