Evaluation of methods for determining the pore size distribution and pore-network connectivity of porous carbons

The pore size distribution (PSD) and the pore-network connectivity of a porous material determine its properties in applications such as gas storage, adsorptive separations, and catalysis. Methods for the characterization of the pore structure of porous carbons are widely used, but the relationship between the structural parameters measured and the real structure of the material is not yet clear. We have evaluated two widely used and powerful characterization methods based on adsorption measurements by applying the methods to a model carbon which captures the essential characteristics of real carbons but (unlike a real material) has a structure that is completely known. We used three species (CH4, CF4, and SF6) as adsorptives and analyzed the results using an intersecting capillaries model (ICM) which was modeled using a combination of Monte Carlo simulation and percolation theory to obtain the PSD and the pore-network connectivity. There was broad agreement between the PSDs measured using the ICM and the geometric PSD of the model carbon, as well as some systematic differences which are interpreted in terms of the pore structure of the carbon. The measured PSD and connectivity are shown to be able to predict adsorption in the model carbon, supporting the use of the ICM to characterize real porous carbons.     

Characterization of the pore area distribution in porous membranes using transport measurements

An approach originally proposed by Mason and coworkers has been applied to model porous membranes to show that transport measurements with small and large solutes can be used to distinguish between porous membranes with the same average pore size but different pore size distributions. In addtion, it is shown that such measurements can be used to account for membrane heteroporosity when predicting the sieving characteristics of a membrane. This is done by applying moment theory to results from flux measurements for a small solute at Pe ≈ 1 or to results from measurements of the reflection coefficient for a large solute at infinite Pe. No a priori assumptions about the nature of the distribution of pore areas are necessary.In this paper, the results from calculations performed with three different model membranes with log-normal pore size distribution are reported. These results show that one can begin to distinguish between membranes by measuring the hydraulic and diffusive permeability and performing at least one additional flux measurement — with either a small, non-hindered solute at Pe ≈ 1 or a large solute at infinite Pe. Results also show that a fairly narrow window can be placed on the sieving curve for a heteroporous membrane without performing any sieving measurements. This is an interesting and encouraging result because it means that many of the problems that arise from measuring and interpreting pore size distributions using more traditional techniques can be avoided by using small solute flux measurements to predict the separation characteristics of many porous membranes.     

Comparison between different presentations of pore size distribution in porous materials

The different presentations of the pore size distribution derived from the gas adsorption method and the mercury porosimetry are connected with some problems. This concerns especially the use of the logarithmically differential pore volume distribution. The incorrect application of this distribution to bimodal pore systems involves the danger of an apparent overemphasizing of larger pores. This effect may also occur using the incremental pore size distribution in case the experimental point spacing considerably increases towards the larger pore radii. The pore volume density distribution defined as the linear derivative of the cumulative pore volume curve with respect to the pore radius has been found the most convenient form among the various kinds of pore volume distribution presentations. It has been shown that the direct comparison between this distribution and the logarithmically differential pore volume distribution is not allowed. Nevertheless, there is a clear connection between these definitions for the pore size distribution so that they are completely equivalent.     

Comparison between different presentations of pore size distribution in porous materials

The different presentations of the pore size distribution derived from the gas adsorption method and the mercury porosimetry are connected with some problems. This concerns especially the use of the logarithmically differential pore volume distribution. The incorrect application of this distribution to bimodal pore systems involves the danger of an apparent overemphasizing of larger pores. This effect may also occur using the incremental pore size distribution in case the experimental point spacing considerably increases towards the larger pore radii. The pore volume density distribution defined as the linear derivative of the cumulative pore volume curve with respect to the pore radius has been found the most convenient form among the various kinds of pore volume distribution presentations. It has been shown that the direct comparison between this distribution and the logarithmically differential pore volume distribution is not allowed. Nevertheless, there is a clear connection between these definitions for the pore size distribution so that they are completely equivalent. Received: 15 May 1998 / Revised: 8 October 1998 / Accepted: 10 October 1998     

