Effective medium theory expressions for the effective diffusion in chromatographic beds filled with porous, non-porous and porous-shell particles and cylinders. Part I: Theory

Using the permeability analogue of the diffusion and partitioning processes occurring in a chromatographic column, the different Effective Medium Theory (EMT) models that exist in literature for the electrical and thermal conductivity have been transformed into expressions that accurately predict the B-term band broadening in chromatographic columns. The expressions are written in such a form that they hold for both fully porous and porous-shell particles, and both spherical and cylindrical particles are considered. Mutually comparing the established EMT-expressions, it has been found that the most basic variant, i.e., the Maxwell-based expression, is already accurate to within 5% for the typical conditions encountered in liquid phase chromatography, independently of the exact microscopic morphology of the packing. For most typical values of the intra-particle diffusion rate and the species retention factors, it is even accurate to within 1%. If even higher accuracies are needed, more elaborate EMT-expressions are available. The modelling accuracy of all explicit EMT-expressions is much better than the residence time weighted (RTW) B-term expressions that have been used up to now in the field of chromatography, where the error is typically on the order of 10% and more. The EMT-models have also been used to establish expressions for the obstruction and tortuosity factor in packings of non-porous particles. The EMT has also been applied to the meso-porous zone only, yielding an expression for the intra-particle diffusion coefficient that can be used without having to specify any obstruction factor. It has also been shown that the EMT also provides a very simple but exact expression to represent the way in which the solid core obstructs the effective intra-particle diffusion in the case of porous-shell particles. This obstruction factor is given by γpart = 2/(2 + ρ3) for spherical particles and γpart = 1/(1 + ρ3) for cylinders. Back-transforming the obtained expressions, a set of simple explicit expressions has been obtained that allow to directly obtain the intra-particle diffusion coefficient (Dpart) from peak parking or B-term constant measurements. Using these expressions, it could be demonstrated that the traditionally employed RTW-model yields Dpart-values that display an erroneous retention factor dependency, even in cases where the RTW-model appears to be able to closely fit the peak parking measurements.