Sliding Dynamics for Bubble Phases on Periodic Modulated Substrates

We analyze a bubble forming system composed of particles with competing long range repulsive and short range attractive interactions driven over a quasi-one-dimensional periodic substrate. We find various pinned and sliding phases as a function of substrate strength and drive amplitude. When the substrate is weak, a pinned bubble phase appears that depins elastically into a sliding bubble lattice. For stronger substrates, we find anisotropic bubbles, disordered bubbles, and stripe phases. Plastic depinning occurs via the hopping of individual particles from one bubble to the next in a pinned bubble lattice, and as the drive increases, there is a transition to a state where all of the bubbles are moving but are continuously shedding and absorbing individual particles. This is followed at high drives by a moving bubble lattice in which the particles can no longer escape their individual bubbles. The transition between the plastic and elastic sliding phases can be detected via signatures in the velocity-force curves, differential conductivity, and noise. When the bubbles shrink due to an increase in the attractive interaction term, they fit better inside the pinning troughs and become more strongly pinned, leading to a reentrant pinning phase. For weaker attractive terms, the size of the bubbles becomes greater than the width of the pinning troughs and the depinning becomes elastic with a reduced depinning threshold.


I. INTRODUCTION
There are a variety of particle systems that can form bubble phases exhibiting two length scales, with the shorter scale determined by the average spacing between particles that are confined inside an individual bubble, and the larger scale arising from the assembly of the bubbles themselves into an ordered lattice.Bubble lattices typically appear when there is a competition between attractive and repulsive interactions, such as short-range attraction and longer-range repulsion 1-9 .The bubbles can distort and break up into smaller bubbles as a function of temperature or as a result of interacting with quenched disorder 2 .Bubble phases known as mesophases arise in colloidal systems that have multiple length scales, such as multi-step repulsive interactions [10][11][12][13] .Bubble phases can also appear in superconducting systems for vortices with competing interactions [14][15][16][17][18][19][20] , as well as for magnetic skyrmionsuperconducting vortex hybrids 21 .Bubbles containing two or more particles can form when charge ordering occurs in two-dimensional election systems [22][23][24][25][26][27] .In a system with competing interactions, the bubbles are often only one of several types of possible phases, including crystals, stripes, and void lattices, that arise as a function of particle density or the ratio of the repulsive and attractive interaction terms 7,11,13,[28][29][30] .The bubble phases occur for lower densities or stronger attractive interaction terms.There are also a variety of bubble-like systems where the bubble textures can distort or the shape can change, including emulsions and magnetic textures such as skyrmions 31 .
Bubble lattices can interact with a substrate, which for soft matter systems could be created using optical trap arrays [32][33][34][35][36][37] or patterned surfaces 38 .In solid state systems, ordered substrates can be made using vari-ous nanostructuring techniques.One of the simplest substrate geometries is a periodic quasi-one-dimensional (q1D) arrangement of troughs, which can induce the formation of various crystalline, smectic, and disordered phases for purely repulsive colloidal particles as a function of the trough strength and the ratio of the number of particles to the number of troughs [39][40][41][42][43][44][45][46] .Different kinds of vortex patterns and depinning phenomena have been studied in superconducting systems for vortices interacting with periodic q1D substrates [47][48][49] .Far less is known about what happens when bubble or pattern-forming systems with competing interactions are coupled to a periodic substrate, and the depinning or sliding dynamics under a drive have received even less study.McDermott et al. 50considered a pattern-forming system with completing repulsion and attraction on periodic q1D substrates, and found that for a parameter regime where the system forms a stripe phase in the absence of the substrate, several stripe and modulated stripe phases as well as kinks appear as a function of substrate strength or substrate lattice spacing.
Here, we examine the statics and dynamics of particles with competing long-range repulsion and short-range attraction interacting with a periodic q1D substrate in the limit where the system forms a bubble phase in the absence of a substrate.For weak substrates, a triangular bubble lattice appears that becomes increasingly anisotropic as the substrate strength increases.When an external drive is applied to the system, the depinning transition can be elastic, where each bubble retains its original set of particles, or plastic, where the bubble lattice remains pinned but individual particles hop from one bubble to the next.At higher drives, a bubble shedding phase can appear in which all of the bubbles are moving but a portion of the particles can break away from one moving bubble and become attached to a differ-ent moving bubble, while a dynamic reordering transition into a moving elastic bubble lattice occurs for sufficiently large driving forces.In the plastic flow regime, we find a phase that we term a sliding bubble track phase, in which stripes of particles remain pinned to the substrate and form tracks along which the bubbles travel.We show that the nature of the depinning transition and the net velocity depend strongly on whether an individual bubble can fit into a single substrate trough, and demonstrate that smaller bubbles are more easily pinned.This leads to the emergence of reentrant pinning as a function of bubble size, where for constant applied drive the bubbles repin as they become smaller.Bubbles that are larger than the width of the substrate troughs can slide easily in an elastic phase.Our results should be relevant to a variety of bubble-forming systems on q1D substrates, including electron bubbles, colloidal particles, and magnetic skyrmions.

