Surface-area fractal dimension

Mandelbrot defines the surface area fractal dimension 8 as... 

The one-dimensional lines, two-dimensional surfaces and three-dimensional solids of Euclidean geometry are concepts so familiar to us that we tend to regard them as common sense . Fractals involve less familiar concepts, such as curves between two points, which have infinite length, and surfaces with infinite surface area. Fractals can be characterized by a dimension, but the dimension is fractional. For example, the triadic Koch curve (see Fractals Fig. 1) has a dimension of approximately 1.26186 and the surface with an infinite succession of regular tetrahedral asperities (ibid. Figure 4), a dimension of approximately 2.58496. In this article, the meaning and calculation of fractal dimension are discussed. Texts on the mathematics of fractals introduce different kinds of fractal dimension, which are beyond the scope of this article. The dimension discussed here is strictly the self-similarity dimension. Further aspects of fractal dimensions are considered in Fractals and surface roughness. 

The currently useful model for dealing with rough surfaces is that of the selfsimilar or fractal surface (see Sections VII-4C and XVI-2B). This approach has been very useful in dealing with the variation of apparent surface area with the size of adsorbate molecules used and with adsorbent particle size. All adsorbate molecules have access to a plane surface, that is, one of fractal dimension 2. For surfaces of Z> > 2, however, there will be regions accessible to small molecules... 

Polycondensation reactions (eqs. 3 and 4), continue to occur within the gel network as long as neighboring silanols are close enough to react. This increases the connectivity of the network and its fractal dimension. Syneresis is the spontaneous shrinkage of the gel and resulting expulsion of Hquid from the pores. Coarsening is the irreversible decrease in surface area through dissolution and reprecipitation processes. 

For a fractal surface D > 2, and usually D < 3. In simple terms the larger D, the rougher the surface. The intuitive concept of surface area has no meaning when applied to a fractal surface. An area can be computed, but its value depends on both the fractal dimension and the size of the probe used to measure it. The area of such a surface tends to infinity, as the probe size tends to zero. 

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

The CFD model described above is adequate for particle clusters with a constant fractal dimension. In most systems with fluid flow, clusters exposed to shear will restructure without changing their mass (or volume). Typically restructuring will reduce the surface area of the cluster and the fractal dimension will grow toward d — 3, corresponding to a sphere. To describe restructuring, the NDF must be extended to (at least) two internal coordinates (Selomulya et al., 2003 Zucca et al., 2006). For example, the joint surface, volume NDF can be denoted by n(s, u x, t) and obeys a bivariate PBE. 

In order to characterize the self-affine scaling properties of the fractal Pt films, the self-affine fractal dimensions of the film surfaces t/Fsa were determined by using the perimeter-area... 

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. 

On the other hand, the surface fractal dimension characterizes the pore surface irregularity the larger the value of surface fractal dimension is, the more irregular and the rougher is the pore surface. Since the pore structure is closely related to the electroactive surface area which plays a key role in the increase of capacity in practical viewpoint, many researchers have investigated the microstructure of the pores by using fractal geometry. 

Note 3 For the surface area of a fractal object, s r m which s is the surface area contained within a radius, r, measured from any site or bond and d is termed the surface fractal dimension. 

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). 

As already mentioned, we chose three different physicochemical properties for studying the influence of the surface area and fractal dimension in the ability of dendritic macromolecules to interact with neighboring solvent molecules. These properties are (a) the differential chromatographic retention of the diastereoisom-ers of 5 (G = 1) and 6 (G = 1), (b) the dependence on the nature of solvents of the equilibrium constant between the two diastereoisomers of 5 (G = 1), and (c) the tumbling process occurring in solution of the two isomers of 5 (G = 1), as observed by electron spin resonance (ESR) spectroscopy. The most relevant results and conclusions obtained with these three different studies are summarized as follows. 

Pure nickel electrodeposits with macropores were prepared from electrolytic solutions of 0.2 mol dm NiCl2 and NH4CI with concentrations varying between 0.25 and 4 mol dm [64]. The effects of the electrodeposition current density and the NH4CI concentration on the surface morphology were determined. Surface area, faradaic efficiency, and fractal dimension... 

One way of measuring the fractal dimension of aggregates is discussed in Chapter 5 (See Section 5.6a and Example 5.4). In the example below, we illustrate the relation between the fractal structure of aggregates and the surface area of the aggregates. 

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