Siarhei Khirevich

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.

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.

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.

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.

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.

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).

Impact of microstructure on the effective diffusivity in random packings of hard spheres

We present results of computer simulations of the effective diffusion coefficient in bulk random packings of hard monosized spheres with solid volume fraction between 0.54 (random-loose packing) and 0.634 (maximally random jammed). Six types of sphere packings were generated with different protocols and parameters resulting in a systematically varied degree of microstructural heterogeneity. The packing morphology is qualitatively characterized by statistical analyses of Voronoi cells obtained from spatial tessellation of the packing space. Diffusive transport of point-like tracers in the pore space of the packings was simulated with a random-walking particle-tracking technique. Our results indicate that the effective transport characteristics of the random sphere packings are not fully defined from the solid volume fraction but also depend on the packing microstructure. For the first time, we compared (i) the values of the effective diffusion coefficient Deff simulated in packings with different morpholog...

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.

Pore-size entropy of random hard-sphere packings

We introduce a method for calculating the entropy of random hard-sphere packings, also referred to as pore-size entropy. The method is applicable to packings of monodisperse or polydisperse spheres as well as non-spherical particles. Pore-size entropy allows us to analyze the packing microstructure and provides deep insight into the traditional concept of pore-size distribution. Specifically, the logarithm of the pore-size distributions tail area is equal to the packing entropy. We reveal a local minimum in the plot of pore-size entropy vs. packing density (φ) for monodisperse frictionless sphere packings at a critical density of φC ≈ 0.65, independent of the employed packing generation protocol (Lubachevsky–Stillinger, Jodrey–Tory, and force-biased algorithms), which is a density with minimal number of available packing configurations. This entropy minimum is followed by an entropy increase as φ increases up to ∼0.68, corresponding to the emergence of crystalline structures in the coexistence region; beyond this packing density the entropy decreases again. In a complementary study we modify the Lubachevsky–Stillinger protocol and reproduce the random-close packing limit at φRCP ≈ 0.64. We conclude that φRCP ≈ 0.64 is the jamming point of the glassy states with the lowest density, whereas φC ≈ 0.65 is the jamming point of the densest glassy state (the ideal glass state).

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 100πd2p (where dp is the sphere diameter) and dimensions of about 20dp (cylinder diameter) by 6553.6dp (length), 25dp by 12.5dp (rectangle sides) by 8192dp or 14.1dp (radius of semicircle) by 8192dp, 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 4×106 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.