Fragmentation fractal dimension

Equation (2.30) amounts to a power-law relationship between N r ) and r , so, as n oo, it defines aprobabilistic fractal. Following Kaye [43], it is occasionally called a fragmentation fractal, and Dftagm is termed the fragmentation fractal dimension. 

Kozak, E., Pachepsky, Y.A., Sokolowski, S., Sokolowska,Z. andSt pniewski,W. (1996). A modified number-based method for estimating fragmentation fractal dimensions of soils. Soil Sci. Soc. Am. J., 60, 1291-1297. 

In order to be normalizable, this distribution can be only valid over the size range / min < R < / max, which is limited by the cut-off values / niin and / max- Such distributions are very important in various applications, and their properties depend strongly on the value of the exponent v. In many situations, the exponent falls into the range of 2 < u < 3, and in such cases it was referred to as fragmentation fractal dimension. Such... 

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). 

Typical examples of these fractals are the Cantor set ( dust ), the Koch curve, the Sierpinski gasket, the Vicsek snowflake, etc. Two properties of deterministic fractals are most important, namely, the possibility of exact calculation of the fractal dimension and the infinite range of self-similarity -°° +°°). Since a line, a plane, or a volume can be divided into an infinite number of fragments in different ways, it is possible to construct an infinite number of deterministic fractals with different fractal dimensions. Therefore, deterministic fractals cannot be classified without introducing other parameters, apart from the fractal dimension. 

The fractal dimension D of a chain fragment between the points of topological fixing (entanglements, clusters, crosslinks) is an important structural parameter, which controls the molecular mobility and deformability of polymers. Crucial factors accounting for the use of the dimension D are clearly defined limits of variation (1super-molecular structure of the polymer. It should be emphasised that all fractal relations contain at least two variables. 

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). 

Aspergillus niger had values of Dbm = 1.3-1.47 [34], Fractal dimension has also been used in analysis of fragment size distribution, when mycelia have been fragmented in a homogeniser to form inoculum [40]. With the wood decay basidiomycete Phanerochaete chrysosporium, D u decreased with increasing shear force [40]. 

Using this method. Park et al. [81] analysed TEM images of diesel particles and showed that the projected area equivalent diameter nearly equals the mobility diameter in the mobility size range from 50 to 220 nm. Doubly charged particles and possible fragments were observed for the DMA-classified particles. The fractal dimension calculated from the TEM images of mobility-classified aggregates was 1.75. The... 

Fig. 4.6 Fractal dimension from growth of Xg = 2 x Rg, from fragmentation, from box-counting, and from pair-correlation versus aggregation number, each time averaged from 100 aggregates left DLCA aggregates, right RLCA aggregates...

Fig. 4.7 Distribution of the fractal dimension within populations of numeric DLCA and RLCA aggregates left cumulative frequencies for df for monodisperse aggregates determined by the fragmentation method right standard deviation of the f/f-distribution determined by fragmentation and box-counting...

Fig. 4.8 Correlation between fractal dimension and fractal prefactor left global values (from fractal growth, Eq. (4.32)) and individual prefactors (from method of fragmentation) right comparison with Sorensen and Roberts (1997) and linear fit...

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.

N = number of fragments of the stated size or larger 5k = Korcak fractal dimension... 

---