Theoretical estimation on the percolation threshold for polymer matrix composites with hybrid fillers

Hybrid particles can improve the performance and reduce the cost of polymer matrix composites (PMCs), and thus increasingly used as fillers in multifunctional and structural materials. For these materials, the percolation threshold is of great importance because it determines whether or not the excellent mechanical, electrical and thermal properties of fillers can be fully utilized in the composites. However, the existing theoretical methods have limitations in percolation threshold predictions, and moreover, they are not applicable for hybrid filler systems. In this paper, an approach to predict the percolation threshold of hybrid filler systems is developed based on the modified average intersection number, which is proposed in this paper and revealed to be an invariant at the threshold. The theoretical expression for the percolation threshold of PMCs with fillers of different shapes is developed, and especially, the analytical expression for systems with spherical and cylindrical particles is obtained and validated by Monte-Carlo simulations. The proposed method on percolation threshold prediction can well explain the synergistic effect of fillers with different shapes, and provides great convenience for the design of conductive PMCs with hybrid fillers.