Fractal structures dimensions

The value a = 1 corresponds to ideal capacitive behavior. The fractal dimension D introduced by Mandelbrot275 is a formal quantity that attains a value between 2 and 3 for a fractal structure and reduces to 2 when the surface is flat. D is related to a by... 

Fig. 41. Typical 2D fractal structure obtained by aggregation of particles in the journal bearing flow. Fractal dimension of the cluster is 1.54 (Hansen and Ottino, 1996b).

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. 

In media of fractal structure, non-integer d values have been found (Dewey, 1992). However, it should be emphasized that a good fit of donor fluorescence decay curves with a stretched exponential leading to non-integer d values have been in some cases improperly interpreted in terms of fractal structure. An apparent fractal dimension may not be due to an actual self-similar structure, but to the effect of restricted geometries (see Section 9.3.3). Another cause of non-integer values is a non-random distribution of acceptors. 

One way of measuring the fractal dimension of aggregates is discussed in Chapter 5 (See Section 5.6a and Example 5.4). In the example below, we illustrate the relation between the fractal structure of aggregates and the surface area of the aggregates. 

An independent x-ray and light scattering analysis (see Section 5.6a and Example 5.4) of a dispersion of the aggregates suggests that the aggregates have a fractal structure with a fractal dimension of 2.65. Is this confirmed by your result ... 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Fractal-Like Dimensions of Graphs 3-2 Correlation with Physical Properties 3-3 Ordering of Structures ... 

Flease note that the packing of fibers also has a fractal structure. Hie pecking fifectal dimension is obtained by measuring the number of holes in the fiber mat of various sizes. A log—log plot of number of hdes versus their size gives a line whose slope is the fi actal dimension of the fiber packing. See Kaye [56]. 

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). 

Mass fractal dimensions are always less than the dimension of the space in which the fractal object exists. Therefore, as a fractal structure grows, its mass increases less rapidly than the volume it occupies. Therefore, the density of a fractal object is not constant but decreases with increasing size. Surface fractal dimensions, on the other hand, must lie in the range of one less than the dimension of space up to the dimension of space. The surface area of these objects increases with increasing mass at a faster rate than for Euclidian objects. As a result, the surfaces are very convoluted. Both types of structures are observed in silica polymers. 

Much of the current interest in fractal geometry stems from the fact that fractal dimensions are experimentally accessible quantities. For polymers and colloids, the measurement techniques of choice are scattering experiments using X-rays, neutrons, or light. These measurements may be made on liquids or solids and can be performed readily as a function of time and temperature. Both mass and surface fractal structures yield scattering curves that are power laws, the exponents of which depend on the fractal dimension (6). For mass fractal structures the relation is... 

The fractal structure of the interconnected network has been confirmed through numerical analysis of the TEM micrographs [279]. For a homogeneous medium, the mass, M(r), increases as a function of distance (r) from the origin in two dimensions M(r) == r. For a fractal structure, however. 

The chapter consists of three main sections. In Section II the elements of fractal theory are given. In Section III the basis of percolation theory is described moreover, a model of fractal structures conceived by us is described. Fractal growth models, constructed using small square or rectangular generating cells as representative structural elements, are considered. Fractal dimensions of structures generated on various unit cells (2x1, 2x2, 2x3, 2x4, 3x1, 3x2, 3x3, 3x4, 4x1, 4x2, 4x3, 4x4) are calculated. Probability... 

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. 

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

Conductivity. Let a(/j be the conductivity of a fragment of the fractal structure of dimension /, where Iq < l < E,. Because of the self-similarity of the structure, the ratio of conductivities on different scales l and / is defined only by the ratio of the scales ... 

According to (314), a material with a negative Poisson s coefficient can be obtained either if it is very rigid (i.e. if its shear modulus fulfills the above condition) or by forming a structure with dimension less than 2 i/K (i.e., d < 2 i/K) or by combining the first and the second methods. Change of dimension of the system is impossible for continuum structures however, it is possible in fractal structures [163]. 

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

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