Rough surfaces fractal geometry

Euclidean geometry fails to describe disordered surfaces such as real solid surfaces. Fractal geometry, which has been developed to overcome this obstacle, covers surface, mass, and pore fractality. It has been pointed out that the diffusion process can be used to characterize the fractal dimension of a rough surface. The impedance response of a rough electrode could be used for the characterization of the roughness and. 

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]. 

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. 

III. CHARACTERIZATION OF ROUGH SURFACES AND INTERFACES BASED UPON FRACTAL GEOMETRY METHODS NEEDED FOR THE DETERMINATION OF THE SURFACE FRACTAL DIMENSION... 

Fractal Approach to Rough Surfaces and Interfaces in Electrochemistry, from a review of Fractal Geometry to the application of Fractal Geometry to the classification of surfaces and Electrochemistry... 

Farm and Avnir [113] were the first to use fractal geometry to determine effects of surface morphology on drug dissolution. This was accomplished by the use of the concept of fractal reaction dimension dr [114], which is basically the effective fractal dimension of the solid particle toward a reaction (dissolution in this case). Thus, (5.7) and (5.8) were modified [113] to include surface roughness effects on the dissolution rate of drugs for the entire time course of dissolution... 

Figure 2. Small-angle X-ray-scattering curves from silica polymers grown in alkaline solution with various ratios (R) of water to TEOS. The data are slit smeared and are plotted as log 1(h) versus log h, with the curves displaced vertically for clarity. The observed slopes are algebraically one greater than would be observed with pinhole geometry. The curve for R = 1 is consistent with a mass fractal structure with D/ = 2.84. The curves for R = 2-4 arise from colloids (Df = 3) with fractally rough surfaces with D, = 2.71, 2.51, and 2.45, respectively. (Reproduced with permission from reference 10. Copyright 1984 Elsevier.)...

Kaye, B.H. Part I rugged boundaries and rough surfaces applied fractal geometry and the fineparticle specialist. Part. Part. Syst. Charact 1993, 10 (3), 99-110. 

In some specific cases other parameters can be considered as important in the characterisation of membrane morphology like the surface roughness, pore anisotropy and porous network connectivity [16,17]. Concepts of percolation and fractal geometry are also of interest to better describe the statistical and random structures of many porous solids [14,18,19]. 

The most modern methods of dealing with irregular, rough surfaces are based on fractal geometry. Examples of the application of the fractal approach to the analysis of silica surfaces are included in the Legrand et al. monograph (17) and in Brinker and Scherer s textbook (8). 

A new development in the approach to both apparently smooth and rough surfaces is based on fractal geometry. According to this approach, only ordered surfaces can be adequately described by Euclidean geometry. 

Experiments on model electrodes of fractal geometry were found to be in accordance with the theoretical prediction (Pajkossy and Nyikos, 1989). The effect of surface roughness and porosity is of great importance, and several impedance models were derived to explain the effect of three-dimensional electrode structures, as reviewed by de Levie (1967). 

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