Numerical analysis of capillarity in packed spheres: planar hexagonal-packed spheres

General solutions of the capillary pressure for liquids as a function of contact angle and volume in planar close-packed spheres have been calculated numerically using Surface Evolver software. Applied pressure differences between liquid and vapor result in undulating (puckered) menisci exhibiting anticlastic curvature in the narrower spaces near particle contacts. The corresponding capillary pressures exhibit maxima with infiltration volume (minima with drainage), corresponding to critical pressures for engulfment of the spheres by the liquid (vapor). The analysis also reveals the formation of residual pendular rings of the wetting phase around particle contacts. Pendular ring formation is explored further by analyzing hexagonally packed spheres separated by 1/10 their radius. The results are discussed relative to the wide range of approximate solutions available in the literature.     

Nodoids and toroids: comparison of two geometries for the meniscus profile of a wetting liquid between two touching isolated spheres and extensions to the model of a collection of packed spheres

In the mid 1960s the present authors published two papers dealing with penetration of nonwetting liquids such as mercury into the interstitial void spaces using the model of uniform packed spheres. A circular arc was used to approximate the liquid-vapor interface in both papers. However, our circular arc-toroid values for the pressure-volume relationship in the pendular ring which exists between two touching spheres was criticized. The authors concluded that our approximation led to unacceptably large differences compared to the values calculated from the exact nodoid shape. This incorrect conclusion was never rebutted and has, in fact, been misinterpreted by subsequent workers to include values calculated for the shape of the access opening and the associated pressure for penetration into the void space of a collection of spheres. This leaves a cloud of uncertainty, not only over our original work on nonwetting fluids, but on the application of our procedures to the field of wetting fluids. The contrast in the geometrical shapes of the toroid and nodoid is depicted and the pressure values are compared at equal volumes. In contrast to the claim of excessive error, we show the toroid geometry, in conjunction with a pressure-volume work derivation, to have a maximum error of 0.06% as compared to the nodoid at all liquid-solid contact angles. The toroid also has the advantage of using a readily derived work versus surface free energy balance rather than requiring the use of incomplete elliptic integrals to evaluate the nodoid. Attempts to use radii of curvature to evaluate the toroid shape are shown to give extremely poor approximations of the exact values for the pressure. Values reported for access to the interior void space of a collection of spheres still need adjustment for the effect of contact angles between 0 degrees and 180 degrees for characterizing assemblies of real solids by computing "equivalent spherical" particle size from porosity and mercury penetration data. However, there is no anticipation that use of the circular arc will introduce large errors in the results. This gives confidence to us and others working with wetting media to test the potential applicability of the packed sphere model to various diverse fields.     

The geometry of primary drainage

We show that arc menisci configuration under primary drainage in capillary tube cross sections and, by extension, in throats in the void structure of rock and soil, can be understood in terms of the computational geometry theory of medial axis analysis. The solution for arc meniscus configuration is developed for cross sections of arbitrary, simply connected polygonal shape at both entry- and over-pressure values during primary drainage for arbitrary values of wetting angle. Using this solution technique, we have obtained highly accurate solutions of entry pressure arc meniscus radius for over 21,500 throats obtained from analysis of computed tomography images in a suite of 4 Fontainebleau core samples ranging from 7.5 to 22% porosity. We show that the ratio A/P, of throat area to throat perimeter, is an excellent predictor of entry pressure meniscus radius (and hence entry pressure) for primary drainage for real pores, while inscribed radius and area equivalent radius over-predict entry pressure meniscus radius by factors of 1.5-3, and are consequently poor predictors.     

Total pore blocking as an alternative method for the on-column determination of the external porosity of packed and monolithic reversed-phase columns

An alternative method to determine the interstitial void volume and the external porosity inside a packed or a monolithic column was developed. The method is based on the total blocking of the mesopores of a porous support by filling them with a hydrophobic solvent. The strong interaction of the latter with the hydrophobic coating inside the pores keeps the solvent in position during the subsequent measurements. With the pores of the stationary phase material completely inaccessible for any type of polar molecules, the method allows to perform interstitial void measurements using small molecular weight (MW) molecules instead of the large MW molecules that need to be used in inverse size exclusion chromatography (ISEC). These small MW molecules are able to penetrate every corner of the interstitial volume and therefore lead to a very accurate determination of the external porosity. Since only one type of molecules needs to be injected, the often troublesome regression analysis needed in ISEC is omitted as well. In the present contribution, the method has been applied to a packed bed and a monolithic column to investigate the optimal conditions of flow velocity, liquid compositions, and unretained marker selection. The robustness and the repeatability of the method are discussed as well.     

