Percolation threshold, conductive

The critical volume fraction of the filler has a different application in the case of conductive materials. As the amount of conductive filler is increased, the material reaches a percolation threshold. Below the percolation threshold concentration, the electric conductivity is similar to that of matrix. Above the percolation threshold conductivity rapidly increases. Above the critical volume fraction of filler which is, in turn, a concentration above the percolation threshold, there is a rapid increase in conductivity. " The critical volume fraction depends on the type of filler and its particles size. For example, for silver powder, it ranges from 5 to 20 vol% for... 

The amount of filler required to reach this threshold value depends on the conductivity of the particular filler, its particle shape, and its interaction with matrix. After percolation threshold, conductivity increases rapidly. The steepness of the increase... 

Ravati and Favis [12] generated a low percolation threshold conductive device prepared through the control of multiple encapsulation and multiple percolation effects in a five-component polymer blend system. 

Scanning electron micrographs (SEMs) of (a,b) polyaniline (PANI) network in 25/25/25/25 polystyrene/polymethyl methacrylate/poly(vinylidene fluoride)/polyaniline (PS/PMMA/ PVDF/PANI) blend after extraction of all phases by dimethylformamide (DMF) followed by freeze drying, and (c,d,e) PANI network in 15/20/15/25/25 PS/PS-co-PMMA/PMMA/PVDF/ PANI blend after extraction of all phases by DMF followed by freeze drying. (Reproduced from Ravati, S., and Favis, B. D. 2010. Low percolation threshold conductive device derived from a five-component polymer blend. Polymer 51 3669-3684 with permission from Elsevier.)... 

Ravati, S., and Favis, B. D. 2010. Low percolation threshold conductive device derived from a five-component polymer blend. Polymer 51 3669-3684. 

Sumita M, Abe H, Kayaki H, Miyasaka K (1986) Effect of melt viscosity and surface tension of polymers on the percolation threshold conductive-particle-filled polymeric composites. J Macromol Sci.-Phys ser B 25 171-184... 

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-... 

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. 

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. 

The defects caused by the high contact resistance especially manifest themselves in the metal-filled composites where the value of the percolation threshold may reach 0.5 to 0.6 [30]. This is caused by the oxidation of the metal particles in the process of CPCM manufacture. For this reason, only noble metals Ag and Au, and, to a lesser extent, Ni are suitable for the use as fillers for highly conductive cements used in the production of radioelectronic equipment [32]. 

In pressing, the threshold concentration of the filler amounts to about 0.5% of volume. The resulting distribution of the filler corresponds, apparently, to the model of mixing of spherical particles of the polymer (with radius Rp) and filler (with radius Rm) for Rp > Rm as the size of carbon black particles is usually about 1000 A [19]. During this mixing, the filler, because of electrostatical interaction, is distributed mainly on the surface of polymer particles which facilitates the forming of conducting chains and entails low values of the percolation threshold. 

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. 

In the case of the filler localization in one of the polymer components of the mixture, an increase of the portion of the second unfilled polymer component in it entails sharp (by a factor of lO10) rise of a in the conducting polymer composite. In this case the filled phase should be continuous, i.e. its concentration should be higher than the percolation threshold. 

As already noted, the main merit of fibers used as a filler for conducting composite materials is that only low threshold concentrations are necessary to reach the desired level of composite conductivity. However, introduction of fiber fillers into a polymer with the help of ordinary plastic materials processing equipment presents certain difficulties which are bound up mainly with significant shearing deformations entailing fiber destruction and, thereby, a decrease of parameter 1/d which determines the value of the percolation threshold. 

Most of the models developed to describe the electrochemical behavior of the conducting polymers attempt an approach through porous structure, percolation thresholds between oxidized and reduced regions, and changes of phases, including nucleation processes, etc. (see Refs. 93, 94, 176, 177, and references therein). Most of them have been successful in describing some specific behavior of the system, but they fail when the... 

Above a critical hller concentration, the percolation threshold, the properties of the reinforced rubber material change drastically, because a hller-hUer network is estabhshed. This results, for example, in an overproportional increase of electrical conductivity of a carbon black-hUed compound. The continuous disruption and restorahon of this hller network upon deformation is well visible in the so-called Payne effect [20,21], as represented in Figure 29.5. It illustrates the strain-dependence of the modulus and the strain-independent contributions to the complex shear or tensUe moduli for carbon black-hlled compounds and sUica-hUed compounds. 

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 ... 

Similar to the percolation threshold, the effective electrical conductivity of a porous Ni-YSZ cermet anode depends on the morphology, particle size, and distribution of the starting materials as well. In general, the effective conductivity increases as the NiO particle size is reduced when other parameters are kept constant. As shown in Figure 2.4 (samples 1 and 2), the cermet conductivity increased from -10 S/cm to 103 S/cm as the NiO particle size was decreased from 16 to 1.8 pm while using the same YSZ powder (primary particle size of -0.3 pm) and the same Ni to YSZ volume fraction [30],... 

Similarly, other studies concluded that the anode effective electrical conductivity increases with the YSZ particle size when other parameters (Ni to YSZ volume ratio and the particle size of NiO) remain constant, as can be seen in Figures 2.5 [13], 2.4 (compare samples 4, 5, and 6) [30], 2.3 [14], and 2.1 [12], This is because it greatly influences the tendency of NiO clustering and downshifts the percolation threshold. 

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. 

Alignment of CNTs markedly affects the electrical properties of polymer/CNT composites. For example, the nanocomposites of epoxy/MWCNTs with MWCNTs aligned under a 25 T magnetic field leads to a 35% increase in electric conductivity compared to those similar composites without magnetic aligned CNTs (Kilbride et al., 2002). Improvements on the dispersion and alignment of CNTs in a polymer matrix could markedly decrease the percolation threshold value. 

When the filling is higher than the percolation threshold, the conductivity increases sharply as conductive paths begin to form [248,249]. 

Tab. 4.3 Percolation threshold and conductivity for electrical transport using different types of nanocarbons in polymer composites.

Carbon structure Percolation threshold Electrical conductivity (S/cm) References... 

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