Cylindrical pores, diffusion through

Among the dynamical properties the ones most frequently studied are the lateral diffusion coefficient for water motion parallel to the interface, re-orientational motion near the interface, and the residence time of water molecules near the interface. Occasionally the single particle dynamics is further analyzed on the basis of the spectral densities of motion. Benjamin studied the dynamics of ion transfer across liquid/liquid interfaces and calculated the parameters of a kinetic model for these processes [10]. Reaction rate constants for electron transfer reactions were also derived for electron transfer reactions [11-19]. More recently, systematic studies were performed concerning water and ion transport through cylindrical pores [20-24] and water mobility in disordered polymers [25,26]. 

A further consideration in porous materials is the shape of the pores. Molecules have to diffuse through the pores to feel the effect of the catalytic groups which exist in the interior and, after reaction, the reaction products must diffuse out. These diffusion processes can often be the slowest step in the reaction sequence, and thus pores which allow rapid diffusion will provide the most active catalysts. It is another feature of the MTSs that they have quite straight, cylindrical pores - ideal for the rapid diffusion of molecules. 

The shape of steady-state voltammograms depends strongly on the geometry of the microhole [13,14], Wilke and Zerihun presented a model to describe diffusion-controlled IT through a microhole [15], In that model, a cylindrical microhole is assumed to be filled with the organic phase, so that a planar liquid-liquid interface is located at the aqueous phase side of the membrane. Assuming that the diffusion is linear inside the cylindrical pore and spherical outside [Fig. 2(a)], the expression for the steady-state IT voltammo-gram is... 

Scanning electron microscopy and other experimental methods indicate that the void spaces in a typical catalyst particle are not uniform in size, shape, or length. Moreover, they are often highly interconnected. Because of the complexities of most common pore structures, detailed mathematical descriptions of the void structure are not available. Moreover, because of other uncertainties involved in the design of catalytic reactors, the use of elaborate quantitative models of catalyst pore structures is not warranted. What is required, however, is a model that allows one to take into account the rates of diffusion of reactant and product species through the void spaces. Many of the models in common use simulate the void regions as cylindrical pores for such models a knowledge of the distribution of pore radii and the volumes associated therewith is required. 

After passing through the boundary layer, the molecules of adsorbate diffuse into the complex structure of the adsorbent pellet, which is composed of an intricate network of fine capillaries or interstitial vacancies in a solid lattice. The problem of diffusion through a porous solid has attracted a great deal of interest over the years and there is a fairly good understanding of the mechanisms involved, at least for gas phase diffusion. Here, diffusion within a single cylindrical pore is considered and, then, the pore is related to the pellet as a whole. 

Transport in a microporous biomedical membrane is described in Fig. 11. Membranes consist of cylindrical liquid-filled pores of length l and radius rp with spherical solute molecules of radius rs diffusing through the pores. The solute... 

This equation applies to uniform cylindrical pores whose length equals the thickness of the catalyst through which the diffusion takes place. The actual diffusivity in common porous catalysts usually is intermediate between bulk and Knudsen. Moreover, it depends on the pore size distribution and on the true length of... 

A third model of iontophoretic flux arises from the idea of a retardation in the entry of solute ions into, and movement via, pores in the skin. Renkin [101] described a pore-restriction model for the diffusion of uncharged molecules through water-filled cylindrical pores. He suggested that diffusion would be restricted relative to the movement in aqueous solution by (a) steric hindrance inhibiting the entry of solute ions into the pore, and (b) hindrance due to frictional forces during the movement of solute ions through the pore. According to this model, Pjo , may be expressed by... 

In view of evidence such as that in Fig. 8-5, it is unlikely that detailed quantitative descriptions of the void structure of solid catalysts will become available. Therefore, to account quantitatively for the variations in rate of reaction with location within a porous catalyst particle, a simplified model of the pore structure is necessary. The model must be such that diffusion rates of reactants through the void spaces into the interior surface can be evaluated. More is said about these models in Chap. 11. It is sufficient here to note that in all the widely used models the void spaces are simulated as cylindrical pores. Hence the size of the void space is interpreted as a radius 2 of a cylindrical pore, and the distribution of void volume is defined in terms of this variable. However, as the example of the silver, catalyst indicates, this does not mean that the void spaces are well-defined cylindrical pores. 

For diffusion through a cylindrical pore, an equation describing the dependence of diffusion rate on the porosity (e), temperature (T), pressure (P), and pore size (r) is given as... 

Figure 10.14 (a) Tetrakaidecahedron model of intermediate-stage sintering, b) Expanded view of one of the cylindrical pore channels. The vacancies can diffuse down the grain boundary (dashed arrow) or through the bulk (solid arrows). Note that in both cases the vacancies are annihilated at the grain boundaries. 

Geometrically, the simplest membranes are thin sheets with cylindrical pores. The diffusion of spherical solutes through cylindrical pores in a membrane can be described quite accurately, even for solutes that are nearly as large as the pore [69]. One commonly used semi-empirical expression for the diffusion of solutes in porous materials was developed by Renkin [70] to describe the diffusion of proteins in cellulose membranes. According to Renkin, the reduced diffusion coefficient is given by ... 

The Pore Flow Model uses the Hagen-Poiseuille Equation to describe solvent flow through cylindrical pores of the membrane. No membrane characteristics other than pore size or pore density are accounted for, and neither limitation of flux due to friction nor diffusion is considered. Flux occurs due to convection under an applied pressure. The equation is derived from the balance between the driving force pressure and the fluid viscosity, which resists flow (Braghetta (1995), Staude (1992)). Solvent flux ( ) is described by equation (3.26) and solute flux (Js) by equation (3.27), where rp is the pore radius, np the number of pores, T the tortuosity factor. Ax the membrane thickness and ct the reflection coefficient. 

For the case of diffusion through cylindrical pores, the restricted-diffusion parameter p is related to the solute Stokes diameter <4 and pore diameter rfpon as (69) ... 

As described in Sec. 35a, there are still many puzzling aspects of conf nra-tional diffusion that remain to be explained. About the only theoretical infcmna-tion available concerns the motion of spherical particles in liquids through cylindrical pores. Anderson and Quinn [71] have shown that the fective difiiirivity in straight, round pores (tortuosity t = 1.0) is given by ... 

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