Fractals pore fractal

Figure 8.13. Different kinds of fractals, (a) Surface fractal, (b) Mass fractal, (c) Pore fractal... 

Porous materials have attracted considerable attention in their application in electrochemistry due to their large surface area. As indicated in Section I, there are two conventional definitions concerning with the fractality of the porous material, i.e., surface fractal and pore fractal.9"11 The pore fractal dimension represents the pore size distribution irregularity the larger the value of the pore fractal dimension is, the narrower is the pore size distribution which exhibits a power law behavior. The pore fractal dimensions of 2 and 3 indicate the porous electrode with homogeneous pore size distribution and that electrode composed of the almost samesized pores, respectively. 

Soil Fraction % Organic Matter PZC Pore Diameter (nm) % Surface Area in <10-nm Pores Fractal Dimension"6 ... 

There are two conventional definitions in describing the fractality of porous material - the pore fractal dimension which represents the pore distribution irregularity56,59,62 and the surface fractal dimension which characterizes the pore surface irregularity.56,58,65 Since the geometry and structure of the pore surfaces are closely related to the electro-active surface area which plays a key role in the increases of capacity and rate capability in practical viewpoint, the microstructures of the pores have been quantitatively characterized by many researchers based upon the fractal theory. 

On the other hand, it is impossible to apply the SP method to the correct description of gas adsorption in the micropores, since the adsorption in the micropores does not occur by multilayer adsorption but by micropore volume filling process. In this case, the pore fractal dimension gives a physical importance for the description of structural heterogeneity of the microporous solids. Terzyk et al.143"149 have intensively investigated the pore fractal characteristics of the microporous materials using gas adsorption isotherms theoretically simulated. 

Recently, Lee and Pyun have focused on the characterization of pore fractality of the microporous carbon powder specimens by using nitrogen gas adsorption method based upon the D-A adsorption theory in consideration of PSD with pore fractality. Figure 5 envisages the nitrogen gas adsorption isotherm obtained from the as-reactivated carbon powder specimen prepared by reactivation of the commercially as-activated carbon powder at 1000 °C in an atmosphere of C02/C0 gas mixture for 2 h. The solid... 

On the other hand, for the microporous carbons with pore size distribution (PSD) with pore fractality, the pore fractal dimensions56,59,62 which represent the size distribution irregularity can be theoretically calculated by non-linear fitting of experimental adsorption isotherm with Dubinin-Astakhov (D-A) equation in consideration of PSD with pore fractality.143"149 The image analysis method54,151"153 has proven to be also effective for the estimation of the surface fractal dimension of the porous materials using perimeter-area method.154"159... 

Apore.sp Pore fractal dimension determined by using the... 

In the context of particle aggregation, the particular fractal structures called mass fractals are usually considered. Mass fractals correlate the location of mass with radius, as opposed to other types of fractal constructs such as boundary fractals, surface fractals, and pore fractals (see Ref. 17). The power law which correlates the size of an aggregate with the number of primary particles in the aggregate is given by... 

Coppens and Froment (1995a, b) employed a fractal pore model of supported catalyst and derived expressions for the pore tortuosity and accessible pore surface area. In the domain of mass transport limitation, the fractal catalyst is more active than a catalyst of smooth uniform pores having similar average properties. Because the Knudsen diffusivity increases with molecular size and decreases with molecular mass, the gas diffusivities of individual species in... 

The selectivity and deactivation processes in pore fractals such as the Sier-pinski gasket were simulated by Gavrilov and Sheintuch (1997) and Sheintuch (1999). Their studies investigated, e.g., the effect of the fractal pore structure on the selectivity of a system that incorporates two parallel reactions. Geometrical factors, which influence dynamic processes in a porous fractal solid media, were also investigated by Garza-Lopez and Kozak (1999). 

Note that the fractal dimensions discussed here are the fractal dimensions of the excitation transfer paths connecting the hydration centers located on the inner surface of the pores. Due to the low humidity, all of the water molecules absorbed by the materials are bound to these centers. The paths of the excitation transfer span along the fractal pore surface and depict the backbone of clusters formed by the pores on a scale that is larger than the characteristic distance between the hydration centers on the pore surface. Thus the fractal dimension of the paths Dp approximates the real surface fractal dimension in the considered scale interval. For random porous structures, Dp can be also associated with the fractal dimension D, of the porous space Dp = Dr. Therefore, the fractal dimension Dp can be used for porosity calculations in the framework of the fractal models of the porosity. 

Figure 10. A hierarchy of stochastic fractal pore development a. Straight cylindrical constant diameter, b. Straight variable diameter, c. Tourtous variable diameter, d. Tortous non-circular cross-section,...

The surface property of a solid is characterized by the nature of the surface boundary. The surface boimdary is expected to be related to the underlying geometric nature of the surface, hence its fractal dimension. Many properties of the solid depend on the scaling behavior of the entire solid and of the pore space. The distribution of mass in the porous solid and the distribution of pore space may also reflect the fractal nature of the surface. If the mass and the surface scale are alike, that is, have the same power-law relationship between the radius of a particle and its mass, then the system is referred to as a mass fractal. In a similar manner if the pore volume of porous material has the same power-law relationship between the pore volume and radius as that of the surface, then it is described as a pore fractal. 

It has long been known that, when the network of pores is fractal, diffusion by molecular movement in this network differs from the transport in media with properties independent of scale. In particular, diffusion of solutes in a fractal pore network does not obey Fick s law, and anomalous, or non-Fickian diffusion takes place instead (Gefen at al., 1983). When Fick s second law... 

Although anomalous diffusion is expected in fractal pore systems, the presence of anomalous diffusion does not prove that the porous media is fractal. A heterogeneity along transport pathways may result in an anomalous transport regardless of the presence or the absence of self-similarity of the pore space (Beven et al., 1993). The physical interpretation of Levy motions does not presume the presence of fractal scaling in the porous media in which the motions occur (Klafter et al, 1990). The applicability of the FADE may be closely related to the distribution of pore-water velocities. In saturated media, the presence of heavy-tailed distributions of the hydraulic conductivity directly implies the validity of the FADE (Meer-schaert et al., 1999 Benson et al., 1999). The heavy-tailed hydraulic conductivity distributions were found in geologic media (Painter, 1996 Benson et al., 1999). Heavy-tailed velocity distributions can also be expected in unsaturated and structured soils, and therefore the FADE may be a useful model in these conditions. 

Perrier, E., M. Rieu, G. Sposito, and G. de Marsily. 1996. Models of the water retention curve for soils with a fractal pore size distribution. Water Resour. Res. 32 3025-3031. 

Santra, S.B., B. Sapoval, and O. Haeberle. 1997. Levy distribution of collisions for Knudsen diffusion in fractal pores, p. 350-355. In J.L. Vehel et al. (ed.) Fractals in engineering. From theory to industrial applications. Springer-Verlag, New York. 

Euclidean geometry fails to describe disordered surfaces such as real solid surfaces. Fractal geometry, which has been developed to overcome this obstacle, covers surface, mass, and pore fractality. It has been pointed out that the diffusion process can be used to characterize the fractal dimension of a rough surface. The impedance response of a rough electrode could be used for the characterization of the roughness and. 

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