Percolation threshold for conductivity

Figure 3 Electrical percolation threshold for conductive particles in an insulating matrix.

The physical model can be used to describe trends seen in experimental data. For example, the interconnectivity of the cluster network is predicted to have a profound effect on a membrane s transport properties. The percolation threshold for conductivity should increase when the clusters become smaller, which could be due to a stiflfer and/or more crystalline polymer matrix. These smaller clusters would also mean that the membrane would exhibit lower electro-osmotic coefficients, larger liquid water uptakes, and a greater dependence of the various properties on water content than in Nafion . In fact, these predictions are what is seen in such systems as sulfonated polyetherketones [19, 72] and Dow membranes [73, 74] or when the equivalent weight [22] or drying temperature [4, 6] of Nafion is increased. 

Making a polysiloxane electrically conductive is best carried out using anisometric particles such as films and fibers, because of their low values of the percolation threshold for conducting pathways.Nonetheless, roughly spherical particles of carbon black have also been used for this purpose.2 2 The use of layer-like particles is illustrated by functionalized graphene sheets, while needle shapes are illustrated by unmodified or... 

In more recent work, Mandal and Mandal [382] described unique IPNs of P(Py) and poly(methyl acrylate) or poly(styrene-c(>-butyl acrylate) which showed very low percolation thresholds for conductivity, ca. 0.023 volume fraction of P(Py). 

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. 

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

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. 

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. 

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... 

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],... 

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 ... 

The percolation threshold has many similarities with phase transitions, since the behavior of the system changes abruptly at this point. Assuming, for instance, that the occupied sites are able to conduct an electric current, whereas the empty ones are not, the conductivity a will be zero when p p. Moreover, close to the percolation threshold, the conductivity will follow a power law... 

Pron and coworkers6 have blended PAn with cellulose acetate and cast from m-cresol to produce highly transparent and conductive ( 1 S cm-1) PAn. Addition of plasticizers (in particular, phenylphosphonic acid) not only resulted in more flexible films but lowered the percolation threshold for PAn to an amazing 0.05% (w/w). 

Interesting data have been obtained for polyamide-HAp composites. The mechanical properties are strongly related to the heterogeneity of the amorphous phase and the localization of HAp in the constrained amorphous phase—a universal behavior. Recent studies have shown promising results for polymer-nanowire composites. Due to their high aspect ratio, these filler particles have a low percolation threshold for electrical conductivity but the enhancement of Young s modulus remains... 

As the number of SWNTs eontinues to inerease, the percolation threshold for metallic SWNTs is reached. At the density of SWNTs continues to increase, the number of metallic pathways increases coimnensurately. The result is thin-111m which conducts electricity with metalhc behavior (further decreasing resistance) and less semiconductive behavior (sensitivity to molecular adsorption events). 

Studies in this group currently involve determining the effect of density and level of alignment of SWNTs on overall device performance. For semiconductive networks of SWNTs, the devices are expected to have a greater response at the number of SWNTs increases. This will occur until the percolation threshold for metalhc SWNTs is reached and then response will level off as metallic pathways will show little change in conductivity upon exposure. 

Pure NBR has conductivity in the region of 10 S/cm. The PAni.DBSA had an electrical conductivity of 1.2 0.5 S/cm. The electrical conductivity of all the blends increased with the proportion of PAni.DBSA. The conductivity percolation threshold for the blends was estimated by fitting the data from the curve of log blend electrical conductivity versus PAni.DBSA content (see Figure 8.6) to a simple percolation model as defined by Equation 8.3 [1, 6-7] ... 

Most often incorporation of nanopartides results in finer and more stable drop dispersion with improved performance. This effect is related to migration of particles to the interphase, which might lead to additional benefits, for example, reduction of the percolation threshold for blends conductivity. 

In addition to percolation in reverse microemulsions being driven by field variables such as (disperse phase) volume fraction, temperature, and salinity, cosurfactant concentration appears also to suffice. Low-frequency electrical conductivity for the aqueous acrylamide-AOT-tol-uene reverse microemulsion system is illustrated in Fig. 6 [42]. The cosurfactant concentration = 1-2% (w/w) is annotated with the arrow (Op) in Fig. 6 and corresponds to the approximate onset of electrical conductivity percolation. The percolation threshold in conductivity occurs at ip = 3.09%. 

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