Degenerate Quasicrystal of Hard Triangular Bipyramids

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 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.

Figure 1

Phases formed by (a) triangular bipyramids (TBPs): (b) TBP crystal, (c) degenerate quasicrystal, (d) regular quasicrystal approximant, (e) degenerate quasicrystal approximant. For visualization purposes, we show member tetrahedra of most TBPs at 30% actual size and connect their centers with bonds. In (c)–(e), tetrahedra and bonds are colored according to their orientation projected on the plane.

Figure 2

(a) TBPs assemble into a dodecagonal quasicrystal in isobaric and isochoric Monte Carlo simulations. (b) The square-triangle tiling obtained by connecting the centers of twelvefold rings of member tetrahedra. Intra- and inter-TBP bonds are depicted in black and gray, respectively. (d),(e) Diffraction patterns with centers of member tetrahedra as scatterers calculated (c) perpendicular to and (d) across the layers. (e) Intra-TBP and (f) total bond order diagrams.

Figure 3

(a) Top and (b) bottom views of the regular approximant. The unit cell has 41 triangular bipyramids and particles are colored according to their environment. (c) Schematics of the unit cell with connections between neighboring rings and between rings and central interstitials shown with double arrows. (d),(f) Intra-TBP and (e),(g) total bond order diagrams for (d),(e) the regular approximant and (f),(g) the degenerate approximant. In the legends, “int.” stands for “interstitial.”

Figure 4

The self part of the van Hove correlation function $G_s(r,t)$ measures the particle motion in the approximant. The separation distance $r(t)$ is calculated between centers of mass of member tetrahedra. (a) Large rearrangements occur at $\varphi =57\%$. (b) There is little motion present at $\varphi =60\%$. The observed dynamics is similar to that observed in the hard tetrahedron system [21].

Figure 5

(a) Equation of state for the TBP crystal, the degenerate quasicrystal, the regular and the degenerate approximants. (b) The free energy difference between the TBP crystal and the approximant. (c) Equilibrium phases of hard TBPs.