Fractal analysis defined

Chapter 13 - It was shown, that limiting conversion (in the given case - imidization) degree is defined by purely structural parameter - macromolecular coil fraction, subjected evolution (transformation) in chemical reaction course. This fraction can be correctly estimated within the framework of fractal analysis. For this purpose were offered two methods of macromolecular coil fractal dimension calculation, which gave coordinated results. 

In recent years much attention has been given to the application of fractal analysis to surface science. The early work of Mandelbrot (1975) explored the replication of structure on an increasingly finer scale, i.e. the quality of self-similarity. As applied to physisorption, fractal analysis appears to provide a generalized link between the monolayer capacity and the molecular area without the requirement of an absolute surface area. In principle, this approach is attractive, although in practice it is dependent on the validity of the derived value of monolayer capacity and the tacit assumption that the physisorption mechanism remains the same over the molecular range studied. In the context of physisorption, the future success of fractal analysis will depend on its application to well-defined non-porous adsorbents and to porous solids with pores of uniform size and shape. 

We are drawn to the conclusion that log-log fractal plots are useful for the correlation of adsorption data - especially on well-defined porous or finely divided materials. A derived fractal dimension can also serve as a characteristic empirical parameter, provided that the system and operational conditions are clearly recorded. In some cases, the fractal self-similarity (or self-affine) interpretation appears to be straightforward, but this is not so with many adsorption systems which are probably too complex to be amenable to fractal analysis. 

Hence, the stated aforementioned results have shown that the elastomeric reinforcement effect is the true nanoeffect, which is defined by the initial nanofiller particle size only. The indicated particle aggregation, always taking place in real materials, changes and reduces reinforcement degree quantitatively. This effect theoretical treatment can be received within the frameworks of fractal analysis. For the considered nanocomposites the nanoparticle size upper limiting value makes up 52 nm. 

Fractals are self-similar objects invariant with respect to local dilatations, i.e., objects that reproduce the same shape during observation at various magnifications. The concept of fractals as self-similar ensembles was introduced by Mandelbrot [1], who defined a fractal as a set for which the Hausdorff-Besicovitch dimension always exceeds the topological dimension. The fractal dimension D of an object inserted in a d-dimensional Euclidean space varies from 1 to d. Fractal objects are the natural filling of the gap between known Euclidean objects with integer dimensions 0,1, 2, 3. .. The majority of naturally existing objects proved to be fractal, which is the main reason for the vigorous development of the methods of fractal analysis. 

Cationic water-dissolved polyelectrolytes find wide application in industry different fields, namely, for the ecological problems solution [34]. For understanding these polymers action mechanism and synthesis processes it is necessary to define their molecular characteristics and water solutions properties too [35]. Therefore, the authors [36] performed description of cationic polyelectrolytes in solution behavior within the framework of fractal analysis on the example of copolymer of acrylamide with trimethylammonium methacrylate chloride (PAA-TMAC). The data [35] for four copolymer PAA-TMAC with TMAC contents of 8, 25, 50 and 100 mol. % were used. In Ref [35] the equations Maik-Kuhn-Hou-wink type were obtained, which linked intrinsic viscosity [q] (the Eq. (1)) and macromolecular coil gyration radius with average weight molecular weight of polymer [35] ... 

Within the framework of fractal analysis the intercommunication between andMlf is defined by the Eq. (8). As well as the exponent the... 

In Fig. 31, the dependences of D on the value A6= 6 6j are adduced, which characterizes as the first approximation the solvent thermo-dynamical affinity in respect to polymer [16]. As it follows from this figure data, certain laws are observed for these parameters relation. At first, A6 increase or solvent thermo-dynamical affinity in respect to polyaner change for the worse in all cases results in increase or macromolecular coil compactness enhancement, that is defined by the Eq. (8) within the framework of fractal analysis. Thus, the larger A5, is the smaller macromolecular coil gyration radius is at the same polymer molecular weight MM (or polymerization degree N). 

The stated above results suppose the neeessity of elaboration of a polymer solutions behavior model, in the basis of whieh the macromolecule strueture is put, defining all its main eharacteristics. Sueh model can be reeeived within the framework of fractal analysis. No less important the problem aspeet is accounting for macromolecule in solution strueture ehange d5mamies at the action of either factors. 

