Fractons

Figure 3.19. Fractonation of casein by Sephadex gel chromatography. (From Nakai et al. 1972. Reprinted with permission of the American Dairy Science Association.)...

Another important parameter which appears in connection with dynamical properties of fractals (such as diffusion) is the spectral (fracton) dimension d. Thus, in the diffusion-limited reactions, one has to replace d in (2.1.78) by d, i.e.,... 

The above-defined df and dt are structural parameters characterizing only the geometry of a given medium. However, when we are interested in processes like diffusion or reactions in disordered media, we need functional parameters, which are associated with the notion of time in order to characterize the dynamic behavior of the species in these media. The spectral or fracton dimension ds and random-walk dimension dw are two such parameters, and they will be defined in Section 2.2. 

To characterize the dynamic movement of particles on a fractal object, one needs two additional parameters the spectral or fracton dimension ds and the random-walk dimension dw. Both terms are quite important when diffusion phenomena are studied in disordered systems. This is so since the path of a particle or a molecule undergoing Brownian motion is a random fractal. A typical example of a random fractal is the percolation cluster shown in Figure 1.5. 

Alexander, S. and Orbach, R., Density of states on fractals Fractons, Journal of Physical Letters, Vol. 43, 1982, pp. L625-631. 

Which, due to its diffusive nature, has different time dependence than that of vibrational fractons [22], Substituting Eq. (27) into Eq. (26) leads to the asymptotic solution ... 

Relaxation rate, which looks into the low-frequency vibrational modes of protein molecules, is considered to play a significant role in biological process. A fractal-related parameter, called the fracton dimension, d-p, has been used to describe the relaxation rate at low temperature.f ... 

The overall value of fracton dimension depends on the strength of interaction between molecules in the main chain and in the cross-links. Fracton dimension would be 2 if the interaction is strong as found in the main chain. However, as the interactions in the cross-links are usually very weak, their fracton dimension would be between 1 (no connectivity) and 2 (strong connectivity). ... 

Fracton dimension was used to fit a photolysis kinetic of denatured HbNO shown below ... 

Anomalous subdiffusion occurs on percolation clusters or on objects that in a statistical sense can be described as fractal, by which we mean that selfsimilarity describes simply the scaling of mass with length. Connections between v, the fractal dimension of the cluster, D, and the spectral dimension, d, have been established, relations that were originally derived by Alexander and Orbach [35], who developed a theory of vibrational excitations on fractal objects which they called fractons. An elegant scaling argument by Rammal and Toulouse [140] also leads to these relations, and we summarize their results. 

The diffusion coefficient is Da = (1 /xa)R2, where l/xa is the phonon-assisted fracton-hopping time and R is the mean hopping distance for fracton modes with frequency near CD,. The rate of phonon-assisted fracton hopping is calculated with Eqs. (34) and (35) specifically for the cases where one fracton decays into another fracton and a phonon, as well as for the reverse process. 

The first two terms arise from modes that are extended over the protein. In proteins the size of cytochrome c, myoglobin, and GFP, which range, respectively, from 103 to 228 amino acids, almost all extended modes are fracton modes. The number of phonon modes in proteins of this size is very small, only 24 modes from 4 to 13 cm 1 for GFP, the largest of the three proteins. We cannot isolate the dynamics of a wave packet in terms of just these modes for an object this small. Thus, the contribution of extended modes to thermal conduction in proteins arises almost entirely from delocalized fractons. We give an expression for this contribution below. 

The third distance, perhaps the most relevant to reactions on surfaces, is the actual distance traversed by a diffusing molecule. This is a very complex issue which we only begin to understand. The diffusional distance reflects not only the geometric considerations made above, but also the facts that the surface is energetically heterogeneous, and that the diffusion is some combination of movements which follow closely the surface features, and of jumps from pore-wall to pore-wall and from one tip to the next. Obviously this diffusional distance is also a function of the temperature and of the solvent interfaced with the solid. Furthermore, since different types of connectedness can yield the same D value, this textural characteristic is an additional parameter to be considered (the fracton or spectral dimension (IS)). In view of this complex picture, what is then the practical advise Under the current state of art, the best one can do is to get a preliminary estimate of d from eq s [4]-[6] the direct observation of actual diffusional process in disordered systems, is still in its infancy. For some recent studies see ref. 16,17. 

To begin, recall that, in general, spaces can be characterized by three quantities dg, the dimension of the embedding Euclidean space, df the Hausdorff or fractal dimension, and ds the spectral or fracton dimension. A key to what follows is that for Euclidean spaces, these three dimensions are equal [57,58]. 

Granek, R. and Klafter, J., Fractons in proteins Can they lead to anomalously decaying time autocorrelations , Phys. Rev. Lett., 95, 098106 (2005). 

Alexander S, Orbach R. 1982. Density of states of fractals Fractons . J. Phys. Lett. 43 L625-L631. 

The complexity of the polymer structure is reflected in the large number of dimensions needed to describe it. Alexander and Orbach [28] proposed the use of spectral or fracton dimension for the description of the density of states on a fractal. The necessity of introducing is due to the fact that the fractal dimension defined by Equation (11.1) does not reflect this parameter. The investigators made use of the fact that anomalous diffusion of particles is expected on a fractal and, hence ... 

The first stage of the considered treatment is the formulation of polymeric fractal dimension ZT and its spectral (fracton) dimension d, which characterizes fractal object coimectivity degree [41], intercommunication. In this case Ihe value linear polymer chain and <7 =1.33 for very branched (cross-linked) chain. Using Flory-de Gennes mean field approximation, Vilgis obtained the following equation for/recalculation [39] ... 

Kozlov, G. V Dolbin, I. V. The physical sense and estimation methods of low-molecular solvent structural parameters in diluted polymer solutions. Proceedings of Higher Educational Institutions, North-caucasus region. Natural Sciences, 2004, (3), 69-71. Alexander, S. Orbach, R. Densily of states on fractals fractons . J. Phys. Lett. (Paris), 1982,43(17), L625-L631. 

Alexander, S. Laermans, C. Orbach, R. Rosenberg, H. M. Fracton interpretation of vibrational properties of cross-linked polymers, glasses and irradiabed quartz. Phys. Rev. B, 1983,28(8), 4615 619. 

One of such tendencies is polymers synthesis in the presence of all kinds of fillers, which serve simultaneously as reaction catalyst [26, 54]. The second tendency is the chemical reactions study within the framework of physical approaches [55-59], from which the fractal analysis obtained the largest application [36]. Within the framework of the last approach in synthesis process consideration such fundamental conceptions as the reaction prodrrcts stracture, characterized by their fractal (Hausdorff) dimension [60] and the reactionary medium connectivity, characterized by spectral (fracton) dimension J [61], were introduced. In its titrrt, diffusion processes for fractal reactions (strange or anomalous) differ principally from those occurring in Euclidean spaces and described by diffusion classical laws [62]. Therefore the authors [63] give transesterification model reaction kinetics description in 14 metal oxides presence within the framework of strange (anomalous) diffusion conception. 

Ardyralds [12] showed that at the study of ohemieal reactions on fractal objects the cor-rections on small clusters availability in the system were necessary. Just such corrections require the usage in theoretical estimations not generally accepted spectral (fracton) dimension ds [13], but its effective value. For percolation system two cases are possible [12] ... 

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