II. SIMULATION
We examine a two-dimensional (2D) system of N particles whose pairwise interactions have a long-range repulsive term and a short-range attractive term.The sample is of size L × L with L = 36 and has periodic boundary conditions in the x and y directions.The particle density is ρ = N/L 2 .The particles interact with a periodic q1D substrate and are subjected to a dc driving force.The following overdamped equation governs the dynamics of particle i: where the damping term η is set to η = 1.0 The particle-particle interaction potential is given by where . The first term is a long range Coulomb repulsion that will favor formation of a triangular lattice of particles in the absence of a substrate.For computational efficiency, we treat the long range Coulomb interaction with a Lekner summation, as in previous work 2, 50 .The second term is a short range attraction that falls off exponentially.At very short range, the repulsive Coulomb interaction becomes dominant again, which prevents the particles from collapsing onto a point.In previous work, it was shown that particles with the interaction potential in Eq. (2) can form crystal, stripe, bubble, and void lattice states depending on the values of ρ, B, and κ 2,3,7,50,51 Here we fix κ = 1.0 and focus on a particle density of ρ = 0.44.In the absence of substrate, this system forms a crystal for B < 2.0, a stripe state for 2.0 < B < 2.25, and bubbles for B > 2.25.Adding a substrate and/or changing the particle density will modify the values of B for which these phases appear.
The particles interact with a q1D substrate with N p minima.The substrate force is given by where x i is the x position of particle i and the spacing between substrate minima is a p = L/N p .Here we focus on substrates with N p = 8.0, corresponding to a p = 4.5.
We obtain the initial particle configuration by performing simulated annealing, where the particles are placed in an initial lattice, subjected to a high temperature, and then slowly cooled.The thermal forces are represented by Langevin kicks, and after the simulated annealing is complete, the temperature is set to zero.After the system has been initialized, we apply a driving force of F D = F D x to all of the particles and measure the time-averaged particle velocity in the direction of drive, V = N i v i • x.We typically wait 10 5 or more time steps until the system has reached a steady state before taking data, and we average the velocity over a similar time frame.Due to the long range interactions, the system can exhibit transient dynamics over relatively long time scales.

