Characteristics surface fractal

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. 

The Pore Structures and Surface Fractal Characteristics of Meso/Macroporous Carbon Specimens I, n, and III Calculated from the Analyses of the N2 Gas Adsorption Isotherms. Reprinted with permission from G. -J. Lee and S. -I. Pyun, Carbon, 43 (2005) 1804. Copyright 2005, with permission from Elsevier. 

This section attempts to examine macromolecular geometry, and in particular dendritic surface characteristics, from the perspectives of self-similarity and surface irregularity, or complexity, which are fundamental properties of basic fractal objects. It is further suggested that analyses of dendritic surface fractality can lead to a greater understanding of molecule/solvent/dendrimer interactions based on analogous examinations of other materials (e.g., porous silica and chemically reactive surfaces such as found in heterogeneous catalysts). 52 ... 

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. 

The box-counting fractal dimension is calculated from the log-log plot of the number of occupied boxes, versus side length, /,. The slope of this line is used to determine the fractal dimension. Both very small and very large box sizes should be exempt from the calculation to reduce surface artifacts (Awad and Marangoni 2005). This method is most sensitive to the degree of fill as well as particle size and it can be expected that if there are void volumes it will have a characteristic low fractal dimension. 

The number ds is the fractal dimension of this two-dimensional surface fractal. The fractal dimension of a three-dimensional surface fractal can be defined in a similar manner. In particular, if S(r) is the surface area measured with a measuring tool of characteristic area r2, it depends on r as... 

Environmental particles, microbial colonies, and even patterns of movement of organisms can be characterized in terms of several different fractal dimensions [14] (Table 1.1). The fractal dimension of the surface Ds (boundary/interface) of a solid structure is obviously an important characteristic, but many of the physical properties of solids also depend on the scaling behavior of the entire solid and/or of its pores. Systems where surface and mass scale similarly are termed mass fractal systems, those where surface and the pore volumes scale similarly are described as pore fractals and systems where only the surface is fractal are designated as surface fractals (Table 1.1 Figure 1.2). 

In contrast to their solution characteristics, humic and fill vie acids in the solid state are surface fractals. Values of A are in the range 2 < A < 3 (Table 7.1). To date, these values are observed even with the varying pH or ionic strength conditions that existed when the samples were isolated. 

Aggregates are further classified as surface fractals, volume fractals or non-lfactal objects. Kolb and Herrmann [37] have introduced a mechanism to model aggregation of clusters at high concentration and investigated it by Monte-Carlo simulations. Surfaces of most materials are fractal on a molecular scale to chemically reactive surfaces. Farin and Avnir define the reaction dimension as characteristic parameter of the reaction which is related to the reaction rate v and particle radius R as... 

Hence, the adduced above data have shown the physical grounds for often cited in literature correlations of impact toughness and fracture surface fractal dimension at any rate for polymers. And on the contrary, such grounds exist for correlations of plasticity characteristics or y) and r/g,. The Eq. (10.27) gives the possibility of the value prediction, since <7, as it follows from the Eq. (4.50), is a function of structure characteristic -Poisson s ratio V. In its turn, G and critical defect characteristics knowledge gives the possibility of p prediction [47]. 

FIGURE 10.17 The dependences of fracture surface fractal dimension d, calculated according to the Eq. (10.28), on characteristic ratio C at fixed dissipated energy fraction Tjj 0.2 (1), 0.5 (2) and 0.8 (3). The shaded Imes mdicate level for ductile fracture ( ) and super plasticity (2 ) [57]. 

Indeed, on the basis of this slope, different kinds of shapes and of fractal structures can be distinguished [111,133,134]. D= 1, 2, or 3 are expected for rods, disks, and spheres, respectively [128-132,134]. A slope D = 4 represents a smooth surface for the scattering particle, whereas 3 < D < 4 is due to rough interfaces, characteristic of surface fractals. 

Another important class of morphology is fractal, which includes mass fractal and surface fractal. Although mass and surface fractals appear remarkably different, they share a common trace essential to fractal objects self-affinity. Self-affinity means that the morphology of interest looks the same under different length scales, i.e., within certain limits, the essential geometric characteristics of a fractal object are unchanged by magnification. 

Fig.6 SAXS, USAXS and SANS measurement son Bakken shale (geological) showing a surface fractal characteristic over six decades of I(Q)-...

Figure 8 (a) Time evolution of the SANS profiles during enzymatic polymerization to cellulose. SANS from the enzyme solution was also included as a reference, (b) Time change in surface characteristics (the surface fractal dimension, Ds, and front factor. A), and monomer conversion. Cm. From Tanaka, H. Koizumi, S. Hashimoto, T. etal. Macromolecules 2007, 40,6304-6315. ... 

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