From random sphere packings to regular pillar arrays: Analysis of transverse dispersion

Harnessing an elastic flow instability to improve the kinetic performance of chromatographic columns

Despite decades of research and development, the optimal efficiency of slurry-packed HPLC columns is still hindered by inherent long-range flow heterogeneity from the wall to the central bulk region of these columns. Here, we show an example of how this issue can be addressed through the straightforward addition of a semidilute amount (500 ppm) of a large, flexible, synthetic polymer (18 MDa partially hydrolyzed polyacrylamide, HPAM) to the mobile phase (1% NaCl aqueous solution, hereafter referred to as "brine") during operation of a 4.6 mm × 300 mm column packed with 10μm BEHTM 125 Å particles. Addition of the polymer imparts elasticity to the mobile phase, causing the flow in the interparticle pore space to become unstable above a threshold flow rate. We verify the development of this elastic flow instability using pressure drop measurements of the friction factor versus Reynolds number. In prior work, we showed that this flow instability is characterized by large spatiotemporal fluctuations in the pore-scale flow velocities that may promote analyte dispersion across the column. Axial dispersion measurements of the quasi non-retained tracer thiourea confirm this possibility: they reveal that operating above the onset of the instability improves column efficiency by greater than 100%. These experiments thereby suggest that elastic flow instabilities can be harnessed to mitigate the negative impact of trans-column flow heterogeneities on the efficiency of slurry-packed HPLC columns. While this approach has its own inherent limitations and constraints, our results lay the groundwork for future targeted development of polymers that can impart elasticity when dissolved in commonly used liquid chromatography mobile phases, and can thereby generate elastic flow instabilities to help improve the resolution of HPLC columns.

A logarithmic law for the velocity- and retention-dependency of the eddy dispersion in chromatographic columns

Applying the recently introduced patchwork model for porous media, we present a new step forward in the modelling of eddy dispersion in chromatographic columns. The logarithmic law describing the velocity dependency emerging from this patchwork model is supplemented with a retention factor dependency via first principles modelling of the variations in flow resistance and retention capacity caused by the packing disorder. Furthermore, it is shown the derived expression is also able to fit the eddy dispersion originating from the wall effect on the packing. When applied to literature data of eddy dispersion, the newly introduced logarithmic law has a goodness of fit that is at least equal to that of Knox' empirical power law (R2>0.98). The main difference is that, whereas Knox' power law requires a separate fit for each component due to the retention factor dependency, the present model simultaneously fits all plate height curves measured on one chromatographic column, using only two parameters with a clear physical meaning.

Physical origin of the peak tailing of monoclonal antibodies in size-exclusion chromatography using bio-compatible systems and columns

The analysis of mixtures containing monoclonal antibody (mAb) (approximately 150 kDa molecular weight) and sub-unit impurities (approximately 100 kDa) is challenging, even when adopting the latest ultra-high-pressure liquid chromatography (UHPLC) columns (4.6 mm [Formula: see text] 150 mm coated hardware, 1.7 [Formula: see text]m 250 BEH[Formula: see text] Surface-modified Particles) and systems (ACQUITY[Formula: see text] UPLC[Formula: see text] I-class Bio Plus). The main issue still encountered is a persistent tail of the mAb peak. Here, the physical origin(s) of such peak tailing in size-exclusion chromatography (SEC) are investigated from both fundamental and practical approaches. Up to five relevant physical origins are analyzed: sample heterogeneity (glycoforms), UHPLC system dispersion, strong residual binding of the mAb to the SEC particles (via hydrophobic and/or electrostatic interactions) and to the stainless steel column/system hardware, slow escape kinetics of the mAb from the SEC particles, and flow heterogeneity caused by the non-ideal slurry packing of SEC columns. Experiments (testing sample heterogeneity, system dispersion, and strong residual interactions) and calculations (predicting the transient absorption/escape kinetics in a single SEC particle and the two-dimensional peak concentration profiles) altogether unambiguously demonstrate that the observed mAb peak tailing is caused primarily by the long-range velocity biases across the SEC column combined with the slow transverse dispersion of mAbs. Therefore, improvement in the resolution between mAb and sub-unit fragment impurities can only be achieved by increasing the column length, e.g., by applying recycling chromatography at acceptable pressures.

