A Geometric-Structure Theory for Maximally Random Jammed Packings

Tian, Jianxiang; Xu, Yaopengxiao; Jiao, Yang; Torquato, Salvatore

doi:10.1038/srep16722

Cited by 25 publications

(22 citation statements)

References 53 publications

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“…where G(r) ≡ u 1 (r + r 0 )u 1 (r 0 ) is the displacement-displacement correlation function, u = 0 due to the isotropy of u, and thus G(0) = |u| 2 /d if |u| 2 exists. Using (280), the structure factor S (k) in (278) for small |k| is given approximately by [137] S (k) ≈ |k| 2 G(0) + 1 − |k| 2 G(0) S 0 (k) + ρ|k| 2G (k) + ρ|k| 2 dre −ik·r h 0 (r)G(r),…”

Section: Effect Of Imperfections On Hyperuniform Statesmentioning

confidence: 99%

Hyperuniform states of matter

2018

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Hyperuniform states of matter are correlated systems that are characterized by an anomalous suppression of longwavelength (i.e., large-length-scale) density fluctuations compared to those found in garden-variety disordered systems, such as ordinary fluids and amorphous solids. All perfect crystals, perfect quasicrystals and special disordered systems are hyperuniform. Thus, the hyperuniformity concept enables a unified framework to classify and structurally characterize crystals, quasicrystals and the exotic disordered varieties. While disordered hyperuniform systems were largely unknown in the scientific community over a decade ago, now there is a realization that such systems arise in a host of contexts across the physical, materials, chemical, mathematical, engineering, and biological sciences, including disordered ground states, glass formation, jamming, Coulomb systems, spin systems, photonic and electronic band structure, localization of waves and excitations, self-organization, fluid dynamics, number theory, stochastic point processes, integral and stochastic geometry, the immune system, and photoreceptor cells. Such unusual amorphous states can be obtained via equilibrium or nonequilibrium routes, and come in both quantum-mechanical and classical varieties. The connections of hyperuniform states of matter to many different areas of fundamental science appear to be profound and yet our theoretical understanding of these unusual systems is only in its infancy. The purpose of this review article is to introduce the reader to the theoretical foundations of hyperuniform ordered and disordered systems. Special focus will be placed on fundamental and practical aspects of the disordered kinds, including our current state of knowledge of these exotic amorphous systems as well as their formation and novel physical properties.

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Section: Effect Of Imperfections On Hyperuniform Statesmentioning

confidence: 99%

Hyperuniform states of matter

2018

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“…Shape φmax simulation φmax experiment φmax theory disks (2d) 0.826 (Atkinson et al, 2014) 0.85 0.874 0.834 (Tian et al, 2015) Sphere 0.645 (Skoge et al, 2006) 0.64 (Bernal and Mason, 1960) 0.634 (Roth and Jaeger, 2016) TABLE V Overview of maximal packing fractions φmax for a selection of regular shapes in disordered packings obtained with a variety of different packing protocols. Note that the φmax value is achieved for the aspect ratio, where φ is maximal, so every value is at a different aspect ratio.…”

Section: Coarse-grained Voronoi Volume Of Non-spherical Shapesmentioning

confidence: 99%

Edwards statistical mechanics for jammed granular matter

et al. 2018

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In 1989, Sir Sam Edwards made the visionary proposition to treat jammed granular materials using a volume ensemble of equiprobable jammed states in analogy to thermal equilibrium statistical mechanics, despite their inherent athermal features. Since then, the statistical mechanics approach for jammed matter -one of the very few generalizations of Gibbs-Boltzmann statistical mechanics to out of equilibrium matter -has garnered an extraordinary amount of attention by both theorists and experimentalists. Its importance stems from the fact that jammed states of matter are ubiquitous in nature appearing in a broad range of granular and soft materials such as colloids, emulsions, glasses, and biomatter. Indeed, despite being one of the simplest states of matter -primarily governed by the steric interactions between the constitutive particles -a theoretical understanding based on first principles has proved exceedingly challenging. Here, we review a systematic approach to jammed matter based on the Edwards statistical mechanical ensemble. We discuss the construction of microcanonical and canonical ensembles based on the volume function, which replaces the Hamiltonian in jammed systems. The importance of approximation schemes at various levels is emphasized leading to quantitative predictions for ensemble averaged quantities such as packing fractions and contact force distributions. An overview of the phenomenology of jammed states and experiments, simulations, and theoretical models scrutinizing the strong assumptions underlying Edwards' approach is given including recent results suggesting the validity of Edwards ergodic hypothesis for jammed states. A theoretical framework for packings whose constitutive particles range from spherical to non-spherical shapes like dimers, polymers, ellipsoids, spherocylinders or tetrahedra, hard and soft, frictional, frictionless and adhesive, monodisperse and polydisperse particles in any dimensions is discussed providing insight into an unifying phase diagram for all jammed matter. Furthermore, the connection between the Edwards' ensemble of metastable jammed states and metastability in spin-glasses is established. This highlights that the packing problem can be understood as a constraint satisfaction problem for excluded volume and force and torque balance leading to a unifying framework between the Edwards ensemble of equiprobable jammed states and out-of-equilibrium spin-glasses.

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“…Intuitively speaking, MRJ packings are the maximally disordered among all mechanically stable packings. More precisely, they minimize among the jammed packings some order metric Ψ [10][11][12][13][14][15][16][17][18]. The MRJ state can be unambiguously identified for a particular choice of the order metric, and a variety of sensible, positively correlated order metrics produce an MRJ state in three dimensions with the same packing fraction 0.64 [11].…”

Section: Introductionmentioning

confidence: 99%

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

Klatt

Torquato

2018

Phys. Rev. E

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In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here, we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing "traps." For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro-and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

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A Geometric-Structure Theory for Maximally Random Jammed Packings

Cited by 25 publications

References 53 publications

Hyperuniform states of matter

Hyperuniform states of matter

Edwards statistical mechanics for jammed granular matter

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

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