Media fractal

Matsuura, S. and Miyazima, S. (1993). Formation of ramified colony of fungus Aspergillus oryzae on agar media. Fractals, 1, 336-345. 

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

Chapter 15 - It was shown, that the reesterification reaction without catalyst can be described by mean-field approximation, whereas introduction of catalyst (tetrabutoxytitanium) is defined by the appearance of its local fluctuations. This effect results to fractal-like kinetics of reesterification reaction. In this case reesterification reaction is considered as recombination reaction and treated within the framework of scaling approaches. Practical aspect of this study is obvious-homogeneous distribution of catalyst in reactive medium or its biased diffusion allows to decrease reaction duration approximately twofold. 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

Keywords Imidization, nanofiller, reactive medium, heterogeneity, fractal free volume. 

Figure 4. The dependence of heterogeneity exponent h of reactive medium on relative fractal volume J for PAA solid state imidization. The notation is the same, that in figure 2.

Let s note three important aspects followed from the model [4] application for description of reesterification reaction. At first as reesterification reactions with TBT and in it absence proceed in identical conditions, then from the comparison of figure 1 kinetic curves follows, that the reaction fractal-like behaviour in TBT presence is due to local fluctuations of catalyst distribution in reactive medium. Secondly the division of reaction duration into short and long... 

In case of reaction course in the Euclidean spaces the value D is equal to the dimension of this space d and for fractal spaces D is accepted equal to spectral dimension ds [6], By plotting p i=( 1 -O) (where O is conversion degree) as a function of t in log-log coordinates the value D from the slope of these plots can be determined. It was found, that the mentioned plots fall apart on two linear parts at t<100 min with small slope and at PT00 min the slope essentially increases. In this case the value ds varies within the limits 0,069-3,06. Since the considered reactions are proceed in Euclidean space, that is pointed by a linearity of kinetic curves Q-t, this means, that the reesterefication reaction proceeds in specific medium with Euclidean dimension d, but with connectivity degree, characterized by spectral dimension ds, typical for fractal spaces [5],... 

Daccord G, Lenormand R (1987) Fractal patterns from chemical dissolution. Nature 325 41 3 Daccord G, Lietard O, Lenormand R (1993) Chemical dissolution of a porous medium by a reactive fluid, 2, Convection vs. reaction behavior diagram. Chem Eng Sci 48 179-186 Darmody RG, Thorn CE, Harder RL, Schlyter JPL, Dixon JC (2000) Weathering implications of water chemistry in an arctic-alpine environment, north Sweden. Geomorphology 34 89-100 Dijk P, Berkowitz B (1998) Precipitation and dissolution of reactive solutes in fractures. Water Resour Res 34 457-470... 

More recently several quantitative analyses of both pencil-beam and wide-angle surveys of galaxy distributions have been performed three examples are given by Joyce et al. [34], who analyzed the CfA2-South catalog to find fractal behavior with D= 1.9 0.1 Labini and Montuori [35] analyzed the APM-Stromlo survey to find fractal behavior with D = 2.1 0.1, while Labini et al. [36] analyzed the Perseus-Pisces survey to find fractal behavior with D = 2.0 0.1. There are many other papers of this nature in the literature, all supporting the view that, out to medium depth, at least, galaxy distributions appear to be fractal with D 2. 

So, to date, evidence that galaxy distributions are fractal with D Pi 2 on small to medium scales is widely accepted, but there is a lively open debate over the existence, or otherwise, of a crossover to homogeneity on large scales. 

The main result arising from the present stationary universe analysis is that a perfectly inertial universe, which arises as an idealized limiting case, necessarily consists of a fractal, D = 2, distribution of material. This result is to be compared with the real universe, which approximates very closely perfectly inertial conditions on even quite small scales, and that appears to be fractal with D k, 2 on the medium scale. 

A characteristic feature of the carbon modifications obtained by the method developed by us is their fractal structure (Fig. 1), which manifests itself by various geometric forms. In the electrochemical cell used by us, the initiation of the benzene dehydrogenation and polycondensation process is associated with the occurrence of short local discharges at the metal electrode surface. Further development of the chain process may take place spontaneously or accompanied with individual discharges of different duration and intensity, or in arc breakdown mode. The conduction channels that appear in the dielectric medium may be due to the formation of various percolation carbon clusters. 

