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A Geometric-Structure Theory for Maximally Random Jammed Packings.

Tian J, Xu Y, Jiao Y, Torquato S - Sci Rep (2015)

Bottom Line: 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.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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, Qufu Normal University, Qufu 273165, China.

ABSTRACT
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.

No MeSH data available.


Related in: MedlinePlus

Comparison of the MRJ packing density estimated using Eq. (4) and obtained from simulations for various binary superdisk packings.For each panel, the packing density is shown as a function of small-to-large semi axis ratio α and deformation parameter p with a fixed small-particle relative concentration x. The predicted values of ϕMRJ are shown as blue curves and the simulated values are shown as red dots with confidence intervals shown.
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f5: Comparison of the MRJ packing density estimated using Eq. (4) and obtained from simulations for various binary superdisk packings.For each panel, the packing density is shown as a function of small-to-large semi axis ratio α and deformation parameter p with a fixed small-particle relative concentration x. The predicted values of ϕMRJ are shown as blue curves and the simulated values are shown as red dots with confidence intervals shown.

Mentions: To verify its accuracy, Eq. (4) is employed to estimate the ϕMRJ for a variety of binary frictionless superdisk packings with x ∈ [0.05, 0.95], α ∈ [0.2, 0.95] and p ∈ [0.85, 4.5]. We find that for all values of α and x considered the estimated ϕMRJ always agrees very well with the corresponding simulation results, with the largest deviations smaller than 1.5%, as shown in Fig. 5 for selected values of x, α and p (see the Supplementary Information for the values of ϕMRJ). For the extreme case where p = 0.5 and ∞, the particle becomes a square with singular curvatures at the corners and our formalism does not hold. In addition, when α is very small (e.g., less than 0.2), even with x ∈ [0.05, 0.95], the large particles can form a jammed backbone with small particles moving freely within cases formed by large particles50, and thus, the organizing principle does not hold for these cases.


A Geometric-Structure Theory for Maximally Random Jammed Packings.

Tian J, Xu Y, Jiao Y, Torquato S - Sci Rep (2015)

Comparison of the MRJ packing density estimated using Eq. (4) and obtained from simulations for various binary superdisk packings.For each panel, the packing density is shown as a function of small-to-large semi axis ratio α and deformation parameter p with a fixed small-particle relative concentration x. The predicted values of ϕMRJ are shown as blue curves and the simulated values are shown as red dots with confidence intervals shown.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4644945&req=5

f5: Comparison of the MRJ packing density estimated using Eq. (4) and obtained from simulations for various binary superdisk packings.For each panel, the packing density is shown as a function of small-to-large semi axis ratio α and deformation parameter p with a fixed small-particle relative concentration x. The predicted values of ϕMRJ are shown as blue curves and the simulated values are shown as red dots with confidence intervals shown.
Mentions: To verify its accuracy, Eq. (4) is employed to estimate the ϕMRJ for a variety of binary frictionless superdisk packings with x ∈ [0.05, 0.95], α ∈ [0.2, 0.95] and p ∈ [0.85, 4.5]. We find that for all values of α and x considered the estimated ϕMRJ always agrees very well with the corresponding simulation results, with the largest deviations smaller than 1.5%, as shown in Fig. 5 for selected values of x, α and p (see the Supplementary Information for the values of ϕMRJ). For the extreme case where p = 0.5 and ∞, the particle becomes a square with singular curvatures at the corners and our formalism does not hold. In addition, when α is very small (e.g., less than 0.2), even with x ∈ [0.05, 0.95], the large particles can form a jammed backbone with small particles moving freely within cases formed by large particles50, and thus, the organizing principle does not hold for these cases.

Bottom Line: 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.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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, Qufu Normal University, Qufu 273165, China.

ABSTRACT
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.

No MeSH data available.


Related in: MedlinePlus