  We noted that the tunneling-percolation framework is quite well understood at
the extreme cases of percolation-like and hopping-like behaviors but that the
intermediate regime has not been previously discussed, in spite of its
relevance to the intensively studied electrical properties of nanocomposites.
Following that we study here the conductivity of dispersions of particle
fillers inside an insulating matrix by taking into account explicitly the
filler particle shapes and the inter-particle electron tunneling process. We
show that the main features of the filler dependencies of the nanocomposite
conductivity can be reproduced without introducing any \textit{a priori}
imposed cut-off in the inter-particle conductances, as usually done in the
percolation-like interpretation of these systems. Furthermore, we demonstrate
that our numerical results are fully reproduced by the critical path method,
which is generalized here in order to include the particle filler shapes. By
exploiting this method, we provide simple analytical formulas for the composite
conductivity valid for many regimes of interest. The validity of our
formulation is assessed by reinterpreting existing experimental results on
nanotube, nanofiber, nanosheet and nanosphere composites and by extracting the
characteristic tunneling decay length, which is found to be within the expected
range of its values. These results are concluded then to be not only useful for
the understanding of the intermediate regime but also for tailoring the
electrical properties of nanocomposites.
