Fractals self-affinity

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. 

Quantitatively, a self-afRne fractal is defined by the fact that a change Ax XAx (and possibly Ay —> XAy) transforms Az into X Az, where H lies between 0 and 1. The case H — 1 corresponds to a self-similar fractal. Self-affine fractal structures are no longer characterised by just one (mass or boundary) fractal dimension they require two. The first is local and can be determined by the box-counting method, for example it describes the local scale invariance and its value lies between 1 and 2. The second is global and its value is a simple whole number describing the asymptotic behaviour of the fractal. In the case of a mountain, this global dimension is simply 2. When viewed from a satellite, even the Himalayas blend into the surface of the Earth. 

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. 

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 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)... 

In order to examine the current response to the imposition of the potential step on the self-affine fractal interface, the current transients were calculated theoretically by random walk simulation.153 The simulation cell was taken as the square area bottom boundary which is replaced by one of the self-affine fractal profiles in Figure 7. The details of the simulation condition were described in their publication.43... 

Figures 8a and 8b present the simulated current transients obtained from the self-affine fractal interfaces of r/ = 0.1 0.3 0.5 and r] = 1.0 2.0 4.0, respectively, embedded by the Euclidean two-dimensional space. It is well known that the current-time relation during the current transient experiment is expressed as the generalized Cottrell equation of Eqs. (16) and (24).154 So, the power exponent -a should have the value of - 0.75 for all the above self-affine fractal interfaces.

However, the simulated current transients in Figure 8 never exhibited the expected power exponent of -0.75 with the exception of the original self-affine fractal interface ( = 1) the... 

Figure 7. Self-affine fractal profiles of various amplitudes with a self-affine fractal dimension of 1.5, obtained from the...

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.

It should be stressed here that the specific power dependences from the above self-affine fractal interfaces are maintained even during the relatively long time (or number of random jumps) interval. This implies that the morphology of the self-affine fractal interfaces tested is possibly characterized by the self-similar fractal dimension within a relatively wide spatial cutoff range. 

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... 

Fractal Dimensions of the Profiles h(x) at Various Morphological Amplitudes rj in h(x) = 77/wsCv) Determined by the Current Transient Technique (2nd Column) and the Triangulation Method (3rd Column). Here, /ws(-v) Means the Weierstrass Function with a Self-Affine Fractal Dimension t/Fsa = 1.5 ... 

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. 

---