Surface fractals

It turns out that many surfaces (and many line patterns such as shown in Fig. XV-7) conform empirically to Eq. VII-20 (or Eq. VII-21) over a significant range of r (or a). Fractal surfaces thus constitute an extreme departure from ideal plane surfaces yet are amenable to mathematical analysis. There is a considerable literature on the subject, but Refs. 104-109 are representative. The fractal approach to adsorption phenomena is discussed in Section XVI-13. 

A fractal surface of dimension D = 2.5 would show an apparent area A app that varies with the cross-sectional area a of the adsorbate molecules used to cover it. Derive the equation relating 31 app and a. Calculate the value of the constant in this equation for 3l app in and a in A /molecule if 1 /tmol of molecules of 18 A cross section will cover the surface. What would A app be if molecules of A were used ... 

A modification of Eq. X-31 has been made for fractal surfaces (see Section VII-4C). Contact angles near 90° on fractal surfaces are well described by [68]... 

Fig. X-6. Drop of water on an alkylketene dimer surface (rt) fractal surface D (b) flat surface. (From Ref. 68.)...

Discuss the paradox in the wettability of a fractal surface (Eq. X-33). A true fractal surface is infinite in extent and a liquid of a finite contact angle will trap air at some length scale. How will this influence the contact angle measured for a fractal surface ... 

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... 

Consider the roughness factor, r (Eq. 19), for such a fractal surface ... 

For a fractal surface D > 2, and usually D < 3. In simple terms the larger D, the rougher the surface. The intuitive concept of surface area has no meaning when applied to a fractal surface. An area can be computed, but its value depends on both the fractal dimension and the size of the probe used to measure it. The area of such a surface tends to infinity, as the probe size tends to zero. 

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

A high modulus gradient at the interface is also be avoided in materials Joined as a result of the interdiffusion of materials to form a fractal surface [32]. The effect is to produce an interfacial composite region. This strengthens the interface and leads to a more gradual change in modulus and avoids the sharp concentrations of stress which would occur at a smooth interface. 

E. V. Albano. Finite-size effects in kinetic phase transitions of a model reaction on a fractal surface Scahng approach and Monte Carlo investigation. Phys Rev B 42 R10818-R10821, 1990. 

E. V. Albano. Monte Carlo simulation of the oxidation of carbon monoxide on fractal surfaces. Surf Sci 255 351-359, 1990. 

Fractal surfaces, Mandelbrot s work, 52 Fredlein and Bockris, use of a... 

Mandelbrot, on fractal surfaces, 52 Mao and Pickup, their work on the oxidation of polypyrrole, 587 Marcus model, inapplicability for interfacial electron transfer, 513 Mechanical breakdown model for passivity, 236... 

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

In order to characterize the self-affine fractal surface, the self-affine fractal dimension dv has been determined by using the... 

The thickness of the layer of the adsorbed molecules is the characteristic distance scale for fractal surface. (3) Van der Waals attraction forces between solid/gas interactions and the liquid/gas surface tension forces are contributed to the grand potential of the system. 

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... 

J. C. Russ, Fractal Surfaces, Plenum Press, New York, 1994. 

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