Dislocation-free growth of quasicrystals from two seeds due to additional phasonic degrees of freedom

We explore the growth of two-dimensional quasicrystals, i.e., aperiodic structures that possess long-range order, from two seeds at various distances and with different orientations by using dynamical phase-field crystal calculations. We compare the results to the growth of periodic crystals from two seeds. There, a domain border consisting of dislocations is observed in case of large distances between the seed and large angles between their orientation. Furthermore, a domain border is found if the seeds are placed at a distance that does not fit to the periodic lattice. In the case of the growth of quasicrystals, we only observe domain borders for large distances and different orientations. Note that all distances do inherently not match to a perfect domain wall-free quasicrystalline structure. Nevertheless, we find dislocation-free growth for all seeds at a small enough distance and for all seeds that approximately have the same orientation. In periodic structures, the stress that occurs due to incommensurate distances between the seeds results in phononic strain fields or, in the case of too large stresses, in dislocations. In contrast, in quasicrystals an additional phasonic strain field can occur and suppress dislocations. Phasons are additional degrees of freedom that are unique to quasicrystals. As a consequence, the additional phasonic strain field helps to distribute the stress and facilitates the growth of dislocation-free quasicrystals from multiple seeds. In contrast, in the periodic case the growth from multiple seeds most likely leads to a structure with multiple domains. Our work lays the theoretical foundations for growing perfect quasicrystals from different seeds and is therefore relevant for many applications.

Figure 1

Examples of initial configurations for (a) the periodic case, $\epsilon =0.018,\overline{\psi }=-0.0775$ and (b) the quasiperiodic case $\epsilon =0.25$, $\overline{\psi }=-0.315$. The periodic crystalline seeds are chosen such that they are symmetric around the center of the seed which contains a density maximum, while the quasicrystalline seeds contain the main symmetry center. The seeds are cut out from equilibrated solid configurations.

Figure 2

Long-time situation of the growth started from two seeds at a distance $d$ and orientations that differ by an angle $\theta$ for the periodic case. (a) Depending on $d$ and $\theta$, circles mark the points in the diagram where we observe defect-free crystals at late times. Squares denote cases where a domain border consisting of dislocations occurs. The red (light gray region with circles) and blue color (dark gray with squares) are used to highlight the parameter regions where defect-free final configurations or domain borders are found, respectively. (b–e) Density fields of the the grown structure at late times long after the two crystalites have been united. In (b, c) for a small angle, $\theta =2^{\circ }$, either the large seed incorporates the small seed if the seeds are at a commensurate distance, e.g., for $d=16a$ as shown in (b) or both seeds are stable and an interface is formed in case of incommensurate distances, e.g., for $d=17a$ shown in (c). For a large angle $\theta ={25}^{\circ }$, the large seed incorporates the small seed at small distances up to about $d=13a$ shown in (d). For larger distances, e.g., $d=14a$ (e), an interface occurs.

Figure 3

(a) The phase diagram for the quasicrystal in the region of interest (for the complete diagram, see Ref. [15]). The fluid, quasicrystalline, and triangular phase are marked with the letters F, Q, and T, respectively. Coexistence regions are observed that occur between the two lines drawn at phase boundaries in the phase diagram. (b, c) Long-time situation of the structures grown from two seeds as a function of distance $d$ and angle $\theta$ for $\epsilon =0.22$ (b) or $\epsilon =0.25$ (c). The circles and red (light gray) regions mark the cases with a dislocation-free quasicrystal at late times. The squares and blue (dark gray) regions denote where a metastable grain boundary is observed. (d–g) Density fields for $\epsilon =0.22$ and (h–k) the corresponding filtered fields where dislocations become visible using the method explained in [48, 49] for a very small angle $\theta =0.{093}^{\circ }$ (d, e, h, i) and a large angle $\theta =13.{625}^{\circ }$ (f, g, j, k). For small distances $d=43a$ (d, h) or $d=35a$ (f, j) no dislocations occur while for a larger distance $d=47a$ (e, i) or $d=39a$ (g, k) domain borders are observed.

Figure 4

Growth of a dislocation-free quasicrystal with $\epsilon =0.25$ from two seeds that are rotated with respect to each other by $\theta =13.{625}^{\circ }$ and are placed at an incommensurate distance $d=16a$. (a–c) Growth process that demonstrates how the quasicrystal that is grown from the small (right) seed is taken over by the structure that originated from the large (left) seed. The white solid circles denote the initial positions of the seeds. The orange (light gray) or blue (dark gray) areas on the right- or left-hand side of the figure indicate the region where the influence from the small or large seed dominates, respectively. The colors are determined by calculating the differences between the density field obtained from the growth from two seeds and the density field obtained by growing a quasicrystal from only the large (orange, light gray) or only the small seed (blue, dark gray). The resulting differences have been squared and smoothened by a Gaussian blur of width $\sigma =4.1a$. Therefore, orange (light gray) or blue (dark gray) denotes the area where one mainly finds the structure that does not fit to the large or small seed, respectively. White, gray, and black areas outside of the grown structures originate from additive mixing of blue and orange, i.e., white denotes areas that do not fit to either seed while black areas fit to both. (d) Zoom view into the area marked by a white box in (c) (i.e., at time $t=500$) in the same representation, except without blurring. Therefore, blue (dark gray) spots denote points that do not fit to the small seed and orange (light gray) spots indicate differences from a quasicrystal grown from only the large seed. (e) Histograms of density values that appear in the density fields in the regions that are dominated by the small seed [orange, light gray areas in (a–c)] for different times $t$. The black solid line indicates the distribution in a perfect quasicrystal (arbitrary scale). Note that for times $t\ge 750$, an orange (light gray) domain maintaining the orientation of the right seed is no longer detectable.