Low percolation threshold of graphene/polymer composites prepared by solvothermal reduction of graphene oxide in the polymer solution

Graphene/polyvinylidene fluoride (PVDF) composites were prepared using in-situ solvothermal reduction of graphene oxide in the PVDF solution. The electrical conductivity of the composites was greatly improved by doping with graphene sheets. The percolation threshold of such composite was determined to be 0.31 vol.%, being much smaller than that of the composites prepared via blending reduced graphene sheets with polymer matrix. This is attributed to the large aspect ratio of the SRG sheets and their uniform dispersion in the polymer matrix. The dielectric constant of PVDF showed a marked increase from 7 to about 105 with only 0.5 vol.% loading of SRG content. Like the other conductor-insulator systems, the AC conductivity of the system also obeyed the universal dynamic response. In addition, the SRG/PVDF composite shows a much stronger nonlinear conduction behavior than carbon nanotube/nanofiber based polymer composite, owing to intense Zener tunneling between the SRG sheets. The strong electrical nonlinearity provides further support for a homogeneous dispersion of SRG sheets in the polymer matrix.     

Lowering the percolation threshold of conductive composites using particulate polymer microstructure

The percolation thresholds of carbon black–polymer composites have been successfully lowered using particulate polymer starting materials (i.e., latex and water‐dispersible powder). Composites prepared using carbon black (CB) and commercial poly(vinyl acetate) (PVAc) latex exhibit a percolation threshold near 2.5 vol % CB. This threshold value is significantly lower than that of a comparable reference composite made from poly(N‐vinylpyrrolidone) (PNVP) solution and the same CB, which exhibits a sharp rise in electrical conductivity near 15 vol % CB. This dramatic difference in critical CB concentration results from the segregated microstructure induced by the latex during composite film formation. Carbon black particles are forced into conductive pathways at low concentration because of their inability to occupy volume already claimed by the much larger latex particles. There appears to be good qualitative agreement between experimental findings and current models dealing with conductive behavior of composites with segregated microstructures. Lack of quantitative agreement with the models is attributed to the polydispersity of the polymer particles in the latex. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 80: 692–705, 2001     

Synthesis and characterization of fullerene based systems for photovoltaic applications: Evidence for percolation threshold

Acceptor polymers for photovoltaic applications were synthesized by grafting fullerene C₆₀ onto polystyrene. The quality of the reaction was verified by various analytical techniques after each of the three steps of the reaction: nuclear magnetic resonance, infrared and UV-visible spectroscopies, and thermo-gravimetric analysis. In order to determine the optimal amount of C₆₀, a series of polymers were prepared containing from 4 to 59 vol.% of fullerene. The optical (absorption, optical gap energy) and electrical (electron mobility, conductivity) properties have been measured. A percolation threshold at around 4 vol.% was identified for both conductivity and mobility measurements. This provides the lowest amount of C₆₀ required for solar cells applications.     

Electrical properties of composites in the vicinity of the percolation threshold

The electrical response of thermoplastic composites composed of carbon black and high‐density polyethylene near the electrical percolation threshold (p_c) has been investigated through the study of the volume resistivity and complex permittivity. The change in conductivity beyond p_c exhibited a critical exponent that was greater than predicted from percolation theory. Composites with carbon black contents slightly larger than p_c exhibited the greatest sensitivity in volume resistivity with temperature variations under the melting point of polyethylene. In addition, percolating composites with low carbon black contents exhibited significant “negative temperature coefficient” (NTC) effects and improvements in conductivity with annealing. Maxwell–Wagner interfacial polarization resulted in moderate increases in both the permittivity (ϵ′) and dielectric loss factor (ϵ″) below p_c, while at percolation, an abrupt and dramatic increase was observed for both components of the complex permittivity. © 1999 John Wiley & Sons, Inc. J Appl Polym Sci 72: 1573–1582, 1999     

The effect of different GNPs addition on the electrical conductivities and percolation thresholds of the SiAlON matrix composites

Insulating SiAlON ceramics may become electrically conductive with the addition of a conductive phase such as GNPs and can be used more widely. However, the differences in the properties of the used GNPs significantly affect the amount of electrical conductivity that they provide to the matrix. In this study, four different GNPs with different properties such as lateral dimension, thickness and aspect ratio were added to SiAlON in the amount of 1.5, 2, 3 and 4 wt. % and the effects of different properties on the conductivity of composites were investigated. The thinnest GNPs with largest dimension and aspect ratio among the used GNPs provided the highest electrical conductivity and lowest percolation thresholds to SiAlON. The decrease in dimension, aspect ratios and the increase in thickness decreased the electrical conductivity of GNPs. Composites exhibited anisotropic behavior with better conductivity and percolation threshold values in the in-plane direction than through-plane direction.     

