Fractal analysis application

It should be noted, that the considered above correction can be obtained and without fractal analysis application as well, using well-known Flory concept, that follows from the Eq. (1) of Chapter 1. However, obtaining of the anal dical correlation between Flory exponent and structural characteristics of polymers in solid phase is very difficult, if possible at all. At the same time this can be made within the framework of fractal analysis, since both macromolecular coil [25] and solid-phase polymer structure [62] are fractal objects. Hence, the possibility of solid-phase polymers properties quantitative prediction appears in such sequence molecular characteristics (for example, ) structure of macromolecular coil in solution polymer condensed state structure— polymer properties. In Section 2.6, this problem will be considered in detail. These considerations predetermined the choice of fractal analysis in Ref. [59] as a mathematical calculus. 

Hence, the stated above results have shown that the change of microgels stmcture, characterized by its fractal dimension, in the system EPS-4/DDM curing reaction course influences on both steric factor value and curing reaction conversion degree. The irreversible aggregation models and fractal analysis application... 

The main aspects of the fractal analysis application for the description of the behavior of macromolecular coils in the diluted solution are also considered to emphasize the intercommunication of the classical and fractal (structural) characteristics of macromolecular coils. Developed in the physical chemistry of polymer solutions, the basic ideas are the basis of oiuimderstanding of the peculiar properties of polymers. Such an approach allows one to receive the direct correlations stmcture-propeities , which is the main task of any physical domain including the physical chemistry of polymer solutiorrs and polymers synthesis. 

Not rejecting correctness of these principles, becoming already classical, we nevertheless distinguished three fundamental factors, the role of which happens obvious owing to the fractal analysis application for the description of polymer composites structure and properties [2-4]. 

Thus, the stated above results demonstrated, that fractal analysis application for polymers fracture process description allowed to give more general fracture concept, than a dilation one. Let us note, that the dilaton model equations are still applicable in this more general case, at any rate formally. The fractal concept of polymers fracture includes dilaton theory as an individual case for nonfractal (Euclidean) parts of chains between topological fixation points, characterized by the excited states delocalization. The offered concept allows to revise the main factors role in nonoriented polymers fracture process. Local anharmonicity ofintraand intermolecular bonds, local mechanical overloads on bonds and chains molecular mobility are such factors in the first place [9, 10]. 

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]. 

As it follows from the plot of Fig. 13.14, the Eq. (13.19) is valid for PASF considered film samples that confirms the correctness of the fractal analysis application for their fracture description [1]. 

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... 

A. Sadana and T. Vo-Dinh, Single- and dual-fractal analysis of hybridization binding kinetics biosensor application. Biotechnol. Prog. 14, 782-790 (1998). 

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. 

In our view, an oversimplified application of fractal analysis may tend to obscure rather than clarify the interpretation of adsorption data. In practice, there are two complicating factors (1) the derived values of nm are not always reliable, and (2) the mechanisms of adsorption and pore filling are dependent on the adsorbent-adsorbate interactions and the ratio of pore width to molecular diameter and may not be the same for all the members of a series of adsorptives on a given adsorbent. 

Barrett, A. H. and Peleg, M. 1995. Applications of fractals analysis to food structure. Lebensm. Wiss. U Technol. 28 553-563. 

Sadana, A. A single- and a dual-fractal analysis of antigen-antibody binding kinetics for different biosensor applications. Biosens. Bioelectron. 1999, 14, 515-531. 

Sadana, A. Ramakrishnan, A. A kinetic study of analyte-receptor binding and dissociation for biosensor applications a fractal analysis for cholera toxin and peptide-protein interactions. Sens. Actuators, B, Chem. 2002, 85, 61-72. 

Gas adsorption is a suitable method for a fractal analysis because it is sensitive to the fine structure of the pores and has negligible adverse affects on the pore system. The results are usually analyzed by using fractal generalizations of the Brunauer-Emmett-Teller (BET) isotherm (30) or of the Frenkel-nalsey-TfiU (FHH) isotherm (31). The latter may also be seen as a fractal generalization of the Kelvin equation and is therefore also applicable in the capillary condensation regime (32). It has been claimed that the fractal BET theory is more appropriate for mass fractals (see sect. Fractals ), whereas surface fractals are to be analyzed using the fractal FHH theory (33). These methods have been applied to cellulose powders (34) and tablets (35). 

