Degenerate Quasicrystal of Hard Triangular Bipyramids

Abstract

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...

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

“…95 While it is easy to imagine the degeneracy of hard sphere di-and n-mers positionally ordered packings, by pairing or more generally linking randomly hard-sphere monomers arranged in their densest fcc or hcp crystal packings or their (Barlow) stacking variants, more remarkable is the observation of the degeneracy in the quasicrystal-like structure of hard triangular bipyramids, as it was not straightforward to envision that pairing tetrahedra into triangular bipyramids did not affect the quasicrystal-like structure. In the conclusions of that work, it was also hypothesized that degenerate dense packings might be common in systems made of hard dimers.…”

Packing and self-assembly of truncated triangular bipyramids

“…All known densest packings of two-and three-dimensional objects are ordered [46], which necessitates a disorder-order transition at some finite pressure and density. However, it is also known that the ordered structures formed by many hard particle systems at intermediate densities can be geometrically unrelated to their corresponding densest pack-ings [34,35,43,47].…”

“…While the emergent order is typically periodic in the form of a crystal, quasiperiodic and degenerate [70] order are also possible. These phases are stabilized by entropy alone and are observed in compu- * Electronic address: sglotzer@umich.edu tational studies of various hard particle shapes including spheres [2,3], spherocylinders [24], thin disks [25,26], ellipsoids [27][28][29][30][31], dumbbells [32], tetrahedra [33,34], triangular bipyramids [35], superballs, cubes and octahedra [36,37], snowman particles [38], squares [39], spacefilling polyhedra [40][41][42] and many other polyhedra [43]. The preponderance of optical, electrical, magnetic and mechanical properties of ordered structures formed from such particles [44,45] makes knowledge and prediction of their expected thermodynamic assemblies of particular current interest.…”

“…[33,35,47,50] to investigate the packing and self-assembly of hard truncated triangular bipyramids. We study the densest packings with small repeating unit cells using Monte Carlo simulations and further analytical and numerical optimization.…”

Monte Carlo simulations of densely-packed athermal polymers in the bulk and under confinement

“…While initial studies focused understandably on dense packing of uniform spheres research has since been extended to objects with complex, anisotropic shapes (Donev et al, 2006;Frenkel et al, 1988;Haji-Akbari et al, 2013, 2011a, 2011bJiao et al, 2009;Jiao and Torquato, 2011;Torquato and Jiao, 2009;Veerman and Frenkel, 1991;Xu et al, 2005;Zeravcic et al, 2009). Athermal polymer packings (of hard-sphere chains) fall in the latter category: while the constituent monomers are well-defined, nonoverlapping spheres (of uniform size), the global shape and size of each molecule are highly non-trivial, fluctuate over time, and are distinctly different from one chain to the other.…”