Growth Modes of Quasicrystals

The growth of quasicrystals, i.e., aperiodic structures with long-range order, seeded from the melt is investigated using a dynamical phase field crystal model. Depending on the thermodynamic conditions, two different growth modes are detected, namely defect-free growth of the stable quasicrystal and a mode dominated by phasonic flips which are incorporated as local defects into the grown structure such that random tilinglike ordering emerges. The latter growth mode is unique to quasicrystals and can be verified in experiments on one-component mesoscopic systems.

Figure 1

Equilibrium phase behavior determined with the PFC model for $k_2=2\cos (\pi /12)$. In the center (d) the phase diagram with stable triangular, 12-fold quasicrystalline, and fluid phases is shown depending on the reduced temperature $\epsilon$ and the reduced average density $\overline{\psi }$. Coexistence regions are shaded in grey. The green line denotes the path considered in Fig. 3. (a),(e) Quasicrystalline density fields with possible tilings, (b),(f) the corresponding structure factors, and (c),(g) the distributions of heights of local maxima of the density field (a)–(c) close to the triple point for $\epsilon =0.25$ and $\overline{\psi }=-0.314$ [orange point in (d)] and (e)–(g) far away from the triple point for $\epsilon =1.90$ and $\overline{\psi }=-0.947$ (red point). The red line in (c) is a power law fit.

Figure 2

Snapshots of structures during the growth. The plotted density is scaled and shifted such that its values in bulk range from 1 for the maximum (depicted black) to 0 for the minimum (white). The blue circles mark the positions of the original seed, which either is (a),(d) around the global symmetry center (red circle) or (b,e) at another position. The parameter sets correspond to the orange and red points in Fig. 1, i.e., (a)–(c) close to the triple point leading to a perfect quasicrystal or (d)–(g) far away from the triple point such that a random tilinglike phase develops. (c),(f) Enlargements of the growth fronts in (b),(e). (g) Comparison of the grown structure (red tiling) and the perfect equilibrium quasicrystal (green tiling). Movies of (a), (b), (d), and (e) are available in the Supplemental Material [38].

Figure 3

(a)–(f) Examples of phasonic flips that occur during growth for parameters far away from the triple point. (a)–(c) Difference of the density field of the equilibrium quasicrystal and the density field of the grown structure. (d)–(f) The corresponding tilings for the equilibrium (green) and the grown structure (red). (g) Number of defects $N_d$ divided by the total number of analyzed particles $N_t$ for structures grown along the green line shown in 1. The inset shows the same plot in log-linear representation. The orange and the red circle mark the results for the parameters used in Fig. 2.

Figure 4

Snapshots of the growth of a quasicrystal with 10-fold symmetry. (a) Close to the triple point, for $\epsilon =0.0685$, a defect-free quasicrystal grows. (b) For $\epsilon =0.52$ the growth front is sharper and the grown structure differs from the equilibrium quasicrystal. Close to the growth front, Archimedean-like tilings are observed that are periodic in one direction and have been obtained for particles that are deposited on quasicrystalline substrates [47]. The insets show comparisons of the tilings corresponding to the equilibrium (green) and the grown structures (red). Movies are included in the Supplemental Material [38].