A new method to determine pore size and its volume distribution of porous solids having known atomistic configuration

A simple method, based on Monte Carlo integration, is presented to derive pore size and its volume distribution for porous solids having known configuration of solid atoms. Because pores do not have any particular shape, it is important that we define the pore size in an unambiguous manner and the volume associated with each pore size. The void volume that we adopt is the one that is accessible to the center of mass of the probe particle. We test this new method with porous solids having well defined pores such as graphitic slit pores and carbon nanotubes, and then apply it to obtain the pore volume distribution of complex solids such as disordered solids, rectangular pores, defected graphitic pores, metal organic framework and zeolite.     

Pore network modelling: determination of the dynamic profiles of the pore diffusivity and its effect on column performance as the loading of the solute in the adsorbed phase varies with time

A three-dimensional pore network model for diffusion in porous adsorbent particles was employed in a dynamic adsorption model that simulates the adsorption of a solute in porous particles packed in a chromatographic column. The solution of the combined model yielded the dynamic profiles of the pore diffusion coefficient of beta-galactosidase along the radius of porous ion-exchange particles and along the length of the column as the loading of the adsorbate molecules on the surface of the pores occurred, and, the dynamic adsorptive capacity of the chromatographic column as a function of the design and operational parameters of the chromatographic system. The pore size distribution of the porous adsorbent particles and the chemistry of the adsorption sites were unchanged in the simulations. It was found that for a given column length the dynamic profiles of the pore diffusion coefficient were influenced by: (i) the superficial fluid velocity in the column, (ii) the diameter of the adsorbent particles and (iii) the pore connectivity of the porous structure of the adsorbent particles. The effect of the magnitude of the pore connectivity on the dynamic profiles of the pore diffusion coefficient increased as the diameter of the adsorbent particles and the superficial fluid velocity in the column increased. The dynamic adsorptive capacity of the column increased as: (a) the particle diameter and the superficial fluid velocity in the column decreased, and (b) the column length and the pore connectivity increased. In preparative chromatography, it is desirable to obtain high throughputs within acceptable pressure gradients, and this may require the employment of larger diameter adsorbent particles. In such a case, longer column lengths satisfying acceptable pressure gradients with adsorbent particles having higher pore connectivity values could provide high dynamic adsorptive capacities. An alternative chromatographic system could be comprised of a long column packed with large particles which have fractal pores (fractal particles) that have high pore connectivities and which allow high intraparticle diffusional and convective flow mass transfer rates providing high throughputs and high dynamic adsorptive capacities. If large scale monoliths could be made to be reproducible and operationally stable, they could also offer an alternative mode of operation that could provide high throughputs and high dynamic adsorptive capacities.     

Evaluation of the porous structures of new polymer packing materials by inverse size-exclusion chromatography

A highly cross-linked porous polymer resin based on styrene-divinylbenzene matrix with pores created by the use of micellar imprinting technique was used as chromatographic packing material. Its performance as a column packing material in inverse size-exclusion chromatography was compared with a non-imprinted resin of the same polymer matrix. The porous structures (the pore size and the porosity) of the resins in the dry and wet states and their relationships with the elution volume of probe solutes (alkanes and polystyrene standards) were established. Characteristic properties of the resins such as specific pore volume, specific surface area and porosity are compared with results obtained by other methods of characterization such as mercury intrusion porosimetry, solvent regain and nitrogen sorption. The results show that the new porous resin can be used in the separation of small molecules. The separation is based on the size of the molecules, and the larger pores (meso- and macropores) in the porous resin can provide a much easier access to the smaller pores (micropores) which are useful in the chromatographic separations.     

Diffusion processes in mesoporous adsorbents probed by NMR

In this contribution, we review the results of our experimental studies on diffusion of guest molecules in mesoporous solids using pulsed field gradient (PFG) NMR technique. Having unique potentials to non-invasively probe the microscopic diffusion processes in pores, this method may provide quintessential information on the character of molecular propagation for different pore morphologies and fluid phase state. In particular, different modes of molecular diffusion in partially filled pores may be separately probed and the overall diffusion process could be analyzed taking account of the details of the inter-phase coexistence. In addition to the dynamic properties, some information concerning the distribution of guest molecules within the porous matrix may also be obtained.     

