Percolation in polydisperse systems of aligned rods: a lattice-based analysis.

A model is developed for percolation in polydisperse systems of oriented cylinders that integrates excluded volume arguments with an analogy to site percolation on a modified Bethe lattice. Results from this treatment are presented for the volume fraction at the percolation threshold (denoted ϕc) as a function of the degree of polydispersity, mixture composition, and degree of orientational ordering. For monodisperse systems, ϕc is found to be a monotonically increasing function of the traditional orientational order parameter that quantifies degree of alignment. The presence of a fraction of isotropically oriented rods of small aspect ratio is shown to lower the percolation threshold for systems in which the longer rods are strongly aligned.