PHYSICAL MATHEMATICS SEMINAR
TOPIC: SPHERE PACKINGS, MAXIMAL DISORDER, AND JAMMING
SPEAKER: SALVATORE TORQUATO
Department of Chemistry and Princeton Materials Institute
Princeton University
ABSTRACT:
Bernal has remarked that "heaps (random close-packed arrangements of
particles) were the first things that were ever measured in the form of
basketfuls of grain for the purpose of trading or of collection of taxes."
Random packings of identical spheres have been studied by biologists,
materials scientists, engineers, chemists, and physicists to understand
the structure of living cells, liquids, composites, granular media,
glasses and amorphous solids, to mention but a few examples [1].
Despite its long history, there are many fundamental issues concerning
random sphere packings that remain elusive, including the nature of the
venerable 50-year old notion of "random close packing" (RCP) state. We
show that the RCP concept is not mathematically precise by introducing
scalar metrics to characterize disorder and using molecular dynamics
simulations. To replace the old notion of the RCP state, we introduce the
new concept of a maximally random jammed (MRJ) state, which can be made
precise [2].
This lays the mathematical groundwork for studying randomness in dense
packings of spheres and initiates the search for the MRJ state in a
quantitative way not possible before. But the MRJ state depends on the
definition of a "jammed" state. We have devised several meaningful
definitions [3], two of which are the ``collectively jammed" state and the
"strictly jammed" state. The latter demands that the equivalent "contact"
network is stable under homogeneous volume-preserving deformations. The
particle packing can be transformed to an equivalent "contact" network by
joining the centers of contacting particles by lines. Once this
equivalent network is determined, we are able to pose the stability
question as a novel optimization problem, namely, a linear programming
problem [4]. This procedure enable us to test if a packing is
collectively or strictly jammed.
1. S. Torquato, "Random Heterogeneous Materials: Microstructure and
Macroscopic Properties," (Springer-Verlag, New York, 2002).
2. S. Torquato, T. M. Truskett and P. G. Debenedetti, Phys. Rev. Lett.
Vol. 84, 2064 (2000).
3. S. Torquato and F. H. Stillinger, "Multiplicity of Generation,
Selection, and Classification Procedures for Jammed Hard-Particle
Packings," J. Physical Chemistry, Vol. 105, 11849 (2001).
4. A. Donev, S. Torquato, and F. H. Stillinger, in preparation.
DATE: TUESDAY, APRIL 2, 2002
TIME: 2:30 PM
LOCATION: Building 2, Room 338
Refreshments will be served at 3:30 PM in Room 2-349
Massachusetts Institute of Technology
Department of Mathematics
Cambridge, MA 02139