Pore size distribution determination General

Lindstrom and Boersma (1971) pioneered the prediction of breakthrough curves from equivalent cylindrical pore size distributions, determined by either the water retention or mercury porosimetry methods. The model developed by these authors includes the effects of bothintra- and interpore dispersion. In general, dispersion due to differences in tube size has a much greater influence on the shape and position of the breakthrough curve than mixing within tubes due to microscopic velocity profiles (Rao et al., 1976). For completeness, however, it is preferable to include both effects. Lindstrom and Boersma (1971) defined a CDE for each tube, so that C/C0 for the bundle as a whole is given by ... 

Rather broad pore size distributions are generally found in this way for the polymeric SEC packings. This property widens the range of molecular sizes that can be determined by the polymeric column. In general, though, SEC columns based on macroporous sdica or sifica bonded phases function equally good. 

Mesoporous materials (with pore widths between 2 and 50 nm) experience pore condensation at pressures below the corresponding saturation pressure of the bulk liquid. The volume pore size distribution is generally determined according to the Barret-Joyner-Halenda model. This model considers that condensation occurs in pores when a critical relative pressure is reached according to the modified Kelvin equation... 

Gel filtration chromatography has been extensively used to determine pore-size distributions of polymeric gels (in particle form). These models generally do not consider details of the shape of the pores, but rather they may consider a distribution of effective average pore sizes. Rodbard [326,327] reviews the various models for pore-size distributions. These include the uniform-pore models of Porath, Squire, and Ostrowski discussed earlier, the Gaussian pore distribution and its approximation developed by Ackers and Henn [3,155,156], the log-normal distribution, and the logistic distribution. 

Table 16-4 shows the IUPAC classification of pores by size. Micropores are small enough that a molecule is attracted to both of the opposing walls forming the pore. The potential energy functions for these walls superimpose to create a deep well, and strong adsorption results. Hysteresis is generally not observed. (However, water vapor adsorbed in the micropores of activated carbon shows a large hysteresis loop, and the desorption branch is sometimes used with the Kelvin equation to determine the pore size distribution.) Capillary condensation occurs in mesopores and a hysteresis loop is typically found. Macropores form important paths for molecules to diffuse into a par-... 

No current theory is capable of providing a general mathematical description of micropore fiUirig and caution should be exercised in the interpretation of values derived from simple equations. Apart from the empirical methods described above for the assessment of the micropore volume, semi-empirical methods exist for the determination of the pore size distributions for micropores. Common approaches are the Dubinin-Radushkevich method, the Dubinin-Astakhov analysis and the Horvath-Kawazoe equation [79]. 

ISEC, which was introduced by Halasz and Martin in 1978 [119], represents a simple and fast method for the determination of the pore volnme, the pore size distribution profile, and the spe-cihc snrface area of porous solids. Generally, ISEC is based on the principle of SEC. SEC, also referred to as gel permeation or gel filtration chromatography, is a noninteractive chromatographic method that separates analytes according to their size by employing a stationary phase that exhibits a well-dehned pore distribution. 

The most common method used for the determination of surface area and pore size distribution is physical gas adsorption (also see 1.4.1). Nitrogen, krypton, and argon are some of the typically used adsorptives. The amount of gas adsorbed is generally determined by a volumetric technique. A gravimetric technique may be used if changes in the mass of the adsorbent itself need to be measured at the same time. The nature of the adsorption process and the shape of the equilibrium adsorption isotherm depend on the nature of the solid and its internal structure. The Brunauer-Emmett-Teller (BET) method is generally used for the analysis of the surface area based on monolayer coverage, and the Kelvin equation is used for calculation of pore size distribution. 

The available transport models are not reliable enough for porous material with a complex pore structure and broad pore size distribution. As a result the values of the model par ameters may depend on the operating conditions. Many authors believe that the value of the effective diffusivity D, as determined in a Wicke-Kallenbach steady-state experiment, need not be equal to the value which characterizes the diffusive flux under reaction conditions. It is generally assumed that transient experiments provide more relevant data. One of the arguments is that dead-end pores, which do not influence steady state transport but which contribute under reaction conditions, are accounted for in dynamic experiments. Experimental data confirming or rejecting this opinion are scarce and contradictory [2]. Nevertheless, transient experiments provide important supplementary information and they are definitely required for bidisperse porous material where diffusion in micro- and macropores is described separately with different effective diffusivities. 

Transmission electron microscopy (TEM) observations, nitrogen adsorption-desorption and mercury porosimetry measurements indicated that increasing EDAS/TEOS ratio results in (a) a decrease of the building block particle size, (b) an increase of the specific surface area (S ), (c) an increase of mesopore volume determined at saturation pressure of N (Vp) and a decrease of the total pore volume (V,) (d) a general shift of the pore size distribution towards smaller pores, (e) an increase of the pressure of transition (P,), above which mercury can intrude the sample without destroying the pore structure [1-3]. To explain this behaviour... 

Permeability and permselectivity of a membrane depend on its pore size distribution. But equally important, they are applications specific and determined by the interactions between the process stream and the pore or membrane surface. However, for general characterization purposes, some model permeants (solvents and molecules) are often used to obtain a generic" permeability and permselectivity for that membrane. Water is... 

Finally, track-etched MF membranes are made from polymers, such as polycarbonate and polyester, wherein electrons are bombarded onto the polymeric surface. This bombardment results in sensitized tracks, where chemical bonds in the polymeric backbone are broken. Subsequently, the irradiated film is placed in an etching bath (such as a basic solution), in which the damaged polymer in the tracks is preferentially etched from the film, thereby forming cylindrical pores. The residence time in the irradiator determines pore density, and residence time in the etching bath determines pore size. Membranes made by this process generally have cylindrical pores with very narrow pore-size distribution, albeit with low overall porosity. Furthermore, there always is the risk of a double hit, i.e., the etched pore becomes wider and could result in particulate penetration. Such filter membranes are often used in the electronic industry to filter high-purity water. 

Gas adsorption. The general procedures for the determination and interpretation of adsorption and desorption isotherms have been reviewed in earlier volumes of this series (129,276). An adsorption-desorption isotherm permits estimation of the surface area, pore volume, average pore diameter, and approximate pore-size distribution. 

A differential scanning calorimeter is used to obtain a solidification (or melting) thermogram, from which a pore size distribution can be extracted. By comparing the solidification and melting processes, thermoporometry can also be used to determine a thermodynamic pore shape factor, which varies generally from 1 (spherical pores) to 2 (cylindrical pores). 

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