Fractal model

FIG. 10 Scaling parameter for the fractal model. (Reprinted with permission of American Institnte of Physics and the anthors from Ref. 183, Copyright 1993, American Institnte of Physics.)... 

Figure 6.16 shows a clear transition between the low volume fraction response and that at higher volume fractions. Perhaps the major problem with a fractal model is that it underestimates the complexity of colloidal... 

T. Vicsek, Fractal models for diffusion-controlled aggregation, J. Phys. A 16 L647 (1983). 

There are three principal approaches used for the description of processes of microporous material formation fractal models [1,2], the thermodynamic approach [3-5] and methods of polymer science [6]. 

Pyrolysis, oxidation FRACTAL MODEL S Low temperature drying... 

Of course, both microporous structure and the rest nonempty solid phase can be considered as fractals. For processes of pyrolytic treatment of organic materials, in most cases it is more useful to apply fractal methods to pores. For polymerization, it is more effective to apply fractal modeling of macromolecules. 

Glenny, R. and Robertson, H., Fractal modeling of pulmonary blood flow heterogeneity, Journal of Applied Physiology, Vol. 70, No. 3, 1991, pp.1024-1030. 

A detailed description of the relaxation mechanism associated with an excitation transfer based on a recursive (regular) fractal model was introduced earlier [47], where it was applied for the cooperative relaxation of ionic microemulsions at percolation. 

Note that the fractal dimensions discussed here are the fractal dimensions of the excitation transfer paths connecting the hydration centers located on the inner surface of the pores. Due to the low humidity, all of the water molecules absorbed by the materials are bound to these centers. The paths of the excitation transfer span along the fractal pore surface and depict the backbone of clusters formed by the pores on a scale that is larger than the characteristic distance between the hydration centers on the pore surface. Thus the fractal dimension of the paths Dp approximates the real surface fractal dimension in the considered scale interval. For random porous structures, Dp can be also associated with the fractal dimension D, of the porous space Dp = Dr. Therefore, the fractal dimension Dp can be used for porosity calculations in the framework of the fractal models of the porosity. 

Disordered porous media have been adequately described by the fractal concept [154,216]. It was shown that if the pore space is determined by its fractal structure, the regular fractal model could be applied [154]. This implies that for the volume element of linear size A, the volume of the pore space is given in units of the characteristic pore size X by Vp = Gg(A/X)°r, where I), is the regular fractal dimension of the porous space, A coincides with the upper limit, and X coincides with the lower limit of the self-similarity. The constant G, is a geometric factor. Similarly, the volume of the whole sample is scaled as V Gg(A/X)d, where d is the Euclidean dimension (d = 3). Hence, the formula for the macroscopic porosity in terms of the regular fractal model can be derived from (65) and is given by... 

In general, in order to include more types of porous media the random fractal model can be considered [2,154,216]. Randomness can be introduced in the fractal model of a porous medium by the assumption that the ratio of the scaling parameters c X/A is random in the interval [c0,1 ], but the fractal dimension I) in this interval is a determined constant. Hence, after statistical averaging, (66) reads as follows ... 

Figure 34 shows the temperature dependencies of the static fractal dimensions of the maximal cluster. Note that at percolation temperature the value of the static fractal dimension Ds is extremely close to the classical value 2.53 for a three-dimensional lattice in the static site percolation model [152]. Moreover, the temperature dependence of the stretch parameter v (see Fig. 34) confirms the validity of our previous result [see (62)] Ds = 3v obtained for the regular fractal model of the percolation cluster [47]. 

Intensive research has continued into the mechanism of snowflake formation [15], This research encompasses the broader question of dendritic crystal growth. New approaches, such as fractal models, and copious use of computer simulation have greatly facilitated these attempts. It is fascinating how dendritic growth penetrates even chemical synthetic work witnessed by the development of dendrimer chemistry of ever increasing complexity, which is an example of nanochemistry par excellence [16], An illustration is given in Figure 2-23. 

Fractal Model of Stability to the Cracking of Modified Polyethylene... 

For a calculation of the diffusivity D the fractal model of transport processes [7] will be use, according to which the value D is equal to ... 

The volume of free volume microvoid vh within the framework of fractal model [10] can be calculated according to the equation ... 

It is shown that the applicability of fractal model of anomalous diffusion for quantitative description of thermogravimetric analysis results in case of high density polyethylene modified by high disperse mixture Fe/FeO (Z). It is shown the influence of diffusion type on the value of sample 5%-th mass loss temperature and was offered structural analysis of this effect. The critical content Z it is determined, at which degradation will be elapse so, as in inert gas atmosphere. 

Despite experimental progress, the mechanisms and kinetics of acid milk gelation are still not fully understood. Theoretical approaches such as the adhesive sphere, percolation or fractal model applied to acid-induced milk gel formation can successfully explain specific aspects of the process (Tuinier and Kruif 1999), but all fail in rationalizing its kinetics (Home 1999). 

History of the Development of the Fractal Model of Fat Crystal Networks... 

Diffusion-limited aggregation of particles results in a fractal object. Growth processes that are apparendy disordered also form fractal objects (30). Sol—gel particle growth has also been modeled using fractal concepts (3,20). The nature of fractals requires that they be invariant with scale, ie, the fractal must look similar regardless of the level of detail chosen. The second requirement for mass fractals is that their density decreases with size. Thus, the fractal model overcomes the problem of increasing density of the classical models of gelation, yet retains many of its desirable features. The mass of a fractal, Af, is related to the fractal dimension and its size or radius, R, by equationS ... 

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