Particle dispersion through porous media with heterogeneous attractions

Porous media used in many practical applications contain natural spatial variations in composition and surface charge that lead to heterogeneous physicochemical attractions between the media and transported particles. We performed Stokesian dynamics (SD) simulations to examine the effects of heterogeneous attractions on quiescent diffusion and hydrodynamic dispersion of particles within geometrically ordered arrays of nanoposts. We find that transport under quiescent conditions occurs by two mechanisms, diffusion through the void space and intermittent hopping between the attractive wells of different nanoposts. As the attraction heterogeneity increases, the latter mechanism becomes dominant, resulting in an increase in the particle trajectory tortuosity, deviations from Gaussian behavior in the particle displacement distributions, and a decrease in the long-time particle diffusivity. Similarly, under flow conditions corresponding to low Péclet number (Pe), increased attraction heterogeneity leads to transient localization near the nanoposts, resulting in a broadening of the particle distribution and enhanced longitudinal dispersion in the direction of flow. At high Pe where advection strongly dominates, however, the longitudinal dispersion coefficient is insensitive to attraction heterogeneity and exhibits Taylor-Aris dispersion behavior. Our findings provide insight into how heterogeneous interactions may influence particle transport in complex 3-D porous media.

Resolution limits of size exclusion chromatography columns identified from flow reversal and overcome by recycling liquid chromatography to improve the characterization of manufactured monoclonal antibodies

The flow reversal (FR) technique consists of reversing the flow direction along a chromatographic column. It is used to reveal the origin (such as poor column packing, active sites, or slow absorption/escape kinetics) for the resolution limit of 4.6 mm × 150 mm long columns packed with 1.7 μm 200 Å Bridge-Ethylene-Hybrid (BEHTM) Particles. These columns are used to separate manufactured monoclonal antibodies (mAb, ∼ 150 kDa) from their close impurities (or IdeS fragments, ∼ 100 kDa) by size exclusion chromatography (SEC). FR unambiguously demonstrates that the resolution limit of these SEC columns is primarily due to long-range flow velocity biases covering distances of at least 500 μm across the column diameter. This confirms the existence of center-to-wall flow heterogeneities which cause undesirable tailing for the mAb peak. Because the transverse dispersion coefficient (Dt=1.1 × 10-6 cm2/s) of mAbs across the column diameter is intrinsically low, the bandspreading of the mAb in a single flow direction is in part reversible upon reversing the flow direction. For the very same residence time in the column, the column efficiency is found to increase by +85% relative to that observed under conventional elution mode. The observed peak tailing of the mAb and its sub-units is not caused by active surface sites or by slow absorption/escape from the BEH Particles. Therefore, the most critical mAb impurities (hydrolytic degradation Fab/c and IdeS [Formula: see text] fragments) can only be successfully separated and quantified with acceptable accuracy by adopting alternate pumping recycling liquid chromatography (APRLC). APRLC enables the full baseline separation of the mAb and 100 kDa mAb fragments and partial separation of Fab/c and [Formula: see text] fragments after increasing the number of cycles to ten. It was made possible to accurately measure the relative abundances of the mAb (99.0 ± 0.1%), [Formula: see text] fragment (0.88 ± 0.03%), and Fab/c immunogenic fragment (0.13 ± 0.02%) in less than 45 min for a total mAb sample load of only 5 μg. Still, further improvements are needed to increase the sensitivity of the APRLC method and to reduce the solvent consumption by adopting narrow-bore 2.1 mm i.d. SEC columns.

An approach to high throughput measurement of accurate retention data in liquid chromatography

