Fractal treatment

As a conclusion of this chapter the fractal treatment of the effect of crosslinked polymers will be considered. There are two modified variants of Equation (10.2), which take into account the course of curing reactions with auto-acceleration or autoslowing-down [45] ... 

Kozlov, G. V Lipatov, Yu. S. The change of polymer matrix structure in particulate-filled composites the fractal treatment. Mechanics of Composite Materials and Structures, 2004, 40(6), 827-834. 

Kozlov, G. V. Beloshenko, V. A. Lipatov, Yu. S. The fractal treatment of cross-linked polymers physical aging process. Ukrainian Chemical Journal, 1998,64(3), 56-59. 

Kozlov, G. V. Malkanduev, Yu. A. Burya, A. I. Burmistr, E. M. Fractal treatment of the dependences of macromolecular coils self-diffusion coefficient on their structure. Problems of Chemistry and Chemical Technology, 2003, 2,69-72. 

Kozlov, G. V Mikitaev, A. K. The fractal treatment of the dependence of molecular weight on reagents concentration at polyhydroxy ether synthesis. Mater, of VI International Sci.-Pract. Conf. New Polymer Composite Materials. Nal chik, KBSU, 2010, 205-210. 

Kozlov, G. V. Afaunova, Z. 1. Zaikov, G. E. Fractal treatment of catalysis in the process of hquid-phase epoxidation of the ethylaUylethylacrylate by tetrabutyl hydroperoxide. Oxidation Commun., 2005,28(4), 863-868. 

Thus, the proposed in the present work stmetnral (fractal) treatment explains quantitatively the coke lesidne formation process at eomposites HDPE/Al(OH)3 combustion. The antipyrene (in the considered case - Al(OH)3) decomposition is realized in the surface (interfacial) layers of fractal aggregates, formed by Al(OH)j particles in their aggregation process. This process in the given case is due to the indicated layers friable structure that allows an easy access to them the air, necessary for decomposition. 

In Fig. 10.13, the dependences of f/ on for HDPE are adduced for three testing temperatures. Despite a definite data scattering (typical for impact tests), it is obvious, that the Eqs. (10.21) and (10.22) in its fractal treatment are approximated by one straight line that allows to determine the only parameter Gj (Jj ) for HDPE, characterizing its plasticity. Such correspondence was to be expected, since the values and characterize the same polymer property. The obtained at T= 293 K G values are typical for polyeftylenes [22, 44]. 

FIGURE 10.13 The dependences of effective fracture energy on samples effective cross-sectional area 5 for HDPE at testing temperatures 293 (1), 313 (2), 333 (3) and 353 K (4). The black symbols correspond to the Eq. (10.21), bright ones - to the Eq. (10.22) fractal treatment [45]... 

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... 

The foregoing treatment can be extended to cases where the electron-ion recombination is only partially diffusion-controlled and where the electron scattering mean free path is greater than the intermolecular separation. Both modifications are necessary when the electron mobility is - 100 cm2v is-1 or greater (Mozumder, 1990). It has been shown that the complicated random trajectory of a diffusing particle with a finite mean free path can have a simple representation in fractal diffusivity (Takayasu, 1982). In practice, this means the diffusion coefficient becomes distance-dependent of the form... 

Sometimes, as in the case of particle segregation on fractals (e.g., the planar Sierpinski gasket discussed in Section 6.1) this effect indeed is self-evident [88-90]. Its analytical treatment for particle accumulation was presented in [91, 92] we reproduce here simple mesoscopic estimates following these papers. Particle concentrations obey the kinetic equations... 

A rigorous treatment of fractal mathematics is beyond the scope of this treatise therefore, only essential concepts relating to a descriptive understanding of fractal shapes and analyses will be discussed. More complete and comprehensive reviews of this growing area are available. 52,56-61l... 

Muthukumar[691 described a theory of a fractal polymer possessing solution viscosity. Solutions containing dilute, semidilute, and high concentrations of fractal polymers were examined intra- and interfractal hydrodynamic interactions as well as excluded volume effects were included in the treatment. 

In some cases, when the polymerization appears, the energy distribution of micropores is negligible in comparison with the energy of polymerization. That is possible when the temperature of the treatment of the primary material (if this one can be polymerized, e.g., silica, alumina) is low (less 300-350 °C). In such cases, traditional methods of nonequilibrium thermodynamics are not effective, and the micropore formation can be considered as the result of the polymerization process which is described by methods of polymer science. However, models of macromolecular systems do not always give enough information about micropores as the empty space between polymers. For such systems, the application of fractal methods can allow us to obtain additional information, while one has to take into account the fact that they cannot be applied to very narrow pores (ultramicropores which are found, for instance, in some silica gels). 

Of course, both microporous structure and the rest nonempty solid phase can be considered as fractals. For processes of pyrolytic treatment of organic materials, in most cases it is more useful to apply fractal methods to pores. For polymerization, it is more effective to apply fractal modeling of macromolecules. 

From the discussion of various simulation methods, it is clear that they will continue to play an important role in further development of aggregation theories as they have advanced the state of knowledge over the last 20 years. The major limitation of the precise methods of Molecular and Brownian Dynamics continues to be difficulty associated with treatment of aggregates with complex geometry the same topic that limits the ability to model these systems using von Smoluchowski s approach. Research needs to be conducted on the hydrodynamics of interactions between fractal aggregates of increasing complexity in order to advance the current ability to describe these types of systems. 

The fractal dimensions of the excitation paths in samples D, F, and G lie between 2 and 3. Thus, percolation of the charge carriers (protons) is also moving through the Si02 matrix because of the availability of an ultra-small porous structure that occurs after special chemical and temperature treatment of the initial glasses [156]. 

Molecular nitrogen will see a significant surface area due to its small size comparable to the dimensions of the surface roughness, while bigger molecules such as pyrene will not be able to see all ridges and valleys and will see a significantly lower effective surface area. These factors have been studied extensively [17] for silica, and authors have found fractal factors to vary between 2 and 3, depending on the silica synthesis, treatment, and so on. 

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