Physical Review Letters 106, 060603 (2011)

Dynamical Ordering and Directional Locking For Particles Moving Over Quasicrystalline Substrates

C. Reichhardt and C. J. Olson Reichhardt

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 19 November 2010; published 10 February 2011)

We use molecular dynamics simulations to study the driven phases of particles such as vortices or colloids moving over a decagonal quasiperiodic substrate. In the regime where the pinned states have quasicrystalline ordering, the driven phases can order into moving square or smectic states, or into states with aligned rows of both square and triangular tiling which we term dynamically induced Archimedean-like tiling. We show that when the angle of the drive is varied with respect to the substrate, directional locking effects occur where the particle motion locks to certain angles. It is at these locking angles that the dynamically induced Archimedean tiling appears. We also demonstrate that the different dynamical orderings and locking phases show pronounced changes as a function of filling fraction.

DOI: 10.1103/PhysRevLett.106.060603
PACS numbers: 05.60.Cd, 05.45.-a, 74.25.Wx, 82.70.Dd



Quasicrystals have received intense study since their discovery due to their unusual property of combining nonperiodicity with long-range order [1]. Recently there has been growing interest in understanding how interacting particles such as vortices in type-II superconductors [2,3,4] or charged colloidal particles [5,6,7,8,9] order in the presence of a quasicrystalline substrate created with nanolithographic or optical techniques. On a fully periodic substrate, these types of particles form states that are commensurate with the periodicity of the substrate at fillings which meet integer or fractional matching conditions [10,11,12,13]. In the case of vortices, the commensurate fillings are associated with peaks in the critical current or the external force required to depin the vortices from the substrate [10,11]. When vortices interact with quasicrystalline Penrose or decagonal pinning site arrays, a new set of peaks in the critical current appear at nonrational fields in addition to the commensurate fields due to a novel type of ordering of the vortices on these substrates [2,3]. For colloids interacting with decagonal substrates, experiments show that strong substrates produce quasicrystalline order of the colloids but weak substrates permit the triangular lattice favored by the colloid-colloid interactions to form. Remarkably, for intermediate substrate strengths a new type of ordering arises consisting of a combination of triangular and square order. This has been termed an Archimedean-like tiling [5,8].
Numerous experimental and simulation studies have been performed on the dynamic phases of a collection of particles driven over a periodic substrate. One of the most striking observations is directional locking in which the particles cease to follow the direction of the drive and instead lock to a symmetry direction of the substrate. The locking effect produces a series of pronounced steps in the velocity versus drive angle curve [14,15]. Directional locking effects were experimentally observed for colloids moving through a periodic optical trap array [16,17] and for vortices driven at differing angles with respect to a pinning substrate [18]. For colloidal systems, directional locking can be used for practical applications such as the fractionation of different species of particles, only one of which locks to the symmetry direction of the substrate [17,19]. An open question is whether directional locking can also occur for particles moving over quasiperiodic substrates. It has already been shown that the nonequilibrium, driven state for particles driven over random substrates can show dynamically induced ordered states which are inaccessible to the static system. For example, on strong random substrates which produce a disordered pinned state, the driven state can partially order to a moving smectic or to a triangular lattice [20,21].
Here we show that directional locking and dynamical ordering can occur for particles moving over quasicrystalline substrates. The dynamical ordering occurs only for certain driving directions, and changes the system from a quasicrystalline pinned state to a moving state with novel dynamical ordering that is similar to a square and triangular tiling. We call this ordering a dynamically induced Archimedean tiling. Other types of ordering to moving smectic, moving square, and moving liquid states are also possible as a function of filling and drive direction. We specifically demonstrate the locking and ordering for vortices in type-II superconductors, and note that the same results appear for charged colloids moving over decagonal substrates [22].
Fig1.png
Figure 1: (a) The locations of the Penrose tiled pinning sites (open circles) for a portion of the sample. Each Penrose tile is shaded. (b) The location of the vortices (filled circles) and pinning sites (open circles) at B/Bϕ=1.61 in the same sample at zero applied drive. Inset: The structure factor S(k) of the vortex positions from (b) has tenfold peaks indicative of quasicrystalline order.
