Percolation of particles on recursive lattices using a nanoscale approach. I. Theoretical foundation at the atomic level

Powdered materials of sizes ranging from nanometers to micrometers are widely used in materials science and are carefully selected to enhance the performance of a matrix. Fillers have been used in order to improve properties, such as mechanical, rheological, electrical, magnetic, thermal, etc., of the host material. Changes in the shape and size of the filler particles are known to affect and, in some cases, magnify such enhancement. This effect is usually associated with an increased probability of formation of a percolating cluster of filler particles in the matrix. In this series of papers, we will consider lattice models. Previous model calculations of percolation in polymeric systems generally did not take into account the possible difference between the size and shape of monomers and filler particles and usually neglected interactions or accounted for them in a crude fashion. In our approach, the original lattice is replaced by a recursive structure on which calculations are done exactly and interactions as well as size and shape disparities can be easily taken into account. Here, we introduce the recursive approach, describe how to derive the percolation threshold as a function of the various parameters of the problem, and apply the approach to the analysis of the effect of correlations among monodisperse particles on the percolation threshold of a system. In the second paper of the series, we tackle the issue of the effect of size and shape disparities of the particles on their percolation properties. In the last paper, we describe the effects due to the presence of a polymer matrix.