Predictive model of solute transport with reversible adsorption in spatially periodic hierarchical porous media

Multiscale simulation of liquid chromatography: Effective diffusion in macro-mesoporous beds and the B-term of the plate height equation

We performed multiscale simulations of analyte sorption and diffusion in hierarchical porosity models of monolithic silica columns for reversed-phase liquid chromatography to investigate how the mean mesopore size of the chromatographic bed and the analyte-specific interaction with the chromatographic interface influence the analyte diffusivity at various length scales. The reproduced experimental conditions comprised the retention of six analyte compounds of low to moderate solute polarity on a silica-based, endcapped, C18 stationary phase with water‒acetonitrile and water-methanol mobile phases whose elution strength was varied via the volumetric solvent ratio. Detailed information about the analyte-specific interfacial dynamics received from molecular dynamics simulations was incorporated through appropriate linker schemes into Brownian dynamics diffusion simulations in three hierarchical porosity models received from physical reconstructions of silica monoliths with a mean macropore size of 1.23 µm and mean mesopore sizes of 12.3, 21.3, or 25.7 nm. The mean mesopore size was found to have a similar influence on the effective mesopore diffusivity as the analyte polarity and the mobile-phase elution strength, which together determine the analyte residence time on a column. A smaller mesopore size attenuated the increase of the effective mesopore diffusivity with increasing mobile-phase elution strength significantly. The effective bed diffusivity was limited by the analyte residence time rather than by morphological details of the mesopore space. The stronger an analyte was retained by the chromatographic interface inside the mesopores, the slower became the mass transfer between the pore space hierarchies and the lower was the effective bed diffusivity. The B-terms of the plate height equation were finally generated with the bed diffusivities and phase-based retention factors derived from the hierarchical porosity models using additional information about the stationary-phase limit obtained from the analysis of analyte-bonded phase contacts. The B-terms highlight analyte- and mobile phase-specific behavior relevant to isocratic and gradient elution conditions in chromatographic practice. In particular, U-shaped B-term curves are observed due to the dominating contribution of the retention factor and the bed diffusivity to the B-term at low and high elution strength of the mobile phase, respectively.

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.

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.

Theoretical computation of the band broadening in micro-pillar array columns

We have investigated the possibility to establish a theoretical plate height expression for the band broadening in the most widely used micro-pillar array column format, i.e., a cylindrical pillar array wherein the pillar walls and the channel bottom are coated with a thin layer of meso‑porous material. Assuming isotropic diffusion in the shell-layer, it was found that the vertical diffusive transport along the porous shell-layer covering the pillar walls significantly suppresses the band broadening originating from the vertical migration velocity gradients. As the vertical transport in the shell-layer increases linearly with the retention equilibrium constant K, this leads to an anomalous dependency on the retention factor. Indeed, instead of increasing with k'' and following the classic (1+ak''+bk''2)/(1 + k'')2-dependency governing a classic Taylor-Aris system, the variation of the mobile zone mass transfer resistance term hCm in a 3D pillar array with bottom-wall retention goes through a maximum (resp. factor 1.5 (k''=4) and 2 (k''=16) difference between observed and classic Taylor-Aris behaviour). This effect increases with increasing pillar heights and increasing reduced velocities. Because of this complex k''-dependency, it proves very cumbersome to establish a general plate height equation covering all conditions. Instead, a plate height expression was established that is limited up to k''=4, but remains accurate for higher k''-values for cases where the ratio of pillar height over inter-pillar distance remains below 5. It can however be anticipated the proposed analytical model is only valid in a rather limited range around the presently considered external porosity of ε=0.5.

