Fractal analysis

Endo, M., Takeuchi, K., Sasuda, Y., Matsubayashi, K., Oshida, K. and Drcsselhaus, M. S., Fractal analysis on pore structure for activated carbon fibers. Electron. Commun. Jpn., Part II Electron., 1994, 77(6), 98 107. 

The first detailed book to describe the practice and theory of stereology was assembled by two Americans, DeHoff and Rhines (1968) both these men were famous practitioners in their day. There has been a steady stream of books since then a fine, concise and very clear overview is that by Exner (1996). In the last few years, a specialised form of microstructural analysis, entirely dependent on computerised image analysis, has emerged - fractal analysis, a form of measurement of roughness in two or three dimensions. Most of the voluminous literature of fractals, initiated by a mathematician, Benoit Mandelbrot at IBM, is irrelevant to materials science, but there is a sub-parepisteme of fractal analysis which relates the fractal dimension to fracture toughness one example of this has been analysed, together with an explanation of the meaning of fractal dimension , by Cahn (1989). 

Clark, N.N., 1986. Three techniques for implementing digital fractal analysis of particle shape. Powder Technology, 46, 45-52. 

In order to describe the geometrical and structural properties of several anode electrodes of the molten carbonate fuel cell (MCFC), a fractal analysis has been applied. Four kinds of the anode electrodes, such as Ni, Ni-Cr (lOwt.%), Ni-NiaAl (7wt.%), Ni-Cr (5wt.%)-NijAl(5wt.%) were prepared [1,2] and their fractal dimensions were evaluated by nitrogen adsorption (fractal FHH equation) and mercury porosimetry. These methods of fractal analysis and the resulting values are discussed and compared with other characteristic methods and the performances as anode of MCFC. 

K. L. Nielsen and J. Lynch, Fractal analysis of bean root systems. Agron. Ahstr. 150 (1994). 

Experimentally accessible is D by means of scattering methods [144], The corresponding fractal analysis of scattering data is gaining special attractivity from its intriguing simplicity. In a double-logarithmic plot of I (s) v.v. s the fractal dimension is directly obtained from the slope of the linearized scattering curve. It follows from the theory of fractals that... 

Figure 8.14 shows a sketch of the plot that is utilized for the purpose of fractal analysis. For the theoretical fractal self-similarity holds for all orders of magnitude - to be measured in units of space (r) or reciprocal space (s)53. In practice, a fractal regime is limited by a superior cut and a lower cut54. In the sketch superior and lower cut limit the fractal region to two orders of magnitude in which self-similarity may be governing the materials structure. 

A. Sadana and T. Vo-Dinh, Single- and dual-fractal analysis of hybridization binding kinetics biosensor application. Biotechnol. Prog. 14, 782-790 (1998). 

Chapter 13 - It was shown, that limiting conversion (in the given case - imidization) degree is defined by purely structural parameter - macromolecular coil fraction, subjected evolution (transformation) in chemical reaction course. This fraction can be correctly estimated within the framework of fractal analysis. For this purpose were offered two methods of macromolecular coil fractal dimension calculation, which gave coordinated results. 

Karperien, A., 2004, FracLac Advanced User s Manual (Frac Lac.jar) Plugins/Fractal Analysis/FracLac.2005/03/23. version 2.0aF. Charles Stuart University, Australia. Available online at http //www.geoeities.eom/akarpe sbeglobal.net/usefraelae.html... 

A. Arneodo, Y. d Aubenton-Carafa, E. Bacry, P. V. Graves, J.-F. Muzy, and C. Thermes, Wavelet based fractal analysis of DNA sequences. Physica D 96, 291-320 (1996). 

Pedreschi, F., Aguilera, J. M., and Brown, C. (2000). Characterization of food surfaces using scale-sensitive fractal analysis. ]. Food Process Eng. 23,127-143. 

Rubnov, M. and Saguy, I. S. (1997). Fractal analysis and crust water diffusivity of a restructured potato product during deep-fat frying. J. Food Sci. 62,135-137,154. 

Ismail, l.M.K. Pfeifer, P. (1994) Fractal analysis and surface roughness of nonporous carbon fibers and carbon blacks. Langmuir 10 1532-1538... 

Barret, A. M., Normand, M.D., Peleg, M. and Ross, E. (1992). Characterisation of the jagged stress-strain relationships of puffed extrudates using the Fast Fourier Transform and fractal analysis. J. Food Sci. 57, 227-232, 235. 

Fini, A., Fernandez-Hervas, M. J., Holgado, M. A., Rodriguez, L., Cavallari, C., Passerini, N., and Caputo, O. 1997. Fractal analysis particles compacted by ultrasoilnB .harm. 

The electro-active surface of the porous carbon electrode for EDLCs is accessible only through the cumulative resistance of the electrolyte inside the pore. Therefore, the porous structure of the porous carbon becomes one of the most important factors influencing the energy/power densities. Fractal analysis has proven to be useful to describe the geometric and structural properties of rough surfaces and pore surfaces.56 66... 

In the late 1980 s and the early 1990 s, two modelling theories have been successfully used in the modelling of irregular surfaces fractal analysis and molecular dynamics theories. 

Surface area and porosity are examples of polysaccharide properties other than gelation that are amenable to fractal analysis. 

Building of the adequate model (nonequilibrium thermodynamic approach, polymeric or pseudopolymeric models, fractal analysis, combinations of various methods, etc.). 

The basic approach of the fractal analysis is to describe quantitatively a complex geometry through a scaling-law relation that characterizes one or more extensive properties of the object ... 

For the characterization of surface roughness of carbon blacks, different experimental techniques have been applied in the past. Beside microscopic investigations, e.g., AFM that give an impressive but more qualitative picture [90-92], scattering techniques such as SANS [93] and SAXS [94-96] as well as gas adsorption techniques [79-84, 97-99] have been used for a fractal analysis of surface roughness. The results discussed in the literature appear somewhat controversy, since almost flat surfaces with ds 2 [98, 99] and also rough surfaces with 2.2[Pg.12]

---