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Applied Mathematics Colloquium
Friday, February 27, 11:30 am
Cullimore Lecture Hall II
New
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Theory of random packings
Hernán Makse
Levich Institute and Physics Department
Abstract
The problem of finding the most efficient way to pack spheres has an illustrious
history, dating back to the crystalline arrays conjectured
by Kepler
and the random geometries explored by Bernal in the 60's. This
problem finds applications spanning from the
mathematician's pencil, the processing of granular
materials, the jamming and glass transitions, all the way to fruit packing in
every
grocery. There are presently
numerous experiments showing that the loosest way to pack spheres gives a
density of ~55% (RLP) while
filling all the loose voids results
in a maximum density of ~63-64% (RCP).
While those values seem robustly true, to this date there is
no physical explanation or
theoretical prediction for them. Here we show that random packings
of monodisperse hard spheres in 3d can pack
between the densities 4/(4 + 2 \sqrt 3) or 53.6% and 6/(6 + 2 \sqrt
3) or 63.4%, defining RLP and RCP, respectively. The reason for these
limits arises from a statistical
picture of jammed states in which the RCP can be interpreted as the ground
state of the ensemble of jammed
matter with zero compactivity, while the RLP arises in the infinite compactivity limit.
Ultimately, our results lead to a phase
diagram that provides a unifying view of the disordered hard sphere packing problem.