I have been long interested in the measurement and theoretical/empirical description of the dielectric properties of heterogeneous mixtures. In the case of non-conducting or semiconducting fillers in an insulating matrix various mixture formulas based on continuum theories give at least a qualitative framework to describe the observed phenomena, sometimes with semiquantitative agreement between observed and predicted properties, such as filler shape and orientation dependence, volume fraction dependence of static and high frequency limiting permittivities. One of the more probelmatic parameters is the MW relaxation time, which is determined by the conductivity of the fillers. Here surface and contamination effects (e.g. the peresence of adsorbed water) can cause serious deviations from the simpliscit model.
Nowadays CNT based composites are widely studied. The majority of the papers deal with the percolation phenomenon under DC or AC conditions. Much less papers (although still many) publish complex permittivities or impedance analysis. Let us assume that we want to study CNT filled polymeric composites BELOW the percolation threshold. (Close to or above the percolation threshold we have additional problems due to contact resistances between the nanofibers). Let us assume that we want at least to try to describe the observed behavior in terms of e.g. the Mawell Garnett theory (I mention this model, as the physicists dealing with metal/insulator composites usually this one better than others). See e.g. an excellent paper:
Article Three-dimensional simulations of the complex dielectric prop...
We face several problems: the first of which: what permittivity or conductivity can be attributed to a single nanotube?
In a very interesting PhD work Andreas Setter investigates the resistivity of carbon nanotubes in terms of wall number and defect structure:
https://www.lehmanns.de/shop/naturwissenschaften/20162313-9783868450767-conductivity-of-multiwall-carbon-nanotubes-role-of-multiple-shells-and-defects
It turns our that the determination of these quantities are far from obvious.
In another paper
Article Optical and electrical properties of preferentially anisotro...
the one dimensional Drude model is applied to conductive CNTs and another mode for semiconducting nanotubes. In the Drude model the permittivity functon is 1 at low frequencies (much lower than the plasma frequency which appears to be in the THz range) and changes close to the plasma resonance. Does it mean that below the plasma freqency the addition of metallic filler can DECREASE the permittivity of a polymer? (The real permittivity of polymers is higher than 2). Of couse, only if the frequency is higher than the Maxwell-Wagner relaxation frequency.
Then comes the next question: what conductivity can be ascribed to a single CNT? (in ohm-1cm-1 units). Is it calcuated from the band theory of an infinite tube? How to account for the end-effects? Do the nanotubes form a macro-dipole by themselves if they do not coantact each other or tunneling is not allowed between them? How to account fo the fact that nanotubes are very far form being straight rods?
Scanning Microwave Microscopy technique will allow to scan and measure the conductivity this type of material down to um (and even nm).
However, the approach (i.e. crystal or bulk etc.) you need to follow depends on the final application and the hypothesis you are trying to prove.
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