Pressure drop in CIM disk monolithic columns

Pressure drop analysis in commercial CIM disk monolithic columns is presented. Experimental measurements of pressure drop are compared to hydrodynamic models usually employed for prediction of pressure drop in packed beds, e.g. free surface model and capillary model applying hydraulic radius concept. However, the comparison between pressure drop in monolith and adequate packed bed give unexpected results. Pressure drop in a CIM disk monolithic column is approximately 50% lower than in an adequate packed bed of spheres having the same hydraulic radius as CIM disk monolith; meaning they both have the same porosity and the same specific surface area. This phenomenon seems to be a consequence of the monolithic porous structure which is quite different in terms of the pore size distribution and parallel pore nonuniformity compared to the one in conventional packed beds. The number of self-similar levels for the CIM monoliths was estimated to be between 1.03 and 2.75.     

Hydrodynamic aspects of slurry packing processes in microcolumn liquid chromatography

A Stokesian dynamics computer simulation based method is presented for the estimation of the bed porosity of slurry-packed capillary liquid chromatography (LC) columns. A colloidally well-described reversed-phase stationary phase-slurry liquid suspension was used as a model system. The applied simulation method takes into account the velocity of the slurry and colloidal interaction forces, as well as inter-particle hydrodynamic interactions. The predicted bed porosities suggest that a lower slurry velocity leads to a denser packing structure due to the increased effect of colloidal repulsion effects. The results of the simulations were compared with the external porosity and chromatographic performance of capillary LC columns that were packed at different filtration and compaction pressures. However, the trends that were observed in the experimental results suggest that hydrodynamic packing parameters have no or little effect on the chromatographic performance of capillary LC columns. Within the experimental parameter window, the chromatographic performance and the column porosity were not influenced by the filtration and compaction pressure, nor by the duration of the compaction process.     

Wetting phase connectivity and irreducible saturation in simple granular media

We present drainage simulations that allow trapping of wetting phase in a simple but nontrivial granular medium, a dense random packing of equal spheres. The basis for the simulations is a network model derived directly from the known locations and dimensions of pore space features. This provides a means of evaluating the morphology of trapped wetting phase. The possible morphologies depend on the assumed connectivity of the wetting phase. At one extreme, we assume that the entire wetting phase except for pendular rings is connected. At the opposite extreme, we illustrate a low level of connectivity by assuming that pendular rings are trapped as soon as the pores surrounding them are drained; any wetting phase not yet drained from pore throats connecting these pores is also assumed to be trapped. Finally we consider a set of criteria involving larger neighborhoods within the network, which allows trapping in individual pores. Irreducible wetting phase saturations obtained in the latter case agree with experimental data. The numbers of pendular rings and liquid bridges are also consistent with observations. Because the agreement does not involve adjustable parameters, we conclude that a relatively simple, local evaluation of trapping criteria can yield physically representative wetting phase configurations.     

Capillary penetration failure of blood suspensions

Blood suspension fails to penetrate a capillary with radius R less than 50 microm even if the capillary is perfectly wettable. This invasion threshold is attributed to three red blood cells (RBCs) segregation mechanisms--corner deflection at the entrance, the intermediate deformation-induced radial migration and shear-induced diffusion within a packed slug at the meniscus. The shear-induced radial migration for deformable particles endows the blood cells with a higher velocity than the meniscus to form the concentrated slug behind the meniscus. This tightly packed slug has a higher resistance and arrests the flow. Rigid particles and rigidified blood cells result in wetting behavior similar to that seen for homogeneous liquids, with decreased RBC migration towards the capillary centerline and reduction of packing. Corner deflection with a radial drift velocity accelerates the radial migration for small capillaries. However, deformation-induced radial migration is the key mechanism responsible for penetration failure. This sequence of mechanisms is confirmed through videomicroscopy and scaling theories were applied to capture the dependence of the critical capillary radius as a function of RBC concentrations.     

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.     

