The paper describes general concepts of mercury porosimetry measurements and provides an overview on the current status of pore-network analysis tools. Practical aspects of the technique are described as well as emphasizing the need for testing of model pore structures and the status on pore network modeling software.

Mercury Porosimetry: A General (Practical) Overview

Progress in the synthesis and engineering of advanced porous materials demands better pore structure characterization. The analysis of pore structure is complicated by (1) the wide range in pore sizes observed, from molecular ( 1 mm) dimensions, (2) complex pore shapes and connectivities, (3) chemical and physical heterogeneities, and (4) pore structure changes that can occur during characterization.The required pore structure information varies with application. Bulk density and the pore-size distribution are needed for thermal insulation. In this case, the dimension of interest is the so-called hydraulic radius since, for small pores, the gas-phase conductivity is proportional to the mean hydraulic radius to the mean free path. A few large but isolated pores will significantly affect conductivity but will go undetected in typical gas-absorption methods. In contrast, for separations, bottlenecks control performance. For transport, such as migration through geologic formations, both the pore-size distribution and pore connectivity are important. For adsorption, surface area and pore size are the relevant factors. Finally, the conventional concepts of pore structure lose meaning as the pore size approaches molecular dimensions, typical of adsorbents and gas-separation membranes.

Characterization of Porous Solids

New perspectives in mercury porosimetry

A model is provided for predicting gel shrinkage during drying by combining the empirical observations that: (1) the bulk modulus of a gel increases with density, ρ, according to K p = K 0 ( ρ ρ y ) m (2.5 ≤ m ≤ 4) ; and (2) the variation in pore radius, r, is approximately proportional to pore volume (contrary to the dependence, r α ρ −1 3 , conventionally assumed). No allowance is made for viscoelastic relaxation, so the model applies only for drying from an inert solvent. The model may be used to guide processing efforts to yield either aerogel-like materials with high porosity or xerogels with high density and small pore size for adsorbents, membranes, etc. The extent of shrinkage is governed by two dimensionless parameters, P and m. The quantity P = As γ cos (θ)mρ y K 0 (where As is specific surface area, γ is surface tension, and θ is the contact angle) represents the relative magnitudes of capillary pressure and gel stiffness, and m describes the variation of stiffness with density. For P values less than 1, the shrinkage upon drying is less than 10%, and is reversible. For shrinkages greater than ⋍ 50%, the density increase is irreversible, and is proportional to P 1/(m - 1). Predicted shrinkage is compared with experimental results for silica gels dried from different surface tension pore fluids (2

Shrinkage during drying of silica gel

NMR measurements of the spin—lattice relaxation decay time for a fluid in a porous solid have been previously demonstrated as a significant tool for pore size/structure analysis. The analysis requires the solution of a Fredholm integral equation of the first kind to extract the desired T1, and therefore, pore size distribution from the relaxation measurements. Earlier work successfully used a nonnegative least-squares (NNLS) technique to solve this problem. However, for stability to be achieved with that approach, the problem must be significantly overconstrained (i.e., more relaxation measurements than pore size distribution points) which limits resolution. In addition, broad distributions are represented as several narrow peaks. In this work, the method of regularization is applied to extracting continuous pore volume distributions from relaxation measurements. The validity of the approach used to calculate the optimum value of the regularization smoothing parameter given an expected noise level and the data is demonstrated using “data sets” generated from known T1 distributions. Varying levels of random noise are added to these data sets and the T1 distribution is calculated for comparison to the starting distribution. For error levels in the range of 0.001 to 0.01, agreement between the starting and the calculated distributions is good. The method has also been applied to relaxation data obtained for water in several controlled pore glasses and “random packing of sphere” solids. Agreement to NMR/regularization and mercury porosimetry-derived pore size distributions was excellent with the only deviations being the result of porosimetry measuring pore neck size and NMR measuring the true pore volume to surface area ratio.

