Fractals, high-elasticity

Fundamental Aspects of Filling of Nanocomposites with High-Elasticity Matrix Fractal Models... 

Keywords Nanocomposite high-elasticity matrix filler aggregation strain localization fractal structure. 

Further the main equation of high-elasticity fractal theory for a calculation in its simplified form will be used [9] ... 

As it is known [83], a glassy polymers behavior on cold flow plateau (part III in Fig. 4.17) is well described within the frameworks of the rubber high-elasticity theory. In Ref [39] it has been shown that this is due to mechanical devitrification of an amorphous polymers loosely packed matrix. Besides, it has been shown [82, 84] that behavior of polymers in rubber-like state is described correctly under assumption, that their structure is a regular fractal, for which the identity is valid ... 

Kozlov, G. V., Mikitaev, A. K. (2007). The Fractal Analysis of Yielding and Forced High-Elasticity Processes of Amorphous Glassy Polymers. Mater. I-th All-Russian Sci-Techn. Conf Nanostructures in Polymers and Polymer Nanocomposites . Nal chik,... 

The experimental data about rubbers deformation are usually interpreted within the frameworks of the high-elastieity entropic theory [1-3], elaborated on the basis of assumptions about high-elastic polymers incompressibility (Poisson s ratio V = 0.5) and polymer chains Gaussian statistics. As it is known [4], the Gaussian statistic is characteristic only for the networks, prepared by chains concentrated solution curing, in the case of their compression or weak (draw ratio A, < 1.2) tension. For such stmctures the fractal dimension d = 2 and in case of v = 0.5 the following classical expression was obtained [3] ... 

Hence, the stated above results show that the classical theory of entropic high-elasticity can be used for the description of stress-draw ratio curves for rubbers with weak strain hardening, but it is incorrect in case of nanocomposites with elastomeric matrix. The correct description of deformation behavior of the latter gives the high-elasticity fractal model that is due to fractal nature of filled rubbers structure [13]. 

Therefore, within the frameworks of fractal analysis an increase in network density with reduction in chain statistical flexibility was obtained. The increase in the number of topological fixation points of macromolecules in the glassy state in comparison with the high-elastic state can be predicted by using fractal analysis methods [29,61]. 

The application of fractal analysis for the description of the behaviour of rubbers is difficult because of the fact that these materials are (or are close to) Euclidean objects. Nevertheless, at present the theory of elasticity and entropic high-elasticity of fractals is developed, which differs principally from the classical theory. The change of molecular mobility, characterised by fractal dimension of a chain part between crosslinking nodes, is of interest for rubbers. Lastly, local order models can be used successfully for quantitative description of the nucleation process of crystalline regions and the melting temperature of rubbers. These and some other questions will be considered in detail in the present chapter. 

As a rule, at present crosslinked polymer networks are characterised within the frameworks of entropic rubber high-elasticity concepts [2, 3]. However, in recent years works indicating a more complex structure of crosslinked rubbers have appeared. Flory [4] demonstrated the existence of dynamic local order in rubbers. Balankin [5] showed principal inaccuracy of the entropic high-elasticity theory and proposed a high-elasticity fractal theory of polymers. These observations suppose that more complete characterisation of these materials is necessary for the correct description of the structure of rubbers and their behaviour at deformation. In paper [6] this was carried out by the combined use of a number of theoretical physical concepts, namely the rubber high-elasticity entropic theory, the cluster model of the amorphous state structure of polymers [7, 8] and fractal analysis [9]. 

Therefore, the complete methods of calculation of the characteristics of crosslinked networks was proposed, which combines the ruhher high-elasticity entropic theory, the cluster model of amorphous state structure of polymers and fractal analysis methods. The proposed method has shown that growth in statistical segment length is observed as the drawing ratio increases. This snpposes that the chain statistical flexibility depends not only on its chemical constitntion, but also on the network deformed state. The considered method can be nsed for computer simulation and prediction of the structure of crosslinked polymer networks [6]. 

Therefore in papers [46-48] the fractal treatment of high-elasticity theory was proposed, which does not possess the indicated deficiencies. In paper [49] description... 

Balankin [46-48] obtained the following eqnation for the description of elastoplastics curves o-A, within the frameworks of entropic high-elasticity fractal theory ... 

Hence, the results obtained above have shown that behaviour at deformation for the considered polyurethanes and nanocomposites on its basis is described within the frameworks of entropic high-elasticity fractal theory or, equivalently, within the frameworks of the classical theory approximation for long polymer chains. The considered polymer networks obey Ganssian statistics due to their preparation method. The inaccuracy of the application of the entropic high-elasticity classical theory (Equation 7.13) is defined by non-fnlfilment in the given case of a postulate about elastoplastics incompressibility [49]. 

Marangoni, A.G. 2000. Elasticity of high-volume-fraction fractal aggregate networks A thermodynamic approach. Phys. Rev. B. 62, 13951-13955. 

The strong-, weak- and intermediate regimes are all a product of the elastic constant of the basic mechanical unit (the floe, the links between the floes, or a combination of both) and the number of these units present in the direction of the externally applied force (Shih et al. 1990). Therefore, the fractal dimension defines to the size of the clusters. A large fractal dimension represents a large cluster that translates to less cluster-cluster interactions per unit volume and a decreased elastic modulus. At high volume fractions, cluster size decreases and the number of cluster-cluster interactions increases, and thus the elastic constant also increases. 

The results of calculations of the effective Poisson s ratio vp dependence on the bulk concentration of a rigid phase p at various values of a = log i/C/Au) are shown in Fig. 53. The calculations were made for Poisson s ratios of the phases ranging from 0.1 to 0.4. It can be seen that at percolation threshold Poisson s ratio of the isotropic fractal composite is vp = 0.2, when K jK > 0 it is also independent of the Poisson s ratios of the individual components of the composite. The Poisson s ratio obtained by us near the percolation threshold is in agreement with computer simulation results and the conjecture of Arbabi and Sahimi [161]. It has been shown that an approximate theoretical treatment of percolation on a cubic lattice exactly reproduces the Poisson s ratio obtained in computer simulation at the percolation threshold. This result may encourage one to use this approximation to describe various elastic properties of composites. It is worth noting that some critical indices have been calculated recently with a high degree of accuracy in the context of the present model. 

And at last, the third and the most fundamental factor is the ehange of nanocomposite structure at the introduction of particulate filler in high-elastieity polymeric matrix. As Balankin showed [9], classical theory of entropic high-elastieity has a number of principal deficiencies due to non-fulfilment for real rubbers of two main postulates of this theory, namely, essentially non-Gaussian statisties of real polymeric networks and lack of coordination of postulates about Gaussian statistics and incompressibility of elastic materials. Last postulate means, that Poisson s ratio v of these materials must be equal to 0.5. As it is known [10], Gaussian statistics of macromolecular coil is correct only in case of its dimension Dj=2.0, i.e., for coil in 0-solvent. Since between value Df and fractal dimension... 

The authors of Ref. [10] considered the elasticity and entropic high-elas-ticity fractal concept (the Eq. (A.2)) [7-9] application for elastomaterials deformation behavior description on the example of styrene-butadiene rubber (SBR) and nanocomposite on its basis with carbon soot content of 34 mas. % (SBR-S). 

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