Nonuniform Pores

An example of the electrical response of a nonuniform pore structure was given in the previous section. An aiternative approach is to use the transmission line approximation discussed in Section 3.2. The model is modified by allowing r and c to be functions of x. Equation (15) Is then replaced by  

One type of behavior is observed experimentally to be particularly widespread. It is described by the constant-phase element (CPE)  

In the complex plane, this element appears as a straight line inclined at the angle a ir/2 to the real axis. Macdonald has shown that, for physical situations in which a relaxation time description is appropriate, CPE behavior may arise from an exponential distribution of activation energies for the relaxation process (30). For porous electrodes, such a description 

The transmission line response itself, in the region where the current penetration is not complete, is an example of CPE behavior. It should perhaps be emphasized here that CPE behavior cannot continue indefinitely to either infinitely high or low frequencies. At low frequencies, the behavior will ultimately become purely capacitative and the phase angle will approach ir/2. At high frequencies, the current will ultimately withdraw from even the shallowest pores, and the phase angle will approach ir/4 in the simple transmission line case or n/2, according to the more realistic calculations based on Laplace s equation described above. 

It has been suggested that fractal pore structures lead to CPE behavior during double-layer charging (31 )(32). In afractal structure, all length scales are present, and for a process, such as diffusion, that contains a well-defined size scale ( [D/w it seems to be true that a CPE with an exponent closely related to the fractal dimension occurs in the response (33). For the case of double-layer charging, however, the situation is not so clear because there is no similar characteristic distance involved (34)(35). It appears that the fractal blocking interface does lead to CPE behavior, but the exponent is not simply related to the fractal dimension. 

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Membrane Morphology—Pores, Symmetric, Composite Only nucleopore and anodyne membranes have relatively uniform pores. Reverse osmosis, gas permeation, and pervaporation membranes have nonuniform angstrom-sized pores corresponding to spaces in between the rigid or agamic membrane molecules. Solute-membrane molecular interactions are very high. Ultrafiltration membranes have nonuniform nanometer sized pores with some solute-membrane interactions. For other microfiltration membranes with nonuniform pores on the submicrometer to micrometer range, solute-membrane interactions are small. 

Activity-versus-time curves shown in Fig. 25 for alumina-supported Ni and Ni bimetallic catalysts show two significant facts (1) the exponential decay for each of the curves is characteristic of nonuniform pore-mouth poisoning, and (2) the rate at which activity declines varies considerably with metal loading, surface area, and composition. Because of large differences in metal surface area (i.e., sulfur capacity), catalysts cannot be compared directly unless these differences are taken into account. There are basically two ways to do this (1) for monometallic catalysts normalize time in terms of sulfur coverage or the number of H2S molecules passed over the catalysts per active metal site (161,194), and (2) for mono- or bimetallic catalysts compare values of the deactivation rate constant calculated from a poisoning model (113, 195). 

It is worth noting that this capacitve response appears also in models, which regard the more realistic situation of nonuniform pore size distributions [12,13]. The frequency range applied in the impedance spectroscopical investigation was between 20 kHz and 8.25 mHz with an amplitude of 30 mV at open circuit. 

These models will be presented in order of increasing complexity in the representation of pore space geometry, from individual uniform pores to interacting nonuniform pore networks. 

Pakula, R.J., and R. A. Greenkom. 1971. An experimental investigation of a porous medium model with nonuniform pores. AIChEJ. 17 1265-1268. 

Despite our use of a capillary model to characterize a porous medium, most porous beds employed for chromatographic purposes are random and generally the medium is isotropic. In such media, the effective solute dispersivity still arises from the nonuniform pore velocity coupled with molecular diffusion... 

IV—Nonuniform pores Chalklike. silicagels obtained from hydrolysis of salts from strong acids in a silicate solution... 

A more realistic picture of the pore neck would be needed to rationalize pore swelling at the PEM surface. This treatment would include a complex, nonuniform arrangement of sulfonic acid groups, a nonuniform pore radius, and a finite proton concentration near the meniscus. Inclusion of these effects would render an analytical treatment of the problem impossible. 

Schematic illustration of cracking resulting from draining of nonuniform pores, (a) Liquid covers surface before drying starts (b) larger pores empty first, after critical point. The higher tension in the smaller pore creates stress that cracks the wall between the pores. From Zarzycki et at. [42).

Micropore Diffusion. In very small pores in which the pore diameter is not much greater than the molecular diameter the diffusing molecule never escapes from the force field of the pore wall. Under these conditions steric effects and the effects of nonuniformity in the potential field become dominant and the Knudsen mechanism no longer appHes. Diffusion occurs by an activated process involving jumps from site to site, just as in surface diffusion, and the diffusivity becomes strongly dependent on both temperature and concentration. 

