Fractal Dimension, DP

The fractal dimension was introduced earlier in section 2.1.1. If the minimum number of d-dimensional boxes of side e needed to eover the attractor A, N e), scales as 

Since N e) w Dp log(l/e). Dp es.sentially tells us how much information is needed to specify the location of A to within a specified accuracy, e. In practice, one obtains values of N e) for a variety of e s and estimates Dp from the slope of a plot of ln(7V (e)) versus ln(l/e). 

Notice that while Dp clearly depends on the metric properties of the space in which the attractor, A, is embedded - and thus provides some structural information about M - it does not take into account any structural iidiomogeneities in the A. In particular, since the box bookkeeping only keeps track of whether or not an overlap exists between a given box and A, the individual frequencies with which each box is visited are ignored. This oversite is corrected for by the so-called information dimension, which depends on the probability measure on A. 

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Measure Entropy In the same way as the information dimension, Dp generalizes the fractal dimension. Dp, of an attractor. 4, by taking into account the relative frequency with which the individual e-boxes of a partition are visited by points on the attractor, so too the finite set entropy generalizes to a finite measure entropy,... 

Table 6. Relationship between the fractal dimension dp the exponent for the molar mass dependence of the second virial coefficient and the expected exponent m for the osmotic modulus when the scaling assumptions of Eqs. (93)-(96) are made. The experimental data were derived from the exponents for the second virial coefficient...

The relationship between the exponent v, (v = lnp/lnfc), and the fractal dimension Dp of the excitation transfer paths may be derived from the proportionality and scaling relations by assuming that the fractal is isotropic and has spherical symmetry. The number of pores that are located along a segment of length Lj on the jth step of the self-similarity is / , — pi. The total number of pores in the cluster is S nj (pJf, where d is the Euclidean dimension... 

The Values of KWW Exponent v, Fractal Dimension Dp, Porosity 4>m Obtained from the Relative Mass Decrement (A, B, C, and D Glasses) and BETA (E, F, and G Glasses) Measurements and Average Porosity (4>p) Estimated from Dielectric Spectra for Porous Glasses Samples. 

Note that the fractal dimensions discussed here are the fractal dimensions of the excitation transfer paths connecting the hydration centers located on the inner surface of the pores. Due to the low humidity, all of the water molecules absorbed by the materials are bound to these centers. The paths of the excitation transfer span along the fractal pore surface and depict the backbone of clusters formed by the pores on a scale that is larger than the characteristic distance between the hydration centers on the pore surface. Thus the fractal dimension of the paths Dp approximates the real surface fractal dimension in the considered scale interval. For random porous structures, Dp can be also associated with the fractal dimension D, of the porous space Dp = Dr. Therefore, the fractal dimension Dp can be used for porosity calculations in the framework of the fractal models of the porosity. 

Thus, the non-Debye dielectric behavior in silica glasses and PS is similar. These systems exhibit an intermediate temperature percolation process associated with the transfer of the electric excitations through the random structures of fractal paths. It was shown that at the mesoscale range the fractal dimension of the complex material morphology (Dr for porous glasses and porous silicon) coincides with the fractal dimension Dp of the path structure. This value can be obtained by fitting the experimental DCF to the stretched-exponential relaxation law (64). 

A relatively new method to determine the fractal dimension is by using oil permeability measurements. In this method, the permeability coefficient, B, is measured for fat samples containing different SFC. This physical fractal dimension, the permeability fractal dimension. Dp, links the volumetric flow rate of liquid oil penetrating a colloidal fat crystal network with its SFC as (Bremer et al. 1989) ... 

Bremer et al. (1989) described a detailed experimental method used to measure the permeability fractal dimension of a fat crystal network. Similarly to the rheology fractal dimension, the permeability fractal dimension, Dp, could be obtained from the nonlinear regression between Q and O as shown in Figure 17.23 (Tang and Marangoni 2005). 

In the above equations, dp values should lie between 2 and 3, corresponding to a planar and a volumic electrode surface, respectively. Following Andrieux and Audebert (2001), both CA and CV experiments would be suitable in determining the fractal dimension dp of a volumic modified electrode. In the first case (CA), plots of In i vs. In t from CA data should provide straight lines of slope -a [= (dp - l)/2]. In CV, successive experiments using different potential scan rates, v, have to be performed so that plots of log/ vs. logv should yield straight lines of slope a. 

