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Spectrum of structure for jammed and unjammed soft disks

A. T. Chieco, M. Zu, A. J. Liu, N. Xu, and D. J. Durian
Phys. Rev. E 98, 042606 – Published 17 October 2018
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  • INTRODUCTION
  • DEFINITIONS AND BOUNDS
  • METHODS
  • RESULTS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We investigate the short-, medium-, and long-range structure of soft-disk configurations for a wide range of area fractions and simulation protocols by converting the real-space spectrum of volume fraction fluctuations for windows of width L to the distance h(L) from the window boundary over which fluctuations occur. Rapidly quenched unjammed configurations exhibit size-dependent super-Poissonian long-range features that surprisingly approach the totally random limit even close to jamming. Above and just below jamming, the spectra exhibit a plateau h(L)=he for L larger than particle size and smaller than a cutoff Lc beyond which there are long-range fluctuations. The value of he is independent of protocol and characterizes the putative hyperuniform limit. This behavior is compared with that for Einstein solids, with and without hyperuniformity-destroying defects. We find that key structural features of the particle configurations are more evident, as well as easier and more intuitive to quantify, using the real-space spectrum of hyperuniformity lengths rather than the spectral density.

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    • Received 19 October 2017
    • Revised 26 June 2018

    DOI:https://doi.org/10.1103/PhysRevE.98.042606

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Jamming
    1. Physical Systems
    Glassy systemsGranular materialsSoft colloids
    Polymers & Soft Matter

    Authors & Affiliations

    A. T. Chieco1, M. Zu2, A. J. Liu1, N. Xu2, and D. J. Durian1

    • 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA
    • 2Department of Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China

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    Vol. 98, Iss. 4 — October 2018

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    Images

    • Figure 1
      Figure 1

      Unjammed bidisperse disks at area fraction ϕ=0.70 made by rapid quench. The area fraction variance is controlled by the number of particles in the shaded region of thickness h(L), averaged over window placements. As depicted here for L=15〈a〉, the hyperuniformity disorder length is h(L)=3〈a〉, where 〈a〉 is the area-fraction weighted average disk area, equal to 0.81 large disk diameters.

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    • Figure 2
      Figure 2

      Configurations of 2048 bidisperse disks with global packing fraction (a)–(c) ϕ=0.70 or (d)–(f) ϕ=0.85 generated by different preparation protocols. (a), (b), (d), and (e) display the entire system but (c) and (f) show only a subset of a system with N=105 particles. (a)–(c) are unjammed configurations and there are noticeably larger voids in (b) and (c) than in (a). (d)–(f) are jammed configuration and are relatively indistinguishable. The differences seen by eye are quantified with h(L) because the normalized spectra are protocol and system-size dependent for ϕ<ϕc but collapse for ϕ≥ϕc.

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    • Figure 3
      Figure 3

      Spectral density and hyperuniformity disorder length spectra for Einstein patterns and jammed soft disks created at different quench rates rq. At small lengths, h(L) matches the separated-particle lower bound L/2(1−ϕL2/〈a〉), but χ(q) deviates irregularly from the Poisson limit. At intermediate lengths, h(L) becomes constant, but for the soft disks χ(q) at first plunges precipitously and then shows no such obvious signature of incipient hyperuniformity. At long lengths, both h(L) and χ(q) show that hyperuniformity is destroyed except for the defect-free Einstein patterns, but only h(L) shows a clear trend toward hyperuniformity as rq is reduced. Putative and actual hyperuniformity are characterized by the value of he, which cannot be extracted from χ(q).

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    • Figure 4
      Figure 4

      Uniformity cutoff length Lc, defined by where h(L) rises 10% above he. The colorized contours interpolate between measured Lc values at quench rates and volume fractions indicated by the light blue symbols. The data are from packings of N=106 particles.

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    • Figure 5
      Figure 5

      Spectral density and hyperuniformity disorder length spectra for unjammed soft-disk configuration created at various area fractions and system sizes, using two different protocols. The sequence of area fractions is ϕ={0.10,0.15,0.20,...,0.75,0.80,0.81,0.82,0.83,0.84} from top to bottom. In (e) and (f) ϕ=0.8412 is also included.

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    • Figure 6
      Figure 6

      Area fraction dependence of the hyperuniformity disorder length evaluated at various L as labeled, scaled to decay from 1 to 0 as ϕ goes from 0 to ϕc, for the same hard-disk configurations shown in Fig. 5. The expression on the y axis is evaluated at (a) L=10〈a〉, (b) L=20〈a〉, (c) L=30〈a〉, and (d) L=40〈a〉, unless otherwise noted. The data have a power-law exponent of the final decay (solid lines) of 1.5 for the thermal configurations and 1.8 for the quench configurations, independent of L. (a)–(c) The curves from top to bottom have a critical packing fraction of ϕc={0.8416±0.0003,0.8409±0.0012,0.8465±0.0005}; (d) follows the same sequence from top to bottom but the top two curves have ϕc=0.8416±0.0003.

