Carbon percolation theories

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). 

The electrical percolation behavior for a series of carbon black filled rubbers is depicted in Fig. 26 and Fig. 27. The inserted solid lines are least square fits to the predicted critical behavior of percolation theory, where only the filled symbols are considered that are assumed to lie above the percolation threshold. According to percolation theory, the d.c.-conductivity Odc increases with the net concentration 0-0c of carbon black according to a power law [6,128] ... 

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. 

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 pore network connectivity is usually determined by gas sorption analysis [2-4] or mercury intrusion [5] based on percolation theory. Recently, Ismadji and Bhatia [6] have successfully employed the liquid phase adsorption isotherms to determine the pore network connectivity and the pore size distribution of three commercial activated carbons. In our recent study [7], the pore network connectivity of three commercial activated carbons was characterized using liquid phase adsorption isotherms of eight different compounds. In that study we used ester molecules with complex structure, as probe molecules. 

Under certain approximations, using the concepts of percolation theory, the basic parameters can be related to the volume portions of the components of the layer. This offers a relationship between the structure of the porous composite catalyst layer and its performance. An optimum composition (in terms of volume fractions of electrolyte material, carbon and carbon-supported catalyst, and pore space) is a Holy Grail here. Albeit this goal can still be far away in view of the simplified character of the models used, these models give at least some rational scheme for... 

The dependence of the apparent resistivity of the toner materials with inorganic loading (i.e., iron oxide and carbon content) is perhaps most appropriately explained in terms of percolation theory,7 where conduction arises due to electron tunneling between islands of free-carriers. The dramatic increase in conductivity at a certain critical volume concentration predicted by the theory has been observed experimentally for metal colloids in ionic crystals and fine metal powders in insulating polymers. In fact, Kolosova and Boitsov showed that for non-agglomerated 0.1/t diameter metal powders dispersed within a polymer, the critical volume concentration was 10%. 

The conductance of compacted carbon materials has been described by the percolation theory [51] and more recently by the effective media theory [52]. It was shown that above the percolation threshold the conductivity o of the compact carbon powder comprising gas-filled voids may be described as... 

Percolation theory (the model supposes that asymmetrically structured carbon-black particles are statistically distributed and results in percolation in accordance with probability laws) [3,27,31,32,33,34,35], Although this theory is the most widespread one, it lacks important experimental fundamentals and cannot describe the multitude of factors affecting percolation behaviour,... 

Percolation theory is a statistical/geometrical approach to explaining the shape of the conductivity curve in carbon-black compounds. The theory does not explicitly embody aspects of thermodynamics, though these are implicit in it. 

Percolation theory proceeds from a statistical distribution of the carbon-black particles, which corre-... 

It will thus be seen that there is a strong relationship between the critical volume concentration and the properties of the matrix. But was it not one of the basic premises of percolation theory that no interactions took place between matrix and carbon-black particles How else could one explain a random distribution ... 

In studies of CO2 adsorption on carbon-black compounds with different concentrations of carbon-black, a sudden increase in adsorption at < >, was observed (studied by Tanioka et al, described in [72]), an observation that cannot be explained with the aid of percolation theory. 

Figure 11.13 shows a specimen with a carbon-black concentration of 0.5 vol.%, i.e. less than Oc. In it there are clearly visible individual spherical carbon-black particles which are present in the polymer matrix in isolated and distributed form. This finding would agree with the percolation theory, although the shape of the carbon-black particles is more similar to a sphere than to a linear stnicture. 

Carbon black or metal powder containing polymer compounds show a similar behavior when dispersed in a matrix above certain different critical concentrations. The percolation theory is thought to be the best tool for the description of this effect [14]. It is believed that metal powder, having a globular particle shape, is distributed in a statistically even manner and the powder particles will make contacts, governed by statistical laws (probability), whenever enough particles are present and close enough to finally form the first continuous conductive pathways. 

The critical volume concentration of metal powders is within the range of what is predicted by the percolation theory (45-64 vol%). For carbon black, however, the theory cannot explain the partially rather low critical concentrations, between 25 and 10 vol% and in well-defined optimal cases even down to 1% and below. Percolation theorists assume that carbon black particles are highly structured, with a high length-to-diameter ratio of their arms, and therefore have a bigger chance of contacting each other. 

Percolation theory can be applied to carbon nanotube networks.(S7-S3) As mentioned earlier, one third of carbon nanotubes or metallic and two thirds or semiconductive. If one has a random arrangement of SWNTs on a nonconducting smface at very low density, there are no electrical pathways. However, as the number of SWNTs increases, one will first reach the percolation threshold for the semiconductive carbon nanotubes. At this point, there is the possibility of having fully semiconductive pathways throughout the thin-film. However metallic pathways are not yet favored as metallic carbon nanotubes compose only l/S of the sample (Figure 2). 

Methods to determine the electronic conductivity of powdered battery materials and their mixtures have been studied intensively. To mathematically describe the electronic conductivity of the active electrode material ntixed with different amounts of conductive carbon, logarithmic equations, the percolation theory (PT), and the effective medium theory (EMT) " may be considered to be rules. 

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]. 

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)...

Ramasubramaniam R, Chen J, Liu H (2003) Homogeneous carbon nanotube/polymer composites for electrical applications. Appl Phys Lett 83 2928 Sahimi M (1994) Applications of percolation theory. Taylor Francis, London Shante VKS, Kirckpatrick S (1971) An introduction to percolation theory. Adv Phys 30 325 Sherman RD, Middleman LM, Jacobs SM (1983) Electron transport processes in conductor-filled polymers. Polym Sci Eng 23 36... 

---