Fractal network

In the case of fat crystal networks, fractal scaling is found between the size of a primary colloidal particle and a floe of primary particles (Marangoni 2005). 

Multifractal analysis provides a complete description of fault-network fractal properties. It could more veritably describe the complexity and essentiality of fault structure compared to single fractal analysis. The multifractal theory has been widely applied in various fields of earth science. More details about the properties of multifractal spectra in earth sciences applications can be found in References (Agterberg et al. 1996, Zhao et al. 2011, Kiyashchenko et al. 2004). Fault tectonic has multifractal structure characteristics (Panahi and Cheng 2004). But it was not found on the coalfield fault tectonics. 

Meakin P 1983 Formation of fractal clusters and networks by irreversible diffusion-limited aggregation Phys. Rev. Lett. 51 1119-22... 

If the dispersion particles are attracted to each other, they tend to flocculate and form a stmcture. At low concentrations the particles form open aggregates, which give a fractal stmcture (93,94). At higher concentrations a network stmcture results, which can be so pronounced that the mixture has a yield point and behaves like a soHd when at rest. Shearing breaks up this stmcture, and viscosity decreases. 

Polycondensation reactions (eqs. 3 and 4), continue to occur within the gel network as long as neighboring silanols are close enough to react. This increases the connectivity of the network and its fractal dimension. Syneresis is the spontaneous shrinkage of the gel and resulting expulsion of Hquid from the pores. Coarsening is the irreversible decrease in surface area through dissolution and reprecipitation processes. 

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

In the case of amphiphilic molecules, characterized by the coexistence of spatially separated apolar (alkyl chains) and polar moieties, both parts cooperate to drive the intermolecular aggregation. This simple but pivotal peculiarity makes amphiphilic molecules soluble in both polar and apolar solvents and able to realize, in suitable conditions, an impressive variety of molecular aggregates characterized by spatially separated apolar and polar domains, local order at short times and fluidity at long times, and differences in size, shape (linear or branched chains, cyclic or globular aggregates, extended fractal-like molecular networks), and lifetime. 

Johansson and coworkers [182-184] have analyzed polyacrylamide gel structure via several different approaches. They developed an analytical model of the gel structure using a single cylindrical unit cell coupled with a distribution of unit cells. They considered the distribution of unit cells to be of several types, including (1) Ogston distribution, (2) Gaussian distribution of chains, and (3) a fractal network of pores [182-184]. They [183] used the equilibrium partition coefficient... 

Fig. 9. Relation between relaxation exponent n and fractal dimension d for a three-dimensional network. In case of complete screening of excluded volume, values of 0 < n < 1 are possible if d is chosen between 1.25 and 2.5...

Optical Storage and Retrieval Memory, Neural Networks, and Fractals, edited by Francis T. S. Yu and Suganda Jutamulia... 

It was shown, that the conception of reactive medium heterogeneity is connected with free volume representations, that it was to be expected for diffusion-controlled sohd phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

The term S represents the strength of the network. The power law exponent m was found to depend on the stochiometric ratio r of crosslinker to sites. When they were in balance, i.e. r = 1, then m - 1/2. From Equations (5.140) and (5.141) this is the only condition where G (co) = G (cd) over all frequencies where the power law equation applies. If the stochiometry was varied the gel point was frequency dependent. This was also found to be the case for poly(urethane) networks. A microstructural origin has been suggested by both Cates and Muthumkumar38 in terms of a fractal cluster with dimension D (Section 6.3.5). The complex viscosity was found to depend as ... 

Lenormand R, Zarcone C (1985) Invasion percolation in an etched network Measurement of fractal dimension. Phys Rev Lett 54 2226-2229... 

Parameter that provides a mathematical description of the fractal structure of a polymer network, an aggregated particulate sol, or of the particles that comprise them. 

Fig. 9.4.21 Electron micrograph of Sb nanoparticles produced by a gas flow-solution trap method. Note that many particles are coalesced with each other, forming a fractal-like network structure.

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