Atomic force microscopy (AFM) has an important application in fractal analysis because it provides data characterizing the material surface in the scale from a few angstroms to hundreds of micrometers with great precision. In the AFM method each sample is scanned in a few different places so as to obtain a representative view of the whole surface topography. For powders a few particles are chosen and a surface fi agment is scanned on each of them. 

The theory of fractals and its application to physical and chemical processes has been developing vigorously in recent years [1-8]. To facilitate the understanding of the results presented in this chapter, we shall introduce some notions and definitions and consider briefly the grounds for applying the principles of synergetics and fractal analysis to the description of structures and properties of polymers. 

Fractal analysis allows consideration of the surface structure of the filler particles, which are characterised by its fractal dimension (d ) and by the self-similarity interval. Because the polymer structure is also described in the framework of the fractal analysis, it becomes possible to consider the interaction between the filler surface and polymer matrix, including the interfacial layers, based on the analysis of their fractal dimensions. Application of the model of irreversible aggregation allows description of the processes of aggregation of the filler particles in a particulate filled composites. This aggregation causes changes... 

The discrepancy indicated requires the application of principally differing approaches to the polymer nanocomposites melt viscosity description. Such an approach can be fractal analysis, within the ffamewoik of which, the authors used the following relationship for fractal liquid viscosity (h) estimation ... 

The ftactal analysis application has demonstrated [3] that the constant is not empirical, but serves as a macromolecular coil stmcture in solution characteristic and it is linked with the coil fractal dimension Dj.hy a simple relationship ... 

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] ... 

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]. 

Hence, the results stated above demonstrated that the cluster model of polymers amorphous state stmcture and fractal analysis allowed quantitative prediction of mechanical properties for pol5miers film samples, prepared from different solvents. Let us note, that the properties prediction over the entire length of the diagram a- was performed within the framework of one approach and with precision, sufficient for practical applications. This approach is based on strict physical substantiation of the analytical intercommunication between structures of a macromolecular coil in solution and pol5miers condensed state [201]. 

The indicated discrepancy requires the application of principally differing approach at the description of pol5nner nanocomposites melt viscosity. Such approach can be the fractal analysis, within the framework... 

The considered circumstances served as a cause for the development of the fractal analysis, particularly practical applications for polymer solutions (melts) properties description, which will be considered in the present monograph. 

Damage tolerance of fibre reinforced thermoplastic composites Processing of polymer matrices using resin transfer moulding Fractal analysis of wear in short-fibre reinforced thermoplastic composites Rheology flow behavior of associative polymers in coating applications Kevlar-thermoplastic composites... 

Multifractal analysis provides a complete description of fault-network fractal properties. It could more veritably describe the complexity and essentiality of fault structure compared to single fractal analysis. The multifractal theory has been widely applied in various fields of earth science. More details about the properties of multifractal spectra in earth sciences applications can be found in References (Agterberg et al. 1996, Zhao et al. 2011, Kiyashchenko et al. 2004). Fault tectonic has multifractal structure characteristics (Panahi and Cheng 2004). But it was not found on the coalfield fault tectonics. 

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 adduced above results on the example of the dependences MM(C(,) for polyaiylates Ph-2 showed the necessity of polycondensation process both static and kinetic aspects consideration for correct dependences of one or another limiting characteristics. The absence of maximum on curves MM(c ) means the absence of optimum value c , at which maximum value MM is reached. The adduced examples demonstrate clearly correctness and expediency of fractal analysis methods application for polycondensation processes description. 

Hence, the stated above results have shown, that fractal analysis and irreversible aggregation models application allows to obtain the clear physical interpretation of copolycondensation process and estimate its quantitative characteristics. The fractal dimension D. of macromolecular coil in solution is the main characteristic, controlling this process [142]. 

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]. 

Kozlov, G. V Shustov, G. B. Zaikov, G. E. The reaction cessation in polycondensation process fractal analysis. In book Progress in Chemistry and Biochemistry. Linetics, Thermodynamics, Synthesis, Properties and Applications. Ed. Pearce, E. Zaikov, G. New York, Nova Science Publishers, Inc., 2009,61-72. 

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