Geometric fractals

The replacement rule we have used so far to generate geometric fractals creates isotropic fractals. In other words, the property of geometric self-similarity is... 

Before we close this section some major, unique kinetic features and conclusions for diffusion-limited reactions that are confined to low dimensions or fractal dimensions or both can now be derived from our previous discussion. First, a reaction medium does not have to be a geometric fractal in order to exhibit fractal kinetics. Second, the fundamental linear proportionality k oc V of classical kinetics between the rate constant k and the diffusion coefficient T> does not hold in fractal kinetics simply because both parameters are time-dependent. Third, diffusion is compact in low dimensions and therefore fractal kinetics is also called compact kinetics [23,24] since the particles (species) sweep the available volume compactly. For dimensions ds > 2, the volume swept by the diffusing species is no longer compact and species are constantly exploring mostly new territory. Finally, the initial conditions have no importance in classical kinetics due to the continuous re-randomization of species but they may be very important in fractal kinetics [16]. 

Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. In this respect, the temporal fractal differs from a geometrical fractal (e.g., Cantor dust) for which only an upper limit (i.e., the initial segment before its subdivision) is assumed to exist. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the... 

Figure 7.7. The Menger sponge, a geometric fractal made of a cube from which three central parts (cube stacks) have been removed. This leaves 20 of the original 27 subcubes in a cubic arrangement. All the remaining subcubes get the same treatment. If this is iterated an infinite number of times the Menger sponge of no weight and infinite surface area is formed. Its dimension is 2.73. From B. B. Mandelbrot. The Fractal Geometry of Nature. W. H. Freeman Co., New York (1983). With kind permission from B. B. Mandelbrot.

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

Fig. 3.3 First, three st( ps in the recursive geometric construction of the large-time pattern induced by R90 when starting from a simple nonzero initial state. The actual final pattern would be given as the infinite time limit of the sequence shown hero, and is characterized by a fractal dimension Df,actai = In 3/In 2.

The electrical double-layer structure and fractal geometry of a pc-Ag electrode have been tested by Se vasty an ov et al.272 They found that the geometrical roughness of electrochemically polished pc-Ag electrodes is not very high (/pz 1.5 to 1.25), but the dependence of Chtr curves on cej, as well as on/pz, is remarkable (C, =30 to 80 fi cm-2 if/pz =1.5 to 1.0). 

In order to describe the geometrical and structural properties of several anode electrodes of the molten carbonate fuel cell (MCFC), a fractal analysis has been applied. Four kinds of the anode electrodes, such as Ni, Ni-Cr (lOwt.%), Ni-NiaAl (7wt.%), Ni-Cr (5wt.%)-NijAl(5wt.%) were prepared [1,2] and their fractal dimensions were evaluated by nitrogen adsorption (fractal FHH equation) and mercury porosimetry. These methods of fractal analysis and the resulting values are discussed and compared with other characteristic methods and the performances as anode of MCFC. 

Fractal theory is a relatively new field of geometry, formulated by Mandelbrot [196] for irregular rough-surfaced objects. The major properties of such objects are the dependence of the measured length (perimeter), surface, or volume on the scale of measurement and geometrical self-similarity... 

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

Electrochemical processes usually take place on rough surfaces and interfaces and the use of fractal theory to describe and characterize the geometric characteristics of surfaces and interfaces can be of significant importance in electrochemical process description and optimization. Drs. Joo-Young Go and... 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

Here, Amici is the microscopic area which is defined within the fractal limits y, the dimensionless geometrical parameter X0, the spatial outer (upper) cutoff /fmacr, the macroscopic area which is defined beyond the fractal limits (= Aex in Eq. 15) and Xi represents the spatial inner (lower) cutoff. 

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