Fractal atomization

We have considered briefly the important macroscopic description of a solid adsorbent, namely, its speciflc surface area, its possible fractal nature, and if porous, its pore size distribution. In addition, it is important to know as much as possible about the microscopic structure of the surface, and contemporary surface spectroscopic and diffraction techniques, discussed in Chapter VIII, provide a good deal of such information (see also Refs. 55 and 56 for short general reviews, and the monograph by Somoijai [57]). Scanning tunneling microscopy (STM) and atomic force microscopy (AFT) are now widely used to obtain the structure of surfaces and of adsorbed layers on a molecular scale (see Chapter VIII, Section XVIII-2B, and Ref. 58). On a less informative and more statistical basis are site energy distributions (Section XVII-14) there is also the somewhat laige-scale type of structure due to surface imperfections and dislocations (Section VII-4D and Fig. XVIII-14). 

Polymer-metal fractal interfaces may result from processes such as vacuum deposition and chemical vapour deposition where metal atoms can diffuse con-... 

Figure 10. Three-dimensional AFM images of (a) Pt/polished AI2O3, (b) Pt/etched Ni, and (c) Pt/unpolished AI2O3 electrodes. Reprinted from J. -Y. Go et al., A study on ionic diffusion towards self-affine fractal electrode by cyclic voltammetry and atomic force microscopy, J. Electroanal. Chem., 549, p. 49, Copyright 2003, with permission from Elsevier Science.

Crystals are often associated with their unique polyhedral shapes. Their smooth, shining facets provide a constant source of fascination to ancient, arid modem beings alike. Advances in microscopy have not, tainted the myth a bit interesting surface stmctures are found down to the atomic level, be it surface reconstmction, surface ripples and dimples, or snow-flakish fractals... 

The same situation is expected for stereoisomers of the dendrimer 6 (G = 1), as revealed by a careful inspection of Figure 10 in which the dependence of the fractal dimension of the 12 isomers on probe radii is depicted. Due to the large number of atoms of this open-shell dendrimer, a low level theoretical method, the Dreiding force-field, was used for the optimizations of the geometries of these 12 isomers. ... 

Thin film technology is becoming one of the important technologies today. While there are infinite varieties of thin film fabrication methods, most amorphous thin films seem to exhibit fractal-like atomic structures. Depending on the fabrication conditions, a thin film grows on the substrate into columnar structures with many voids interdispersed in the thin film.80 These structures can be seen in the field ion microscope, and compositional variation can be analyzed with the atom-probe. In addition, formation of atomic clusters inside the thin film can be substantiated with the observation of a large fraction of cluster ions in field evaporation by the atom-probe. 

Fractals are mathematically defined self-similar structures (Fig. 1.11) [26]. The scaffold of cascade or dendritic molecules is fractal if the atoms are considered to be points and the bonds to be strictly one-dimensional lines. Self-similarity... 

For a limited discussion of fractal geometry, some simple descriptive definitions should suffice. Self-similarity is a characteristic of basic fractal objects. As described by Mandelbrot 58 When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar. Another term that is synonymous with self-similarity is scale-invariance, which also describes shapes that remain constant regardless of the scale of observation. Thus, the self-similar or scale-invariant macromolecular assembly possesses the same topology, or pattern of atomic connectivity, 62 in small as well as large segments. Self-similar objects are thus said to be invariant under dilation. 

Fig. 11 Surface fractal dimensions ds on atomic length scales of furnace blacks and graphitized blacks in dependence of specific surface. The data are obtained from nitrogen adsorption isotherms in the multilayer regime...

These three cases are classical examples of fractal curves and dimensions as pioneered by Mandelbrot [96, 97] for macroscopic objects. They suggested to us possible extensions for the creation and analysis of unique spatial organizations of atoms within monomer repeat units, within branch cells, within dendrons, within a dendrimer, to give molecular-level surfaces in three dimensions of nanoscopic properties. 

---