Fractals and Self-Similarity

The power-law scaling of the cluster properties shown in Table 5-1 arises from their /ractoZ or self-similar character. Self-similarity implies that the huge clusters formed near the gel point look the same at any magnification, as long as elementary units making up the cluster are too small to see. Furthermore, the cluster size distribution at one value of (fi) is the 

Scaling Exponents for Classical and Percolation Theories of Gelation 

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Hutchinson, J.E. (1981). Fractals and self-similarity. Indiana Univ. Math J., 30,713-747. 

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. 

An analogous procedure can be applied to a plane surface. The surface can be roughened by the successive application of one or another recipe, just as was done for the line in Fig. VII-6. One now has a fractal or self-similar surface, and in the limit Eq. VII-20 again applies, or... 

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. 

Mansfield 79 performed Monte Carlo calculations on model dendrimers and determined that as a result of the unique architecture of the branches, even when similar chemically, they are well segregated. Further, he concluded that dendrimers are fractal (D ranges from 2.4 to 2.8) and self-similar only over a rather narrow scale of lengths. 

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 equations derived from the dynamic scaling theory are valid for self-afiBne and self-similar surfaces. Accordingly, the theory provides information about fractal properties and growth mechanisms of rough surfaces. 

Hence, the aforementioned results have shown that nanofiller particle (aggregates of particles) chains in elastomeric nanocomposites are physical fractal within self-similarity (and, hence, fractality [41]) range of -500-1,450 nm. In this range, their dimension can be estimated accord-... 

According to the Family classification [9], fractal objects can be divided into two main types, namely, deterministic and statistical objects. Deterministic fractals are self-similar objects that can be precisely constructed on the basis of several fimdamental laws. 

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). 

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. 

According to Family s classification [8], fractal objects can be divided into two main types deterministic and statistical. The deterministic fractals are self-similar objects, which are precisely constructed on the basis of some basic laws. Typical examples of such fractals are the Cantor set ( dust ), the Koch curve, the Serpinski carpet, the Vichek snowflake and so on. The two most important properties of deterministic fractals are the possibility of precise calculation of their fractal dimension and the unlimited range (- o +°°) of their self-similarity. Since a line, plane or volume can be divided into an infinite number of fragments by various modes then it is possible to construct an infinite number of deterministic fractals with different fractal dimensions. In this connection the deterministic fractals are impossible to classify without introduction of their other parameters in addition to the fractal dimension. 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

What does the orbit look like for Ooo It is an infinite (and therefore aperiodic) self-similar point set with fractal dimensionality, Dfractai 0.5388 [grass86c]. Figure 4.5 shows the first six stages in the Cantor-set like construction. 

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