Rigidity Percolation in Disordered 3D Rod Systems

Abstract

In composite materials composed of a soft polymer matrix and rigid, high aspect-ratio particles, the composite undergoes a transition in mechanical strength when the incorporated particle phase surpasses a critical density. This phenomenon, termed rheological percolation, is well known to occur in many polymer-rod and polymer-platelet composites at a critical density that exceeds the conductivity percolation threshold (which occurs when the conducting particles form a large connected component that spans the composite). Contact percolation in rod-like composites has been routinely exploited to engineer thermal or electrical conductivity in otherwise nonconducting polymers, and the characterization of contact percolation is well established. Mechanical or rheological percolation, however, has evaded a complete theoretical explanation and predictive description. A natural hypothesis is that rheological percolation is due to a rigidity phenomenon, whereby a large rigid component of inclusions spans the composite. Here we build an algorithm to detect the rigidity percolation threshold in rod-polymer composites. We model the composites as systems of randomly distributed, soft-core (intersecting at contact) rods and study the emergence of a giant (i.e., spanning) rigid component. Building on our previous results for two-dimensional composites, we develop an approximate algorithm that identifies spanning rigid components through hierarchically identifying and compressing provably rigid motifs---equivalently, decomposing a giant rigid component into rigid assemblies of a hierarchy of successively smaller rigid components. We apply this algorithm to random monodisperse systems that are generated in Monte Carlo simulations to estimate a rigidity percolation threshold (critical density) and explore its dependence on rod aspect ratio. We show that this transition point---like its contact percolation analogue---scales inversely with the excluded volume of a rod. Moreover, this implies that the critical contact number (i.e., the number of contacts per rod at the rigidity percolation threshold) is constant for aspect ratios above some relatively low value and is lower than the prediction from Maxwell's isostatic condition.