Fractal dimension experimental determination

Figure 5.29 The dependences of structure fractal dimension, experimentally determined d (1, 2) and calculated according to Equation 5.42 (3, 4) on the value of for epoxy polymers EP-1 (1, 3) and EP-2 (3, 4) [100]...

The first method of dimension experimental determination uses the following fractal relationship [40, 41] ... 

With the experimental results about the wetting ability and the fractal dimension of four kinds of anode electrodes, we could conclude the following. The addition of NisAl could make the electrolyte wet the electrode very well. The pore structures of all the electrodes prepared in this study were highly irregular and rough. Finally, the chemical properties of the surfaces were as important as the physical properties in determining the wetting ability of the electrodes in this study. 

Bearing in mind that diffusing ions move randomly in all directions, it is reasonable to say that the diffusing ions sense selfsimilar scaling property of the electrode surface irrespective of whether the fractal surface has self-similar scaling property or self-affine scaling property. Therefore, it is experimentally justified that the fractal dimension of the self-affine fractal surface determined by using the diffusion-limited electrochemical technique represents the apparent self-similar fractal dimension.43... 

Figure 2.14. Employing the log of experimentally determined dendritic diameters the fractal dimension D — 2.29 was found for a series of acid terminated dendrimers. [See the table presented in Figure 2.8]...

Techniques have been developed to determine the fractal dimension of experimental aggregate particles in solution using small-angle scattering techniques, from x-ray, light, or even neutron sources. In these techniques the scattering intensity, I q), is proportional to the scattering vector, q, raised to the mass fractal dimension by ... 

The inter-particle force F can be computed as / = A]idol l2Hl), where Ah is the Hamaker parameter for the liquid-particle system and is the distance between two primary particles. The coordination number is based on experimental observation and can be calculated as kc 150p, where 0p is the volume fraction of solid within the aggregates. In the case of compact (or solid) particles 0p is close to unity, whereas in the case of fractal aggregates 0p can be determined once the fractal dimension T)f of the aggregates is known 0p = (0.414T)f - 0.21 l)(r/p/(io) , where dp is the size of the particle and do is the size of the primary particle (Vanni, 2000b). 

We recommend that future research in this area focus on establishing theoretical and/or empirical relations between the prefactor and exponent in the power law relating K to Pn, and quantitative pore characteristics, such as the mass fractal dimension and Betti numbers. In particular, there is an urgent need for additional experimental studies, in which K is determined over a wide range of Pn on undisturbed samples of natural porous media that are well-characterized in terms of their pore space geometry. Such data are required for model testing. 

It has been suggested [65, 66] that the exponent in the equation for p(co) is determined by the fractal dimension. However, Alexander and Orbach [28] showed that in this case, the spectral dimension is involved, which is not equal to df [see Equation (11.8) and Table 11.1]. Note that the publication by Stapleton and co-workers [65], appeared earlier than the study by Alexander and Orbach [28]. In the case of self-avoiding random walk, dj = 1 substitution of this value into the equation for p((o) gives a result which, as shown by Young [67], is at variance with experimental results. 

At present there are several methods of filler structure (distribution) determination in the polymer matrix, both experimental and theoretical. All the indicated methods describe this distribution by a fractal dimension... 

The change of the fractal dimension D with temperature reflects the corresponding changes in sizes, degree of compactness and asymmetiy of shape of a macromolecular coil in solution [89]. The importance of the temperature dependence of study is determined by strong influence of this parameter on the processes of synthesis [28], catalysis [90], flocculation [91], so forth. At present as far as we know experimental evaluations of the temperature dependence of a macromolecular coil are absent. Theoretical estimations [13] suppose that the temperature enhancement makes the fractal less compact, that is, leads to Devalue reduction. Therefore, the authors [92] performed the experimental study of dependence on temperature for the macromolecular coils of polyarylate F-1 [5] in diluted solutions and evaluation of change influence on s nithesis processes. 

The necessary for calculations experimental data (solubility parameters of polymers 6 and solvents Flory-Huggins interaction parameters Xi, the exponents in Mark-Kuhn-Houwink equation) were accepted according to the data of chapters [1, 14, 16,29, 32]. The experimental values of fractal dimension of a macromoleeular coil in diluted solution were determined according to the equation (4). 

The values a for seven polyarylates of different chemical structure, obtained by high-temperature (equilibrium) and interfacial polycondensation, determined in three solvents (sirmn-tetrachloroethane, tetrahydrofuran and 1,4-dioxane) are accepted according to the data of work [53]. The fractal dimension Dj. experimental values (o ) in the indicated solvents were determined according to the Eq. (4). The values of solubility parameter s for these solvents are taken from hterary sources [25, 36, 56]. The fractal dimension 5 of solvent molecules stracture was determined according to the equation [71] ... 

At present there are several methods of filler structure (distribution) determination in polymer matrix, both experimental [10, 35] and theoretical [4]. All the indicated methods describe this distribution by fractal dimension of filler particles network. However, correct determination of any object fractal (Hausdorff) dimension includes three obligatory conditions. The first from them is the indicated above determination of fiiactal dimension numerical magnitude, which should not be equal to object topological dimension. As it is known [36], any real (physical) fractal possesses fiiactal properties within a certain scales range. Therefore, the second condition is the evidence of object self-similarity in this scales range [37]. And at last, the third condition is the correct choice of measurement scales range itself As it has been shown in Refs. [38, 39], the minimum range should exceed at any rate one self-similarity iteration. 

Mathematical or nonrandom fractals are scale invariant, i.e. the pattern is the same at all scales (self-similar). Natural, real or random fractals are quasi or statistically self-similar over a finite length scale that is most often determined by the characterization technique that is employed. An object or process can be classified as fractal when the length scale of the property being measured covers at least one order of magnitude. Fractal structures obey a power law, allowing the fractal dimension D to be determined from experimental data ... 

These kinds of experiments are without exception carried out in a column of fluid, usually of the same composition as that from which the aggregates were sampled. The aggregates are introduced into the top of the column and one or more microscopes are used to measure the settling velocity. Farrow and Warren describe a floe density analyser [69] which may be used to determine the fractal dimension. Nobbs et al. [70] review many of the practical aspects involved in performing this type of experimental investigation. 

For X 0, Equations (6.17) and (6.24) coincide and are independent of the surface fractal dimension. However, in contrast to Equation (6.17), the term x resulting from Equation (6.24) is independent of the fractal dimension. The region x < 0.15 is of interest for the determination of the monolayer capacity. Thus, for any value of A, monolayer capacities An can be determined from the standard BET method as long as the experimental measurements are restricted to the range of low relative pressures. 

The dependence of the Equations (6.17) and (6.24) and Equations (6.22) and (6.25) on the surface fractal dimension A offers a method for measuring A from a single experimental isotherm. This may be done by fitting Equations (6.22) or (6.25) to the data obtained at relative pressures close to unity. In such a case, it is necessary to determine how large x should be in order for the asymptotic Equations (6.17) and (6.25) to be valid. A simple estimate proposed by Pfeifer et al. [3] follows from the requirements that N/N > 2, since the adsorption energy of molecules adsorbed in the second and higher layers is independent of the energy of interaction with the solid substrate. This estimate yields x > 1 — as a minimal condition for the... 

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