The stretched exponent u depends essentially on the temperature (Fig. 26). At 14°C, u has a value of 0.5. However, when the temperature approaches the percolation threshold = 27°C, u reaches its maximum value of 0.8, with an error margin of less than 0.1. Such rapid decay of the KWW function at the percolation threshold reflects the increase of the cooperative effect of the relaxation in the system. At temperatures above the T, the value of the stretched exponent v decreases, and indicates that the relaxation slows down in the interval 28°-34°C. At temperatures above 34°C, the increase in u with the rise in temperature suggests that the system undergoes a structural modification. Such a change implies a transformation from anZ.2 phase to lamellar or bicontinuous phases (132, 133). On the other hand, the temperature behavior of the fractal dimension Dp (Fig. 26) shows that below the percolation... 

As it has been shown above, polymers macromolecular coils in solution are fractal objects, i.e., self-similar objects, having dimension, which differs from their topological dimension. The coil fractal dimension Dp characterizing its structure (a coil elements distribution in space), can be determined according to the Eq. (4). The exponent ax values for polyarylate Ph-2 solutions in three solvents (tetrachloroethane, tetrahydrofuran and 1,4-dioxane) are adduced in [36]. The values ar] for the same polyarylate are also given in paper [37]. This allows to use the Eq. (4) for the macromolecular coil of Ph-2 Devalue estimation in the indicated solvents. The estimations showed D variation from 1.55 in tetrachloroethane (good solvent for Ph-2) up to 1.78 in chloroform. As it is known [38],... 

TABLE 1 Comparison of the macromolecular coil fractal dimensions Dp obtained by different methods, for polyarylate Ph-2, synthesized in different solvents [33],... 

In Fig. 62 the dependence of glass transition temperature on fractal dimension Dp calculated according to the Eq. (22), is adduced for the five considered copolymers PAASO. As it follows from the adduced plot, linear decay at increasing is observed, that perfectly justifies the proposed by Kargin idea about polymers properties encodion on molecular level [170], Let us remind, that the value Df characterizes stracture of macromolecular coil in diluted solution, i.e., as a matter of fact, an isolated macromolecule. A solvent variation results to change, that explains polymers properties change at their samples production from solutions in different solvents [171]. 

Hence, the proposed technique allows to estimate variation of macromolecu-lar coil structure of DMDAACh, characterized by its fractal dimension Dp during the entire polymerization reaction. It has been shown, that exactly this factor, is not taken into account in conventional theories, and defines the most important characteristics of polymerization process rate, conversion degree, molecular weight. Besides, the fractal kinetics methods allow the quantitative description of polymers synthesis, particularly, they give the correct shape of the kinetic curve... 

Therefore, the stated above results have confirmed again that D, values distribution is the main reason of microgels stracture variation, characterized by its fractal dimension Dp Dp change at reaction duration growth is well described quantitatively within the frameworks of aggregation mechanism cluster-cluster. Fractal space, in which curing reaction proceeds, is formed by the stracture of the largest cluster in system [55],... 

For example, a 1-dimensional curve such as a coastline may have one length according to one definition, but may have arbitrarily small irregularities giving rise to a different, larger length by another definition. This is a fractal curve. The fractal dimension Dp is defined by Dp = [(log Lq - log L )/log S] + D where Lq is a constant, L is a measured length,... 

The roughness versus scanned area pattern is characteristic of a given material and defines a fractal dimension, dp., which is evaluated as dp.= 3—a where a is the so-called roughness exponent that can be calculated as the slope of roughness versus scan size in a double log plot [42]. 

P.G. de Gennes This is discussed in the paper by F. Brochard. In conditions of complete thermal equilibrium, the crucial point anraunts to compare the fractal dimension Dp of the polymer (here Dp = 5/3) and the fractal dimension of the absorbing surface D (2 < D5 < 3). Whenever Dp < Ds the polymer should cover all pores (even those which are smaller than the coil size Rp). But, in many practical cases, there is a kinetic barrier = adsorbed polymers do not enter easily in small pores, and the large pores (of size > Rp) are the only accessible pores. 

Adsorption of large objects We study the coating of a fractal with a) latex spheres, b) rigid or flexible macromolecules, c) a layer of preferential adsorption of one species in a binary mixture (near criticality). In all these cases, on a "surface fractal (DJ another fractal object of fractal dimension Dp and size R was adsorbed. For macromolecules. Dp ranges from 3 for globular proteins to unity for rods. 

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