Amorphous polymers fractal structure

Kozlov, G. V., Ozden, S., Dolbin, I. V. (2002). Small Angle X-Ray Studies of the Amorphous Polymers Fractal Structure. Russian Polymer News, 7(2), 35-38. 

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

It has been noted previously that the structures and properties of polymers are studied using the principles of synergetics and methods of fractal analysis. This is based on several prerequisites. Firstly, amorphous glassy polymers have thermodynamically nonequilibrimn structure [32]. Schaefer and Keefer [33] showed that fractal structures are formed in nonequilibrium processes. Therefore, there is good reason to believe that there are fractal structures in glassy polymers and that they can be described using the methods of synergetics. These assumptions have been repeatedly confirmed by experiments [32, 34-40]. 

For example, the low-freqnency region of the spectra of inelastic light scattering of amorphous polymers is a broad structureless plateau [22]. This is due to the fractal structure of polymers on small linear scales [22, 36]. 

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

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]. 

In Fig. 1.3 amorphous polymers nanostructure cluster model is presented. As one can see, within the limits of the indicated above dimensional periodicity scales Fig. 1.2 and 1.3 correspond each other, that is, the cluster model assumes p reduction as far as possible from the cluster center. Let us note that well-known Flory felt model [20] does not satisfy this criterion, since for it p const. Since, as it was noted above, polymeric mediums structure fractality was confirmed experimentally repeatedly [14-16], then it is obvious, that cluster model reflects real solid-phase polymers structure quite plausibly, whereas felt model is far from reality. It is also obvious, that opposite intercommunication is true - for density p finite values change of the latter within the definite limits means obligatory availability of structure periodicity. 

As it is known [83], a glassy polymers behavior on cold flow plateau (part III in Fig. 4.17) is well described within the frameworks of the rubber high-elasticity theory. In Ref [39] it has been shown that this is due to mechanical devitrification of an amorphous polymers loosely packed matrix. Besides, it has been shown [82, 84] that behavior of polymers in rubber-like state is described correctly under assumption, that their structure is a regular fractal, for which the identity is valid ... 

The studies carried out earlier have shown that polymer film samples strength to a considerable extent is defined by growth parameters of stable crack in local deformation zone (ZD) at a notch tip [1-3], As it has been shown in Refs. [4, 5], the fiactal concept can be used successfully for the similar processes analysis. This concept is used particularly successfully for the relationships between fracture processes on different levels and subjecting fracture material microstructure derivation [5]. This problem is of the interest in one more respect. As it has been shown earlier, both amorphous polymers structure [7] and Griffith crack [4] are fractals. Therefore, the possibility to establish these objects fractal characteristics intercommunication appears. The authors of Refs. [8, 9] consider stable cracks in polyarylatesul-fone (PASF) film samples treatment as fractals and obtain intercommunication of this polymer structure characteristics with samples with sharp notch fracture parameters. 

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]. 

The authors of Ref [9] conducted cross-linked polymers microhardness description within the frameworks of the fractal (structural) models and the indicated parameter intercommunication with structure and mechanical characteristics clarification. The epoxy polymers structure description is given within the frameworks of the cluster model of polymers amorphous state structure [10], which allows to consider polymer as natural nanocomposites, in which nanoclusters play nanofiller role (this question will be considered in detail in chapter fifteen). 

Yech, G. S. (1979). The General Notions on Amorphous Polymers Structure. Local Order and Chain Conformation Degrees. N sokomolek.SoedA, 21 (Nil), 2433-2446. Perepechko, 1.1. (1978). Introduction in Physics of Polymers. Moscow, Khimiya, 312p. Kozlov, G. V, Zaikov, G. E. (2001). The Generalized Description of Local Order in Polymers. In Fractals and Local Order in Polymeric Materials. Kozlov, G. V., Zaikov, G. E., Ed., New York, Nova Science Pubhshers Inc. 55-63. 

A sol is a dispersion of solid particles or polymers in a liquid. It is possible to precipitate particles that are amorphous or crystalline, or to make amorphous particles that become crystalline through dissolution and reprecipitation. The latter process can produce particles that differ little from ordinary ceramic oxides, except that the sol particles are small (submicron). If the solubility of the solid phase in the liquid is limited, monomers may attach irreversibly to a growing cluster, so that rearrangement into the equilibrium structure is impossible. In that case, polymeric clusters appear with fractal structures that are quite different from ceramics they typically have much lower connectivity (i.e., fewer bridging oxygen bonds) and consequently contain many hydroxyl and organic ligands. 

However, one should not forget that fractal analysis gives only a common mathematic description of a polymer s structure, i.e., it does not identify those structural units (elements) that any real polymer consists of. The cluster model of polymer amorphous state structure allows one to obtain a physical description of a thermodynamically nonequilibrium polymer s structure with local (short-range) order representations drawing and molecular characteristics usage, which identifies its element quantitatively. Since these models consider polymer structure from different positions, they are a very good complement of one another [7, 29]. 

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]. 

In the present section a number of modern physical concepts for the description of the structure of crosslinked polymers is used the thermodynamic concept, the cluster model of amorphous state structure of polymers, fractal analysis, irreversible aggregation models and the thermal cluster model. Within the frameworks of the thermodynamic approach the interconnection of structural and molecular characteristics of crosslinked polymers with disorder parameter 8 is considered [69]. According to the concept [69] the indicated parameter, connected with the thermal mobility of molecules near the melting temperature, is expressed by Formula 1.28. Since p. is given by Equation 1.29 then Relationship 1.30 can be received from combination of Equations 1.28 and 1.29. 

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] ... 

The treatment of an epoxy polymer as a natural nanocomposite or quasi-two-phase system [8] puts in the foregroimd the interaction of such system components, which for nanocomposites is expressed first of all in an interfacial regions formation [2-4]. Let us note that in the reinforcement process (increase in the elasticity modulus of the nanocomposite in comparison with the matrix polymer) interfacial regions play the same role as the nanofiller [2-4], Such a reinforcement mechanism is due to the formation of nanocomposites with inorganic nanofiller [1-A and the structure of natural nanocomposites (linear amorphous polymers) [9] in three-dimensional Euclidean space. Therefore in paper [10] the study of structure formation conditions for crosslinked epoxy polymers, treated as natural nanocomposites, was carried out within the frameworks of fractal analysis. 

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... 

The authors of paper [60] gave the description of the microhardness of crosslinked epoxy polymers within the frameworks of fractal (structural) models and elucidated the indicated parameter interconnection with structure and mechanical characteristics. Description of the structure of the epoxy polymers is given within the frameworks of the cluster model of the amorphous state structure of polymers [5-7], which allows polymers to be considered as natural nanocomposites in which nanoclusters play the role of nanofiller. 

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