Electrical conductivity and percolation threshold of hybrid carbon/polymer composites

The electrical conductivity and percolation threshold of single and hybrid carbon filled composites are experimentally investigated. Polystyrene, carbon fiber (CF) and carbon black (CB) at three CF/CB ratios of 1.67, 3.33, 6.67 were compounded in a twin screw extruder micro‐compounder and compression molded into sheets. The through‐plane and in‐plane electrical conductivity of the composites are measured by 2 and 4 probe techniques. The percolation threshold of the single filler and hybrid composites are determined from the experimental results using a percolation model. The hybrid composites have a higher value of electrical conductivity and lower percolation threshold than the single CF filler composite except for the CF/CB ratio of 6.67. The percolation threshold for the cases of single filler and hybrid composites are modeled. The hard core / soft shell model is used and it is assumed that the percolation in a particle filled system depends on the ratio of tunneling distance to particle diameter. This ratio is determined by modeling single filler composites using the experimental data and kept constant in the modeling of the hybrid system. Finite size scaling is used to determine the percolation threshold for the infinite size hybrid system containing (nanosize) particles and micron size fibers for three CF/CB ratios. The simulation results show that the percolations of hybrid composites have the same trends observed in the experimental results. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2015 , 132, 41744.     

Surface modification-based three-phase nanocomposites with low percolation threshold for optimized dielectric constant and loss

Integration of the excellent attributes of high dielectric constant and low dielectric loss in flexible polymer-based nanocomposites has attracted increased research attention because of their extensive applications in modern electronic and electric industry. In this study, to obtain the optimized dielectric constant and loss, the fabrication and properties of a three-phase nanocomposites, including poly(vinylidene fluoride) (PVDF) and two nanofillers, namely, surface-modified multi-wall carbon nanotubes (mCNTs) and barium titanate nanoparticles (mBTs), are investigated in detail. The mCNTs and mBTs were obtained via the hydrolysis of 3-aminopropyltriethoxysilane (AMEO) and condensation reactions between the AMEO and nanofillers. The three-phase nanocomposites are fabricated by a phase-separation and hot-pressing process. The mCNTs and mBTs can be uniformly dispersed within the PVDF polymer matrix because of the enhanced hydrogen bonding interaction and compatibility with the polymer matrix. The percolation threshold (as low as 0.50 vol%) of the two-phase mCNTs/PVDF nanocomposites is adopted to optimize the dielectric properties of the three-phase mCNTs/mBTs/PVDF nanocomposites. At the frequency of 10² Hz, a high dielectric constant of 109 and low loss of 0.06 are obtained for the three-phase nanocomposites with only 0.41 vol% mCNTs and 2.8 vol% mBTs, respectively. Meanwhile, owing to the low percolation threshold and enhanced surface compatibility between the nanofillers and PVDF, the tensile strength of the three-phase nanocomposites is greater than that of PVDF by a factor of greater than 1.5. Owing to their high dielectric constant, low dielectric loss and good mechanical properties, these PVDF-based ternary nanocomposites show potential for applications in electronic devices and energy storage systems.     

Temperature and time dependence of conductive network formation: Dynamic percolation and percolation time

Temperature and time dependence of conductive network formation in vapor-grown carbon fiber (VGCF) filled high-density polyethylene (HDPE)/poly(methyl methacrylate) (PMMA), VGCF and ketjenblack (KB) filled HDPE/isotactic polypropylene (iPP) blends have been investigated. It is found that the filled conductive polymer composites are thermodynamically non-equilibrium systems, in which the conductive network formation is temperature and time dependent, a concept named as dynamic percolation is proposed. When the composites are annealed at a temperature above the melt point of polymer matrix, the dynamic process of conductive network formation can be monitored in a real time way. Such an in situ characterization method provides more interesting information about the dispersion of conductive particles in the polymer matrix. Furthermore, a thermodynamic percolation model is modified to predict the percolation time for VGCF and KB filled HDPE/iPP multi-phase systems during the annealing treatment, and it expresses experimental results well.     

AC conductivity of carbon fiber–polymer matrix composites at the percolation threshold

This paper reports results on experimental investigation of the conductivity behavior of carbon fiber filled polymer composites at the percolation threshold. Two types of carbon fiber‐epoxy matrix composites have been studied and comparison of the measured data has been made. These two types of composites differ in the surface modification of carbon fibers (in one case the surface of carbon fibers is covered with polymer beads using the microencapsulation technology, in the other their surface stayed unmodified). Experimental data reveal that surface modification of carbon fibers influences greatly the DC conductivity (percolation threshold moves to higher concentrations) but does not influence the AC electrical properties. From the frequency dependence of conductivity upon fiber concentration it becomes clear that it is not possible to predict the high frequency conductivity (electromagnetic interference shielding properties) based on the DC conductivity. Percolation behavior of conductivity as a function of conductive filler concentration is typical only for DC or low frequency AC conductivity. The percolation threshold gradually vanishes for high frequencies of electromagnetic field. The temperature dependence of electrical properties has also been studied. Composites with concentration near the percolation threshold show the switch‐off effect (at the specific temperature the DC conductivity drops by several orders of magnitude). This switch‐off effect does not occur for high frequency AC conductivity.     

