Fractal porosity

Another example of a fractal surface is anodically etched p-type silicon. It has a fractal porosity and shows photoluminescence and even electroluminescence, which would be impossible in compact bulk silicon. Electrons behave differently in fractally distributed solids than in three-dimensional extended lattices they are supposed to be more localized. The density of states p in a three-dimensional box is... 

K tnxcb fractal porosity, duxoatography, interferouetiy, porosity oi orgaoic tissue, Imited selfsimlarity... 

The dissolution channels (wormholes), obtained under certain conditions of attack of carbonate rocks by hydrochloric acid, have been recently proven to have a fractal geometry. An equation was proposed, relating the increase of the equivalent wellbore radius (i.e. the decrease of the skin) to the amount of acid injected, in wellbore geometry and in undamaged primary porosity rocks. This equation is herein extended to damaged double porosity formations through minor modifications. 

In damaged double porosity reservoirs, the influence of the fractal dimension df is masked by the severity ratio kpp/kpp (Equation 12). ... 

Specimen BET surface area - I l .rr Micro- Meso- Macro- porosity" porosity porosity" Average pore diameter, Dave Standard Pore radius of the deviation of maximum differential PSD, a pore volume, Surface fractal dimension, tfsurf,SP Low-end thickness, High-end thickness, ad. max... 

Surface area and porosity are examples of polysaccharide properties other than gelation that are amenable to fractal analysis. 

A. Static Percolation in Porous Materials, Fractal Concept, and Porosity Determination... 

The Values of KWW Exponent v, Fractal Dimension Dp, Porosity 4>m Obtained from the Relative Mass Decrement (A, B, C, and D Glasses) and BETA (E, F, and G Glasses) Measurements and Average Porosity (4>p) Estimated from Dielectric Spectra for Porous Glasses Samples. 

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. 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

Taking into account that E,0 0 (since topologically Dy < 3, and d = 3), we obtain a simple approximate relationship between the average porosity of a glass and the fractal dimension of the pore space, which reads... 

Note that in our approximation, due to the randomized character of the fractal medium the average porosity of the disordered porous glasses determined by (70) depends only on the fractal dimension Dr and does not exhibit any scaling behavior. In general, the magnitude of the fractal dimension may also depend on the length scale of a measurement extending from X to over... 

Porosity within a particle is a manifestation of the shape of a particle. Fractal particles will have internal porosity as a result of their shapes. Fractal particles with low fractal dimensions (i.e., <2.0) will have a broad pore size distribution, where the largest pore approaches the size of the aggregate. Fractal particles with large fractal dimensions (i.e., >2.0) will have narrower pore size distributions with most of the porosity occurring at a size much smaller than that of the aggregate. Calcination of metal salt particles or metal hydroxides to produce oxides is another common method to produce internal porosity. In the gas evolution that takes place in transformation to the oxide, pores are opened up in the particle structure. The opening of pores in a hydrous alumina powder can increase its surface area from 0.5 m /gm (its external area) to 450 mVgm (its internal pore area). 

Once the average shape of the particles in a powder has been established by one of several means and its ramifications on shape factors, fractal dimensions, surface area, and porosity are determined, the distribution of particle sizes is the next piece of information necessary to characterize the powder. 

Da Rocha, B. R. P. Habashy, T.M., 1997. Fractal geometry, porosity and complex resistivity from hand specimens to field data, from Lovell, M.A. Harvey, P.K., in Developments in Petrophysics, Geology Society Special Publication, N° 122, pp. 287-297. 

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