A New Insight in Determining the Percolation Threshold of Electrical Conductivity for Extrinsically Conducting Polymer Composites through Different Sigmoidal Models

The electrical conductivity of extrinsically conducting polymer composite systems passes through a transition state known as percolation threshold. A discussion has been made on how different Sigmoidal models (S-models), such as Sigmoidal–Boltzmann (SB), Sigmoidal–Dose Response (SD), Sigmoidal–Hill (SH), Sigmoidal–Logistic (SL), and Sigmoidal–Logistic-1 (SL-1), can be applied to predict the percolation threshold of electrical conductivity for ethylene vinyl acetate copolymer (EVA) and acrylonitrile butadiene copolymer (NBR) conducting composite systems filled with different carbon fillers. An interesting finding that comes from these observations is that the percolation threshold for electrical conductivity determined by SB and SD models are similar, whereas, the other models give different result when estimated for a particular composite system. This similarity and discrepancy in the results of percolation threshold have been discussed by considering the strength, weakness, and limitation of the models. The percolation threshold value for the composites has also been determined using the classical percolation theory and compared with the sigmoidal models. Moreover, to check the universal applicability, these Sigmoidal models have also been tested on results from some published literature. Finally, it is revealed that, except SL-1 model, the remaining models can successfully be used to determine the percolation threshold of electrical conductivity for extrinsically conductive polymer composites.

black has higher value compared to printex black one. The present of volatile content reduce the electrical conductivity of carbons but to a very lesser extent. For the present composite systems, the overall variation in electrical conductivity is in the order of 10 12 S/cm. Consequently, the effect of volatile content on electrical conductivity for the present composite systems is totally negligible, and hence cannot be accounted for. The average length and aspect ratio of carbon fiber in different composites are presented in Table S2. The average length of carbon inside the polymer composites was measured using the optical microscopy imaging of the composites [Polymer Composites 32(11), 1790-1805, 2011. The diameter of the fiber was 6.0 micron.

Characterizations
The morphological analysis of carbon blacks and their composites has been carried out through transmission electron microscopy (TEM), scanning electron microscopy (SEM), field emission scanning electron microscopy (FESEM), and optical microscopy.
The instrument used to study the morphology of carbons and their composites through SEM analysis was JEOL JSM 5800 scanning electron microscope (Tokyo, Japan). All samples were gold coated prior to analysis using vacuum gold-sputter machine. The SEM study has carried out on cryo-fractured surface of vulcanized samples and etched surface of unvulcanized samples.
The distribution and morphology of carbons into the polymer matrix were studied using a high Austria). Freshly sharpened diamond knives with cutting edges of 45° were used to obtain cryosections of 50-60 nm thickness specimens at ambient temperature. The cut samples were supported on a copper mesh before observation under the microscope. To study the shape and size of carbon particle, the particle were dispersed in acetone for one hour using bath type sonicator and then it was placed on copper mesh to perform the morphological study.
The morphology of carbons and polymer/carbon composites was evaluated using field emission scanning electron microscope (model Supra 40, Carl Zeiss SMT AG, Oberkochen, Germany). Samples were gold coated by means of manually operated sputter coater (model SC7620, Polaron Brand, Quorum Technologies Ltd., East Sussex, UK) unit.
For calculating fiber average length (already reported in the above section) and their distribution in composite materials, optical microscopy was used. To perform this test, carbon fibers were extracted from unvulcanized samples of different composites through solvent extraction method.

Morphology of Carbons and their Composites
The morphological analysis of carbon blacks and their composites, carried out through TEM, SEM) FESEM) and optical microscopy, are shown in Fig. S1. The morphology of conductex black and its composites is shown in Fig. S1 (a-f), printex black and its composites is shown in Fig. S1 (g-l), and SCF and its composites is shown in Fig. S1 (m-r). It is seen from the TEM images that the carbon black particles are existed as aggregated form called its structure (Fig. S1 a and h). The particle size/structure of printex carbon black is much higher than that of conductex black. This is one of the reason of getting higher conductivity and low percolation threshold of printex black filled composites compared to conductex black filled one. The carbon fibers are long sticks and are having high length to diameter ratio as is observed from optical microscopy, SEM, and TEM images. Hence, we get same type of observation that is higher electrical conductivity and low percolation threshold for carbon fiber filled composites compared to conductex filled composite one. FESEM images also show that the particle size of printex black is higher compared to conductex black within the composites. At low filler loading, the particles aggregates are isolated from each other (Fig. S1 d and j) but for higher loaded composite, the aggregates are compact with each other and form conductive continuous network (Fig. S1 e and k). Similar thing is observed for carbon fiber filled composites. The cryo-fractured surface of carbon fiber composite shows breakage of fibers (Fig. S1 p); whereas, surface itched sample of carbon fiber composites shows interconnected fiber network within the polymer matrix (Fig. S1 q). This interconnected network helps in the conduction of charge carriers throughout the composite system and hence results in higher conductivity. The fibers are having very low bending strength. As a result, there is the breakage of it during cryo-fracture process. Also, it is seen that there is the holes within the polymer matrix which has happened due to pulling out of the fibers. This indicates low fiber-polymer interaction within the composite. Figure S1. TEM image of (a) Structure of conductex black particle, (b) aggregated conductex black particle; FESEM image of (c) conductex black particle, (d) EC composite at low loading, and (e) EC composite at high loading; TEM image of (f) EC composite at high loading; TEM image of ( g) a single printex black particle, (h) structure/aggregated printex black particle; FESEM image of (i) printex black particle, (j) EP composite at low loading, and (k) EP composite at high loading; TEM image of (l) EP composite at high loading; (m) and (n) are the optical microscopy of SCF; SEM image of (o) SCF, (p) EF composite of cryo-fractured sample, and (q) EF composite of solvent itched sample; TEM image of (r) EF composite showing a single fiber.

Critical Exponent (slope value) and Percolation Threshold
The value of critical exponent t in the classical percolation theory is determined from the slope of the linear plot of log σ vs log (Vf -Vfc) as mentioned earlier within the manuscript. Actually, this linear plot [plotting log σ vs log (Vf -Vfc)] corresponds to the exponential nature of conductivity curve [when plotted log σ vs Vf] at and beyond percolation threshold. Hence, a correlation curve is shown in illustrative Fig.   S2. In this figure two slopes with different value (black line less value; whereas, red line high value) are plotted with their corresponding percolation threshold value. It is seen that with the lowering of slope value, the percolation point has shifted to higher value and hence exhibited higher percolation threshold value. Figure S2. Correlation of critical exponent (slope value) with percolation threshold.