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    • Figure 7
      Figure 7

      Potential energy (left axis) and pressure (right axis) versus packing fraction for many configurations below, near, and above the jamming transition. The configurations are generated with the infinite quench protocol and have N=105 particles. The values of potential energy (dark circles) and pressure (pluses) also depend on the maximum allowed force for any given particle which is given as Fmax. There is a discontinuity in both the potential and pressure within 0.840≤ϕ≤0.842 (vertical lines) and power-law fits give an estimate for the critical packing fraction as ϕc=0.8416±0.0003.

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    • Figure 8
      Figure 8

      The left column shows the spectral density versus wave vector and the right column shows the hyperuniformity disorder length versus measuring window side length, for global packing fractions: (a) and (b) ϕ=0.85, (c) and (d) ϕ=0.86, (e) and (f) ϕ=0.88, and (g) and (h) ϕ=0.90. The data are obtained from N=106 particle packings for all of the quench rates we study. The spectral density plots have a solid triangle which demonstrates linear behavior in q and a dot-dashed curve χ(q)=χ0+κq2, where χ0=0.0012±0.0002 and κ=0.084±0.003, which was obtained by fit to the average of the ϕ=0.90 data. In plots of the hyperuniformity disorder length some of the curves are labeled by their quench rate and the colors are the same as in the plots for the spectral density. In the h(L) plots the dotted curve at small L the separated-particle lower bound h(L)=(L/2)(1−ϕL2/〈a〉). In all plots the dashed lines labeled “Poisson” represent the expectation for a completely random arrangement of particles.

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    • Figure 9
      Figure 9

      Hyperuniformity disorder length versus measuring window side length for several values of the maximum allowed force residue at global packing fractions (a) ϕ=0.82, (b) ϕ=0.83, (c) ϕ=0.84, (d) ϕ=0.842, (e) ϕ=0.85, and (f) ϕ=0.86. The data are for Fmax={10−4,10−8,10−12,10−16} and each curve is an average from five configurations of N=105 particles. The systems are either (a)–(c) unjammed or (d)–(f) jammed with a critical packing fraction of ϕc=0.8416±0.0003. The data overlap except near the jamming transition where the affected data are labeled by Fmax. Our data in the main text are gathered with a choice of Fmax=10−12 which is unaffected at all ϕ.

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    • Figure 10
      Figure 10

      Hyperuniformity disorder length versus measuring window side length for packings quenched from different initial seeding. Data for a standard quench configuration like those described in the main text are labeled “Random” and data for Einstein quench configurations are labeled by the rms particle displacement σ/〈a〉. The Einstein quench curves show that changing the magnitude of the particle displacement does not change the uniformity of the final configuration. Regardless of initial seeding, none of the final configurations are hyperuniform, but the Einstein quench patterns are slightly more ordered.

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    • Figure 11
      Figure 11

      (a) Area fraction variance and (b) hyperuniformity disorder length versus relative measuring window size calculated for either square or circular measuring windows. Data are collected from systems with quench rate, global packing fraction, and number of particles as labeled. Here X is either the window side length L for square windows or the window diameter D for circular windows. In (a) the relative variance for circular windows lies slightly above that for the square windows for all X, but in (b) the spectra of hyperuniformity disorder lengths collapse regardless of window shape for large X. Expectations for Poisson patterns (dashed lines) depend on window shape for the relative variance but not for the hyperuniformity disorder length. In both panels the data depend on window shape for small X; this is expected because in the separated particle limit (where only 1 or 0 particles land in a window) differences in the ratio between the window area and average particle size matter [24]. In (b) the expectations for the separated particle limit (dotted curves) for different window shapes match the data exactly until X exceeds the limit.

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    • Figure 12
      Figure 12

      Hyperuniformity disorder length spectra for soft-disk packings above jamming generated by the infinite quench rate protocol. Data for N={2048,105,106} particles are an average of {200,5,1} configurations, respectively. These spectra overlap until finite-size artifacts pull h(L) noticeably downward at L greater than about half the system width. All data are truncated beyond 0.7 times the system width.

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    • Figure 13
      Figure 13

      Spectral density versus wave number, obtained by FFT of Limage×Limage pixelated image representations of particle positions for a configuration generated by the infinite quench protocol for N=106 particles at ϕ=0.86. At intermediate q the data trend down and to the right as image size increases, but at large and small q the data overlap regardless of image size. The curves for Limage={214,215} are nearly identical, so the latter curve is dashed to make both curves distinguishable. The dashed line is the expectation for a completely random system.

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