Exact scaling laws for electrical conductivity properties of nematic polymer nano-composite monodomains

The purpose of this paper is to connect two critical aspects of nano-composite materials engineering. The nano-elements considered here derive from the class of high aspect ratio nematic polymers, either rod-like or platelet spheroids. First, the overall electrical properties of polymer nano-composites are well approximated by the effective electrical conductivity tensor in the low volume fraction regime of the included phase. In turn, the effective conductivity is strongly influenced by the orientation distribution of the nano-inclusions. Second, we recall results of Doi-Hess kinetic theory or mesoscopic model approximations of the orientational probability distribution, for quiescent and sheared nematic polymers, at both isotropic and ordered volume fractions. Putting the two features together, we derive the effective electrical conductivity tensor in closed form. Scaling properties of enhanced conductivity versus volume fraction and weak shear rate become explicit. The most dramatic effect is that the effective conductivity tensor inherits hysteresis, bi-stability and discontinuous jumps from the isotropic-nematic first order phase transition. These formulas reveal finer estimates depending on a competition between two inherent large parameters in nematic polymer nano-composites: the molecular aspect ratio and the conductivity ratio of the inclusions and matrix. For this first paper we restrict to steady monodomain orientational distributions at rest and in weak shear flows, which serve as benchmarks and guides for future extensions and numerical approaches.