Contribution to the Percolation Threshold Study of Silicon Carbide Filled Polydimethylsiloxane Composites Used For Field Grading Application

Nonlinear V-I characteristics of particulate composite prepared from dispersion of silicon carbide in a siloxane elastomer have been measured as a function of filler concentration up to the maximum allowable of about 32 vol.%. Two critical concentrations (percolation thresholds) are obtained at volume fractions of about 17 and 24 vol.% for low and high electric fields. These values are consistent with the critical concentrations predicted by the theory: 14 and 31 vol.% respectively and may come from edge and face contacts between SiC semi-conducting particles.


Introduction
SiC composites are used as corona discharge protection materials in stator winding coils. 1,2(Corona discharge refers to a phenomenon that occurs in high-voltage systems, where a localized ionization of the surrounding air takes place near sharp or pointed conductors.)These semiconductor materials have a nonlinear current/voltage relationship. 3This characteristic can be taken advantage of to the active control of the distribution of the electric field within or around a device by self-adapting the conductivity of the composite in such a way that its conductivity decreases enough to decrease locally the electric field under an acceptable value.This prevents partial electrical discharge caused by the ionization of the air.At a low SiC volume fraction, current flows through the least resistant path, which, in this case, is within the continuous phase: the matrix.Above a threshold volume fraction, f v , the current/voltage relation is no longer linear. 4In this phase, electric current is believed to be carried by the network of percolated SiC particles, which is more conductive than the silicone matrix.
In the present paper specimens of SiC-Silicone composites have been prepared and their electrical conductivity, σ, reported as a function of the electric field (Figure 1).
Regarding the conductivity vs volume fraction characteristic of the silicone-SiC composites, two percolation thresholds were identified.The first threshold was found at a volume fraction of 17.45%, and the second at 24.14%.These specific SiC concentration separate three conductive zones (Figure 1).The first threshold volume fraction agrees with the classical theory.The value of 16 vol.% is commonly accepted for spherical conductive particles in a continuous path in a three-dimensional network. 5he second threshold is however more exotic.6][7][8] The experimental work presented in [6][7][8] shows only a percolation at a fraction of 40 vol.% at 300 V/mm and only one percolation threshold at a fraction of about 14 vol.% at 1 kV/mm, failing at identifying both thresholds for a given electric field.The observation of two percolation thresholds was explained by the irregular shape of the fillers.The fillers being non-spherical, this leads to two types of contacts between fillers, namely face contact and edge contact, which in turn lead to two percolation thresholds.In order to shade light on this interpretation, we tried to correlate the microstructure to the electrical properties thanks to X-ray microtomography analysis.This investigation was performed at the BM05 beamline at the European Synchrotron Radiation Facility (ESRF) in Grenoble.

Material and method
Composites were formed from a two-component silicone elastomer supplied by Bluestar™ silicones which cures by a polyaddition reaction catalyzed by a chlorine platinum salt at room temperature after about 24 hours.The elastomer is supplied as a two-part liquid component kit.These parts need to be mixed at 10:1 by weight respectively to make a silicone polymer matrix defined by its inorganic main-chain backbones: -[Si-O] n -.The SiC employed was a doped industrial β-crystal structure grade with a particle size and distribution assesed by laser diffractometry to 26 ± 1 µm and 20 ± 1 µm respectively. 4The errors due to the weighing of viscous materials are of the order of ± 0.05 g, i.e. less than 0.25%.The SiC powders were used as received in the composite materials.The desired filler concentrations were first mixed into the base resin by dual asymmetric centrifugal mixing which ensures a highly efficient mixing at about 3-5 mPa vacuum.Silicone-SiC composite materials were prepared by compression molding at 150°C under pressure 150 bar (15 MPa) for 15 minutes in a plate vulcanization machine.Various Silicone-SiC composite materials were fabricated using a steel die with a thickness of 1 mm and a lateral size of 114 x 80 mm as backing and confining material.The composite samples were then stored in a vacuum drier for testing (24h maturation at room temperature).Table 1 lists the prepared specimens.
DC volume conductivity investigations were performed using an electrometer (Keithley 6517B) associated to a resistivity chamber which shields the sample from electrostatic interference.The chamber is equipped with two conductive flexible electrodes enhancing surface contact between the sample and the electrode.The cathode is guarded to prevent surface leakage currents from being added into the measurement following ASTM D257 requirements.The resistance was measured by applying DC 200V voltage steps followed by short-circuit (samples were subjected to zero voltage) while the polarization and the depolarization currents were continuously monitored during 1800 s.The resistance was calculated using the stable portion of the charging current curve and the average electrical field was assumed to be the applied voltage divided by the inter electrode distance.The Keithley 6517B Electrometer is a high-performance instrument designed for measuring extremely low currents down to several femtoamperes (10 À15 A).However, the accuracy depends on the range of measurement and conditions.Considering the electrical measurement bench implemented and the 18-28°C and 200-1000V DC ranges considered in this study, the accuracy of the voltage is given to ± (0.06% + 3 mV) with a calibration performed eleven months ago.The accuracy of current, when measure in the nA range, is about ± (0.2% + 5 pA).The common mode rejection ratio (CMRR) is > 120 dB at DC.
The X-rays generated by ESRF are produced by highenergy electrons circulating in a 844 m circumference annular accelerator.The brilliance of the produced light is several orders of magnitude higher than that of medical X-ray generators.
To relate these thresholds to the microstructure of the composites, several scans were performed on samples of a few mm 3 (2.5x2.5x1mm 3 ) with a voxel size of approximately 0.4 µm.For each formulation, two volumes of approximately 750 μm in diameter and height were acquired, reconstructed, corrected for major artifacts, and then analysed.Figure 2 depicts the dispersion and distribution of SiC particles in such volume.The particles are well dispersed without apparent formation of agglomerates.
The processing of these acquisitions was performed using internal computer tools based on Python libraries.Two indicators were defined: (i) the interconnection corresponding to the percentage that represents the largest network of interconnected particles relative to the total volume of particles; (ii) the tortuosity defined as the ratio of the actual path within the conductive phase to the minimum distance between the two ends of the sample that was traversed.The correction coefficient to obtain the effective diffusivity (D eff ) from the intrinsic diffusivity (D) of the conductive phase was also determined as follows: D eff D = f τ ,   where f is the volume fraction of the latter and τ is the tortuosity.These two parameters were calculated using the Taufactor library (Python).The calculation methods are detailed in. 9