Efforts to model and simulate various aspects of liquid chromatography (LC) separations (e.g., retention, selectivity, peak capacity, injection breakthrough) depend on experimental retention measurements to use as the basis for the models and simulations. Often these modeling and simulation efforts are limited by datasets that are too small because of the cost (time and money) associated with making the measurements. Other groups have demonstrated improvements in throughput of LC separations by focusing on "overhead" associated with the instrument itself - for example, between-analysis software processing time, and autosampler motions. In this paper we explore the possibility of using columns with small volumes (i.e., 5 mm x 2.1 mm i.d.) compared to conventional columns (e.g., 100 mm x 2.1 mm i.d.) that are typically used for retention measurements. We find that isocratic retention factors calculated for columns with these dimensions are different by about 20%; we attribute this difference - which we interpret as an error in measurements based on data from the 5 mm column - to extra-column volume associated with inlet and outlet frits. Since retention factor is a thermodynamic property of the mobile/stationary phase system under study, it should be independent of the dimensions of the column that is used for the measurement. We propose using ratios of retention factors (i.e., selectivities) to translate retention measurements between columns of different dimensions, so that measurements made using small columns can be used to make predictions for separations that involve conventional columns. We find that this approach reduces the difference in retention factors (5 mm compared to 100 mm columns) from an average of 18% to an average absolute difference of 1.7% (all errors less than 8%). This approach will significantly increase the rate at which high quality retention data can be collected to thousands of measurements per instrument per day, which in turn will likely have a profound impact on the quality of models and simulations that can be developed for many aspects of LC separations.

Experimental Quantification of Gas Dispersion in 3D-Printed Logpile Structures Using a Noninvasive Infrared Transmission Technique

3D-printed catalyst structures have the potential to broaden reactor operating windows. However, the hydrodynamic aspects associated with these novel catalyst structures have not yet been quantified in detail. This work applies a recently introduced noninvasive, instantaneous, whole-field concentration measurement technique based on infrared transmission to quantify the rate of transverse gas dispersion in 3D-printed logpile structures. Twenty-two structural variations have been investigated at various operating conditions, and the measured transverse gas dispersion has been correlated to the Péclet number and the structures’ porosity and feature size. It is shown that staggered configurations of these logpile structures offer significantly more tunability of the dispersion behavior compared to straight structures. The proposed correlations can be used to facilitate considerations of reactor design and operating windows.

Nanoparticle dispersion in porous media: Effects of attractive particle-media interactions

We investigate the effects of physicochemical attractions on the transport of finite-sized particles in three-dimensional ordered nanopost arrays using Stokesian dynamics simulations. We find that weak particle-nanopost attractions negligibly affect diffusion due to the dominance of Brownian fluctuations. Strong attractions, however, significantly hinder particle diffusion due to localization of particles around the nanoposts. Conversely, under flow, attractions significantly enhance longitudinal dispersion at low to moderate Péclet number (Pe). At high Pe, by contrast, advection becomes dominant and attractions weakly enhance dispersion. Moreover, attractions frustrate directional locking at moderate flow rates, and shift the onset of this behavior to higher Pe.

On the road towards highly efficient and large volume three-dimensional-printed liquid chromatography columns ?

The current performance of commercially packed LC columns is limited by the random structure of the packed bed and by the wall-to-center heterogeneity of its structure. The minimum reduced plate heights observed are not smaller than 1.4 whereas they could theoretically be as low as 0.1 for dense and perfectly ordered packings of spheres. To bridge this gap, a wide inner diameter column with an ordered macroporous structure is printed in three dimensions by stereolithography of poly(ethylene glycol diacrylate) resin. Feature sizes below 100 μm are achieved by combining conventional polymer stereolithography with photolithography using photomasks. A layer-by-layer polymerization is performed by alternating two distinct photomasks having horizontally and vertically oriented patterns. Despite the inevitable printing imperfections, minimum reduced plate heights around unity are measured for non-retained analytes. The next challenges for the successful printing of highly efficient and large volume LC columns are threefold: reducing the feature size down to below 10 μm, keeping minimum the unevenness of the flow channel dimensions, and tackling additive manufacturing of silica aerogels at such small feature sizes for higher mechanical stability and broader range of retention/selectivity than those delivered by polymer materials. This article is protected by copyright. All rights reserved.

Nanoparticle dispersion in porous media: Effects of array geometry and flow orientation

We investigate the effects of array geometry and flow orientation on transport of finite-sized particles in ordered arrays using Stokesian dynamics simulations. We find that quiescent diffusion is independent of array geometry over the range of volume fraction of the nanoposts examined. Longitudinal dispersion under flow depends on the direction of incident flow relative to the array lattice vectors. Taylor-Aris behavior is recovered for flow along the lattice directions, whereas a nonmonotonic dependence of the dispersion coefficient on the Péclet number is obtained for flow orientations slightly perturbed from certain lattice vectors, owing to a competition between directional locking and spatial velocity variations.

