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 (? = 0.366–0.46). With the precise knowledge of particle positions, size, and shape we conducted a Voronoï tessellation of all packings and correlated the statistical moments of the Voronoï volume distributions (standard deviation and skewness) with the porosity and the protocol-dependent microstructural disorder. The deviation of the Voronoï 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 Voronoï 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.     

Numerical analysis of electroosmotic flow in dense regular and random arrays of impermeable, nonconducting spheres

We present a numerical scheme for analyzing steady-state isothermal electroosmotic flow (EOF) in three-dimensional random porous media, involving solution of the coupled Poisson, Nernst-Planck, and Navier-Stokes equations. While traditional finite-difference methods were used to resolve the Poisson-Nernst-Planck problem, the (electro)hydrodynamics has been addressed with high efficiency using the lattice-Boltzmann method. The developed model allows simulation of electrokinetic transport under most general conditions, including arbitrary value and distribution of electrokinetic potential at the solid-liquid interface, electrolyte composition, and pore space morphology. The approach provides quantitative information on a spatial distribution of simulated velocities. This feature was utilized to characterize EOF fields in regular and random, confined and bulk packings of hard (i.e., impermeable, nonconducting) spheres. Important aspects of pore space morphology (sphere size distribution), surface heterogeneity (mismatch in electrokinetic potentials at confining wall and sphere surface), and fluid phase properties (electrical double layer thickness) were investigated with respect to their influence on the EOF dynamics over microscopic and macroscopic spatial domains. Most important is the observation of a generally nonuniform pore-level EOF velocity profile in the sphere packings (even in the thin double layer limit) which is caused by pore space morphology and which is in contrast to the pluglike velocity distribution in a single, straight capillary under the same conditions.     

Geometrical cluster ensemble analysis of random sphere packings

We introduce a geometric analysis of random sphere packings based on the ensemble averaging of hard-sphere clusters generated via local rules including a nonoverlap constraint for hard spheres. Our cluster ensemble analysis matches well with computer simulations and experimental data on random hard-sphere packing with respect to volume fractions and radial distribution functions. To model loose as well as dense sphere packings various ensemble averages are investigated, obtained by varying the generation rules for clusters. Essential findings are a lower bound on volume fraction for random loose packing that is surprisingly close to the freezing volume fraction for hard spheres and, for random close packing, the observation of an unexpected split peak in the distribution of volume fractions for the local configurations. Our ensemble analysis highlights the importance of collective and global effects in random sphere packings by comparing clusters generated via local rules to random sphere packings and clusters that include collective effects.     

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.     

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.     

How does column packing microstructure affect column efficiency in liquid chromatography?

Full three-dimensional computer simulations of the fluid flow and dispersion characteristics of model nonporous chromatographic packings are reported. Interstitial porosity and packing defects are varied in an attempt to understand the chromatographic consequences of the packing microstructure. The tracer zone dispersion is calculated in the form of plate height as a function of fluid velocity for seven model particle packs where particles are selectively removed from the packs in clusters of varying size and topology. In an attempt to examine the consequences of loose but random packs, the velocities and zone dispersion of seven defect-free packs are simulated over the range 0.36< or =epsilon< or =0.50, where epsilon is the interstitial porosity. The results indicate that defect-free loose packings can give good chromatographic efficiency but the efficiency can vary depending on subtle details of the pack. When the defect population increases, the zone dispersion increases accordingly. For a particle pack where 6% of the particles are removed from an epsilon=0.36 pack, approximately 33% of the column efficiency is lost. These results show that it is far more important in column packing to prevent defect sites leading to inhomogeneous packing rather than obtaining the highest density pack with the smallest interstitial void volume.     

Statistical analysis of random sphere packings with variable radius distribution

Many solid state systems can be modelled by means of packings of hard spheres of variable size. Therefore the spatial-statistical analysis of the geometrical structure of such packings is of great scientific interest. The present paper starts with the application of classical characteristics such as packing fraction, coordination number and pair correlation function for the characterization of sphere packings. Then, the application of tessellation-based methods follows, which includes the analysis of correlations between cell face and coordination numbers. Finally, a search method for crystalline sub-structures is presented.     

Some properties of three-periodic sphere packings

Some properties of sphere packings are reviewed. Candidates for the least-dense packings of one kind of sphere for different coordination numbers are identified and described. Some topological properties of sphere packing nets are also described.     

