Fractal dimensions of structure

The choice of dimension D, depends on the value of relation dhldm [9], At dm<0,6dh interaction of diffusant molecules with walls of free volume microvoid is small and transport process is controlled by fractal dimension of structure (structural transport). At dm<0,6dh on transport processes has strong influence interaction of diffusant molecules with walls of free volume microvoid, which are polymeric macromolecules surface with dimension >/(/)/ is the dimension of excess energy localization regions) [10], In this case Dt=Df (molecular transport) [9] is adopted. 

Let s consider structural aspect of t50 change due to introduction Z. As it is known [3, 15], for compositions HDPE+Z is observed the extreme rise of relative fraction of local order regions (clusters) cpci, that results to decrease of fractal dimension of structure df according to the equation [15] ... 

The chapter consists of three main sections. In Section II the elements of fractal theory are given. In Section III the basis of percolation theory is described moreover, a model of fractal structures conceived by us is described. Fractal growth models, constructed using small square or rectangular generating cells as representative structural elements, are considered. Fractal dimensions of structures generated on various unit cells (2x1, 2x2, 2x3, 2x4, 3x1, 3x2, 3x3, 3x4, 4x1, 4x2, 4x3, 4x4) are calculated. Probability... 

In Fig. 8.3, the relation between fractal dimensions of structure and stable crack for PASF samples is adduced. As it was to be expected from the most general considerations, the intercommunication existed between di-... 

With the experimental results about the wetting ability and the fractal dimension of four kinds of anode electrodes, we could conclude the following. The addition of NisAl could make the electrolyte wet the electrode very well. The pore structures of all the electrodes prepared in this study were highly irregular and rough. Finally, the chemical properties of the surfaces were as important as the physical properties in determining the wetting ability of the electrodes in this study. 

Unlike the simulations which only consider particle-cluster interactions discussed earlier, hierarchical cluster-cluster aggregation (HCCA) allows for the formation of clusters from two clusters of the same size. Clusters formed by this method are not as dense as clusters formed by particle-cluster simulations, because a cluster cannot penetrate into another cluster as far as a single particle can (Fig. 37). The fractal dimension of HCCA clusters varies from 2.0 to 2.3 depending on the model used to generate the structure DLA, RLA, or LTA. For additional details, the reader may consult Meakin (1988). 

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).

The fractal nature of the structures is also of interest. Because of the wide range of flow in the journal bearing, a distribution of fractal clusters is produced. When the area fraction of clusters is 0.02, the median fractal dimension of the clusters is dependent on the flow, similar to the study by Danielson et al. (1991). The median fractal dimension of clusters formed in the well-mixed system is 1.47, whereas the median fractal dimension of clusters formed in the poorly mixed case is 1.55. Furthermore, the range of fractal dimensions is higher in the well-mixed case. 

The structure of this review is composed of as follows in Section II, the scaling properties and the dimensions of selfsimilar and self-affine fractals are briefly summarized. The physical and electrochemical methods required for the determination of the surface fractal dimension of rough surfaces and interfaces are introduced and we discuss the kind of scaling property the resulting fractal dimension represents in Section III. 

Simulation of structure formation on a lattice [7,100] demonstrated that randomly formed branched clusters also fulfill self-similarity conditions and gave fractal dimensions of [7,104,105] ... 

One way of measuring the fractal dimension of aggregates is discussed in Chapter 5 (See Section 5.6a and Example 5.4). In the example below, we illustrate the relation between the fractal structure of aggregates and the surface area of the aggregates. 

An independent x-ray and light scattering analysis (see Section 5.6a and Example 5.4) of a dispersion of the aggregates suggests that the aggregates have a fractal structure with a fractal dimension of 2.65. Is this confirmed by your result ... 

In principle, however, the effective surface area for heat transfer from the single monomer is decreased by the contact area between the particles. The fractal dimension of the aggregates varies in different systems and depends on the evolution process. For dense aggregates, heat transfer rates are decreased due to the reduced heat exchange area compared to primary particles only connected by point contact in a chain like structure (Liu et al., 2006a-c). This leads to an overestimation of... 

For example, the fractal dimension of the Koch curve is 1.2619 since four (m = 4) identical objects are observed (cf. levels i = 0 and i = 1 in Figure 1.1) when the length scale is reduced by a factor r = 3, i.e., dj- = In4/ln3 1.2619. What does this noninteger value mean The Koch curve is neither a line nor an area since its (fractal) dimension lies between the Euclidean dimensions, 1 for lines and 2 for areas. Due to the extremely ramified structure of the Koch curve, it covers a portion of a 2-dimensional plane and not all of it and therefore its dimension is higher than 1 but smaller than 2. 

Observations of the liver reveal an anatomically unique and complicated structure, over a range of length scales, dominating the space where metabolism takes place. Consequently, the liver was considered as a fractal object by several authors [4,248] because of its self-similar structure. In fact, Javanaud [275], using ultrasonic wave scattering, has measured the fractal dimension of the liver as approximately df 2 over a wavelength domain of 0.15-1.5mm. 

The fractal dimensions of the excitation paths in samples D, F, and G lie between 2 and 3. Thus, percolation of the charge carriers (protons) is also moving through the Si02 matrix because of the availability of an ultra-small porous structure that occurs after special chemical and temperature treatment of the initial glasses [156]. 

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