Self-similarity, fractal structure

Within the frameworks of this formalism to account for consistently nonlinear phenomenon complex nature is a success, such as memory effects and spatial correlations. In addition the earlier known solutions are not only reproduced, but their nontrivial generalization is given. Another important feature is connected with fractal structures self-similarity using. Unlike the traditional methods of system description on the basis of averaging different procedures, when microscopic level erasing occurs, in fractal conception medium self-affine structure and thus within the frameworks of this conception system micro and macroscopic description levels are united. Exactly such method is important for complex multicomponent systems, discovered far from thermodynamic equilibrium state [35], which are polymers [12], The authors of Refs. [31, 32] are attempted two indicated trends combination. 

Fractals are self-similar objects, e.g., Koch curve, Menger sponge, or Devil s staircase. The self-similarity of fractal objects is exact at every spatial scale of their construction (e.g., Avnir, 1989). Mathematically constructed fractal porous media, e.g., the Devil s staircase, can approximate the structures of metallic catalysts, which are considered to be disordered compact aggregates composed of imperfect crystallites with broken faces, steps, and kinks (Mougin et al., 1996). 

Note that power-law behaviour is prevalent at gelation. This has been proposed to be due to a fractal or self-similar character of the gel. Note that the exponent )f is termed the fractal dimension. For any three-dimensional structure D = 3) the exponent Df<3 (where Df < 3 indicates an open structure and Df= 3 indicates a dense strucmre). Also Muthu-kumar (Muthukumar and Winter, 1986, Muthukumar, 1989) and Takahashi et al. (1994) show explicitly the relationship between fractal dimension (Df) and power-law index of viscoelastic behaviour (n). Interestingly, more recent work (Altmann, 2002) has also shown a direct relationship between the power-law behaviour and the mobility of chain relaxations, which will be discussed further in Chapter 6. 

The term fractal and the concept of fractal dimension were introduced by Mandelbrot [1]. Since Mandelbrot s work, many scientists have used fractal geometry as a means of quantifying natural structures and as an aid in understanding physical processes occurring within these structures. Fractals are objects that appear to be scale invariant. Mandelbrot defines them as shapes whose roughness and fragmentation neither tend to vanish, nor fluctuate up and down, but remain essentially unchanged as one zooms in continually and examination is refined . The above property is called scale invariance . If the transformations are independent of direction, then the fractal is self-similar if they are different in different directions, then the fractal is self-afflne (see Chapter 2). 

This has made the growth phenomenon of complex structure an interesting area of research for a long time [113-115]. Several models have been presented to understand the phenomenon out of which DLA model had received much attention as this is very common in namre [116], Fractals are self-similar objects with non-integer dimension they are also important to determine the macroscopic properties of the system by microscopic dynamics of system, which has been an area of scientific interest for a long time. Electrochemical deposition and some polymerization processes are the most well-known examples (Fig. 1.13). 

FIGURE 4.18 The dependence of measmement scale L on structure fractal dimension d ior PC at r = 293 K. Horizontal shaded lines indicate nondeformed PC structure self-similarity boimdaries (/, and and the shaded region - deformation fractal behavior range [73]. 

Typical solid state contacts are, in general, not ideal planes. In particular, the fundamental work of Mandelbrot [532] permits us to go to another again ideal limit, that of fractal geometry. Let us imagine a rough interface, whose individual segments are, when inspected at greater resolution, structured just as the overall structure. If this is fulfilled over a certain size scale (the atomic structure naturally sets a limit), this structure is referred to as fractal and self-similar over this size scale. 

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

We focus on aggregation in model, regular and chaotic, flows. Two aggregation scenarios are considered In (i) the clusters retain a compact geometry—forming disks and spheres—whereas in (ii) fractal structures are formed. The primary focus of (i) is kinetics and self-similarity of size distributions, while the main focus of (ii) is the fractal structure of the clusters and its dependence with the flow. 

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. 

Figure 8.14 shows a sketch of the plot that is utilized for the purpose of fractal analysis. For the theoretical fractal self-similarity holds for all orders of magnitude - to be measured in units of space (r) or reciprocal space (s)53. In practice, a fractal regime is limited by a superior cut and a lower cut54. In the sketch superior and lower cut limit the fractal region to two orders of magnitude in which self-similarity may be governing the materials structure. 

In t-DVB/S copolymerization, Antonietti and Rosenauer isolated microgels slightly below the macrogelation point [221]. Using small angle neutron scattering measurements they demonstrated that these microgels exhibit fractal behavior, i.e. they are self-similar like the critically branched structures formed close to the sol-gel transition. 

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. 

Simulation of structure formation on a lattice [7,100] demonstrated that randomly formed branched clusters also fulfill self-similarity conditions and gave fractal dimensions of [7,104,105] ... 

Fractals are mathematically defined self-similar structures (Fig. 1.11) [26]. The scaffold of cascade or dendritic molecules is fractal if the atoms are considered to be points and the bonds to be strictly one-dimensional lines. Self-similarity... 

Owing to their self-similar (fractal) structure, the number of terminal groups of a dendrimer of any generation can be calculated with the aid of the following equation ... 

Rule 1 Dendritic structures consist of self-similar units (fractals). 

The simplest fractals are mathematical constructs that replicate a given structure at all scales, thus forming a scale-invariant structure which is self-similar. Most natural phenomena, such as colloidal aggregates, however, form a statistical self-similarity over a reduced scale of applicability. For example, a colloidal aggregate would not be expected to contain (statistical) self-similarity at a scale smaller than the primary particle size or larger than the size of the aggregate. 

Observations of the liver reveal an anatomically unique and complicated structure, over a range of length scales, dominating the space where metabolism takes place. Consequently, the liver was considered as a fractal object by several authors [4,248] because of its self-similar structure. In fact, Javanaud [275], using ultrasonic wave scattering, has measured the fractal dimension of the liver as approximately df 2 over a wavelength domain of 0.15-1.5mm. 

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. 

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