Estimation of diffusion parameters in functionalized silicas with modulated porosity. Part I: chromatographic studies

The diffusion parameters of binary gas mixture He (tracer gas)-N2 (carrier gas) in hybrid organic-inorganic SiO2-X porous solids which have suffered gradual functionalization with functional groups X of increasing length (X = psi, [triple bond]Si-H, [triple bond]Si-CH2OH, [triple bond]Si-(CH2)3OH, [triple bond]Si-(CH2)11CH3) are reported. The effective diffusivities Deff, the Henry law constants K as well as the tortuosity factors tau for the examined solids were estimated by a typical pulse gas chromatographic method. Analysis of the experimental results was carried out by the well-known method of linearization of moments. The moments s analysis provides a powerful means for extracting diffusion parameters from the experimental response curves The proposed methodology is simple compared to other similar studies and provides rapidly reliable data. The results of this work indicate that the effective diffusivity Deff in porous networks drops markedly as the initial porosity of the parent SiO2 sample is blocked by the functionalization of the pore surfaces with functional groups of increasing size, [triple bond]Si-H, [triple bond]Si-CH2OH, [triple bond]Si-(CH2)3OH and [triple bond]Si-(CH2)11CH3. The low values of the Henry law constants K found indicate that the adsorption of He on the porous surfaces for all the solids is weak. Also, the tortuosity factor r is proportionally correlated to the pore blocking effects and the percolation phenomena of gases flowing into the porous network.     

Inverse size exclusion chromatography and universal calibration

The mechanism of size-exclusion chromatography (SEC) is reciprocal in respect to the properties of the porous material and of the solutes immersed. The porosimetric interpretation (inverse SEC) yields information about the morphology of porous solids. It depends on a number of assumptions and in particular on the steric properties of the probing macromolecules whose ‘Universal Calibration’ remains disputed to this day. The central question concerns the equilibrium distribution of molecules in confined spaces of complex geometry, and is, as such, not limited to the particulars of chromatographic technique. This survey hopes to help focus future research activities in this field of analytical chemistry and material science.     

Pore structure characterization of organic-inorganic materials by inverse size exclusion chromatography

Summary In Inverse Size (or Steric) Exclusion Chromatography (ISEC) measurements, the investigated material is used as a stationary phase in a chromatographic column and the elution volumes of a series of standard solutes with different molecular size are measured. By an appropriate choice of mobile phase and type of standard solutes, specific (enthalpic) interactions between investigated material and solutes are eliminated so that the elution volumes depend on the porous structure of the column filling only. Then, a mathematical treatment of elution data can provide detailed information on both the macropores and the microporous structure of e.g. a swollen polymer gel in a polymer that is grafted onto silica. The basic principle of the evaluation of porosimetric information from chromatographic data is the assumption that the real porous structure of an investigated sample can be modelled as a collection of discrete pore fractions, each containing pores of different but uniform size and of simple geometrical shape. Problem then is to determine the combination of volumes of these fractions which yields the best agreement between computed and experimental values of the elution volumes of standard solutes. It is possible to perform the ISEC measurements either in an organic solvent or non-solvent (e.g. tetrahydrofuran or methanol) or in water, depending on the compatibility of the investigated material with the respective environment. For non-polar polymers, like copolymers of styrene and divinylbenzene, the use of tetrahydrofuran as the mobile phase with alkanes and polystyrenes as standard solutes has been recommended. Alternative ISEC investigation of the same material in an organic and an aqueous environment can provide additional information on its lipo- or hydrophilicity. This method has provided specific information, not obtainable by mercury porosimetry, when modifying silica by a coupling agent, polymerizing different monomers to different extents.     

Differential pressure-jump method for routine measurement of sorption kinetics

A sensitive pressure-jump method has been developed for the measurement and characterization of material transport in porous solids. The apparatus is automatically operated and the data evaluation is performed by means of numerical methods based on the Crank-Nicholson procedure. As an example, the sorption ofn-butane and ethene in spherical mesoporous silica gel pellets has been studied. In this way, it is possible to obtain uptake curves and in case of known particle shape to derive values for effective diffusion coefficients. From the pressure and temperature dependence ofD _e , information on the transport mechanism by means of Knudsen and surface diffusion can be obtained as well as the tortuosity factor characterizing the pore network.     