Numerical Elucidation of Flow and Dispersion in Ordered Packed Beds: Nonspherical Polygons and the Effect of Particle Overlap on Chromatographic Performance

Spherical particles are widely considered as the benchmark stationary phase for preparative and analytical chromatography. Although this has proven true for randomly packed beds in the past, we challenge this paradigm for ordered packings, the fabrication of which are now feasible through additive manufacturing (3D printing). Using computational fluid dynamics (Lattice Boltzmann Method) this work shows that nonspherical particles can both reduce mobile-phase band broadening and increase permeability compared with spheres in ordered packed beds. In practice, ordered packed beds can only remain physically stable if the particles are fused to form a contiguous matrix, thus creating a positional overlap at the points of fusion between what would otherwise be discrete particles. Overlap is shown to decrease performance of ordered packed beds in all observed cases, thus we recommend it should be kept to the minimum extent necessary to ensure physical stability. Finally, we introduce a metric to estimate column performance, the mean deviated velocity, a quantitative description of the spread of the velocity field in the column. This metric appears to be a good indicator of mobile-phase dispersion in ordered packed bed media, including overlapped beds, and is a useful tool for screening new stationary-phase morphologies without having to perform computationally expensive simulations.

Unusual van Deemter plots of optical isomers on a chiral brush-type liquid chromatography column

Unusual dynamic behavior of the enantiomers of 9-bromo-11b-(tert-butyl)-2,3,6,11b-tetrahydrooxazolo[3',2':1,5]pyrrolo[3,4-b]quinoline-5,11-dione (I) was observed on a Nautilus-R column packed with silica grafted with antibiotic ristocetin. It consisted in (i) antibatic behavior of the van Deemter plots of the enantiomers and (ii) high and strongly enantiomer dependent values of the A- and B-terms of the van Deemter equation. Although rare, such a pattern has been found earlier in chiral chromatography, with all reported cases limited to brush-type chiral stationary phases. Adsorption dynamics in this system was studied by means of the moment method and the peak parking technique; hydrodynamic properties of the column were explored by using unretained tracers. It was shown that the peculiar shape of the van Deemter curves for the enantiomers of I is conditioned by imperfect packing of the stationary phase, which result in high transcolumn eddy dispersion, and by slow adsorption/desorption kinetics. It was proven that the whole void volume of the column available to an eluent is not accessible to the studied analyte because it cannot penetrate the space between neighboring grafted ligands. Its mass transfer in pores is also affected by the fact that the stagnant layer of the binary water-acetonitrile mobile phase differs in composition from the bulk liquid due to preferential adsorption of water that influences the apparent molecular diffusivity of solutes. An effect of the structures of analyte and chiral selector on the adsorption kinetics is also briefly discussed.

Simulation and theory of open-tube dispersion in short and long capillaries with slip boundaries and retention

Using random walk techniques, high resolution simulations of zone shape are conducted in open capillary tubes for short and long tube conditions. Finite size solutes are used as tracers in this treatment. Slip flow boundary conditions and wall retention are utilized as needed. These simulations are able to reproduce previous work in short and long tubes. For the short tube case where dispersion does not asymptotically approach the classic Taylor-Aris and Golay solutions, the effect of slip flow boundaries in the transient region shows zone shapes with abbreviated tails where the larger slip flow values cause zone compression. The use of slip flow to lower dispersion in capillary-based, wall-coated separations is shown to favor long tube behavior. This is because slip flow is relevant for cases where slip lengths are fractions of small capillary tube diameters. Incorporating slip flow into transport in capillaries favors a very small capillary radius where the cross-sectional diffusion length is very small and sampling times are fast. The purely convective zone shape with slip flow boundaries is derived analytically. Applications for this type of separation, guided by both analytical theory and simulation, show the potential for nano-sized capillary tubes less than 1 μm in diameter and favor very fast isocratic separations. Using long tube retention theory with slip boundaries shows that the dispersion-reducing region is most important in the range 0 ≤ k' ≤ 1, a relatively small retention window. Further discussion of the gradient elution technique and dispersion in packed beds suggests that the general usage of slip flow boundaries is restricted in liquid phase separation systems.

