Fractal dimension ideal linear

A useful (also extreme) counterpart to the also idealized linear geometry is fractal geometry which plays a key role in many non-linear processes.280 281 If one measures the length of a fractal interface with different scales, it can be seen that it increases with decreasing scale since more and more details are included. The number which counts how often the scale e is to be applied to measure the fractal object, is not inversely proportional to ebut to a power law function of e with the exponent d being characteristic for the self-similarity of the structure d is called the Hausdorff-dimension. Diffusion limited aggregation is a process that typically leads to fractal structures.283 That this is a nonlinear process follows from the complete neglect of the back-reaction. The impedance of the tree-like metal in Fig. 76 synthesized by electrolysis does not only look like a fractal, it also shows the impedance behavior expected for a fractal electrode.284... 

This description is a manifestation of the self-similarity (fractal nature) of polymers, discussed in Section 1.4. The fractal nature of ideal chains leads to the power law dependence of the pair correlation function g(r) on distance r. This treatment for the ideal chain can be easily generalized to a linear chain with any fractal dimension V. The number of monomers... 

The quality of solvent, reflected in the excluded volume v, enters only in the prefactor, but does not change the value of the scaling exponent u for any v > 0. The Flory approximation of the scaling exponent isu = 3/5 for a swollen linear polymer. For the ideal linear chain the exponent = 1/2. In the language of fractal objects, the fractal dimension of an ideal polymer is V — l/i/ = 2, while for a swollen chain it is lower T> — I/u = 5/3. More sophisticated theories lead to a more accurate estimate of the scaling exponent of the swollen linear chain in three dimensions ... 

The reciprocal of the fractal dimension of the polymer (see Section 1.4) is p. For an ideal linear chain p= j2 and the fractal dimension is l/i = 2. The Rouse time of such a fractal chain can be written as the product of... 

In the evolution of solids from solution, a wide spectrum of structures can be formed. In Fig. 4, a simple schematic representation of the structural boundary condition for gel formation is presented. At one extreme of the conditions, linear or nearly linear polymeric networks are formed. For these systems, the functionality of polymerization /, is nearly 2. This means there is little branching or cross-linking. The degree of cross-linking is nearly 0. In silica, gels of this type can be readily formed by catalysis with HCl or HNO3 under conditions of low water content (less than 4 mol water to 1 mol silicon alkoxide). The ideal fractal dimension for such a linear chain structure is 1. The phe-... 

The fractal dimension of purely statistical models, i.e., models without the effect of excluded volume, can be determined accurately [see Equation (11.9a)]. For linear polymers, this model corresponds to phantom random-walk. In the case of branched statistical fractals, the corresponding model is a statistical branched cluster, whose branching obeys the random-walk statistics. Since the root-mean-square distance between the random-walk ends is proportional to the number of walk steps N, then D = 2 irrespective of the space dimension. These types of structures have been studied [61, 75-77]. The value D = 4 irrespective of d was obtained for a branched fractal. Unlike ideal statistical models, models with excluded volume, i.e., those involving correlations, cannot be accurately solved in the general case. The Df values for these systems are usually found either using numerical methods such as the Monte Carlo method or taking into account the spatial position of a renormalisation group. 

The use of fractal analysis makes it possible to relate molecular parameters to characteristics of supermolecular structure of polymers. Figure 11.12 illustrates the linear correlation between D and df [dj was estimated from Equation (11.27)] for epoxy polymers. When the molecular mobility is suppressed (D = 1), the structure of the polymer has the fractal dimension df = 2.5, which corresponds to p. = 0.25. The given value of the Poisson coefficient corresponds to the boundary of ideally brittle structure at p< 0.25, the polymer is collapsed without viscoelastic or plastic dissipation of energy [3]. This is fnlly consistent with the Kansch conclnsion [117] stating that any increase in the molecular mobility enhances dissipation of the mechanical energy supplied from the outside and, as a conseqnence, increases plasticity of the polymer. When D = 2 the df value is equal to 3, which corresponds to p = 0.5, typical of the rubbery state. 

Linear polymer macromolecules are known to occur in various conformational and/or phase states, depending on their molecular weight, the quality of the solvent, temperature, concentration, and other factors [1]. The most trivial of these states are a random coil in an ideal (0) solvent, an impermeable coil in a good solvent, and a permeable coil. In each of these states, a macromolecular coil in solution is a fractal, i.e., a self-similar object described by the so-called fractal (Hausdorff) dimension D, which is generally unequal to its topological dimension df. The fractal dimension D of a macromolecular coil characterises the spatial distribution of its constituent elements [2]. 

This is now the fractal dimension of the ideal polymeric fractal containing Gaussian chains. Hence for d = I the results for linear chains are correctly reproduced as they must be. 

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