Estimation of diffusion parameters in functionalized silicas with modulated porosity. Part II: pore network modeling

In this work, the pore structure of those five (5) silicas SiO2-X (see Part I) which have suffered gradual functionalization with functional groups X of increasing length (X = psi, [triple bond]Si-H, [triple bond]Si-CH2OH, [triple bond]Si-(CH2)3OH, [triple bond]Si-(CH2)11CH3), is modeled as a three-dimensional cubic network of cylindrical pores. Those hybrids organic-inorganic SiO2-X samples are characterized by different extent of pore blocking effects. The pores of samples are represented in a 9 x 9 x 9 lattice by the nodes as well the bonds that are interconnected in a so-called dual site-bond model, DSBM, network. The pore network is developed using a Monte Carlo statistical method where the cylindrical pores (nodes and bonds) are randomly assigned into the lattice, until matching of the theoretical results to the experimental data of N2 adsorption-desorption measurements. Thus, a visual picture of the porous solid is possible. This realistic network is used next in order to study the steady-state gas transport (Knudsen gas-phase and viscous diffusion) properties for the examined materials and how flow processes depend on the morphology of the pore structure. The pore diffusivity Dp and total permeability P of each porous medium is determined based on theoretical calculations and the structural statistical parameters, such as porosity epsilonp, tortuosity factor tau and connectivity c of pores is discussed with the corresponding experimental data described in Part I of this work. The results indicate clearly that the diffusion model made it possible to predict pore effective diffusivity in these porous media in very good agreement with the corresponding experimental results for all the examined solids (Part I). The pore diffusivity increases significantly as the value of the pore connectivity increases but the transport properties of the network are influenced strongly at lowest connectivity. Also the predicted tortuosity factor is related inversely to the extent of interconnection of pores in these solids, which indicates that the influence of pore branching to the tortuosity factor of the pore network decreases, as connectivity increases.     

Determination of Morphological Properties of Swellable Solids by Size Exclusion Chromatography

Many technically interesting porous solids, e.g. ion exchangers or adsorbents for catalysis, are swellable polymers, i.e. the pore structure depends on the solvent medium. A method based on exclusion chromatography, permits determination of the pore size and pore size distribution in the swollen state.—Size exclusion chromatography, also referred to as gel permeation, gel filtration, or molecular sieve chromatography, is a widely employed method for the separation of dissolved substances—mostly polymer mixtures—according to their molecular size. Porous solids are used as stationary phase. Conversely, pore sizes and other structural data can be determined by exclusion chromatography. This application requires a series of standards (polymer samples) of known molecular weight. As a simple and rapid method, it has already proven valuable for such determinations in the case of rigid solids; in the case of swellable solids, this constitutes the sole method by which the pore structure can be characterized: classical methods require dry samples.     

Pore network modelling of the behaviour of a solute in chromatography media: transient and steady-state diffusion properties

A pore network model is presented, that is a geometrical simplification of a porous medium. The network consists of pore chambers interconnected by pore throats. A recursive algorithm for the simulation of mercury intrusion porosimetry in the network is presented. Calculations indicate that it is possible to fit simulated mercury intrusion data to experimental data, and thereby obtain parameters of the pore size distribution and pore topology (pore connectivity). A time-dependent material balance equation for diffusion on the pore level is set up and solved for the pore network. By calculating the concentration evolution in the network, the transient diffusivity and the steady-state diffusivity are found. When the network is well connected, those two diffusivities are equal, but for poorly-connected networks they differ. For migrating solutes that are non-negligibly small compared to the pore throats, considerable differences between the transient and steady-state diffusivities were found.     

Liquid chromatography of dextrans on porous silica beds

Kinetics, equilibrium isotherms and chromatography retention times for sorption of dextrans T-10, T-20, T-40, T-70, T-110, T-161, T-250 and T-500 on porous silica were measured at 25 degrees C. The Henry constant and retention factors for the dextrans were obtained. The values of the partition coefficient for the distribution of the dextrans between the bulk solution and the pore space were calculated within the framework of a pore volume filling model with consideration given to the ratio between the sizes of the macromolecular coils and the pore inlet. The measurements showed that this parameter depends on the structure of the sorbent and the molecular mass distribution of the dextran. The interaction of aqueous dextran solution with porous silica is characterized by the sieve effect. Large macromolecular coils of dextran T-161 cannot penetrate into the pore space of the silica sorbent with pore diameter 14 nm. The difference in Henry law constants calculated from adsorption and chromatographic data for dextrans T-70 and T-110 can be explained by the slow diffusion of dextran macromolecules into silica pores under chromatographic conditions.     

