Session U2: Jamming and Geometric Constraints

Jammed Ellipsoids Beat Jammed Spheres: Experiments with Candies and Colloids

Packing problems, how densely objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. For equal spheres, it has only recently been proved that the face-centered cubic lattice has the highest possible packing fraction $\phi \quad \sim $ 0.74. It is also well-known that the corresponding random (amorphous) jammed packings have $\phi \quad \sim $0.64. The density of packings in lattice and amorphous forms is intimately related to the existence of liquid and crystal phases and is responsible for the melting transition. Geometrical aspects of packing different shapes and the thermodynamic consequences are most readily observed in colloidal systems. Colloids are also useful for building micromachines and there is much more flexibility in colloidal architecture if the building blocks are non-spherical particles. A first step is to understand how such systems densely pack. Here we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely; up to $\phi \quad \sim $0.68 - 0.71 for spheroids with an aspect ratio close to that of M{\&}M's$^{\mbox{{\textregistered}} }$Candies, and even approach $\phi \quad \sim $0.75 for general ellipsoids. The higher density relates directly to the higher number of degrees of freedom per particle, d, and then to the number of contacts per particle Z. We find Z $\sim $10 for our spheroids as compared to Z $\sim $ 6 for spheres, confirming the isostatic conjecture Z=2d. Our results lead to the question as to ellipsoids, or any shaped particle will pack denser randomly than crystalline. In our studies we have found the crystal packings of ellipsoids to a density, $\phi \quad \sim $.7707 which exceeds the highest previous packing.   

Multiscaling at Point J: Jamming is a Critical Phenomenon

We analyze the jamming transition that occurs as a function of increasing packing density in a disordered two-dimensional assembly of disks at zero temperature for "Point J" of the recently proposed jamming phase diagram. Using numerical simulations, we drag a single disk through an increasingly dense assembly of hard disks. We measure the total number of moving disks and the transverse length of the moving region, and find a power law divergence as the packing density increases toward a critical jamming density. This provides evidence that the zero-temperature jamming transition as a function of packing density is a second order phase transition. Additionally, we find evidence for multiscaling, indicating the importance of long tails in the velocity fluctuations of the driven particle. [1] J.A. Drocco, M.B. Hastings, C.J. Olson Reichhardt, and C. Reichhardt, cond-mat/0310291.   

Soft modes and the onset of jamming

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 {\it 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 $\omega^*$, and c) power-law increase of $\omega^*$ with compression. We predict a length scale $l^*$ below which the boundary conditions strongly affect the system. $l^*$ diverges at the jamming transition when the system becomes isostatic.   

The onset of jamming as the sudden emergence of an infinite k-core cluster

A theory is constructed to describe the zero-temperature jamming transition as the density of repulsive soft spheres is increased. Local mechanical stability imposes a constraint on the minimum number of bonds per particle; we argue that this constraint suggests an analogy to $k$-core percolation. The latter model can be solved exactly on the Bethe lattice, and the resulting transition has a mixed first-order/continuous character. The exponents characterizing the continuous part appear to be the same as for the jamming transition. Finally, numerical simulations suggest that in finite dimensions the $k$-core transition can be discontinuous.   

Spatial structures and dynamics of kinetically constrained models of glasses

In glass-formers and more general jamming systems the microscopic motion is highly constrained because of the interaction with the surrounding particles. An example is the cage effect in glass-forming liquids. Kinetically constrained models encode this in a simple way. They are lattice models in which particles evolve by a stochastic dynamics with kinetic constraints: particles cannot move if surrounded by too many others. We shall show that from these simple dynamical rules highly non trivial physical phenomena emerge as super-Arrhenius behavior, dynamical heterogeneity and finite dimensional glass-jamming transitions.