Results
From a volume fraction of 17.45% SiC, there is an increase in the volume of the largest interconnected particle network.By estimating the percentage of volume occupied by this object relative to the total volume of particles (defined as the interconnectivity parameter), it is observed that the size of this network increases rapidly, and the remaining particles represent a decreasingly negligible fraction of all SiC particles (<1% in the case of the 27.95 vol.%).
The Figure 3 is a dual-axis graph that demonstrates the existence of a correlation between the percolation threshold at 17.45 vol.% and the microstructure interconnectivity.Indeed, the interconnectivity abruptly increases from 20 vol.% to approximately 90% and then saturates for the most concentrated formulation: 27.95 vol.%.
Figure 4 shows the tortuosity as a function of the SiC concentration.A significant decrease in tortuosity is observed after the concentration of 20.59 vol.%.This concentration corresponds to the second percolation threshold observed in the σ (E) curves.
The normalized effective diffusivity ( D eff D ) is proportional to τ, and an inverse relationship with tortuosity is observed, exhibiting a threshold behaviour when plotting the effective diffusivity against the interconnection parameter (Figure 5).
The study conducted using X-ray microtomography provides a physical interpretation of the two percolation thresholds observed in the σ(E) characteristic.The first threshold is attributed to a significant increase in interconnectivity, while the second threshold is associated with a significant decrease in tortuosity.Figure 6 summarizes the four regions of the σ(E) characteristic.At low field strength, the particles are dispersed and without contact in the insulating matrix.Beyond the first percolation threshold (17.45 vol.%), a path through the particles appears.Above 20.59 vol.%, the percolation paths between electrodes become more direct, resulting in a substantial reduction in tortuosity (from 3.3 to 2.5).With higher filler concentrations, a shorter conduction path is more likely to    form i.e. conduction paths become more and more straightforward.The tortuosity leap is therefore attributed to a general increase of the overall surface contact between particles rather than a transition between edge contact and face contact of aligned parallelepiped particles.Over a certain threshold an increase in surface contact does not impact much the percolation path once this one is close to the shortest possible length and the electrical conductivity plateau.

Conclusion
In the voltage field range 200-1000 V/mm, the conductivity of SiC-polydimethylsiloxane composite exhibits two leaps with the increase of the volume fraction of SiC.The microtomography analysis performed in the framework of the current study shows that the first gap is related to the onset of particles interconnection.Within this gap, between 17.5 and 20.6 vol% of SiC, most of the particles are connected while the tortuosity remains relatively high.Then the analysis reveals that for a volume fraction that varies between 20.6 and 28% vol, the tortuosity of the formed SiC network significantly drops.This result suggests that the second conductivity gap can be attributed to a sudden appearance of significantly short conductive paths with the increase of the SiC volume fraction.Further work at smaller length scales would be needed to also investigate the evolution of surface contact between particles to assess the role of conduction path thickening.

Figure 1 .
Figure 1.Conductivity versus SiC volume fraction (In hatched frames, the formulation studied by microtomography).

Figure 2 .
Figure 2. Volume cropping after artefact and noise reduction for the three critical SiC concentrations.

Figure 3 .
Figure 3. Percentage of the largest agglomerate relative to the total volume of particles in the volume studied by microtomography, or % interconnectivity (■), and apparent electrical conductivity measured at 40°C, 1 kVmm À1 , RH<20%, 1800:1800s (Á), as a function of volume fraction.

Table 1 .
Weight and volume experimental concentrations (±0.02) of the prepared specimens.