III. RESULTS
In Fig. 1(a-c), we illustrate the particle positions and substrate potential for a system with F p = 2.0, ρ = 0.44, and a p = 4.5 at F D = 0.0.For B = 2.25 in Fig. 1(a), the particles form stripes that are aligned with the substrate.In Fig. 1(b) at B = 2.85, bubbles appear that have an anisotropic shape due to the confinement from the substrate.At B = 6.0 in Fig. 1(c), the bubbles are much smaller and can fit better into the sinusoidal substrate minima.Figure 1(d) shows the bubble phase sample with B = 2.85 from Fig. 1(b) at a higher particle density of ρ = 1.38,where the bubbles are more compact and each contain a larger number of particles.
We next focus on the depinning of the bubble phase for B ≥ 2.25 by measuring the average velocity as a function of external drive for fixed ρ = 0.44 and B = 2.85.In Fig. 2(a), we plot the velocity-force curve for the system in Fig. 1(b).Near F D = 0.85, the bubbles depin elastically, with all of the particles remaining in their original bubble both during the depinning process and at higher drives.Figure 2(b) shows that the corresponding d V /dF D versus F D curve has a sharp peak near depinning, while at higher drives the velocity-force curve becomes linear and V ∝ F D .In systems that depin elastically, there is generally only a single peak in the differential velocity-force curves 52 .When we increase the substrate strength, we see a transition to plastic flow where the motion above depinning consists of individual particles hopping from one pinned bubble to the next, followed at higher drives by a state where moving bubbles shed and reabsorb individual particles as they travel through the system.In Fig. 2(c,d), we plot the velocityforce and differential velocity-force curves for the same system from Fig. 2(a) but at F p = 5.0, where the depinning is plastic.Here, there are two peaks in the differential velocity-force curve, and the velocity-force curve does not exhibit linear behavior until F D > 8.5.
In Fig. 3(a), we show the particle locations and trajectories for the system in Fig. 2(c,d) at F D = 3.75, just above the first peak in the d V /dF D curve.The system forms a rectangular lattice of pinned bubbles, and individual particles are able to hop from one bubble to the next.As F D increases, the number of particles participating in this hopping process increases until F D /F p > 1.0, at which point all of the particles are able to move at the same time.When this happens, the bubble structure is partially broken up, as shown in Fig. 3(b) at F D /F p = 1.1.The second peak in the d V /F D curve thus corresponds to the drive at which all of the particles become able to flow simultaneously.As F D further increases, the bubble structure reassembles, as shown in Fig. 3(c) at F D /F p = 1.4,and each bubble can shed individual particles that proceed to jump from one moving bubble to another, where the particles are reabsorbed.In general, the individual particles move more slowly than the bubbles.For F D > 1.7, the system forms a bubble lattice where the hopping process no longer occurs and all particles remain trapped within individual bubbles, as shown in Fig. 3(d) at F D /F c = 2.2.The bubble lattice formation occurs at the same drive that marks the transition to linear behavior of the velocity-force curve.
In the moving bubble phase, the individual bubbles are anisotropic and are elongated in the driving direction.The nonlinear velocity-force curve and the double peak feature in the differential velocity-force curve is a general feature of systems that exhibit plastic depinning 52 .In Fig. 4(a,b), we plot the velocity-force and differential velocity-force curves for the system from Fig. 2 but at F p = 4.0, while Fig. 4(c,d) shows V versus F D and d V /F D versus F D for the same system at F p = 6.0,where the multiple peak feature in d V /F D is even more pronounced.From the images and the features in the transport curves, we can identify four dynamical phases that appear as a function of applied drive and substrate strength.These are: a pinned bubble phase where the velocity is zero; an intra-bubble hopping phase where the bubbles are pinned but individual particles can hop from one bubble to the next; a disordered moving partial bubble phase where all of the particles are moving but the bubble structure has been disrupted by a shedding and reabsorption process; and a high drive elastically moving bubble lattice.
In Fig. 5(a), we show the depinning threshold F c versus pinning strength F p for the system in Figs 2 and 4 periodic substrates that exhibit depinning phenomena 52 .Additionally, for F p ≥ 2.25, we find that the depinning threshold increases linearly according to F c ≈ F p , as expected for plastic depinning 52 .
From the particle structures and the features in the transport curves, we can construct a dynamical phase diagram highlighting the different dynamical regimes, as shown in Fig. 6 as a function of F D versus F p for the system from Figs. 2 and 4. For F p < 2.25, there is an elastic depinning transition directly from a pinned bubble (PB) lattice to a moving bubble (MB) lattice, while for F p ≥ 2.25, an initial depinning transition takes the system into the intra-bubble hopping (IBH) phase, a second depinning transition results in the appearance of the disordered moving partial bubble (MPB) phase, and at higher drives a dynamic reordering transition occurs into the moving bubble (MB) lattice.Dynamical ordering at high drives in systems with plastic depinning transitions has been observed in systems with both random and ordered substrates 52 .

