Computational investigation of longitudinal diffusion, eddy dispersion, and trans-particle mass transfer in bulk, random packings of core-shell particles with varied shell thickness and shell diffusion coefficient.

In recent years, chromatographic columns packed with core-shell particles have been widely used for efficient and fast separations at comparatively low operating pressure. However, the influence of the porous shell properties on the mass transfer kinetics in core-shell packings is still not fully understood. We report on results obtained with a modeling approach to simulate three-dimensional advective-diffusive transport in bulk random packings of monosized core-shell particles, covering a range of reduced mobile phase flow velocities from 0.5 up to 1000. The impact of the effective diffusivity of analyte molecules in the porous shell and the shell thickness on the resulting plate height was investigated. An extension of Giddings' theory of coupled eddy dispersion to account for retention of analyte molecules due to stagnant regions in porous shells with zero mobile phase flow velocity is presented. The plate height equation involving a modified eddy dispersion term excellently describes simulated data obtained for particle-packings with varied shell thickness and shell diffusion coefficient. It is confirmed that the model of trans-particle mass transfer resistance of core-shell particles by Kaczmarski and Guiochon [42] is applicable up to a constant factor. We analyze individual contributions to the plate height from different mass transfer mechanisms in dependence of the shell parameters. The simulations demonstrate that a reduction of plate height in packings of core-shell relative to fully porous particles arises mainly due to reduced trans-particle mass transfer resistance and transchannel eddy dispersion.

Comparison of first and second generation analytical silica monoliths by pore-scale simulations of eddy dispersion in the bulk region.

We present the first quantitative comparison of eddy dispersion in the bulk macropore (flow-through) space of 1st and 2nd generation analytical silica monoliths. Based on samples taken from the bulk region of Chromolith columns, segments of the bulk macropore space were physically reconstructed by confocal laser scanning microscopy to serve as models in pore-scale simulations of flow and dispersion. Our results cover details of the 3D velocity field, macroscopic Darcy permeability, transient and asymptotic dispersion behavior, and chromatographic band broadening, and thus correlate morphological, microscopic, and macroscopic properties. A complete set of parameters for the individual eddy dispersion contributions in the bulk was obtained from a Giddings analysis of the simulated plate height data. The identified short-range structural heterogeneities correspond to the average domain size of the respective monoliths. Our plate height curves show that structural improvements in the bulk morphology of 2nd generation monoliths play only a minor role for the observed improvement in overall column efficiency. The results also indicate a topological dissimilarity between 1st and 2nd generation analytical silica monoliths, which raises the question how the domain size of silica monoliths can be further decreased without compromising the structural homogeneity of the bed.

Geometrical and topological measures for hydrodynamic dispersion in confined sphere packings at low column-to-particle diameter ratios.

At low column-to-particle diameter (or aspect) ratio (d(c)/d(p)) the kinetic column performance is dominated by the transcolumn disorder that arises from the morphological gradient between the more homogeneous, looser packed wall region and the random, dense core. For a systematic analysis of this morphology-dispersion relation we computer-generated a set of confined sphere packings varying three parameters: aspect ratio (d(c)/d(p)=10-30), bed porosity (ɛ=0.40-0.46), and packing homogeneity. Plate height curves were received from simulation of hydrodynamic dispersion in the packings over a wide range of reduced velocities (v=0.5-500). Geometrical measures derived from radial porosity and velocity profiles were insufficient as morphological descriptors of the plate height data. After Voronoi tessellation of the packings, topological information was obtained from the statistical moments of the free Voronoi volume (V(free)) distributions. The radial profile of the standard deviation of the V(free) distributions in the form of an integral measure was identified as a quantitative scalar measure for the transcolumn disorder. The first morphology-dispersion correlation for confined sphere packings deepens our understanding of how the packing microstructure determines the kinetic column performance.