Determination of pore size distribution in hollow fibre membranes

Isopropanol displacement under nitrogen pressure was used for the determination of pore size distribution in microfiltration polypropylene hollow fibres. Applying various assumptions about gas transport process two completely different characteristics of pore sizes were obtained. To verify these results an analysis of SEM images of the investigated membrane was conducted concerning its porous structure (pore diameters, surface occupied by pores). According to the SEM analysis the mean coverage of membrane surface by pore entrances should be about 20% of total area. For the distribution which accounted for pore evacuation according to Young–Laplace equation with contact angle θ=67° surprisingly dense coverage amounting to over 70% of total surface (by calculated total pore number over 10¹³ per m²) was predicted. Results for the distribution which accounted for gas bubble formation at the membrane surface (equivalent to θ=0°) fit into the expected range of pore numbers and membrane coverages (about 10¹¹ per m² and about 10%, respectively). It is concluded that the mechanism of bubble formation, determined by an actual pressure, liquid surface tension and pore size, is the crucial process while the value of contact angle θ does not play any role in the determination of pore size distribution.     

Evolution of pore size distribution and average pore size of porous ceramic membranes during modification by counter-diffusion chemical vapor deposition

The modification of porous ceramic membranes by counter-diffusion chemical vapor deposition (CVD) is studied theoretically and experimentally. Numerical simulations of the evolution of the membrane permeance, average pore size and pore size distribution as a function of extent of modification are presented and compared with experimetal data. It is found that the change of the average pore size of the membranes after modification strongly depends on the initial pore size distribution of the membrane, CVD reaction kinetics and characterization method. Experimental data suggest that CVD of zirconia (and yttria) inside porous ceramic membranes by reaction of zirconium (and yttrium) chlorides with steam/air at elevated temperatures proceeds by quasi-zero reaction kinetics with respect to the oxidant, typical of non-stoichiometric supply of the reactants from opposite sides of the membrane. Under such conditions, CVD modification may result in a modest increase of the average pore size of coarse-pore ceramic membranes as suggested by numerical calculations and experimental data.     

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.     

Characterization of pore size distribution in porous silicon by NMR cryoporosimetry and adsorption methods

The results of studying the pore size distribution of mesoporous silicon by NMR cryoporosimetry are described. These data are compared with the results obtained by adsorption methods.     

Modified liquid displacement method for determination of pore size distribution in porous membranes

A new method called constant pressure liquid displacement method (CPLM) was developed and tested to measure the pore size distribution of porous membranes. The permeability, defined as a ratio of the flow rate to the pressure applied, used to be assumed constant either for a conventional liquid displacement method or for a bubble point method, leading to the erroneous interpretation of the pore size distribution. However, it was possible to eliminate such an assumption by measuring the flow rates experimentally at a standard low pressure through the pores penetrated with a permeating liquid according to the proposed method. The pore size distribution for a hydrophobic PVDF membrane was successfully measured by the CPLM and compared with those measured by two different methods such as the conventional liquid displacement method and the mercury intrusion method.     

Determination of equilibrium constants from chromatographic and electrophoretic measurements

Chemical interactions, such as acid-base, complex-forming, ion association and other equilibria, are widely exploited to improve the separation efficiency in liquid chromatography as well as in electrophoresis. On the other hand, these techniques can be advantageously used to study the chemical equilibria affecting the separations. If the equilibium is sufficiently fast in comparison with the separation process, then the retention characteristics in chromatography (retention factors) or the migration characteristics in electrophoresis (effective mobilities) may be expressed as functions of the composition of mobile phase or background electrolyte (BGE), respectively. Using a proper experimental arrangement, the dependencies of retention (migration) characteristics on the mobile phase (background electrolyte) composition can be measured and utilized to calculate the equilibrium constants for equlibria taking place in the mobile phase (background electrolyte). Although principles of these measurements have been known for a long time, only more recent studies utilizing HPLC and capillary electrophoretic techniques are reviewed in this paper.     

