Fractals theory

Fractal theory is a relatively new field of geometry, formulated by Mandelbrot [196] for irregular rough-surfaced objects. The major properties of such objects are the dependence of the measured length (perimeter), surface, or volume on the scale of measurement and geometrical self-similarity... 

Electrochemical processes usually take place on rough surfaces and interfaces and the use of fractal theory to describe and characterize the geometric characteristics of surfaces and interfaces can be of significant importance in electrochemical process description and optimization. Drs. Joo-Young Go and... 

Since Avnir and Pfeifer s pioneer works83"86 regarding the characterization of the surface irregularity at the molecular level by applying the fractal theory of surface science, the molecular probe method using gas adsorption has played an important role in the determination of surface fractal dimension of the porous and particulate materials. 

Pfeifer, P. Avnir, D. (1983) Chemistry in noninteger dimensions between two and three. I. Fractal Theory of heterogeneous surfaces. 

The electron-ion recombination in high-mobility systems has also been analyzed in terms of the fractal theory [24,25]. It was postulated that even when the fractal dimension of particle trajectories is not equal to 2, the motion of particles is still described by diffusion but with a distance-dependent effective diffusion coefficient D r) = D(H-// where the parameter / is proportional to the mean free path X [24]. However, when the fractal dimension of trajectories is not equal to 2, the motion of particles is not described by orthodox diffusion. 

Weber, L. Fractal theory applied to structure-activity relationships. Euro-QSAR 2004, Istanbul, Turkey, September 5-10, 2005. 

There are two conventional definitions in describing the fractality of porous material - the pore fractal dimension which represents the pore distribution irregularity56,59,62 and the surface fractal dimension which characterizes the pore surface irregularity.56,58,65 Since the geometry and structure of the pore surfaces are closely related to the electro-active surface area which plays a key role in the increases of capacity and rate capability in practical viewpoint, the microstructures of the pores have been quantitatively characterized by many researchers based upon the fractal theory. 

The structure of this review is as follows in Section II, the activation and templating methods for preparing the porous carbons are briefly summarized. Section III surveys the structural characteristics of the porous carbons by using gas adsorption method. In Section IV, the molecular probe method and the image analysis method for quantitative characterization of the pore surface irregularity and the size distribution irregularity based upon the fractal theory are discussed in detail. Section V is devoted to... 

See, for instance, M. Lax, in Multiple Scattering and Waves in Random Media, P. L. Chow, W. E. Kohler, and G. C. Papanicolaou, Eds., North Holland, 1981, and references therein J. Klafter and M. S. Schlesinger, Proc. Nat. Acad. Sci. U.S.A. 83, 848 (February 1986), and references therein R. Brown, Thesis, University of Bordeaux 1,1987 R. Brown et al. J. Phys. C20, L649 (1987) 21 (1988) in press. (This last work thoroughly discusses the applicability of fractal theory to isotopically mixed crystals as disordered system. Serious criticism is presented both of the analysis of the experimental data and of the fundementals of their description in the present theory of fractals. For this reason we omit all works treated there. 

In 1992, Vreeker et al. presented rheological data for aggregated fat networks in the framework of previous fractal theories. These authors indicated that the elastic modulus varied with particle concentration according to a power law, in keeping with the proposed models for the elasticity of colloidal gels. 

Marangoni s group has since advanced the fractal theory applying Shih et al. s weak link regime with Vreeker s rheological findings to develop a fractal theory for fat crystal networks. Fat crystal networks are considered as cross-linked fractal clusters formed by aggregating fat crystals. Self-similarity is assumed to exist within the clusters, from the primary fat crystals to the clusters. If the force-constant of the links between micro structures was expressed as k/, then the macroscopic elastic constant K (in one dimension) of the network could be modeled as ... 

The objective of this research work was to use the surface fractal theory to analyze the adsorbent from industrial sludge at various conditions. Effects of activated temperature, activated time and activated agent on the specific surface area and pore volume surface of adsorbents were investigated. 

Fractal theories of random walks developed a different expression [6] ... 

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... 

Algorithms for the formation of fractal sets and the determination of fractal dimension have a significant place in fractal theory. Simple models of fractal sets are considered below to illustrate formation algorithms and the calculation of the fractal dimension. 

The chemical reaction between a solid and a reactive fluid is of interest in many areas of chemical engineering. The kinetics of the phenomenon is dependent on two factors, namely, the diffusion rate of the reactants toward the solid/fluid interface and the heterogenous reaction rate at the interface. Reactions can also take place within particles, which have accessible porosity. The behavior will depend on the relative importance of the reaction outside and inside the particle. Fractal analysis has been applied to several cases of dissolution and etching in such natural occurring caves, petroleum reservoirs, corrosion, and fractures. In these cases fractal theory has found usefulness for quantifying the shape (line or surface) with only a few parameters the fractal dimension and the cutoffs. There have been some attempts to use a fractal dimension for reactivity as a global parameter. Finally, fractal concepts have been used to aid in the interpretation of experimental results, if patterns quantitatively similar to DLA are obtained. 

In the mid-1980s, there was an explosion of research seeking applications for fractal theory to various aspects of physical phenomenon in physics, chemistry. 

Go, J.-Y., and Pyun, S.-L 2004. A study on lithium transport through fi-actal l.p.sCoOj film electrode by analysis of current transient based upon fractal theory. Electrochimica Acta 49, 2551-2562. 

Gutierrez, L.G.F., Chanona, P.J., Alamilla, B.L., Hernandez, P.A., Jimenez, A.A., Farrera, R.R., and Ondorica, V.C. A proposal of anaUsis the drying phenomena by means of fractal theory. Engineering and food for the 21st century, J. Welti-Chanes, G.V. Barbosa-Canovas and J.M.C.R. Aguilera, eds., CRC Press, Boca Raton, pp. 269-287, 2002. 

Chanona-Perez, J.J., Alamilla-Beltran, L., Farrera, R.R.R., Quevedo, R., Aguilera, J.M., and Gutierrez-Lopez, G.F. Description of the convective air drying of a food model by means of the fractal theory. Food Sci. Technol. Int., 9, 207, 2003. 

Mayer, A.Y., Farin, D. and Avair, D. (1986) Cross-sectional areas of alkanoic adds. A comparative study applying fractal theory of adsorption and considerations of molecular shape. J. Am. Chem. Soc., 108, 7897-7905. 

Kopelman, R. (1986). Rate processes on fractals theory, simulations, and experiments. J. Stat. Physics 42, 185-200. 

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