Course of dislocation lines in templated three-dimensional colloidal quasicrystals

We present an approach based on colloidal epitaxy to obtain three-dimensional colloidal quasicrystals, i.e., structures with long-range order but no periodicity. In contrast to other attempts to obtain soft-matter quasicrystals by design, no fine tuning of the colloid-colloid interactions is required. By using Monte Carlo simulations, we explore the development of multiple layers with the desired quasicrystalline symmetry when the colloidal particles sediment onto suitable quasicrystalline substrates. In addition, well-defined defect lines can be induced in the grown colloidal quasicrystal by constructing substrates with well-chosen defects that lead to induced defect lines. We present a study of the three-dimensional course of such dislocation lines by inducing and analyzing stable lines that bend, unstable ones that split up, or defect lines that merge with other defects. Dislocations are characterized by Burgers vectors that are composed of a phononic part as in a periodic crystal and an additional phasonic part that is unique to quasicrystals. As in a periodic crystal, the sum of all Burgers vectors is conserved along defect lines and at all forks. However, unlike in a periodic crystal, the course of a dislocation line in a quasicrystal depends on its position. Furthermore, although in a quasicrystal all Burgers vectors can always be written as sum of Burgers vectors with a smaller phononic part, we find dislocations that do not decompose into dislocations with smaller phononic Burgers vectors. For quasicrystals with eightfold and twelvefold rotational symmetry, we identify basis Burgers vectors that denote stable dislocations. Our results can be verified in experiments and are important for applications of colloidal quasicrystals, e.g., for the development of photonic quasicrystals.

Figure 1

Illustration of the colloidal epitaxy model. (a) The substrate possesses a plane surface with circular slots of equal depth arranged with quasicrystalline order. (a) The colloids are deposited onto the substrate. The color of the colloidal particles indicates their vertical position such that the layer-wise structure in the adsorbate is visible.

Figure 2

Schematic illustration of dislocations in a crystal and a quasicrystal in two dimensions. (a) Defect-free square lattice. (b) A single dislocation (surrounded by a yellow shading) in a square lattice is obtained by an injection or a removal of half of a line of nodes. By considering a closed loop (green line) around the dislocation and a corresponding path in a defect-free lattice, the Burgers vector is given by the difference from the end and starting point in the perfect crystal [cf. green path and magenta vector in (a)]. (c) A defect-free eightfold quasicrystal. (d) After omitting half a row of particles, it is not possible to connect the neighboring rows in a straightforward way due to the aperiodicity of a quasicrystal. Therefore, there is a gap (green) in a quasicrystal with a partial dislocation (marked yellow) that contains only the phononic contribution of the Burgers vector. The gap cannot be filled with the prototiles of the tiling. (e) Quasicrystal with a dislocation (marked yellow) whose Burgers vector contains a phononic and a phasonic part. The additional phasonic contribution corresponds to rearrangements of the tiles adjacent to the cut line such that the previously empty gap area can be covered with prototiles.

Figure 3

(a) Real-space pattern of an eightfold laser intensity field with dislocations. The dislocations are hard to find in this pattern. However, isolated density modes can be employed to localize the dislocations and to determine their Burgers vectors. (b) Pattern denoting one isolated mode of the field shown in (a). The dislocations can be easily identified in this presentation. The pattern that shows one isolated mode is obtained by an inverse Fourier transformation of two selected peaks of the Fourier transformed density (see inset where the selected peaks are marked by red circles).

Figure 4

Comparison between the growths on an eightfold substrate without phasonic flips and a substrate containing local phasonic excitations. The tilings that correspond to the substrates without and with flipped tiles are shown in (a) and (c), respectively. The flipped tiles in (c) are depicted orange and dark green. The density modes for the seventh layer of the colloidal structure grown on the substrate without phasonic flips are shown in (b), for the structure grown on a substrate where local phasonic flips were introduced in (d). In both cases, during the deposition process some strain accumulates as apparent by the green areas without leading to the formation of dislocations. However, concerning the global order that is analyzed here, there is no significant difference between the defect-free substrate and the substrate with local disorder.

Figure 5

Intensity field of the interference pattern of eight symmetrically arranged laser beams. The holes of the substrate are placed at local maxima of the intensity that have at least a distance of the colloidal diameter $\sigma$ between them. As a consequence, an Ammann-Beenker–type tiling is formed (drawn in white, see also [46]). The cores of two dislocations are marked in red. The upper one is given by the Burgers vector $\mathbf{B}=({\mathbf{b}}_1^{\left(u\right)},{\mathbf{b}}_1^{\left(w\right)})$ where ${\mathbf{b}}_1^{\left(u\right)}$ and ${\mathbf{b}}_1^{\left(w\right)}$ are given by Eq. (3), the bottom one by the opposite Burgers vector.

