Surface fractal dimensions

Fig. 2. Plot to calculate surface fractal dimension by mercury porosimetry data.

Table 1 Surface fractal dimension determined by nitrogen adsorption...

Su-Il Pyun provide a comprehensive review of the physical and electrochemical methods used for the determination of surface fractal dimensions and of the implications of fractal geometry in the description of several important electrochemical systems, including corroding surfaces as well as porous and composite electrodes. 

Diffusion-limited electrochemical techniques as well as physical techniques have been effectively used to determine the surface fractal dimensions of the rough surfaces and interfaces made by electrodeposition, " fracture, " vapor deposition, ... 

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. 

Several algorithms have been used to determine surface fractal dimension from SPM images. The most popular ones are... 

Since Avnir and Pfeifer s pioneer works83"86 regarding the characterization of the surface irregularity at the molecular level by applying the fractal theory of surface science, the molecular probe method using gas adsorption has played an important role in the determination of surface fractal dimension of the porous and particulate materials. 

There are two molecular probe methods available for the determination of surface fractal dimension. One is the multiprobe method (MP method),83,84,87-100 which uses several kinds of multiprobe molecules with different molecular sizes and requires the number of adsorbed molecules to form a monolayer Nmoao for each probe molecule. If the probe molecule is varied through a series of spheres with radius rm, the surface fractal dimension is given by Eq. (7) ... 

Here, the surface fractal dimension determined by using the MP method is denoted by dF MP. The probe molecules need not be... 

The other molecular probe method is the single-probe method (SP method), which is separately proposed by Avnir and Jaroniec,93 and Pfeifer et al.108-112 In the SP method, a single adsorption isotherm is analyzed using a modified FHH theory. The FHH model was developed independently by Frenkel,113 Halsey,114 and Hill,115 and describes the multilayer adsorption coverage. Since the SP method uses only one probe molecule, this method is more convenient than the MP method. However, there are many theoretical limitations in applying the SP method to determination of the surface fractal dimension. Therefore, it is really necessary to discuss whether the SP method is an adequate tool to investigate the surface fractal dimension or not before applying the SP method to certain system. 

On the basis of the theory of Pfeifer et al.,108-112 the surface fractal dimension can be determined from the gas adsorption isotherms according to the following equation ... 

Van der Waals forces between solid/gas interactions and the liquid/gas surface tension forces represent the limiting cases, but in general both the forces competitively affect the adsorption process. Therefore, in determining the surface fractal dimension of the carbon specimen, it is very important to use appropriate relation between C and dFSF. According to Ismail and Pfeifer,111... 

It is reported116,117 that as more adsorbed layers are built up, the interface between the adsorbent and the adsorbed molecules becomes smooth, and hence the surface fractal dimension would no longer describe the interface but would describe the adsorbed molecule agglomerates. Also, Eq. (9) is only valid when the adsorbed layer exceeds monolayer coverage. Therefore, for the correct calculation, dFSP should be determined from the linear... 

On the other hand, the surface fractal dimension characterizes the pore surface irregularity the larger the value of surface fractal dimension is, the more irregular and the rougher is the pore surface. Since the pore structure is closely related to the electroactive surface area which plays a key role in the increase of capacity in practical viewpoint, many researchers have investigated the microstructure of the pores by using fractal geometry. 

Besides the molecular probe method using gas adsorption,107 162 recently, the TEM image analysis method163"167 has been applied to evaluate the surface fractal dimension of porous materials. The most attractive fact in this method is that the pores in different size ranges can be extracted from the TEM images which include contributions from many different pore sizes by the inverse fast Fourier transform (FFT) operation by selecting the specific frequency range.165 167... 

The present article summarized the fractal characterization of the rough surfaces and interfaces by using the physical and the electrochemical methods in electrochemistry. In much research, both the physical and the electrochemical methods were used to evaluate the fractal dimension and they are complementary to each other. It should be stressed that the surface fractal dimension must be determined by using the adequate method, according to the inherent scaling properties of the rough surfaces and interfaces. 

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. 

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

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