Surface self-similar

An analogous procedure can be applied to a plane surface. The surface can be roughened by the successive application of one or another recipe, just as was done for the line in Fig. VII-6. One now has a fractal or self-similar surface, and in the limit Eq. VII-20 again applies, or... 

Figure XVI-1 and the related discussion first appeared in 1960 [1], and since then a very useful mathematical approach to irregular surfaces has been applied to the matter of surface area measurement. Figure XVI-1 suggests that a coastline might appear similar under successive magnifications, and one now proceeds to assume that this similarity is exact. The result, as discussed in Section VII-4C and illustrated in Fig. VII-6, is a self-similar line, or in the present case, a self-similar surface. Equation VII-21 now applies and may be written in the form...

The chemical reactivity of a self-similar surface should vary with its fractional dimension. Consider a reactive molecule that is approaching a surface to make a hit. Taking Fig. VII-6d as an illustration, it is evident that such a molecule can see only a fraction of the surface. The rate of dissolving of quartz in HF, for example, is proportional to where Dr, the reactive... 

Various means of constructing self-similar surfaces are known.33 Some of them do not allow one to produce different realizations of surface profiles, for example, by making use of the Weierstrass function. These methods should be avoided in the current context because it would be difficult to make statistically meaningful statements without averaging over a set of statistically independent simulations. An appropriate method through which to construct self-similar surfaces i s to use a representation of the height profile h(x) via its Fourier transform h(q). 

In reciprocal space, self-similar surfaces described by Eqs. [8] and [9] are typically characterized by the spectrum S(q) defined as... 

Alternatively, one may simply write h(x) as a sum over terms h(q) cos(qx +

random number with the proper second moment of b(q) with zero mean and one random number for each phase

uniformly distributed between 0 and ji, and filter the absolute value of b x) in the same way as described in the previous paragraph. Other methods exist with which to generate self-similar surfaces, such as the midpoint technique, described in Ref. 24. 

The equations derived from the dynamic scaling theory are valid for self-afiBne and self-similar surfaces. Accordingly, the theory provides information about fractal properties and growth mechanisms of rough surfaces. 

Many of the adsorbents used have rough surfaces they may consist of clusters of very small particles, for example. It appears that the concept of self-similarity or fractal geometry (see Section VII-4C) may be applicable [210,211]. In the case of quenching of emission by a coadsorbed species, Q, some fraction of Q may be hidden from the emitter if Q is a small molecule that can fit into surface regions not accessible to the emitter [211]. 

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

Fractal theory is a relatively new field of geometry, formulated by Mandelbrot [196] for irregular rough-surfaced objects. The major properties of such objects are the dependence of the measured length (perimeter), surface, or volume on the scale of measurement and geometrical self-similarity... 

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. 

In order to characterize the three-dimensional self-similar fractal surface, the self-similar fractal dimension d ss has been... 

Figure 3. Process of determination of the self-similar fractal dimension of the three-dimensional surface by the triangulation method.

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. 

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... 

Regarding the electrochemical method, the generalized forms of the Cottrell relation and the Randles-Sevcik relation were theoretically derived from the analytical solutions to the generalized diffusion equation involving a fractional derivative operator under diffusion-controlled constraints and these are useful in to determining the surface fractal dimension. It is noted that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent self-similar fractal dimension rather than the self-affine fractal dimension. This means the fractal dimension determined by using the diffusion-limited electrochemical method is the self-similar fractal dimension irrespective of the surface scaling property. 

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