Physical Review E 87, 022308 (2013)

Colloidal Lattice Shearing and Rupturing with a Driven Line of Particles

A. Libál1,2, B.M. Csíki2, C. J. Olson Reichhardt1, and C. Reichhardt1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2Department of Mathematics and Computer Science, Babes-Bolyai University, RO-400591 Cluj-Napoca, Romania
3Department of Physics, Babes-Bolyai University, RO-400591 Cluj-Napoca, Romania

(Received 8 October 2012; published 25 February 2013)

We examine the dynamics of two-dimensional colloidal systems using numerical simulations of a system with a drive applied to a thin region in the middle of the sample to produce a local shear. For a monodisperse colloidal assembly, we find a well defined decoupling transition separating a regime of elastic motion from a plastic phase where the driven particles break away or decouple from the bulk particles and produce a shear band. For a bidisperse assembly, the onset of a bulk disordering transition coincides with the broadening of the shear band. We identify several distinct dynamical regimes that are correlated with features in the velocity-force curves. As a function of bidispersity, the decoupling force shows a nonmonotonic behavior associated with features in the noise fluctuations, power spectra, and bulk velocity profiles. When pinning is added in the bulk, we find that the shear band regions can become more localized, causing a decoupling of the driven particles from the bulk particles. For a system with thermal noise and no pinning, the shear band region becomes more extended and the average velocity of the driven particles drops at the thermal disordering transition of the bulk system.
I. INTRODUCTION
II. SIMULATION AND SYSTEM
III. MONODISPERSE SYSTEM
IV. BIDISPERSE SYSTEMS
A. Dynamic phases
V. QUENCHED DISORDER AND SHEAR LOCALIZATION
VI. THERMAL EFFECTS
VII. SUMMARY
References


I.  INTRODUCTION

As the ability to manipulate small particles or groups of small particles has advanced, there have been a growing number of studies examining the effect of local perturbations such as driving a single probe particle with an external force through a particle assembly in colloidal [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and granular [19,20,21,22,23] media. The fluctuations and transport characteristics of the probe particle can undergo pronounced changes depending on the ordering of the surrounding media, the particle-particle interactions, the temperature, and the magnitude of the external force. In systems with short range interactions such as disks or granular particles [19,23,20,21,22], the probe particle velocity can drop under even a small change in the bulk media density as the jamming transition is approached. Below the jamming density, the threshold force needed to move the probe particle is small or absent and the probe particle velocity distribution is bimodal [23]. Close to jamming, the threshold force becomes finite and rapidly increases, while the velocity fluctuations are intermittent and have a power law distribution that has been interpreted as arising due to criticality associated with the jamming transition [22,19,23,20,2].
For systems with longer range interactions such as charged colloidal particles, there is an elastic or coupled flow regime at low drives where the probe particle drags the entire assembly of particles without any local tearing [24]. At higher drives there can be a well defined transition to a plastic response regime where tearing rearrangements occur in the surrounding media and the probe and bulk particles move at different velocities. Velocity-force curves can be used to identify the critical driving force separating the elastic and plastic regimes. The velocity increases as a power law with increasing external force in the plastic regime, similar to the behavior observed for plastic depinning of colloidal particles [25] in the presence of quenched disorder. The case of driving a single particle through a surrounding medium containing quenched disorder has been studied by dragging individual vortices in type-II superconductors [10,26,27,28,29]. Here, it is possible for the bulk pinning to counterintuitively lower the effective drag on the probe particle [10]. Shear thinning [1] and directional locking effects can occur for probe particles driven through crystalline structures in the absence of quenched disorder. In this case, the probe particle velocity depends on the orientation of the drive with respect to the symmetry directions of the crystalline structure [30,8].
In studies of single driven probe particles, interpretation of the induced shear response of the bulk is complicated by the fact that the motion of the bulk particles differs in front and in back of the probe particle. Additionally, in order to obtain clean measurements in experiments and simulations, numerous realizations must be performed. Here, we generalize the driven probe to a localized quasi-one-dimensional region of particles driven externally along a line. The drive is parallel to the orientation of the line, which coincides with a symmetry direction of the background lattice. With the driven line geometry we find that a much smaller number of runs is required to produce clean results. This geometry could be created using a laser beam directed along the sample edge, with magnetic particles driven by a magnetic strip, or with particles coupled to a mechanical external drive or driven with one-dimensional arrays of optical traps [31,32,33]. Driven line geometries have already been realized for particles with Yukawa interactions in dusty plasmas, where a laser beam pushes particles only along a line [34,35,36,37,38]. It is also possible to create systems with bulk pinning and an easy flow channel in confined geometries [39,40,41,42]. A key difference between a point probe and the line probe considered here is that the line probe more directly mimics a shear response. When the system is strongly coupled, the response is elastic and the bulk and probe particles move at the same velocity. Here we consider a line of particles driven through monodisperse ordered, bidisperse ordered, and bidisperse disordered systems, both for pin-free samples and for samples that have pinning present in the bulk. The driven line geometry makes analyzing and understanding shear banding effects and coupling-decoupling transitions as a function of drive easier than for a single driven probe particle. Additionally, the driven line may be easier to implement in certain experiments than the driven single probe particle.
Fig1.png
Figure 1: Illustration of the sample geometry for the monodisperse case. Black dots: Colloidal particles interacting via a pairwise Yukawa potential to form a triangular lattice. Arrows indicate the region within which an external drive in the positive x direction is applied to the particles.

