Statistical fractals, defined

Yet another factor which also influences the shape of the cluster is whether or not attractive interactions are present. As the temperature increases, the attractive interactions diminish. For example, no interactions of this sort are found in isolated macromolecular coils in good solvents at high temperature [77]. Family [9] defined this state of a statistical fractal as an uncoiled state because in this case, it is characterised by the smallest fractal dimension. 

Since we consider statistical TV-mers (at very large TV) as random fractal-like objects, we need to define the characteristic dimension (size) of the fractal statistical polymer. Let us define the characteristic size of fractal statistical TV-mer (at very large TV) as follows ... 

The fractal consideration of statistical polymer allows us to define the radius of gyration for clusters constructed as statistical polymers. 

FIGURE 13.7 Fractal aggregates, (a) Side view of a simulated aggregate of 1000 identical spherical particles of radius a (courtesy of Dr. J. H. J. van Opheusden). (b) Example of the average relation between the number of particles in an aggregate Np and the aggregate radius R as defined in (a). The fractal dimensionality D = tan 6 its value is 1.8 in this example. The region between the dotted lines indicates the statistical variation to be encountered (about 2 standard deviations). 

Family [6] defined three states of polymeric statistical fiactals depending on the system statistics extended, compensated and collapsed ones. The two main factors, influencing fiactal branching degree, were pointed out. The first Irom them is cluster concentration in system—if there are many clusters in the system, and then they occupy the entire volume. Therefore, other clusters availability restricts a fractal branching degree it is more branched in isolation (very diluted solution), than in concentrated solution. 

However, as it has been shown in Refs. [73, 74], polymer macromolecule caimot reproduce such high roughness of the surface by virtue of its final rigidity, defined by statistical segment final size, and therefore real (effective) surface fractal dimension is determined as follows [73] ... 

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. 

F often has some form of self-similarity, perhaps approximate or statistical usually, the fractal dimension of F (defined in some way) is greater than its topological dimension ... 

There are two main approaches for the deduction of the stretched exponential law. The first is based on the fractal properties of the characteristic relaxation times spectrum. In order to give a short mathematical description of these variants the so-called forsteric model of direct transfer [5] can be applied. The above-mentioned model was the result of investigations on excitation transfer from donor to statistical defects in condensed media. A law of excitation decrease of the selected donor, which is located in the origin of coordinates, at the expense of direct energy transfer to a defect located in the junction Rj of a lattice with the defined structure is considered. The relaxation function V llO the... 

Fractals are defined in general as objects made of similar parts to the whole in some way either exactly the same except for scale or statistically the same. In short, fractals are self-similar or scaling, that is, invariance against changes in scale or size (scale-invariance). 

The main factor that defines interconnection of local order and the fractal nature of the structure of solid polymers is the fact that both these features are a reflection of the key property of these polymers- their thermodynamical non-equilibrium nature. The scales of fractal behaviour and indicated above correspond very well to cluster structure border sizes - to statistical segment length - to distance... 

Let us note one more important aspect. The treatment of the structure of amorphous polymers adduced above belongs to elastomers [56]. Transference of these notions on amorphous glassy polymers assumes the description of densely packed domains freezing , i.e., a sharp increase in their life time. In addition, fractal forms of macromolecules (statistical macromolecular coils), formed in non-equilibrium physical-chemical processes, are preserved ( frozen ) in polymers. This assumes that in a glassy state the mobility of chain parts between their fixation points will be the main factor defining molecular mobility [57]. 

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]. 

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