Fractal skeleton structure

The similar approach was used by Webman [3], according to which under external force Faction fractal deformation occurs only on scales, exceeding some definite length Z. In the general case elastic moduli of fiactal part with specific size depend on fractal skeleton structure, which in space with small d consists of both nonduplicated and repeatedly duplicated bonds. Assuming, that domain elasticity is defined by nonduplicated bonds only (i.e., considering parts, consisting of duplicated bonds, as absolutely stiff ones), can be found the lower limit of the exponent [3] ... 

The calculation of the real part of the effective shear modulus p of a composite with fractal structure is illustrated. According to this calculation (Fig. 58) the percolation transition appears after go < 10 4 and at doping concentration p 0.12, i.e. for p > 0.12 in a composite with a continuous and strong skeleton composed of particles of a doping compound connected by a boundary stratum of a polymetric compound. 

Most gels, however, form structures between these two extremes. The hydrolysis and polycondensation of metal alkoxide precursors yield an initial formation of fractal clusters, which upon reaching a critical size begin to form a continuous network or skeleton via cluster-cluster aggregation. Both the initial cluster formation and the subsequent aggregation of clusters are fractal in nature. There is a very distinct difference between these two processes, however, as is more evident in Sec. X. 

The critical indices estimated from these relations fall into the admissible ranges of variation P = 0.39-0.40, V = 0.8-0.9, and t = 1.6-1.8, determined in terms of the percolation model for three-dimensional systems. The researchers [7] noted that not only numerical values but also the meanings of these values coincide. Thus the index P characterises the chain structure of a percolation cluster. The 1/p value, which serves as the index of the first subset of the fractal percolation cluster in the model considered [7], also determines the chain structure of the cluster. The index v is related to the cellular texture of the percolation cluster. The 2/df index of the second subset of the fractal percolation cluster is also associated with the cellular structure. By analogy, the index t defines the large-cellular skeleton of the fractal percolation cluster. The relationship between the critical percolation indices and the fractal dimension of the percolation cluster for three-dimensional systems and examples of determination of these values for filled polymers are considered in more detail in the book cited [7]. Thus, these critical indices are universal and significant for analysis of complex systems, the behaviour of which can be interpreted in terms of the percolation theory. 

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. 

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