II.  SIMULATION AND SYSTEM

In Fig. 1 we illustrate the geometry of the sample, which consists of a two-dimensional assembly of Nc colloidal particles with periodic boundary conditions in the x− and y-directions. The arrows in the middle of Fig. 1 indicate the region of width d within which the particles experience an external drive FD=Fd∧x. Particles outside this region are undriven and have FD=0. The particle density is ρ = Nc/L2, where L is the size of the simulation cell. The dynamics of particle i are obtained from overdamped dynamics as previously used to model driven colloidal systems [4,8,12,24,43]. The equation of motion for particle i is
η d Ri
dt
 = − Ni
∑
i ≠ j 
 ∇V(Rij)
^
R
 
ij 
 + FPi + FDi + FTi.
(1)
Here η is the damping constant, Ri(j) is the position of particle i(j), Rij = |Ri − Rj|, and ∧Rij=(Ri−Rj)/Rij. The particle-particle interaction potential is of Yukawa or screened Coulomb form, V(Rij) = q2E0exp(−κRij)/Rij. Here E0 = Z*2/4πϵϵ0a0, where q is the dimensionless interaction strength, Z* is the effective charge of the colloidal particle, ϵ is the solvent dielectric constant, and 1/κ is the screening length. Lengths are measured in units of a0, time in units of τ = η/E0, and forces in units of F0 = E0/a0. We increase the external drive in increments of δFd, then hold the drive at a constant value for a fixed period of time and measure the average velocity of the particles within the driven channel 〈Vc〉, normalized by the number of particles in the driven channel, and of the particles in the bulk 〈Vb〉, normalized by the number of particles outside the driven channel. The pinning force arises from Np non-overlapping parabolic pinning sites that are placed outside the driven channel. The pinning force has the form FPi = Fp(Rik/Rp)Θ(Rp− Rik)∧Rik, where Rik = |Ri − Rk| is the distance between particle i and the center of pinning site k, and ∧Rik = (Ri − Rk)/Rik. Here Rp is the pinning site radius, Fp is the maximum force of the pinning site, and Θ is the Heaviside step function. The effects of thermal fluctuations come from the Langevin noise term FT with the properties 〈FTi(t)〉 = 0 and 〈FTi(t)FjT(t′)〉 = 2ηkBTδijδ(t − t′), where kB is the Boltzmann constant. The initial particle configurations are obtained by placing the particles in a triangular arrangement, and unless otherwise noted, the average lattice constant is a=2 and the width of the driven channel d = 2.
Fig2.png
Figure 2: (Color online) The velocities of the particles in (a) the driven channel 〈Vc〉 and (b) the bulk outside the driven channel 〈Vb〉 vs Fd for a monodisperse system with a=2 and varied interaction strength q. From left to right, q=0.1 (black), 0.3 (red), 0.5 (light green), 0.7 (blue), 0.8 (orange), 1.0 (brown), 1.2 (maroon), 1.4 (purple), and 1.6 (dark green). The q=1.6 sample remains in the coupled regime with 〈Vc〉 = 〈Vb〉 over the entire range of Fd. The inset in (a) shows that the driving force Fc at which the decoupling transition occurs increases with increasing q.