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

We study the impact of microscopic order on transverse dispersion in the interstitial void space of bulk (unconfined) chromatographic beds by numerical simulations of incompressible fluid flow and mass transport of a passive tracer. Our study includes polydisperse random sphere packings (computer-generated with particle size distributions of modern core-shell and sub-2 µm particles), the macropore space morphology of a physically reconstructed silica monolith, and computer-generated regular pillar arrays. These bed morphologies are analyzed by their velocity probability density distributions, transient dispersion behavior, and the dependence of asymptotic transverse dispersion coefficients on the mobile phase velocity. In our work, the spherical particles, the monolith skeleton, and the cylindrical pillars are all treated as impermeable solid phase (nonporous) and the tracer is unretained, to focus on the impact of microscopic order on flow and (particularly transverse) hydrodynamic dispersion in the interstitial void space. The microscopic order of the pillar arrays causes their velocity probability density distributions to start and end abruptly, their transient dispersion coefficients to oscillate, and the asymptotic transverse dispersion coefficients to plateau out of initial power law behavior. The microscopically disordered beds, by contrast, follow power law behavior over the whole investigated velocity range, for which we present refined equations (i.e., Eq.(13) and the data in Table 2 for the polydisperse sphere packings; Eq.(17) for the silica monolith). The bulk bed morphologies and their intrinsic differences addressed in this work determine how efficient a bed can relax the transverse concentration gradients caused by wall effects, which exist in all confined separation media used in chromatographic practice. Whereas the effect of diffusion on transverse dispersion decreases and ultimately disappears at increasing velocity with the microscopically disordered chromatographic beds, it dominates in the pillar arrays. The pillar arrays therefore become the least forgiving bed morphology with macroscopic heterogeneities and the engendered longitudinal dispersion in chromatographic practice. Wall effects in pillar arrays and the monolith caused by their confinement impact band broadening, which is traditionally observed on a macroscopic scale, more seriously than in the packings.

From random sphere packings to regular pillar arrays: effect of the macroscopic confinement on hydrodynamic dispersion.

Flow and mass transport in bulk and confined chromatographic supports comprising random packings of solid, spherical particles and hexagonal arrays of solid cylinders (regular pillar arrays) are studied over a wide flow velocity range by a numerical analysis scheme, which includes packing generation by a modified Jodrey-Tory algorithm, three-dimensional flow field calculations by the lattice-Boltzmann method, and modeling of advective-diffusive mass transport by a random-walk particle-tracking technique. We demonstrate the impact of the confinement and its cross-sectional geometry (circular, quadratic, semicircular) on transient and asymptotic transverse and longitudinal dispersion in random sphere packings, and also address the influence of protocol-dependent packing disorder and the particle-aspect ratio. Plate height curves are analyzed with the Giddings equation to quantify the transcolumn contribution to eddy dispersion. Confined packings are compared with confined arrays under the condition of identical bed porosity, conduit cross-sectional area, and laterally fully equilibrated geometrical wall and corner effects on dispersion. Fluid dispersion in a regular pillar array is stronger affected by the macroscopic confinement and does not resemble eddy dispersion in random sphere packings, because the regular microstructure cannot function as a mechanical mixer like the random morphology. Giddings' coupling theory fails to preserve the nature of transverse dispersion behind the arrays' plate height curves, which approach a linear velocity-dependence as transverse dispersion becomes velocity-independent. Upon confinement this pseudo-diffusive behavior can outweigh the performance advantage of the regular over the random morphology.

Structure-transport correlation for the diffusive tortuosity of bulk, monodisperse, random sphere packings.

The mass transport properties of bulk random sphere packings depend primarily on the bed (external) porosity ε, but also on the packing microstructure. We investigate the influence of the packing microstructure on the diffusive tortuosity τ=D(m)/D(eff), which relates the bulk diffusion coefficient (D(m)) to the effective (asymptotic) diffusion coefficient in a porous medium (D(eff)), by numerical simulations of diffusion in a set of computer-generated, monodisperse, hard-sphere packings. Variation of packing generation algorithm and protocol yielded four Jodrey-Tory and two Monte Carlo packing types with systematically varied degrees of microstructural heterogeneity in the range between the random-close and the random-loose packing limit (ε=0.366-0.46). The distinctive tortuosity-porosity scaling of the packing types is influenced by the extent to which the structural environment of individual pores varies in a packing, and to quantify this influence we propose a measure based on Delaunay tessellation. We demonstrate that the ratio of the minimum to the maximum void face area of a Delaunay tetrahedron around a pore between four adjacent spheres, (A(min)/A(max))(D), is a measure for the structural heterogeneity in the direct environment of this pore, and that the standard deviation σ of the (A(min)/A(max))(D)-distribution considering all pores in a packing mimics the tortuosity-porosity scaling of the generated packing types. Thus, σ(A(min)/A(max))(D) provides a structure-transport correlation for diffusion in bulk, monodisperse, random sphere packings.

