Entanglement transition in rod packings

Random packings of stiff rods are self-supporting mechanical structures stabilized by their geometrical and topological complexity. To understand why, we deploy X-ray computerized tomography to unveil the structure of the packing. This allows us to define and directly visualize the spatial variations in "entanglement," a mesoscopic field that characterizes the local average crossing number, a measure of the topological complexity of the packing. We show that the entanglement field has information that is distinct from the density, orientational order, and contact distribution of the packing. We find that increasing the aspect ratio of the constituent rods in a packing leads to an abrupt change in the entanglement, correlated with a sharp transition in the mechanical response of the packing. This leads to an entanglement phase diagram for the mechanical response of dense rod packings that is likely relevant for a broad range of problems that goes beyond our specific study.