III.  MONODISPERSE SYSTEM

In Fig. 2(a,b) we plot the average velocity of the particles inside (〈Vc〉) and outside (〈Vb〉) the driven channel region versus external drive Fd for a system with a monodisperse triangular crystalline arrangement of particles and different values of the interaction coefficient q. For low Fd, all the curves for both 〈Vc〉 and 〈Vb〉 increase linearly and 〈Vc〉 = 〈Vb〉, corresponding to the elastic flow regime where the bulk and driven particles are locked together. Here there is no tearing or rearrangements, so all the particles keep their same neighbors and move together. At higher drives, a transition occurs to a regime where 〈Vb〉 decreases and 〈Vc〉 increases with increasing Fd when the driven particles partially decouple from the bulk particles and are able to slip past them. Due to the coupling to the driven particles, the bulk particles continue to move as a unit; however, as Fd increases, this coupling weakens and 〈Vb〉 decreases. The inset in Fig. 2(a) shows the force Fc at which the decoupling occurs as a function of q. As the strength of the particle-particle interactions increases, a larger force must be applied before decoupling can occur.
The velocity-force curves in Fig. 2 are very similar to those found in other systems that exhibit decoupling and drag effects. For example, in the Giaever transformer geometry for coupled superconducting layers in the presence of a magnetic field [44,45], when only the vortices in the top layer are driven they can drag the vortices in the bottom layer, producing a well defined coupling-decoupling transition. In the case of repulsively interacting particles in coupled one-dimensional wires, such as classical electrons [46] and particles with Yukawa interactions [47,48], a commensurate state can form when the number of particles in each wire is the same, producing a well defined coupling-decoupling transition when only one wire is driven. In the case of our driven line of particles, the system can be viewed as a single driven one-dimensional layer interacting with an array of non-driven layers. At the decoupling transition, the particles outside of the driven region remain locked together so that a very small shear band occurs only at the driven line. For more disordered systems, a much broader shear band forms.
Fig3.png
Figure 3: (Color online) (a) 〈Vc〉 and (b) 〈Vb〉 vs Fd for a monodisperse system with q=1.0 for varied particle density, measured in terms of the particle lattice constant a. From left to right, a = 3.0 (dark green), 2.5 (purple), 2.3 (maroon), 2.1 (brown), 2.0 (orange), 1.9 (blue), 1.7 (green), 1.6 (red), and 1.5 (black). Samples with larger a are less dense and decouple at a lower Fd. Inset of (a): Fc vs a.
In Fig. 3 we show 〈Vc〉 and 〈Vb〉 vs Fd for a monodisperse sample with fixed q = 1.0 but varied particle density, measured in terms of the lattice constant a. The inset of Fig. 3(a) shows the dependence of the decoupling force Fc on a. We observe a trend similar to that of changing q, where the samples with lower density have weaker particle-particle interactions and lower Fc. The decoupling transition should also depend on the commmensurablity of the particles in the driven channel with the surrounding media. The bulk particle lattice creates an effective periodic potential for the driven particles. If the particle density in the driven channel differs from the bulk density, incommensurations such as vacancies or interstitials appear in the effective periodic potential, depressing the decoupling force relative to the commensurate, density-matched case. The incommensurations depin below the bulk decoupling transition and shift the transition to lower Fd. Studies of one-dimensional coupled channels also found that the decoupling transition drops at incommensurate channel filling ratios [47].

