On the modelling of the effective longitudinal diffusion in bi-continuous chromatographic beds

We report on the possibility to extend to bi-continuous packings the two models for the effective longitudinal diffusion D_eff, or B-term band broadening, recently proposed for discontinuous chromatographic beds. In bi-continuous packings, like monolithic columns, solutes experience a connected end-to-end pathway in both the mobile and stationary zones, as opposed to discontinuous packings, wherein the stationary adsorptive zone is distributed over a set of isolated elements. Since it is unclear whether a densely packed bed of spherical particles should be treated as a continuous or a bi-continuous medium, this extension is also crucial to fully understand the behaviour of packed particle beds. The proposed models for the effective longitudinal diffusion D_eff originate from the adoption of the Two Zone Moment Analysis (TZMA) method by which D_eff can be expressed as a linear combination of two essential quantities γ_m and γ_s, referred to as effective zone-diffusion factors. In the present work we propose two analytical models for γ_m and γ_s that now cover both the discontinuous and the bi-continuous case. To validate the theory, several bi-continuous packings are investigated, including the tetrahedral skeleton model (TSM), six different Triple Periodic Minimal Surface (TPMS) monoliths and randomly packed beds of spheres. For all of these, the models provide highly accurate results for D_eff over a wide range of porosities and zone retention factors k^″. The comparison with literature experimental data for both monolithic silica columns and columns packed with fully porous and porous-shell particles is also presented.