Leonardo E. Silbert

Granular flow down an inclined plane: Bagnold scaling and rheology

We have performed a systematic, large-scale simulation study of granular media in two and three dimensions, investigating the rheology of cohesionless granular particles in inclined plane geometries, i.e., chute flows. We find that over a wide range of parameter space of interaction coefficients and inclination angles, a steady-state flow regime exists in which the energy input from gravity balances that dissipated from friction and inelastic collisions. In this regime, the bulk packing fraction (away from the top free surface and the bottom plate boundary) remains constant as a function of depth z, of the pile. The velocity profile in the direction of flow vx(z) scales with height of the pile H, according to vx(z) proportional to H(alpha), with alpha=1.52+/-0.05. However, the behavior of the normal stresses indicates that existing simple theories of granular flow do not capture all of the features evidenced in the simulations.

Geometry of frictionless and frictional sphere packings

We study static packings of frictionless and frictional spheres in three dimensions, obtained via molecular dynamics simulations, in which we vary particle hardness, friction coefficient, and coefficient of restitution. Although frictionless packings of hard spheres are always isostatic (with six contacts) regardless of construction history and restitution coefficient, frictional packings achieve a multitude of hyperstatic packings that depend on system parameters and construction history. Instead of immediately dropping to four, the coordination number reduces smoothly from z=6 as the friction coefficient mu between two particles is increased.

Vibrations and Diverging Length Scales Near the Unjamming Transition

We numerically study the vibrations of jammed packings of particles interacting with finite-range, repulsive potentials at zero temperature. As the packing fraction phi is lowered towards the onset of unjamming at phi(c), the density of vibrational states approaches a nonzero value in the limit of zero frequency. For phi >phi(c), there is a crossover frequency, omega* below which the density of states drops towards zero. This crossover frequency obeys power-law scaling with phi-phi(c). Characteristic length scales, determined from the dominant wave vector contributing to the eigenmode at omega*, diverge as power laws at the unjamming transition.

Effects of compression on the vibrational modes of marginally jammed solids

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.

Confined granular packings: structure, stress, and forces.

The structure and stresses of static granular packs in cylindrical containers are studied by using large-scale discrete element molecular dynamics simulations in three dimensions. We generate packings by both pouring and sedimentation and examine how the final state depends on the method of construction. The vertical stress becomes depth independent for deep piles and we compare these stress depth profiles to the classical Janssen theory. The majority of the tangential forces for particle-wall contacts are found to be close to the Coulomb failure criterion, in agreement with the theory of Janssen, while particle-particle contacts in the bulk are far from the Coulomb criterion. In addition, we show that a linear hydrostaticlike region at the top of the packings unexplained by the Janssen theory arises because most of the particle-wall tangential forces in this region are far from the Coulomb yield criterion. The distributions of particle-particle and particle-wall contact forces P(f) exhibit exponential-like decay at large forces in agreement with previous studies.

Granular flow down a rough inclined plane: Transition between thin and thick piles

The rheology of granular particles in an inclined plane geometry is studied using three dimensional molecular dynamics simulations. The flow–no-flow boundary is determined for piles of varying heights over a range of inclination angles θ. Three angles determine the phase diagram: θr, the angle of repose, is the angle at which a flowing system comes to rest; θm, the maximum angle of stability, is the inclination required to induce flow in a static system; and θmax is the maximum angle for which stable, steady state flow is observed. In the stable flow region θr<θ<θmax, three flow regimes can be distinguished that depend on how close θ is to θr: (i) θ≫θr: Bagnold rheology, characterized by a mean particle velocity vx in the direction of flow that scales as vx∝h3/2, for a pile of height h, (ii) θ≳θr: The slow flow regime, characterized by a linear velocity profile with depth, and (iii) θ≈θr: Avalanche flow characterized by a slow underlying creep motion combined with occasional free surface events and large ...

Statistics of the contact network in frictional and frictionless granular packings

Simulated granular packings with different particle friction coefficient mu are examined. The distribution of the particle-particle and particle-wall normal and tangential contact forces P(f) are computed and compared with existing experimental data. Here f identical with F/(-)F is the contact force F normalized by the average value (-)F. P(f) exhibits exponential-like decay at large forces, a plateau/peak near f=1, with additional features at forces smaller than the average that depend on mu. Additional information beyond the one-point force distribution functions is provided in the form of the force-force spatial distribution function and the contact point radial distribution function. These quantities indicate that correlations between forces are only weakly dependent on friction and decay rapidly beyond approximately three particle diameters. Distributions of particle-particle contact angles show that the contact network is not isotropic and only weakly dependent on friction. High force-bearing structures, or force chains, do not play a dominant role in these three-dimensional, unloaded packings.

Jamming of frictional spheres and random loose packing

The role of the friction coefficient, μ, on the jamming properties of disordered, particle packings is studied using computer simulations. Compressed, soft-sphere packings are brought towards the jamming transition—the point where a packing loses mechanical stability—by decreasing the packing fraction. The values of the packing fraction at the jamming transition, ϕμc, gradually decrease from the random close packing point for zero friction, to a value coincident with random loose packing as the friction coefficient is increased over several orders of magnitude. This is accompanied by a decrease in the coordination number at the jamming transition, zμc, which varies from approximately six to four with increasing friction. Universal power law scaling is observed in the pressure and coordination number as a function of distance from the generalised, friction-dependent jamming point. Various power laws are also reported between the ϕμc, zμc, and μ. Dependence on preparation history of the packings is also investigated.

Structural signatures of the unjamming transition at zero temperature

We study the pair correlation function g(r) for zero-temperature, disordered, soft-sphere packings just above the onset of jamming. We find distinct signatures of the transition in both the first and split second peaks of this function. As the transition is approached from the jammed side (at higher packing fraction) the first peak diverges and narrows on the small-r side to a delta function. On the high-r side of this peak, g(r) decays as a power law. In the split second peak, the two subpeaks are both singular at the transition, with power-law behavior on their low-r sides and step-function drop-offs on their high-r sides. These singularities at the transition are reminiscent of empirical criteria that have previously been used to distinguish glassy structures from liquid ones.

The rheology and microstructure of concentrated, aggregated colloids

The rheology of concentrated, aggregated colloidal suspensions is determined through particulate simulations. Aggregating systems experience a large viscous enhancement over nonaggregating systems, this being due to the increase in the component of the viscosity arising from the repulsive colloid (thermodynamic) forces when attractive forces are present. The shear behavior of aggregating systems, for colloid volume fraction 0.47⩽φc⩽0.57, is characterized in the steady state regime over a wide range in shear rate, and is found to be power law, shear thinning η∼f(φc)γ−α, where the shear thinning index α=0.84±0.01. The effect of volume fraction enters as f(φc)=(1−φc/φmax)−1, with φmax=0.64, the value of random close packing; similarly, the viscosity also scales with the potential well depth as a power law, of index α. Consequently, we are able to deduce the full constitutive relation for this power law behavior. The associated structural features which emerge as a result of the imposed shear are identified ...