Review of recent insights in the measurement and modelling of the B-term dispersion and related mass transfer properties in liquid chromatography

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 Deff, 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 Deff originate from the adoption of the Two Zone Moment Analysis (TZMA) method by which Deff can be expressed as a linear combination of two essential quantities γm and γs, referred to as effective zone-diffusion factors. In the present work we propose two analytical models for γm and γs 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 Deff 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.

On the contribution of the top and bottom walls in micro-pillar array columns and related high-aspect ratio chromatography systems

We report on a steady-state based, and hence highly accurate numerical modelling study of the effect of the top and bottom wall in the current generation of micro-pillar array columns. These have a mesoporous retention layer that not only covers the pillar walls but also the bottom wall. Our results show that the performance of these columns can in general not be improved by also covering the top wall with the same layer, despite the increased column symmetry this approach would offer. The reason for this is that the local species retardation caused by a retentive layer is much stronger than the pure flow arresting effect of an uncovered wall. At least, this has a crucial impact in high aspect-ratio systems such as micro-pillar array columns because these require a small inter-pillar distance to promote mass transfer together with a large channel depth to enable a sufficiently high flow rate. On the other hand, a notable improvement could be made if micro-pillar array would be produced without having a retentive layer at the bottom. At Péclet number Pe = 50 and aspect ratio AR = 5 for flow-channels, this gain amounts up to about 4.5 h-units at a zone retention factor k'' = 2 and 1.75 h-units at k'' = 16 (gain scales almost linearly with Pe). To verify these results, we also considered another high aspect-ratio system with a simplified geometry: the open-tubular channel with a flat-rectangular cross-section. This led to very similar observations, thus confirming the findings for the micro-pillar array. The results produced in the present study also allow us to conclude that the classic modelling paradigm adopted in chromatography, which is based on the independency and hence additivity of the hCm- and hCs-contributions, can lead to large modelling errors in chromatographic systems with a high aspect-ratio, even when their geometry is so simple as that of a straight open-tubular channel with constant cross-section. Indeed, when both zones are treated independently, the analysis misses how the vertical diffusion through the retentive layer helps suppressing the vertical gradients in the mobile zone. The diffusion through this layer occurs in a ratio of k''Ds/Dm (Dm being the diffusion coefficient in mobile phase zone and Ds being the diffusion coefficient in stationary phase zone), such that at high retention factors this diffusion contribution even becomes the dominant one.

Novel insights into the dependence of adsorption-desorption kinetics on particle geometry in chiral chromatography

The existence of slow adsorption-desorption kinetics in chiral liquid chromatography is common knowledge. This may significantly contribute to worsening the efficiency and kinetic performance of a chromatographic run, especially when high flow rates are employed. Many attempts and protocols have been proposed to access this term, the so-called c ads , but they are based on different (theoretical) assumptions. As a consequence, no official method is available for the estimation of the adsorption-desorption kinetics term. In this work, a novel approach to access c ads is presented. This procedure combines experimental results obtained with kinetic and thermodynamic measurements. The investigations have been performed on two zwitterionic teicoplanin chiral stationary phases (CSPs) based on 1.9 μ m fully porous and 2.0 μ m superficially porous particles (FPPs and SPPs), using Z-D,L-Methionine as probe molecule. Kinetic studies have been performed through the combination of both stop-flow and dynamic measurements, while adsorption isotherms have been calculated through Inverse Method. This study has confirmed that, on both particle formats, analyte diffusion on the surface of the particle is negligible, meaning that adsorption is localized, and it has been demonstrated that adsorption-desorption kinetics is strongly dependent on particle geometry and, in particular, on the loading of chiral selector. These findings are fundamental not only to unravel novel aspects of the complex enantiorecognition mechanism but also to optimize the employment of CSPs for ultra-fast and preparative applications.

An alternative general model for the effective longitudinal diffusion in chromatographic beds filled with ordered porous particles

The two-zone moment-analysis method for the determination of the dispersion tensor in hierarchical retentive porous media has been adopted to compute and model the effective longitudinal diffusion Deff, or equivalently the B-term band broadening, in chromatographic beds filled with ordered porous particles. On the one hand, this approach offers accurate numerical results for Deff while keeping computational expenses low. On the other hand, it also gives direct insight for the analytical modelling, readily revealings the two main essential quantities (resp. referred to as the mobile-zone and stationary-zone effective diffusion factors γm and γs) that contribute to Deff. Modelling these two main parameters provided us with two new analytical models for Deff: a general one, valid for diluted and concentrated packings and accurate in the whole range of relevant intra-particle diffusion coefficient Dpz, and an approximate one, reliable for diluted packings and accurate also for concentrated packings with low to intermediate values of Dpz. The large advantage of both models is that they do not need any fitting parameter because all the required information is incorporated into the experimentally accessible geometric obstruction factor in the mobile phase originating from the tortuosity of the through-pore space (limiting case of fully solid particles without any retention). These models hence serve as an alternative to the Effective Medium Theory (EMT) models used so far in the literature. To validate the theory, five ordered geometries have been investigated. The accuracy of the general model proposed has been quantified and found to be comparable with that of the 3rd order approximate Torquato model for four geometries, even for macro-porosities close to the close-packing limit. The case of a 2-d triangular array of ellipsoidal particles with different elongations is also investigated to show the general validity and applicability of the models.

