Percolation threshold for

Composilion has a marked effect on p. Dilution can cause (i to drop by orders of magnitude. The functional dependence is often expressed in terms of an exponential dependence on intersite distance R=at m as suggested by the homogeneous lattice gas model, p c)exponential distance dependence of intersite coupling precludes observing a percolation threshold for transport. 

The composites with the conducting fibers may also be considered as the structurized systems in their way. The fiber with diameter d and length 1 may be imagined as a chain of contacting spheres with diameter d and chain length 1. Thus, comparing the composites with dispersed and fiber fillers, we may say that N = 1/d particles of the dispersed filler are as if combined in a chain. From this qualitative analysis it follows that the lower the percolation threshold for the fiber composites the larger must be the value of 1/d. This conclusion is confirmed both by the calculations for model systems [27] and by the experimental data [8, 15]. So, for 1/d 103 the value of the threshold concentration can be reduced to between 0.1 and 0.3 per cent of the volume. 

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. 

As stated, the particle size also influences the distribution of phases and the percolation threshold. In general, the small particles tend to cluster around the large particles to form a continuous path (lower percolation threshold for the smaller particles). Thus, if NiO particles are smaller than YSZ particles, we would expect high electrical conductivity. In contrast, if the YSZ particles are smaller, electrical conductivity would be lower because the small YSZ particles tend to cluster around the larger Ni particle, making them electrically isolated. 

Fig. 8.3 Percolation threshold for different aspect ratio fillers and...

The side chain separation varies in a range of 1 nm or slightly above. The network of aqueous domains exhibits a percolation threshold at a volume fraction of 10%, which is in line with the value determined from conductivity studies. This value is similar to the theoretical percolation threshold for bond percolation on a face-centered cubic lattice. It indicates a highly interconnected network of water nanochannels. Notably, this percolation threshold is markedly smaller, and thus more realistic, than those found in atomistic simulations, which were not able to reproduce experimental values. 

If S is smaller than the site percolation threshold for the square lattice Sc = 0.59275 we obtain a system which consists only of finite clusters. In principle these clusters can be completely occupied by one-kind species. For the case that no desorption is allowed this represents a poisoned state for which the production rate Rco2 goes to zero as t —> oo. Then the whole system consists of finite clusters poisoned by particles A or B. For this state the condition Ca + Cb = S holds where C is the density of particles of type A (C a + Cb + Co = S). 

We have studied above a model for the surface reaction A + 5B2 -> 0 on a disordered surface. For the case when the density of active sites S is smaller than the kinetically defined percolation threshold So, a system has no reactive state, the production rate is zero and all sites are covered by A or B particles. This is quite understandable because the active sites form finite clusters which can be completely covered by one-kind species. Due to the natural boundaries of the clusters of active sites and the irreversible character of the studied system (no desorption) the system cannot escape from this case. If one allows desorption of the A particles a reactive state arises, it exists also for the case S > Sq. Here an infinite cluster of active sites exists from which a reactive state of the system can be obtained. If S approaches So from above we observe a smooth change of the values of the phase-transition points which approach each other. At S = So the phase transition points coincide (y 1 = t/2) and no reactive state occurs. This condition defines kinetically the percolation threshold for the present reaction (which is found to be 0.63). The difference with the percolation threshold of Sc = 0.59275 is attributed to the reduced adsorption probability of the B2 particles on percolation clusters compared to the square lattice arising from the two site requirement for adsorption, to balance this effect more compact clusters are needed which means So exceeds Sc. The correlation functions reveal the strong correlations in the reactive state as well as segregation effects. 

Fig. 30a behaves similarly to that of the NBR/N220-samples shown in Fig. 29, i.e., above a critical frequency it increases according to a power law with an exponent n significantly smaller than one. In particular, just below the percolation threshold for 0=0.15 the slope of the regression line in Fig. 30a equals 0.98, while above the percolation threshold for 0=0.2 it yields n= 0.65. This transition of the scaling behavior of the a.c.-conductivity at the percolation threshold results from the formation of a conducting carbon black network with a self-similar structure on mesoscopic length scales. 

In another study Slobodian et al. (57) found that the percolation threshold for electrical conductivity of MWCNT-PMMA composites depends on the solvent used. The lowest percolation threshold was achieved for toluene where percolation was found to be at 4 wt% of MWCNT, for chloroform at 7 wt% and for acetone at 10 wt%. The highest conductivity was obtained at 20 wt% of MWCNT at values around 4x 10 5 Sc nr1 for composite prepared from toluene solution. They observed that the Hansen solubility parameters of individual solvent play an important role in the dispersion of MWCNT in PMMA. 

Fig. 41. Percolation probability for finite-sized lattices. Computer calculations of the percolation probability, P(p), as a function of the site-filling probability, p, for two-dimensional square lattices of varied dimension O, 10 x 10 , 20 x 20 , 40 x 40. Each curve is an average over a set of site percolation simulations for a lattice size. The site percolation threshold for an infinite two-dimensional square lattice is 0.593. Nonzero values of P p) below the infinite lattice threshold reflect the variance of the threshold value for finite lattices (unpublished results).

Table 1.1 The site and bond percolation thresholds for different lattice types...

If we consider a random bond network, where the bonds are conductors with concentration p and insulators with concentration 1 — p, then such a network has a macroscopic conductivity E(p) (measured by the ratio of the net current across the two ends of the network to the voltage across it) for P Pc) the percolation threshold for the lattice. Obviously E(p) = 0 for p < Pc, and one observes the conductivity E(p) to grow with p above pc following a power law... 

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8)... 

Conducting polymer blends based upon polyaniline (PANI) are a new class of materials in which the percolation threshold for the onset of electrical conductivity can be reduced to volume fractions well below that required for classical percolation (16% by volume for globular conducting objects dispersed in an insulating matrix in three dimensions) [277,278], The origin of this remarkably low threshold for the onset of electrical conductivity is the self-assembled network morphology of the PANI poly blends, which forms during the course of liquid-liquid phase separation [61],... 

At the most fundamental level the percolation threshold for a model, be it as a representation of the whole pore space or merely the macroscopic scale, must equate to the percolation threshold for the real material. For the general case for non-infinite lattices, the percolation threshold has been [10] fundamentally defined as ... 

In the following discussion the subscript o refers to the experimentally observed percolation threshold, whilst the subscript a refers to the percolation threshold for the model derived from NMR images representing only the macroscopic scale. It is proposed that to relate the macroscopic model to reality then the following relation is taken to hold ... 

With a = 53.7 and b = 3.2, Eq. (90) reproduces well experimental and simulation data for face centered cubic (fee) lattice (with no claim on reproduction of critical exponent at the percolation threshold) for other structures the parameters could be different. The first term in curly brackets in Eq. (89) takes into account the probability that the two particles have at least one pore as their bond neighbor that belongs to the infinite pore network, cf. [120]. The parameter M is the average number of bond neighbors (for fee lattice M = 4). 

Reference 70 provides the first quantitative test of the random resistor network model. In Ref. 121 the authors employed the random resistor network model to determine the behavior of the low-field Hall effect in a 3D metal-nonmetal composite near the percolation threshold. For the following power laws of effective values of ohmic conductivity a, Hall coefficient R, and Hall conductivity a 12, Bergman et al. 121 have obtained the critical exponents ... 

Assuming that Eq. (333) is fulfilled also at the percolation threshold for random lattices with Z replaced by the average coordination number (Z,) (where Z, means the coordination number of the th lattice site), one can obtain negative Poisson s ratios from percolation systems of particles—for example, polymer molecules, which can be due to a sufficiently large number of neighbors, < Z > > Ad. 

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