  The properties of polymer composites with nanofiller particles change
drastically above a critical filler density known as the percolation threshold.
Real nanofillers, such as graphene flakes and cellulose nanocrystals, are not
idealized disks and rods but are often modeled as such. Here we investigate the
effect of the shape of the particle cross section on the geometric percolation
threshold. Using connectedness percolation theory and the second-virial
approximation, we analytically calculate the percolation threshold of hard
convex particles in terms of three single-particle measures. We apply this
method to polygonal rods and platelets and find that the universal scaling of
the percolation threshold is lowered by decreasing the number of sides of the
particle cross section. This is caused by the increase of the surface area to
volume ratio with decreasing number of sides.
