SUPPLEMENTARY MATERIAL
for Dynamical Ordering and Directional Locking for Particles Moving Over Quasicrystalline Substrates



  Directional Locking for Vortices on Tetradecagonal Quasicrystalline Substrates

Here, we show that the directional locking effects described in the main text for driven vortices moving over decagonal quasicrystalline substrates also occur when the vortices are driven over tetradecagonal or sevenfold-symmetric quasicrystalline substrates such as that shown in Suppl. Fig. 1. We use the same pinning parameters as for the decagonal substrate with Fp = 1.85 and rp = 0.35λ, and apply a drive FD=2.0 at B/Bϕ = 3.9. In Suppl. Fig. 2(a) we plot 〈Vy〉 versus the drive angle θ and in Suppl. Fig. 2(b) we show the fraction of sixfold coordinated particles P6 versus θ. There is a clear set of steps in 〈Vy〉 which are accompanied by enhanced sixfold ordering as shown by the increases in P6. For the tetradecagonal substrate potential, the major locking steps occur at drive angles that are integer multiples of 360°/14, producing the 1/1 step at 25.7°, the 2/1 step at 51.42°, and the 3/1 step at 77°. For the decagonal substrate the locking steps fall at integer multiples of 360°/10, as described in the main text. Fractional locking steps are nearly absent for the tetradecagonal substrate; however, for lower fillings it is possible to resolve some fractional steps. This is shown in Suppl. Fig. 3(a,b) for B/Bϕ = 2.96. The fractional steps m/4 with m integer are the most pronounced and are associated with enhanced sixfold ordering in P6. This result indicates that directional locking is a generic feature for particles moving over different types of quasicrystalline substrates. The width of the locking steps is generally much narrower for the tetradecagonal substrate than for the decagonal substrate. Additionally, although the decagonal substrate produced pronounced five-fold ordering of the particles for some fillings, we do not find strong seven-fold particle ordering on the tetradecagonal substrate. Instead, on the locking steps, we observe smectic ordering composed of sixfold-coordinated particles and a limited number of dislocations.
EPAPS_Fig1.png
Figure 1: The locations of the pinning sites (open circles) for a portion of the sample with seven-fold or tetradecagonal ordering. Dashed lines indicate the positions of the tiles used to define the pinning locations.
EPAPS_Fig2.png
Figure 2: (a) The average velocity in the y-direction 〈Vy〉 vs drive angle θ for the system in Suppl. Fig. 1 with a sevenfold quasicrystalline pinning array for Fp = 1.85, rp = 0.35λ, B/Bϕ = 3.9, and FD=0.2. Several steps appear at the directional locking angles which are integer multiples of 360°/14. The 1/1, 2/1, 3/1, and 4/1 lockings are clearly visible. (b) The corresponding P6 vs θ shows that on the locking steps the system develops a considerable amount of sixfold ordering.
EPAPS_Fig3.png
Figure 3: (a) 〈Vy〉 vs drive angle θ for the system in Suppl. Fig. 2 but for a filling of B/Bϕ = 2.96 where additional fractional locking steps are visible, particularly at fractions of m/4 where m is an integer. Some representative steps are marked. (b) The corresponding P6 vs θ.

