Rigidity percolation in random 3D rod systems

TL;DR

This paper models rigidity percolation in 3D random rod systems to explain the transition in mechanical strength of composite materials, developing an algorithm to identify rigid components and analyzing how the percolation threshold depends on rod aspect ratio.

Contribution

It introduces an approximate algorithm for identifying spanning rigid components in 3D rod systems and analyzes the dependence of rigidity percolation thresholds on aspect ratio.

Findings

Rigidity percolation threshold scales inversely with rod excluded volume.

The critical contact number is constant above a certain aspect ratio.

Scaling behavior is valid for relatively low aspect ratios.

Abstract

In composite materials composed of soft polymer matrix and stiff, high-aspect-ratio particles, the composite undergoes a transition in mechanical strength when the inclusion phase surpasses a critical density. This phenomenon (rheological or mechanical percolation) is well-known to occur in many composites at a critical density that exceeds the conductivity percolation threshold. Conductivity percolation occurs as a consequence of contact percolation, which refers to the conducting particles' formation of a connected component that spans the composite. Rheological percolation, however, has evaded a complete theoretical explanation and predictive description. A natural hypothesis is that rheological percolation arises due to rigidity percolation, whereby a rigid component of inclusions spans the composite. We model composites as random isotropic dispersions of soft-core rods, and study rigidity percolation in such systems. Building on previous results for two-dimensional systems, we develop an approximate algorithm that identifies spanning rigid components through iteratively identifying and compressing provably rigid motifs -- equivalently, decomposing giant rigid components into rigid assemblies of successively smaller rigid components. We apply this algorithm to random rod systems to estimate a rigidity percolation threshold and explore its dependence on rod aspect ratio. We show that this transition point, like the contact percolation transition point, scales inversely with the average (aspect ratio-dependent) rod excluded volume. However, the scaling of the rigidity percolation threshold, unlike the contact percolation scaling, is valid for relatively low aspect ratio. Moreover, the critical rod contact number is constant for aspect ratio above some relatively low value; and lies below the prediction from Maxwell's isostatic condition.