Transient and asymptotic dispersion in confined sphere packings with cylindrical and non-cylindrical conduit geometries.

We study the time and length scales of hydrodynamic dispersion in confined monodisperse sphere packings as a function of the conduit geometry. By a modified Jodrey-Tory algorithm, we generated packings at a bed porosity (interstitial void fraction) of ε=0.40 in conduits with circular, rectangular, or semicircular cross section of area 100πd(p)(2) (where d(p) is the sphere diameter) and dimensions of about 20d(p) (cylinder diameter) by 6553.6d(p) (length), 25d(p) by 12.5d(p) (rectangle sides) by 8192d(p) or 14.1d(p) (radius of semicircle) by 8192d(p), respectively. The fluid-flow velocity field in the generated packings was calculated by the lattice Boltzmann method for Péclet numbers of up to 500, and convective-diffusive mass transport of 4×10(6) inert tracers was modelled with a random-walk particle-tracking technique. We present lateral porosity and velocity distributions for all packings and monitor the time evolution of longitudinal dispersion up to the asymptotic (long-time) limit. The characteristic length scales for asymptotic behaviour are explained from the symmetry of each conduit's velocity field. Finally, we quantify the influence of the confinement and of a specific conduit geometry on the velocity dependence of the asymptotic dispersion coefficients.

Influence of the particle size distribution on hydraulic permeability and eddy dispersion in bulk packings.

The narrow particle size distribution (PSD) of certain packing materials has been linked to a reduced eddy dispersion contribution to band broadening in chromatographic columns. It is unclear if the influence of the PSD acts mostly on the stage of the packing process or if a narrow PSD provides an additional, intrinsic advantage to the column performance. To investigate the latter proposition, we created narrow-PSD and wide-PSD random packings based on the experimental PSDs of sub-3 µm core-shell and sub-2 µm fully porous particles, respectively, as determined by scanning electron microscopy. Unconfined packings were computer-generated with a fixed packing protocol at bed porosities from random-close to random-loose packing to simulate fluid flow and advective-diffusive mass transport in the packings' interparticle void space. The comparison of wide-PSD, narrow-PSD, and monodisperse packings revealed no systematic differences in hydraulic permeability and only small differences in hydrodynamic dispersion, which originate from a slightly increased short-range interchannel contribution to eddy dispersion in wide-PSD packings. The demonstrated intrinsic influence of the PSD on dispersion in bulk packings is negligible compared with the influence of the bed porosity. Thus, the reduced eddy dispersion observed for experimental core-shell packings cannot be attributed to a narrow PSD per se.

Statistical analysis of packed beds, the origin of short-range disorder, and its impact on eddy dispersion.

We quantified the microstructural disorder of packed beds and correlated it with the resulting eddy dispersion. For this purpose we designed a set of bulk (unconfined) monodisperse random sphere packings with a systematic, protocol-dependent degree of microstructural heterogeneity, covering a porosity range from the random-close to the random-loose packing limit (ε = 0.366-0.46). With the precise knowledge of particle positions, size, and shape we conducted a Voronoiï tessellation of all packings and correlated the statistical moments of the Voronoiï volume distributions (standard deviation and skewness) with the porosity and the protocol-dependent microstructural disorder. The deviation of the Voronoiï volume distributions from the delta function of a crystalline packing describes the origin of short-range disorder of the investigated random packings. Eddy dispersion was simulated over a wide range of reduced velocities (0.5 ≤ ν ≤ 750) and analyzed with the comprehensive Giddings equation. Transient dispersion was found to correlate with the spatial scales of heterogeneity in the packings. The analysis of short-range disorder based on the Voronoiï volume distributions revealed a strong correlation with the short-range interchannel contribution to eddy dispersion, whereas transchannel dispersion was relatively little affected. The presented approach defines a strictly scientific route to the key morphology-transport relationships of current and future chromatographic supports, including their morphological reconstruction, statistical analysis, and the correlation with relevant transport phenomena. It also guides us in our understanding, comparison, and optimization of the diverse packing algorithms and protocols used in simulations and experimental studies.

