Fractal structures

Rouw P W and de Kruif C G 1989 Adhesive hard-sphere colloidal dispersions fractal structures and fractal growth in silica dispersions Phys. Rev. A 39 5399-408... 

Wool, R.P, Dynamics and fractal structure of polymer interfaces. In Lee, L.-H. (Ed.), New Trends in Physics and Physical Chemistry of Polymers. Plenum Press, New York, 1989, p. 129. 

So far we have described the growth of a compact cluster. But, as we have seen in Fig. 5, some clusters show more fllamental structures under speciflc circumstances, and this type of form is called fractal. The simplest growth process leading to fractal structures is the so-called diffusion-limited... 

Furthermore, it turns out that noise, which always exists in the system (for example, the thermodynamic noise), appears to play a crucial role in the formation of fractal structures but is not so important for compact patterns. 

For some rules (such as R4 and R94) the patterns obtained from single seeds are stable under the addition of further nonzero sites. For other rules (such as R218) the pattern is unstable. Although the patterns for R18 and R150 retain much of their fractal structure, some of the details appear to be washed away. A similar contrast between the evolutionary behaviors of single and small nonzero region seeded rules can be observed for the r = 2 totalistic rules shown in figures 3.7 and 3.8. 

The value a = 1 corresponds to ideal capacitive behavior. The fractal dimension D introduced by Mandelbrot275 is a formal quantity that attains a value between 2 and 3 for a fractal structure and reduces to 2 when the surface is flat. D is related to a by... 

Figure 21. Noise spectrum of detector amplifiers. Note that both axes have logarithmic scale. There are two main components of noise - the white noise which is present at all frequencies, and the 1// noise that is dominant at low frequencies. 1// noise has a fractal structure and is seen in many physical systems. The bandpass of a measurement decreases for slower readout, and the readout noise will correspondingly decrease. A limit to reduction in readout noise is reached at the knee of the noise spectrum (where white noise equals l/f noise) - reading slower than the frequency knee will not decrease readout noise.

Matsushita, M Hayakawa, Y. and Sawada, Y. (1985) Fractal structure and cluster statistics of zinc-metal trees deposited on a line electrode. Phys. Rev. A, 32, 3814-3816. 

Fig. 31. Fractal structures obtained experimentally at different stages of aggregation of a colloidal monolayer of 1 /im sulfonated polystyrene particles on the surface of an aqueous calcium chloride solution, initially uniformly distributed (Robinson and Earnshaw, 1992).

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

We focus on aggregation in model, regular and chaotic, flows. Two aggregation scenarios are considered In (i) the clusters retain a compact geometry—forming disks and spheres—whereas in (ii) fractal structures are formed. The primary focus of (i) is kinetics and self-similarity of size distributions, while the main focus of (ii) is the fractal structure of the clusters and its dependence with the flow. 

Fig. 41. Typical 2D fractal structure obtained by aggregation of particles in the journal bearing flow. Fractal dimension of the cluster is 1.54 (Hansen and Ottino, 1996b).

Illustration Aggregation of fractal structures in chaotic flows. In a... 

In situ SAXS investigations of a variety of sol-gel-derived silicates are consistent with the above predictions. For example, silicate species formed by hydrolysis of TEOS at pH 11.5 and H20/Si = 12, conditions in which we expect monomers to be continually produced by dissolution, are dense, uniform particles with well defined interfaces as determined in SAXS experiments by the Porod slope of -4 (non-fractal) (Brinker, C. J., Hurd, A. J. and Ward, K. D., in press). By comparison, silicate polymers formed by hydrolysis at pH 2 and H20/Si = 5, conditions in which we expect reaction-limited cluster-cluster aggregation with an absence of monomer due to the hydrolytic stability of siloxane bonds, are fractal structures characterized by D - 1.9 (Porod slope — -1.9) (29-30). 

Different kinds of heterogeneity can be imagined. In the most simple case only a few differing structural entities are found to coexist without correlation inside the volume irradiated by the primary beam. In this case it is the task of the scientist to identify, to separate and to quantify the components of such a multimodal structure. In an extreme case heterogeneity may even result in a fractal structure that can no longer be analyzed by the classical methods of materials science. 

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. 

It is unreasonable to assess the significance of a fractal structure by resorting to the number of magni-tudes covered on the intensity scale. 

Nevertheless, fractal structure is an issue in porous materials. 

Keywords Macromolecules m Dendrimers m Fractal Structure m Hyperbranched Polymers m Nanostructure m Divergent Synthesis m Convergent Synthesis m Dendrylation... 

Illumination generates holes within the material of PS and causes photo corrosion of PS that is much faster than that in the dark. Depending on illumination intensity and time, the pore walls in a PS can be thinned to various extents by the photo induced corrosion. This corrosion process is responsible for the etched crater between the initial surface and the surface of PS as illustrated in Figure 28. It is also responsible for the fractal structure of the micro PS formed under illumination. 

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