  Directional Locking for Colloids on Decagonal and Tetradecagonal Substrates

We next show that the same directional locking effects described in the main text for vortices driven over quasicrystalline substrates also occurs for colloidal particles driven over quasicrystalline substrates. We model a two-dimensional system of Nc interacting colloids in the presence of fivefold and sevenfold quasicrystalline substrates. The substrate is composed of localized pinning traps with maximum strength Fp and radius rp=0.35. We simulate the motion of the colloids using the same procedure used previously to model colloid dynamics on random substrates [1] and periodic substrates [2], by integrating the following equation of motion: ηdRi/dt = Fcci + Fsi + FD. Here Ri is the location of colloid i and η is the damping coefficient. The colloid-colloid interaction potential has the Yukawa form V(Rij) = (E0/Rij)exp(−κRij), where Rij = |Ri − Rj|, E0 = Z*2/(4πϵϵ0), ϵ is the solvent dielectric constant, Z* is the effective charge of each colloid, and 1/κ is the screening length. The colloid-colloid interactions are repulsive and are given by Fcci = −∑Ncj ≠ i∇V(Rij). Lengths are measured in units of a0 and forces in units of F0 = E0/a0. The substrate force term Fis comes from Np pinning sites placed in a decagonal or tetradecagonal pattern and has the same form as described for the vortex system. The colloid density relative to the pinning density is Nc/Np. We neglect hydrodynamic interactions in the colloidal system and assume that the colloids are in the low volume fraction, highly charged, electrophoretically driven limit. The external force FD is the same as that used for the vortex case and we measure 〈Vy〉 and P6 as a function of the drive angle θ.
EPAPS_Fig4.png
Figure 4: The directional locking for charged colloidal particles driven over a decagonal substrate in the same manner as the vortices in the main text. Here Nc/Np = 2.9 and Fp = 0.75. In general, the same features observed for the vortex system also appear for the colloidal system. (a) 〈Vy〉 vs θ with the main locking steps at integer multiples of 360°/10 highlighted at 1/1 and 2/1. Several fractional steps also appear at rational fractions of the integer steps such as at 1/4, 3/4, and 3/2. (b) The corresponding P6 vs θ shows that along the steps the system has higher ordering. On the main integer matching steps, the system forms the dynamically induced Archimedean ordering also observed in the vortex system (as described in the main text).
In Fig. 4(a) we plot 〈Vy〉 versus θ for a colloidal system on a decagonal substrate with Fp = 0.75 at Nc/Np = 2.9, and in Fig. 4(b) we show the corresponding P6 versus θ. Here, all the general features of the directional locking in the vortex system also appear for the colloidal system, including the dominant locking steps at integer multiples of 360°/10 such as 1/1 and 2/1 and additional fractional lockings at 1/2, 3/2, 5/2, 1/4, and 3/4. Each locking step is associated with enhanced ordering of the colloidal lattice when the system forms a dynamically ordered Archimedean tiling state.
EPAPS_Fig5.png
Figure 5: Directional locking for charged colloidal particles driven over a sevenfold or tetradecagonal substrate for the same parameters in Suppl. Fig. 4. (a) 〈Vy〉 vs θ and (b) P6 vs θ. The same features found for the vortex system in Suppl. Fig. 3 appear for the colloid system, including locking steps falling at integer multiples of 360°/14. We also observe the strongest fractional locking steps at fractions m/4 with m integer.
In Fig. 5(a,b) we show 〈Vy〉 and P6 versus θ for colloids moving over a tetradecagonal substrate. We find the same features illustrated in Suppl. Fig. 3 for vortices moving over a tetradecagonal substrate. The dominant locking steps occur at integer multiples of 360°/14. In general, the locking steps are weaker than for the decagonal substrate and the dominant fractional steps occur at ratios of m/4 with m integer. The fractional steps are also associated with a series of peaks in P6, where additional fractions beyond m/4 appear as smaller peaks that do not show full ordering. These results indicate that the directional locking effects on quasicrystalline periodic substrates can be observed for various types of interacting particles including colloids and vortices, and that the directional locking is a robust feature in these systems.

References

[1]
C. Reichhardt and C.J. Olson, Phys. Rev. Lett. 89, 078301 (2002); J. Chen, Y. Cao, and Z. Jiao, Phys. Rev. E 69, 041403 (2004); C. Reichhardt and C.J. Olson Reichhardt, Phys. Rev. Lett. 103, 168301 (2009).
[2]
A. Libál, C. Reichhardt, B. Jankó, and C.J. Olson Reichhardt, Phys. Rev. Lett. 96, 188301 (2006); C. Reichhardt and C.J. Olson Reichhardt, Phys. Rev. E 79, 061403 (2009).


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