A jamming plane of sphere packings

The concept of jamming has attracted great research interest due to its broad relevance in soft matter such as liquids, glasses, colloids, foams, and granular materials, and its deep connection to the sphere packing problem and phase transitions. Here we show numerically that the phase space of frictionlessly jammed states can be extended from the well-known jamming-point along the density axis to a jamming plane spanned by the density and shear-strain axes,. We discuss how this jamming plane can be explored by standard athermal and thermal jamming protocols, and relate the protocol-dependence of the explored phase space to the reversibility of the routes to the jammed states. While all jammed states are isostatic and in the same universality class, their anisotropies and amorphous orders depend on the density and the shear strain. Among all points on the jamming plane, the jamming-point is special: it sets a sharp lower bound in the thermodynamic limit for the jamming density of all frictionless random packings, and the configurations there have the minimum amorphous order.