For t > 1 at any 2, we obtain p (2, t) oc t-1/2. This behavior in a homogeneous medium corresponds to (2.8), giving the probability density in a fractal medium with spectral dimension ds. 

Before we close this section some major, unique kinetic features and conclusions for diffusion-limited reactions that are confined to low dimensions or fractal dimensions or both can now be derived from our previous discussion. First, a reaction medium does not have to be a geometric fractal in order to exhibit fractal kinetics. Second, the fundamental linear proportionality k oc V of classical kinetics between the rate constant k and the diffusion coefficient T> does not hold in fractal kinetics simply because both parameters are time-dependent. Third, diffusion is compact in low dimensions and therefore fractal kinetics is also called compact kinetics [23,24] since the particles (species) sweep the available volume compactly. For dimensions ds > 2, the volume swept by the diffusing species is no longer compact and species are constantly exploring mostly new territory. Finally, the initial conditions have no importance in classical kinetics due to the continuous re-randomization of species but they may be very important in fractal kinetics [16]. 

The release problem can be seen as a study of the kinetic reaction A+B —> B where the A particles are mobile, the B particles are static, and the scheme describes the well-known trapping problem [88]. For the case of a Euclidean matrix the entire boundary (i.e., the periphery) is made of the trap sites, while for the present case of a fractal matrix only the portions of the boundary that are part of the fractal cluster constitute the trap sites, Figure 4.11. The difference between the release problem and the general trapping problem is that in release, the traps are not randomly distributed inside the medium but are located only at the medium boundaries. This difference has an important impact in real problems for two reasons ... 

In this context, Berry [277] studied the enzyme reaction using Monte Carlo simulations in 2-dimensional lattices with varying obstacle densities as models of biological membranes. That author found that the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial concentration increase. In addition, the rate constant controlling the rate of the complex formation was found to be, in essence, a time-dependent coefficient since segregation effects arise due to the fractal structure of the reaction medium. In a similar vein, Fuite et al. [278] proposed that the fractal structure of the liver with attendant kinetic properties of drug elimination can explain the unusual... 

It has been stated that heterogeneous reactions taking place at interfaces, membrane boundaries, or within a complex medium like a fractal, when the reactants are spatially constrained on the microscopic level, culminate in deviant reaction rate coefficients that appear to have a sort of temporal memory. Fractal kinetic theory suggested the adoption of a time-dependent rate constant , with power-law form, determined by the spectral dimension. This time-dependency could also be revealed from empirical models. 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

In the fractal porous medium, the diffusion is anomalous because the molecules are considerably hindered in their movements, cf. e.g., Andrade et al., 1997. For example, Knudsen diffusion depends on the size of the molecule and on the adsorption fractal dimension of the catalyst surface. One way to study the anomalous diffusion is the random walk approach (Coppens and Malek, 2003). The mean square displacement of the random walker (R2) is not proportional to the diffusion time t, but rather scales in an anomalous way ... 

The third relaxation process is located in the low-frequency region and the temperature interval 50°C to 100°C. The amplitude of this process essentially decreases when the frequency increases, and the maximum of the dielectric permittivity versus temperature has almost no temperature dependence (Fig 15). Finally, the low-frequency ac-conductivity ct demonstrates an S-shape dependency with increasing temperature (Fig. 16), which is typical of percolation [2,143,154]. Note in this regard that at the lowest-frequency limit of the covered frequency band the ac-conductivity can be associated with dc-conductivity cio usually measured at a fixed frequency by traditional conductometry. The dielectric relaxation process here is due to percolation of the apparent dipole moment excitation within the developed fractal structure of the connected pores [153,154,156]. This excitation is associated with the selfdiffusion of the charge carriers in the porous net. Note that as distinct from dynamic percolation in ionic microemulsions, the percolation in porous glasses appears via the transport of the excitation through the geometrical static fractal structure of the porous medium. 

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 ... 

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