Conductive composites percolation theory

Experimental dependences of conductivity cr of the CPCM on conducting filler concentration have, as a rule, the form predicted by the percolation theory (Fig. 2, [24]). With small values of C, a of the composite is close to the conductivity of a pure polymer. In the threshold concentration region when a macroscopic conducting chain appears for the first time, the conductivity of a composite material (CM) drastically rises (resistivity Qv drops sharply) and then slowly increases practically according to the linear law due to an increase in the number of conducting chains. 

Where a is the composite conductivity, a0 a proportionally coefficient, Vfc the percolation threshold and t an exponent that depends on the dimensionality of the system. For high aspect ratio nanofillers the percolation threshold is several orders of magnitude lower than for traditional fillers such as carbon black, and is in fact often lower than predictions using statistical percolation theory, this anomaly being usually attributed to flocculation [24] (Fig. 8.3). 

An explanation of the observed relaxation transition of the permittivity in carbon black filled composites above the percolation threshold is again provided by percolation theory. Two different polarization mechanisms can be considered (i) polarization of the filler clusters that are assumed to be located in a non polar medium, and (ii) polarization of the polymer matrix between conducting filler clusters. Both concepts predict a critical behavior of the characteristic frequency R similar to Eq. (18). In case (i) it holds that R= , since both transitions are related to the diffusion behavior of the charge carriers on fractal clusters and are controlled by the correlation length of the clusters. Hence, R corresponds to the anomalous diffusion transition, i.e., the cross-over frequency of the conductivity as observed in Fig. 30a. In case (ii), also referred to as random resistor-capacitor model, the polarization transition is affected by the polarization behavior of the polymer matrix and it holds that [128, 136,137]... 

If the estimated fitting parameters are compared to the predicted values of percolation theory, one finds that all three exponents are much larger than expected. The value of the conductivity exponent ji=7A is in line with the data obtained in Sect. 3.3.2, confirming the non-universal percolation behavior of the conductivity of carbon black filled rubber composites. However, the values of the critical exponents q=m= 10.1 also seem to be influenced by the same mechanism, i.e., the superimposed kinetic aggregation process considered above (Eq. 16). This is not surprising, since both characteristic time scales of the system depend on the diffusion of the charge carriers characterized by the conductivity. 

The all or nothing feature of metal powder composites is very much a feature of conductive composite systems. In order to understand this behaviour, most theories borrow from percolation theory (Broadbent and Hamersley, 1957), which was originally developed as a model for predicting fluid permeation through porous media. The percolation model is based on having a medium... 

Armand (1994) has briefly summarised the history of polymer electrolytes. A more extensive account can be found in Gray (1991). Wakihara and Yamamoto (1998) describe the development of lithium ion batteries. Sahimi (1994) discusses applications of percolation theory. Early work on conductive composites has been covered by Norman (1970). Subsequent edited volumes by Sichel (1982) and Bhattacharya (1986) deal with carbon- and metal-filled materials respectively. Donnet et al. (1993) cover the science and technology of carbon blacks including their use in composites. GuF (1996) presents a detailed account of conductive polymer composites up to the mid-1990s. Borsenberger and Weiss (1998) discuss semiconductive polymers with non-conjugated backbones in the context of xerography. Bassler (1983) reviews transport in these materials. 

The following simple experiment demonstrates the essential concept of the critical threshold for percolation. A mixture of small plastic and metal balls of equal size is poured into a beaker with a crumpled-foil electrode at the bottom, another crumpled-foil electrode is pressed onto the top, and the electrodes are connected to a battery through an ammeter. Current is measured as a function of the composition (i.e., fraction of metal balls) of the conductor/insulator mixture. There is a critical composition below which no current flows and above which the conductivity increases nearly exponentially. At this threshold the two electrodes suddenly become spatially connected along a statistical pathway originating in the random medium. Percolation theory tells us that the critical composition is 0.25 fraction metal balls, a remarkably low concentration. This is perhaps not an intuitive result. 