On the caging number of two- and three-dimensional hard spheres

Local structural arrest in random packings of colloidal or granular spheres is quantified by a caging number, defined as the average minimum number of randomly placed spheres on a single sphere that immobilize all its translations. We present an analytic solution for the caging number for two-dimensional hard disks immobilized by neighbor disks which are placed at random positions under the constraint of a nonoverlap condition. Immobilization of a disk with radius r = 1 by arbitrary larger neighbor disks with radius r > or = 1 is solved analytically, whereas for contacting neighbors with radius 0 < r < 1, the caging number can be evaluated accurately with an approximate excluded volume model that also applies to spheres in higher Euclidean dimension. Comparison of our exact two-dimensional caging number with studies on random disk packing indicates that it relates to the average coordination number of random loose packing, whereas the parking number is more indicative for coordination in random dense packing of disks.     

Solvation forces versus the nano-colloidal structural forces under the film confinement: Layer to in-layer structural transition in wetting solids

This work reviews the origins, similarities, measurement techniques, and differences of the solvation and nano-colloidal oscillatory structural forces in confined domains. With an increasing confinement, the particles’ structural transition changes from 2D random layering to 2D crystalline packing, as observed experimentally and revealed by the radial distribution function. The 2D in-layer structural energy transition was estimated to be 1.8 kT using the Boltzmann normal distribution law. The transition from a 2D random structure away from the vertex to 2D cubic/hexagonal domains near the vortex was discontinuous and was confirmed by the particle density profile computed by an integral equation. The critical roles of the solvation and nano-colloidal oscillatory structural forces on the wetting and spreading of the simple liquids and complex liquids on solid surfaces are elucidated.     

浅谈质心分数坐标在确定等径圆球密堆积空隙中的应用

理解等径圆球密堆积的性质和特点是学习金属晶体结构和性质的基础,密堆积中的空隙问题对学习和理解离子晶体的结构和性质非常重要。但晶体结构的多样性和复杂性,导致这部分内容成为结构化学课程中讲解和学习的一个难点问题。本文将以A1,A2,A3这三种最常见的密堆积结构为例,详细介绍一种利用质心分数坐标计算推导密堆积结构中四面体和八面体空隙中心的方法,以及如何通过坐标计算求解中心到顶点的距离和中心到堆积球面的最短距离。与立体几何方法相比较,质心分数坐标法不仅更加简洁易学,而且更有助于理解空隙在晶胞中的位置和分布问题。     

Dispersion in retentive pillar array columns

The method of volume averaging is applied to estimate the Taylor–Aris dispersion tensor of solute advected in columns consisting of ordered pillar arrays with wall retention of the type used in chromatographic separation. The appropriate closure equations are derived and solved in a unit cell with periodic boundary conditions to obtain the dispersion tensor (or the reduced plate height) as a function of the Peclet number (reduced velocity); pillar pattern, shape and size; partition coefficient; and resistance to mass transfer. The contributions of the velocity profile, the wall adsorption, and the mass transfer resistance to the dispersion tensor are identified and delineated. The model is verified by comparing its predictions and obtaining favorable agreement with results of direct numerical simulations and with experimental data for columns containing ordered pillars. The model is then used to study the effect of pillars’ shape and pattern on the longitudinal dispersion coefficient (plate height).     

Colloidal crystallization and banding in a cylindrical geometry

Colloidal crystallization takes advantage of the strong interfacial forces and tunable interactions that organize particles into regular structures at small scales. Thus, colloidal crystallization and patterning provide a powerful and simple method to functionalize planar surfaces with applications to optical, catalytic, sensing, and cleansing materials. Nevertheless, the ability to pattern topologically more complex surfaces such as curved, confined, or soft substrates can open new avenues for novel, "intelligent", and responsive materials. We present one step in this direction by characterizing colloidal crystallization inside circular capillaries: a nearly periodic banding is observed, and the colloidal packing is dictated by confinement produced by the wedge-like region formed by a capillary confined meniscus. The packing consists of a succession of hexagonally close-packed regions, which are separated by narrow regions of "buckled phase crystals".     

Electrochemical reactivity of atomic and molecular species under solid-state confinement

The nanoconfinement of electrochemically-active guest species in host solid state electrode materials provides opportunities to tune mass transport between the bulk electrolyte and inner surface of the electrode, enhance electron-transfer rates, and/or improve the stability and dispersion of active material. This review summarizes recent experimental and theoretical electrochemical studies of three types of nanoconfined guest species: (1) ion adsorption of electrolyte ions, (2) confined redox-active molecules, and (3) electrocatalytic reactions of confined ions/solvents and catalytic particles. The examples discussed in this review illustrate how the confinement of guest species within enclosed spaces with nanoscale dimensions – such as pores, pockets, channels, and interlayers – can lead to improved electrochemical performance.     

