On the rigidity of a hard sphere glass near random close packing

– We study theoretically and numerically the microscopic cause of the rigidity of hard sphere glasses near their maximum packing. We show that, after coarse-graining over time, the hard sphere interaction can be described by an effective potential which is exactly logarithmic at the random close packing φc. This allows to define normal modes, and to apply recent results valid for elastic networks: rigidity is a non-local property of the packing geometry, and is characterized by some length scale l∗ which diverges at φc [1, 2]. We compute the scaling of the bulk and shear moduli near φc, and speculate on the possible implications of these results for the glass transition. Hard spheres present a glass phase between φ0, where the glass transition occurs and structural relaxation becomes unobservable, and φc where the pressure p diverges. In this region this system is solid and resists to shear on any measurable time scales. Although a large amount of works focused on the super-cooled liquid, the glass itself received less attention. In particular, there is no undoubted microscopic theory to explain its mechanical properties and its rigidity. In the cage-escape picture [3], the cage formed by the neighboring particles tighten as φ increases, and the typical time for a particle to escape its cage grows and eventually diverges. Nevertheless, Maxwell showed that the stability against collective motions of particles is more demanding than against individual particle displacements: in particular z = d+ 1 inter-particle contacts are sufficient to pin one particle in d dimensions, whereas zc = 2d contacts in average are required to guarantee mechanical stability [4]. Thus considering a priori rigidity as a local property may be inappropriate. Recently several works [1, 2, 5, 6, 8, 7] studied the mechanical properties of weaklyconnected elastic networks with an average contacts number —the coordination number— z close to the critical value zc = 2d, such as those encountered for athermal repulsive shortrange particles above φc [5, 6]. In particular it was shown that (i) these systems present an