The fractal analysis main rules in reference to polymer solutions description can be found in the reviews [4, 5]. The common remark should be made in respect to the Eq. (1). The fractal dimension Df characterizes macromolecular coil stmcture, defining its elements distribution in space. The increase of Df means Rg decreasing at N = const, i.e., a coil compactness enhancement. 

Since the introduction in analysis of macromolecular coil stmcture, characterized by its fractal dimension Df, is the key moment of polycondensation process fractal physics, then the value Df determination methods are necessary for practical application of polycondensation fractal analysis for solutions. This parameter for macromolecular coil in solution is defined by two groups of interactions interactions polymer-solvent and interactions of coil elements among them [6]. At... 

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. 

Hence, the stated above results assume, that polycondensation in solution products (macromolecular coils) stracture defines polymers in condensed state stmcture and properties. The application of fractal analysis and cluster model ideas allows to both to point out these changes tendencies and to obtain polymers properties quantitative estimation [158]. 

Hence, the results quoted above demonstrated clearly, that the transition of transesterification reaction kinetic curves from an autodecelerated regime to an autoaccelerated one was defined by a structural factor, namely, by reaching of fiactal-like molecule (reaction product) fractal dimension critical value, at which the number of reactive sites in volume and on the surface of molecule became equal. Within the fiamewoik of the fractal analysis the analytic relationship is obtained, confirming this hypothesis. 

Thus, the present chapter results have been shown the applicability and usefullness of fractal analysis and strange (anomalous) diffusion conception for description of polymerization reactions, catalyzed by nanofillers. The nanofiller introduction in reactionary mixtnre results in two-phase system formation, where a decisive role will be played by interfacial interactions. The polymerization conversion degree is defined by its active (fractal) time. Hence, the ability to control active time gives the possibility of reaction conrse operation. 

As it is known, autohesion strength (coupling of the identical material surfaces) depends on interactions between some groups of polymers and treats usually in purely chemical terms on a qualitative level [1, 2], In addition, the structure of neither polymer in volume nor its elements (for the example, macromolecular coil) is taken into consideration. The authors [3] showed that shear strength of autohesive joint depended on macromolecular coils contacts number A on the boundary of division polymer-polymer. This means, that value is defined by the macromolecular coil structure, which can be described within the frameworks of fiactal analysis with the help of three dimensions fractal (Hausdorff) spectral (fraction) J and the dimension of Euclidean space d, in which ifactal is considered [4]. As it is known [5], the dimension characterizes macromolecular coil connectivity degree and varies from 1.0 for linear chain up to 1.33 for very branched macromolecules. In connection with this the question arises, how the value influences on autohesive joint strength x or, in other words, what polymers are more preferable for the indicated joint formation - linear or branched ones. The purpose of the present communication is theoretical investigation of this elfect within the frameworks of fractal analysis. 

Hence, the offered techniques, using the methods of fractal analysis and cluster model of polymers amorphous state structure, allow the theoretical estimation of both deformability and strength of polymers. The side groups influence on these characteristics was considered in detail. It has been shown that the polymer strength is defined by both strength resource and deformability resource. 

The adduced results have shown that loosely packed matrix of devit-rificated amorphous phase and disordered in deformation process crystalline phase part are structural components, defining impact energy dissipation and hence, impact toughness of semicrystalline polymers. The fractal analysis allows correct quantitative description of processes, occurring at HDPE impact loading. It is important, that the intercommunication exists between polymer initial structure characteristics and its changes in deformation process [2, 3]. 

Hence, the stated above results have demonstrated, that intercomponent adhesion level in natural nanocomposites (polymers) has structural origin and is defined by nanoclusters relative fraction. In two temperature ranges two different reinforcement mechanisms are realized, which are due to large friction between nanoclusters and loosely packed matrix and also perfect (by Kemer) adhesion between them. These mechanisms can be described successfully within the frameworks of fractal analysis. 

Within the frameworks of fractal analysis the parameter e is defined with the aid of the Eq. (17.4) ... 

Huggins interaction parameter which is defined within the framework of fractal analysis with the aid of the Equation S ... 

Therefore, the fractal analysis application stated above allows elucidation of the interconnection of parameters defining the value of the Kolmogorov-Avrami exponent n. The increase in the tension extent X always results in a reduction in chain molecular mobility, characterised by its fractal dimension In turn, reduction in results in a linear decrease in n. Change in the nucleation mechanism defines the parallel displacement of the straight lines The fractal concept stated in the present... 

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