Geometry fractal

This section attempts to examine macromolecular geometry, and in particular dendritic surface characteristics, from the perspectives of self-similarity and surface irregularity, or complexity, which are fundamental properties of basic fractal objects. It is further suggested that analyses of dendritic surface fractality can lead to a greater understanding of molecule/solvent/dendrimer interactions based on analogous examinations of other materials (e.g., porous silica and chemically reactive surfaces such as found in heterogeneous catalysts). 52  

Intuitively, objects such as lines, squares, and cubes possess dimensionalities of 1.0, 2.0, and 3.0, respectively. It is also rational to expect that many natural, as well as manmade, objects possess non-integral, or fractional, dimensionalities due to complicated patterns. Classical examples of common fractal patterns and forms include naturally-occurring objects such as coastlines, clouds, mountains, and snowflakes. 53-55  

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 

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B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, New York, 1982 Fractals Form, Chance, arul Dimension, Freeman, New York, 1977. 

Many of the adsorbents used have rough surfaces they may consist of clusters of very small particles, for example. It appears that the concept of self-similarity or fractal geometry (see Section VII-4C) may be applicable [210,211]. In the case of quenching of emission by a coadsorbed species, Q, some fraction of Q may be hidden from the emitter if Q is a small molecule that can fit into surface regions not accessible to the emitter [211]. 

Finally, as might be expected, a fractal geometry approach has been made. 

Schaefer D W, Martin J E, Wiltzius P and Cannell D S 1984 Fractal geometry of colloidal aggregates Phys. Rev. Lett 52 2371-4... 

The electrical double-layer structure and fractal geometry of a pc-Ag electrode have been tested by Se vasty an ov et al.272 They found that the geometrical roughness of electrochemically polished pc-Ag electrodes is not very high (/pz 1.5 to 1.25), but the dependence of Chtr curves on cej, as well as on/pz, is remarkable (C, =30 to 80 fi cm-2 if/pz =1.5 to 1.0). 

Mandelbrot, BB, The Fractal Geometry of Nature WH Freeman San Francisco, 1983. 

The dissolution channels (wormholes), obtained under certain conditions of attack of carbonate rocks by hydrochloric acid, have been recently proven to have a fractal geometry. An equation was proposed, relating the increase of the equivalent wellbore radius (i.e. the decrease of the skin) to the amount of acid injected, in wellbore geometry and in undamaged primary porosity rocks. This equation is herein extended to damaged double porosity formations through minor modifications. 

K. Falconer, Fractal Geometry Mathematical Foundations and Application, Wiley, New York, 1990. 

Su-Il Pyun provide a comprehensive review of the physical and electrochemical methods used for the determination of surface fractal dimensions and of the implications of fractal geometry in the description of several important electrochemical systems, including corroding surfaces as well as porous and composite electrodes. 

In Section IV, from the studies on diffusion towards self-affine fractal interface, the surface fractal dimension as determined by the electrochemical method is characterized as being self-similar, even though the rough surfaces and interfaces show the self-affine scaling property. Finally, in Section V, we exemplified the application of fractal geometry in electrochemical systems in view of the characterization of rough surfaces and interfaces by the surface fractal dimension. 

Here, the actual thickness of adsorbed molecule layers /ad is obtained by multiplying n by adsorbed molecule diameter a0. The actual thickness range in which the fractal geometry is satisfied represents the length-scale cutoff range of fractality.116,118 Here, the value of the lower (inner) cutoff length tadjmfa is equal to... 

Figure 14. Typical SEM micrographs of pit morphology on the surface of alloy 600 subjected to a constant anodic potential of 0.8 V(Ag/AgCl) for 300s in aqueous 0.1 M Na2S2C>3 + 0.1 M NaCl solution at (a) 25°, (b) 60°, (c) 100°, (d) 150°, (e) 200°, (f) 250°, and (g) 300°C. Reprinted from J. -J. Park and S. -I. Pyun, Pit formation and growth of alloy 600 in CF ion-containing thiosulphate solution at temperatures 298-573 K using fractal geometry, Corros. Sci., 45, p. 995, Copyright 2003, with permission from Elsevier Science.

Figure 15. KFM images obtained from the PVDF-bonded composite made from (a) the as-received SFG50 graphite and from (b) the surface-modified SFG50 graphite. Reprinted from S.-B. Lee and S.-I. Pyun, Determination of the morphology of surface groups formed and PVDF-binder materials dispersed on graphite composite electrodes in terms of fractal geometry, J Electroanal. Chem. 556, p. 75, Copyright 2003, with permission from Elsevier Science.

On the other hand, the surface fractal dimension characterizes the pore surface irregularity the larger the value of surface fractal dimension is, the more irregular and the rougher is the pore surface. Since the pore structure is closely related to the electroactive surface area which plays a key role in the increase of capacity in practical viewpoint, many researchers have investigated the microstructure of the pores by using fractal geometry. 

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