Statistics percolation

Due to their high aspect ratio, nanocarbons dispersed in a polymer matrix can form a percolating conductive network at very low volume fractions (< 0.1 %). The conductivity of a composite above the transition from an insulator can be described by the statistical percolation using an excluded volume model [22,23] to yield the following expression ... 

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). 

Statistical fractals are generated by disordered (random) processes. An element of disorder is typical of most real physical phenomena and objects. The fact that disorder, i.e., the absence of any spatial correlation, is a sufficient condition for the formation of fractals was first noted by Mandelbrot [1]. A typical example of this type of fractal is the random-walk path. However, real physical systems are often inadequately described by purely statistical models. Among other reasons, this is due to the effect of excluded volume. The essence of this effect lies in the geometric restriction that forbids two different elements of a system to occupy the same volume in space. This restriction is to be taken into account in the corresponding modelling [10, 11]. The best-known examples of this type of models are self-avoiding random walk, lattice animals and statistical percolation. 

Many models have been proposed (117) to explain the electrical conductivity of mixtures composed of conductive and insulating materials. Percolation concentration is the most interesting of all of these models. Several parameters, such as filler distribution, filler shape, filler/matrix interactions, and processing technique, can infiuence the percolation concentration. Among these models, the statistical percolation model (118) uses finite regular arrays of points and bonds (between the points) to estimate percolation concentration. The thermodynamic model (119) emphasizes the importance of interfacial interactions at the boimdary between individual filler particles and the polymeric host in the network formation. The most promising ones are the structure-oriented models, which explain condnctivity on the basis of factors determined from the microlevel stmctin-e of the as-produced mixtures (120). 

Given that all of the CNT-filled polymer blends in this study are prepared with 50 vol% of the CNT-filled PET phase (with 12 vol% CNTs) plus 50 vol% of the second immiscible polymer phase (with no CNTs), it is reasonable to assume that both the CNT-fiUed PET phase and the second immiscible neat polymer phase have formed self-continuous 3D networks in the polymer blends. This expectation is confirmed by the microstructure examination (see Sect. 12.1), which reveals that the area fractions of the CNT-filled region and the CNT-free region are both near 50%. Furthermore, the electrical conductivity data suggest that the carbon nanotubes within the PET phase have also formed a 3D conductive path because the electrical resistivity has been reduced from the neat polymer blends to the CNT-filled polymer blends by about 12 orders of magnitude. With such a triple-continuous structure, the conductive CNT-filled PET network and the non-conductive second polymer phase can be treated as parallel conductors, and the resulting resistivity, p, of the CNT-filled polymer blend can be estimated using the statistical percolation model proposed by Bueche [61] ... 

The percolation thresholds observed for composites prepared with a PS matrix and SDS- and PEDOT PSS-stabilized SWCNTs are shown in Figure 6.4 [Q. The linear fittings oft and cpp (ii) using the statistical percolation law are given in Figure 6.4 (ii). The statistical percolation law is given as ... 

The maximum values of the percolation threshold are characteristic of matrix systems in which the filler does not form the chain-like structures till large concentrations are obtained. In practice, statistical or structurized systems are apparently preferable because they become conductive at considerably smaller concentrations of the filler. The deviation of the percolation threshold from the values of Cp to either side for a statistical system ( 0.15) can be used to judge the nature of filler distribution. 

For the second method the threshold concentration of the filler in a composite material amounts to about 5 volume %, i.e. below the percolation threshold for statistical mixtures. It is bound up with the fact that carbon black particles are capable (in terms of energy) of being used to form conducting chain structures, because of the availability of functional groups on their surfaces. This relatively sparing method of composite material manufacture like film moulding by solvent evaporation facilitates the forming of chain structures. 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

The percolation simulations clearly allow ring formation it also represents self-avoiding statistics, i.e., includes the excluded volume effects in good solvents. Finally, the probability of placing units on lattice sites becomes more and more dependent on whether a site in the neighborhood is already occupied. In other words the percolation experiment becomes a non-mean field approach when the occupation reaches the critical percolation threshold. Therefore, strong deviations were expected between the more accurate percolation and the Flory-Stockmayer mean field approaches. Physicists were of the opinion that the mean field results must be basically wrong. 

In this section, we describe the role of fhe specific membrane environment on proton transport. As we have already seen in previous sections, it is insufficient to consider the membrane as an inert container for water pathways. The membrane conductivity depends on the distribution of water and the coupled dynamics of wafer molecules and protons af multiple scales. In order to rationalize structural effects on proton conductivity, one needs to take into account explicit polymer-water interactions at molecular scale and phenomena at polymer-water interfaces and in wafer-filled pores at mesoscopic scale, as well as the statistical geometry and percolation effects of the phase-segregated random domains of polymer and wafer at the macroscopic scale. 

The structure-related statistical factors include the surface-to-volume atom ratio of Pf nanoparficles, fhe effecfiveness factor of cafalysf ufilizafion of mesoscopic agglomerates, T, and percolation and wetting effecfs af fhe macroscopic level, represented by the functions/(Xpjc,Xei) and g Sf, where S, is the liquid saturation. 

Eikerling et al. ° used a similar approach except that they focus mainly on convective transport. As mentioned above, they use a pore-size distribution for Nafion and percolation phenomena to describe water flow through two different pore types in the membrane. Their model is also more microscopic and statistically rigorous than that of Weber and Newman. Overall, only through combination models can a physically based description of transport in membranes be accomplished that takes into account all of the experimental findings. 

The theoretical model describes the break up of the coal macromolecular network under the influence of bond cleavage and crosslinking reactions using a Monte Carlo statistical approach (32-38). A similar statistical approach for coal decomposition using percolation theory has been presented by Grant et al. (39). 

Grant, D.M., Pugmire, R.J., Fletcher, T.H., and Kerstein, A.R., "A Chemical Model to Coal Deyolatilization Using Percolation Lattic Statistics , submitted for publication. 

The percolation model, which can be applied to any disordered system, is used for an explanation of the charge transfer in semiconductors with various potential barriers [4, 14]. The percolation threshold is realized when the minimum molar concentration of the other phase is sufficient for the creation of an infinite impurity cluster. The classical percolation model deals with the percolation ways and is not concerned with the lifetime of the carriers. In real systems the lifetime defines the charge transfer distance and maximum value of the possible jumps. Dynamic percolation theory deals with such case. The nonlinear percolation model can be applied when the statistical disorder of the system leads to the dependence of the system s parameters on the electrical field strength. 

Statistically defined structures may also arise from the formation of crosslinks in a melt the resulting gels are described within a percolation framework which predicts the existence of definite meshes [7, 8]. Contact-lenses, jellies or even jellyfish are common examples of gels. Latex beads with specific functionalities attached, such as antigens, are used in biodiagnostics. 

A description of the percolation phenomenon in ionic microemulsions in terms of the macroscopic DCF will be carried out based on the static lattice site percolation (SLSP) model [152]. In this model the statistical ensemble of various... 

Since percolation is a property of macroscopic many-particle systems, it can be analyzed in terms of statistical mechanics. The basic idea of statistical mechanics is the relaxation of the perturbed system to the equilibrium state. In general the distribution function p(p,q t) of a statistical ensemble depends on the generalized coordinates q, momentum p, and time t. However, in the equilibrium state it does not depend explicitly on time [226-230] and obeys the equation... 

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