Some studies of potential commercial significance have been made. For instance, deposition of catalyst some distance away from the pore mouth extends the catalyst s hfe when pore mouth deactivation occui s. Oxidation of CO in automobile exhausts is sensitive to the catalyst profile. For oxidation of propane the activity is eggshell > uniform > egg white. Nonuniform distributions have been found superior for hydrodemetaUation of petroleum and hydrodesulfuriza-tion with molybdenum and cobalt sulfides. Whether any commercial processes with programmed pore distribution of catalysts are actually in use is not mentioned in the recent extensive review of GavriUidis et al. (in Becker and Pereira, eds., Computer-Aided Design of Catalysts, Dekker, 1993, pp. 137-198), with the exception of monohthic automobile exhaust cleanup where the catalyst may be deposited some distance from the mouth of the pore and where perhaps a 25-percent longer life thereby may be attained. 

Heterogeneity, nonuniformity and anisotropy are defined as follows. On a macroscopic basis, they imply averaging over elemental volumes of radius e about a point in the media, where e is sufficiently large that Darcy s law can be applied for appropriate Reynolds numbers. In other words, volumes are large relative to that of a single pore. Further, e is the minimum radius that satisfies such a condition. If e is too large, certain nonidealities may be obscured by burying their effects far within the elemental volume. 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

However, in most cases the AW(D) dependencies are distinctly nonlinear (Fig. 9), which gives impulse to further speculations. Clearly, dependencies of this type can result only from mutual suppression of the hydrogel particles because of their nonuniform distribution over the pores as well as from the presence of a distribution with respect to pore size which does not coincide with the size distribution of the SAH swollen particles. A considerable loss in swelling followed from the W(D) dependencies, as shown in Fig. 9, need a serious analysis which most probably would lead to the necessity of correlating the hydrogel particle sizes with those of the soil pores as well as choice of the technique of the SAH mixing with the soil. Attempts to create the appropriate mathematical model have failed, for they do not give adequate results. 

FIGURE 18.4 Concerning the derivation of equations for nonuniform current distribution (a) in a flat electrode (b) in a cylindrical pore. 

One of the main reasons for a lower specific activity resides in the fact that electrodes with disperse catalysts have a porous structure. In the electrolyte filling the pores, ohmic potential gradients develop and because of slow difiusion, concentration gradients of the reachng species also develop. In the disperse catalysts, additional ohmic losses will occur at the points of contact between the individual crystallites and at their points of contact with the substrate. These effects produce a nonuniform current distribution over the inner surface area of the electrode and a lower overall reaction rate. 

It follows from previous discussion that the destabilizing electrostatic contribution grows in absolute value with x (with increasing A.). But the influence of the nonuniform electrical force is overwhelmed by the stabilizing bending and stretching contributions. As a result, the traditional smectic model cannot explain how a small transmembrane voltage can lead to membrane breakdown. The obvious solution is to abandon this approach and to develop an alternative, such as the pore formation model. However, as we noticed before, this approach postulates rather than proves the appearance of hydrophobic pores. 

However, this expression assumes that the total resistance to flow is due to the shear deformation of the fluid, as in a uniform pipe. In reality the resistance is a result of both shear and stretching (extensional) deformation as the fluid moves through the nonuniform converging-diverging flow cross section within the pores. The stretching resistance is the product of the extension (stretch) rate and the extensional viscosity. The extension rate in porous media is of the same order as the shear rate, and the extensional viscosity for a Newtonian fluid is three times the shear viscosity. Thus, in practice a value of 150-180 instead of 72 is in closer agreement with observations at low Reynolds numbers, i.e.,... 

For an unconsolidated porous medium with uniform pore size distribution, the ratio of VJHa approaches its maximum value much more rapidly than that found in a medium where pore size distribution is nonuniform, leading to the significant variation of water saturation in the contaminated zone. 

In general, pores swell nonuniformly. As a simplification, fhe random network was assumed to consist of fwo types of pores. In fhis fwo-stafe model, nonswollen or "dry" pores (referred to later as "red" pores) permit only a small residual conductance due to tightly bound surface water, which solvates the charged surface groups. Swollen or "wet" pores (referred to later as "blue" pores) contain extra water in the bulk, allowing them to promote the high bulk-like conductance. Water uptake by the membrane corresponds to the swelling of wef pores and to the increase of their relative fraction. 

For any battery applications, the separator should have uniform pore distribution to avoid performance losses arising from nonuniform current densities. The submicrometer pore dimensions are critical for preventing internal shorts between the anode and the cathode of the lithium-ion cell, particularly since these separators tend to be as thin as 25 /[Pg.192]

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