IV.  BIDISPERSE SYSTEMS

Fig4.png
Figure 4: (Color online) (a,b,c) Particle positions (dots) and trajectories (lines) over a fixed time interval for bidisperse systems. Dark blue particles have charge q1 and light red particles have charge q2. (d,e,f) Corresponding Voronoi construction for instantaneous particle positions showing 6-fold (white), 5-fold (dark blue), and 7-fold (light red) coordinated particles. (a,d) At q1/q2 = 0.7 and Fd=0.05, the system is mostly ordered. (b,e) At q1/q2 = 1.6, the system is partially disordered in the bulk and there is a mixing of the trajectories as the shear band widens. (c,f) At q1/q2 = 2.4, the system is strongly disordered and the width of the shear band is increased.
We next consider a bidisperse system containing a 50:50 ratio of particles with q1 and q2, where we set q2=1.0. In Fig. 4(a,b,c) we plot the particle positions and trajectories during a fixed period of time, and in Fig. 4(d,e,f) we show the Voronoi tessellations of the particle positions at one instant. For q1/q2 = 0.7 and Fd = 0.05, Fig. 4(a,d) shows that the system is crystalline with sixfold ordering in the bulk. The topological defects in Fig. 4(a) are concentrated near the driven line and have their Burgers vectors aligned to permit slipping of the driven particles relative to the bulk particles. There is also no transverse diffusion in the system, as indicated by the nearly one-dimensional, non-mixing trajectories of the particles in Fig. 4(d). At q1/q2 = 1.6, the bulk is disordered and there is a proliferation of 5-7 paired defects uniformly throughout the system, as shown in Fig. 4(e). With the disappearance of structural order, the motion in the bulk changes, and Fig. 4(b) illustrates that the particles start mixing or exchanging neighbors within the bulk. The mixing is strongest near the driven line where the particles are moving most rapidly. Away from the driven line, mixing still occurs but at a slower rate and the trajectories are more ordered. Fig. 4(c,f) shows that for q1/q2 = 2.4, the system is even more disordered and contains an increased number of 5-7 defects. In addition, the region of strong mixing, denoted by the region with crossing trajectories, is wider in the y-direction. We find that for 0.6 < q1/q2 < 1.6, the bulk is ordered, while for q1/q2 < 0.6 the system disorders and the trajectories are similar to those shown in Fig. 4(b).
Fig5.png
Figure 5: (Color online) 〈Vc〉 vs Fd for bidisperse systems. (a) q1/q2 = 0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9, 1.0, and 1.1, from left to right. The system is disordered for q1/q2 ≤ 0.6. (b) q1/q2 = 1.3 (center right), 1.4 (upper right), and 1.8 (lower right), showing a crossing of the velocity-force curves. (c) The disordered states at q1/q2 = 2.2, 2.4, 2.6, 2.8 and 3.0, from top to bottom. Here 〈Vc〉 for fixed Fd decreases with increasing q1/q2.
The different phases can also be identified via velocity-force curve features. In Fig. 5(a) we plot 〈Vc〉 vs Fd for samples with q1/q2 = 0.1 to 1.1. As q1/q2 increases, the decoupling force Fc increases. The concavity of the velocity-force curves can be fit to 〈Vc〉 ∝ (F−Fc)β, where β > 1.0 for the disordered systems with q1/q2 ≤ 0.6 and β < 1.0 in the ordered systems. This is shown more clearly in Fig. 5(b) where we plot 〈Vc〉 vs Fd for systems with q1/q2 = 1.3, 1.4, and 1.8. For the disordered case of q1/q2=1.8, there are stronger fluctuations in 〈Vc〉. A crossing of the velocity-force curves occurs since Fc for the q1/q2=1.8 system is lower than for the q1/q2=1.3 system, but at higher drives 〈Vc〉 is lower in the q1/q2=1.8 sample. This occurs since the ordered state at q1/q2 = 1.3 produces a sharp shear band, while the disordered state at q1/q2 = 1.8 has a wide shear band region similar to that illustrated in Fig. 4(b), meaning that more particles are being dragged by the line particles, causing the driven particles to move more slowly.
Fig6.png
Figure 6: (Color online) (a) 〈Vb〉 vs Fd − Fc for bidisperse systems. (a) q1/q2 = 2.6, 2.8, and 3.0, from top right to bottom right. The dashed lines are power law fits (vertically offset for clarity) with exponents of β = 1.26, 1.29, and 1.32, respectively. (b) q1/q2 = 0.7, 0.9, and 1.0, from bottom right to top right, in the ordered regime, with dashed lines indicating power law fits (vertically offset for clarity) of β = 0.746, 0.84, and 0.8, respectively.