Large-scale simulation of flow and transport in reconstructed HPLC-microchip packings.

Flow and transport in a particle-packed microchip separation channel were investigated with quantitative numerical analysis methods, comprising the generation of confined, polydisperse sphere packings by a modified Jodrey-Tory algorithm, 3D velocity field calculations by the lattice-Boltzmann method, and modeling of convective-diffusive mass transport with a random-walk particle-tracking approach. For the simulations, the exact conduit cross section, the particle-size distribution of the packing material, and the respective average interparticle porosity (packing density) of the HPLC-microchip packings was reconstructed. Large-scale simulation of flow and transport at Peclet numbers of up to Pe = 140 in the reconstructed microchip packings (containing more than 3 x 10(5) spheres) was facilitated by the efficient use of supercomputer power. Porosity distributions and fluid flow velocity profiles for the reconstructed microchip packings are presented and analyzed. Aberrations from regular geometrical conduit shape are shown to influence packing structure and, thus, porosity and velocity distributions. Simulated axial dispersion coefficients are discussed with respect to their dependence on flow velocity and bed porosity. It is shown by comparison to experimental separation efficiencies that the simulated data genuinely reflect the general dispersion behavior of the real-life HPLC-microchip packings. Differences between experiment and simulation are explained by differing morphologies of real and simulated packings (intraparticle porosity, packing structure in the corner regions).

Time and length scales of eddy dispersion in chromatographic beds.

Time and length scales as well as the magnitude of individual contributions to eddy dispersion in chromatographic beds are resolved. We address this issue by a high-resolution numerical analysis of flow and mass transport in computer-generated bulk (unconfined) packings of monosized, nonporous, incompressible, spherical particles and complementary confined cylindrical packings with a cylinder-to-particle diameter ratio of d(c)/d(p) = 20. The transient behavior of longitudinal and transverse dispersion is analyzed and correlated with the spatial scales of heterogeneity in the bulk and confined packings. Simulations were carried out until complete transcolumn equilibration in the confined packings was achieved to facilitate a quantitative study of the geometrical wall effect. Longitudinal plate height data calculated over a wide range of reduced velocities (0.1 < or = nu < or = 500) were fitted to the comprehensive Giddings equation. The determined transition velocities for individual contributions to eddy dispersion were found to be widely disparate. As a consequence, the total effect of eddy dispersion on the plate height curves can be approximated in the practical range of chromatographic operational velocities (5 < or = nu < or = 20) by a composite expression in which only the short-range interchannel contribution retains its coupling characteristics, while transchannel and transcolumn contributions appear as simple mass transfer velocity-proportional terms.

Structure-transport analysis for particulate packings in trapezoidal microchip separation channels.

