What Phasons Look Like: Particle Trajectories in a Quasicrystalline Potential

Among the distinctive features of quasicrystals—structures with long-range order but without periodicity—are phasons. Phasons are hydrodynamic modes that, like phonons, do not cost free energy in the long-wavelength limit. For light-induced colloidal quasicrystals, we analyze the collective rearrangements of the colloids that occur when the phasonic displacement of the light field is changed. The colloidal model system is employed to study the link between the continuous description of phasonic modes in quasicrystals and collective phasonic flips of atoms. We introduce characteristic areas of reduced phononic and phasonic displacements and use them to predict individual colloidal trajectories. In principle, our method can be employed with all quasicrystalline systems in order to derive collective rearrangements of particles from the continuous description of phasons.

Figure 1

Colloidal particles located in the minima of a potential with decagonal symmetry when a phasonic drift is applied in the $x$ direction [(a1), (a2), and (c1)–(d4)] and in the $y$ direction [(b1) and (b2)] (cf. movies in the Supplemental Material [19]). Colloids are plotted in a similar color when they move, on average, in the same direction. (a1) Snapshots for $\mathbf{w}=(0.05,0)$ and (b1) $\mathbf{w}=(0,0.05)$ (solid circles). The previous positions of the colloids at $\mathbf{w}=(0,0)$ are marked by stars, the future positions by open circles at (a1) $\mathbf{w}=(0.1,0)$ and (b1) $\mathbf{w}=(0,0.1)$. The right parts show the Delaunay triangulation for the current and future colloidal positions by gray solid and blue dashed lines, respectively. Trajectories of the colloids for a phasonic drift from $\mathbf{w}=(0,0)$ (solid circles) to (a2) $\mathbf{w}=(1,0)$ or to (b2) $\mathbf{w}=(0,1)$. (c1)–(d4) Single colloids in the decagonal landscape when a phasonic drift in the $x$ direction is applied. Due to different starting positions, the trajectories in (c1)–(c4) are straight and point in the $+x$ direction, while the ones in (d1)–(d4) have the zigzag shape and point in the $-x$ direction. First, the colloids are in a local minimum and hardly move [see, e.g., (c1) and (c2) for the straight path]. However, with increasing phasonic displacement the minimum disappears and the colloids slide into a newly appearing minimum [see (c2), (c3), (d1), and (d2)].

Figure 2

Area of (a) reduced positions ${\mathbf{r}}^{(\mathrm{red})}$ and (b) reduced phasonic displacements ${\mathbf{w}}^{(\mathrm{red})}$. The borders of the characteristic area to which all colloids can be mapped are the black decagons marked with an $a$. The reduced displacements of all potential minima, ${\mathbf{r}}^{(\mathrm{red})}$ and ${\mathbf{w}}^{(\mathrm{red})}$, lie in the region which is bordered by the solid colored lines labeled $b$. A minimum disappears when it reaches the lines $b$ from inside the region. A colloid sitting in such a minimum slides from the lines $b$ to the colored segments $c$ outside the decagon. It is then mapped back into the characteristic decagonal area in a region bordered by lines marked with $d$. With the help of both diagrams, the trajectory of a particle in response to a phason displacement in the quasicrystalline potential can be predicted as described in the text and demonstrated in Fig. 3 for specific examples. All colored lines in the figure are determined by numerically analyzing the potential for all possible directions of the reduced phasonic displacement.

Figure 3

Selected trajectories and how they can be predicted by employing the diagrams of Fig. 2 for particles in a potential with a phasonic drift in the $x$ direction [(a1)–(b3)] and in the $y$ direction [(c1)–(d3)]. Each column describes a trajectory for a starting condition where in the beginning ${\mathbf{w}}^{(\mathrm{red})}$ is within the crosshatched area. The first line [(a1), (b1), (c1), and (d1)] shows the path of the colloid, the second line [(a2), (b2), (c2), and (d2)], the corresponding reduced positions ${\mathbf{r}}^{(\mathrm{red})}$, and the last line [(a3), (b3), (c3), and (d3)], the reduced phasonic displacements ${\mathbf{w}}^{(\mathrm{red})}$. First, ${\mathbf{r}}^{(\mathrm{red})}$ changes only slightly while the phasonic displacement (either $w_x$ or $w_y$) is increased (see arrows labeled $A$, $A^{\prime }$, $A1$, or $A2$). The local minimum that is occupied by the colloid disappears when ${\mathbf{w}}^{(r\mathrm{red})}$ reaches the colored solid border. As a consequence, the colloid slides into another local minimum depicted by the same color (see labels $B$, $B^{\prime }$, $B1$, or $B2$). Since the new ${\mathbf{r}}^{(\mathrm{red})}$ is outside the characteristic decagonal area, in the next step (see $C$, $C^{\prime }$, $C1$, or $C2$) ${\mathbf{r}}^{(\mathrm{red})}$ and ${\mathbf{w}}^{(\mathrm{red})}$ are mapped back such that ${\mathbf{r}}^{(\mathrm{red})}$ is close to the origin. Note that the real position $\mathbf{r}$ does not change due to this mapping. Afterwards, all steps are repeated for the new reduced quantities. In the case of the zigzag paths, ${\mathbf{r}}^{(\mathrm{red})}$ and ${\mathbf{w}}^{(\mathrm{red})}$ end up at their starting values after two slides (six steps: $A1$, $B1$, $C1$, $A2$, $B2$, $C2$) and in case of the straight path [(a1)–(a3)], after one slide (steps $A$, $B$, $C$). For some starting positions the very first steps might differ from the following ones; e.g., the steps $A^{\prime }$, $B^{\prime }$, and $C^{\prime }$ shown in (a3) do not repeat but are followed by the usual steps of a straight path. We present paths (b1)–(b3) and (c1)–(c3) in movies in the Supplemental Material [19].