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A variational approach to predict percolation thresholds of nanomaterials

Study on the properties of inhomogeneous media plays a central role throughout the history of physics, e.g.
Possion’s theory on magnetism of composites, Faraday’s dielectric model of composites, Maxwell’s and later
Rayleigh’s work on conductivity of composites, and Einstein’s thesis on viscosity of fluid-particle media. As
a special branch of this field, percolation research on phase transition and critical phenomena was originated
with lattice percolation theories in 1950s [1]. The idealization of a lattice structure is convenient for analytical
and numerical manipulations; however, it is far from reality of most inhomogeneous materials. Continuum
percolation theories were then proposed [2], which mainly consist of the interpenetrating model, e.g. [3,4],
and the potential model, e.g. [5,6]. There still remains a major theoretical question on rigorous determination
of percolation thresholds, especially for those fillers with large aspect ratios that have significant applications
in the fast-growing field of nanotechnology. This study aims to fill the gap between the lack of theoretical
prediction and the gigantic amount of experimental results produced each year on percolation. Universal formulae
of percolation thresholds for various transport properties of composites are for the first time rigorously
presented. New bounds of transport properties and percolation thresholds estimated enable the geometry of
fillers or cavities, the most direct and obvious microstructure information, to be explicitly taken into account
for both engineering composites and natural media (rocks, soils, sands) containing spheroidal particles/voids,
fibers, cracks, nanotubes, etc.

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