Figure 6

Bending of defect lines with Burgers vector $\mathbf{B}=({\mathbf{b}}_1^{\left(u\right)},{\mathbf{b}}_1^{\left(w\right)})$ (upper defect in all cases) where ${\mathbf{b}}_1^{\left(u\right)}$ and ${\mathbf{b}}_1^{\left(w\right)}$ are given by Eq. (3) and the corresponding inverse Burgers vector $-\mathbf{B}$ (bottom defects). The phononic and phasonic displacement fields of the substrate are given according to Eq. (7) with (a) $\alpha =0$ and (b) $\alpha =1$. In the left column, a schematic representation of the Burgers vector of the upper dislocation with phononic (red) and phasonic contribution (blue) is given in context with the basis Burgers vectors. The middle column contains the mode analysis for the mode in direction of ${\mathbf{b}}_1^{\left(u\right)}$ for each layer such that the position and topology of the dislocations is revealed in each layer. In the right column, this analysis is summarized into a three-dimensional illustration for the course of defect lines through the grown colloidal quasicrystal. (a) For $\alpha =0$, a vertical line is observed. (b) For $\alpha =1$, and the same position of the dislocation seed as in (a) the lines bent away from each other (green lines). However, the bending of defect lines not only depends on $\alpha$, but also on their starting position, e.g., for different starting positions straight dislocation lines can also be observed for $\alpha =1$ (cf. light blue lines).

Figure 7

(a) Annihilation and (b) forking of defect lines. The dislocations in the substrate are given by the Burgers vectors (a) $\mathbf{B}=({\mathbf{b}}_3^{\left(u\right)},{\mathbf{b}}_3^{\left(w\right)})$ and (b) $\mathbf{B}=({\mathbf{b}}_1^{\left(u\right)}-{\mathbf{b}}_4^{\left(u\right)},{\mathbf{b}}_1^{\left(w\right)}-{\mathbf{b}}_4^{\left(w\right)})$ (upper dislocations) and their inverse counterparts (bottom dislocations). In the left column, a schematic representation of the Burgers vector is given, the center images display the mode analysis along a selected mode, and in the right column a three-dimensional picture of the course of the dislocation lines is shown. In (a), the dislocations bend towards each other and ultimately anneal. In (b), the dislocation lines split up into dislocation lines given by basis Burgers vectors, i.e., $({\mathbf{b}}_1^{\left(u\right)},{\mathbf{b}}_1^{\left(w\right)})$ and $(-{\mathbf{b}}_4^{\left(u\right)},-{\mathbf{b}}_4^{\left(w\right)})$ or the corresponding inverse vectors. The sum of Burgers vectors is preserved both at the annihilation point and the forks.

Figure 8

Coalition of defect lines on a twelvefold substrate. The dislocation pairs in the substrate are given by the Burgers vectors ${\mathbf{B}}_1=({\mathbf{b}}_1^{\left(u\right)},{\mathbf{b}}_1^{\left(w\right)})$ and ${\mathbf{B}}_2=({\mathbf{b}}_5^{\left(u\right)},{\mathbf{b}}_5^{\left(w\right)})$ (upper dislocation pair) and their inverse counterparts (lower dislocation pair). In the left column, a schematic representation of the phononic contribution of the Burgers vectors is depicted, the center images show excerpts of the mode analysis for the first and fifth layers, and in the right column the three-dimensional course of the dislocation lines is illustrated. Note that all dislocations are visible in two modes (denoted by the smaller vectors in the left panel). In all columns, the respective color coding for Burgers vector, associated density modes, and defect lines is the same. The dislocations ${\mathbf{B}}_1$ (yellow) and ${\mathbf{B}}_2$ (blue) bend towards each other. By partially annealing in one mode, they unite in the third layer and form a defect line with Burgers vector $({\mathbf{b}}_3^{\left(u\right)},{\mathbf{b}}_3^{\left(w\right)})$ (green). The sum of Burgers vectors is preserved throughout the coalition.

Figure 9

Development of a defect plane on a twelvefold substrate. The dislocations in the substrate are given by the Burgers vectors $\mathbf{B}=({\mathbf{b}}_3^{\left(u\right)},{\mathbf{b}}_3^{\left(w\right)})$ (upper dislocation) and the inverse counterpart $-\mathbf{B}$ (lower dislocation). In the left column, a schematic representation of the phononic contribution of the Burgers vectors and the mode directions is depicted. The defect is visible in two modes (light green and dark green). The mode analysis in the center images contains the two modes associated with the dislocation in the odd-numbered layer. Unlike in the previous figures, the right panel does not show the course of the dislocation lines, but the three-dimensional course of the defects that are observed in the individual modes, i.e., the light-green and the dark-green lines together correspond to a dislocation line. Until the fourth layer, the induced dislocations develop vertical in both modes preserving a well-defined and well-localized dislocation within the resolution of our analysis. Then, in the fifth layer, the defects partially anneal in one mode (dark green), while they progress vertically in the second one (light green). This leads to the formation of a defect plane which eventually vanishes in the seventh layer.