Pore shape cylindrical

Fig. 3.14 (a) A cone-shaped pore with hemispherical meniscus, (b) A wedge-shaped pore with cylindrical meniscus. 

The pore shape affects the pressure of mercury intrusion in ways not contemplated by the usual Washbum-Laplace or Kloubek-Rigby-Edler models. These models have been developed for cylindrical pores and correctly account for the penetration of mercury in the cylindrical pores of MCM-41. The uneven surface of the cylindrical pores of SBA-15 is responsible for a significant increase of the pressure of mercury intrusion and, thereby, for a corresponding underevaluation of the pore size if the classical pressure-size correlations are applied. 

The pore shape is determined by the particle shape. Plate-shaped particles lead to plate-shaped pores in the case of regular packing. Sphere-shaped particles favor cylindrical or sometimes ink-bottle-type pores. 

TITANIA WITHOUT BINDERS pore model cylindrical shaped... 

TITANIA WITH PVA AND HPC (Mw = 10E5) pore model cylindrical shaped... 

It has to be noted that this relation is only valid for pores, possessing cylindrical shape. From Equation 1.1, it gets apparent that under zero pressure, none of the nonwetting liquid will enter the pores of the immersed material. If now the pressure is raised to a certain level, the liquid will penetrate pores possessing radii greater than that calculated from Equation 1.1. Consequently, the higher the pressure that is applied, the smaller the pores that are penetrated by the liquid. 

The ratio of volume to area within a pore depends upon the pore geometry. For example, the volume to area ratios for cylinders, parallel plates and spheres are, respectively, r/2, r/2 and r/3, where r is the cylinder and sphere radii or the distance of separation between parallel plates. If the pore shapes are highly irregular or consist of a mixture of regular geometries, the volume to area ratio can be too complex to express mathematically. In these cases, or in the absence of specific knowledge of the pore geometry, the assumption of cylindrical pores is usually made, and equation (8.6) becomes... 

The exact pore shape is usually unknown and cylindrical pores are generally assumed. Mikhail, Brunauer, and Bodor show in their paper that equation (9.17) is equally valid for parallel plate or cylindrical pores and that the mean hydraulic radius in Table 9.1, is the same as the separation between plates or the cylindrical radius. 

Equations (12.5) through (12.8) assume no specific pore shape. By assuming cylindrical pore geometry the validity of these equations can be established. For a cylindrical pore the area in any pore interval is given by... 

The pore volume and the pore size distribution can be estimated from gas adsorption [83], while the hysteresis of the adsorption isotherms can give an idea as to the pore shape. In the pores, because of the confined space, a gas will condense to a liquid at pressures below its saturated vapor pressure. The Kelvin equation (Eq. (4.5)) gives this pressure ratio for cylindrical pores of radius r, where y is the liquid surface tension, V is the molar volume of the liquid, R is the gas constant ( 2 cal mol-1 K-1), and T is the temperature. This equation forms the basis of several methods for obtaining pore-size distributions [84,85]. 

The pore shape is generally assumed to be either cylindrical or slit-shaped in the former case, the meniscus is hemispherical, and hence r, is equal to r2 in the latter case, the meniscus is hemicylindrical, and thus r, is equal to the width of the slit and r2 is... 

Assuming that the external surface of a porous substrate is negligible, and that the pores are cylindrically shaped, the average pore diameter of a substrate can be calculated by the Wheeler formula 21... 

Based on the above general principles, quite a number of models have been developed to estimate pore size distributions.29,30,31-32,33 They are based on different pore models (cylindrical, ink bottle, packed sphere,. ..). Even the so-called modelless calculation methods do need a pore model in the end to convert the results into an actual pore size distribution. Very often, the exact pore shape is not known, or the pores are very irregular, which makes the choice of the model rather arbitrary. The model of Barett, Joyner and Halenda34 (BJH model) is based on calculation methods for cylindrical pores. The method uses the desorption branch of the isotherm. The desorbed amount of gas is due either to the evaporation of the liquid core, or to the desorption of a multilayer. Both phenomena are related to the relative pressure, by means of the Kelvin and the Halsey equation. The exact computer algorithms35 are not discussed here. The calculations are rather tedious, but straightforward. 

In this particular case of a cylindrical pore shape, it seems reasonable to assume that the condensate has a meniscus of spherical form and radius rK. However, as some physisoiption has already occurred on the mesopore walls, it is evident that rE and rf are not equal. If the thickness of the adsorbed multilayer is t, and die contact angle is assumed to be zero, the radius of the cylindrical pore is simply... 

Dollimore and Heal (1964) and Roberts (1967). In the early work, it was customary to assume the pore shape to be cylindrical, but now the slit-shaped and packed sphere models are considered to be more appropriate for some systems. 

The applied pressure is related to the desired pore size via the Washburn Equation [1] which implies a cylindrical pore shape assumption. Mercury porosimetry is widely applied for catalyst characterization in both QC and research applications for several reasons including rapid reproducible analysis, a wide pore size range ( 2 nm to >100 / m, depending on the pressure range of the instrument), and the ability to obtain specific surface area and pore size distribution information from the same measurement. Accuracy of the method suffers from several factors including contact angle and surface tension uncertainty, pore shape effects, and sample compression. However, the largest discrepancy between a mercury porosimetry-derived pore size distribution (PSD) and the actual PSD usually... 

On postmodification the pore width (Dbjh) was found to decrease from 2.4 nm to 1.6 nm. Assuming a cylindrical pore shape the predicted decrease in pore volume would be expected to correspond to a decrease in pore volume from 0.39 cm g to 0.17 cm g on postmodification. However, the measured pore volume for postmodified sample is only 0.02 cm g, supporting the hypothesis that considerable pore-blocking has occurred upon postmodification. 

Assumptions on particles or pores shape for raw data acquisition non porous particles of any shape none no assumption on shape of particles, cylindrical pores Non porous spherical particles... 

Fig. 2.1. Schematic picture of pore shapes. A and B are single wall, symmetric and asymmetric membranes respectively with straight cylindrical (a) or conical (b) pore shape (c) represents a ceramic asymmetric multilayered membrane with intercormected pores.

The difference between the solidification and melting temperatures in cylindrical pores is due to the fact that the shapes of the interfaces present during these transitions are different. In spherical shaped pores however, there is no difference and the same thermodynamic equation can be used to describe both solid —> liquid and liquid -> solid transitions. Consequently by analysing both the melting and solidification curves, one determines a pore shape factor. In thermoporometry the shape factor for a porous material [68,69] can vary generally between 1 (spherical pores) and 2 (cylindrical pores). 

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