Axial development and radial non-uniformity of flow in packed columns

Flow inhomogeneity and axial development in low-pressure chromatographic columns have been studied by magnetic resonance imaging velocimetry. The columns studied included (a) an 11.7-mm I.D. column packed with either 50 microm diameter porous polyacrylamide, or 99 or 780 microm diameter impermeable polystyrene beads, and (b) a 5-mm I.D. column commercially packed with 10 microm polymeric beads. The packing methods included gravity settling, slurry packing, ultrasonication, and dry packing with vibration. The magnetic resonance method used averaged apparent fluid velocity over both column cross-sections and fluid displacements greater than one particle diameter and hence permits assessment of macroscopic flow non-uniformities. The results confirm that now non-uniformities induced by the conical distributor of the 11.7-mm I.D. column or the presence of voids at the column entrance relax on a length scale of the column radius. All of the 11.7-mm I.D. columns examined exhibit near wall channeling within a few particle diameters of the wall. The origins of this behavior are demonstrated by imaging of the radial dependence of the local porosity for a column packed with 780 microm beads. Columns packed with the 99-microm beads exhibit reduced flow in a region extending from ten to three-to-five particle diameters from the wall. This velocity reduction is consistent with a reduced porosity of 0.35 in this region as compared to approximately 0.43 in the bulk of the column. Ultrasonicated and dry-packed columns exhibit enhanced flow in a region located between approximately eight and 20 particle diameters from the wall. This enhancement maybe caused by packing density inhomogeneity and/or particle size segregation caused by vibration during the packing process. No significant non-uniformities on length scales of 20 microm or greater were observed in the commercially packed column packed with 10 microm particles.     

Ultrahigh-pressure liquid chromatography using a 1-mm id column packed with 1.5-microm porous particles

The evolution of chromatography has led to the reduction in the size of the packing materials used to fabricate HPLC columns. The increase in the backpressure required has led to this technique being referred to as ultrahigh-pressure liquid chromatography (UHPLC) when the column backpressure exceeds 10000 psi (approximately 700 bar). Until recently, columns packed with sub-2-microm materials have generally fitted into two classes; either short (less than 5 cm) columns designed for use on traditional HPLC systems at pressures less than 5000 psi (350 bar), or capillary columns (inner diameters less than 100 microm). By using packing materials with diameters <2 microm to fabricate UHPLC columns, there is an increase in efficiency and a decrease in the analysis time that are directly proportional to the size of the packing material. In order to realize and exploit the increase in efficiency, however, the columns must maintain lengths typically associated with analytical columns (15-25 cm). We have packed 1 mm diameter, 150 mm in length columns with 1.5 microm packing material, and evaluated their performance in UHPLC. The pressure required to achieve optimum linear velocities in plots of plate height versus linear velocity was in the vicinity of 1104 bar (16000 psi). The 1.5 microm particle-packed column was compared with the more traditional 150 mm long analytical columns packed with 3 microm materials. This column showed an efficiency that was approximately twice that observed with the 3 microm packed column and a concomitant reduction in the analysis time, theoretically predicted.     

Understanding and design of existing and future chromatographic support formats

The present contribution reviews the use of alternative support formats as a means to surpass the chromatographic performance of the packed bed of spheres. First, a number of idealized structures are considered to obtain a general insight in how the performance of a chromatographic support depends on its shape and size, using the isocratic peak-capacity generation speed as the main performance indicator. Using this criterion, it is found that the packing density or, equivalently, the external porosity, is the most important of all geometrical shape factors. Depending on whether the sample consists of weakly or strongly retained components, the optimal external porosity can be expected to vary between 60% and a value near 100%. The optimal exploitation of a high external porosity, however, also requires overall shrinkage of the domain size, towards and into the sub-micron range. With the current fabrication technologies, this requirement seems difficult to achieve. In the presence of a lower limit on the characteristic support size, each range of desired plate numbers or peak capacities has its own optimal external porosity, ranging from a very low value (high packing density) for high speed, small peak capacity applications, to very high external porosities (low packing density) for applications requiring a very large peak capacity. Subsequently, the obtained theoretical insights are used to review and discuss the past and current research on alternative support formats. Finally, a number of emerging micro- and nano-fabrication technologies are introduced and their potential for the future production of supports with improved shape and homogeneity is discussed.     