We model a two-dimensional system of Nv interacting particles in the presence of a pinning array with fivefold Penrose ordering, as shown in Fig. 1(a). In the superconducting system we consider a sample of size 24λ× 24λ with periodic boundary conditions, where λ is the London penetration depth. The vortex motion evolves according to the following overdamped equation: η[(dRi)/(dt)] = Fvvi + Fpi + Fexti . Here Ri is the location of vortex i and η is the damping coefficient. The repulsive vortex-vortex interaction force is Fivv = ∑Nvj ≠ if0K1(Rij/λ)∧Rij, where K1 is the modified Bessel function, f0=ϕ02/(2πμ0λ3), ϕ0=h/2e is the elementary flux quantum, Rij=|Ri−Rj|, and ∧R=(Ri−Rj)/Rij. The substrate force from Np parabolic traps of radius rp=0.35λ and maximum strength Fp=1.85 has the form Fpi=∑k=0Npf0 (Fp/rp) Θ(1−Rik/Rp)∧Rik, where Θ is the Heaviside step function. At the matching field of Bϕ, Nv/Np=1. Fexti is the driving force from an external current applied uniformly to all particles. We initialize the particle positions using simulated annealing, then slowly turn on an external drive Fext = FD(cosθ∧x+sinθ∧y) with θ reported in degrees. We measure the velocity in the y-direction Vy=∑i=0Nvvi ·∧y as we increment the drive angle θ. All forces and lengths are measured in units of f0 and λ.
In Fig. 1(a) we plot the pinning site locations showing the fivefold Penrose tiling and in Fig. 1(b) we show the positions of the non-driven pinned particle state at B/Bϕ = 1.61. The structure factor S(k) of the particle positions in this sample plotted in the inset of Fig. 1(b) has a tenfold peak structure, indicative of quasicrystalline order such as that found in the strong substrate limit for colloidal systems [5]. At this Fp, pinned states at different filling fractions generally have quasicrystalline ordering.
Fig2.png
Figure 2: (a) The average velocity in the y-direction 〈Vy〉 vs drive angle θ for the system in Fig. 1(a) with B/Bϕ=3.225 at FD=2.0. A series of pronounced steps appear at multiples of 36°, marked 0/1, 1/1, and 2/1, when the particle motion locks to certain symmetry directions of the substrate. Locking at fractional multiples of 36° produces many smaller steps. Inset: A blowup of the main panel near the 1/1 step showing some of the fractional steps. (b) The corresponding P6 vs θ showing that the system is more ordered on the locking steps.
Figure 2(a) shows 〈Vy〉 versus the drive angle θ for a sample with B/Bϕ=3.225 and FD = 2.0. At this drive, all the particles are in motion. The clear set of steps in 〈Vy〉 is a signature of the directional locking that occurs when the particles move along a single direction over a range of drive angles [14,15,16,17,19]. For the same set of parameters but with randomly placed pinning, no steps occur in 〈Vy〉 [22]. The locking angles are related to the tenfold orientational ordering, and occur at multiples of θ = 360°/10 = 36°, as highlighted in Fig. 2(a) for θ = 0° (the 0/1 step), 36° (the 1/1 step), and 72° (the 2/1 step). There are numerous smaller locking steps associated with rational fractions of 36°, including 1/4, 1/2, 2/3, 3/4, 5/4, and 5/2. Some of these smaller steps are visible in the inset of Fig. 2(a). This result proves that it is possible for directional locking to occur on quasiperiodic substrates, indicating that orientational order rather than translational order of the substrate is the essential ingredient for directional locking. This opens the possibility of creating novel fractionation devices using quasiperiodic substrates, and may also be relevant to frictional studies performed with quasicrystalline substrates. To show the presence of a higher amount of order in the particle lattice on the locking steps, in Fig. 2(b) we plot the fraction of sixfold coordinated particles P6=∑iNvδ(zi−6) as a function of θ, where the particle coordination number zi is obtained from a Voronoi construction that is not sensitive to square orderings. In the pinned state P6 = 0.525, while for the driven systems P6 reaches its highest values on the locking steps and is lower in the nonlocking ranges of θ.
Fig3.png
Figure 3: Delaunay triangulations of the particle positions in the sliding state. Bonds longer than 1.1a0 are omitted, where a0 is the lattice spacing of a triangular lattice with the same density. Yellow (light) polygons are triangular and blue (dark) polygons are square or nontriangular. For a sample with B/Bϕ=3.225 and FD=2.0 we show (a) the 0/1 step at θ = 0 exhibiting the oriented triangles and squares of the dynamically induced Archimedean tiling, (b) the 1/1 step at θ = 32° where the Archimedean ordering aligns with the direction of motion, and (c) θ = 41° in the unlocked region just above the 1/1 step. (d) The triangulation for the 1/1 step at θ = 32° for a sample with B/Bϕ=1.61 and FD=2.0, where most of the tiles are square. Smectic-like ordering appears in S(k) corresponding to the Archimedean tilings of (e) panel (a) and (f) panel (b). (g) S(k) for the unlocked state shown in panel (c) has a ring structure indicating liquid ordering. (h) At the lower field of B/Bϕ=1.61, S(k) corresponding to panel (d) has square ordering.
