Matrix fractal

The Eq. (27) application gave an exact enough (within the limits of 6%) description of composites polymier matrix fractal dimension change in fractal space, created by filler particles (aggregates of particles) network [45]. 

Fundamental Aspects of Filling of Nanocomposites with High-Elasticity Matrix Fractal Models... 

The A1 AI2O3 composite grown at low temperatures (450-500 °C) and low pressure (10 -10 mbar) consists of aluminum particles (diameters ranging from 1-50 nm depending on reaction time), which are embedded in an almost amorphous AI2O3 matrix. The sizes of the particles seem to follow a fractal distribution with a fractal exponent of 2.4 [24] which we have already found for other metal/metal-oxide composites grown by similar CVD processes [22,29]. The amorphous aliuninum oxide is transformed to the crystalhne 7-AI2O3 at temperatures aroimd 550-600 °C. 

The considered model of a straight line of M nanoparticles illustrates only general features of dielectric losses caused by an M nanoparticle cluster in polymer matrix. Actually such cluster is a complex fractal system. Analysis of dielectric relaxation parameters of this process allowed the determination of fractal properties of the percolation cluster [104], The dielectric response for this process in the time domain can be described by the Kohlrausch-Williams-Watts (KWW) expression... 

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 ... 

The drug molecules move inside the fractal matrix by the mechanism of diffusion, assuming excluded volume interactions between the particles. The matrix can leak at the intersection of the percolation fractal with the boundaries of the square box where it is embedded, Figure 4.11. 

Figure 4.12 shows simulation results (line) for the release of particles from a fractal matrix with initial concentration Co = 0.50, on a lattice of size 50 x 50. The simulation stops when more than 90% of the particles have been released from the matrix. This takes about 20, 000 MCS. In the same figure the data by... 

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]... 

It was shown recently that disordered porous media can been adequately described by the fractal concept, where the self-similar fractal geometry of the porous matrix and the corresponding paths of electric excitation govern the scaling properties of the DCF P(t) (see relationship (22)) [154,209]. In this regard we will use the model of electronic energy transfer dynamics developed by Klafter, Blumen, and Shlesinger [210,211], where a transfer of the excitation... 

Apparently the fractal dimension of the excitation paths in sample A is close to unity. Topologically, this value of Dp corresponds to the propagation of the excitation along a linear path that may correspond to the presence of second silica within the pores of the sample A. Indeed, the silica gel creates a subsidiary tiny scale matrix with an enlarged number of hydration centers within the pores. 

The fractal dimensions of the excitation paths in samples D, F, and G lie between 2 and 3. Thus, percolation of the charge carriers (protons) is also moving through the Si02 matrix because of the availability of an ultra-small porous structure that occurs after special chemical and temperature treatment of the initial glasses [156]. 

Bonny, J.D. Leuenberger, H. Determination of fractal dimensions of matrix type solid dosage forms and their relation with drug dissolution kinetics. Eur. J. Pharm. Bio-pharm. 1993, 39 (1), 31-37. 

Fernandez-Hervas, M.J. Vela, M.T. Fini, A. Rabasco, A.M. Fractal and reactive dimension in inert matrix systems. Int. J. Pharm. 1995,130, 115-119. 

Section II provides a summary of Local Random Matrix Theory (LRMT) and its use in locating the quantum ergodicity transition, how this transition is approached, rates of energy transfer above the transition, and how we use this information to estimate rates of unimolecular reactions. As an illustration, we use LRMT to correct RRKM results for the rate of cyclohexane ring inversion in gas and liquid phases. Section III addresses thermal transport in clusters of water molecules and proteins. We present calculations of the coefficient of thermal conductivity and thermal diffusivity as a function of temperature for a cluster of glassy water and for the protein myoglobin. For the calculation of thermal transport coefficients in proteins, we build on and develop further the theory for thermal conduction in fractal objects of Alexander, Orbach, and coworkers [36,37] mentioned above. Part IV presents a summary. 

In the majority of numerical calculations of the anomalous frequency behavior of such composites (in particular, near the percolation threshold pc) under the action of an alternating current, lattice (discrete) models have been used, which were studied in terms of the transfer-matrix method [91,92] combined with the Frank-Lobb algorithm [93], Numerical calculations and the theoretical analysis of the properties of composites performed in Refs. 91-109 have allowed significant progress in the understanding of this phenomenon however, the dielectric properties of composites with fractal structures virtually have not been considered in the literature. 

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