Planarity of Force Tilings in Jammed Packings of Disks

We propose a new order parameter for load induced jamming transitions in disk packings based on the planarity of force tilings. Contact forces between disks in mechanical equilibrium can be arranged in a dual space to form a network (tiling) represented by a set of vertices and edges $\mathcal{G} = (V,E)$. A Delaunay triangulation of these vertices then forms a related network $\mathcal{G}_D = (V,E_D)$. We define a planarity order parameter $\psi$ as the overlap of these two graphs $\psi = \langle \mathcal{G}_D| \mathcal{G} \rangle$. We use this parameter to characterize jamming transitions in two dimensional granular systems. We find clear signatures of the existence of non-planar and planar phases as a function of external load. We study this behaviour using simulation data of frictionless soft disks and experimental data of frictional disk packings.