Diffusion in nanoporous phases: size dependence and levitation effect

Self-diffusivity, D, of diffusants in widely differing mediums such as liquids (e.g., solution), porous solids (e.g., guests in zeolites), or ions in polar solvents exhibit strong size dependence. We discuss the nature of the size dependence observed in these systems. Altogether, different theoretical approaches have been proposed to understand the nature of size dependence of D not only across these widely differing systems but even in just one medium or class of systems such as, for example, ions in polar solvents. But molecular dynamics investigations in the past decade have shown that the size dependence of self-diffusion in guest-porous solids could have origins in the mutual cancellation of forces that occurs when the size of the diffusant is comparable to the size of the void. The effect leading to the maximum in D is known as the levitation effect (LE). Such a cancellation is a consequence of symmetry. This effect exists in all porous solids irrespective of the geometrical and topological details of the pore network provided by the solid. Recent studies show that the levitation effect and size-dependent diffusivity maximum exists for uncharged solutes in solvents. One of the consequences of this is the breakdown in the Stokes-Einstein relationship over a certain range of solute-solvent size ratio. Experimental measurements of ionic conductivity over the past hundred years have found the existence of a size-dependent diffusivity maximum leading to violation of the Walden's rule for ions in polar solvents. Molecular dynamics simulations and experimental data suggest that even this maximum has its origin in LE. Simulation studies of impurity atom diffusion in close-packed solids as well as ions in superionic and other solids suggest the existence of a size-dependent diffusivity maximum in these materials as well. The levitation effect is a universal effect leading to a maximum in diffusivity of a diffusant in a variety of condensed matter phases. The only condition for its existence appears to be the presence of van der Waals or electrostatic interactions.     

Role of the Dissolution Mechanism in Determining Chromatographic Separation of Biogenic Amines from Their Precursors and Metabolites by HPLC-EC

Abstract

The current study was designed to elucidate the theoretical basis for chromatographic separation of biogenic amines on an octadecyl-silica (C-18) reverse phase column by determining the intermolecular forces between the solute and the stationary phase. The solutes mass transfer diffusion and the heat effect between solutes and stationary phase were assessed by a convenient method. This study demonstrates that the dissolution mechanism plays a major role in the process of chromatographic resolution of biogenic amines and their precursors and metabolites by HPLC-EC.     

The use of moment theory to interpret diffusion and sieving measurements in terms of the pore size distribution in heteroporous membranes

Moment theory has been applied to model porous membranes to show that one can place reasonable bounds on the cumulative pore size distribution, the hindered diffusivity or the reflection coefficient of large solutes in a heteroporous membrane by measuring the diffusive permeability to a small solute, the hydraulic permeability and one or two additional transport characteristics. These additional measurements involve either the flux of a small solute at Pe1, the hindered diffusivity of a large solute or the reflection coefficient of a large solute at Peå1. Membrane heteroporosity is incroporated in the predicted bounds without requiring one to make any a priori assumptions about the nature of the pore size distribution. In this paper, the results from calculations performed with different model membranes containing log-normal pore size distributions are reported. A comparison of the results obtained with three different membranes shows that one can distinguish between membranes with the same average pore size but different pore size distributions by measuring either the hindered diffusion coefficient or the reflection coefficient of two different sized solutes. A comparison of the bounds on D and the bounds on σ predicted from different types of transport measurements shows that, under certain conditions, one can place tighter bounds on one transport characteristic by measuring a different one.     

Effects of Multisolute Steric Interactions on Membrane Partition Coefficients

A key parameter in membrane and chromatographic separations is the partition coefficient, the equilibrium ratio of the solute concentration in a porous or fibrous material to that in bulk solution. The theoretical effects of solute size on partition coefficients in straight pores or randomly oriented fiber matrices have been investigated previously for very dilute solutions, where solute-solute interactions are negligible, and also for more concentrated solutions consisting of spherical solutes of uniform size. For concentrated solutions it has been found that steric and other repulsive interactions among solutes increase the partition coefficient above the dilute limit. To extend the results for porous or fibrous media to include concentrated mixtures of solutes with different sizes or shapes, we used an excluded volume approach. In this formulation, which describes steric interactions only, partition coefficients were computed by summing all volumes excluded to a solute molecule by virtue of its finite size, the finite size of other solutes, and the presence of fixed obstacles (pore walls or fibers). For a mixture of two spherical solutes, the addition of any second solute at finite concentration increased the partition coefficient of the first solute. That increase was sensitive to the size of the second solute; for a given volume fraction of the second solute, the smaller its radius, the larger the effect. When the total volume fraction of solutes was fixed, an increase in the amount of a second, smaller solute increased the partition coefficient of the first solute, whereas an increase in the amount of a second, larger solute had the opposite effect. Results were obtained also for oblate or prolate spheroidal solutes and for fibrous media containing fibers of different radii. For constant total fiber volume fraction, an increase in the amount of a second, smaller fiber decreased the partition coefficient of a spherical solute, whereas an increase in the amount of a second, larger fiber had the opposite effect. Overall, the theory suggests that the introduction of heterogeneities, whether as mixtures of solute sizes or mixtures of fiber sizes, may cause partition coefficients to differ markedly from those of uniform systems. Copyright 2000 Academic Press.