Figure 5(c) shows 〈Vc〉 versus Fd for 2.2 ≤ q1/q2 ≤ 3.0, where the system remains disordered with large fluctuations in 〈Vc〉. For a fixed drive, 〈Vc〉 drops with increasing q1/q2. Here the velocity-force curves can be fit to the form 〈Vc〉− Vdc ∝ (Fd − Fc)β with β ≈ 1.3, where Vdc is the velocity at Fc. This fit is illustrated in Fig. 6(a) for q1/q2 = 2.6, 2.8, and 3.0. A power law fit of this type is observed for a single particle moving through a disordered medium and creating a large amount of distortion, including particle rearrangements [24,2]. Similar scaling is found at depinning for assemblies of particles that undergo plastic flow upon depinning; such tearing behavior generates an exponent ranging from β = 1.25 to β = 2.0 [49,25]. In contrast, near an elastic depinning transition a power law fit produces a much smaller exponent β < 1.0 [50,25]. For samples in the ordered regime, we can fit 〈Vc〉− Vdc to a power law with β ≈ 0.8, as shown in Fig. 6(b) for q1/q2 = 0.7, 0.9, and 1.0. This indicates that the decoupling transition in the ordered phase resembles the elastic depinning of a one-dimensional coupled chain of particles from a periodic substrate created by the periodic ordering of the bulk particles. We find similar scalings for the other fillings we have considered, where for the ordered systems β < 1.0 and for the disordered systems β > 1.0.
Fig7.png
Figure 7: (Color online) (a,b) Velocity profiles ―Vx vs y for bidisperse systems. The driven region is centered at yd=23. (a) In a sample with q1/q2=1.0 at Fd=1.0, a sharp shear band forms. (b) Samples with q1/q2 = 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, and 3.0, from top center to bottom center. (c) ―Vx in only the top half of the sample vs y−yd for samples with q1/q2 = 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, and 3.0, from top to bottom. Dotted line is a power law fit for the q1/q2=2.2 system with an exponent of −1.4 in the region of strong mixing. At large y−yd, the bulk becomes locked again and moves as a solid.
The shear banding effect can be better seen by examining profiles of the average x velocity ―Vx taken at different points along the y-direction. Here, ―Vx(y) = ∑i=1Nc (· Ri ·∧x)[Θ(Ryi−y+δy/2)Θ(y+δy/2 −Ryi)] where Ryi is the y coordinate of particle i and δy is the width of the averaging region. Figure 7(a) shows ―Vx(y) for q1/q2 = 1.0 at Fd = 1.0. A sharp spike in ―Vx appears at y=yd, the center of the driven region, while ―Vx ≈ 0 in the bulk undriven portion of the sample. This type of profile is observed in the decoupled phase for all samples where the system remains ordered. In Fig. 7(b) we plot ―Vx(y) for disordered samples with q1/q2 = 1.8 to 3.0. In all cases, ―Vx is maximum at y=yd and falls off for larger |y−yd|. As q1/q2 increases, ―Vx at y=yd decreases while in the bulk ―Vx increases. This coincides with the decrease in 〈Vc〉 for increasing q1/q2 shown in Fig. 5(c). The increase of the response in the bulk occurs when a larger number of particles are dragged by the driven line due to the increased particle-particle interaction strength. In Fig. 7(c) we plot ―Vx for only the upper half of the sample versus y−yd on a log-log scale, to better show where the shear banding is occurring. The dashed line in Fig. 7(c) is a power law fit to the data from the q1/q2=2.2 sample in the shear band region, and has an exponent of −1.4. From the trajectory images in Fig. 4(c), it is clear that there is a region of strong mixing near the driven line and a region of less mixing at larger values of y−yd, correlated with a saturation of ―Vx in Fig. 7(c). As q1/q2 increases, the saturation region shifts out to larger values of y−yd. This results shows that the system has liquid-like behavior near the shear band region, but acts more like a disordered elastic solid away from the driven region.
Fig8.png
Figure 8: (Color online) Summary of the behavior of bidisperse systems. (a) Fc vs q1/q2. (b) 〈Vb〉 vs q1/q2 at Fd = 0.05. The dashed line indicates a fit to 1/(q1/q2) in the disordered regime. (c) The standard deviation of the fluctuations in the bulk velocity δVb vs q1/q2 for Fd = 0.05. (d) δVb vs q1/q2 for Fd = 0.1.
We summarize the behavior of Fc vs q1/q2 for the bidisperse samples in Fig. 8(a). In the disordered regime 0.1 < q1/q2 ≤ 0.6, Fc increases with increasing q1/q2. Above q1/q2=0.6, when the system enters the ordered regime, Fc increases more rapidly and the crystalline order that appears near q1/q2=1.0 enhances the coupling transition, as indicated by the peak in Fc near q1/q2 = 1.0. As the system becomes disordered away from q1/q2=1.0, weak or defected spots appear that lower the decoupling transition and reduce Fc. A minimum in Fc appears near q1/q2 = 1.8 in the disordered region, while Fc increases again for larger q1/q2 due to the increasing strength of the particle-particle interactions, which also cause the system to depin plastically. The inset of Fig. 2(a) shows that Fc in the monodisperse system also increases with increasing q.
