Dense Periodic Packings of Tetrahedra with Small Repeating Units

Kallus, Yoav; Elser, Veit; Gravel, Simon-Pierre

doi:10.1007/s00454-010-9254-3

Cited by 52 publications

(67 citation statements)

References 10 publications

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“…30 Comparatively much less is known about dense packings, equilibrium phase behavior, and non-equilibrium glassy states of hard non-spherical particle systems. There are many ways to generalize the hard sphere to hard non-spherical particle models, e.g., ellipsoids, [31][32][33][34][35][36][37][38][39] spherocylinders, [40][41][42][43][44] cutspheres, 45,46 polyhedra, [47][48][49][50][51][52][53][54][55][56][57][58] and superballs, a shape that interpolates between the cube and the octahedron via the sphere. 59 Together with all these hard convex particle models, increasing attention is also being paid to hard non-convex particle models such as bent-shaped particles, 60 (hollowed) spherical caps, 61 helices, 62 and tori.…”

Section: Introductionmentioning

confidence: 99%

Hard convex lens-shaped particles: Densest-known packings and phase behavior

Cinacchi

Torquato

2015

The Journal of Chemical Physics

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By using theoretical methods and Monte Carlo simulations, this work investigates dense ordered packings and equilibrium phase behavior (from the low-density isotropic fluid regime to the highdensity crystalline solid regime) of monodisperse systems of hard convex lens-shaped particles as defined by the volume common to two intersecting congruent spheres. We show that, while the overall similarity of their shape to that of hard oblate ellipsoids is reflected in a qualitatively similar phase diagram, differences are more pronounced in the high-density crystal phase up to the densest-known packings determined here. In contrast to those non-(Bravais)-lattice two-particle basis crystals that are the densest-known packings of hard (oblate) ellipsoids, hard convex lens-shaped particles pack more densely in two types of degenerate crystalline structures: (i) non-(Bravais)-lattice two-particle basis body-centered-orthorhombic-like crystals and (ii) (Bravais) lattice monoclinic crystals. By stacking at will, regularly or irregularly, laminae of these two crystals, infinitely degenerate, generally non-periodic in the stacking direction, dense packings can be constructed that are consistent with recent organizing principles. While deferring the assessment of which of these dense ordered structures is thermodynamically stable in the high-density crystalline solid regime, the degeneracy of their densest-known packings strongly suggests that colloidal convex lens-shaped particles could be better glass formers than colloidal spheres because of the additional rotational degrees of freedom. C 2015 AIP Publishing LLC. [http://dx

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Section: Introductionmentioning

confidence: 99%

Hard convex lens-shaped particles: Densest-known packings and phase behavior

Cinacchi

Torquato

2015

The Journal of Chemical Physics

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“…In both of the known ordered phases of hard, regular tetrahedra, each tetrahedron is in almost-perfect face-to-face contact with at least one other tetrahedron. The densest known packing of tetrahedra (φ = 4000 4671 ≈ 85.63%) is a parallel arrangement of two dimers (four tetrahedra) -that is, two TBPs -in a triclinic unit cell to form a dimer crystal [17,18], which we refer to in the present paper as the TBP crystal (Fig. 1b).…”

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Degenerate Quasicrystal of Hard Triangular Bipyramids

2011

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We report a degenerate quasicrystal in Monte Carlo simulations of hard triangular bipyramids each composed of two regular tetrahedra sharing a single face. The dodecagonal quasicrystal is similar to that 1 recently reported for hard tetrahedra [Haji-Akbari et al., Nature (London) 462, 773 (2009)] but degenerate in the pairing of tetrahedra, and self-assembles at packing fractions above 54%. This notion of degeneracy differs from the degeneracy of a quasiperiodic random tiling arising through phason flips. Free energy calculations show that a triclinic crystal is preferred at high packing fractions.Hard disks and spheres order into hexagonal and facecentered cubic crystals, respectively, above a certain packing fraction. A more complex phase behavior is observed if the disks or spheres are rigidly bonded into dimers (dumbbells) [1][2][3][4]. A solid phase, disordered in the orientation of dimers while ordered on the monomer level, forms if the distance between monomers within a dimer is roughly the diameter of a monomer. This equilibrium solid phase can be alternatively understood as a random pairing of neighboring monomers within the native monomer crystal. The resulting thermodynamic ensemble of ground states is degenerate and the structure is therefore called a degenerate crystal. As shown by Wojciechowski et al.[1] for hard disks, the entropy associated with the degeneracy exceeds the entropy from excluded volume effects, which by itself is sufficient to drive the crystallization of hard monomers. Other consequences of the pairing of monomers into dimers include topological defects [5], a restricted, glassy dislocation motion [6,7], and unusual elastic properties [8]. Similar degenerate phases have also been observed for freely-joined chains of hard spheres [9,10].Although degenerate crystals can potentially assemble from dimers of hard shapes other than disks and spheres, few examples have been reported. One reason is the competition between degenerate crystals and the liquid crystalline phases frequently observed for particles with large aspect ratios. For example, elongated tetragonal parallelepipeds, which for an aspect ratio of 2:1 can be viewed as dimers of face-sharing cubes, form a degenerate parquet phase at intermediate densities before transforming into a smectic liquid crystal that eventually crystallizes [11]. Another simple dimer is the triangular bipyramid (TBP), which consists of two face-sharing, regular tetrahedra (Fig. 1a). The TBP is the simplest face-transitive bipyramid and the twelfth of the 92 Johnson solids. The lack of inversion symmetry of the TBP, however, makes lattice packings non-optimal [12], and thus it is potentially more interesting as a dimer than dimers of spheres and cubes. Moreover, the recent synthesis of TBP-shaped nanoparticles and colloids [13][14][15][16] relevance. In both of the known ordered phases of hard, regular tetrahedra, each tetrahedron is in almost-perfect face-to-face contact with at least one other tetrahedron. The densest known packing of tetrahedr...

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“…Yet this statement does not guarantee that such structures are thermodynamically stable at finite pressure, nor that they can be readily accessed through random thermal motion. Indeed, computer simulations by Haji-Akbari et al show, in the related case of tetrahedra, that these factors result in self-assembly of an intriguing quasi-crystal that is not the densest packing [5,20,21,22].…”

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confidence: 99%