Fractal surface fractal

In the context of particle aggregation, the particular fractal structures called mass fractals are usually considered. Mass fractals correlate the location of mass with radius, as opposed to other types of fractal constructs such as boundary fractals, surface fractals, and pore fractals (see Ref. 17). The power law which correlates the size of an aggregate with the number of primary particles in the aggregate is given by... 

For w 4, unhydrolyzed sites are incorporated into the growing cluster. The probability of condensation at these sites is less than at hydrolyzed sites. Under these conditions, growth is described by a poisoned Eden model. Depending on the number and distribution of the poisoned sites, mass fractals, surface fractals or uniformly porous objects can result. The addition of more water in a second hydrolysis step is expected to completely hydrolyze the clusters and further growth should be described by the Eden model. 

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... 

The Surface Fractal Investigatioii of Anode Electrode of Molten Carbonate Fuel Cell... 

FHH (Frenkel-Halsey-Hill) theory is valid for multi molecules adsorption model of the flat surfrtce material. When this model is applied for the surface fractal in the range of capillary condensation, in other words, in the state of interface which was controlled by the surface tension between liquid and gas, the modified FHH equation can be expressed as Eq. (3). 

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

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

T. Zavada, R. Kimmich 1999, (Surface fractal probed by adsorbate spin-lattice relaxation dispersion), Phys. Rev. E 59, 5848. 

Figure 8.13. Different kinds of fractals, (a) Surface fractal, (b) Mass fractal, (c) Pore fractal... 

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. 

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