A percolation threshold model that effectively characterizes the full concentration range for electrical-conducting polymer composites

The new percolation threshold model introduced in this study is a modification of Clingerman's model, which in turn is a modification of the model originally developed by Mamunya et al. Several of the original constants from Clingerman's and Mamunya's were consolidated, and most importantly, the concentration of the percolation threshold is not a required constant in the model. One extraordinary characteristic of this new model is that it is possible to separate this model into two different equations that separately describe the conducting filler component and the insulating matrix component. In general, this new percolation threshold model introduces five new calculated quantities on the S-shaped curve, including the calculated volume fraction of the percolation threshold. The capability of this new model was evaluated with three of Clingerman's polymer composite percolation threshold data sets involving electrical conductivity measurements. This new model did an excellent job of fitting the data extremely well over the whole concentration range. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2019 , 136, 47184.     

Microstructure evolution and simulation of copolymerization reaction using a percolation kinetic gelation model

A new kinetic gelation model that incorporates the kinetics of non-linear free radical copolymerization is presented. Copolymerization of bi- and tetrafunctional monomer mixtures is simulated to characterize kinetic effects on polymerization statistics and microstructures. An algorithm for random next step selection in a self-avoiding random walk and efficient mechanisms of component's mobility are introduced to improve the universality of the predictions by removing commonly occurring simulation deficiencies due to early trapping of radicals. The model has the capability to predict the onset of the sol-gel transition, and the effect of chemical composition on the transition point. It is shown that a better understanding of microstructure evolution during polymerization and chemical gelation is attained. Lastly, one important benefit of the simulation method is the ability to simulate very highly packed random chains or microgels within a polymer network.     

Tensile modulus of polymer/CNT nanocomposites containing networked and dispersed nanoparticles

The properties of three‐dimensional networks of nanoparticles in polymer/carbon nanotubes (CNT) nanocomposites (PCNT) are particularly interesting from fundamental and application views. In this article, a new model is suggested for predicting the tensile modulus of PCNT using the Ouali and Paul models. The Ouali model considers the network of CNT in a polymer matrix, while the Paul model predicts the tensile modulus of samples containing dispersed nanoparticles. The predictions of the suggested approach are compared with experimental data from several samples. Also, the roles of the main parameters in the tensile modulus of PCNT are evaluated. The predictions agree with the experimental results at different filler concentrations. The roles of these parameters on the tensile modulus of PCNT are discussed based on the properties of CNT networks. © 2017 American Institute of Chemical Engineers AIChE J, 63: 220–225, 2018     

Effect of clay orientation on the tensile modulus of polypropylene-nanoclay composites

We present an experimental investigation of the effect of clay orientation, as produced by melt extrusion, on the tensile modulus of compatibilized and uncompatibilized syndiotactic polypropylene nanoclay composites. The orientation of the clay tactoids in extruded tape samples was quantified using 2D X-ray diffraction data. It was found that in the case of the tapes made from compatibilized nanocomposites the orientation of the clay tactoids increased with extrusion shear rate, while in the case of tapes extruded from uncompatibilized hybrids the clay orientation was independent of the shear rate. Tensile modulus of the extruded tapes along the flow direction was measured and was found to correlate well with the average orientation of the clay tactoids. In the case of the compatibilized hybrids the modulus increased with the extrusion shear rate until a saturation value, whereas for the uncompatibilized hybrids the modulus was nearly independent of the shear rate. Semi-quantitative predictions of the effect of clay orientation on the tensile modulus of the compatibilized tape samples were obtained using a micromechanical model.     

Characterization of the granular packing and percolation threshold of reactive powder concrete

Reactive powder concrete (RPC) is a new cement-based material developed through microstructural engineering. RPC is composed of very fine powders: sand, crushed quartz and silica fume, all with particle sizes comprised between 300 and 0.02 μm, and a low water content, W/C<0.20. A very dense matrix is achieved by optimizing the granular packing of these powders. This compactness confers to RPC ultrahigh strength and durability.