Practical examination of flow rate effects and influence of the stationary phase water layer on peak shape and retention in hydrophilic interaction liquid chromatography

The effect of flow velocity on retention and peak shape of neutral, acidic and basic probe compounds was studied using seven different UHPLC hydrophilic interaction chromatography (HILIC) columns. Surprisingly on some columns, the retention factor k was found to vary somewhat with flow velocity, due to the combined effects of pressure and of frictional heating on retention. The selectivity of different HILIC columns was much greater than typically found with RP columns. The volume of the water layer on the HILIC columns was measured using the toluene exclusion procedure. For the neutral solute uridine, a good correlation was found between the volume of the water layer and retention, indicating the likely domination of a partition mechanism. For the ionogenic solutes, the correlation was generally poor, due to the presence of strong additional mechanisms such as ionic retention and repulsion. Reduced Van Deemter plots for uridine showed a negative correlation between the reduced b coefficient and the volume of the water layer, which can be attributed to reduced surface diffusion in this viscous layer. Once again, the behaviour of ionic solutes was complex on some columns making detailed interpretation difficult.

Detailed analysis of the effective and intra-particle diffusion coefficient of proteins at elevated pressure in columns packed with wide-pore core-shell particles

To determine the efficiency that can be obtained in a packed-bed liquid-chromatography column for a particular analyte, a correct determination of the molecular and effective diffusion coefficients (Dm and Deff) of the analyte is required. The latter is usually obtained via peak parking experiments wherein the flow is stopped. As a result, the column pressure rapidly dissipates and the measurement is essentially conducted at ambient pressure. This is problematic for analytes whose retention depends on pressure, such as proteins and potentially other large (dipolar) molecules. In that case, a conventional peak parking experiment is expected to lead to large errors in Deff. To obtain a better estimate ofDeff, the present study reports on the use of a set-up enabling peak parking measurements under pressurized conditions. This approach allowed us to report, for the first time, Deff for proteins at elevated pressure under retained conditions. First, Deff was determined at a (average) pressure of about 105 bar for a set of proteins with varying size, namely: bradykinin, insulin, lysozyme, β-lactoglobulin, and carbonic anhydrase in a column packed with 400 Å core-shell particles. The obtained data were then compared to those of several small analytes: acetophenone, propiophenone, benzophenone, valerophenone, and hexanophenone. A clear trend between Deff and analyte size was observed. The set-up was then used to determine Deff of bradykinin and lysozyme at variable (average) pressures ranging from 28 bar to 430 bar. These experiments showed a decrease in intra-particle and surface diffusion with pressure, which was larger for lysozyme than bradykinin. The data show that pressurized peak parking experiments are vital to correctly determine Deff when the analyte retention varies significantly with pressure.

Measurement of the molecular diffusion coefficient and the effective longitudinal diffusion under supercritical fluid chromatography conditions in packed bed columns

The improvement of supercritical fluid chromatography (SFC) instrumentation enhanced its reliability and utility over the past decade. The further development of high speed and high resolution separations is however obstructed by the lack of accurate models for axial dispersion in SFC. This work is a first step to tackle this by developing more reliable methods to measure molecular (D_mol) and longitudinal diffusion (D_eff) in SFC, as these affect all aspects of separation efficiency. In the present contribution, we report on an improved method, to enable more flexible, reliable and accurate measurements of D_mol in SFC using commercial instrumentation. A two-column variant of the stopped-flow experiment is proposed as an adapted set-up for measuring the effective longitudinal diffusion coefficient D_eff in SFC-conditions. Using the set-ups for a number of test-compounds, it has been found that D_eff, and the coefficients describing its constituent sub-processes (cf. particle diffusion D_part and surface diffusion γ_sD_s), all vary in a linearly proportional way with the bulk diffusion coefficient D_mol within a high degree of accuracy. It has also been found that D_eff decreases much more sharply with increasing retention factor compared to LC. By applying the effective medium theory, it was found that the relative surface diffusion coefficient γ_sD_s/D_mol decreases strongly with retention factor for the investigated solutes and column, in contrary to what is typically observed in reversed phase liquid chromatography. Results indicate that this might be related to a change in retention behavior of the analytes. Obviously, more analytes and conditions need to be explored to complete this picture and the extend range of applicability of these observations.

Liquid Chromatography Column Design and Dimensional Analysis of the van Deemter Equation

The fundamental mechanisms of band broadening are usually introduced to students through the van Deemter equation. Dimensional analysis of this equation can give physical meaning to the equation coefficients and enhance our understanding relative to qualitative descriptions. This approach can also guide improvements to future liquid chromatography (LC) column designs.