Percolation thresholds

Electrical properties of polymer/CNT composites 6.4.1. Percolation threshold 

At low filler loading, CNTs are dispersed independently as individual fillers in the polymer matrix. The electrical behavior of composites is limited by the polymer matrix, having conductivity in the order of 10 S.cm . When sufficient nanotube content is loaded, the distance between the fillers is reduced considerably, leading to the formation of a network of connected nanotube paths through the insulating matrix. This critical filler content is commonly termed as the percolation threshold . At this filler content, the conductivity rises drastically by several orders of magnitude. In this respect, the insulating polymer becomes electrically conductive. 

Percolation theory gives a phenomenological description of the conductivity of a system near an insulator-conductor transition. According to the percolation theory, the conductivity and permittivity of a composite follow the power law near the percolation threshold ( f J [80-82]  

It is well known that the conductivity and permittivity of dielectrics are frequency dependent. Above the percolation threshold, the variations of o and e with frequency (f) can be described by the following equations [81,82] 

Prom the literature, a wide range of values have been reported for the percolation threshold and electrical conductivity of polymer/CNT composites, depending on the type of CNTs, nanotube functionalization, composite processing method and polymer employed [85,86]. Bauhofer and Kovacs analyzed the effects of experimental conditions, types of CNTs and polymers on the percolation threshold of CNT/polymer nanocomposites. They reported that the type of polymer and CNT dispersion play a more important role than the type and synthesis method of CNTs. Moreover, the electrical conductivity of nanocomposites with fully dispersed and exfoliated MWNTs in the matrix is 50 times higher than that of entangled ones [86]. 

Percolation models are roughly classified into percolation on regular lattices and percolation in continuum space. Both derive the scaling laws near the percolation threshold by focusing on the self-similarity of the connected objects. The percolation theory is suitable for the study of fluctuations in the critical region, but has a weak point in that the analytical description of the physical quantities in wider regions is difficult. 

When p exceeds a certain threshold value pc, a connected cluster of infinite size appears. This critical value pc of the fraction depends on the space dimensions d and lattice structure (symbolically described as A), and hence it is indicated by Pc A,s). In two-dimensional space (d = 2), there are square lattices (A = S), triangular lattices (A = T), honeycomb lattices (A = H), etc. In three-dimensional space (d = 3), there are simple cubic lattices (A = sc), face-centered cubic lattices (A = fee), body-centered cubic lattices (A = bcc), and hexagonal close packed lattices (A=hcp) (Table 8.1). 

Dual lattice and dual transformation, (a) Square lattice is self-dual S = S. (b) The dual lattice of a honeycomb lattice is a triangular lattice H = T and vice versa T = H. 

Because bonds on the lattice A and bonds on its dual lattice A cross each other, percolation problems on them are related by 

The state on A where bonds are connected to infinity at pc corresponds to the infinitely connected vacancies that appear on A at 1 — Pc- This relation is called the matching relation. We find immediately that the critical value of the percolation on a square lattice 

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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... 

If the sites of a lattice are randomly colored either black (with probability Pbiack or white (with probability Pwhite = 1 — Pbiack) then there exists a percolation threshold Pc, such that if, say, Pbiack < Pc, the lattice consists of isolated clusters of black sites immersed in a white sea , and if pbiack > Pc a connected black structure percolates (i.e. spans) the entire lattice. 

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 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. 

Fillers with extremely high aspect ratios (1000-10,000) such as carbon nanotubes (Figure 32.5) have a much lower percolation threshold (lower amount is required for equivalent reinforcement). 

The configuration of a system in which percolation may occur is classically treated as one in which the ingredients do not move. Considerable work has been devoted to these static models, leading to numerical solutions of the critical concentrations and cluster sizes associated with a percolation threshold. 

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]. 

Effects of additives (electrolytes, surfactants, nonelectrolytes) on the volume fraction and temperature percolation thresholds of a water/AOT/n-heptane system have been investigated [280,281]. 

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 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

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 ... 

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. 

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