Percolation of particles on recursive lattices: (I) a new theoretical   approach
  Powdered materials of sizes ranging from nanometers to microns 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, among the others, mechanical, rheological, electrical, magnetic and thermal properties 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. Previous model calculations of percolation in polymeric systems generally did not take the possible difference between the size and shape of monomers and filler particles into account 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, we describe how to derive the percolation threshold as a function of the various parameters of the problem and we apply the new approach to the analysis of the effect of correlations among monodisperse particles on the percolation threshold of a system. Then 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 while in the last part we describe the effect of the presence of a polymer matrix. 
Andrea Corsi and P. D. Gujrati (The Department of Physics and The   Department of Polymer Science, The University of Akron)