Extending the phase space of jamming from a jamming point to a jamming plane

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 frictionless jammed states can be extended from the well-known jamming point, to a jamming line by using deeply super-cooled liquid states as initial configurations, which can be further extended into a jamming plane by adding shear strains. While all jammed states are isostatic and belong to the same universality class, their various anisotropy and amorphous order can be mapped out on the jammed plane. The jamming point is isotropic, with the minimum amorphous order (or maximum randomness), which, in the thermodynamic limit, sets a sharp lower bound for the jamming density of frictionless packings.