Polymer multifractal structure

The data of Figs. 4.10 and 4.11 comparison shows, that the value obtained according to the dependences D- (q) and D (q), corresponds to the value X, at which fracture stress o -drop (Fig. 14.10) or interfacial boundaries polymer-filler fracture begins [2]. Thus, within the frameworks of multifractal formalism interfacial boundaries fracture of componors in solid-phase extrusion process is realized by polymer matrix structure regular fractal state achievement [56]. 

Yet another important aspect is the change in the fractal dimension of polymers when they are simulated on fractal rather than Euclidean lattices. This fact is also important from the practical standpoint for multicomponent polymer systems. The introduction of a dispersed filler into a polymer matrix results in structure perturbation in terms of fractal analysis, this is expressed as an increase in the fractal dimension of this structure. As shown by Novikov and co-workers [25], the particles of a dispersed filler form in the polymer matrix a skeleton which possesses fractal (in the general case, multifractal) properties and has a fractal dimension. Thus, the formation of the structure of the polymer matrix in a filled polymer takes place in a fractal rather than Euclidean space this accounts for the structure modifications of the polymer matrix in composites. 

It has been found that the basic elements of structure of initial and deformed polymers are homogeneous fractals that can be characterised by their fractal dimension. Examples of such elements are macromolecular coil, supermolecular organisation as seen in the cluster structure and a stable crack in film samples of polymers. These examples are given as examples of the possible variants of the term "multifractality with reference to polymers. 

The structural analysis of nanocomposites polymer/organoclay flame-resistance was performed within the framework of percolation and multifractal models. The possibility of flame-resistance characteristics prediction on the basis of the indicated approach has been shown. 

These qualitative effects can be described quantitatively within the framework of percolation model of reinforcement and multifractal model of gas transport processes for nanocomposites polymer/organoclay [3, 4]. It has been supposed that two structural components are created for a barrier effect to fire spreading actually organoclay and densely packed regions on its surface with relative volume fractions (p and (p respectively. In other words, it has been supposed, that the value should be a diminishing function of the sum ((p -l-(pp. For this supposition verification let us estimate the values (p and (p The value (p is determined according to the well-known equation [5] ... 

In the general case polymers structure is multifractal, for behavior description of which in deformation process in principle its three dimensions knowledge is enough fractal (Hausdorff) dimension d informational one d and correlation one d [82]. Each from the indicated dimension describes multifractal definite properties change and these dimensions combined application allows to obtain more or less complete picture of yielding process [73]. 

Since the polymers structure is multifractal [53], then, following to Williford [42], the fracture surface can be considered as the first subfractal, having dimension d (information dimension, see the Eq. (4.49)) [48]. In this case within the fiiameworks of the indicated above formalism [42] a=f, where/is dimension of singularities a, equal to [47] ... 

Kozlov, G. V., Misra, R. D. K., Aphashagova, Z. Kh. (2009). A Particulate-Filled Polymer Nanocomposites Impact Toughness Structural Model. Nanotekhnika, 2,71-74. Williford, R. E. (1988). Multifractal Fracture. Scripta Metal, 22(11), 1749-1754. 

Hence, the cited above results shown correctness and expediency of multifractal formalism in it s the simplest variant for analysis of structure changes at polymerization-filled compositions on the basis of UHMPE during solid-phase extrusion. The observed experimentally during extrusion process effects were received the quantitative description within the frameworks of this formalism. Let us note purely geometrical character of main multifractal characteristics calculation, independent on polymer matrix and filler properties [56]. 

Kozlov, G. V, Ovcharenko, E. N. (2001). The Intercommunication of Multifractal Characteristics and Structure Parameters for Particulate-Filled Polymer Composites. Izvestiya KBNC RAN, 2, 81-85. 

In the present monograph these problems were studied from the principally new point of view, when object structure, either macromolecular coil (microgel) or condensed state structure, was considered in terms of chemical or physical processes. Such an approach is possible with the availability of appropriate structural models, of which fractal (multifractal) analysis and the cluster model of the amorphous state structure of polymers were used. The first of the indicated approaches is a general mathematical calculus, whereas the second represents itself as a purely polymeric concept. This circumstance defines the excellent combination and addition to one another of crosslinked polymers. 

In Figure 1.15 the dependence of the density of the macromolecular entanglements clnster network on is shown. As might be expected, chaos intensification X increase) reduces the value, i.e., the local ordering degree in the polymer amorphous state structure [68]. More precise interpretation of the polymer structure chaotic character within the frameworks of multifractal formalism will be given below. 

At present it is known [92, 93, 159] that the polymer structure is fractal (generally multifractal) with fractal dimension d 2 fluctuation free volume is its mirror, that it also possesses fractal properties. If this assumption is correct, the general relation should be fulfilled [18] ... 

At present it is established that the structures of both natural and many model objects cannot be described with the aid of only one value of fractal dimension. For more precise description of disordered structures, including polymers, it is necessary to calculate a spectrum of different dimensions, i.e., to use the multifractal formalism [23-25]. At present a number of papers exist that show correspondence of either... 

Hence, the results stated above have shown that analysis of structural properties for epoxy polymers, which are considered as natural nanocomposites, can be carried out within the frameworks of multifractal formalism in its most simple variant. The structure adaptability resource is reduced as the crosslinking density increases and is defined by the relative fraction of the loosely packed matrix. The properties of epoxy polymers are a function of their structure adaptability. 

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