Double percolation system

In these systems, it is possible to obtain low percolation thresholds if a double percolation is present, that is, particle and phase percolation. This effect may be observed when the conductive particles, localized preferentially in one polymer phase, have a concentration equal or larger than the electric percolation threshold, and when the host polymer phase is the matrix or continuous phase of the polymer blend [155]. There are several models that describe the electroconductivity of these systems the effective medium theory, the onset for percolation theory, and thermodynamic models. Sumita s model considers the formation of chainlike conductive structures [151, 156]. 

The study of electroconductive polymer systems, based on conductive particles and polymer blends, has been quite intensive during the recent past. Gubbels et al. [149] studied the selective localization of CB particles in multiphase polymeric materials (PS and PE). According to these results, the percolation threshold may be reduced by the selective localization of CB. The minimum resistivity was obtained when double percolation (phase and particle percolation) exists in the PS-PE blend. In addition, it was found that the percolation threshold may be obtained at very low particle concentrations, provided that CB is selectively localized at the interface of the blend components. Soares et al. [150] found that the type of CB (i.e., different surface areas) does not affect the conductivity of the blend with 45/55 PS/PIP (polyisoprene) composition. 

It may be mentioned here that two stage (or double) percolation has been reported by Ray and Moulik [92] and Maiti et al. [93] for (water/AOT/decanol) and water/DTAB (octadecyltrimethylammonium bromide)-butanol/heptane microemulsion systems, respectively. The double percolation process for AOT/ decane/NaCl (0.5%) was also reported by Eicke et al. by conductivity, viscosity, and electro-optical Kerr effect [94]. The two processes demarcated three structural regimes viz, o/w, w/o, and oil or water continuous. 

Here, <5 is the percolation exponent (we take 6 = 2.5), and is the percolation threshold for the hard phase. The percolation exponent, 6, typically ranges between 1.5 and 2 (see, e.g., ref. [52]), depending on the type of the system and the property described by a percolation model (modulus, conductivity, etc.). There are instances, however, when the percolation exponent could be larger than 2 (see, e.g., ref. [53]). Various models (e.., double percolation -see ref. [54]) have been proposed to explain these high percolation exponents. In our analysis, we refrain firom ascribmg any specific meaning to exponent 5 = 2.5, and treat it simply as an adjustable parameter that is found fi-om the best fit to experimental data. 

In this multiphase system, the term double percolation is defined to describe the conductive mechanism of polymer nanocomposites with a percolated network of nanofiller in one phase, which enables the formation of the conductive network through the whole polymer matrix. It has been proved that addition of conductive nanofillers into an immiscible polymer blend allows for the formation of cocon-tinuous structure and efficiently decreases the percolation threshold of nanofiUers due to the selective localization of the conductive networks. For example, Petra Poitschke et al. [89] introduced CNTs into Polycarbonate/Poly(styrene-acrylonitrile) (PC/SAN) to prepare CPCs. The percolation threshold of CNTs was less than 1 wt %, which is lower than those of CNTs in single PC matrix (1.2 wt%) and in single SAN phase (2.0 wt%).The localization of the conductive fiUer in polymer matrix depends on the interfacial energies of components and can be predicted by following Eq. (2) [86]. 

The percolation threshold can be further reduced, and conductivity increased, if one makes use of the concept of double percolation, first theoretically studied by Levon, Margolina, and Patashinsky [164], and experimentally observed by Sumita and coworkers [161], as well as other researchers [157, 158, 162] for the dispersion of carbon black in binary polymer blends. To obtain double percolation, one needs to have a ternary system, with two phase-separating polymers and conducting filler with strong affinity to one of the polymers (we denote it as A, and the second polymer as B). Then, the system could be conducting if (i) filler loading in the A-domain is above percolation threshold and (ii) the volume fraction of the filled A-domains is above... 

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