Extending the Total Pore Blocking method to normal phase high performance liquid chromatography

It has been demonstrated that it is possible to extend the Total Pore Blocking (TBP)-method for the determination of the external porosity (ε(e)) of packed bed columns from reversed-phase (RP) chromatography to normal-phase (NP) chromatography conditions by switching the nature of the blocking agent and the interstitial void flushing liquid, i.e., by using a hydrophilic blocking agent (pure or buffered water at pH 3.0 or pH 7.0) and a hydrophobic flushing liquid (linear alkanes such as decane). Several parameters that might influence the accuracy of the method, such as the applicable range of flow rates and the meso-pore size of the particles have been investigated. The influence of several different parameters on the obtained external porosity value has been investigated. From a wide selection of possible t(0)-markers, the class of linear alkanes has been shown to be the single possible one. This brings along the need to use refractive index detection to measure the signal of the linear alkane tracer (e.g., dodecane) in a stream of (another) linear alkane. The results of the newly established NP-TPB method have been compared to the values of the external porosity obtained by ISEC and proved to generate the same results, however with a much smaller read-out error (being only of the order of 0.1%).     

Effect of extra-column volume on practical chromatographic parameters of sub-2-μm particle-packed columns in ultra-high pressure liquid chromatography

Effects of extra-column volume on apparent separation parameters were studied in ultra-high pressure liquid chromatography with columns and inlet connection tubings of various internal diameters (id) using 50-mm long columns packed with 1.8-μm particles under isocratic conditions. The results showed that apparent retention factors were on average 5, 11, 18, and 41% lower than those corrected with extra-column volumes for 4.6-, 3.0-, 2.1-, and 1.0-mm id columns, respectively, when the extra-column volume (11.3 μL) was kept constant. Also, apparent pressures were 31, 16, 12, and 10% higher than those corrected with pressures from extra-column volumes for 4.6-, 3.0-, 2.1-, and 1.0-mm id columns at the respective optimum flow rate for a typical ultra-high pressure liquid chromatography system. The loss in apparent efficiency increased dramatically from 4.6- to 3.0- to 2.1- to 1.0-mm id columns, less significantly as retention factors increased. The column efficiency was significantly improved as the inlet tubing id was decreased for a given column. The results suggest that maximum ratio of extra-column volume to column void volume should be approximately 1:10 for column porosity more than 0.6 and a retention factor more than 5, where 80% or higher of theoretically predicted efficiency could be achieved.     

Kinetic plot equations for evaluating the real performance of the combined use of high temperature and ultra-high pressure in liquid chromatography. Application to commercial instruments and 2.1 and 1 mm I.D. columns

The use of ultra-high pressure liquid chromatography (UHPLC) with pressures up to 1000bar and columns packed with sub-2-mum particles combined with high-temperature mobile phase conditions (up to 90 degrees C) is assessed according to the current available instrumentation via constrained kinetic plot equations. It is shown that the gain in separation speed, theoretically expected from high-temperature UHPLC (HT-UHPLC), is significantly reduced when taking into account the existing instrumental constraints (extra-column band broadening, flow-rate and column length limitations). This study also shows that significant improvements could be expected on the current commercial instruments by increasing the flow-rate limit and/or using packing columns with particle size in the range 2.5-3.5mum instead of the current sub-2mum. These particles should obviously withstand very high pressure.     

Simultaneous spreading and evaporation: Recent developments

The recent progress in theoretical and experimental studies of simultaneous spreading and evaporation of liquid droplets on solid substrates is discussed for pure liquids including nanodroplets, nanosuspensions of inorganic particles (nanofluids) and surfactant solutions. Evaporation of both complete wetting and partial wetting liquids into a nonsaturated vapour atmosphere are considered. However, the main attention is paid to the case of partial wetting when the hysteresis of static contact angle takes place. In the case of complete wetting the spreading/evaporation process proceeds in two stages. A theory was suggested for this case and a good agreement with available experimental data was achieved. In the case of partial wetting the spreading/evaporation of a sessile droplet of pure liquid goes through four subsequent stages: (i) the initial stage, spreading, is relatively short (1–2 min) and therefore evaporation can be neglected during this stage; during the initial stage the contact angle reaches the value of advancing contact angle and the radius of the droplet base reaches its maximum value, (ii) the first stage of evaporation is characterised by the constant value of the radius of the droplet base; the value of the contact angle during the first stage decreases from static advancing to static receding contact angle; (iii) during the second stage of evaporation the contact angle remains constant and equal to its receding value, while the radius of the droplet base decreases; and (iv) at the third stage of evaporation both the contact angle and the radius of the droplet base decrease until the drop completely disappears. It has been shown theoretically and confirmed experimentally that during the first and second stages of evaporation the volume of droplet to power 2/3 decreases linearly with time. The universal dependence of the contact angle during the first stage and of the radius of the droplet base during the second stage on the reduced time has been derived theoretically and confirmed experimentally. The theory developed for pure liquids is applicable also to nanofluids, where a good agreement with the available experimental data has been found. However, in the case of evaporation of surfactant solutions the process deviates from the theoretical predictions for pure liquids at concentration below critical wetting concentration and is in agreement with the theoretical predictions at concentrations above it.     

