Self-affinity

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 question as to which rule has the greatest influence on the dissolution, Tb(SS) or Tb(WS), was addressed in another study. Again using the fo (solute) attribute as an indication of the extent of dissolution, the results from variations in these two rule sets are compared. The studies indicated that the T b(SS) and J(SS) rules had the greatest influence. This result suggest that the self-affinity of a solute, reflected by the Pb(SS) value, is a greater determinant of the solubility than the hydropathic state, reflected by the Pb(WS) value [5]. The validity of these findings is open to debate. 

Study 4.4a. Variation of solute self-affinity (SS) effect on dissolution... 

Figure 9 Flat elastic manifold pressed against a self-affine rigid surface for different loads L per atom in top wall.

Element Analysis of Contact between Elastic Self-Affine Surfaces. 

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. 

Now we will introduce briefly the concept of self-similar and self-affine fractals by considering the assumption that fractals are sets of points embedded in Euclidean E-dimensional space. 

Figure 2. (a) A deterministic self-similar fractal, i.e., the triadic Koch curve, generated by the similarity transformation with the scaling ratio r = 1/3 and (b) a deterministic self-affine fractal generated by the affine transformation with the scaling ratio vector r = (1/4, 1/2). 

In a similar way, the set S is statistically self-affine when S is the union of N non-overlapping subsets each of which is scaled down by r from the original and is congruent to r(.S ) in all statistical respects. 

In contrast to the self-similar case, the self-affine fractal dimension dEsa of even the simplest self-affine fractal is not... 

It is the local dimension that describes the irregularity of the self-affine fractal. The local dimension can be determined by such methods as the box-counting method1,61,62,65 and the dividerwalking method.61,66 The box dimension dEB is defined by the... 

In order to characterize the self-affine fractal surface, the self-affine fractal dimension dv has been determined by using the... 

IV. INVESTIGATION OF DIFFUSION TOWARDS SELF-AFFINE FRACTAL INTERFACE... 

Since diffusing species move randomly in all directions, the diffusing species may sense the self-affine fractal surface and the self-similar fractal surface in quite different ways. Nevertheless a little attention has been paid to diffusion towards self-affine fractal electrodes. Only a few researchers have realized this problem Borosy et al.148 reported that diffusion towards self-affine fractal surface leads to the conventional Cottrell relation rather than the generalized Cottrell relation, and Kant149,150 discussed the anomalous current transient behavior of the self-affine fractal surface in terms of power spectral density of the surface. 

Recently, Pyun et al.43,45 gave a clear interpretation of diffusion towards self-affine fractal interface. They investigated theoretically how the diffusing ions sense the self-affine fractal interface during the diffusion-limited process43 and then provided successful experimental evidence of the theoretical investigation.45 Here, let us explore their works in detail. 

In their theoretical work,43 the various self-affine fractal interfaces were mathematically constructed employing the Weierstrass function /ws(x), 151>152... 

A self-affine fractal profile with dFsR =1.5 was obtained by taking a, b, and N as 0.8165, 1.5, and 50, respectively. Next, various affine functions h(x) were made by multiplying /ws (x)... 

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