Fractal density

In Fig. 2, the relative fractal density is plotted as a function of increasing radius for selected values of fractal dimension. A fractal dimension of 1.0 represents linear chain growth, whereas a fractal dimension of 3.0 represents three-dimensional nonfractal growth. True fractal growth is bounded by these two extremes in fractal dimension. As can be seen in Fig. 2, the decrease in density can be severe. For example, a fractal growing with a dimension of 2.0 has a relative density of 0.25 its core density at only 4 times its core radius. Fractal dimensions of approximately 2.0 have been observed for silica gels evolved under both mildly acid and mildly basic conditions [5,12]. 

As a result of the rapid decrease in density as a function of increasing fractal radius, the fractal density may become so low as to be non-self-supporting. As a consequence, there is be a limiting, or critical, fractal radius, r beyond which particle growth is not favored or greatly retarded. 

Correspondingly, there is a critical fractal density F that defines this radius, such that... 

L. A. Kuhn, M. A. Siani, M. E. Pique, C. L, Fisher, E. D, Getzoff, and J. A. Tainer, /. Mol. Biol., 228, 13 (1992). The Interdependence of Protein Surface Topography and Bound Water Molecules Revealed by Surface Accessibility and Fractal Density Measures. 

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. 

One characterizes a fractal cluster by its fractal dimension Df which can be determined from the integration of the radius-dependent mass density p r) from the center up to a large radius R as follows ... 

A normal diffusion process, however, runs at a finite concentration of particles different from zero. In this situation it was found [101] that a fractal character (73) of the resulting structure is restricted to an interval a < R < if), where d is the diffusion length (67). Larger clusters have a constant density on a length scale larger than They are no longer fractal there. These observations have various consequences for crystal growth, and will be discussed in the next section. 

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... 

Lahey (1990) indicated the applications of fractal and chaos theory in the field of two-phase flow and heat transfer, especially during density wave oscillations in boiling flow. 

Since diffusing species move randomly in all directions, the diffusing species may sense the self-affine fractal surface and the self-similar fractal surface in quite different ways. Nevertheless a little attention has been paid to diffusion towards self-affine fractal electrodes. Only a few researchers have realized this problem Borosy et al.148 reported that diffusion towards self-affine fractal surface leads to the conventional Cottrell relation rather than the generalized Cottrell relation, and Kant149,150 discussed the anomalous current transient behavior of the self-affine fractal surface in terms of power spectral density of the surface. 

In their works,51"54 the self-similar fractal dimension dF>ss of the two-dimensional distribution of the pits was determined by the analysis of the digitized SEM images using the perimeter-area method. The value of dF>ss increased with increasing solution temperature,51 and it was inversely proportional to the pit shape parameter and the pit growth rate parameter.53 Keeping in mind that dr>ss is inversely proportional to the increment of the pit area density, these results can be accounted for in terms of the fact that the increment of the pit area density is more decelerated with rising solution temperature. 

It is known [3], that macromolecular coil in various polymer s states (solution, melt, solid phase) represents fractal object characterized by fractal (Hausdorff) dimension Df. Specific feature of fractal objects is distribution of their mass in the space the density p of such object changes at its radius R variation as follows [4] ... 

The clusters which obey Eq. (61) are self similar to each other. Sometimes, however, the curve flattens at large molar masses and may form another straight line with a different exponent. Such behavior is an indication of a limitation in the separation capability of the column (or some other artifacts) or it is the result of large particles with a different fractal behavior. These particles can be aggregates or clusters of a higher branching density. Similar behavior can be observed also from the molar mass dependence of the viscosity. An example will be shown in the next section. 

---