Fractal analysis methods

Hence, the considered above scheme of genetic intercommunication of synthesis products and condensed state structures with polymer properties demonstrates clearly, that the foundation of the latter lays (encodes) quite at the synthesis stage and cannot be set by only singular polymer macromolecule chemical structure. The using of the concrete quantitative model of solid polymers structure together with fractal analysis methods gives the possibility of prediction of not only glass transition temperature T, but also polymers other important properties munber [120]. 

Hence, the stated above results demonstrated once more the fractal analysis methods applicabihty for the quantitative description of branched polymer macromolecular coil structure. The type of interactions between macromolecule elements (attraction or repulsion) is an important aspect of the proposed treatment. The indicated macromolecule structure, characterized by its fiactal dimension, is one more important factor. 

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]. 

Thus, the fractal analysis methods were used above for treatment of comb-like poly(sodiumoxi) methylsylseskvioxanes behavior in solution. It has been shown that the intrinsic viscosity reduction at transition from a linear analog to a branched one is due to the sole factor, namely, to a macromolecule connectivity degree enhancement, characterized by spectral dimension. This conclusion is confirmed by a good correspondence of the experimental and calculated according to Mark-Kuhn-Houwink equation fiactal variant intrinsic viscosity values. It has been shown that qualitative transition of the stmcture of branched polymer macromolecular coil from a good solvent to 0-solvent can be reached by a solvent change. 

Hence, the perfonned within the framewoik of fractal approach analysis of behavior of polystirene, modified by Dendron s, in diluted solutions gave the same conclusions, as the analysis within the fiamewoik of classical approaches. The main distinction of the indicated approaches is the fact, that the structural model, allowing to describe quantitatively macromolecules structural state and conformation, was placed in the fractal approach base. Other characteristics (gyration radius, Kuhn segment length and so on) are the function of the indicated structural state of a macromolecule. The fractal analysis methods, used for the description of linear flexible-chain polymers behavior, can be applied successfiilly also in case of polymers with more complex macromolecular architecture. 

Temiraev, K. B. The evaluation of aromatic copolyethersulfoneformals molecular weight by the fractal analysis methods. Manuscript deposited to VINITl RAS, Moscow, 20.07.1998, 22, V91-V98. 

Hence, the stated above results have shown, that the main cause of different macromolecular coil structure of PAr with the same chemical stmcture, but obtained by different polycondensation modes, is participation (or imparticipation) of solvent molecules with 5 >0 in synthesis process. Obtained in this process coil stmcture maintains at subsequent polymer dilution and solvents influence in this case is similar to their influence in polymer synthesis process. Fractal analysis methods give mathematical apparatus for this problem quantitative study. 

Hence, the adduced above results on the example of the dependences MM(C(,) for polyaiylates Ph-2 showed the necessity of polycondensation process both static and kinetic aspects consideration for correct dependences of one or another limiting characteristics. The absence of maximum on curves MM(c ) means the absence of optimum value c , at which maximum value MM is reached. The adduced examples demonstrate clearly correctness and expediency of fractal analysis methods application for polycondensation processes description. 

However, in virtue of high enough commrmity of this tendency of MM reduction it can be assumed, that it has much more key reasons, that it was supposed earlier [92]. Since at present some general description of synthesis processes does not exist, then the authors [94] used the fractal analysis methods for this purpose. They proposed key physical treatment of APESF molecular weight change at their composition variation. 

Hence, the stated above results have shown, that conversion degree and the reduced viscosity, obtained in PUAr synthesis process, are a funetion of copolymer chain statistical flexibility the more rigid chain is, the higher Q and tired are. The fractal analysis methods allow to make this correlation quantitative treatment. From the ehemieal point of view the values Q and tired depend on eomonomers functional groups activity % The higher % is, the larger the values Q and tired are. The value also defines a synthesized copolymer type. 

As it is known [2], the macromolecular coil, which is the main structural unit at polymers synthesis in solution, represents a fractal and its structure (coil elements distribution in space) can be described by the fractal dimension Df. Proceeding from this, the authors [1] used the fractal analysis methods for the description of T effect on PHE synthesis course and its main characteristics. 

Hence, the fractal analysis methods are efficient for clear structural identification of both chemical and physical factors, controlling a chain branching degree. 

Jahn, R. and Truckenbrodt, H. (2004) A simple fractal analysis method of the surface roughness. Journal of Materials Processing Technology, 145 40-5. 

The correctness of fractal analysis methods application to chain part between entanglements (chemical cross-linking) is proved by the indicated parts fractality experimental confirmation [33-35]. In this case for the macromolecule, simulated by freely formed chain from statistical segments, the Eq. (2.12) was obtained, where and L and R are chain part... 

The authors of Ref [3] substantiated in very general terms and with fractal analysis methods application the described above behavior of polymer materials at uniaxial tension. 

The dependence p(A,) and mechanism of fracture knowledge allow to use the fractal analysis methods [47] for theoretical calculation of limiting draw ratio which is coimected with the value p according to the following relationship [47] ... 

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. 

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]. 

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]. 

For oriented polymers it is shown that the value of n depends also on the drawing ratio [18,27]. Reduction in n with growth in A, is a general tendency. A considerable number of factors influencing the value of n makes its description within the frameworks of structnral and molecular models difficult. Therefore the authors of papers [34-36] generalised the influence of the indicated factors on the value of n with the application of fractal analysis methods in the example of uniaxially stretched PCP. Molecular characteristics of PCP crosslinked networks are adduced in Table 4.2. 

The authors of paper [14] used fractal analysis methods for the description of the glass transition process of crosslinked polymers. For this purpose they used the expression for the estimation of beginning time T. of the accumulation of avalanche-type defects obtained in paper [15] ... 

In the present chapter the formation of the structure of these materials and the properties defined by it will be studied within the frameworks of such a representation of the structure of crosslinked epoxy polymers using fractal analysis methods. 

However, the application of the fractal analysis methods to synthesis process for today becomes a vital problem. Such necessity is not due to the convenience of the fractal analysis as mathematical approach that supposes the existence of approaches, alternate to it. The necessity of the indicated problem solution is defined only by physical reasons. The basic object during synthesis of polymers in solutions is the macromolecular coil, which represents a fractal object [16, 17]. The description of... 

Therefore, the authors [60] carried out the description of the dependence or MM c for PHE with the usage of irreversible aggregation models and fractal analysis methods. 

Hence, the fractal analysis methods are efficient for clear structural identification of both chemical and physical factors, controlling a chain branching degree. The number of effective branching centers per one macromolecule m is controlled by four factors polymer molecular weight MM, maximum chemical density of reactive centers dimension of unscreened surface cl of macromolecular coil, and its fractal dimension D. The Equation (45) allows to determine the critical value D DJ, below of whichg = 0 (i.e., branching does not occur) = 1.10 [64],... 

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