TY - JOUR

T1 - A Geometric-Structure Theory for Maximally Random Jammed Packings

AU - Tian, Jianxiang

AU - Xu, Yaopengxiao

AU - Jiao, Yang

AU - Torquato, Salvatore

N1 - Funding Information:
J.T. gratefully acknowledges the support of Grant NSFC No. 11274200 and NSFSD No. ZR2011AM017. Y.J. thanks Arizona State University for his start-up fund. S.T. was supported in part by the National Science Foundation under Grants No. DMR-0820341 and No. DMS-1211087.

PY - 2015/11/16

Y1 - 2015/11/16

N2 - Maximally random jammed (MRJ) particle packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The prediction of the MRJ packing density φMRJ, among other packing properties of frictionless particles, still poses many theoretical challenges, even for congruent spheres or disks. Using the geometric-structure approach, we derive for the first time a highly accurate formula for MRJ densities for a very wide class of two-dimensional frictionless packings, namely, binary convex superdisks, with shapes that continuously interpolate between circles and squares. By incorporating specific attributes of MRJ states and a novel organizing principle, our formula yields predictions of φMRJ that are in excellent agreement with corresponding computer-simulation estimates in almost the entire α-x plane with semi-axis ratio α and small-particle relative number concentration x. Importantly, in the monodisperse circle limit, the predicted φMRJ = 0.834 agrees very well with the very recently numerically discovered MRJ density of 0.827, which distinguishes it from high-density "random-close packing" polycrystalline states and hence provides a stringent test on the theory. Similarly, for non-circular monodisperse superdisks, we predict MRJ states with densities that are appreciably smaller than is conventionally thought to be achievable by standard packing protocols.

AB - Maximally random jammed (MRJ) particle packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The prediction of the MRJ packing density φMRJ, among other packing properties of frictionless particles, still poses many theoretical challenges, even for congruent spheres or disks. Using the geometric-structure approach, we derive for the first time a highly accurate formula for MRJ densities for a very wide class of two-dimensional frictionless packings, namely, binary convex superdisks, with shapes that continuously interpolate between circles and squares. By incorporating specific attributes of MRJ states and a novel organizing principle, our formula yields predictions of φMRJ that are in excellent agreement with corresponding computer-simulation estimates in almost the entire α-x plane with semi-axis ratio α and small-particle relative number concentration x. Importantly, in the monodisperse circle limit, the predicted φMRJ = 0.834 agrees very well with the very recently numerically discovered MRJ density of 0.827, which distinguishes it from high-density "random-close packing" polycrystalline states and hence provides a stringent test on the theory. Similarly, for non-circular monodisperse superdisks, we predict MRJ states with densities that are appreciably smaller than is conventionally thought to be achievable by standard packing protocols.

UR - http://www.scopus.com/inward/record.url?scp=84947239619&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947239619&partnerID=8YFLogxK

U2 - 10.1038/srep16722

DO - 10.1038/srep16722

M3 - Article

C2 - 26568437

AN - SCOPUS:84947239619

VL - 5

JO - Scientific Reports

JF - Scientific Reports

SN - 2045-2322

M1 - 16722

ER -