Schematic drawings of the composite systems considered. (a) Random two-phase material shown as white (polymer matrix) and black (filler particles) regions: no interaction exists between the filler particles and the host matrix. (b) Random three-phase composite system in which the interphase (dotted region) volume is dependent upon the characteristics of the filler component. Because of the physical adsorption of the polymer chains to the filler surface, the filler particles are “glued” in the matrix. Conductivity is determined by resistance of direct contact or by the properties of the inter-particle layer where the tunnel effect allows electrons to pass through the thin layers. This contact resistance causes also a region of smearing to appear. The region of smearing represents the area of critical transition from the insulator sate to the conductor state.
Experimental (symbols) dc conductivity data as a function of CB volume fraction (normalized to ) and corresponding best fits (lines) of the data to the percolation equations, i.e., Eqs. ((1))–((3)), for CB filled DGEBF and different types of CB: (solid star) Raven 2000, (circle) Raven 5000, (square) Raven 7000, and (open star) Monarch 1100. Room temperature.
(Color online) Calculated values of the region of the complex plane inside in which the allowed values (cross-hatched region in the inset) of the intrinsic complex (relative) permittivity of the carbonaceous phase exist from the HS bounds. Raven 7000 filled DGEBF sample with ϕ2 = 5.5 vol.% (> ). F = 15 MHz. Room temperature.
(Color online) The allowed values (indicated by vertical bars) of as a function of frequency for Raven 7000 filled DGEBF sample and ϕ2 = 2 vol.% (< ). Each symbol refers to the average value of and . (b) Same as in (a) for for ϕ2 = 2 vol.%. (c) Same as in (a) for for ϕ2 = 4.5 vol.% (> ). (d) Same as in (a) for ϕ2 = 4.5 vol.%. (e) Same as in (a) for for ϕ2 = 5.5 vol.% (>). (f) (d) Same as in (a) for ϕ2 = 5.5 vol.%.
Same as in Fig. 4 for the Raven 2000 filled DGEBF sample with ϕ2 = 3 vol.%.
Same as in Fig. 4 for the Raven 5000 filled DGEBF sample with ϕ2=5.5 vol.%.
Same as in Fig. 4 for the Monarch 1100 filled DGEBF sample with ϕ2 = 3 vol.%.
(Color online) (a) A fit of the frequency dependence of the average values (symbols) of the estimates of obtained from HS bounds with the Drude metal model, for the Raven 7000 filled DGEBF sample with ϕ2 = 4.5 vol.% (>). (b) Same as in (a) for ϕ2 = 5.5 vol.% (>).
The specifications of the CB materials examined in the current study from manufacturer product literature (Ref. 47) and extracted parameter data of the carbonaceous phase obtained from the comparison of the experimental and theoretical values using Eqs. ((1))–((3)). The neat DGEBF has a dc conductivity ≈10−14 Ω−1 m−1. The curves in Fig. 2 show that the effective conductivity data of the CB filled samples measured at room temperature can be fitted using Eqs. ((1))–((3)).
The values of the plasma frequency ωp and scattering frequency ωd estimated from the Drude model in the case of a percolative morphology of the carbonaceous phase (Raven 7000 filled DGEBF samples with 4.5 and 5.5 vol. %). For comparison, we have also indicated typical values of ωp and ωd for an ideal metal (gold) and SiC.
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