Percolation geometrical

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

The percolation argument is based on the idea that with an increasing Cr content an insoluble interlinked cliromium oxide network can fonn which is also protective by embedding the otherwise soluble iron oxide species. As the tlireshold composition for a high stability of the oxide film is strongly influenced by solution chemistry and is different for different dissolution reactions [73], a comprehensive model, however, cannot be based solely on geometrical considerations but has in addition to consider the dissolution chemistry in a concrete way. 

Geometrical Analysis of the Structure of Simple Liquids Percolation Approach. 

A characteristic feature of the carbon modifications obtained by the method developed by us is their fractal structure (Fig. 1), which manifests itself by various geometric forms. In the electrochemical cell used by us, the initiation of the benzene dehydrogenation and polycondensation process is associated with the occurrence of short local discharges at the metal electrode surface. Further development of the chain process may take place spontaneously or accompanied with individual discharges of different duration and intensity, or in arc breakdown mode. The conduction channels that appear in the dielectric medium may be due to the formation of various percolation carbon clusters. 

The third relaxation process is located in the low-frequency region and the temperature interval 50°C to 100°C. The amplitude of this process essentially decreases when the frequency increases, and the maximum of the dielectric permittivity versus temperature has almost no temperature dependence (Fig 15). Finally, the low-frequency ac-conductivity ct demonstrates an S-shape dependency with increasing temperature (Fig. 16), which is typical of percolation [2,143,154]. Note in this regard that at the lowest-frequency limit of the covered frequency band the ac-conductivity can be associated with dc-conductivity cio usually measured at a fixed frequency by traditional conductometry. The dielectric relaxation process here is due to percolation of the apparent dipole moment excitation within the developed fractal structure of the connected pores [153,154,156]. This excitation is associated with the selfdiffusion of the charge carriers in the porous net. Note that as distinct from dynamic percolation in ionic microemulsions, the percolation in porous glasses appears via the transport of the excitation through the geometrical static fractal structure of the porous medium. 

For the catalyst particle modeled by the network this ratio corresponds to the ratio of the geometric area to overall surface area, which for most catalysts is essentially zero as is also the case for three-dimensional networks. Accounting for the activity of the surface pores in Bethe networks tends to smooth out all the abrupt changes in the activity that would be otherwise observed at the percolation threshold. 

The growth of the conductivity or elasticity of such networks near their respective percolation threshold points can be expressed as powers (known as exponents) of the interval (of random concentration of the conducting or elastic material) from the percolation threshold. These powers or the exponents are observed to be universal, in the sense that they do not depend on many details of the problem or of the lattice, but depend on only some subtle geometric features of the problem e.g., the exponents often depend only on the lattice dimensionality. 

Percolation describes the geometrical transition between disconnected and connected phases as the concentration of bonds in a lattice increases. It is the foundation for the physical properties of many disordered systems and has been applied to gelation phenomena (de Gennes, 1979 Stauffer et al., 1982). At just above gelation threshold, denoting the fraction of reacted bonds as p and p=Pc + A/ , pc the critical concentration (infinite cluster), the scaling laws (critical exponents) for gel fraction (5oo) and modulus E) are ... 

The electrical conductivity of hard-sphere-like microemulsions increases smoothly as the volume fraction < > is increased. On the contrary, the conductivity of microemulsions with attractive interaction between droplets increases steeply around pM).08-0.14 (Figure 3). The behavior of the conductivity may be accounted for by percolation theories and < >p is identified to the percolation threshold. However, in such systems one must distinguish between geometrical percolation and conductivity percolation. 

Usually geometrical connectivity and concentration fluctuations are not related. However, in our case, the electrical percolation is not a simple geometrical connectivity. This fact can explain that electrical percolation and critical points seem to be associated and that (jp % [Pg.80]

In the Scala-Shklovsky model [64] it was assumed that the structure of an infinite cluster is a net with the characteristic geometric distance between knots being the percolation length... 

In this range, the connecting set is a fractal that is, it is geometrically similar to a percolating cluster, and its properties depend on the linear scale. Therefore, both the correlation length and the P s of the connecting set (the upper index oo means that the limit / —> oo is taken) should scale with distance from the critical point (i.e., percolation threshold pc = p ) as... 

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