Transverse solute dispersion in microfluidic paper-based analytical devices (μPADs)

Key rules for the design of analytical operations based on the transverse solute dispersion in paper are provided.

Modelling ordered packed beds of spheres: The importance of bed orientation and the influence of tortuosity on dispersion

Ordered packing has previously been considered for porous media applications in the industrial and analytical worlds, with implementation constrained only by the lack of feasible fabrication methods. Additive manufacturing now provides the answer to this limitation, which leads to the novel domain of customized ordered packing and a variety of optimized geometries. In this work, the chromatographic behaviour of ordered configurations of particles was described using computational fluid dynamics methods based on the Lattice Boltzmann Model. The model was first validated by matching van Deemter trends for ordered and random packings shown in previous research. The influence of rotations of the ordered configurations was then considered, indicating that orientational changes with respect to the main flow axis can strongly affect minimum plate height. In particular, it is demonstrated that targeted rotations of ordered packings can reduce axial dispersion while improving transverse dispersion, thus improving chromatographic performance. This principle is clearly shown in a strong linear correlation between tortuosity and plate height, offering an additional parameter to enable a priori control of the performance of ordered packings. Furthermore, rotation of the packing does not change porosity or surface area and has a relatively small effect on permeability. Thus, highly permeable packings with poor dispersion can be improved in terms of chromatographic impedance by simple rotation of the packing orientation. This work further demonstrates the advantages of ordered packings over randomly packed beds, and introduces new perspectives on the development of chromatographic structures with improved performance.

Particle dispersion in porous media: Differentiating effects of geometry and fluid rheology

We investigate the effects of geometric order and fluid rheology on the dispersion of micron-sized particles in two-dimensional microfluidic porous media. Particles suspended in a mixture of glycerol and water or in solutions of partially hydrolyzed polyacrylamide (HPAM) polymers were imaged as they flowed through arrays of microscale posts. From the trajectories of the particles, we calculated the velocity distributions and thereafter obtained the longitudinal and transverse dispersion coefficients. Particles flowed in the shear-thinning HPAM solution through periodic arrays of microposts were more likely to switch between streamlines, due to elastic instabilities. As a result, the distributions of particle velocity were broader in HPAM solutions than in glycerol-water mixtures for ordered geometries. In a disordered array of microposts, however, there was little difference between the velocity distributions obtained in glycerol-water and in HPAM solutions. Correspondingly, particles flowed through ordered post arrays in HPAM solutions exhibited enhanced transverse dispersion. This result suggests that periodic geometric order amplifies the effects of the elasticity-induced velocity fluctuations, whereas geometric disorder of barriers effectively averages out the fluctuations.

Impact of diffusion on transverse dispersion in two-dimensional ordered and random porous media

Solute dispersion in fluid flow results from the interaction between advection and diffusion. The relative contributions of these two mechanisms to mass transport are characterized by the reduced velocity ν, also referred to as the Péclet number. In the absence of diffusion (i.e., when the solute diffusion coefficient D_{m}=0 and ν→∞), divergence-free laminar flow of an incompressible fluid results in a zero-transverse dispersion coefficient (D_{T}=0), both in ordered and random two-dimensional porous media. We demonstrate by numerical simulations that a more realistic realization of the condition ν→∞ using D_{m}≠0 and letting the fluid flow velocity approach infinity leads to completely different results for ordered and random two-dimensional porous media. With increasing reduced velocity, D_{T} approaches an asymptotic value in ordered two-dimensional porous media but grows linearly in disordered (random) structures depending on the geometrical disorder of a structure: a higher degree of heterogeneity results in a stronger growth of D_{T} with ν. The obtained results reveal that disorder in the geometrical structure of a two-dimensional porous medium leads to a growth of D_{T} with ν even in a uniform pore-scale advection field; however, lateral diffusion is a prerequisite for this growth. By contrast, in ordered two-dimensional porous media the presence of lateral diffusion leads to a plateau for the transverse dispersion coefficient with increasing ν.