IV. EFFECT OF BUBBLE SIZE ON DYNAMICS
We next consider the effect of changing the strength B of the attractive interaction term.For larger B, the bubbles shrink in size and become more rigid, as shown  V vs B for a system with ap = 4.5, ρ = 0.44, and Fp = 0.1 at FD = 0.875, 1.0, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875, 2.0, and 2.25, from bottom to top.Here, V is nonmonotonic, and there is a reentrant pinning transition at higher B.
in Fig. 1(c).We note that individual bubbles can capture an increased number of particles even as the bubble radius decreases with increasing B; however, because the Coulomb interaction becomes dominant again at small length scales, the particles are unable to assemble into a single large bubble but instead form a collection of bubbles.
We first consider the case for F p = 2.0, where the depinning transition is elastic.In Fig. 7, we plot V versus B in a sample with ρ = 0.44 and F p = 0.1 at F D = 0.875, 1.0, 1.125, 1.25, 1.375, 1.5, 1.625, 1.75, 1.875, 2.0, and 2.25.When F D < 1.375, the velocity drops to zero for low values of B. In this low B pinned state, a strongly pinned stripe phase appears, and the depinning and sliding dynamics of this stripe phase will be described in a separate paper.For each value of F D , V has a nonmonotonic dependence on B, with a maximum appearing near B = 3.0 followed by a velocity decrease.When F D < 1.875, at higher B the velocity drops completely to zero, indicating that the bubbles have become pinned.This result indicates that there can be both a low B pinned state and a high B reentrant pinned state.The non-monotonic behavior of the velocity and the reentrant pinning occur because of two effects: the ability of the bubble to distort, and the change in size of the bubble.For B < 3.0, the bubbles are large but can partially distort to fit inside the pinning troughs as elongated or anisotropic bubbles.As B increases, the bubbles shrink but also become stiffer, and are thus less able to accommodate anisotropic distortions.Near B = 3.0, the bubbles become too round to elongate enough to fit completely into an individual pinning trough, reducing the effectiveness of the pinning.For B > 3.0, the bubbles remain round, but the bubble radius decreases, as shown in Fig. 1(c).These smaller bubbles can fit more easily into the pinning troughs, increasing the effectiveness of the pinning, and the system starts to act more like a lattice of point particles.Thus, the smaller bubbles are better pinned even when the number of particles in each individual bubble remains the same.In Fig. 8, we plot a dynamic phase diagram as a function of F D versus B showing the pinned bubbles and moving bubble lattice phases.A dip in the transition point appears near B = 3.0, where the pinning effectiveness of the substrate is the most greatly reduced by the shape of the bubbles.For B ≤ 2.25, the system forms a moving stripe phase.
In Fig. 9 we plot V versus B for the same system from Fig. 7 but with F p = 5.0 at F D = 3.5, 4.0, and 5.0.For this system, the depinning is plastic for B < 3.25 and elastic for B ≥ 3.25.The velocities are lower within the plastic or IBH regime and show a jump up at the entrance to the elastic regime, with a peak velocity appearing near B = 3.25 followed by a velocity drop at larger B as the bubble size decreases.There is a reentrant pinning regime at high values of B. The smaller bubbles are less likely to undergo plastic deformations or permit bubble-to-bubble hopping because the attractive interaction forces generate a greater barrier for individual particles to jump out of a bubble.From the flow patterns and the features in the transport curves, in Fig. 10 we construct a dynamic phase diagram for the system in Fig. 9 as a function of F D versus B. The depinning force is lowest near B = 3.5, close to the elastic to plastic transition, and the drive that must be applied to the plastic phase in order to dynamically reorder the system increases with decreasing B. As B increases in the elastic regime, the depinning threshold increases because the bubble size is diminishing.Another way to characterize the plastic and elastic flow regimes is to measure the power spectra of the velocity noise fluctuations from time series data.In Fig. 11(a), we plot the time dependent velocity for the system in Figs. 9 and 10 at B = 2.75, which corresponds to the inter-bubble hopping plastic flow phase where the velocity signal has no significant features.We also plot V as a function of time for a sample with B = 3.5 in the moving bubble phase, where there is a strong periodic velocity signal consistent with the washboard signature expected for an elastic solid moving over a periodic potential.Figure 11(b) shows the power spectra S(ω) = | V (t)e −iωt | 2 for the two velocity signals.In the intra-bubble hopping phase, there is 1/f noise at lower frequencies, while a strong narrow band noise signal appears for the moving bubble phase.This indicates that moving bubbles should produce a washboard signal, while the intra-bubble hopping flow is more disordered.We find similar noise features for plastic versus elastic flow for other parameters.
