Geometrical Frustration in Amorphous and Partially Crystallized Packings of Spheres

N. Francois, M. Saadatfar, R. Cruikshank, and A. Sheppard

Phys. Rev. Lett. 111, 148001 – Published 2 October 2013

Abstract

We study the persistence of a geometrically frustrated local order inside partially crystallized packings of equal-sized spheres. Measurements by x-ray tomography reveal previously unseen grain scale rearrangements occurring inside large three-dimensional packings as they crystallize. Three successive structural transitions are detected by a statistical description of the local volume fluctuations. These compaction regimes are related to the disappearance of densely packed tetrahedral patterns of beads. Amorphous packings of monodisperse spheres are saturated with these tetrahedral clusters at Bernal’s limiting density ( $ϕ \approx 64 %$ ). But, no periodic lattice can be built upon these patterns; they are geometrically frustrated and are thus condemned to vanish while the crystallization occurs. Remarkably, crystallization-induced grain rearrangements can be interpreted in terms of the evolution of key topological features of these aggregates.

Received 16 March 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.148001

Authors & Affiliations

N. Francois^*, M. Saadatfar^†, R. Cruikshank, and A. Sheppard

Department of Applied Mathematics, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Capital Territory 0200, Australia

^*nicolas.francois@anu.edu.au
^†mos110@physics.anu.edu.au

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Vol. 111, Iss. 14 — 4 October 2013

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Images

Figure 1
(a) Tomographic slice of a partially crystallized packing ( $ϕ \approx 70.5 %$ ) in a spherical container. (b) PDF of the local volume fluctuations for increasing global volume fraction $ϕ$ . The PDF for $ϕ = 0.598$ is fitted by a gamma distribution (red line). Inset: The Voronoi partition of a local configuration of beads. The bead and the surrounding space closest to its center define the Voronoi elementary brick. (c) The granular “specific heat” $k_{g} = (\bar{V} - V_{\min })^{2} / σ^{2}$ versus $ϕ$ . Three successive transitions are highlighted at $Φ = 0.64$ , 0.68, 0.72. 34 subsets [26] of roughly 4000 beads extracted from six different packings (indicated by different markers) have been analyzed: two initial jammed packings obtained by pouring $+$ four partially crystallized packings. Inset: Variance $σ^{2}$ of the Voronoi volume versus $ϕ$ .Reuse & Permissions
Figure 2
Evolution of the densest simplices (quasiregular tetrahedra) of the Delaunay tessellation versus the volume fraction. (a) Fraction of tetrahedra ( $δ < 0.25 \times d$ ) versus $ϕ$ . Inset: The Delaunay representation of a local configuration of beads. It is based on local simplices (generalized tetrahedra) built on the centers of the locally four closest beads. (b) Fraction of tetrahedra that are face adjacent and are involved in aggregates (polytetrahedral clusters) containing more than 3 components. (c) Fraction of beads which are part of at least one dense tetrahedron (blue line) or that are part of a polytetrahedral cluster (red line).Reuse & Permissions
Figure 3
Visualizations of the topology of polytetrahedral aggregates. Polytetrahedral clusters are represented as networks in which each vertex (blue sphere) is a dense tetrahedron and each bond (white tube) represents two tetrahedra sharing a face. The inset of (a) shows the centers of beads forming a pair of face-adjacent dense tetrahedra and their topological representation. (a) Polytetrahedral clusters in a packing at $ϕ \approx 0.64$ . A loop composed of five face-adjacent tetrahedra (five-ring) is highlighted in red. (b) Close-up on an “incomplete” icosahedral configuration. A cluster of 12 five-rings would form an icosahedron; here, this structure contains two five-rings adjacent to an eight-ring. (c) Close-up on a “chain” of rings composed of five-rings (red lines) and a six-ring (green lines).Reuse & Permissions
Figure 4
(a) Evolution of the polytetrahedral topological classes versus the volume fraction. Tetrahedra involved in rings (red line, $N_{rings}$ ) and branched tetrahedra (orange line, $N_{branched}$ ) form the polytetrahedral family $N_{polytetra} = N_{rings} + N_{branched}$ ; the fraction of tetrahedra in bipyramids ( $N_{bipyr}$ ) is the difference between the orange and the blue lines. The fractions are normalized by the number of face-adjacent tetrahedra $N_{facetetra} = N_{polytetra} + N_{bipyr}$ . (b) Size distribution (in tetrahedra) of the ring patterns normalized by $N_{facetetra}$ . (c) Outgoing average connectivity of the five-rings (see the Supplemental Material 25); this connectivity is five for a five-ring part of an isolated icosahedron.Reuse & Permissions

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