Entropic formation of a thermodynamically stable colloidal quasicrystal with negligible phason strain

Abstract

Quasicrystals have been discovered in a variety of materials ranging from metals to polymers. Yet, why and how they form is incompletely understood. In situ transmission electron microscopy of alloy quasicrystal formation in metals suggests an error-and-repair mechanism, whereby quasiperiodic crystals grow imperfectly with phason strain present, and only perfect themselves later into a high-quality quasicrystal with negligible phason strain. The growth mechanism has not been investigated for other types of quasicrystals, such as dendrimeric, polymeric, or colloidal quasicrystals. Soft-matter quasicrystals typically result from entropic, rather than energetic, interactions, and are not usually grown (either in laboratories or in silico) into large-volume quasicrystals. Consequently, it is unknown whether soft-matter quasicrystals form with the high degree of structural quality found in metal alloy quasicrystals. Here, we investigate the entropically driven growth of colloidal dodecagonal quasicrystals (DQCs) via computer simulation of systems of hard tetrahedra, which are simple models for anisotropic colloidal particles that form a quasicrystal. Using a pattern recognition algorithm applied to particle trajectories during DQC growth, we analyze phason strain to follow the evolution of quasiperiodic order. As in alloys, we observe high structural quality; DQCs with low phason strain crystallize directly from the melt and only require minimal further reduction of phason strain. We also observe transformation from a denser approximant to the DQC via continuous phason strain relaxation. Our results demonstrate that soft-matter quasicrystals dominated by entropy can be thermodynamically stable and grown with high structural quality--just like their alloy quasicrystal counterparts.

Keywords: colloidal crystal; entropic crystallization; phason; quasicrystal growth; tilings.

Fig. 1.

Tiling hierarchy in the DQC from hard regular tetrahedra. (A) Thick gray lines connect the centers of nearest-neighbor tetrahedra. The DQC can be described as a decorated tiling on different hierarchy levels as indicated by colors. On each hierarchy level, tile vertices are located at the centers of motifs marked by translucent colored circles. Connecting tile vertices gives square tiles, triangle tiles, and rhombus tiles (as phason defects (50, 51), SI Appendix, Fig. S2) arranged into a quasiperiodic tiling. Four hierarchical tilings are shown within the yellow square tile. (B) Left column: PD (blue), interdigitating 22-T (green), noninterdigitating 22-T (red), and large dodecahedral cluster (yellow) motifs. Middle and right columns: Arrangement of the motifs and relationship to the tetrahedron network for a triangle tile. In this work, we analyze the DQC using the green scale.

Fig. 2.

Evolution of the quasicrystalline tiling during DQC growth. (A) Fractions of tetrahedra that are part of an icosahedron, a 22-T, both an icosahedron and a 22-T, a PD but not an icosahedron or a 22-T, and none of these motifs (“None”) during DQC growth. All five labels add up to 100% (SI Appendix, Fig. S4). (B) Distribution of local density ${\varphi }_{\text{loc}}$ sampled at four different MC checkpoints as marked by the four vertical lines in A. The distribution changes from unimodal to bimodal and back to mostly unimodal, indicating first the appearance and then the partial disappearance of solid–fluid coexistence. Because the simulation is conducted in the isochoric ensemble at $\varphi =0.49$, peaks shift toward lower densities as the solid grows. (C) The growing solid is identified by clustering 22-Ts that share tetrahedra as shown after $10\times {10}^6$ and $15\times {10}^6$ MC sweeps. Tetrahedra belonging to the fluid are translucent gray. Diffraction patterns of the solid (Lower Right Inset) exhibit 12-fold symmetry indicating that the solid is a DQC. (D) Networks of 22-T centers in C, which define the quasicrystalline tiling of DQC (green hierarchy level in Fig. 1). Spots in bond orientational order diagrams (Upper Right Inset) and diffraction patterns (Lower Right Inset) of the tilings gradually sharpen as DQC growth proceeds.

Fig. 3.

Phason strain analysis during DQC growth starting from the hard tetrahedron fluid. (A) The growing DQC solid (Upper Row) and its tiling (Lower Row). Diffraction patterns at $9\times {10}^6$ MC sweeps for tetrahedron centers (Upper Right) and tiling vertices (Lower Right) show many peaks with 12-fold symmetry, indicating a well-formed quasicrystal. (B) Evolution of system pressure P* (magenta) and phason strain α measured from the tiling (blue). The tiling size is large enough to measure phason strain reliably after $4.5\times {10}^6$ MC sweeps (start of interpolation). Pressure converges after $9\times {10}^6$ MC sweeps (magenta arrow). Phason strain converges more slowly after $18\times {10}^6$ MC sweeps (Inset). (C) Phason displacement field analysis at the times when pressure (green) and phason strain (cyan) each converge to equilibrium values. Here, the times are marked by green and cyan arrows in B, respectively. Phason displacement is the average perpendicular space distance $r^{\perp }/\text{δ}$ as a function of parallel space distance $r^{\parallel }/\text{δ}$, where $\text{δ}$ is the tile edge length. Phason displacement grows linearly. The slope of this growth is the phason strain $\alpha$, which is measured from $r^{\perp }\left(r^{\parallel }\right)/\text{δ}$ after removing background noise $\overline{r^{\perp }}\left(r^{\parallel }\right)/\text{δ}$ (SI Appendix, Fig. S11) and is scaled by phason strain of the first-order approximant ${\alpha }_{1st}$. Further details are found in SI Appendix.

Fig. 4.

Continuous transformation from the first-order approximant to the DQC during a long MC simulation. (A) Evolution of the diffraction pattern from fourfold symmetry in the approximant ($0\times {10}^6$ MC sweeps) to 12-fold symmetry in the DQC ($80\times {10}^6$ MC sweeps). (B) Phason strain $\alpha /{\alpha }_{1st}$ gradually relaxes to zero during the transformation. (C) Radial density in perpendicular space $r^{\perp }$ sharpens over time toward a compact occupation domain as expected for a high-quality DQC. (D) Snapshots of projected tile vertices in the perpendicular space sampled at $0\times {10}^6$, $10\times {10}^6$, and $80\times {10}^6$ MC sweeps. When the transformation is complete ($80\times {10}^6$ MC sweeps), the positions form a single roughly circular domain with radius $\text{δ}$, where $\text{δ}$ is the tile edge length. Due to random phason fluctuations, the boundary of the domain is blurred.