Modeling and simulation of the dynamic behavior of monoliths. Effects of pore structure from pore network model analysis and comparison with columns packed with porous spherical particles

A mathematical model is presented that could be used to describe the dynamic behavior, scale-up, and design of monoliths involving the adsorption of a solute of interest. The value of the pore diffusivity of the solute in the pores of the skeletons of the monolith is determined in an a priori manner by employing the pore network modeling theory of Meyers and Liapis [J. Chromatogr. A, 827 (1998) 197 and 852 (1999) 3]. The results clearly show that the pore diffusion coefficient, Dmp, of the solute depends on both the pore size distribution and the pore connectivity, nT, of the pores in the skeletons. It is shown that, for a given type of monolith, the film mass transfer coefficient, Kf, of the solute in the monolith could be determined from experiments based on Eq. (3) which was derived by Liapis [Math. Modelling Sci. Comput., 1 (1993) 397] from the fundamental physics. The mathematical model presented in this work is numerically solved in order to study the dynamic behavior of the adsorption of bovine serum albumin (BSA) in a monolith having skeletons of radius r(o) = 0.75x10(-6) m and through-pores having diameters of 1.5x10(-6)-1.8x10(-6) m [H. Minakuchi et al., J. Chromatogr. A, 762 (1997) 135]. The breakthrough curves of the BSA obtained from the monolith were steeper than those from columns packed with porous spherical particles whose radii ranged from 2.50x10(-6) m to 15.00x10(-6) m. Furthermore, and most importantly, the dynamic adsorptive capacity of the monolith was always greater than that of the packed beds for all values of the superficial fluid velocity, Vtp. The results of this work indicate that since in monoliths the size of through-pores could be controlled independently from the size of the skeletons, then if one could construct monolith structures having (a) relatively large through-pores with high through-pore connectivity that can provide high flow-rates at low pressure drops and (b) small-sized skeletons with mesopores having an appropriate pore size distribution (mesopores having diameters that are relatively large when compared with the diameter of the diffusing solute) and high pore connectivity, nT, the following positive results, which are necessary for obtaining efficient separations, could be realized: (i) the value of the pore diffusion coefficient, Dmp, of the solute would be large, (ii) the diffusion path length in the skeletons would be short, (iii) the diffusion velocity, vD, would be high, and (iv) the diffusional response time, t(drt), would be small. Monoliths with such pore structures could provide more efficient separations with respect to (a) dynamic adsorptive capacity and (b) required pressure drop for a given flow-rate, than columns packed with porous particles.     

Nondestructive technique for the characterization of the pore size distribution of soft porous constructs for tissue engineering

Polymer scaffolds tailored for tissue engineering applications possessing the desired pore structure require reproducible fabrication techniques. Nondestructive, quantitative methods for pore characterization are required to determine the pore size and its distribution. In this study, a promising alternative to traditional pore size characterization techniques is presented. We introduce a quantitative, nondestructive and inexpensive method to determine the pore size distribution of large soft porous solids based on the on the displacement of a liquid, that spreads without limits though a porous medium, by nitrogen. The capillary pressure is measured and related to the pore sizes as well as the pore size distribution of the narrowest bottlenecks of the largest interconnected pores in a porous medium. The measured pore diameters correspond to the narrowest bottleneck of the largest pores connecting the bottom with the top surface of a given porous solid. The applicability and reproducibility of the breakthrough technique is demonstrated on two polyurethane foams, manufactured using the thermally induced phase separation (TIPS) process, with almost identical overall porosity (60-70%) but very different pore morphology. By selecting different quenching temperatures to induce polymer phase separation, the pore structure could be regulated while maintaining the overall porosity. Depending on the quenching temperature, the foams exhibited either longitudinally oriented tubular macropores interconnected with micropores or independent macropores connected to adjacent pores via openings in the pore walls. The pore size and its distribution obtained by the breakthrough test were in excellent agreement to conventional characterization techniques, such as scanning electron microscopy combined with image analysis, BET technique, and mercury intrusion porosimetry. This technique is suitable for the characterization of the micro- and macropore structure of soft porous solids intended for tissue engineering applications. The method is sensitive for the smallest bottlenecks of the largest continuous pores throughout the scaffold that contributes to fluid flow.     

Relationship between the particle size distribution of commercial fully porous and superficially porous high-performance liquid chromatography column packings and their chromatographic performance

The separation efficiency and kinetics of several commercial HPLC particle types (both fully porous and superficially porous) have been investigated using a pharmaceutical weakly basic N-containing compound as a test molecule. A strong trend between the particle size distribution (PSD) of the particles and the typically employed “goodness of packing”-parameters was observed. The relative standard deviation of the PSD of the tested particles ranged between 0.05 and 0.2, and in this range, a near linear relationship between the A-term constant, the h_min-value and the minimal separation impedance was found. The experimental findings hence confirm the recent observations regarding the relationship between the narrow PSD of the recently commercialized porous-shell particles and their superior efficiency and kinetic performance. The outcome also suggests that the performance of the current generation of fully porous particle columns could be significantly improved if the PSD of these particles could be reduced.     

