Expressions for the C-term in the presence of pore flow

Abstract

In the course of our work on capillary electrochromatography (CEC) we, as others, have found strong evidence that flow in pores of particles can be significant. Its magnitude relative to the interstitial flow is characterized by the flow reduction factor, ω. Indirect evidence for pore flow was obtained much earlier by others, when it was noted that plate height, especially the C-term part, was significantly smaller in electrically driven (ED) than in pressure drive (PD) systems. This was interpreted as enhanced mass transfer, for which the intra-particle flow was held responsible. More direct evidence was produced by us when the size-exclusion (SEC) behaviour of polymers was studied in ED systems. It was found that the effect of exclusion on migration velocity could vanish entirely, and large and small molecules were co-eluted. This can only be explained if ω approaches 1; flow within the pores being as large as the interstitial flow. Indeed, consideration of double layer overlap indicated that ω-values close to 1 can often be expected in CEC. These large values ω inspired us to reconsider the effect of pore flow on the mass transfer term. We have arrived at the conclusion that enhanced mass transfer cannot explain in itself the extremely small values for the reduced plate height, h, (<1) observed especially for weakly retained solutes. In fact, when the pore flow is equal in magnitude to the interstitial flow, an unretained solute moves as fast within the particle as in the interstices; there is no non-equilibrium generated and a mass transfer term in h is not expected. For the migration of the solute the system is essentially uniform. Thus, apart from the mass transfer enhancement, another factor plays a role in the decrease of the h-values. We have attempted to derive a suitable expression for this effect. Some results are presented here. In one approach the situation is compared to that of an open tubular column with moving pseudo-stationary phase on the wall, an experiment that has actually been carried out by Krejci et al., or with micellar electrokinetic chromatography. In that case the plate height is easily derived. The result says that the plate height is proportional to the square of velocity difference between the two zones. However, the analogy is not perfect, and another approach suggests a direct proportionality rather than a square law one. Finally, a more refined treatment could be made only for a slab, not for a sphere. Extrapolation of this result to a sphere is put forward as a tentative expression for this effect.