Fractal hierarchy

The three dimensions introduced above, Dp, Di and Dq, are actually three members of an (uncountably infinite) hierarchy of generalized fractal dimensions introduced by Heiitschel and Procaccia [hent83]. The hierarchy is defined by generalizing the information function /(e) (equation 4.88) used in defining Dj to the i/ -order Renyi information function, Io e) -... 

Generalized Renyi Entropies and Dimensions A hierarchy of generalized entropies and dimensions, SQ B,t), s[ B,t), S B,t),. .. - analogous to the hierarchy of fractal dimensions, Dq, Di,. .., introduced earlier in equation 4.94 for continuous systems may also be defined ... 

Figure 10. A hierarchy of stochastic fractal pore development a. Straight cylindrical constant diameter, b. Straight variable diameter, c. Tourtous variable diameter, d. Tortous non-circular cross-section,...

Finally, we note that the materials with fractal structure exhibiting viscoelastic properties similar to those of the model described here can be manufactured. For example, at the first step (the lowest size level) one produces tablets—for example, of a polymer with required inclusions. At the next step, the tablets obtained at the preceding level are used as inclusions in larger tablets. The process is continued and the hierarchy shown in Fig. 63 is obtained [160],... 

Van Damme, H. 1998. Structural hierarchy and molecular accessibility in clayey aggregates, p. 55-74. In Fractals in soil science. Adv. Soil Sci., Boca Raton, FL. 

Meakin, R Coniglio, A. Stanley, E. H. Witten, T. A. Scaling properties for the surfaces of fractal and nonfractal objects an infinite hierarchy of critical exponents. Phys. Rev. A, 1986, 34(4), 3325-3340. 

Hierarchy can be described in analogy to rope (stretched polymer molecules in domains that make up nanofibers, combined to microwhiskers, bundled into fibers that are spun into yarn that is twined to make up the rope). Wood and tendon are biological examples that have six or more hierarchical levels. Compared to these, fiber-reinforced matrix composites made up of simple massive fibers embedded in a metallic, ceramic, or polymer matrix are primitive. Hierarchical inorganic materials, as discussed in Chapter 7, can be made with processes for fractal-like solid products spinodal decomposition, diffusion-limited growth, particle precipitation from the vapor, and percolation. Fractal-like solids have holes and clusters of all sizes and are therefore hierarchical if the interactions... 

Shlesinger, M. F., Klafter, J. (1986). Nature of Temporal Hierarchies, Defining Relaxation in Disordered Systems. In Fractals in Physics. Ed. Pietronero, L., Tosatti, E. Amsterdam, Oxford, New York, Tokyo. North-HoUand, 553-560. 

The first from the indicated points was considered in detail above. The authors of Refs. [20,21] showed that nanoclusters surface fractal dimension changes within the range of 2.15 2.85 that is their well developed surface sign. And at last, let us consider quantum (wave) aspect of nanoclusters nature on the example of PC [22]. Structural levels hierarchy formation and development scenario in this case can be presented with the aid of iterated process [23] ... 

In this chapter we define the structural hierarchy of the network formed by a crystallizing fat (after Ref. 1), highlights the major developments in quantification of the various levels of structure, and summarizes the application of fractal geometry to the quantification of the microstructural level of structure. Furthermore, we make the case that the microstructure, by virtue of its proximity to the macroscopic world, provides important predictive information on the value of the elastic modulus of the network. Models we have developed to relate the struc-... 

In conclusions, we must reiterate that fat crystal networks demonstrate distinct structural hierarchies. A consideration of the microstructural level and constrac-tion of models based on elastic rheological measurements implicate all levels of structure in the determination of the macroscopic elastic moduli. Consideration of the fractal arrangement of the microstructural elements is integral to the construction of models that fit well with experimental data. Furthermore, it is seen that the fractal dimension and other network characteristics such as microstructural element size and microstructural size are indicators of macroscopic hardness, and that these factors can be manipulated by processing conditions to achieve tailored elastic characteristics of fat crystal networks. 

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