Journal of Physics: Condensed Matter 30, 244005 (2018)

Clogging and Transport of Driven Particles in Asymmetric Funnel Arrays

C. J. O. Reichhardt and C. Reichhardt

Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, 87545, United States of America

E-mail: cjrx@lanl.gov
Received 27 February 2018, revised 23 April 2018
Accepted for publication 3 May 2018
Published 24 May 2018
Abstract
We numerically examine the flow and clogging of particles driven through asymmetric funnel arrays when the commensurability ratio of the number of particles per plaquette is varied. The particle-particle interactions are modeled with a soft repulsive potential that could represent vortex flow in type-II superconductors or driven charged colloids. The velocity-force curves for driving in the easy flow direction of the funnels exhibit a single depinning threshold; however, for driving in the hard flow direction, we find that there can be both negative mobility where the velocity decreases with increasing driving force as well as a reentrant pinning effect in which the particles flow at low drives but become pinned at intermediate drives. This reentrant pinning is associated with a transition from smooth one-dimensional flow at low drives to a clogged state at higher drives that occurs when the particles cluster in a small number of plaquettes and block the flow. When the drive is further increased, particle rearrangements occur that cause the clog to break apart. We map out the regimes in which the pinned, flowing, and clogged states appear as a function of plaquette filling and drive. The clogged states remain robust at finite temperatures but develop intermittent bursts of flow in which a clog temporarily breaks apart but quickly reforms.
Keywords: clogging, channel flow, jamming
Simulation
Results
Summary
References
A wide variety of both hard and soft matter systems can be modeled as particles that exhibit depinning phenomena under an external drive [1]. Specific examples include vortices in type-II superconductors [2,3,4], sliding charge density waves [5], the depinning of electron crystals [6,7] or colloidal assemblies [8,9,10], and skyrmions in chiral magnets [11,12]. In such systems, the depinning and subsequent sliding dynamics can be characterized using features in the velocity-force (V−F) curves. The particle velocity is zero in the pinned phase and becomes finite at a critical drive Fc. Within the moving phase the velocity can increase linearly with the driving force or it can show a series of steps corresponding to different effective depinning thresholds [1]. It is also possible to observe negative differential mobility in which the velocity drops with increasing driving force due to changes in the flow pattern or the onset of different modes of dissipation [1,13,14,15,16]. In some cases, the particles move in the direction opposite to the applied drive, producing absolute negative mobility [17,18]. Many studies involve particles moving over an effectively two-dimensional (2D) substrate; however, depinning phenomena and nonlinear transport can also arise for particle flow in confined geometries such as quasi-one dimensional (1D) channels, constrictions [19,20,21,22,23], bottlenecks [24,25,26,27], or asymmetric funnel arrays [28,29,30,31,32,33]. Pinning in these systems is produced by a confining effect of the walls, and particles that get stuck along the walls can reduce or stop the flow of other particles due to the particle-particle interactions. This pinning mechanism generates strong commensuration effects in which the critical drive Fc is enhanced when the number of particles is an integer multiple of the number of plaquettes or of any other periodicity in the system. Confined flow also appears in the motion of particulate matter through confining regions such as hoppers and bottlenecks, where clogging effects occur due to steric particle-particle interactions [34,35,36,37,38,39,40,41]. Similar effects arise in the flow of pedestrians or active matter through constrictions [42,43,44].
In this work we consider an assembly of particles driven through a linear array of asymmetric funnels. The soft repulsive interactions between particles are modeled as a Bessel function K1(r) which decays exponentially at large r. This specific interaction potential describes vortices in type-II superconductors; however, a variety of simulation studies have shown that many of the behaviors observed for vortices interacting with pinning landscapes are generic to other systems of particles with similar interactions, such as charge-stabilized colloids with Yukawa interactions or magnetic colloids with dipolar interactions [1,4,7,14,45,46]. Previous work on particles driven through funnel arrays focused on motion in the easy flow direction where the depinning force is low, and showed that a clear commensurability effect occurs whenever the number of particles is an integer multiple of the number of funnel plaquettes, while there is an overall increase in the depinning force with increasing commensurability ratio since crowding effects make it more difficult to force particles through the funnels [30]. Here we consider particles driven in the hard direction of the funnel asymmetry. We find distinctive V−F relations, including a clogging effect associated with reentrant pinning for increasing drive. In most systems that exhibit depinning phenomena, the velocity increases with increasing drive; however, we find that there are extended regions of drive and filling ratios in which reentrant pinning occurs. An initial depinning occurs at a low driving threshold Fc1 when the particle density is uniform throughout the system, but at higher drives clogging events occur in which particles pile up in one or more of the funnel plaquettes and block the flow, so that the velocity drops back to zero. As the drive increases further, particles can eventually force their way through the clog and allow the system to flow again above a second depinning threshold Fc2.
