Physical fractal density

In Fig. 1.2, the dependence of physical fractal density p on measurement scale L in double logarithmic coordinates was shown. ForL L Euclidean behavior is observed and within the range ofL = L. h- L - fractal one [40]. Let us pay attention to the complete analogy of the plots of Figs. 1.2 and 4.17. 

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

The fractal dimension can be estimated by several techniques, including structured walk (Richardson s plot), bulk density-particle diameter relation, sorption behavior of gases, pore size distribution, and viscoelastic behavior. The fiactal dimension obtained by each method has its own physical meaning (Rahman, 1997). 

Fractal dimension, D is considered as an effective number that characterises the irregular electrode surface. The term has been related to physical quantities such as mass distribution, density of vibrational stages, conductivity and elasticity. If we consider a 2-D fractal picture in its self-similar multi-steps, one can draw various spheres of known radii at various points of its structure and may thus count the number of particles, N inside the sphere by microscope, following relation will then hold good ... 

The physical significance of this last relationship is that, as the fractals grow larger, they become less dense. Even though the mass concentration of oxide precursor is constant, the volume concentration of fractal particles is increasing because of the decreasing density of the fractals themselves ... 

In the end the conclusion can be made, that reaction cessation in low-temperature polycondensation process is limited by pnrely physical factor, namely, by macromolecular coil density reaching of reactive medium density (mixture monomer — polymer solution) [121], that is possible for fractal objects only. Such conclusion follows from the simple analysis, adduced below. As it is known [122], the fractal object density pfr can be calculated according to the Eq. (73) ... 

Physical properties, such as floe densities, settling velocities or the diffusion of compounds inside aggregates and floes, depend on their fractal dimensions, which have to be calculated. The floe fractal dimensions can be determined directly from simulation, since each monomer and particle position is repeatedly computed during a simulation, allowing the mathematical definition of a fractal to be applied directly. 

The main conclusion of the present paper is that, what reaction cessation in polycondensation process is limited by purely physical factor, namely, the achievement of macromolecular coil density of solution density in which reaction proceeds. The possibility of such treatment is due fully to macromolecular coil fractality. The indicated processes are simulated within the framework of an irreversible aggregation models over mechanism cluster-cluster. The critical values of molecular weight and reaction duration, above which a S5mthesis reaction ceases, exist. 

This strategy is common in atomic fluid theory at low to moderate densities, and for Coulombic systems, and corresponds to the reference idea ubiquitous in liquid state theory [5], The purely hard core problem is treated using the accurate [27] PY closure. (2) The construction of a closure approximation for the tail part of the potential is subject to the constraint of exactly describing the weak coupling limit. In physical terms, for fractal-like interpenetrating molecules these indirect processes may strongly couple the direct correlation functions associated with those pairs of sites which are in simultaneous contact The number of such two-molecule pair contacts. Me. scales with N as [23] ... 

It is shown, that the fractal geometry of chain part between its fixation points to a great extent is determined by the carbon plastics deformability. This is trae also in relation to the density of cluster network of physical entanglements, which is also influenced on the value of fractal dimension of the mentioned chain part. The carbon plastics plasticity is controlled by state of polymeric matrix stracture characterized by its fractal dimension. 

Here (Po ) is an approximation for the average density (p) of the sorptive layer, cp. Fig. 1.20, a quantity which at least for technical porous sorbents with a complicated, fractal-like surface structure cannot be measured up to date, cp. Figure 1.21, [1.3]. Actually, there are two possibilities so far to get physically reasonable numerical values for (Po) ... 

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