Fractal Shapes

When a particle s surface is not uniformly smooth like that of a sphere but has a texture like that of broccoli or catdiflower, the particle is said to be fractal [12]. With the two-dimensional projection of a fractal particle, we find that the perimeter of the particle increases without limit as the size of the ruler used as a measure decreases in length. The circumference, C, estimated with a ruler of size x is proportional to 

FIGURE 2JS Two-dimensianal computer generated fractal aggregate. Taken fiom Hurd [13], cop3Tight 1986 by John Sons, bic. Reprinted by permission of John Wiley Sons, Inc. 

The properties of fractal ceramic powders depend on their fractal dimensions. The density, p, of a fractal particle depends on its radius, R, and the fractal dimension  

Fractal particles modify external phenomena in the surrounding space (e.g., a conducting fractal modifies the electric field around it, an absorbing fractal modifies the concentration of the diffusing species around it, and a fi ctal immersed in a fluid modifies fluid flow around 

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Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

The present volume gives a general and at the same time rather detailed review on main research developments in the field of dendrimers (oligomer and polymer) during the past several years, but also offers views and visions of the future - of what could soon be achieved in this area at the interface between small organic molecules and macromolecules (polymers). We are sure that the rapid development of fractal-shaped molecules will continue in academic institutes as well as in industry - there is still more to come. 

A rigorous treatment of fractal mathematics is beyond the scope of this treatise therefore, only essential concepts relating to a descriptive understanding of fractal shapes and analyses will be discussed. More complete and comprehensive reviews of this growing area are available. 52,56-61l... 

Beyond that, fractal geometry is of direct relevance for the transport in inhomogeneous systems, since the percolation cluster (immediately at the percolation threshold, c.f. Fig. 77) assumes a fractal shape. In two-phase mixtures of an insulating and a conducting phase this predicts a power law dependence of the conductivity on q>— q>c (qx. volume fraction).281 A relevant example in the context of our considerations is the conduction behavior of AgCl a-AgI composites.118... 

For fractal-shaped particles, this volume cannot be used to compute the mass of the particle. Instead, the mass is related to the length through the fractal dimension. 

N. V. Kantartzis, T. T. Zygiridis, and T. D. Tsiboukis, A nonstandard higher-order FDTD algorithm for 3-D arbitrarily and fractal-shaped antenna structures on general curvilinear lattices, IEEE Trans. Magn., vol. 38, no. 2, pp. 737-740, Mar. 2002. doi 10.1109/20.996191... 

A fractal possesses a dilation symmetry, that is, it retains a self-similarity under scale transformations. In other words, if we magnify part of the structure, the enlarged portion looks just like the original. Figure 5.15 shows a fractal shape, the Koch curve. If we magnify by three the part of the Koch curve in the interval (0, 1/3), it becomes exactly identical to the whole shape. The same is true if the part in (0,1/9) is enlarged... 

Recently, a silicon wafer was also hydrophohized hy electrochemical etching [33], in which a silicon wafer was used as the working electrode and a mixture of HF acid and ethanol was used as the electrolyte. After anodization at a high current density (170 mA/cm ), fractal-shaped hierarchical silicon surface was formed. The WCA was about 135°. After a further wet etching with a mixture of NH4F and HF (7 1, buffered oxide etcher, BOB), the hierarchical sificon surface was converted into a pillar-like structure, whose WCA increased to 160° and sliding angle decreased to a value smaller than 1°. 

The next step they have to generalize the mathematical formulation of the statistics of linear polymer to general fractal objects by assuming that the clusters formed by the filler particles can be described by a fractal shape. This assumption allows them to predict certain specific forms of the reinforcement, such as to work out the probability distribution for the filler clusters. By appropriate modelling of the filler structure they arrive at the generalization distribution to calculate the self-energy function which corresponds directly to the reinforcement factor. They then derive a new form of the Green function G which contains the effects of the filler particles. In this way, they are able to take into account of all the effects the shape of the filler particles, the spatial distribution of the particles, etc. 

It is important to consider whether the domains observed are growth forms or are closer to being eqnilibrinm domain shapes. An appreciation of the significance of observing dendritic or fractal shapes... 

At present, the most effective and versatile chemical antifoamers are mixtures. In fact, modem commercial formulations often contain mixtures of silicone oils, silicone surfactants or silica gels, with possibly two or more types of dispersed hydrophobic aggregates (with sizes around 1 micron, specific gravity 1.0-1.3, with rough fractal shapes). These mixed-type antifoamers are very effective at low concentrations (10-1000 ppm) and are widely used. The (hydrophobic) particles may be hydrophobized silica or glass and are often referred to as the activator, with the hydrocarbon or poly-dimethylsiloxane (PDMS) liquid or oil being referred to as the carrier. However, less expensive single-system antifoamers are commonly used, and these exist in many different forms such as soluble liquids, insoluble... 

In most cases where particle shapes are included in the investigations, the shape of particles is registered only in two dimensions. Up to now it is not possible to state what kind of error is induced by this proceeding. Only for fractal shapes of a special kind Gentry [i] proved by simulations that a systematical error in the calculated fractal dimension arises from analyzing the projected images of particles instead of their three dimensional shape (Gestalt). 

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