This article investigates the efficiency of particulate beds confined in quadrilateral microchannels by analyzing the three-dimensional fluid flow velocity field and accompanying hydrodynamic dispersion with quantitative numerical simulation methods. Random-close packings of uniform, solid (impermeable), spherical particles of diameter d(p) were generated by a modified Jodrey-Tory algorithm in eighteen different conduits with quadratic, rectangular, or trapezoidal cross-section at an average bed porosity (interparticle void fraction) of epsilon = 0.48. Velocity fields were calculated by the lattice Boltzmann method, and axial hydrodynamic dispersion of an inert tracer was simulated at Péclet numbers Pe = u(av)d(p)/D(m) (where u(av) is the average fluid flow velocity through a packing and D(m) the bulk molecular diffusion coefficient) from Pe = 5 to Pe = 30 by a Lagrangian particle-tracking method. All conduits had a cross-sectional area of 100d(p)(2) and a length of 1200d(p), translating to around 10(5) particles per packing. We present lateral porosity distribution functions and analyze fluid flow profiles and velocity distribution functions with respect to the base angle and the aspect ratio of the lateral dimensions of the different conduits. We demonstrate significant differences between the top and bottom parts of trapezoidal packings in their lateral porosity and velocity distribution functions, and show that these differences increase with decreasing base angle and increasing base-aspect ratio of a trapezoidal conduit, i.e., with increasing deviation from regular rectangular geometry. Efficiencies are investigated in terms of the axial hydrodynamic dispersion coefficients as a function of the base angle and base-aspect ratio of the conduits. The presented data support the conclusion that the efficiency of particulate beds in trapezoidal microchannels strongly depends on the lateral dimensions of the conduit and that cross-sectional designs based on large side-aspect-ratio rectangles with limited deviations from orthogonality are favorable.

Impact of conduit geometry and bed porosity on flow and dispersion in noncylindrical sphere packings.

The three-dimensional velocity field and corresponding hydrodynamic dispersion in pressure-driven flow through fixed beds of solid (impermeable), uniformly sized, spherical particles are studied by quantitative numerical analysis for conduits with different cross-sectional geometries. Packings with average interparticle porosities (bed porosities) of 0.40 < or = epsilon < or = 0.50 were generated in conduits with circular, quadratic, rectangular, and semicircular cross sections utilizing a parallel collective-rearrangement algorithm. The lateral dimensions of the generated packings were chosen to represent typical values encountered in miniaturized liquid chromatography (LC) systems. The interparticle velocity field was calculated using the lattice-Boltzmann (LB) method, and a random-walk particle-tracking method was employed to model advective-diffusive transport of an inert tracer in the LB velocity field. We present the morphologies and corresponding flow patterns for these packings and demonstrate that the porosity distribution and velocity profiles of noncylindrical packings deviate significantly from those of conventional cylindrical packings. This deviation becomes more pronounced at higher bed porosities. Extended regions of high local porosity in the corners of noncylindrical conduits give rise to the formation of fluid channels of advanced flow velocity. The differences in the flow velocity distributions of cylindrical and noncylindrical packings are analyzed, and their impact on the axial hydrodynamic dispersion coefficient is shown. The presented data support the conclusion that LC performance depends critically on the conduit geometry and bed porosity. Our results have particular relevance for microchip-LC, where noncylindrical conduit geometries are prevalent.

Pore-scale dispersion in electrokinetic flow through a random sphere packing.

The three-dimensional velocity field and corresponding hydrodynamic dispersion in electrokinetic flow through a random bulk packing of impermeable, nonconducting spheres are studied by quantitative numerical analysis. First, a fixed bed with interparticle porosity of 0.38 is generated using a parallel collective-rearrangement algorithm. Then, the interparticle velocity field is calculated using the lattice-Boltzmann (LB) method, and a random-walk particle-tracking method is finally employed to model advection-diffusion of an inert tracer in the LB velocity field. We demonstrate that the pore-scale velocity profile for electroosmotic flow (EOF) is nonuniform even under most ideal conditions, including a negligible thickness of the electrical double layer compared to the mean pore size, a uniform distribution of the electrokinetic potential at the solid-liquid interface, and the absence of applied pressure gradients. This EOF dynamics is caused by a nonuniform distribution of the local electrical field strength in the sphere packing and engenders significant hydrodynamic dispersion compared to pluglike EOF through a single straight channel. Both transient and asymptotic dispersion behaviors are analyzed for EOF in the context of packing microstructure and are compared to pressure-driven flow in dependence of the average velocity through the bed. A better hydrodynamic performance of EOF originates in a still much smaller amplitude of velocity fluctuations on a mesoscopic scale (covering several particle diameters), as well as on the microscopic scale of an individual pore.