Implementation of high slurry concentration and sonication to pack high-efficiency, meter-long capillary ultrahigh pressure liquid chromatography columns

Slurry packing capillary columns for ultrahigh pressure liquid chromatography is complicated by many interdependent experimental variables. Previous results have suggested that combination of high slurry concentration and sonication during packing would create homogeneous bed microstructures and yield highly efficient capillary columns. Herein, the effect of sonication while packing very high slurry concentrations is presented. A series of six, 1m×75μm internal diameter columns were packed with 200mg/mL slurries of 2.02μm bridged-ethyl hybrid silica particles. Three of the columns underwent sonication during packing and yielded highly efficient separations with reduced plate heights as low as 1.05.

Larger voids in mechanically stable, loose packings of 1.3μm frictional, cohesive particles: Their reconstruction, statistical analysis, and impact on separation efficiency

Lateral transcolumn heterogeneities and the presence of larger voids in a packing (comparable to the particle size) can limit the preparation of efficient chromatographic columns. Optimizing and understanding the packing process provides keys to better packing structures and column performance. Here, we investigate the slurry-packing process for a set of capillary columns packed with C18-modified, 1.3μm bridged-ethyl hybrid porous silica particles. The slurry concentration used for packing 75μm i.d. fused-silica capillaries was increased gradually from 5 to 50mg/mL. An intermediate concentration (20mg/mL) resulted in the best separation efficiency. Three capillaries from the set representing low, intermediate, and high slurry concentrations were further used for three-dimensional bed reconstruction by confocal laser scanning microscopy and morphological analysis of the bed structure. Previous studies suggest increased slurry concentrations will result in higher column efficiency due to the suppression of transcolumn bed heterogeneities, but only up to a critical concentration. Too concentrated slurries favour the formation of larger packing voids (reaching the size of the average particle diameter). Especially large voids, which can accommodate particles from>90% of the particle size distribution, are responsible for a decrease in column efficiency at high slurry concentrations. Our work illuminates the increasing difficulty of achieving high bed densities with small, frictional, cohesive particles. As particle size decreases interparticle forces become increasingly important and hinder the ease of particle sliding during column packing. While an optimal slurry concentration is identified with respect to bed morphology and separation efficiency under conditions in this work, our results suggest adjustments of this concentration are required with regard to particle size, surface roughness, column dimensions, slurry liquid, and external effects utilized during the packing process (pressure protocol, ultrasound, electric fields).

Morphological Analysis of Physically Reconstructed Silica Monoliths with Submicrometer Macropores: Effect of Decreasing Domain Size on Structural Homogeneity

Silica monoliths are increasingly used as fixed-bed supports in separation and catalysis because their bimodal pore space architecture combines excellent mass transport properties with a large surface area. To optimize their performance, a quantitative relationship between morphology and transport characteristics has to be established, and synthesis conditions that lead to a desired morphology optimized for a targeted application must be identified. However, the effects of specific synthesis parameters on the structural properties of silica monoliths are still poorly understood. An important question is how far the macropore and domain size can be reduced without compromising the structural homogeneity. We address this question with quantitative morphological data derived for a set of eight macroporous-mesoporous silica monoliths with an average macropore size (d(macro)) of between 3.7 and 0.1 μm, prepared following an established route involving the sol-gel transition and phase separation. The macropore space of the silica monolith samples is reconstructed using focused ion beam scanning electron microscopy followed by a quantitative assessment of geometrical and topological properties based on chord length distributions (CLDs) and branch-node analysis of the pore network, respectively. We observe a significant increase in structural heterogeneity, indicated by a decrease in the parameter k derived from fitting a k-gamma function to the CLDs, when d(macro) reaches the submicrometer range. The compromised structural homogeneity of silica monoliths with submicrometer macropores could possibly originate from early structural freezing during the competitive processes of sol-gel transition and phase separation. It is therefore questionable if the common approach of reducing the morphological features of silica monoliths into the submicrometer regime by changing the point of sol-gel transition can be successful. Alternative strategies and a better understanding of the involved competitive processes should be the focus of future research.

Synthesis and morphological characterization of phenyl-modified macroporous–mesoporous hybrid silica monoliths

We present a comprehensive approach to characterize the one-pot synthesis, macropore space morphology, and chromatographic performance of phenyl-modified macro–mesoporous silica monoliths.