Derivatives fractal structure analysis

The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208-215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200-203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. 

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. 

Secondly, polymers are known to possess multilevel structures (molecular, topological, supermolecular, and floccular or block levels), the elements of which are interconnected [43, 44]. In addition, an external action on a polymer can induce the formation of new (secondary) structural elements — cracks, fractured surfaces, plastic flow regions, etc. These primary and secondary structural elements as well as the processes forming them are characterised by miscellaneous parameters therefore, only empirical correlations have been obtained, at best, between these parameters. If each of the above-mentioned elements (processes) is described by a standard parameter, for example, fractal dimension, one can derive analytical equations relating them to one another and containing no adjustable parameters. This is very significant for the computer synthesis of structure and for the prediction of properties and behaviour of polymeric materials during performance. Note that fractal analysis has been used successfully to describe the phenomena of rubber elasticity [16, 45, 46] and fluidity [25, 47-49]. 

It is worth noting that at the minimum value of d = 2, value A becomes negative. That means, in accordance with Equation (12.1), that tan < tan 8m- In other words, at small d (smooth surfaces of the filler particles) the packing of the polymer molecules at the interface may be more dense compared with the bulk. This fact leads to the diminishing molecular mobility in the interfacial layer [1, 37, 38] Extrapolation of the dependence of A on d to maximum value of d = 3 gives the limiting value of A 4.5. This value meets the value A = 4.2, that was derived from the extrapolation of the volume of the interfacial layer on d to d = 3 [39]. Thus, this analysis allows an estimation of the structural factors influencing formation of the adhesion joints. The main factors are fractal dimensions of the particle surface, d and of the polymer df, which determine adhesion at the polymer-filler particle interface. 

The authors of Ref. [53] have shown, that frictional properties of fractal clusters can be different essentially for the usual results for compact (Euclidean) structures. It is known through Ref. [54], that the polymer melt structure can be presented as a macromolecular coils sets, which are fractal objects. Therefore, the authors [55] proposed general structural treatment of polymer melt viscosity within the framework of fractal analysis, using the model [53]. Within the framework of the indicated model the derivations for translational friction coefficient f(N) of clusters from N particles in three-dimensional Euclidean space were received, calculated according to Kirkwood-Riseman theory in the presence of hydrodynami-cal interaction between the cluster particles. The fundamental relationship of this theory is the following equation [53] ... 

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 polymer-metal interface shown in Fig. was derived from an electron micrograph obtained by Mazur and Reich.They electrodeposited silver from a silver ion solution diffusing through a polyimide film. Particles not connected to the diffusion source were removed by computer analysis. The deposited silver particles essentially "decorate the concentration profile and permit the diffusion front to be observed. A 1000-A thin slice was used to aproximate two dimensional diffusion. The fractal dimension of this interface was determined by computer analysis to be approximately 1.7. Similar ramified interface fronts are created by vapor deposition of metal atoms on polymers and by certain ion bombardment treatments of polymer surfaces. The fractal front is fairly insensitive to the details of the concentration profile. However, strong chemical potential gradients in asymmetric interfaces may promote a more planar, less ramified structure. The fractal characteristics of polymer interfaces... 

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