Percolation theory conductivity

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 theory [5, 20-23] is the most adequate for the description of an abstract model of the CPCM. As the majority of polymers are typical insulators, the probability of transfer of current carriers between two conductive points isolated from each other by an interlayer of the polymer decreases exponentially with the growth of gap lg (the tunnel effect) and is other than zero only for lg < 100 A. For this reason, the transfer of current through macroscopic (compared to the sample size) distances can be effected via the contacting-particles chains. Calculation of the probability of the formation of such chains is the subject of the percolation theory. It should be noted that the concept of contact is not just for the particles in direct contact with each other but, apparently, implies convergence of the particles to distances at which the probability of transfer of current carriers between them becomes other than zero. 

The main notion of the percolation theory is the so-called percolation threshold Cp — minimal concentration of conducting particles C at which a continuous conducting chain of macroscopic length appears in the system. To determine this magnitude the Monte-Carlo method or the calculation of expansion coefficients of Cp by powers of C is used for different lattices in the knots of which the conducting par-... 

Experimental dependences of conductivity cr of the CPCM on conducting filler concentration have, as a rule, the form predicted by the percolation theory (Fig. 2, [24]). With small values of C, a of the composite is close to the conductivity of a pure polymer. In the threshold concentration region when a macroscopic conducting chain appears for the first time, the conductivity of a composite material (CM) drastically rises (resistivity Qv drops sharply) and then slowly increases practically according to the linear law due to an increase in the number of conducting chains. 

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

The common disadvantage of both the free volume and configuration entropy models is their quasi-thermodynamic approach. The ion transport is better described on a microscopic level in terms of ion size, charge, and interactions with other ions and the host matrix. This makes a basis of the percolation theory, which describes formally the ion conductor as a random mixture of conductive islands (concentration c) interconnected by an essentially non-conductive matrix. (The mentioned formalism is applicable not only for ion conductors, but also for any insulator/conductor mixtures.)... 

The main conclusion of the percolation theory is that there exists a critical concentration of the conductive fraction (percolation threshold, c0), below which the ion (charge) transport is very difficult because of a lack of pathways between conductive islands. Above and near the threshold, the conductivity can be expressed as ... 

Diffusion of cations in a Nation membrane can formally be treated as in other polymers swollen with an electrolyte solution (Eq. (2.6.21). Particularly illustrative here is the percolation theory, since the conductive sites can easily be identified with the electrolyte clusters, dispersed in the non-conductive environment of hydrophobic fluorocarbon chains (cf. Eq. (2.6.20)). The experimental diffusion coefficients of cations in a Nation membrane are typically 2-4 orders of magnitude lower than in aqueous solution. 

Where a is the composite conductivity, a0 a proportionally coefficient, Vfc the percolation threshold and t an exponent that depends on the dimensionality of the system. For high aspect ratio nanofillers the percolation threshold is several orders of magnitude lower than for traditional fillers such as carbon black, and is in fact often lower than predictions using statistical percolation theory, this anomaly being usually attributed to flocculation [24] (Fig. 8.3). 

In some of the metal-insulator transitions discussed here the use of classical percolation theory has been used to describe the results. This will be valid if the carrier cannot tunnel through the potential barriers produced by the random internal field. This may be so for very heavy particles, such as dielectric or spin polarons. A review of percolation theory is given by Kirkpatrick (1973). One expects a conductivity behaving like... 

Various authors (Fuchs 1965, Lightsey 1973, Webman et al 1976) have discussed the conductivity in terms of classical percolation theory, but we think this is unlikely to be correct for particles for which there is no evidence for mass enhancement. 

In all of these cases, it does seem that the law that you do not get something for nothing applies. The conductivity decreases as would be expected from percolation theory, if the conducting polymer is dispersed, and according to the volume fraction if it forms a network. The mechanical properties improve as one might expect from an additivity rule applied to the two components. 

Percolation is widely observed in chemical systems. It is a process that can describe how small, branched molecules react to form polymers, ultimately leading to an extensive network connected by chemical bonds. Other applications of percolation theory include conductivity, diffusivity, and the critical behavior of sols and gels. In biological systems, the role of the connectivity of different elements is of great importance. Examples include self-assembly of tobacco mosaic virus, actin filaments, and flagella, lymphocyte patch and cap formation, precipitation and agglutination phenomena, and immune system function. 

The conductivity of the alloys as a function of nanotube concentration is shown in Fig. 5.9. P30T is an insulator with a conductivity 10-8 S/m. The conductivity increases monotonically with nanotube concentration. However the rate of increase is relatively small at low concentrations. It increases dramatically above 12% concentration. Above this concentration the nanotubes are close enough so that percolation becomes possible. The fit of the percolation theory is shown by the solid line. 

The electrical percolation behavior for a series of carbon black filled rubbers is depicted in Fig. 26 and Fig. 27. The inserted solid lines are least square fits to the predicted critical behavior of percolation theory, where only the filled symbols are considered that are assumed to lie above the percolation threshold. According to percolation theory, the d.c.-conductivity Odc increases with the net concentration 0-0c of carbon black according to a power law [6,128] ... 

For a quantitative analysis of the scaling and cross-over behavior of the a.c.-conductivity above the percolation threshold we refer to the predictions of percolation theory [128, 136, 137] ... 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

If the estimated fitting parameters are compared to the predicted values of percolation theory, one finds that all three exponents are much larger than expected. The value of the conductivity exponent ji=7A is in line with the data obtained in Sect. 3.3.2, confirming the non-universal percolation behavior of the conductivity of carbon black filled rubber composites. However, the values of the critical exponents q=m= 10.1 also seem to be influenced by the same mechanism, i.e., the superimposed kinetic aggregation process considered above (Eq. 16). This is not surprising, since both characteristic time scales of the system depend on the diffusion of the charge carriers characterized by the conductivity. 

To illustrate, solution of problems of this kind in the percolation theory, Quemada [71] cites the analysis of the dependence of electrical conductivity on concentration for a mixture of conducting and nonconducting spherical particles carried out by Fitzpatrick [80] or Clerk [81]. According to Clerk, this dependence is described by an equation similar to Eq. (66) or other similar formulas ... 

The all or nothing feature of metal powder composites is very much a feature of conductive composite systems. In order to understand this behaviour, most theories borrow from percolation theory (Broadbent and Hamersley, 1957), which was originally developed as a model for predicting fluid permeation through porous media. The percolation model is based on having a medium... 

Armand (1994) has briefly summarised the history of polymer electrolytes. A more extensive account can be found in Gray (1991). Wakihara and Yamamoto (1998) describe the development of lithium ion batteries. Sahimi (1994) discusses applications of percolation theory. Early work on conductive composites has been covered by Norman (1970). Subsequent edited volumes by Sichel (1982) and Bhattacharya (1986) deal with carbon- and metal-filled materials respectively. Donnet et al. (1993) cover the science and technology of carbon blacks including their use in composites. GuF (1996) presents a detailed account of conductive polymer composites up to the mid-1990s. Borsenberger and Weiss (1998) discuss semiconductive polymers with non-conjugated backbones in the context of xerography. Bassler (1983) reviews transport in these materials. 

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