Fractals structural properties

A typical situation is realized in such cases - fractal structure properties appear on a range scale which is limited by the dimensions of the particles forming the aggregate at the beginning, and at the end, by the dimensions of the initial fractal clusters. Typical particle dimensions are 1-10 nm aggregate dimensions are 10-1000 nm. The fractal dimension, df, depends on the conditions of aggregate formation and, as a rule, lies within a range of df = 2 — 2.9. 

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. 

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

The electro-active surface of the porous carbon electrode for EDLCs is accessible only through the cumulative resistance of the electrolyte inside the pore. Therefore, the porous structure of the porous carbon becomes one of the most important factors influencing the energy/power densities. Fractal analysis has proven to be useful to describe the geometric and structural properties of rough surfaces and pore surfaces.56 66... 

In this context, Berry [277] studied the enzyme reaction using Monte Carlo simulations in 2-dimensional lattices with varying obstacle densities as models of biological membranes. That author found that the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial concentration increase. In addition, the rate constant controlling the rate of the complex formation was found to be, in essence, a time-dependent coefficient since segregation effects arise due to the fractal structure of the reaction medium. In a similar vein, Fuite et al. [278] proposed that the fractal structure of the liver with attendant kinetic properties of drug elimination can explain the unusual... 

Conversely, the relationship (7.2) expresses a time-scale invariance (selfsimilarity or fractal scaling property) of the power-law function. Mathematically, it has the same structure as (1.7), defining the capacity dimension dc of a fractal object. Thus, a is the capacity dimension of the profiles following the power-law form that obeys the fundamental property of a fractal self-similarity. A fractal decay process is therefore one for which the rate of decay decreases by some exact proportion for some chosen proportional increase in time the self-similarity requirement is fulfilled whenever the exact proportion, a, remains unchanged, independent of the moment of the segment of the data set selected to measure the proportionality constant. 

Now various structures—for example, aggregates of particles in colloids, certain binary solutions, polymers, composites, and so on—can be conceived as fractal. Materials with a fractal structure belong to a wide class of inhomogeneous media and may exhibit properties differing from those of uniform matter, like crystals, ordinary composites, or homogeneous... 

The coupled dipole equations (CDE) have been used in calculating the optical properties of composite media, including larger particles, where the dipoles are arranged to mimic a more complicated system, such as those used in DDA [38], [39], as well as fractal structures [40], which could be applied to model aggregation, surface composition, or percolation. The general nature of the solution allows for calculation of optical properties, as well as enhanced Raman and electric fields at any point in space. 

The scaling properties of fractal structures, both surface and mass structures, are quantified by their fractal dimensions. A mass fractal dimension (Df) may be defined in the following way. The mass (M) of an object may be expressed as a power law of its radius (R). 

Essential characteristics of fine-grained solids, such as the Fe oxides, include the specific surface area, the porosity and the fractal structure. The standard procedure for measuring these properties is the Brunauer-Em-met-Teller (BET) method (Gregg Sing, 1991) This depends on the fact... 

This implies that the structure of the central part of each span can be different from the end parts. In the case just considered, the central part of each span is straight, because it is influenced by only two control points, while the part from 3/4 along one span to 1/4 along the next has a fractal structure. In fact the basis function has the interesting combination of properties that it is entirely made up of pieces of linear functions at different scales, but it has a continuous first derivative. 

The hierarchical structure model is generalized and applied to study the viscoelastic properties of a two-component inhomogeneous medium with chaotic, fractal structure. It is shown that just as the results obtained recently using the Hashin-Strikman model, the present model predicts the possibility of obtaining composites with an effective shear and dumping coefficient much higher than those characterizing the individual component phases. The viscoelastic properties of the fractal medium, however, differ qualitatively from the properties of the Hashin-Strikman medium. 

Another class of materials with fractal structure are amorphous polymers. Here fractal properties manifest themselves on scales exceeding the dimensions... 

The main characteristic of fractal structures is the dependence of their properties, C, on some linear scale, L ... 

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