Nonadditivity and Nonlinearity of Mobile and Stationary Zone Mass Transfer Resistances in Chromatography

Using a two-zone moment analysis (TZMA) method based on Brenner's generalized dispersion theory for two-dimensional (2D) and three-dimensional (3D) periodic media, we investigated the mechanisms for dispersion in particulate media for liquid chromatography. This was done using a set of plate height data covering an unprecedented wide range of retention factors, diffusion coefficients, and velocities, all computed with unequaled accuracy. Applying Giddings' additivity test, based on alternatingly making the diffusion coefficient in the mobile and stationary zones infinitely large, the dispersion data clearly indicate a lack of additivity. Although this lack could be directly understood by identifying the existence of multiple parallel mass transfer paths, the additivity assumption interestingly overestimates the true C term band broadening (typically by more than 10%, depending on conditions and dimensionality of the system). However, Giddings originally asserted the occurrence of parallel paths would always lead to an underestimation of the dispersion. The origin of the lack of additivity is analyzed in detail and qualitatively explained. Finally, we also established a generic framework for the modeling of the effect of the reduced velocity and the retention coefficient on the C term in ordered chromatographic media. This led to the introduction of a new expression for the mobile zone mass transfer term, which, unlike the currently used literature expression, contains the complete k″ dependency.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

An explicit expression for the retention factor and velocity dependency of the mobile zone mass transfer band broadening in packed spheres beds used in liquid chromatography

The present study proposes a ready-to-use analytical expression to calculate the mobile zone mass transfer contribution (h_Cm) in packed bed columns. For this purpose, first high-accuracy computations of the band broadening in a perfectly ordered sphere array (fcc-arrangement, external porosity ε=0.40) were made using computational fluid dynamics (CFD), covering a broad range of zone retention factors (2≤k''≤18) and reduced velocities (0≤ν_i≤48). Subsequently, these data were used to determine the value of the geometrical constants in a number of possible analytical expressions for the h_Cm-contribution. This fitting exercise showed the traditional literature approach, using the Wilson-Geankoplis correlation to calculate the dimensionless Sherwood (Sh) number for the mass transfer, leads to fitting errors on the h_Cm-term as large as 150%. Instead, a new correlation for Sh is established. In addition, we also explored the difference in fitting accuracy between h_Cm-expressions based on either a plug-flow or a laminar flow profile assumption. Surprisingly, no significant difference in fitting accuracy between both assumptions was observed. Finally, a best-fit analytical expression is proposed that can represent the CFD-computed band broadening data with an average absolute fitting error of Δh=0.005, corresponding to a relative error of 2.5% on the h_Cm-term and of only 0.3% on the total plate height in a perfectly ordered sphere packing. Defining the presently investigated fcc-ordered sphere array with external porosity=40% as the reference geometry for a perfect sphere packing, the established expression can be used as a new yardstick expression against which the degree of eddy-dispersion can be measured.

The checkerboard model for the eddy-dispersion in laminar flows through porous media. Part I: Theory and velocity field properties

The additivity assumption underlying Giddings' coupling model for the eddy-dispersion in laminar flows through heterogeneous media is critically analyzed and a potential solution for its non-additivity in the high velocity limit is presented. Whereas the unit cell in Giddings' model only consists of a single velocity bias step, the unit dispersion cell of the newly proposed model comprises two consecutive velocity bias steps. Consequently, the unit cell of this new model allows to account for the occurrence of an internal velocity bias rectification at high reduced velocities and is therefore additive in both the low and high velocity limit. First, a mathematical expression for the velocity- and diffusion-dependency of the model's dispersion characteristics has been established. Subsequently, the physical behavior of the model is discussed. It is shown the relation between the eddy-dispersion plate height h and the reduced velocity ν can be expected to display a local maximum in systems where the transversal dispersion purely occurs by molecular diffusion, as is the case in perfectly ordered flow-through media. In disordered media, where the transversal dispersion also contains a significant advective component, the model predicts a velocity-dependency that is qualitatively similar to that described by Giddings' coupling model but, all other conditions being equal, converges to a significantly smaller horizontal asymptote at high reduced velocity. The latter might shed new light on earlier eddy-dispersion studies pursuing a quantitative agreement between experimental data and the Giddings model.