Fillers percolation threshold

With an increase in filler concentration that exceeds the filler percolation threshold, a loose filler network is formed that would show elastic behavior and. 

Polymer Filler Percolation threshold (wt%) Matrix conductivity (S cm / Maximum composite conductivity (S cm-/ Processing method Reference... 

The maximum values of the percolation threshold are characteristic of matrix systems in which the filler does not form the chain-like structures till large concentrations are obtained. In practice, statistical or structurized systems are apparently preferable because they become conductive at considerably smaller concentrations of the filler. The deviation of the percolation threshold from the values of Cp to either side for a statistical system ( 0.15) can be used to judge the nature of filler distribution. 

The composites with the conducting fibers may also be considered as the structurized systems in their way. The fiber with diameter d and length 1 may be imagined as a chain of contacting spheres with diameter d and chain length 1. Thus, comparing the composites with dispersed and fiber fillers, we may say that N = 1/d particles of the dispersed filler are as if combined in a chain. From this qualitative analysis it follows that the lower the percolation threshold for the fiber composites the larger must be the value of 1/d. This conclusion is confirmed both by the calculations for model systems [27] and by the experimental data [8, 15]. So, for 1/d 103 the value of the threshold concentration can be reduced to between 0.1 and 0.3 per cent of the volume. 

The defects caused by the high contact resistance especially manifest themselves in the metal-filled composites where the value of the percolation threshold may reach 0.5 to 0.6 [30]. This is caused by the oxidation of the metal particles in the process of CPCM manufacture. For this reason, only noble metals Ag and Au, and, to a lesser extent, Ni are suitable for the use as fillers for highly conductive cements used in the production of radioelectronic equipment [32]. 

In pressing, the threshold concentration of the filler amounts to about 0.5% of volume. The resulting distribution of the filler corresponds, apparently, to the model of mixing of spherical particles of the polymer (with radius Rp) and filler (with radius Rm) for Rp > Rm as the size of carbon black particles is usually about 1000 A [19]. During this mixing, the filler, because of electrostatical interaction, is distributed mainly on the surface of polymer particles which facilitates the forming of conducting chains and entails low values of the percolation threshold. 

For the second method the threshold concentration of the filler in a composite material amounts to about 5 volume %, i.e. below the percolation threshold for statistical mixtures. It is bound up with the fact that carbon black particles are capable (in terms of energy) of being used to form conducting chain structures, because of the availability of functional groups on their surfaces. This relatively sparing method of composite material manufacture like film moulding by solvent evaporation facilitates the forming of chain structures. 

In the case of the filler localization in one of the polymer components of the mixture, an increase of the portion of the second unfilled polymer component in it entails sharp (by a factor of lO10) rise of a in the conducting polymer composite. In this case the filled phase should be continuous, i.e. its concentration should be higher than the percolation threshold. 

As already noted, the main merit of fibers used as a filler for conducting composite materials is that only low threshold concentrations are necessary to reach the desired level of composite conductivity. However, introduction of fiber fillers into a polymer with the help of ordinary plastic materials processing equipment presents certain difficulties which are bound up mainly with significant shearing deformations entailing fiber destruction and, thereby, a decrease of parameter 1/d which determines the value of the percolation threshold. 

Fillers with extremely high aspect ratios (1000-10,000) such as carbon nanotubes (Figure 32.5) have a much lower percolation threshold (lower amount is required for equivalent reinforcement). 

Polymer Filler type Percolation threshold Volume or surface realativity Processing method... 

Hu et al. showed a decrease in electrical resistivity of PVA by four orders of magnitude with a percolation threshold of 6 wt% [68], while biodegradable polylactide-graphene nanocomposites were prepared with a percolation threshold as low as 3 5wt% [46]. For polystyrene-graphene composites, percolation occurred at only 0.1 °/o of graphene filler, a value three times lower than those for other 2D-filler [69]. Figure 6.7(b) shows the variation of conductivity of the polystyrene-graphene composite with filler content. A sharp increase in conductivity occurs at 0.1 % (the percolation threshold) followed by a saturation. The inset shows the four probe set up for in-plane and trans-... 

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). 

Fig. 8.3 Percolation threshold for different aspect ratio fillers and...

This behavior becomes more transparent in Fig. 30a,b, where the a.c.-con-ductivity a and relative dielectric constant (permittivity e ), respectively, for a series of less polar S-SBR-samples filled with various amounts of the coarse black N550 are show at 20 °C in a broader frequency range up to 107 Hz. For filler concentrations below the percolation threshold (O<0.15), the conductivity behaves essentially as that of an isolator and increases almost linearly with frequency. Above the percolation threshold (5>>0.2), it shows a characteristic conductivity plateau in the small frequency regime. Since at low frequencies the value of the conductivity a agrees fairly well with the d.c.-con-ductivity, the plateau value exhibits the characteristic percolation behavior considered above. In the high frequency regime the conductivity depicted in... 

The reduced value of the scaling exponent, observed in Fig. 29 and Fig. 30a for filler concentrations above the percolation threshold, can be related to anomalous diffusion of charge carriers on fractal carbon black clusters. It appears above a characteristic frequency (O when the charge carriers move on parts of the fractal clusters during one period of time. Accordingly, the characteristic frequency for the cross-over of the conductivity from the plateau to the power law regime scales with the correlation length E, of the filler network. 

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]... 

Zhang et al. studied the effect of conductive network formation in a polymer melt on the conductivity of MWNT/TPU composite systems (91). An extremely low percolation threshold of 0.13 wt% was achieved in hot-pressed composite film samples, whereas a much higher CNT concentration (3-4 wt%) is needed to form a conductive network in extruded composite strands. This was explained in terms of the dynamic percolation behavior of the CNT network in the polymer melt. The conductivity of extruded strand showed a hopping resistivity dominated behavior at low concentrations and a dynamic percolation induced network dominated behavior at higher concentrations. It was shown that a higher temperature can reduce the filler concentration required for the dynamic percolation to take effect. 

This paper represents an overview of investigations carried out in carbon nanotube / elastomeric composites with an emphasis on the factors that control their properties. Carbon nanotubes have clearly demonstrated their capability as electrical conductive fillers in nanocomposites and this property has already been commercially exploited in the fabrication of electronic devices. The filler network provides electrical conduction pathways above the percolation threshold. The percolation threshold is reduced when a good dispersion is achieved. Significant increases in stiffness are observed. The enhancement of mechanical properties is much more significant than that imparted by spherical carbon black or silica particles present in the same matrix at a same filler loading, thus highlighting the effect of the high aspect ratio of the nanotubes. 

The recognition of the unique properties of carbon nanotubes (CNTs) has stimulated a huge interest in their use as advanced filler in composite materials. In particular, their superior mechanical, thermal and electrical properties are expected to provide much higher property improvement than other nanofillers (18-22). For example, as conductive inclusions in polymeric matrices, CNTs shift the percolation threshold to much lower loading values than traditional carbon black particles. 

Typical strain dependences of volume resistivity are shown in Figure 12.7 the results are related to an ethylene-propylene-diene rubber (EPDM) (supplied by ExxonMobil Chemical under the trade name Vistalon 5601) filled with 6 phr of MWNTs that is at a filler content well above the percolation threshold (determined around... 

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