Materials Fractal

A. Static Percolation in Porous Materials, Fractal Concept, and Porosity Determination... 

Park, Y. (2000). Eractal geometry of porous materials. Fractals, 8, 301-306. 

Material Fractal dimension Synthesis method Measurement method Reference... 

The porosity and pore systems in cement-based composites may be one example of fractality, which is characteristic for these materials. Fractal quantity depends on the scale used to measure it for example, the fracture area of a concrete element cannot be determined in an unambiguons way if the method and scale of its determination are not given (cf. Chapter 10.5). The classification and analysis of a pore system and all quantitative results derived depend among other things on the method of observation and magnification of microscopic images (Mandelbrot 1982 Guyon 1988). 

Yavuz, O. Ram, M. K. Supramolecular Engineering of Conducting Materials Fractal System Inc. Safety Harbor, FL, 2005 p 303. 

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). 

A high modulus gradient at the interface is also be avoided in materials Joined as a result of the interdiffusion of materials to form a fractal surface [32]. The effect is to produce an interfacial composite region. This strengthens the interface and leads to a more gradual change in modulus and avoids the sharp concentrations of stress which would occur at a smooth interface. 

The fractal-like organization led, therefore, to conductivity measurements at three different scales (1) the macroscopic, mm-size core of nanotube containing material, (2) a large (60 nm) bundle of nanotubes and, (3) a single microbundle, 50 nm in diameter. These measurements, though they do not allow direct insights on the electronic properties of an individual tube give, nevertheless, at a different scale and within certain limits fairly useful information on these properties. 

FHH (Frenkel-Halsey-Hill) theory is valid for multi molecules adsorption model of the flat surfrtce material. When this model is applied for the surface fractal in the range of capillary condensation, in other words, in the state of interface which was controlled by the surface tension between liquid and gas, the modified FHH equation can be expressed as Eq. (3). 

Physically, the wetting abdity increases (the contact angle decreases) as the values of the fractal dimension of the electrode increases if the electrode material is same. However, in this study, we could not obtain a good correlation between the fractal dimensions and the wetting abilities as shown in Table 1. It means that not only the physical properties such as the surface irregularity and roughness but also the chemical interaction between electrolyte and electrode were important in wetting ability. 

From the most general point of view, the theory of fractals (Mandelbrot [1977]), one-, two-, three-, m-dimensional figures are only borderline cases. Only a straight line is strictly one-dimensional, an even area strictly two-dimensional, and so on. Curves such as in Fig. 3.11 may have a fractal dimension of about 1.1 to 1.3 according to the principles of fractals areas such as in Fig. 3.12b may have a fractal dimension of about 2.2 to 2.4 and the figure given in Fig. 3.14 drawn by one line may have a dimension of about 1.9 (Mandelbrot [1977]). Fractal dimensions in analytical chemistry may be of importance in materials characterization and problems of sample homogeneity (Danzer and Kuchler [1977]). 

Different kinds of heterogeneity can be imagined. In the most simple case only a few differing structural entities are found to coexist without correlation inside the volume irradiated by the primary beam. In this case it is the task of the scientist to identify, to separate and to quantify the components of such a multimodal structure. In an extreme case heterogeneity may even result in a fractal structure that can no longer be analyzed by the classical methods of materials science. 

Figure 8.8. Important components of diffuse scattering in the SAXS of polymer materials with two or more phases. Only for fractals Porod s law is fundamentally changed...

Characteristic for a fractal structure is self-similarity. Similar to the mentioned pores that cover all magnitudes , the general fractal is characterized by the property that typical structuring elements are re-discovered on each scale of magnification. Thus neither the surface of a surface fractal nor volume or surface of a mass fractal can be specified absolutely. We thus leave the application-oriented fundament of materials science. A so-called fractal dimension D becomes the only absolute global parameter of the material. 

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. 

Nevertheless, fractal structure is an issue in porous materials. 

The properties characteristic to fractal objects were mentioned first by Leonardo da Vinci, but the term fractal dimension appeared in 1919 in a publication by Felix Hausdorff [197], a more poetic description of fractals was given by Lewis Richardson in 1922 [198] (cited by [199]), but the systematic study was performed by Benoit B. Mandelbrot [196], Mandelbrot transformed pathological monsters by Hausdorff into the scientific instrument, which is widely used in materials science and engineering [200-202]. Geometrical self-similarity means, for example, that it is not possible to discriminate between two photographs of the same object taken with two very different scales. 

Illumination generates holes within the material of PS and causes photo corrosion of PS that is much faster than that in the dark. Depending on illumination intensity and time, the pore walls in a PS can be thinned to various extents by the photo induced corrosion. This corrosion process is responsible for the etched crater between the initial surface and the surface of PS as illustrated in Figure 28. It is also responsible for the fractal structure of the micro PS formed under illumination. 

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. 

Composite electrodes used in the electrochemical processes are often partially active since they are composed of the active powder material and the inactive binder and conductor. The partially blocked active electrode can be characterized by the contiguous fractal with dy < 2.0. In the case of the electrodes composed of the active islands on an inactive support, they are characterized by the non-contiguous fractal with dy < 2.0.121... 

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.

Porous materials have attracted considerable attention in their application in electrochemistry due to their large surface area. As indicated in Section I, there are two conventional definitions concerning with the fractality of the porous material, i.e., surface fractal and pore fractal.9"11 The pore fractal dimension represents the pore size distribution irregularity the larger the value of the pore fractal dimension is, the narrower is the pore size distribution which exhibits a power law behavior. The pore fractal dimensions of 2 and 3 indicate the porous electrode with homogeneous pore size distribution and that electrode composed of the almost samesized pores, respectively. 

Besides the molecular probe method using gas adsorption,107 162 recently, the TEM image analysis method163"167 has been applied to evaluate the surface fractal dimension of porous materials. The most attractive fact in this method is that the pores in different size ranges can be extracted from the TEM images which include contributions from many different pore sizes by the inverse fast Fourier transform (FFT) operation by selecting the specific frequency range.165 167... 

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