To further explore how the bubble size affects the pinning effectiveness, we consider a substrate with a smaller lattice constant at the same particle density and the same range of B values.In Fig. 12 we plot V versus B for a system with ρ = 0.44, fixed F D = 0.5, and a substrate lattice constant of a p = 2.11, half as large as what was used for the results presented up to this point, at substrate strengths of F p = 2.0, 3.0, 4.0, 5.0, and 6.0.For F p = 2.0, the system is pinned when B < 2.4375, and the particles form a 1D stripe-like pattern as shown in Fig. 13(a) for B = 2.25.A new plastic flow state, distinct from that illustrated earlier, appears for 2.4375 < B < 2.6, where the system forms 12. V vs B for a system with FD = 0.5, ρ = 0.44, and a smaller pinning lattice constant of ap = 2.11 for Fp = 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0, from top to bottom.At low B, the system forms a pinned stripe phase as illustrated in Fig. 13(a) for B = 2.25 and Fp = 2.0.When the velocity becomes finite, the system depins plastically into the sliding bubble track phase shown in Fig. 13(b) for B = 2.5 and Fp = 2.0.At high velocities, the system forms a bubble phase where the bubbles are close to twice the size of the pinning lattice constant, as shown in Fig. 13(c) for B = 2.75 and Fp = 2.0.For higher B, the system reenters a pinned state when the bubbles become small enough to fit inside a single pinning trough, as shown in Fig. 13(d) for B = 4.5 and Fp = 2.0.
the sliding bubble track phase illustrated in Fig. 13(b) at B = 2.5.Here, pinned particles remain trapped in stripes that form tracks parallel to the driving direction, and the bubbles travel along these pinned tracks.For B ≥ 2.6, the system forms a moving bubble lattice, as shown in Fig. 13(c) at B = 2.75, where it can be seen that when the bubbles form, they do not fit into the substrate troughs.The transition to the moving bubble lattice occurs at the large jump up in V in Fig. 12.As F p is varied, the same phases appear for shifted values of B.
In general, the bubbles are less strongly pinned for the smaller substrate lattice spacing because the size of the bubbles is larger than the width of the substrate troughs, causing the bubbles to sit partially on the potential maxima separating adjacent pinning troughs.As B increases, the bubbles shrink, and the velocity goes to zero when B becomes large enough to permit each bubble to fit entirely within a single substrate trough, as shown in Fig. 13(d) for B = 4.5.When the driving force is fixed to F D = 0.5, the moving bubble phase continues to flow as the pinning force increases up until F p = 6.0, when the bubbles become pinned.In Fig. 14, we construct a dynamic phase diagram as a function of F p versus B for the system in Fig. 12, highlighting the regime in which the moving bubble lattice can occur.For B < 3.0, the pinned phase consists of a strongly anisotropic bubble or stripe arrangement, while for B > 3.0, the system forms a pinned bubble lattice.We note that right along the  showing the pinned anisotropic bubble/stripe phase (PAB/S), the moving bubble lattice (MB), and the pinned bubble lattice (PB) that appears for higher B. V vs ρ for a system with Fp = 4.0, ap = 4.5, and B = 2.85 at FD = 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0, from bottom to top, showing a general increase in the velocity with increasing ρ as the bubbles grow in size.The system is more strongly pinned for the lower densities, where 1D anisotropic structures appear.
boundary between the pinned anisotropic bubble/stripe state and the moving bubble state, the system is in the plastic sliding bubble track phase, which is too narrow to highlight in the diagram.
To further demonstrate that the larger bubbles are less well pinned, we fix B = 2.85 while increasing the particle density in order to produce large bubbles similar to those shown in Fig. 1(d).In general, we find that smaller particle densities result in higher effective pinning.In Fig. 15(a,b) we show the bubble configurations for a system with a p = 4.5 and F p = 4.0 at lower particle densities, where we find a dimer bubble state for ρ = 0.0617 and a lattice-like arrangement of 1D stripes at ρ = 0.126.In general, the 1D-like structures are more strongly pinned than the 2D bubble phases that form at higher particle densities.In Fig. 16 we plot V versus ρ for a system with F p = 4.0, a p = 4.5, and B = 2.85 at F D = 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0.When F D < 4.0 and ρ < 0.75, the system is a pinned 1D bubble state similar to those illustrated in Fig. 15.In general, the average particle velocity increases with increasing ρ as the bubbles become larger.

