Index of /~tten/Glasperlenspiel

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Glasperlenspiel: granular materials research

This page collects material on granular research in the Witten group.

Disclaimer -- Many items in this directory are working files, whose accuracy has not been checked.



Slides for a colloquium: Solidity without Elasticity. Given at Indiana University by TW January 10, 2007. (pdf 1.6 megabytes)


Signal transmissibility in marginal granular materials

Matthew Pinson and Thomas A. Witten

We examine the 'transmissibility' of a simulated two-dimensional pack of frictionless disks formed by confining dilute disks in a shrinking, periodic box to the point of mechanical stability. Two opposite boundaries are then removed, thus allowing a set of free motions. Small free displacements on one boundary then induce proportional displacements on the opposite boundary. Transmissibility is the ability to distinguish different perturbations by their distant responses. We assess transmissibility by successively identifying free orthonormal modes of motion that have the smallest distant responses. The last modes to be identified in this 'pessimistic' basis are the most transmissive. The transmitted amplitudes of these most transmissive modes fall off exponentially with mode number. Similar exponential falloff is seen in a simple elastic medium, though the responsible modes differ greatly in structure in the two systems. Thus the marginal pack's transmissibility is qualitatively similar to that of a simple elastic medium. We compare our results with recent findings based on the projection of the space of free motion onto interior sites. Journal of Physics: Condensed Matter, Volume 28, Number 49 (2016)

Force propagation in isostatic granular packs

Nathan Krapf

In which we probe the anomalous ray-like force propagation expected seen in some marginally jammed packs. We extend the study to isotropic marginal jamming systems developed by Nagel, Liu and collaborators. By altering the boundary conditions so that all forces are absorbed on one unique boundary, we apply point forces on individual beads and investigate the spatial distribution of the resulting stress. By studying variants of these packs plus a set of regular lattices, we find that ray-like force propagation is observed when the distribution of contact directions is anisotropic. For the isotropic packs of Nagel and Liu, only diffuse stress propagation like that of an elastic solid is observed, even very close to the jamming threshold, where many other features differ strongly from conventional elastic solids.

Compressing nearly hard sphere fluids increases glass fragility

Ludovic Berthier and T. A. Witten

We use molecular dynamics to investigate the glass transition occuring at large volume fraction, $\phi$, and low temperature, $T$, in assemblies of soft repulsive particles. We find that equilibrium dynamics in the ($\phi$, $T$) plane obey critical scaling in the proximity of a critical point at $T=0$ and $\phi=\phi_0$, which corresponds to the ideal glass transition of hard spheres. This glass critical point, `point $G$', is distinct from athermal jamming thresholds. A remarkable consequence of scaling behaviour is that the dynamics at fixed $\phi$ passes smoothly from that of a strong glass to that of a very fragile glass as $\varphi$ increases beyond $\varphi_0$. We explore correlations between fragility and various physical properties.

Effects of compression on the vibrational modes of marginally jammed solids

Matthieu Wyart, Leonardo E. Silbert, Sidney R. Nagel, Thomas A. Witten

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(w) of vibrational modes of angular frequency w 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(w) toward a non-zero constant as w goes to 0, b) appearance of a low-frequency cutoff w*, and c) power-law increase of w* with compression. Differences between these two systems occur at lower frequency. The density of states of the modified system displays an abrupt plateau that appears at w*, 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 dz: dz > p^(1/2), 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.

Geometric origin of excess low-frequency vibrational modes in amorphous solids
Matthieu Wyart, Sidney R. Nagel, T.A. Witten

Glasses have a large excess of low-frequency vibrational modes in comparison with crystalline solids. We show that such a feature is a necessary consequence of the geometry generic to a marginally connected solid. In particular, we analyze the density of states of a recently simulated system, comprised of weakly compressed spheres at zero temperature. We account for the observed a) constancy of the density of modes with frequency, b) appearance of a low-frequency cutoff, and c) power-law increase of this cutoff with compression. We predict a length scale below which the boundary conditions strongly affect the system.

Robust propagation direction of stresses in a minimal granular packing
D. A. Head, A. V. Tkachenko, T. A. Witten

By employing the adaptive network simulation method, we demonstrate that the ensemble-averaged stress caused by a local force for packings of frictionless rigid beads is concentrated along rays whose slope is consistent with unity: forces propagate along lines at 45 degrees to the horizontal or vertical. This slope is shown to be independent of polydispersity or the degree to which the system is sheared. Further confirmation of this result comes from fitting the components of the stress tensor to the null stress constitutive equation. The magnitude of the response is also shown to fall off with the -1/2 power of distance. We argue that our findings are a natural consequence of a system that preserves its volume under small perturbations.

Stress Propagation through Frictionless Granular Material
Stress in frictionless granular material: adaptive network simulations


Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation, which provided partial support for the research mentioned above.