In Fig. 8(b) we plot 〈Vb〉 vs q1/q2 in the bidisperse samples at Fd = 0.05. Here the bulk velocity decreases at a moderate rate with increasing q1/q2 until the system becomes disordered for q1/q2 ≥ 1.6 and large shear bands form, leading to a more rapid drop in 〈Vb〉 with increasing q1/q2. The dashed line is a fit to 1/(q1/q2) in the disordered regime. Just before the system disorders, for 1.4 ≤ q1/q2 < 1.6, there is a slight increase in 〈Vb〉 that is correlated with a drop in the standard deviation of the velocity fluctuations δVb, as shown in Fig. 8(c). For Fd = 0.05, Fig. 8(c) indicates that δVb reaches a maximum for q1/q2=1.1, the same value at which there is a peak in Fc in Fig. 8(a). At this point, when the velocity oscillations are the largest, the particles are moving in a one-dimensional ordered pattern and each particle exhibits an oscillating velocity component due to the effective periodic potential created by the ordered particles in the bulk. There is a dip in δVb at the transition from the ordered to the disordered region at q1/q2=1.6; for q1/q2 > 1.6, δVb decreases with increasing q1/q2. At the disordering transition, the particles no longer move together but have different sliding velocities, resulting in a cancellation of the large oscillations that appeared in the ordered phase. For larger q1/q2, deeper into the disordered regime, the velocity fluctuations decrease with increasing q1/q2 due to a decrease in the amount of motion in the system as the fixed value of Fd gets closer to Fc, which increases with increasing q1/q2. At Fd = 0.1, Fig. 8(d) shows that the same general trends in δVb persist; however, in this case the velocity fluctuations for larger q1/q2 increase with increasing q1 since the higher drive causes a large shear banding effect. There is a saturation of δVb near q1/q2 = 3.0, and we expect that the velocity fluctuations will decrease for higher q1/q2 as Fc increases and approaches the fixed value Fd=0.1.
Fig9.png
Figure 9: (Color online) (a,b) Power spectra S(ν) for bidisperse samples at Fd=0.05. (a) In ordered systems, q1/q2 = 0.6 (black), 0.8 (red), 1.0 (green), and 1.4 (blue), there is a narrow band noise signature. (b) In disordered systems, q1/q2 = 2.4 (black), 2.6 (red), 2.8 (green), and 3.0 (blue), a broad band noise signal occurs. The dashed line is a power law fit with an exponent of −1.5. (c) The noise power S0 vs q1/q2. At the transition into the disordered regime, S0 jumps to a much higher value.
The velocity fluctuations can also be characterized using the power spectrum S(ν) = |∫Vb(t)e−2πiνtdt|2. In the ordered regime, the noise fluctuations have a narrow band noise feature indicating that there is a characteristic frequency, similar to the response of particles moving over a periodic substrate [51]. In Fig. 9(a) we plot S(ν) in the ordered regime for samples with q1/q2 = 0.6 to 1.4 for a fixed drive of Fd = 0.05. The peaks in the spectra indicate the presence of narrow band noise. In the disordered regime, we find broad band noise or a 1/fα noise signature with α = −1.5, as shown in Fig. 9(b) for q1/q2=2.4 to 3.0. Similar broad band noise has been observed for dragging a single particle through granular media at the jamming transition [10] and for plastic depinning of particles on disordered substrates [52]. We can also analyze the noise power S0=∫ν1ν2S(ν) for a fixed frequency octave, as shown in Fig. 9(c) for ν1=10 and ν2=100. Here S0 is small in the ordered regime 0.6 < q1/q2 < 1.6 and undergoes a pronounced increase to a maximum near the onset of the disordered regime. For q1/q2 ≤ 0.6, the system is also disordered and S(ν) shows broad band noise; however, the noise power remains very low for these small values of q1/q2.
Fig10.png
Figure 10: (Color online) High drive behavior for a bidisperse system with q1/q2 = 1.6. (a) 〈Vc〉 vs Fd. For Fd > 0.075, there is a transition to a state with strongly localized shear and reduced velocity fluctuations. (b) The corresponding 〈Vb〉 vs Fd. (c,d) Particle positions (filled circles) and trajectories (lines) in the same system for (c) the disordered decoupled state at Fd=0.05 and (d) the strongly localized shear state at Fd=0.09. Arrows in (a) and (b) indicate the drives at which the images in (c) and (d) were obtained.