Fabrication of ZIF‐8@SiO2 Core–Shell Microspheres as the Stationary Phase for High‐Performance Liquid Chromatography

The unique features of high porosity, shape selectivity, and multiple active sites make metal–organic frameworks (MOFs) promising as novel stationary phases for high‐performance liquid chromatography (HPLC). However, the wide particle size distribution and irregular shape of conventional MOFs lead to lower column efficiency of such MOF‐packed columns. Herein, the fabrication of monodisperse MOF@SiO₂ core–shell microspheres as the stationary phase for HPLC to overcome the above‐mentioned problems is reported. Zeolitic imidazolate framework 8 (ZIF‐8) was used as an example of MOFs due to its permanent porosity, uniform pore size, and exceptional chemical stability. Unique carboxyl‐modified silica spheres were used as the support to grow the ZIF‐8 shell. The fabricated monodisperse ZIF‐8@SiO₂ packed columns (5 cm long × 4.6 mm i.d.) show high column efficiency (23 000 plates m^?1 for bisphenol A) for the HPLC separation of endocrine‐disrupting chemicals (bisphenol A, β‐estradiol, and p‐(tert‐octyl)phenol) and pesticides (thiamethoxam, hexaflumuron, chlorantraniliprole, and pymetrozine) within 7 min with good relative standard deviations for 11 replicate separations of the analytes (0.01–0.39, 0.65–1.7, 0.70–1.3, and 0.17–0.91 % for retention time, peak area, peak height, and half peak width, respectively). The ZIF‐8@SiO₂ microspheres combine the advantages of the good column packing properties of the uniform monodisperse silica microspheres and the separation ability of the ZIF‐8 crystals.     

填充条件对半制备柱性能影响的机理研究

对稀匀浆法填充半制备柱(Ф10 mm×100 mm)的过程进行研究,并用正交法对柱填充条件进行了优化,考察了压力、匀浆液组成、匀浆液体积对半制备柱填充效果的影响.由柱效和分离度定义式中的参数入手,讨论填充条件对柱性能的影响.通过计算总孔隙率ε_t、内部孔隙率ε_p、外部孔隙率ε_e、Van-Deemter曲线、吸附等温线等参数,发现填充压力对填充密度的作用最大,是最重要的影响因素;匀浆配比和匀浆体积共同影响其填充的均匀性.在压力11 MPa,流速275 mL/min,120 mL纯乙醇为匀浆液的条件下得到了最好的填充效果.     

Quantification of the Relationship between Porosity and Interparticle Forces for the Packing of Wet Uniform Spheres

This paper presents an attempt to quantify the relationship between porosity and interparticle forces. In particular, it concerns the packing of wet mono-sized spheres where capillary force is dominant. The interrelationships between capillary force, porosity, and liquid content are first analyzed based on well-established theories and experimental observations. The resultant data are then used to correlate porosity with the force ratio defined as the average capillary force to the gravity of a particle. Application of this correlation is finally demonstrated by examples of porosity prediction. Copyright 2000 Academic Press.     

Models of turbulent viscosity and mixing in channels and packed flow-through mixers

The use of packed flow-through mixers for intense mixing of liquid media and of liquids with a finely dispersed phase is considered. Ways to determine the mixing and turbulent exchange coefficients for evaluating the mixing efficiency in channels and in packed flow-through mixers are presented. The results are consistent with the experimental data on turbulent viscosity in pipes. An expression is obtained for calculating the turbulent viscosity coefficient in a packing bed. Comparative characteristics of mixers with Raschig rings and Inzhekhim packing are presented. The calculated values of the Peclet number for various packings in relation to the Reynolds number are given.