The hydration kinetic of the cementitious matrix using electrical conductivity and isothermal calorimetry is presented. A linear relationship between the logarithm of conductivity ln(σ) and the degree of hydration , has been found:

ln(σ)=A_v−B_v

where A_v depends on the nature and the dosage of the different mineral and organic components, and B_v is an invariant related to the granular packing. From 19 different RPC compositions, an average value of 12.1 with a standard deviation of 1.7 were found for B_v. The capillary porosity percolation threshold was also deduced from this relationship. It has been found that when the degree of hydration is equal to 26%, the capillary pore space is discontinuous. This last result is in agreement with the prediction of the NIST microstructural model by Bentz and Garboczi [D., Bentz, E. Garboczi, Percolation of phases in a three-dimensional cement paste microstructural model, Cem Concr Res 21 (2) (1991) 325–344]. A continuous determination of the degree of hydration based on electrical conductivity is proposed.     

Numerical simulation of the effect of nanotube orientation on tensile modulus of carbon‐nanotube‐reinforced polymer composites

By correlating the curvature of carbon nanotubes to the orientation of fibers in a polymer, the effect of the curvature of nanotubes on the tensile modulus of carbon‐nanotube‐reinforced polymer composites was investigated with a numerical simulation method. The simulation results showed that the tensile modulus of a nanotube‐reinforced composite drops sharply when the nanotubes diverge from their orientation in the axial direction, and the presence of curved nanotubes in the polymer matrix significantly decreases the modulus of the composite. This finding could explain, partly, why in most cases, the predicted tensile modulus of a carbon‐nanotube‐reinforced composite, based on the assumption that the nanotubes are fully isolated and aligned in the polymer matrix, is much higher than the value obtained from experiments. Copyright © 2004 Society of Chemical Industry     

Particle size dependence of the elastic modulus of particulate filled PMMA near its Tg

Models for composition dependence of elastic modulus of particulate filled polymers inherently assume modulus of the matrix independent of particle content and size. In this letter, experimental evidence is presented for existence of a critical particle size below which elastic modulus of the matrix becomes strongly dependent on the particle content due to the extensive chain stiffening. It is also suggested that below the critical particle size, specific interface area should replace volume fraction as the structural variable.     

Effects of the filler size on the electrical percolation threshold of carbon black–carbon nanotube–polymer composites

The Monte Carlo simulation of filled conductive polymer materials is a method of continuously generating a random conformation and averaging the interesting results to simulate the random dispersion of fillers in space. In our simulation, the irregular shapes of the filler were abstracted into regular ones. Carbon black (CB) aggregates were modeled as spheres, and carbon nanotubes (CNTs) were modeled as capped cylinders. The connection of the fillers was estimated via the calculation of the shortest distance between them; this determined whether a percolation pathway was formed or not. Numerical results were obtained, and these highlight the effects of the filler size, including the aspect ratio of CNTs and the ratio of the diameter of the CB aggregates to the diameter of the CNTs on the electrical percolation threshold (EPT). We found that the EPT decreased with increasing CNT aspect ratio and decreased the diameter ratio of the CB aggregates to CNTs. The simulation results were obtained by a simple pathway‐finder algorithm, which was proven to be effective compared with existing numerical, theoretical, and experimental simulations. © 2018 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2018 , 135, 46517.     

The elastic modulus of filled polymer composites

The elastic modulus of polyepichlorohydrin (PECH) filled with glass beads and wollastonite was studied. It was found that the elastic modulus of the composites depends not only on the volume fraction of the fillers but also on their size. Percolation theory was used to explain the experimental results. © 1993 John Wiley & Sons, Inc.     

Study on percolation phenomena and mechanism of enhanced asphalt by polymers

Abstract

The properties of asphalt modified by styrene–butadiene–styrene, polyvinyl acetate and waste crumb rubber separately in a wide range of polymer loadings were studied in this paper. In analysing the relationship between polymer loading and modified asphalt properties, we found that the ductility of polymer modified asphalt measured at 5°C exhibited a percolation phenomenon, that is, as the loading of polymer exceeded a critical value, the ductility of modified asphalt changed abruptly. Furthermore, a percolation threshold differential existed. A rubber processing analyser, an optical microscope and a fluorescence microscope were used to characterise the microstructure of polymer modified asphalt and validate the occurrence of a polymer network in modified asphalt as a function of polymer loading. The percolation phenomenon of polymer modified asphalt was found to be associated with the polymer network in modified asphalt and can be well explained by the percolation mechanism of rubber toughened plastics.     

Fractality and the percolation transition in complex networks

In this work we study the percolation phase transition of a family of scale-free networks previously introduced to explain the emergence of fractality and self-similarity in complex networks. This model introduces a parameter, e, that allows for tuning the level of fractality in the network. When e=0 the network is a pure fractal, when e=1 it is a pure small-world network. We examine link percolation for several intermediate values of e (0?e?1), and find that the transition follows a continuous function of e that converges to exactly 0 when the network exhibits the small-world property.