A.  Dynamic phases

For bidisperse samples with large q1/q2, we find an additional dynamical phase at high drives where the shear band region becomes strongly localized again and the bulk particles lock together, similar to the behavior found in the ordered regime. In Fig. 10(a) we plot 〈Vc〉 versus Fd for a bidisperse sample with q1/q2 = 1.6. For low Fd, the system is crystalline and strongly coupled, so that all the particles move together. As Fd increases, the system becomes disordered and produces a large shear band, illustrated in Fig. 10(c) for Fd = 0.05, associated with large velocity fluctuations. At higher drives, however, there is a transition to a strongly shear localized state where only a single line of particles are moving while the bulk particles lock together and move at a very slow velocity, as shown in Fig. 10(d) for Fd = 0.09. The onset of this phase is accompanied by a drop in the velocity fluctuations in Fig. 10(a). In Fig. 10(b) we plot 〈Vb〉 versus Fd for the same system, showing that at the shear localization transition, the bulk velocity drops to a value slightly above zero. At the transition, 〈Vc〉 increases since the drag from the bulk particles on the particles in the driven channel is reduced. The shear localization occurs when the particles in the driven line are moving sufficiently rapidly that they can no longer couple effectively to the bulk particles. This transition is a general feature that occurs in the disordered state. Since the transition is dominated by fluctuations, the drive at which it occurs can vary significantly from one sample realization to another; however, on average the transition occurs at higher values of Fd for increasing q1/q2 or decreasing a. The shear localized state has small velocity fluctuations with low noise that is white (α = 0).
Fig11.png
Figure 11: (Color online) The particle positions (filled circles) and trajectories (lines) at Fd = 0.05 for a bidisperse system with q1/q2 = 1.8 and quenched disorder. The open circles are the locations of the pinning sites that each capture one particle. (a) Np = 4 and dp=3a. (b) Np = 20 and dp=3a. (c) Np=10 and dp=2a. (d) Np=10 and dp=18a.

V.  QUENCHED DISORDER AND SHEAR LOCALIZATION

We next examine the effects of adding quenched disorder or pinning sites in the bulk region. In Fig. 11 we plot the particle trajectories in the disordered regime for a bidisperse sample with q1/q2=1.8 at Fd=0.05. The pinning sites are placed an average distance dp from the driven region. Each pin captures a single particle and the pinning force is sufficiently strong that the particles do not depin over the range of Fd we consider. The repulsive nature of the particle-particle interactions prevents unpinned particles from closely approaching pinned particles, leading to a reduction in the density of particle trajectories in the vicinity of each pinned particle. For a small number of pinning sites Np=4, shown in Fig. 11(a) for a sample with dp=3a, the overall particle trajectories do not differ significantly from the pin-free system; however, as the number of pinning sites increases, the mixing region surrounding the driven line is reduced in width, as illustrated in Fig. 11(b) for a sample with Np=20 and dp=3a. Here, there is no longer any net motion of the bulk particles, indicating a complete screening of the shear banding effect by the pinning. This result shows that quenched disorder can produce strong shear localization. In Fig. 11(c,d) we show the particle positions and trajectories for samples where the number of pinning sites is fixed at Np=10 for different pin spacings of dp=2a [Fig. 11(c)] and dp=18a [Fig. 11(d)]. The motion is more localized for the smaller value of dp.
Fig12.png
Figure 12: (a) 〈Vc〉 vs Fd for a bidisperse system with q1/q2=1.8 and with Np=16 and dp=2a. The pinned, coupled disordered, and high drive decoupled regimes are labeled. (b) Dynamic phase diagram for Fd vs Np in bidisperse samples with q1/q2 = 1.8. The pinned, coupled disordered, and high drive decoupled regimes are labeled.
In Fig. 12(a) we plot 〈Vc〉 versus Fd for a system with Np=16 and dp=2a where we observe a pinned regime, a coupled disordered flow regime, and a decoupled regime for high drive similar to that found for the pin-free system in Fig. 10. In the absence of pinning, at low drives the particles all move as a locked solid; however, when pinning is added to the system, there is a true pinned phase at low drive where particle motion does not occur. In Fig. 12(b) we plot the dynamic phase diagram for Fd versus Np. The transitions between the different phases show strong fluctuations from one realization to another, so we average the transition line over several different disorder realizations. The transition between the coupled and decoupled regimes drops to lower Fd with increasing Np. For a fixed drive, we find the interesting effect that the onset of the decoupled regime is correlated with an increase in the velocity of the particles in the driven line. Thus, by increasing the number of pinning sites, it is possible to cause the driven particles to move at a higher velocity which arises due to the effective screening of the driven particles from the bulk particles by the pinned particles.
Fig13.png
Figure 13: (Color online) Velocity profiles ―Vx vs y for bidisperse samples with q1/q2=1.8 at Fd=0.024. (a) dp=2a and Np = 0, 4, 6, 8, 10, 12, 14, 16, 18, and 20, from top center to bottom center. (b) Np = 10 and dp=2a, 4a, 6a, 8a, 10a, 12a, 14a, 16a, and 18a, from bottom center to top center.
In Fig. 13(a) we show ―Vx(y) for bidisperse samples with q1/q2=1.8, dp=2a, and Fd=0.024 for different numbers of pinning sites ranging from Np=0 to Np=20. For the pin-free system with Np=0, ―Vx far from the driven line reaches a finite value since all of the particles in the system move. As the number of pinning sites increases, ―Vx decreases at all length scales, but it decreases less rapidly in the driven line region than in the region far from the driven line. For a sufficiently large number of pinning sites, ―Vx drops to zero for distances of 6a or greater away from the driven line, indicating a complete screening of the shear band by the pinned particles. In Fig. 13(b) we show ―Vx for the same system with fixed Np=10 and varied dp. For the smallest dp, the shear band is strongly localized.
Fig14.png
Figure 14: (Color online) Temperature effects in a monodisperse pin-free system with Fd=0.05. (a,b,c) Particle positions (dots) and trajectories (lines) over a fixed time interval. (d,e,f) Corresponding Voronoi construction for instantaneous particle positions showing 6-fold (white), 5-fold (dark blue), and 7-fold (light red) coordinated particles. (a,d) T=0.6. (b,e) T = 1.2. (c,f) T = 1.8, above the bulk melting temperature.

