Fractal Dimension D

Fractal dimension, D, is another crucial property that is used to describe fractal objects and shapes. It is a measure of the amount of irregularity, or complexity, possessed by an object. For lines, 1 D 2 and for surfaces, 2 D 3. The greater the value of D, the more complex the object. As described by Avnir, 63 D is obtained from a resolu- 

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Roughness factor calculated for a fractal surface, according to the fractal dimension D and probe area a... 

The value a = 1 corresponds to ideal capacitive behavior. The fractal dimension D introduced by Mandelbrot275 is a formal quantity that attains a value between 2 and 3 for a fractal structure and reduces to 2 when the surface is flat. D is related to a by... 

The power, [P], in the fractal power-law regime gives as the fractal dimension, d(. P = —df for each level of the fit, the parameters obtained using the unified model are G, Rg, B, and P. P is the exponent of the power-law decay. When more than one level is fitted, numbered subscripts are used to indicate the level—i.e., G —level 1 Guinier pre-factor. The scattering analysis in the studies summarized here uses two-level fits, as they apply to scattering from the primary particles (level 1) and the aggregates (level 2). 

FIGURE 17.4 USAXS and SALS results for samples Al, A2, and A3 described in the text. Each sample shows four structural levels with the i g for some of the levels indicated in the graph. The power-law value for the second level, corresponding to the mass-fractal dimension, d(, is also indicated. 

Fig. 9. Relation between relaxation exponent n and fractal dimension d for a three-dimensional network. In case of complete screening of excluded volume, values of 0 < n < 1 are possible if d is chosen between 1.25 and 2.5...

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

Because of its strong coupling with MW, its good adsorbent properties towards organic molecules [12], and its layer structure which enables it to form intercalated compounds [13], graphite has great potential in MW-assisted synthetic applications in organic chemistry, despite its weak fractal dimension (D x 2) [14]. 

In order to characterize the three-dimensional self-similar fractal surface, the self-similar fractal dimension d ss has been... 

Under the assumption that the morphology of the self-affine interface has the self-similar scaling property, the apparent selfsimilar fractal dimension d ss of the electrode was calculated... 

Figure 9. (a) Dependence of molecular surface area on probe radii for Dj and isomers of dendrimer 5 (C = 1). (b) Dependence of fractal dimension, D, on probe radii for the same isomers. The derivative in Eq. 3 was numerically approximated from the data illustrated in (a). 

Laszlo7 pointed out that solids of fractal dimension D near 2.0 are usually more efficient catalysts than those of D near 3.0. An adsorbed species diffusing across the surface of a catalyst will find a target much more quickly (i.e., catalysis will be more efficient) in space of dimension near 2 than of D near 3. We find, for example, that catalytic activities of variously prepared activated charcoals increase as we go from D = 3.0 to D = 1.9. 

The fractal dimension D of an ideal solid is 3.00, of a perfectly flat surface 2.00, and of a straight line 1.00—the familiar geometric dimensions. Porous solids and rough surfaces, however, have effective dimensions corresponding to fractional values of D. 

A useful parameter is the fractal dimension, D, which is the exponent in the relation between the mass, M, to a linear dimension, R ... 

Abad-Zapatero and Lin172 examined globular protein surfaces and suggested that the exponent of the Box-Cox transformation173 is a function of the fractal dimension D and the shape parameter S. D was approximated at 2.2 for two lysozymes and 2.4 for superoxide dismutase, which agrees well with previously reported values of 2.19[741 and 2.43,[681 respectively. 

Rabouille, Cortassa, and Aon[81 dried protein, glycoprotein, or polysaccharide containing brine solutions that resulted in dendritic-like fractal patterns. The fractal dimension, D = 1.79, was determined for the pattern afforded by an ovomucin-ovalbumin mixture (0.1 M NaCl). Similar D values were obtained for dried solutions of fetuin, ovalbumin, albumin, and starch the authors subsequently suggest that fractal patterning is characteristic of biological polymers. 

Determination of the fractal nature of a dendritic surface was first reported by Avnir and FarinJ911 Employing data obtained from solvent accessible surface area analysis (see Sect. 2.3.3.3) for the PAMAM dendrimers, a surface fractal dimension (D) was derived. Two methods were used. The first method applied the relation,... 

Figure 2.13. Determination of the fractal dimension (D — 2.40 and D = 2.43) for 3rd (top) and 4th (bottom) generation amide-based dendrimers. [See the table presented in Figure 2.8]...

These starburst dendrimers have been subjected 47 to two different fractal analyses I48 49 (a )A c/2 D)/2, where A is the surface area accessible to probe spheres possessing a cross-sectional area, o, and the surface fractal dimension, D, which quantifies the degree of surface irregularity and (b) A = dD, where d is the object size. Both methods give similar results with D = 2.41 0.04 (correlation coefficient = 0.988) and 2.42 0.07 (0.998), respectively. Essentially, the dendrimers at the larger generations are porous structures with a rough surface. For additional information on dendritic fractality, see Section 2.3. 

We have argued above that in ensemble of strongly contracted unknotted chains (paths) most of them have the fractal dimension d/ = 3. 

Figure 7.18. Microstructural analysis Rheological determination of fractal dimension (D) and exponential term (A) for milk fat (A) slowly (0. l°C/min) or (B) rapidly cooled (5.0°C/min) to 5°C.

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

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. 

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