Scaling 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.)... 

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 ... 

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 ... 

Using the methods of classical statistical physics one may more or less rigorously solve problems where the system on a microscopic level is either in a state of complete chaos (perfect gas) or total order (solid perfectly crystalline bodies). In contrast, disordered media and processes in which there is neither crystalline order nor complete chaos on the microscopic level have not yet had an adequate description. This problem is connected with the condition that the macroscopic variables must considerably exceed the correlation scales of microscopic variables, a condition which is not met by disordered media. Consequently in order to describe such systems, fractal models and phased averaging on different scale levels (meso-levels) should be effective. 

Strictly speaking, all naturally occurring power-laws in fractal or dynamic patterns are finite. Scale-free models nevertheless provide an efficient description of a wide variety of processes in complex systems [16,20,46,106]. This phenomenological fact is corroborated by the observation that the power-law properties of Levy processes persist strongly even in the presence of cutoffs [99]... 

The main conclusion is that the fractal dimension of the distribution profile of acceptors around a donor is inversely dependent on the pore size. It is also important to notice that the same D values are obtained with all three donor/acceptor pairs. We interpret these D values as reflecting the geometry of the support as seen by an adsorbed molecule, and in particular that these D values are the surface fractal dimensions for adsorption, for the following reasons (a) The fact that the D values were found to be insensitive to the different Ro values of the three pairs and to the concentrations employed, is in keeping with the scale-invariance of the fractal model, (b) In a number of studies (21,43) it has been shown that for the same material, higher... 

Pachepsky, Y.A., R.A. Scherbakov, and L.P. Korsunskaia. 1995b. Scaling of soil water retention using a fractal model. Soil Sci. 159 99-104. 

Fractal models for soil structure and rock fractures are becoming increasingly popular (e.g., Sahimi, 1993 Baveye et al., 1998). The primary appeal of these models is their ability to parsimoniously parameterize complex structures. Scale symmetry or scale invariance, in which an object is at least statistically the same after magnification, is a fundamental property of fractals and can also be observed in numerous natural phenomena. Thus, it is logical that some investigators have examined theoretical transport in known prefractals. 

Fractals have fascinating properties that are present in many natural objects and had not been incorporated in previous models of nature (5, 6). If any small piece of a fractal is magnified, it appears similar to a larger piece. This property is called self-similarity and is illustrated by the fractal in Figure 2. Self-similarity can occur only if structures at a small scale are related to structures at a larger scale. Fractal objects include the repeated bifurcations of the airways in the lung (7), the distribution of blood flow in the ever-smaller vessels in the heart (8), and the ever-finer infoldings of cellular membranes (9). 

The development of new molecular closure schemes was guided by analysis of the nature of the failure of the MSA closure. In particular, the analytic predictions derived by Schweizer and Curro for the renormalized chi parameter and critical temperature of a binary symmetric blend of linear polymeric fractals of mass fractal dimension embedded in a spatial dimension D are especially revealing. The key aspect of the mass fractal model is the scaling relation or growth law between polymer size and degree of polymerization Ny cr. The non-mean-field scaling, or chi-parameter renormalization, was shown to be directly correlated with the average number of close contacts between a pair of polymer fractals in D space dimensions N /R if the polymer and/or... 

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

---