Applied Fractal Geometry

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. 

Applying fractal geometry to description of disordered media allows one to use the properties of scaling invariance—that is, to introduce macroscopic... 

APPLYING FRACTAL GEOMETRY TO QUANTIFY GROWTH, POLYMERIZATION AND AGGREGATION... 

Figure 2.9. The cumulative undersize distribution of fineparticle size is an important way of displaying size distribution data. Shown above, plotted on log-log scales, are the size distributions of the fragments produced when two different amorphous materials were shattered by impact after being cooled to low temperatures [18]. From the perspective of chaos theory and applied fractal geometry explained in more detail in a later chapter, the slope of this type of data line is described as a fractal dimension in data space.

B. H. Kaye, Applied Fractal Geometry and the Fineparticle Specialist, Part. Part. Syst. Charact., 10 3 (1993) 99-111. 

Figure 2 Describing the structure of carbonblack profiles originally studied by Medalia constituted the first successful use of applied fractal geometry.

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

Fractal geometry has been used to describe the structure of porous solid and adsorption on heterogeneous solid surface [6-8]. The surface fractal dimension D was calculated from their nitrogen isotherms using both the fractal isotherm equations derived from the FHH theory. The Frenkel-Halsey-Hill (FHH) adsorption isotherm applies the Polanyi adsorption potential theory and is expressed as ... 

Very often, common natural processes involving diffusion and reaction are found to obey power laws which for most of the time have been described within the domain of Euclidean space and hence restricted to integer powers. Table 2. gives a comparison of the Euclidean and fractal geometries. On the other hand, it is observed that a large number of heterogeneous reactions follow fractional-order kinetics under different process conditions [13]. But most classical transport theories, valid for Euclidean structures, fail when applied to transport processes... 

---