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Shaking-induced crystallization of dense sphere packings

D. P. Shinde, Anita Mehta, and G. C. Barker
Phys. Rev. E 89, 022204 – Published 27 February 2014
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  • INTRODUCTION
  • METHODS
  • GLOBAL ORDER ANALYSIS
  • LOCAL ORDER ANALYSIS
  • CONCLUSIONS
  • ACKNOWLEDGEMENTS
    • References

    Abstract

    We use a hybrid Monte Carlo algorithm to simulate the shaking of spheres at different vibrational amplitudes and find that spontaneous crystallization occurs in specific dynamical regimes. Several crystallizing transitions are typically observed, leading to end states which can be fully or partially ordered, depending on the shaking amplitude, which we investigate using metrics of global and local order. At the lowest amplitudes, crystallization is incomplete, at least for our times of observation. For amplitude ranges where crystallization is complete, there is typically a competition between hcp and fcc ordering. It is seen that fcc ordering typically predominates; in fact for an optimal range of amplitudes, spontaneous crystallization into a pure fcc state is observed. An interesting feature is the breakdown of global order when there is juxtaposition of fully developed hcp and fcc order locally: we suggest that this is due to the interfaces between the different domains of order, which play the same role as dislocations.

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    • Received 10 June 2013

    DOI:https://doi.org/10.1103/PhysRevE.89.022204

    ©2014 American Physical Society

    Authors & Affiliations

    D. P. Shinde* and Anita Mehta†

    • Department of Theoretical Sciences, S. N. Bose National Centre for Basic Sciences, Calcutta, Calcutta 700098, India

    G. C. Barker‡

    • Institute of Food Research, Norwich Research Park, Colney, Norwich NR4 7UA, United Kingdom

    • *shinde@bose.res.in
    • †anita@bose.res.in
    • ‡gary.barker@ifr.ac.uk

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    Issue

    Vol. 89, Iss. 2 — February 2014

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    Images

    • Figure 1
      Figure 1

      Plots of radial distribution functions g(r) as a function of normalized distance r/d for various packing fractions ϕ. The number of peaks shows the development of spatial order from (a) low to (d) high.

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    • Figure 2
      Figure 2

      Variation of Q6,global against packing fraction ϕ. Note the slight jump (kink) at 0.62 and steady rise after 0.64. The global order shows breakdown at 0.69 [(c), (d), (g), (h), and (i)]. The vertical line markers at ϕ=0.62 (dashed lines) and 0.64 (dot-dashed lines) serve as a guide to the eye.

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    • Figure 3
      Figure 3

      Plots of Q6,local vs Q4,local. The scattered values show disordered states of ϕ=0.61 (stars), ϕ=0.62 (open circles), and ϕ=0.63 (open triangles). The horizontal line markers at 0.485 (blue solid line for hcp) and 0.575 (magenta dashed line for fcc) serve as a guide for the eye.

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    • Figure 4
      Figure 4

      Probability density plots of Q6,local for ϕ∼0.61 (black solid line), 0.62 (red dash-dotted line), and 0.63 (green dashed line). The peaks are broad and robust. The vertical line markers at 0.485 (blue solid line for hcp) and 0.575 (magenta dashed line for fcc) serve as a guide for the eye.

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    • Figure 5
      Figure 5

      Plots of Q6,local vs Q4,local for the densities ϕ∼0.64 (stars) and 0.65 (open circles). Both states are disordered. The ϕ∼0.65 state has less scatter than the 0.64 state. The horizontal line markers at 0.485 (green solid line for hcp) and 0.575 (red dashed line for fcc) serve as a guide for the eye.

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    • Figure 6
      Figure 6

      Probability density plots of a Q6,local for ϕ∼0.64 (magenta solid line) and 0.65 (blue dashed line). The distributions have sharper peaks than before with some predominance of the second peak. The vertical line markers at 0.485 (green solid line for hcp) and 0.575 (red dashed line for fcc) serve as a guide for the eye.

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    • Figure 7
      Figure 7

      Q6,local vs Q4,local for densities ϕ∼0.68 (open circles) and 0.69 (stars). The state of ϕ∼0.68 has more scatter than 0.69. Notice the sharp division into two distinct groups for ϕ∼0.69. The special noticeable situations are for A=0.10 [(b)] and 0.15 [(c)]. The horizontal line markers at 0.485 (blue solid line for hcp) and 0.575 (magenta dashed line for fcc) serve as a guide for the eye.

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    • Figure 8
      Figure 8

      Probability density plots of Q6,local for ϕ∼0.68 (cyan solid line) and 0.69 (orange dashed line). The peaks of ϕ∼0.69 are sharper than those of 0.68. For ϕ∼0.69 at A=0.10 and 0.15 only one sharp peak at Q6,local∼0.575 is visible [(b) and (c)]. The vertical line markers at 0.485 (green solid line for hcp) and 0.575 (black dashed line for fcc) serve as a guide for the eye.

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    • Figure 9
      Figure 9

      Plots of the maximum densities for A=0.05, 0.08, and 0.10. Scatterplots of Q6,local vs Q4,local are broadly distributed into two groups. The probability density plots [(b), (d), and (f)] indicate a second peak which is sharper than the first. The horizontal and vertical lines at 0.485 (for hcp) and 0.575 (for fcc) serve as a guide for the eye.

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    • Figure 10
      Figure 10

      Scatterplots of Q6,local vs Q4,local show a single fcc cluster for a maximum density of ϕ∼0.72 for A=0.15 0.18, 0.20, and 0.25.

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    • Figure 11
      Figure 11

      Graphs of Q6,local vs Q4,local show the coexistence of two fcc and hcp sphere clusters. The data of fcc and hcp clusters are again plotted in (c) and (d) and (e) and (f) for the respective amplitudes. The horizontal line markers at 0.485 (green dashed line for hcp) and 0.575 (red solid line for fcc) in (a) and (b) serve as a guide for the eye.

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