Surface fractal definition

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. 

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. 

Studies have shown that the definition formula of surface fractal dimension is as follows ... 

Hence, the results stated above demonstrated the important role of catalyst (montmorillonite) surface fractal geometry in its eatalytie properties definition [4]. 

The classical definition of pressnre as the ratio of a force on an area is not always of easy application. For instance, in a gas, in a cloud of vapor, or in plasma, it is difficult to find a well-defined area unless one uses a differential definition, which amounts to the local pressure. Second, if one needs to have a surface systematically available to be able to define a pressure inside a volume, one comes up sometimes against serious difficulties, for instance, in the case of finely divided surfaces (fractal medium) or randomly varying ones (fluctuations, chaos). 

There are some very special characteristics that must be considered as regards colloidal particle behavior size and shape, surface area, and surface charge density. The Brownian motion of particles is a much-studied field. The fractal nature of surface roughness has recently been shown to be of importance (Birdi, 1993). Recent applications have been reported where nanocolloids have been employed. Therefore, some terms are needed to be defined at this stage. The definitions generally employed are as follows. Surface is a term used when one considers the dividing phase between... 

Mandelbrot introduced a new geometry in a book, which was first published in French in 1975, with a revised English edition [64] in 1977. In 1983 he published an extended and revised edition that he considered to be the definitive text [65]. Essentially he stated that there are regions between a straight line that has a dimension of 1, a surface that has a dimension of 2 and a volume that has a dimension of 3, and these regions have fractional dimensions between these integer limits. Kaye has presented an excellent review of the importance of fractal geometry in particle characterization [66]. 

Note that fractals (self-similar sets with fractal dimension) were first studied and described by mathematicians long before the publications of Mandelbrot, when such fundamental definitions as function, line, surface, and shape were analyzed. 

Figure 28 Definition of the angle 6 from the Poincare section of a torus attractor derived from experimental data. The index labels the order with which points appear in this section as the trajectory winds its way over the surface of the torus. This definition can be generalized to a wrinkled or fractal torus.

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