Determination of the intraparticle electroosmotic volumetric flow-rate, velocity and Peclet number in capillary electrochromatography from pore network theory

The results obtained from the pore network model employed in this work, clearly show that the magnitudes of the intraparticle electroosmotic volumetric flow-rate, Qintrap, and velocity, (v(intrap,x)), in the pores of the charged porous silica particles considered in this study are greater than zero. The intraparticle Peclet number, Pe(intra, of a solute in these charged porous silica particles would be greater than zero, and, in fact, the magnitude of the intraparticle Peclet number, Pe(intrap), of lysozyme is greater than unity for all the values of the pore connectivity, nT, of the intraparticle pores and of the applied electric potential difference per unit length, Ex, along the axis of the capillary column considered in this work. Furthermore, the values of the intraparticle electroosmotic volumetric flow-rate, Qintrap, and velocity, (v(intrap,x)), as well as the magnitude of the pore diffusion coefficient, Dp, of the solute increase as the value of the pore connectivity, nT, of the intraparticle pores increases. The intraparticle electroosmotic flow can contribute significantly, if the appropriate chemistry is employed in the mobile liquid phase and in the charged porous particles, in (i) decreasing the intraparticle mass transfer resistance, (ii) decreasing the dispersive mass transfer effects, and (iii) increasing the intraparticle mass transfer rates so that high column efficiency and resolution can be obtained.     

Quantitative information on pore size distribution from the tangents of comparison plots

The comparison plot obtained from the nitrogen adsorption data has a similar shape to that of the curve of accumulating pore volume of a solid. The intrinsic nature of this relation is discussed. It is known that the derivatives of the accumulating pore volume with respect to the pore size are the pore size distribution (PSD) of the solid. Thus, the tangent curve of the comparison plot can display, at least qualitatively, the PSD of a solid, over a wide range of pore sizes (from approximately 1 to 50 nm) because the comparison plot is applicable to both micropores and mesopores. Quantitative pore structure information can be derived from the comparison plots by establishing a relationship between the t value and the pore size from the samples with uniform pore structure and known pore sizes, such as MCM-41 and alumina pillared clay samples. A calculation procedure to derive quantitative PSD from the comparison plots is suggested, giving reasonable results. This study proposes concise and reliable methods based on the comparison plots to derive information on pore structure in porous solids.     

Modelling of the pore size distribution of ultrafiltration membranes

A dilute aqueous solution of polydisperse neutral dextrans was used to determine the sieving properties (flux and rejection) of porous polyacrylonitrile membranes. Gel ermeation chromatography was used to measure the solute mole and concentration in the permeate. From these data, rejection coefficients were calculated as a function of solute molecular size. A mathematical model was then developed to relate the flux and solute rejection to pore size distribution and the total number of pores, based upon the assumption that solute rejection was the result of purely geometric considerations. As a first approximation, a solute molecule was considered either too large to enter a membrane pore, or if it entered, its concentration in the permeate from that pore, as well as the solvent flux through the pore, were not affected. This model also considered the effects of steric hindrance and hydrodynamic lag on the convection of solute through a membrane. The shape and sharpness of pore size distributions were found to be useful in comparisons of ultrafiltration membranes.     

Determination of the size of particles of highly dispersed materials by low temperature nitrogen adsorption

The average size of particles of highly dispersed materials (Al₂O₃, TiO₂, Y₂O₃) is calculated from their specific surface areas. The latter is determined with the analysis of sorption capacity of materials with nitrogen at ?196°C in the ranges of relative equilibrium pressures of 0.05–0.35 (the Brunauer-Emmett-Teller method, BET) and 0.4–0.7 (the comparative method). Good agreement is obtained between results produced with the comparative method and familiar methods of analysis such as scanning and transmission electron microscopy and small-angle X-ray scattering.