V. DISCUSSION
There are many further directions to study in this system.For example, the plastic flow we observe consists of bubble-to-bubble hopping, so the bubbles have to break apart; however, there could also be plastic flow states in which individual bubbles remain intact, but some bubbles remain pinned while others move, similar to the plastic flow observed in non-bubble particle-like systems.We expect that this type of bubble lattice plasticity could appear in systems where the pinning substrate is more heterogeneous.We have only considered external driving applied parallel to the substrate modulation direction or x axis, but the drive could be applied at angles with respect to the x axis, which would likely produce a combination of sliding along the y direction and hopping in the x direction.The depinning forces could also exhibit commensuration effects based on how well the bubble size matches the substrate lattice spacing, so that bubbles whose radii are integer multiples of the substrate lattice constant are more strongly pinned.We have only considered dc driving, but if the dc drive were combined with ac driving, we would expect additional Shapiro steplike phenomena to arise, especially in the sliding bubble lattice state where there is a strong washboard frequency.If thermal effects were included, it is likely that the intrabubble hopping phase would develop an extended range of creep-like behavior, and all of the depinning thresholds would shift to lower values.In this case, it would be interesting to compare intra-bubble creep with bubble lattice creep.

VI. SUMMARY
We have investigated a pattern-forming system of particles with competing long-range repulsion and shortrange attraction driven over quasi-one-dimensional periodic substrates, and have focused on the bubble regime.For a fixed driving force, stripe states are strongly pinned because they can align with the pinning troughs.In the bubble phase, there can be elastic depinning, where the moving bubbles maintain their original complement of particles and the differential velocity force curves show only a single peak, or plastic depinning, where individual particles hop from one pinned bubble to an adjacent pinned bubble.In the plastic depinning regime, the differential velocity force curves exhibit two peaks, with the second peak corresponding to the drive at which all of the bubbles begin to flow but the motion remains plas-tic since individual bubbles continuously shed and reabsorb particles.At higher drives, the system can dynamically reorder into a moving bubble lattice where no particle shedding occurs.When the pinning substrate lattice constant is reduced, we also find a plastic sliding bubble track phase in which a portion of the particles form tracks consisting of pinned stripes oriented with the driving direction, and the remaining particles form bubbles that move over the tracks.The effectiveness of the pinning, which is visible in both the depinning threshold and the velocity of the moving bubbles, depends on the size and flexibility of the individual bubbles.When the attractive interaction term is weak and the bubbles are highly flexible, the bubbles can distort anisotropically in order to fit between the pinning troughs and become better pinned.Increasing the attractive interaction term stiffens the bubbles, making them more round in shape, but also shrinks their radius.The stiffer bubbles cannot accommodate themselves to the shape of the pinning troughs, but once the bubbles drop below the width of the pinning troughs, they can be well pinned by the substrate.This leads to a non-monotonic dependence of the veloc-ity on the strength of the attractive interaction term, where there is a pinned state for flexible bubbles that can distort into stripe-like shapes, as well as a reentrant pinned state that appears when the size of an individual bubble decreases enough that it can fit inside a pinning trough.Our results should be relevant to a variety of bubble-forming systems, including electron bubbles, colloidal particles, and magnetic skyrmions.
FIG.1.The particle positions (red circles) and the q1D substrate potential (green shading) for a bubble-forming system at a particle density of ρ = 0.44 with a substrate lattice constant of ap = 4.5 for Fp = 2.0 and FD = 0.0.(a) A stripe state at B = 2.25.(b) A bubble phase at B = 2.85.(c) More compact bubbles at B = 6.0.(d) The B = 2.85 sample from (b) at ρ = 1.38,where larger bubbles appear.

FIG. 2 .
FIG. 2. (a) The average velocity per particle V vs FD for the system in Fig. 1(b) with ap = 4.5, ρ = 0.44, B = 2.85, and Fp = 2.0, where the depinning is elastic.(b) The corresponding d V /dFD vs FD curve.(c) V vs FD for the same system but with Fp = 5.0, where the depinning is plastic.(d) The corresponding d V /dFD vs FD curve indicates that there is a two-step depinning process.

