Cross-Section of the Pores

Based on the gas adsorption behavior, Ko-zawa and Yamashita proposed a hypothesis [20, 21] that the cross—section of the fine pores of EMD is a cavity shape as shown in Fig. 16. The cross-section of the pore by computer calculation is a circle (Fig. 16A). Nobody knows the real shape of the cross-section as yet. Kozawa s belief in the cavity shape (Fig. 16B) is based on the results of experiments involving the oxygen adsorbed and desorbed from the pore walls [20]. 

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Let C be the circumference of the cross section of the pores and R be the equivalent radius of a circle having the same circumference, i.e.,... 

A somewhat different relationship between the amount of vapor sorbed and the pressure is often experimentally obtained upon decreasing rather than increasing the pressure. This is explained by the different curvature of the meniscus during adsorption and desorption. For example, consider a cylindrical pore. The adsorption is determined by the curvature of the liquid film on the pore surface. If the film thickness is much lower than the pore radius r, the curvature of the surface is l/(2r) and the process is described by Equation 3.10. During desorption the liquid covers the entire cross-section of the pore and has a spherical meniscus of curvature 1/r, and Equation 3.9 holds. Comparison of Equations 3.9 and 3.10 shows that a higher pressure is required to fill pores. 

The associated boundary conditions for the problem may be explained in the following physical sense. At the exterior surface of the pore the reactant concentration equals the bulk concentration of the gaseous species at the central cross-section of the pore, considering the symmetric demand, there is no reactant concentration gradient, i.e. 

The size of the pore is determined by the number of tetrahedra that make up the cross-section of the pore walls within a zeolite framework. For example, if four tetrahedra are linked together in a ring, a very small pore about 0.26 nm in diameter is produced (Figure 2.5a). If the cross-section is made up of six tetrahedra linked together in a ring then the diameter increases to about 0.33 nm (Figure 2.5b). 

Fortunately, it is not necessary to trace the complex individual paths which molecules are forced to take in diffusing down a pore. It will be sufficient to determine the net rate of flow of molecules past a given cross-section of the pore when the concentration gradient at this point is known. The particular form of this rate equation depends on three factors (a) The magnitude of the pore radius as compared to the length of the mean free path of molecules between intermolecular collisions, (b) The presence or absence of total pressure differences along the pore which can lead to a mass flow of molecules into or out of the pore, (c) Under certain specialized conditions it appears the presence of physically adsorbed layers of the pore wall may affect the rate of transport via two dimensional surface diffusion. 

Big PbS04 crystals of regular shape are formed here and there on the surface of the 4BS crystals (Fig. 9.27b). Probably, they have formed as a result of recrystallization of the small PbS04 particles. These PbS04 crystals reduce the cross-section of the pores in the surface plate layer [20]. 

As the pressure of the gas or the vapor is inaeased, the thickness of the multimolecular layers in the transitional pores inaeases imtil the layer on the opposite walls combine in the narrowest cross-section of the pore and form a meniscus of condensed adsorbate. This meniscus is concave when the adsorbate wets the surface. The molecules of the adsorbate then condense on the meniscus at a pressure lower than the saturation vapor pressure. The lowering of the eqnihbrium vapor pressure over a concave meniscus, as compared with that over a flat surface at the same temperature, is due to the molecules in a concave surface being held by a large number of neighboring molecules, rather than if they were held on a flat surface. The quantitative relationship between the lowering of vapor pressure and the radius of the capillary, known as the Kelvin equation, was given by Thomson (later Lord Kelvin). The Kelvin equation can be written as... 

A single cylindrical pore of length L and radius of r (=d) located in a microscopic section of the catalyst particle is generally used for modeling the diffusion-reaction process (Figure 2.3). The steady-state component mass balance for a control volume extending over the cross section of the pore includes diffusion of reactant into and out of the control volume as well as reaction on the inner wall surface. The simple case taken as an example is that of an isothermal, irreversihle first-order reaction ... 

In the previous section, we described how size selectivity was generated by the limited cross section of the pores in 3D systems. In molecular systems such as... 

Essentially all these experimental methods apply when we deal with high-porosity multipore membranes, which are the majority of the cases. However, polymer brush decoration of single nanometric pores on macroscopic membranes has been examined lately [8]. A quite interesting method for the characterization of this system is the measmement of the ionic current that flows through the channel in an electrolyte solution under different electric potential differences. The current-voltage (TV) response of the system can be correlated to the conductance of the polymer-decorated channel and consequently to the effective free cross section of the pore-brush structure. 

If the pore is wide enough to allow multiple monomers at a cross-section of the pore, the chain can fold back and forth, and it can assume deformed conformations as sketched in Figure 5.1c. When a chain is confined in a... 

Even in the absence of any tangential shear, droplets can be spontaneously detached from the pore outlets at small disperse phase fluxes (Figure 6.2d), particularly in the presence of fast-adsorbing emulsiflers in the continuous phase and for a pronounced noncircular cross section of the pores, such as for narrow slots (41 x 11 pm) shown on the micrograph in Figure 6Ad (Kobayashi et al., 2003). Kobayashi et al. found that slot aspect ratios exceeding a threshold ratio of approximately 3 were needed to successfully prepare monodisperse emulsions with a CV below 2%. 

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