Amorphous polymers fractal dimension

Another class of materials with fractal structure are amorphous polymers. Here fractal properties manifest themselves on scales exceeding the dimensions... 

Figure 11.5 presents the dependence of on R, for five amorphous vitreous and five amorphous-crystalline polymers [8]. It can be seen that this correlation is fully consistent with relationship (11.26). The df value of the cluster structure determined from the slope of the straight line is 2.75. The fractal dimension of the cluster in the WS model lies in the range of 2.25-2.75 [91], which is in agreement with the above estimate. 

The fractal dimension for the structure of the amorphous polymer can be found from the relationship [97] ... 

Figure 11.6 shows the plots for the variation of versus r j for two amorphous polymers the plots correspond to relationship (11.28). In other words, a stable crack in polymer film samples is a stochastic fractal with the dimension 1.48. The linearity of plots shown in... 

In Figure 14.3, the double logarithmic dependence is presented by = /(Rsp) for five amorphous and five semicrystalline polymers, where R p = 0.5. From this correlation it can be seen that Equation (14.3) is obeyed and from its slope it is possible to determine the value, df which appears to be equal to 2.75. It is known [3] that the fractal dimension of a cluster in WS-model lies within the limits 2.25-2.75, which is in excellent agreement with the estimation obtained from Figure 14.3. 

In Figure 14.4 double logarithmic dependences 2 In 8 = /(In r) are presented for two amorphous polymers which have appeared to be linear and by virtue of it, correspond to Equation (14.7). Otherwise, the stable crack in film polymeric samples is a stochastic fractal with dimension 1.48. Linearity of the diagrams presented on Figure 14.4 reflects the self-similarity of a crack at different stages of its growth. Thus, at the macroscopic level polymers the fractal properties are also displayed. 

It has been established that the structure of an amorphous polymer is fractal with a fractal dimension (df) (2 < df < 3). Therefore we ought to expect, that the fluctuation free volume should also have fractal properties. The purpose of this chapter is to investigate the fractality of the fluctuation free volume in glassy polymers and epoxy polymers are... 

As can be seen in Figure 15.2, the dependence (15.3) is correct for the epoxy polymers investigated and confirms the self-similarity of a cluster of microvoids of fluctuation free volume. The interval of the scale of the self-similarity, with allowance for correlations between Df and fractal dimension of the structure of the polymer df may be assumed. This interval coincides with a similar interval for the structure of an amorphous polymer which is distributed from several units up to several tens of Angstrom (5-50 A) [1, 4]. 

Yamauchi et al. investigated the structures of NR/high density polyethylene (HDPE) thermoplastic elastomers (TPEs) and their composites with carbon black using SANS. The extremely low contrast between crystalline and amorphous HDPE for SANS enabled us to measure the interface thickness between NR and HDPE in the TPE (5 mn) as well as that between NR and carbon black in the composite (2.4 mn). The interface thickness and fractal dimension of the blends and composites were earefully analysed using SANS even though both polymers were not deuterated. It was revealed that NR and HDPE are immiscible blends. 

The interrelation of elasticity modulus and amorphous chain s tightness characterized by fractal dimension of chain part between its fixation points for nanocomposites based on the polypropylene is shown. This assumes the polymeric matrix stmcture change in comparison with initial polymer the role of densely-packed regions for it is played by interphase areas. An offered fractal model allows estimation of elasticity modulus limiting values. 

Kozlov, G. V., Ozden, S., Krysov, V. A., Zaikov, G. E. (2001). The Experimental Determination of a Fractal Dimension of the Structure of Amorphous Glassy Polymers. In Fractals and Local Order in Polymeric Material. Kozlov, G. V., Zaikov, G. E., Ed., New York, Nova Science Publishers Inc., 83-88. 

Lately the mathematical apparatus of fractional integration and differentiation [58, 59] was used for fractal objects description, which is amorphous glassy polymers structure. It has been shown [60] that Kantor s set fractal dimension coincides with an integral fractional exponent, which indicates system states fraction, remaining during its entire evolution (in our case deformation). As it is known [61], Kantor s set ( dust ) is considered in onedimensional Euclidean space d = ) and therefore, its fractal dimension obey the condition d Euclidean spaces with d > 2 (d = 2, 3,. ..) the fractional part of fractal dimension should be taken as fractional exponent [62, 63] ... 

In paper [126] it was shown that universality of the critical indices of the percolation system was connected directly to its fractal dimension. The self-similarity of the percolation system supposes the availability of the number of subsets having order n (n = 1, 2, 4,. ..), which in the case of the structure of amorphous polymers are identified as follows [125]. The first subset (n = 1) is a percolation cluster frame or, as was shown above, a polymer cluster network. The cluster network is immersed into the second loosely packed matrix. The third (n = 4) topological structure is defined for crosslinked polymers as a chemical bonds network. In such a treatment the critical indices P, V and t are given as follows (in three-dimensional Euclidean space) [126] ... 

The authors [26, 27] used Relationship 6.6 for description of the behaviour of the shear modulus G in the case of linear amorphous polymers. They found out that for the correct description of G the indicated relationship required two modifications. Firstly, in Equations 6.7-6.9 the dimension d should be replaced with the polymer structure fractal dimension d Secondly, it is required to introduce a variable percolation threshold p, accounting for the deviation from the quasi-equilibrium state of the loosely packed matrix [27] ... 

In Figure 9.1 the comparison of dimensions and for the studied EP is adduced. Their good correspondence indicates unequivocally that their loosely packed matrix, which serves simultaneously as a natural nanocomposite matrix, is the fractal space where the nanocluster structure of epoxy polymers is formed. Since for linear amorphous polymers = 3 [9], i.e., their nanostructure formation is realised in three-dimensional Euclidean space, then the conclusion that chemical crosslinking network availability in the considered EP serves as the indicated distinction cause is obvious enough. In Figure 9.2 the dependence of on crosslinking density is... 

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