Exploring the Maxwell-Stefan description of ion exchange

Abstract

An ion exchange, water and several ions diffuse simultaneously, with different velocities. They are driven by activity, electrical and pressure gradients. We describe these complicated processes with the Maxwell-Stefan equation. This equation for multi-component diffusion requires one diffusivity or friction coefficient for each pair of components in the mixture. In this article, we explore the behaviour of these coefficients in the liquid and solid phases of ion exchange. Friction coefficients between ions and liquid are similar to those between uncharged species. They increase with the size of the ions, and depend only weakly on composition. Friction coefficients between positive and negative ions are much larger, especially between ions with multiple charges. They decrease with increasing concentrations. Ions of like charge usually have negative friction coefficients; their behaviour mirrors that 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. Friction between counter-ions and the charged matrix is important. It is much larger than the (free solution with tortuosity) prediction. This is especially so for ions with multiple charges. This type of friction also seems to depend on the polymer morphology. At the high concentrations in the matrix, friction coefficients between different counter-ions with a like charge are usually positive. The friction can be appreciable. Our knowledge of friction coefficients within the solid is still incomplete; we end with a few remarks on this.