Fractal dimension Subject

Chapter 13 - It was shown, that limiting conversion (in the given case - imidization) degree is defined by purely structural parameter - macromolecular coil fraction, subjected evolution (transformation) in chemical reaction course. This fraction can be correctly estimated within the framework of fractal analysis. For this purpose were offered two methods of macromolecular coil fractal dimension calculation, which gave coordinated results. 

These starburst dendrimers have been subjected 47 to two different fractal analyses I48 49 (a )A c/2 D)/2, where A is the surface area accessible to probe spheres possessing a cross-sectional area, o, and the surface fractal dimension, D, which quantifies the degree of surface irregularity and (b) A = dD, where d is the object size. Both methods give similar results with D = 2.41 0.04 (correlation coefficient = 0.988) and 2.42 0.07 (0.998), respectively. Essentially, the dendrimers at the larger generations are porous structures with a rough surface. For additional information on dendritic fractality, see Section 2.3. 

The different lengths of these paths mandate that some terminal groups will be tethered closer to the center of the branched assembly than others and be subject to more steric interactions. This suggests that this dendrimer class has surfaces with higher fractal dimensions [13, 97-99] and interior branching zones which are less ordered than those of the symmetrical Starburst dendrimers (Fig. 17). 

B) the physical significance of ink bottle theories of mercury intrusion have always been the subject of debate (C) a possible reinterpretation of the data in (A) as fractal data. a. and P are the slopes of particular regions of the resulting curve, which may be interpreted as fractal dimensions. 

Successive increments of mathematical fractal random processes are independent of the time step. Here D = 1.5 corresponds to a completely uncorrelated random process r = 0, such as Brownian motion, and D = 1.0 corresponds to a completely correlated process r= 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 

In this section are displayed graphically the numerically exact results that have been obtained for unbiased, nearest-neighbor random walks on finite d = 2,3 dimensional regular. Euclidean lattices, each of uniform valency v, subject to periodic boundary conditions, and with a single deep trap. These data allow a quantitative assessment of the relative importance of changes in system size N, lattice dimensionality d, and/or valency v on the efficiency of diffusion-reaction processes on lattices of integral dimension, and provide a basis for understanding processes on lattices of fractal dimension or fractional valency. 

Although there are fractal systems in the environment which are not aggregates, such as the fractal surface roughness of fractures and wear particles, the texture of clouds and the distribution of stars in the night sky, these are less often of interest in the study of environmental systems. Surface fractal dimensions are also considered important in environmental systems and, although they are subject to many of the same principles discussed later, are not treated explicitly here. 

One of the easiest ways to measure fractal dimension with this technique is to capture images of slices through the structure and measure the fractal scaling of the image as discussed above. The dimension measured in this way is not the projected area dimension Dp, discussed earlier, because the image is a slice not a projection. It turns out that the dimension measured in this way is numerically equal to the mass fractal dimension minus one, by virtue of the codimension rule [58]. The measurement of fractal dimensions by this technique is not subject to the restriction of geometric transparency, as is the case with the analysis of projected images, and so fractal dimensions well over two can be measured. 

Figure 9.6 Mean values of the fractal dimension for butane smoke aged for 4 h and then subject to various relative humidities and supersaturations (data from [115]).

Ramachandran, G. and Reist, P.C. (1995). Characterization of morphological changes in aggregates subject to condensation and evaporation using multiple fractal dimensions. Aerosol Sci. TechnoL, 23, 431-442. 

One quantitative measure of the structure of such objects is their fractal dimension D. Mathematicians calculate the dimension of fractal to quantify how it fills space. The familiar concept of dimensions applies to the object of classical or Euclidian geometry. Fractals have non-integer (fractional) dimensions whereas a smooth Euclidean line precisely fills a one-dimensional space. A fractal line spills over a two-dimensional space. Figure 13.2 shows subjects with increasing fractal dimension. 

Figure 13.2. Subjects with increasing fractal dimensions.

The concept of diffusion-limited cluster-cluster aggregation (DLCA) is very useful and applied in many simulations. In this type of simulation process, particles are placed in a box and subjected to Browni m (random walk) movements. Aggregation (clustering) may occur when two or more particles/clusters come within the vicinity of each other and the combined cluster continues the random walk. The simulation is stopped at the gelation point (percolating system) or when all particles are combined in one final aggregate. The fractal dimension of the DLCA aggregates is approximately 1.8. 

Anisometry and fractal dimension are also used as parameters for characterizing fillers, the former to predict various mechanical properties of filled compounds and vulcanizates and the latter to investigate the surface structure. Anisometry, which is always subject to distribution, is found to widen with the fillers investigated, with an increase in the dimensions of the objects detected. 

The porosity of electroactive polymers has also been the subject of many investigations. An experiment has been described in which the ion exchange property of the polymer was used to modify the morphology [29]. In this study polypyrrole with tosylate as the counterion was ion exchanged with KCl to take up chloride as the counterion. Replacement of the bulky tosylate ion with the much smaller chloride ion resulted in greater porosity of the polymer and an increase of its fractal dimension. 

The tests of concrete specimens subjected to Mode II fracture were aimed at a further investigation of relations between fractal dimension and roughness of the fracture surface after Mode II crack propagation and are presented below based on paper by Brandt and Prokopski (1993). 

The above results are at least partly confirmed by Roh and Xi (2001) who tested concrete specimens with different aggregate subjected to compressive and splitting loads with different loading rates. Fractal dimensions of fractured surfaces were determined. Interesting conclusions are ... 

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