A patchy-particle 3-dimensional octagonal quasicrystal

TL;DR

This work demonstrates a path to realize a 3D octagonal quasicrystal using patchy particles designed from an ideal Ammann-Beenker-based structure. A binary mixture of 5- and 8-patch particles (P5/P8) forms an octagonal QC, and remarkably a one-component P5 system also yields an essentially identical QC, indicating 8-patch particles are not strictly required. The study shows that torsional, angular, and patch-width constraints are critical for assembling the desired symmetry, and that the resulting QCs exhibit edge dislocations and domain structure consistent with entropy-driven stabilization. These findings broaden the possible symmetries accessible to patchy-particle QCs and point toward experimental realizations with DNA origami or protein design, while raising questions about thermodynamic versus kinetic stability of these phases.

Abstract

We devise an ideal 3-dimensional octagonal quasicrystal that is based upon the 2-dimensional Ammann-Beenker tiling and that is potentially suitable for realization with patchy particles. Based on an analysis of its local environments we design a binary system of 8- and 5-patch particles that in simulations assembles into a 3-dimensional octagonal quasicrystal. The local structure is subtly different from the original ideal quasicrystal possessing a narrower coordination-number distribution; in fact, the 8-patch particles are not needed and a one-component system of the 5-patch particles assembles into an essentially identical octagonal quasicrystal. We also consider a one-component system of the 8-patch particles; this assembles into a cluster with a number of crystalline domains, but which, because of the coherent boundaries between the crystallites, has approximate eight-fold order. We envisage that these systems could be realized using DNA origami or protein design.

Figures (7)

• Figure 1: The ideal target octagonal quasicrystal. (a) The structure projected down the $c$-axis is that of an Ammann-Beenker tiling with particles at the corners of the square and rhomboidal tiles. The particles are coloured by their coordination number (See Fig. \ref{['fig:ideal_env']}). (b) A side-view along one of the two-fold axes. (c) In 3-dimensions, the square is puckered and the rhombus corresponds to a right- (illustrated) or left-handed helix with a pitch length of the $c$ repeat.
• Figure 2: (a) The coordination environments in the ideal quasicrystal labelled by their coordination number and their local point group symmetry. The two four-coordinate and six-coordinate environments are enantiomeric. (b) The two patchy particles. (c) The interaction matrix between the four different types of patches on the above two particles.
• Figure 3: (a) Close-up of a cut through a binary octagonal quasicrystal viewed down the 8-fold axis. Bonds are drawn between particles within $5\,\sigma_\mathrm{LJ}$ of the cut surface, and these particles are visualized as small green (P5) or cyan (P8) spheres. Particles further away from this cut plane are visualized as larger grey spheres. The yellow ellipse highlights an "overlapping-squares" motif (see Fig. \ref{['fig:crystal']}(a)). (b) BOOD for the binary quasicrystal. (c) BOOD for a one-component quasicrystal made of 5-patch particles. (d) Diffraction patterns of the binary octagonal QC viewed down the 8-fold axis and a 2-fold axis. (e) Radial distribution functions for the assembled binary, P5, and P8 clusters and the ideal OQC and $C2/c$ crystal. (f) Coordination number distributions for the same systems as (e) (calculated for the assembled QCs using the same energy and distance criteria as for the BOODs).
• Figure 4: (a) A slab of the ideal $C2/c$ crystal viewed along the pseudo 8-fold axis. The 4-coordinate environments are coloured in red and the 5-coordinate environments in green. A primitive unit cell is shown in blue (the six particles in this cell are shown as spheres). An "overlapping squares" motif is highlighted. The coordinates for the particles in the conventional unit cell of the crystal are given in Supplementary Table S3. The diffraction patterns for this crystal are shown in Fig. S9. (b) Side views of this crystal and the ideal octagonal QC. (Further side views of the crystal are given in Fig. S4.) (c) A cut of thickness 5 $\sigma_\mathrm{LJ}$ through the P5 quasicrystal with the periodic direction vertical. Note that the features that look like bow ties in (b) and (c) are side views of the square motifs (Fig. \ref{['fig:ideal']}(c)).
• Figure 5: (a) A slab of dimensions of $20\times 20\times 4$ (in units of $\sigma_\mathrm{LJ}$) from the binary quasicrystal with the particles represented as points (P5: green; P8: cyan). If viewed at a low angle, it becomes easier to see the lines of points and hence to spot the dislocations. (b) Inverse Fourier-transformed image of two equal and opposite diffraction spots in the first intense ring of the Fourier transform of the above slab. Edge dislocations are indicated by a 'T'. Circuits around the dislocations are drawn in blue. If these are followed the number of lines on one side of the dislocation will be different from the other. (Plots for the three other pairs of equivalent diffraction spots are shown in Fig. S10 and enable further dislocations to be identified.)