Figure 6.8 Electrical conductivity and surface resistivity comparison. Upper panel electrical conductivity results of P3HT/SWNT composite films depending on (left) different amounts of pre-separated ( ) and separated metallic (O) nanotube samples, and (right) their corresponding effective metallic SWNT contents in the films (dashed line the best fit in terms of the percolation theory equation). Lower panel Surface resistivity results of PEDOT PSS/SWNT films on glass substrate with the same 10 wt% nano tube content (O pre-separated purified sample and T separated metallic SWNTs and for comparison, blank PEDOT PSS without nano tubes) but different film thickness and optical transmittance at 550 nm. Shown in the inset are representative films photographed with tiger paw print as background.

As Balberg notes in a review The electrical data were explained for many years within the framework of interparticle tunneling conduction and/or the framework of classical percolation theory. However, these two basic ingredients for the understanding of the system are not compatible with each other conceptually, and their simple combination does not provide an explanation for the diversity of experimental results [17]. He proposes a model to explain the apparent dependence of percolation threshold critical resistivity exponent on structure of various carbon black composites. This model is testable against predictions of electrical noise spectra for various formulations of CB in polymers and gives a satisfactory fit [16]. 

It has been possible to directly image the percolation network at the surface of a CB-polymer composite. An early report is that of Viswanathan and Heaney [24] on CB in HOPE in which it was shown that there are three regions of conductivity as a function of the length L, used as a metric for the image analysis. Below I = 0.6pm, the fractal dimension D of the CB aggregates is 1.9 0.1. Between 0.8 and 2 pm, the data exhibit D = 2.6 0.1 while above 3 pm, D = 3 corresponding to homogeneous behavior. Theory predicts D = 2.53. It is not obvious that the carbon black-polymer system should be explainable in terms of standard percolation theory, or that it should be in the same universality class as three-dimensional lattice percolation problems [24]. Subsequent experiments of this kind were made by Carmona [25, 26]. 

CNTs in polymer-CNT composites are efficiently debundled and isotropically dispersed in polymer matrices, the efficient interaction between CNT and polymer provides good dispersion and a low percolation threshold, but only relatively low conductivity near and above percolation, frequently around 10 s cm is achieved at close to 2wt% CNT loading [70, 71], The polymer layer in the intemanotube connections is supposed to be the highest resistance section in the electrical pathway. This polymer layer is a barrier to efficient carrier transport between CNTs, and models for conductivity based on fluctuation-induced tunneling have been proposed [72]. A power law related to percolation theory can be used to model conductivity in the following form ... 

Percolation theory [53] is also used to calculate the effective properties such as the ionic conductivity in the SOFC electrodes. The effective conductivity of a composite electrode is less than that of the pure material due to the composite structure and porosity of the electrode. Percolation theory calculates an effective ionic conductivity that accounts for the tortuous path of the electrolyte phase in the electrodes and is based on the probability of finding a percolated chain of the electrolyte phase through the electrode [53]. 

Fig. 1.1 Relative differential resistance change, AR/R, predicted by percolation theory as a function of the relative volume change, AV/V, of a carbon black-polymer composite upon swelling. The volume of carbon black is assumed to be unaffected by swelling, and the polymer matrix is assumed to have a conductivity 11 orders of magnitude lower than that of carbon black. The three separate lines are for composites with differing initial volume percentages of carbon black, as indicated. The percolation threshold for the system is at CB content=0.33. The total volume change results in a change in the effective carbon black content. When, this value drops below the percolation threshold, a sharp increase in response is observed. Of course, the position of this sharp increase depends on the value of the percolation threshold (Reprinted with permission from Lonergan et al. 1996, Copyright 1996 American Chemical Society)...

In recent years metallic particles have also been considered as fillers to increase the electrical and thermal conductivities of epoxy systems. The electrical and thermal conductivities of epoxy systems filled with metal (i.e. copper and nickel) powders have been studied (Mamunya et al., 2002). In this work it was shown that the composite preparation conditions allow the formation of a random distribution of metallic particles in the polymer matrix. The percolation theory equation holds true for systems with a random distribution of dispersed filler, while in contrast to the electrical conductivity, the dependence of thermal conductivity on concentration shows no jump in the percolation threshold region. 

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