Glasses have a large excess of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature is a necessary consequence of the weak connectivity of the solid, and that the frequency of modes in excess is very sensitive to the pressure. We analyze, in particular, two systems whose density D(omega) of vibrational modes of angular frequency omega display scaling behaviors with the packing fraction: (i) simulations of jammed packings of particles interacting through finite-range, purely repulsive potentials, comprised of weakly compressed spheres at zero temperature and (ii) a system with the same network of contacts, but where the force between any particles in contact (and therefore the total pressure) is set to zero. We account in the two cases for the observed (a) convergence of D(omega) toward a nonzero constant as omega-->0, (b) appearance of a low-frequency cutoff omega*, and (c) power-law increase of omega* with compression. Differences between these two systems occur at a lower frequency. The density of states of the modified system displays an abrupt plateau that appears at omega*, below which we expect the system to behave as a normal, continuous, elastic body. In the unmodified system, the pressure lowers the frequency of the modes in excess. The requirement of stability despite the destabilizing effect of pressure yields a lower bound on the number of extra contact per particle deltaz:deltaz> or =p1/2, which generalizes the Maxwell criterion for rigidity when pressure is present. This scaling behavior is observed in the simulations. We finally discuss how the cooling procedure can affect the microscopic structure and the density of normal modes.

Glasses have an excess number of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature necessarily occurs in solids with low coordination. In particular, we analyze the density D(ω) of normal-mode frequencies ω and the nature of the low-frequency normal modes of a recently simulated system [1], comprised of weakly compressed spheres at zero temperature. We account for the observed a) convergence of D(ω) toward a non-zero constant as the frequency goes to zero, b) appearance of a lowfrequency cutoff ω * , and c) power-law increase of ω * with compression. We introduce a length scale l * which characterizes the vibrational modes that appear at ω * .There is something universal and mystifying about the low-energy behavior of amorphous solids [2,3]. In comparison to most crystals, amorphous solids have a large excess number of low-frequency vibrational modes. In glasses these excitations are seen in the low-temperature specific heat as well as in the spectroscopy of the vibration modes in the terahertz range. These excitations affect the heat transport [3] and might well play an important role at the liquidglass transition [4] . Nevertheless, little is understood about the cause of such excitations. Whereas in a crystal the vibrations are simply plane waves, in a glass, even at low angular frequency ω, they are much more complicated. In this Letter, we show that an excess density of vibrational states is a necessary feature of weakly-connected amorphous solids such as systems with repulsive, short-range interactions. Our analysis elucidates the cause and the peculiar nature of these low-frequency excitations. This gives a new approach for studying some of the ubiquitous phenomena found in glasses.A dramatic illustration of excess low-frequency vibrations was found in recent computer simulations [5,1] of soft-spheres with repulsive, finite range potentials at zero temperature and zero applied shear stress. These simulations were carried out as a function of the packing fraction, φ above the jamming threshold, φ c , where the liquid acquires rigidity and becomes an amorphous solid [6]. O'Hern, Silbert et al. found that the average number of contacting neighbors per particle, z, the pressure, and the shear modulus vary as a power of (φ − φ c ). Moreover, these simulations reveal unexpected features in the density of vibrational mode c EDP Sciences

A consensus is emerging that discontinuous shear thickening (DST) in dense suspensions marks a transition from a flow state where particles remain well separated by lubrication layers, to one dominated by frictional contacts. We show here that reasonable assumptions about contact proliferation predict two distinct types of DST in the absence of inertia. The first occurs at densities above the jamming point of frictional particles; here, the thickened state is completely jammed and (unless particles deform) cannot flow without inhomogeneity or fracture. The second regime shows strain-rate hysteresis and arises at somewhat lower densities, where the thickened phase flows smoothly. DST is predicted to arise when finite-range repulsions defer contact formation until a characteristic stress level is exceeded.

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