Fractal analysis notions

Hence, the stated above results showed, that fractal analysis notions allowed to give principally new treatment of phenomena, occurring at PHE synthesis at... 

The second method is based on the fractal analysis notions, where the value Af is calculated according to the Eq. (7.11). 

The theory of fractals and its application to physical and chemical processes has been developing vigorously in recent years [1-8]. To facilitate the understanding of the results presented in this chapter, we shall introduce some notions and definitions and consider briefly the grounds for applying the principles of synergetics and fractal analysis to the description of structures and properties of polymers. 

Polymers mechanical properties are some from the most important, since even for polynners of different special purpose functions this properties certain level is required [199], However, polymiers structure complexity and due to this such structure quantitative model absence make it difficult to predict polymiers mechanical properties on the whole diagram stress-strain (o-e) length—fi-om elasticity section up to failure. Nevertheless, the development in the last years of fractal analysis methods in respect to polymeric materials [200] and the cluster model of polymers amorphous state structure [106, 107], operating by the local order notion, allows one to solve this problem with precision, sufficient for practical applications [201]. 

Hence, the results stated above have shown that the structural analysis of polyesterification reaction in melt, using fiiactal analysis and irreversible aggregation models notions, allows to give precise enough description of this reaction even witliout applying purely chemical aspects. Let us note that fractal analysis is a more strict mathematical calculus than often used for synthesis kinetics description empirical equations. 

Within the frameworks of fractal analysis the notion of particles or cluster accessibility to active (reactive) sites of other particles or clusters for either reaction realization was introduced. This notion is finked with cluster fractafity notion, which is macromolecular coil in solutiom The macromolecular coil fractal dimension Df characterizes its structure opermess degree — the smaller the more intensive clusters penetration (or particles penetration) into another then cluster (it is more accessible ). This postulate is expressed analytically by the Eq. (27). Differentiating the indicated relationship by time t, let us obtain reaction realization rate (the Eq. (69)) and then it can be written [123] ... 

The fractal analysis of polymerization kinetics in nanofiller presence was performed. The influence of catalyst stmctural features on chemical reaction course was showa The notions of strange (anomalous) diffusion conception was applied for polymerization reactions description. 

It was also found [6], that in this case experimentally determined values of athermic fracture stress turn out to be essentially (2 3 times) smaller than theoretically calculated ones. A small values k 0.2 1.0) is one more important feature of nonoriented polymers fracture in impact tests. This means, that the stress on breaking bonds is essentially lower than nominal fracture stress of bulk sample. And at last, it was found out [7], that the value k reduces at testing temperature growth and the transition from brittle fiactuie to ductile (plastic) one. These effects explanation was proposed in Refs. [4-7], but development of fractal analysis ideas in respect to polymers lately and particularly, Alexander and Orbach woik [8] appearance, which introduced the fraction notion, allows to offer the major treatment of polymer fracture process [9, 10], including the dilaton concept [1-3] as a constituent part. 

Hence, the stated above results shown the conformity of fractional derivatives method and traditional fractal analysis, using Hausdorff dimension d notion. The physical significance of fractional exponent and its determination method were elucidated. The theoretical analysis is given the good quantitative correspondence to experiment [31]. 

The fractals theory and its application to various physical and chemical processes have recently undergone a large amount of development [1-7]. For simplification of understanding of the results represented in subsequent chapters some main notions and definitions are briefly considered and reasons for the application of fractal analysis (and connected with it irreversible aggregation models) for description of the structure and properties of polymer materials and composites on this basis are shown. 

Self-similar objects, invariant about local dilatations, i.e., objects which in observation processes at various magnifications repeat the same form, are called fractals. Mandelbrot [1] introduced the notion of fractals as self-similar sets, defining a fractal as a set for which the Hausdorff-Bezikovich dimension always exceeds the topological dimension. The fractal dimension d oi the object, adopted in d-dimensional Euclidean space, varies from 1 to d. Fractal objects are natural fillings of sets between known Euclideans with whole number dimensions 0, 1, 2, 3,. .. The majority of objects existing in nature turn out to be fractal ones, which is the main reason for the vigorous development of fractal analysis methods. 

Mechanical properties of polymers are among the most important, since a certain level of these properties is always required even for polymers of different special-purpose functions [50]. In papers [38, 51] it has been shown that the curing process of the chemical network of epoxy polymers with the formation of nodes of various density results in a change in the molecular characteristics, particularly the characteristic ratio C. If such an effect actually exists, then it should be reflected in the deformation-strength characteristics of crosslinked epoxy polymers. Therefore the authors [49] offered methods of prediction of the limiting properties (properties at fracture), based on the notions of fractal analysis and the cluster model of the amorphous state structure of polymers, with reference to a series of sulfur-containing epoxy polymers [52, 53] (see also Section 5.4). 

The second method is based on the notions of fractal analysis and the final formula has the form [55] ... 

In this section the notion of an allometric relation is generalized to include measures of time series. In this view, y is interpreted to be the variance and x the average value of the quantity being measured. The fact that these two central measures of a time series satisfy an allometric relation implies that the underlying time series is a fractal random process and therefore scales. It was first determined empirically that certain statistical data satisfy a power-law relation of the form given by Taylor [17] in Eq. (1), and this is where we begin our discussion of the allometric aggregation method of data analysis. 

It has been recognized that one of the fundamental properties of frequency-independent antennas is their abihty to retain the same shape under certain scahng transformations. It has been demonstrated that this property of self-similarity is also shared by many fractals (Mandelbrot, 1983). This commonality has led to the notion that fractal geometric principles be used to provide a natural extension to the traditional approaches for classification, analysis, and design of frequency-independent antennas (D.H. Werner and P.L.Werner). This theory allows the classical interpretation of frequency-independent antennas to be generalized to include the radiation from structures that are not only self-similar in the smooth or discrete sense but also in the rough sense. 

Hence, the condition (3 0, which follows from the data of Table 1.1, also testifies to also polymer matrix structure fractality. It is obvious, that polymer medium structure fractality can be determined by the dependence of (p j on temperature, that is, using analysis of local order thermofluctua-tiond effect (see Fig. 1.4). Let us note that structure fractality and freezing below r local order (cp j = const) from the physical point of view are interexepting notions. 

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