Levels of Fractality in Polymers

It has been found that the basic elements of structure of initial and deformed polymers are homogeneous fractals that can be characterised by their fractal dimension. Examples of such elements are macromolecular coil, supermolecular organisation as seen in the cluster structure and a stable crack in film samples of polymers. These examples are given as examples of the possible variants of the term multifractality with reference to polymers. 

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Floes pack in a space-filling, Euclidean fashion hence, at the floe level of structure, the material can be considered as an orthodox amorphous substance. Within the floes, however, particles pack in a non-Euclidean, fractal fashion. For such a structural arrangement, the volume fraction of particles in a floe (O ) is equivalent to the volume fraction of particles in the entire system (O), namely = O. This well-known relation of polymer physics (11) has been experimentally shown to also apply to colloidal aggregates above their gelation threshold (12). 

Figure 11.3a shows the variation of versus Rg. Taking accoimt of this dependence, it was found that df 1.60, which is close to the corresponding value for a macromolecular coil in a 0-solvent (df 1.66, see Table 11.2). The elements of Koch figures (Figure 11.3b) resemble most closely the freely jointed chain model, which is normally used to simulate macromolecules [94]. For this version, df = 1.61. Thus, the fractality of a block polymer at a molecular level can be regarded as proven. 

As an example of fractality displayed in polymers at a macroscopic level (a secondary structural element), we shall consider the growth of a crack in a film of amorphous vitreous polyarylate-sulfone [100] (see Figure 11.6). 

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]. 

As it was noted above, at present it becomes clear, that polymers in all their states and on different structural levels are fractals [16, 17]. This fundamental notion in principle changed the views on kinetics of processes, proceeding in polymers. In case of fractal reactions, that is, fractal objects reactions or reactions in fractal spaces, their rate fr with time t reduction is observed, that is expressed analytically by the Eq. (106) of Chapter 2. In its turn, the heterogeneity exponent h in the Eq. (106) of Chapter 2 is linked to the effective spectral dimension d according to the following simple equation [18] ... 

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. 

Thus, the stated above results demonstrated the fractal analysis possibilities at polymers local deformation description. In each from the described cases fractal dimension of either element has simple and clear physical significance that allows to obtain both empirical and analytic correlations between different structural levels in polymers and also describe their evolution in polymers deformation and failure processes. 

Hence, the stated results demonstrated undoubted profit of fractal analysis application for polymer structure analytical description on molecular, topological and supramolecular (suprasegmental) levels. These results correspond completely to the made earlier assumptions (e.g., in Ref [31]), but the offered treatment allows precise qualitative personification of slowing down of the chain in polymers in glassy state causes [32]. 

Polymer mechanical properties are one from the most important ones, since even for polymers of different special-purpose function a definite level of these properties always requires [20]. Besides, in Ref [48] it has been shown, that in epoxy polymers curing process formation of chemical network with its nodes different density results to final polymer molecular characteristics change, namely, characteristic ratio C, which is a polymer chain statistical flexibility indicator [23]. If such effect actually exists, then it should be reflected in the value of cross-linked epoxy polymers deformation-strength characteristics. Therefore, the authors of Ref [49] offered limiting properties (properties at fracture) prediction techniques, based on a methods of fractal analysis and cluster model of polymers amorphous state structure in reference to series of sulfur-containing epoxy polymers [50]. 

Within the frameworks of this formalism to account for consistently nonlinear phenomenon complex nature is a success, such as memory effects and spatial correlations. In addition the earlier known solutions are not only reproduced, but their nontrivial generalization is given. Another important feature is connected with fractal structures self-similarity using. Unlike the traditional methods of system description on the basis of averaging different procedures, when microscopic level erasing occurs, in fractal conception medium self-affine structure and thus within the frameworks of this conception system micro and macroscopic description levels are united. Exactly such method is important for complex multicomponent systems, discovered far from thermodynamic equilibrium state [35], which are polymers [12], The authors of Refs. [31, 32] are attempted two indicated trends combination. 

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. 

The combined use of fractal analysis and cluster models for the structure of the condensed state of crosslinked polymers allows their quantitative treatment on different structural levels, molecular, topological and suprasegmental, to be obtained for the first time and also the interconnection between the indicated levels to be determined. In turn, elaboration of solid-phase crosslinked polymer structure quantitative models allows structure-properties relationships to be obtained for the first time, which is one of the main goals of polymer physics. 

Let us pay attention to one more important circumstance. The comparison of Equations 5.13 and 5.27 demonstrates that the greatest rate of formation of clusters is reached at p = 2 or = 2. In other words, it is supposed that the smallest fractal dimension of polymer structure corresponds to the greatest level of local order [89]. 

Mechanical properties of polymers are among the most important, since a certain level of these properties is always required even for polymers of different special-purpose functions [50]. In papers [38, 51] it has been shown that the curing process of the chemical network of epoxy polymers with the formation of nodes of various density results in a change in the molecular characteristics, particularly the characteristic ratio C. If such an effect actually exists, then it should be reflected in the deformation-strength characteristics of crosslinked epoxy polymers. Therefore the authors [49] offered methods of prediction of the limiting properties (properties at fracture), based on the notions of fractal analysis and the cluster model of the amorphous state structure of polymers, with reference to a series of sulfur-containing epoxy polymers [52, 53] (see also Section 5.4). 

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