Self-affine properties

Except for perfect crystals, the fracture surfaces in solids are never very smooth. In fact, it seems now to be established experimentally that for weakly disordered solids, the surfaces formed during fracture processes are very rough, and on a mesoscopic scale (which is much larger than the atomic scale but smaller than the macroscopic sample size) they are observed to have self-affine properties (Mandelbrot et al 1984, Bouchaud et al 1993 a,b, Roux 1994). These have recently been observed using various fractographic investigations. By self-affinity of the fracture surface, we mean that the surface coordinate in the direction perpendicular to the crack or fracture x — y) plane has the scaling property such that... 

The technology of silicon and germanium production has developed rapidly, and knowledge of die self-diffusion properties of diese elements, and of impurity atoms has become reasonably accurate despite die experimental difficulties associated widi die measurements. These arise from die chemical affinity of diese elements for oxygen, and from die low values of die diffusion coefficients. 

Experiments on transport, injection, electroluminescence, and fluorescence probe the spatial correlation within the film, therefore we expect that their response will be sensitive to the self-affinity of the film. This approach, which we proved useful in the analysis of AFM data of conjugated molecular thin films grown in high vacuum, has never been applied to optical and electrical techniques on these systems and might be an interesting route to explore. We have started to assess the influence of different spatial correlations in thin films on the optical and the electro-optical properties, as it will be described in the next section. 

The structure of this review is composed of as follows in Section II, the scaling properties and the dimensions of selfsimilar and self-affine fractals are briefly summarized. The physical and electrochemical methods required for the determination of the surface fractal dimension of rough surfaces and interfaces are introduced and we discuss the kind of scaling property the resulting fractal dimension represents in Section III. 

In Section IV, from the studies on diffusion towards self-affine fractal interface, the surface fractal dimension as determined by the electrochemical method is characterized as being self-similar, even though the rough surfaces and interfaces show the self-affine scaling property. Finally, in Section V, we exemplified the application of fractal geometry in electrochemical systems in view of the characterization of rough surfaces and interfaces by the surface fractal dimension. 

The word fractal was coined by Mandelbrot in his fundamental book.1 It is from the Latin adjective fractus which means broken and it is used to describe objects that are too irregular to fit into a traditional geometrical setting. The most representative property of fractal is its invariant shape under self-similar or self-affine scaling. In other words, fractal is a shape made of parts similar to the whole in some way.61 If the objects are invariant under isotropic scale transformations, they are self-similar fractals. In contrast, the real objects in nature are generally invariant under anisotropic transformations. In this case, they are self-affine fractals. Self-affine fractals have a broader sense than self-similar fractals. The distinction between the self-similarity and the selfaffinity is important to characterize the real surface in terms of the surface fractal dimension. 

H. -C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem., 531 p. 101, Copyright 2002, with permission from Elsevier Science. 

Figure 8. Simulated current transients obtained from the self-affine fractal profiles h(x) of various morphological amplitudes rj of (a) 0.1, 0.3, and 0.5 (b) 1.0, 2.0, and 4.0 in h(x) = 7]fws(x). Reprinted from H.-C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem., 531, p. 101, Copyright 2002, with permission from Elsevier Science.

Under the assumption that the morphology of the self-affine interface has the self-similar scaling property, the apparent selfsimilar fractal dimension d ss of the electrode was calculated... 

Figure 9 demonstrates the dependence of the scaled length SL on the segment size SS obtained from the self-affine fractal profiles in Figure 7 by using the triangulation method for the Euclidean two-dimensional space. The linear relation was clearly observed for all the self-affine fractal curves, which is indicative of the self-similar scaling property of the curves. 

From the above results, it is noted that the self-similar scaling property investigated by the triangulation method can be effectively utilized to analyze the diffusion towards the self-affine fractal interface. This is the first attempt to relate the power dependence of the current transient obtained from the self-affine fractal curve to the self-similar scaling properties of the curve. 

In order to characterize the self-affine scaling properties of the fractal Pt films, the self-affine fractal dimensions of the film surfaces t/Fsa were determined by using the perimeter-area... 

Bearing in mind that diffusing ions move randomly in all directions, it is reasonable to say that the diffusing ions sense selfsimilar scaling property of the electrode surface irrespective of whether the fractal surface has self-similar scaling property or self-affine scaling property. Therefore, it is experimentally justified that the fractal dimension of the self-affine fractal surface determined by using the diffusion-limited electrochemical technique represents the apparent self-similar fractal dimension.43... 

In summary, from the above theoretical and experimental results, it is concluded that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent selfsimilar fractal dimension rather than the self-affine fractal dimension. In addition, the triangulation method is one of the most effective methods to characterize the self-similar scaling property of the self-affine fractal electrode. 

Panella and Krim suggested that the high value n = 4.7 was more consistent with the properties of a self-affine surface rather than a self-similar one. Self-affine fractals are associated with asymmetric scaling, that is different scaling relations in different directions (Avnir, 1997). 

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