The Percolation Threshold

In this section, we describe the basic features of the o(x) dependences found experimentally in conductor-insulator composites and we try to explain them in view of the expectations that were outlined in Sections 5.3 and 5.4. Considering the fact that the two basic features of the a x) dependence are the abrupt jumps of o(x) at some x - values and the particular mathematical o(x) dependence for x x, we start with the typical x values of the sharp rise in the conductivity that is commonly called (though as we saw above not always well defined) the percolation threshold, x -. The other part of the discussion vdll be concerned with the fundamental cr(x ) dependence and its relation to the predictions of percolation and hopping theories. Since in the literature people quantify the x and Xc values in terms of vol.% and fractional volume contents (i.e., vol.%/100), we will in this section use either of these quantities as required by the context of the discussion. In particular, in the comparison of experimental data with the relations (5.6) and (5.15), we will assume that X stands for the fractional volume content. Also, we note in passing that while the volume is the relevant physical parameter, in many cases the wt% is the much easier to determine parameter. Correspondingly, we will use here the vol.% noting that the difference in the value of vol.% and that of the wt% does not alter the discussion. 

While annealing the granular metals brings about the formation of larger particles (or aggregates (Balberg, I., unpublished.), [39]) that amounts to shifting  

In carbon black-polymer composites, the value of Xc varies from about 40 vol.% in the case of spherical CB (e.g., N990) particles [51] to about 5 vol.% (e.g., for XC-72 [51] and Ketjenblack [52]) following the increase of the aspect ratio of the particles to the order of 10 [34]. The latter result is in excellent agreement with the excluded volume theory that predicts such a dependence of Xc on the aspect ratio (see Eq. (5.11)) if we identify the shell thickness d with a critical average distance, dc, that the tuimeling electrons should cross [29], This trend is continued when we consider the x values for CNTs [43], where x values on the order of 1 vol.% are observed for aspects ratios on the order of 10, and 0.1 vol.% for aspect ratios on the order of lO -lO.  

---

Thus, fracture occurs by first straining the chains to a critical draw ratio X and storing mechanical energy G (X — 1). The chains relax by Rouse retraction and disentangle if the energy released is sufficient to relax them to the critically connected state corresponding to the percolation threshold. Since Xc (M/Mc) /, we expect the molecular weight dependence of fracture to behave approximately 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. 

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. 

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. 

On the other hand, dodecylmethylbutylammonium bromide- and benzyldymethyl-headecylammonium chloride-based w/o microemulsions, which consist of reversed micelles below the percolation threshold, form a bicontinuous stracture above the percolation threshold [279]. 

A kinetic study of the basic hydrolysis in a water/AOT/decane system has shown a change in the reactivity of p-nitrophenyl ethyl chloromethyl phosphonate above the percolation threshold. The applicability of the pseudophase model of micellar catalysis, below and above the percolation threshold, was also shown [285],... 

A somewhat different water, decane, and AOT microemulsion system has been studied by Feldman and coworkers [25] where temperature was used as the field variable in driving microstructural transitions. This system had a composition (volume percent) of 21.30% water, 61.15% decane, and 17.55% AOT. Counterions (sodium ions) were assigned as the dominant charge transport carriers below and above the percolation threshold in electrical... 

Figure 41. The percolation threshold determination for polymer blends undergoing the phase separation. Minority phase volume fraction, fm, is plotted versus the Euler characteristic density for several simulation runs at different quench conditions, /meq- = 0.225,..., 0.5. The bicontinuous morphology (%Euier < 0) has not been observed for fm < 0.29, nor has the droplet morphology (/(Euler > 0) been observed for/m > 0.31. This observation suggests that the percolation occurs at fm = 0.3 0.01.

Thus, one could expect to find a droplet morphology at those quench conditions at which the equilibrium minority phase volume fraction (determined by the lever rule from the phase diagram) is lower than the percolation threshold. However, the time interval after which a disperse coarsening occurs would depend strongly on the quench conditions (Fig. 40), because the volume fraction of the minority phase approaches the equilibrium value very slowly at the late times. 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

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. 

Catalytic activity and electrochemical performance generally increase as the NiO and YSZ particle sizes are reduced. However, ultrafine powders are prone to agglomeration during the milling and mixing process the distributions of the phases (and hence the percolation threshold and many other important properties) are determined by the agglomeration size, not by the primary particle size. 

Generally, percolation media can be characterized not only by the percolation probability, but also by several important quantities [8,121,234], Near the percolation threshold, r/lh, a number of... 

The degree of polymerization of hard clusters increases with evolution of the system as a whole. The hard clusters already exist in pregel molecules. Before the macroscopic gel point of the system is reached they remain usually small. Later on, the hard clusters grow faster and eventually a gel point (percolation threshold) of the hard structure is reached. Below this point, clusters are embedded in the soft matrix beyond the percolation threshold, the hard and soft structures interpenetrate (Figure 5.7). Below the percolation threshold, hard clusters are essentially dendritic when the percolation threshold is surpassed, circuits (cycles) develop within the hard structure. 

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. 

---