Fractals, homogeneous

For compact, homogeneous objects in tliree dimensions, p= 3. Colloidal aggregates, however, tend to be ratlier open, fractal stmctures, witli 3. For a general introduction to fractals, see section C3.6 and [61]. 

E. V. Albano. Critical exponents for the irreversible surface reaction A + B AB with B desorption on homogeneous and fractal media. Phys Rev Lett 69 656-659, 1992. 

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional figures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the figure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]). 

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. 

Chapter 15 - It was shown, that the reesterification reaction without catalyst can be described by mean-field approximation, whereas introduction of catalyst (tetrabutoxytitanium) is defined by the appearance of its local fluctuations. This effect results to fractal-like kinetics of reesterification reaction. In this case reesterification reaction is considered as recombination reaction and treated within the framework of scaling approaches. Practical aspect of this study is obvious-homogeneous distribution of catalyst in reactive medium or its biased diffusion allows to decrease reaction duration approximately twofold. 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

In figure 1 the kinetic curves of reesterification reactions without catalyst and in the presence of TBT are shown. The attention is draw by itself both quantitative and qualitative differences of these Q(t) curves. The quantitative difference is expressed by much faster growth Q at t increase due to catalyst presence that was expected. The qualitative change is reflected in the Q(t) curve form change. If in the absence of TBT linear dependence was obtained, which indicates on the reaction proceeding in Euclidean (homogeneous) space [7], then in TBT presence a typical curvilinear 0(1) dependence was obtained with reaction rate dQ / dt decrease with t increase. Such reactions are typical for heterogeneous (fractal)... 

A reaction occurring at an interface of two phases. Some heterogeneous processes also display homogeneous aspects for example, a particular reaction may occur in only one of the system s phases (e.g., the rapid dissolution of a gas into a liquid followed by a particular reaction). See Fractal Reaction Kinetics... 

So, to date, evidence that galaxy distributions are fractal with D Pi 2 on small to medium scales is widely accepted, but there is a lively open debate over the existence, or otherwise, of a crossover to homogeneity on large scales. 

Cluster fractals that are created by diffusion-limited flocculation processes are described mathematically by power-law relationships like those in Eqs. 6.1 and 6.5. These relationships are said to have a scaling property because they satisfy what in mathematics is termed a homogeneity condition 22... 

See, for example, Section2.7 of J. Feder, Fractals, Plenum Press, New York, 1988, for a discussion of homogeneity conditions. Homogeneous functions are described in detail in Section 11.1 of H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, New York, 1971. 

The fractal concept is based on the assumption of reproduction of the general elements of structure of porous materials at all levels—from microscopic to macroscopic ones. This assumption is valid for numerous macroporous materials, while it is too difficult to check its validity for microporous ones. However, based on general thermodynamic considerations, one may assume that fractal concepts also apply to some of microporous materials. As it is shown below, the main condition of the applicability of the fractal approach to microporous materials consists in their homogeneity. However, one has to take into account that this strict analysis does not allow the assumption of homogeneity of any microporous system, not least, because the subsystem micropore-wall of micropore is obviously heterogeneous. Therefore, the fractal concept is probably not applicable to very narrow micropores (ultramicropores, according to Dubinin s classification). 

Of course, such explanation of the fractal formation is valid only if the properties of the species and the exterior conditions of preparation under which the species are treated assure homogeneity of interior conditions of formation. 

For t > 1 at any 2, we obtain p (2, t) oc t-1/2. This behavior in a homogeneous medium corresponds to (2.8), giving the probability density in a fractal medium with spectral dimension ds. 

Under homogeneous conditions (e.g., vigorous stirring), A = 0 and therefore k (t) is a constant giving back the classical kinetics result. The previous equation has been applied to the study of various reactions in fractals as well as in many other nonclassical situations. For instance, theory, simulations, and experiments have shown that the value of A for A + A reactions is related to the spectral dimension ds of the walker (species) as follows [9,18] ... 

These properties are likely to have an important influence on the behavior of intact biochemical systems, e.g., within the living cell, enzymes do not function in dilute homogeneous conditions isolated from one another. The postulates of the Michaelis-Menten formalism are violated in these processes and other formalisms must be considered for the analysis of kinetics in situ. The intracellular environment is very heterogeneous indeed. Many enzymes are now known to be localized within 2-dimensional membranes or quasi 1-dimensional channels, and studies of enzyme organization in situ [26] have shown that essentially all enzymes are found in highly organized states. The mechanisms are more complex, but they are still composed of elementary steps governed by fractal kinetics. 

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