We tessellate the positions of the particles in the moving state as in Refs. [5,8] in order to reveal the nature of the dynamical ordering. In Fig. 3(a) we show the tessellation of the moving state on the 0/1 step from Fig. 2(a), where the flow is locked in the x-direction. Both square and triangular tiles are present, and the square tiles are aligned with the flow along the x direction. This state is very similar to the Archimedean type tiling found for colloids pinned by substrates of intermediate strength [5,8]. The smectic features of the corresponding structure factor S(k) in Fig. 3(e) indicate that unlike the static colloidal tiling, the dynamical tiling does not have a one-dimensional (1D) quasiperiodic structure, which would appear as a series of peaks in S(k) [8]. More recent colloid experiments reveal the development of smeared smecticlike peaks in S(k) similar to those in Fig. 3(e) when the strength of the laser-induced substrate is increased [9]. Moving smectic states also appear in systems with random pinning, but these states have a purely triangular tiling containing some dislocations [20,21].
We find that oriented Archimedean tilings appear on each locking step, as illustrated for the 1/1 step in Fig. 3(b) and (f). The particles move along 1D channels on the locked steps [22], but follow winding and rapidly changing trajectories when not on the steps. In Fig. 3(c) we show the tessellation for driving just above the 1/1 step in a disordered flow regime. The orientational ordering is lost and the corresponding S(k) in Fig. 3(g) has a ring structure characteristic of a moving liquid [20]. We find that the relative number of square and rectangular tiles appearing on the locking steps varies as a function of filling. For example, at B/Bϕ = 1.61 on the 1/1 locking step, Fig. 3(d) shows that the tiling is predominantly square, and S(k) in Fig. 3(h) has square ordering.
Fig4.png
Figure 4: (a) The depinning threshold FTrc indicating the width of the 0/1 locking step vs B/Bϕ. (b) P6 vs B/Bϕ. The broad peak in both quantities for 1.7 < B/Bϕ < 4.3 appears when the system forms the dynamical Archimedean tiling illustrated in Fig. 3(a), while the dip at B/Bϕ = 1.6 corresponds to the moving square structure shown in Fig. 3(h). Additional peaks in P6 occur at B/Bϕ=1.0 and B/Bϕ=1/τ, where τ is the golden mean. Dashed lines and the labels a-d indicate the fields shown in Fig. 3(a-d).
To understand where the different moving phases occur as well as the changes in the effectiveness of the directional locking, in Fig. 4(a) we plot the transverse depinning force FTrc determined by FTrc=FDsinθe, where θe is the end of the 0/1 locking step, versus B/Bϕ and in Fig. 4(b) we show the corresponding P6 versus B/Bϕ. There is a broad maximum in FTrc associated with the Archimedean type ordering for 1.7 < B/Bϕ < 4.3 and another sharper peak in FTrc at B/Bϕ = 0.62. Both of these features are accompanied by peaks in P6. Our transverse depinning measurements, in which the vortices are depinning from a moving state, differ in several ways from the longitudinal depinning from a static state previously measured in simulations and experiments [2,3,23]. For example, FTrc shows no peaks at B/Bϕ = 0.81 or 1.0, although we do find a weak peak in P6 at B/Bϕ = 1.0. In agreement with the experiment of Ref. [3], we observe a prominent peak in FTrc at B/Bϕ=0.62=1/τ where τ = (1+√5)/2 is the golden mean. The lowest value of FTrc occurs at B/Bϕ = 1.61 ≈ τ. At this field, the widths of the locking steps are suppressed and the system has the square ordering shown in Figs. 3(d) and 3(h). For B/Bϕ > 4.0, the ordering of the locked phases is reduced and S(k) develops a ring structure on the locking steps, while the vortex-vortex interactions are strong enough in the nonlocking regimes to produce sixfold ordering. For 0.5 < B/Bϕ < 1.5 the system generally exhibits a smectic ordering containing few square tiles corresponding to a highly defected triangular lattice similar to that found for driving over random substrates.
We find that the ordered moving phases only occur when FD is sufficiently large. For FD < 1.25 with Fp=1.85, significant plastic flow occurs, S(k) is always liquidlike, and the step structures in the velocity curves disappear. For FD >> Fp the width of the steps slowly decreases with increasing FD. We also find the same results reported here for driven colloids interacting with a screened Coulomb interaction and a quasiperiodic substrate [22]. Sevenfold or tetradecagonal quasiperiodic substrates also produce directional locking, but the locking steps are significantly reduced in width [22]. The addition of a finite temperature does not destroy the ordered moving phases and may produce new phases, as found for a periodic substrate [24]; the effect of temperature will be described elsewhere.
We have shown that interacting particles such as vortices or colloids driven over a quasiperiodic substrate exhibit rich dynamical behaviors including pronounced directional locking where the particles prefer to move at particular angles with respect to the substrate. Directional locking has already been studied in periodic systems, but this study is the first demonstration of directional locking on quasiperiodic substrates. For strong substrates, the pinned state has quasicrystalline ordering, but the moving state can have square or smectic ordering depending on the orientation of the drive. At certain filling fractions the system forms a moving Archimedean tiling ordered state similar to the pinned Archimedean tiling ordering observed for colloids on decagonal substrates, but with smectic rather than quasiperiodic character. The dynamically ordered states produce distinct signatures in the transverse depinning threshold.
This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

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