Gapped gapless packing structures

The assumption of a gapless packing structure has previously been used to obtain the density and partial coordination numbers of a random mixture of hard spheres in the maximally dense regime. Here we extend the notion of a gapless packing structure to allow the determination of the characteristics of a packing away from maximal density by adding an appropriate number of void spherical elements. A gapless packing is then considered in which the void and solid spherical elements are assumed to be indistinguishable except for the purposes of calculating packing fraction and coordination number. We utilize the notion of specific volume to generate a one-parameter family of void distributions to obtain a set of coupled integral equations, which are solved numerically. Monodisperse and bi-disperse packings are investigated for packing fractions ranging from rho=0.26 to 0.78. Results are shown to be comparable to experiments and the effect of varying packing fraction on coordination numbers is shown to be invariant with respect to number distribution. A linear relationship between coordination number and packing fraction is elucidated for moderate to low packing fractions. Maximum and minimum random packing fractions are also discussed.     

Effects of confinement on the order-disorder transition of diblock copolymer melts

The effects of confinement on the order-disorder transition of diblock copolymer melts are studied theoretically. Confinements are realized by restricting diblock copolymers in finite spaces with different geometries (slabs, cylinders, and spheres). Within the random phase approximation, the correlation functions are calculated using the eigenvalues and eigenfunctions of the Laplacian operator inverted Delta(2) in the appropriate geometries. This leads to a size-dependent scattering function, and the minimum of the inverse scattering function determines the spinodal point of the homogeneous phase. For diblock copolymers confined in a slab or in a cylindrical nanopore, the spinodal point of the homogeneous phase (chiN)(s) is found to be independent of the confinement. On the other hand, for diblock copolymers confined in a spherical nanopore, (chiN)(s) depends on the confinement and it oscillates as a function of the radius of the sphere. Further understanding of the finite-size effects is provided by examining the fluctuation modes using the Landau-Brazovskii model.     

Chromatographic performance of monolithic and particulate stationary phases. Hydrodynamics and adsorption capacity

Monolithic chromatographic support structures offer, as compared to the conventional particulate materials, a unique combination of high bed permeability, optimized solute transport to and from the active surface sites and a high loading capacity by the introduction of hierarchical order in the interconnected pore network and the possibility to independently manipulate the contributing sets of pores. While basic principles governing flow resistance, axial dispersion and adsorption capacity are remaining identical, and a similarity to particulate systems can be well recognized on that basis, a direct comparison of sphere geometry with monolithic structures is less obvious due, not least, to the complex shape of theskeleton domain. We present here a simple, widely applicable, phenomenological approach for treating single-phase incompressible flow through structures having a continuous, rigid solid phase. It relies on the determination of equivalent particle (sphere) dimensions which characterize the corresponding behaviour in a particulate, i.e. discontinuous bed. Equivalence is then obtained by dimensionless scaling of macroscopic fluid dynamical behaviour, hydraulic permeability and hydrodynamic dispersion in both types of materials, without needing a direct geometrical translation of their constituent units. Differences in adsorption capacity between particulate and monolithic stationary phases show that the silica-based monoliths with a bimodal pore size distribution provide, due to the high total porosity of the material of more than 90%, comparable maximum loading capacities with respect to random-close packings of completely porous spheres.     

Three-dimensional confinement-related size changes to mixed-surfactant vesicles

The effect of three-dimensional confinement on the size and morphology of a vesicular surfactant mesophase obtained by mixing micellar solutions of cetyltrimethylammonium bromide and dodecylbenzenesulfonic acid has been studied using small-angle neutron scattering (SANS). The confined spaces were generated by the random close packing of polystyrene beads of radius Rb=1.5, 0.25, and 0.1 microm, creating voids of characteristic dimensions R approximately 0.22 Rb=3300, 550, and 220 A, respectively. These void length scales were comparable to or less than the radii of vesicles formed in the system under conditions of no confinement. Vesicles, made by mixing 0.8 wt % micellar solutions of surfactant in a water/D2O mixture that is contrast-matched with the polystyrene beads, were added in a SANS scattering cell without beads, as well as three cells with the different sized beads. The SANS data from the sample without confinement was best fitted by a core-shell model and not by spheres or disks, confirming the presence of vesicles. The data from samples in the confined domains also showed vesicles as the dominant structure. The most important result is that the mean size of these vesicles decreases as the confinement length scale is reduced. A simple thermodynamic model accounting for the balance between increased enthalpy when vesicles with curvature higher than the preferred one are formed, and increased free volume entropy for smaller vesicles supports the experimental data. While these results are focused on a specific vesicle system, the broad principles behind changes in microstructure produced by confinement are applicable to other surfactant aggregates. The results of this study are potentially important for understanding the flow of drug delivery vehicles through microcapillaries, in the recovery of oil from fine pores in rocks using surfactant containing fluids, micellar enhanced ultrafiltration, or in other situations where the size of surfactant aggregate structures approach the length scales between confining walls.