Jamming on convex deformable surfaces

Jamming is a fundamental transition that governs the behavior of particulate media, including sand, foams and dense suspensions. Upon compression, such media change from freely flowing to a disordered, marginally stable solid that exhibits non-Hookean elasticity. While the jamming process is well established for fixed geometries, the nature and dynamics of jamming for a diverse class of soft materials and deformable substrates, including emulsions and biological matter, remains unknown. Here we propose a new scenario, metric jamming, where rigidification occurs on a surface that has been deformed from its ground state. Unlike classical jamming processes that exhibit discrete mechanical transitions, surprisingly we find that metric jammed states possess mechanical properties continuously tunable between those of classically jammed and conventional elastic media. The compact and curved geometry significantly alters the vibrational spectra of the structures relative to jamming in flat Euclidean space, and metric jammed systems also possess new types of vibrational mode that couple particle and shape degrees of freedom. Our work provides a theoretical framework that unifies our understanding of solidification processes that take place on deformable media and lays the groundwork to exploit jamming for the control and stabilization of shape in self-assembly processes.