FIG. 3 .
FIG.3.The particle positions (red circles), trajectories (lines), and q1D substrate potential (green shading) for the system in Fig.2(c,d) with ap = 4.5, ρ = 0.44, B = 2.85, and Fp = 5.0.(a) At FD = 3.75 or FD/Fp = 0.75, the bubbles are pinned, but individual particles are hopping from one bubble to the next.(b) At FD/Fp = 1.1, all of the particles are moving but the bubble structure is disordered.For clarity, the trajectory lines are not shown.(c) At FD/Fp = 1.4,organized bubbles reform but can shed and reabsorb individual particles, which are able to use the shedding mechanism to jump from one bubble to another.(d) At FD/Fp = 2.2, the bubbles are fully formed and there is no shedding.For clarity, the trajectory lines are not shown.

FIG. 4 .
FIG. 4. (a) V vs FD for the system from Fig. 2 with ap = 4.5, ρ = 0.44, and B = 2.85 but at Fp = 4.0.(b) The corresponding d V /dFD vs FD curve showing a multiple step depinning process.(c) V vs FD for the same system but with Fp = 6.0.(d) The corresponding d V /dFD vs FD curve.

FIG. 5 .
FIG. 5. (a) The depinning force Fc vs Fp for the system in Figs. 2 and 4 with ap = 4.5, ρ = 0.44, and B = 2.85.(b) The corresponding dFc/dFp vs Fp curve.The peak corresponds to the transition from elastic depinning to plastic depinning with increasing Fp.

FIG. 6 .
FIG.6.Dynamic phase diagram as a function of FD vs FP for the system from Fig.4with ap = 4.5, ρ = 0.44, and B = 2.85, where we highlight the moving bubble lattice (MB) phase, an intra-bubble hopping (IBH) phase, a moving partial bubble (MPB) phase, and a pinned bubble (PB) phase.

FIG. 8 .
FIG. 8. Dynamic phase diagram as a function of FD vs Bshowing the pinned bubble and moving bubble lattice regimes for the system in Fig.7with ap = 4.5, ρ = 0.44 and Fp = 0.1.A dip appears near B = 3.0 where the bubbles are less well pinned.

FIG. 9 .
FIG. 9. V vs B for the system from Fig.7with ap = 4.5 and ρ = 0.44 at Fp = 5.0 for FD = 3.5, 4.0, and 5.0, from bottom to top.The depinning is plastic for B < 3.25 and elastic for B ≥ 3.25, and there is a reentrant pinning regime at high B.

FIG. 11 .
FIG. 11.Velocity time series V vs time for the system in Fig. 10 with ap = 4.5, ρ = 0.44, Fp = 5.0, and FD = 4.0 at B = 2.75 (red) in the IBH or plastic flow phase and B = 3.5 (blue) in the moving bubble lattice or MB phase.There is a strong time periodic signal in the MB phase.(b) The corresponding power spectra S(ω) shows strong peaks in the moving bubble phase.
FIG.12.V vs B for a system with FD = 0.5, ρ = 0.44, and a smaller pinning lattice constant of ap = 2.11 for Fp = 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0, from top to bottom.At low B, the system forms a pinned stripe phase as illustrated in Fig.13(a) for B = 2.25 and Fp = 2.0.When the velocity becomes finite, the system depins plastically into the sliding bubble track phase shown in Fig.13(b) for B = 2.5 and Fp = 2.0.At high velocities, the system forms a bubble phase where the bubbles are close to twice the size of the pinning lattice constant, as shown in Fig.13(c) for B = 2.75 and Fp = 2.0.For higher B, the system reenters a pinned state when the bubbles become small enough to fit inside a single pinning trough, as shown in Fig.13(d) for B = 4.5 and Fp = 2.0.

FIG. 13 .
FIG.13.The particle positions (red circles) and q1D substrate potential (green shading) for the system in Fig.12with ap = 2.11, ρ = 0.44, Fp = 2.0, and FD = 0.5.(a) A pinned stripe phase at B = 2.25.(b) A plastic sliding bubble track phase at B = 2.5.Pinned particles form stripe tracks and the bubbles move along these stripe tracks.(c) A moving bubble phase at B = 2.75 where the bubble size is greater than the width of the substrate troughs.(d) A pinned bubble lattice at B = 4.5 where the bubbles are small enough to fit within individual pinning troughs.

FIG. 14 .
FIG. 14. Dynamic phase diagram as a function of Fp vs B for the system in Fig. 12 with FD = 0.5, ρ = 0.44, and ap = 2.11showing the pinned anisotropic bubble/stripe phase (PAB/S), the moving bubble lattice (MB), and the pinned bubble lattice (PB) that appears for higher B.