VI.  THERMAL EFFECTS

To study thermal effects, we focus on the monodisperse system under fixed drive Fd=0.05, which forms a crystalline state. In Fig. 14 we plot the particle positions and trajectories as well as Voronoi constructions of snapshots of the particle positions for increasing T. For low temperatures T < 0.6, the system remains ordered and the particle trajectories are one-dimensional. At T = 0.6, Fig. 14(a,d) shows that the trajectories near the driven line begin to mix, producing roughly aligned 5-7 paired defects close to the driven region, while the bulk particles remain ordered. At T = 1.2 in Fig. 14(b,e), the number of 5-7 defect pairs has increased and some pairs begin to migrate from the driven line into the bulk region, although the bulk remains mostly ordered. For T = 1.56 the bulk begins to disorder and a widened shear band appears. Figure 14(c,f) shows that at T = 1.8 the bulk is strongly disordered and topological defects have proliferated.
Fig15.png
Figure 15: (a) 〈Vc〉 vs T for a pin-free monodisperse system at Fd = 0.05. (b) The corresponding fraction of six-fold coordinated particles P6 vs T. The system melts near T = 1.5 and the velocity of the particles in the driven line drops at temperatures just below the bulk melting transition.
In Fig. 15(a), the plot of 〈Vc〉 versus T shows that the velocity drops when the dislocations begin to migrate out from the driven line region. There is a minimum in 〈Vc〉 at the bulk melting temperature. Figure 15(b) shows the corresponding fraction of six-fold coordinated particles, P6=∑i=1Ncδ(zi−6), where zi is the coordination number of particle i. For a perfect triangular lattice, P6 = 1.0. We find that the drop in P6, indicating bulk melting, occurs at a higher temperature than the drop in 〈Vc〉. In previous numerical studies of single driven probe particles moving through a crystalline background, it was also found that the velocity drops when the driven probe generates local dislocations, which typically begins for temperatures below the bulk melting temperature [8]. At much higher temperatures, the velocities gradually increase again.

VII.  SUMMARY

We study the effects of driving a quasi-one-dimensional region of particles through a two-dimensional system of Yukawa interacting particles. In a monodisperse system a crystalline state forms, and as a function of increasing external drive there is a well defined transition from elastic flow, where all the particles move together, to a decoupled flow, where the particles in the driven region decouple from the bulk particles and the bulk particles remain in a locked crystalline state. The properties of this elastic to decoupled transition are similar to those found in studies of layered systems where only one layer is driven and the other layer is dragged, such as a transformer geometry for bilayer superconductors or bilayer Wigner crystals. For bidisperse systems, the bulk becomes disordered, and the driven line produces a local shear band and velocity gradient. The bulk particles near the driven line can be dragged with the driven particles, and the average velocity of the particles in the driven line decreases as the width of the shear band region increases. In the disordered regime, for increasing drive we identify another decoupling transition where the bulk particles become locked, producing an elastic disordered solid, coinciding with the shear band becoming very sharp. We show how the decoupling force, noise fluctuations, and average velocities can be correlated with bulk disordering transitions. With the addition of quenched disorder or pinning, we find that the shear band region becomes increasingly localized for increasing pinning density. This results in an interesting effect where the velocity of the driven particles can be increased by adding more pinning sites to the system. Such systems could be realized experimentally using colloidal particles or dusty plasmas with an optical drive applied along a one-dimensional channel, such as a laser applied along the edge of the sample. Variations of this geometry could also be constructed in superconducting vortex systems by embedding a single weak pinning channel in a bulk sample.
This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396. The work of A. Libál was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2011-3-0114.

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