Physical fractal dimension

A relatively new method to determine the fractal dimension is by using oil permeability measurements. In this method, the permeability coefficient, B, is measured for fat samples containing different SFC. This physical fractal dimension, the permeability fractal dimension. Dp, links the volumetric flow rate of liquid oil penetrating a colloidal fat crystal network with its SFC as (Bremer et al. 1989) ... 

Physically, the wetting abdity increases (the contact angle decreases) as the values of the fractal dimension of the electrode increases if the electrode material is same. However, in this study, we could not obtain a good correlation between the fractal dimensions and the wetting abilities as shown in Table 1. It means that not only the physical properties such as the surface irregularity and roughness but also the chemical interaction between electrolyte and electrode were important in wetting ability. 

With the experimental results about the wetting ability and the fractal dimension of four kinds of anode electrodes, we could conclude the following. The addition of NisAl could make the electrolyte wet the electrode very well. The pore structures of all the electrodes prepared in this study were highly irregular and rough. Finally, the chemical properties of the surfaces were as important as the physical properties in determining the wetting ability of the electrodes in this study. 

Su-Il Pyun provide a comprehensive review of the physical and electrochemical methods used for the determination of surface fractal dimensions and of the implications of fractal geometry in the description of several important electrochemical systems, including corroding surfaces as well as porous and composite electrodes. 

Diffusion-limited electrochemical techniques as well as physical techniques have been effectively used to determine the surface fractal dimensions of the rough surfaces and interfaces made by electrodeposition, " fracture, " vapor deposition, ... 

The structure of this review is composed of as follows in Section II, the scaling properties and the dimensions of selfsimilar and self-affine fractals are briefly summarized. The physical and electrochemical methods required for the determination of the surface fractal dimension of rough surfaces and interfaces are introduced and we discuss the kind of scaling property the resulting fractal dimension represents in Section III. 

The present article summarized the fractal characterization of the rough surfaces and interfaces by using the physical and the electrochemical methods in electrochemistry. In much research, both the physical and the electrochemical methods were used to evaluate the fractal dimension and they are complementary to each other. It should be stressed that the surface fractal dimension must be determined by using the adequate method, according to the inherent scaling properties of the rough surfaces and interfaces. 

Computer simulations combined with experiments have also shown that one can deduce from the fractal dimension the nature of nucleation and growth of particles and what chemical and physical mechanisms control the formation of particle aggregates. We consider this briefly before proceeding to other topics. 

On the other hand, it is impossible to apply the SP method to the correct description of gas adsorption in the micropores, since the adsorption in the micropores does not occur by multilayer adsorption but by micropore volume filling process. In this case, the pore fractal dimension gives a physical importance for the description of structural heterogeneity of the microporous solids. Terzyk et al.143"149 have intensively investigated the pore fractal characteristics of the microporous materials using gas adsorption isotherms theoretically simulated. 

The power-law relation in Eq. 6.1 can be interpreted physically as indicative of a cluster fractal.12 The exponent D is then termed the cluster fractal dimension. Some basic concepts about cluster fractals are introduced in Special Topic 3 at the end of this chapter. Suffice it to say here that Eq. 6.1 can be pictured as a generalization of the geometric relation between the number of primary particles in a cluster that is d-dimensional (d = 1, 2, or 3) and the d-dimensional size of the cluster. For example, if a cluster is one-dimensional (d = 1), it can be portrayed as a straight chain of, say, circular primary particles of diameter L0. The number of particles in a chain of length L is... 

Daoud M, Stanley HE and Stauffer D, "Scaling, Exponents and Fractal Dimensions", In Mark JE (Ed), "Physical Properties of Polymers Handbook", 2nd Ed, Springer-Verlag, Berlin, 2007, Chap. 6. 

Table 17.7 summarizes the effects of the microstructural factors on the microscopy fractal dimensions, Dj, y, and Zlpr- Different fractal dimensions reflect different aspects of the microstructure of the fat crystal networks and thus have different meanings. It is necessary to define which structural characteristic is most closely related to the macroscopic physical property of interest (mechanical strength, permeability, diffusion) and then use the fractal dimension that is most closely related to the particular structural characteristic in the modeling of that physical property. 

The Cantor set C, as well as the fractal shown in Fig. 2.10, are selfsimilar fractals with simple construction rules. Many fractals encountered in physical and mathematical apphcations are not at all that simple. In order to compute their dimensions, one has to use numerical methods. The following are two frequently employed numerical methods for computing fractal dimensions. 

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