The concept of fractal geometry

Let us now explain at the end of this section the important concept of fractal geometry that we have already mentioned several times. 

Typical solid state contacts are, in general, not ideal planes. In particular, the fundamental work of Mandelbrot [532] permits us to go to another again ideal limit, that of fractal geometry. Let us imagine a rough interface, whose individual segments are, when inspected at greater resolution, structured just as the overall structure. If this is fulfilled over a certain size scale (the atomic structure naturally sets a limit), this structure is referred to as fractal and self-similar over this size scale. 

It is useful, even though not usual, to designate this as zoom symmetry or enlargement symmetry. It is evident that the length, surface area or volume of these structures is a function of the scale used, since the degree to which the details of the structure contribute differs. Let us consider Koch s curve, shown in Fig. 6.85 

Let us consider the same point in a somewhat different manner and refer to Fig. 6.86. In addition to Koch s curve (c), the figure displays also two other zoom- 

Power laws attributable to fractal geometry are also known for electrode impedances [534]. This is (not surprisingly) the case for the example in Fig. 6.83 (see also Chapter 7). 

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Kaye, B.H. Characterizing the flowability of powder using the concepts of fractal geometry and chaos theory. Part. Part. Syst. Charact. 1997, 14, 53-66. 

The concept of fractal geometry was first introduced by Mandelbrot and it refers to a rough or fragmented geometric shape that is composed of many smaller copies that have the same shape but different sizes of the whole figure and fractal dimension is a statistical tool to measure how the fractal object rills the space (44). 

One of the most exciting developments of the past decade in the study of colloidal silica is the application of the fractal approach to the study of sols and gels. Fractals are disordered systems for which disorder can be described in terms of nonintegral dimension. The concept of fractal geometry, developed by Mandelbrot (II) in the early 1980s, provides a means of quantitatively describing the average structure of certain random objects. The fractal dimension of an object of mass M and radius r is defined by the relation... 

Whereas attractor in the steady state has zero dimensions and Euclidian dimension of the limit cycle is two, it is not possible to define the Euclidian dimension of the strange attractor. However, using the concept of fractal geometry, it is possible to define the dimension of such an attractor, which is not an integer. 

Chemical gelation has been extensively investigated by sophisticated experiments (7) and accounted for by different theories from the original mean-field theory of Flory (2) to the concept of fractal geometry and the connectivity transition... 

B. H. Kaye, Characterizing the Structure of Fumed Pigments Using the Concepts of Fractal Geometry. Part. Part. Syst. Charact., 9 (1991) 63—71. 

Figure 9.8. Data generated by mercury intrusion porosimetry can be reinterpreted using the concepts of fractal geometry, by replotting the data on log-log scales. The slopes of the linear regions of the resulting graph can be viewed as fractal dimensions in data space, a) Traditional presentation of mercury intrusion porosimetry data, b) Data of (a) plotted on log-log scales.

Another way in which the concepts of fractal geometry have been applied to the description of a porous body is to use what is known as a Sierpinski fractal. Mandelbrot, in his development of the concepts of fractal geometry, drew attention to some mathematical models originally developed by Sierpinski (accessible discussion of the original work of Sierpinski is to be found in Mandelbrot s book 41]. (See also reference 31.) Sierpinski s work dealt with the structure of porous bodies and the mathematical curve which is known as the Sierpinski s carpet. The basic concepts involved in draw-... 

The publication B.H. Kaye, "Characterizing The Structure Of Fumed Pigments Using The Concepts Of Fractal Geometry", in press. Particle and Particle Systems Characterization contains many references leading into the specialist literature on Size Characterization. 

Farm and Avnir [113] were the first to use fractal geometry to determine effects of surface morphology on drug dissolution. This was accomplished by the use of the concept of fractal reaction dimension dr [114], which is basically the effective fractal dimension of the solid particle toward a reaction (dissolution in this case). Thus, (5.7) and (5.8) were modified [113] to include surface roughness effects on the dissolution rate of drugs for the entire time course of dissolution... 

The emerging concept of fractal geometry has opened a wide area of research. The review will begin with a brief description on the methods for obtaining fractal dimension, followed by the use of fractal concept on pharmaceutical and biological applications with specific examples. 

A different procedure for describing the structure of rugged profiles has been developed from the theorems of fractal geometry. " " The basic concept used in fractal geometry is to add a fractional number to the... 

The term fractal and the concept of fractal dimension were introduced by Mandelbrot [1]. Since Mandelbrot s work, many scientists have used fractal geometry as a means of quantifying natural structures and as an aid in understanding physical processes occurring within these structures. Fractals are objects that appear to be scale invariant. Mandelbrot defines them as shapes whose roughness and fragmentation neither tend to vanish, nor fluctuate up and down, but remain essentially unchanged as one zooms in continually and examination is refined . The above property is called scale invariance . If the transformations are independent of direction, then the fractal is self-similar if they are different in different directions, then the fractal is self-afflne (see Chapter 2). 

There have been approximately two dozen papers published in the peer-reviewed literature since 1992 that have used the concept of fractals to characterize humic materials [16-38, 42], Most of the studies were done in a burst of interest during a 7-year period in the 1990s there have been five papers published in the peer-reviewed literature on the application of fractals to the study of humic materials other than reviews [15, 32, 43, 44] since 1999. Two common applications of fractal geometry have been to study the mass distribution of colloidal humic materials and the aggregation processes that produced them. 

Figure 9.3 Concepts of fractal geometry (a) profile of aggregate (b) structured walk explorations (c) Richardson plot of a series of explorations of the aggregate...

In the earlier chapter we have discussed the formation of colour bands in moving wave fronts, stationary structures which are governed by coupling of reaction and diffusion. In this chapter we will be concerned with pattern formation governed by the process of mass flow and diffusion related to precipitation as crystals, electro-deposits, bacterial colonies and diffusion. Just as in the former case, in the present case also we come across very complex patterns, depending on the experimental conditions. However in the present case it is possible to rationalize the complex structure with use of new mathematical concepts of fractal geometry. 

Originally, Greek philosophers thought that the universe was continuous and that the world could be described by lines, areas, and volumes, according to the geometry (literally earth-measurement ), set down by Euclid, for example, around 300 BC. It became evident a few centuries ago that shapes are not continuous but are composed of similar but smaller shapes as they become more magnified. Thus a tree looks more complex the more it is studied on a finer scale, as shown in Fig. 5.7. This concept is the basis of fractal geometry which has been described by Mandelbrot. ... 

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