VRML Viewers

This page contains VRML animations related to the paper ``Testing for Jamming in Hard-Sphere Packings''. Viewing the models on this page requires a VRML plugin such as Download CosmoPlayer (recommended) orDownload CosmoPlayer . These viewers at present work only on Windows and Mac platforms. I recommend that you download the relevant files to your hard-drive before viewing them.

The VRML animations have a play (triangle) and stop button (square) in the upper right corner that you can click on. The slider on then right determines the animations speed. There is also a slider on the left of the viewing window which can be used to manually select the frame and move the animation along by dragging the slider with the mouse.
Please contact Aleksandar Donev at adonev@princeton.edu if you need help.

VRML Models for Jamming in Hard-Sphere Packings

The main VRML file with all the prototypes for the models on this webpage is Models.wrl. Please make sure this is in the same directory as any files that you download! We have many animations related to sphere packings. The ones shown on this page are related to the above paper. Therefore it is recommended that you read the paper before viewing the animations.
Important note: Newer versions (2004) of VRML models that can be used to render sphere and ellipsoid packings can be found here.

Lubachevsky-Stillinger Algorithm

These animations illustrate the Lubachevsky-Stillinger compression algorithm and the kinds of random packings it produces.

2D Lattices: Periodic BCs

These animations illustrate unjamming motions (both collective and strict, if they exist) for some simple 2D lattice packings.

3D Lattices: Periodic BCs

And here are unjamming motions for some 3D lattice packings.

Random Packings: Periodic BCs

The following set of animations illustrate unjamming motions for some random 2D packings produced via the Lubachevsky-Stillinger algorithm.
  1. 2D Monodisperse Packings: At low densities (around 83%) these packings are not even collectivelly jammed, as the first animation illustrates. At high densities (88-89%) they are collectivelly, but not strictly jammed, as is shown for a 500 and 1000 disk packing below. Since it may not be obvious from these animations that strict unjamming involves shearing the packing, look at the 100 disk packing shown below, which shows several replications of the unit cell and zoom out to see what happens to the "infinite" periodic packing during this unjamming motion.
  2. 2D Bidisperse Amorphous Packings: These have a density of about 84% and are collectivelly jammed for small packings, but not strictly, as the first two animations show (these are packings obtained from Corey O'Hern from Yale University). Again we replicate the unit cell several times for better visualization. But larger packings also become strictly jammed, as the later two animations show.
  3. 3D Monodisperse Packings. These are amorphous and behave much like the 2D bidisperse packings. Their typical density is 64%. Below I show that they are not strictly jammed, but also that for larger packings the magnitude of the possible displacements of the spheres decreases.

Shrink-and-Bump Heuristic

The paper discusses alternative heuristics for testing for jamming not based on linear programming, and the fact that they are very sensitive to some parameter (in this case the amount of initial shrinking of the particles) and thus unreliable. These animations illustrate these LS-based heuristics.

Special non-randomized LPs

The paper advocates a randomized LP approach to testing for jamming and finding unjamming motions in sphere packings. It also points out that there are mathematically rigorous non-randomized ways of testing for jamming too. However, the unjamming motions found by these methods are rather a-typical, as these animations illustrate.