A NMR technique for the analysis of pore structure: Determination of continuous pore size distributions

The application of NMR spin—lattice relaxation measurements of water contained in porous solids is investigated as a possible tool for the determination of pore size distributions. NMR methods have several potential advantages over conventional porosimetry/adsorption techniques including the study of wet porous solids, a wide range of sample sizes that may be studied, and the fact that no pore shape assumption is required. In principle, water contained in a pore will relax faster than bulk water. This decrease in the observed relaxation rate decay constant, T1, is directly related to the pore volume to surface area ratio (i.e., hydraulic radius) according to the “two-fraction, fast-exchange model.” However, the suitability of this model and NMR spin—lattice relaxation measurements for the determination of pore size distributions has not been previously demonstrated. Pore size distributions have been determined for two series of porous solids by mercury porosimetry and a NMR method. A series of four controlled-pore glasses with very narrow pore size distributions and a mean radius in the range 3.4 to 17.6 nm were studied at a frequency of 20 MHz and at four temperatures. By using four samples with similar surface chemistry/ pore shape, the validity of the two-fraction, fast-exchange model was assessed. Experiments were conducted using a 180°–τ-90° pulse sequence and the distribution of T1 was calculated using a nonnegative leastsquares routine (NNLS) presented in a previous paper. The two-fraction model described the change of mean T1 with pore radius at a given temperature. The change of the surface relaxation decay constant, T1s, fit an Arrhenius expression with an activation energy of 1.8 kcal/mole. Agreement between the mercury porosimetry and NMR derived pore size distributions is excellent. T1 measurements at both 20 and 300 MHz and 303 K were also made on a series of porous solids fabricated by pelleting submicron silica spheres. These solids exhibit afairly wide pore size distribution (⋍ one order of magnitude). The two-fraction, fast-exchange model satisfactorily described the relationship between pore size and T1. Agreement between mercury porosimetry and NMR pore size distributions is good. However, the NMR derived distributions consisted of a series of narrow peaks overlapping the single broad porosimetry peak. The surface relaxation effect, and, hence, the overall pore size sensitivity, is increased by a factor of 3 when the proton frequency is decreased from 300 to 20 MHz.

A NMR technique for the analysis of pore structure: Application to materials with well-defined pore structure

A new analysis for the application of mercury porosimetry to porous solids comprised of random packings of uniform spheres (ϵ ∼- 0.36–0.39) is presented. The filling of pore volume in the region where two spheres are near neighbors (distance of separation ⪡ sphere size) is modeled by solving the Young-Laplace equation to determine the degree of curvature of the meniscus as a function of sphere spacing, contact angle, and filling angle. The correct meniscus shape for mercury intrusion into the toroidal void is a nonoid. From the known distribution of sphere spacings in a random packing and the calculated pressure-volume relationship, the unfilled void fraction as a function of dimensionless pressure ( PR /γ) and contact angle is determined. Mercury porosimetry experiments have been conducted using porous solids fabricated from monodisperse, nonporous silica spheres (radius = 51, 125, 165,226 nm). Agreement between the toroidal void filling theory and all four experiments was excellent when the measured mercury contact angle of 138.7° was used in the theory. It is demonstrated that very large values of PR /γ (∼-100) are necessary to completely fill the toroidal void region and therefore, determine the total sample porosity.

Mercury porosimetry: Theoretical and experimental characterization of random microsphere packings

Etude de l'adsorption de differents composes (gaz rare, hydrocarbure) sur de la silice pour determiner la rugosite de la surface

Adsorbate shape effects on surface roughness determination

Donnees sur la diffusion de Knudsen, d'un gaz non adsorbant, dans un modele de milieu poreux, non consolide. Utilisation dans ce but des microspheres, monodispersees (130-400 nm de diametre) de silice

Dorothy L Stermer

Papers

A NMR technique for the analysis of pore structure: Application to materials with well-defined pore structure

Mercury porosimetry: Theoretical and experimental characterization of random microsphere packings

Adsorbate shape effects on surface roughness determination

Knudsen diffusion in bidisperse mixtures of uniform spheres

Mercury porosimetry in random sphere packings: Breakthrough pressure and pore structure determination