Fractal dimension roughness

Table 63. Value of the roughness fractal dimension derived from fractal plots of log a versus, log R (Farin and Avnir, 1989)...

The currently useful model for dealing with rough surfaces is that of the selfsimilar or fractal surface (see Sections VII-4C and XVI-2B). This approach has been very useful in dealing with the variation of apparent surface area with the size of adsorbate molecules used and with adsorbent particle size. All adsorbate molecules have access to a plane surface, that is, one of fractal dimension 2. For surfaces of Z> > 2, however, there will be regions accessible to small molecules... 

Fractal objects are quantified by their fractal dimension, dj. For linear-like stmctures, 1 < <2. FractaUy rough stmctures have a mass fractal... 

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

Obviously the roughness factor is similarly arbitrary, but it is of interest to use Eq. 25 to compute its value for some trial values of D and a. This is done in Table 2. In order to map the surface features even crudely, the probe needs to be small. It can be seen that high apparent roughness factors are readily obtained once the fractal dimension exceeds 2, its value for an ideal plane. 

Roughness factor calculated for a fractal surface, according to the fractal dimension D and probe area a... 

Physically, the wetting abdity increases (the contact angle decreases) as the values of the fractal dimension of the electrode increases if the electrode material is same. However, in this study, we could not obtain a good correlation between the fractal dimensions and the wetting abilities as shown in Table 1. It means that not only the physical properties such as the surface irregularity and roughness but also the chemical interaction between electrolyte and electrode were important in wetting ability. 

With the experimental results about the wetting ability and the fractal dimension of four kinds of anode electrodes, we could conclude the following. The addition of NisAl could make the electrolyte wet the electrode very well. The pore structures of all the electrodes prepared in this study were highly irregular and rough. Finally, the chemical properties of the surfaces were as important as the physical properties in determining the wetting ability of the electrodes in this study. 

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. 

Therefore, we can define a length scale range where the rough surface shows the fractal behavior by determining the inner and outer cutoffs using Eq. (28).121 In other words the fractal dimensions determined from the power relation between 7peak and... 

Keeping in mind that the dc sputter-deposited Pt films have the completely electrochemical-active surface, the fractal dimensions of the rough film surfaces were calculated from Eq. (27) according to the peak-current method by taking the slopes of the (log 7peak - log v) plots within the scan rate range of v0 to... 

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. 

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

The fractal dimension D of an ideal solid is 3.00, of a perfectly flat surface 2.00, and of a straight line 1.00—the familiar geometric dimensions. Porous solids and rough surfaces, however, have effective dimensions corresponding to fractional values of D. 

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