We argue that our results are consistent with clogging rather than jamming behavior. In a jammed state, particle-particle interactions cause the system to behave like a uniform rigid solid above a unique jamming density [47]. In our system, although the initially pinned state and the flowing states are generally uniform, the reentrant pinned state is heterogeneous with local regions of high particle density. The reentrant pinning can occur for a wide range of fillings, as has been observed for clogging transitions [48], rather than only at a specific filling, as in the jamming transition. We discuss how our results could be relevant to vortices in superconductors, colloids in constriction geometries, Wigner crystals, and skyrmion systems, as well as how the behavior we observe is different from that found for particles with hard disk interactions such as grains or bubbles.
Fig1.png
Figure 1: (a) Illustration of a portion of the system showing the asymmetric funnel walls. In the initial state, each funnel plaquette holds nc or nc+1 particles where Nc=nc+f with integer nc and 0 ≤ f < 1. A drive FD is applied in the easy (right blue arrow) or hard (left green arrow) direction along the x axis. The full sample contains Npl=16 funnel plaquettes. (b,c,d) Illustrations of representative FD=0 particle configurations at commensurate fillings of Nc = (b) 3.0, (c) 4.0, and (d) 5.0.
Simulation - We model a two-dimensional system with periodic boundaries in the x direction containing walls that form an asymmetric funnel array, as illustrated in Fig. 1(a). There are a total of Npl=16 funnel plaquettes, each of which initially contains Nli particles so that the total number of particles is Np=∑i=1NplNli. We define Nc=nc+f with integer nc and 0 ≤ f < 1 to be equal to the filling fraction, Nc=Np/Npl. For integer Nc with f=0, all plaquettes initially contain Nli=nc particles, while for noninteger Nc with nonzero f, a fraction of the plaquettes initially contain Nli=nc particles while the remaining plaquettes initially contain Nli=nc+1 particles. The dynamics of particle i is governed by the following overdamped equation of motion:
η dRi
dt
 = − Np
∑
 i ≠ j 
 ∇Ucc(Rij) − Fiwall + FD + FTi  .
(1)
We set the damping constant η = 1.0. The particle-particle interactions are repulsive and have the form Ucc(Rij) = A0K0(Rij/λ), where Rij=|Ri−Rj|, Ri(j) is the position of particle i(j), and K0 is a Bessel function. We take λ to be the unit of distance in our simulation. The particle-particle interaction force is given by K1, which decays exponentially for Rij > λ, allowing us to cut off the interactions for Rij > 6λ. The unit of force in our simulation is A0. This model was previously used to study static vortex configurations in a funnel geometry as well as depinning and dynamics for driving in the easy flow direction [30].
The confining walls of each funnel plaquette are composed of 4 repulsive elongated potential barriers so that the entire system contains Nb=4Npl barriers. Each barrier consists of a rectangular repulsive region with a half-parabolic repulsive cap on each end. Particle-barrier interactions are described by Fiwall = (fp/rp)∑Nbk=1[Rik±Θ(rp − Rik±)Θ(Rik|| −lk)∧Rik± + Rik⊥Θ(rp−Rik⊥)Θ(lk − Rik||)∧Rik⊥], where fp=15A0, Rik± = |Ri − Rpk ±lk∧pk|||, Rik⊥,|| = |(Ri − Rpk) ·∧pk⊥,|||, Rpk is the position of the center point of barrier k, rp=0.4λ is the barrier radius or half width, and ∧pk|| and ∧pk⊥ are unit vectors parallel and perpendicular, respectively, to the axis of barrier k. The central rectangular barrier is of size 2lk=2.8λ for the vertical walls and 2lk=18λ/√3 for the slanted walls. The barriers are arranged to form mirror-symmetric sawtooth shapes that produce the funnel array illustrated in Fig. 1(a). The two important energy scales in the system are the particle-particle repulsive interaction energy which goes as K0(r) and the particle-wall interaction energy which goes as fpr2. For the parameters we consider, the particle-particle spacing is always greater than r=0.1 so that we are in the regime where fpr2/K0(r) >> 1.0, corresponding to the hard wall limit. We note that since the particle-particle interaction force K1(r) grows more rapidly than linearly for small r, only one particle at a time can move through the apertures separating adjacent funnel plaquettes.
Initially, Npl particles are evenly distributed across the funnel plaquettes, and the particles are then allowed to thermally relax under simulated annealing using Langevin kicks FT that obey 〈FiT(t)〉 = 0 and 〈FTi(t)FTj(t′)〉 = 2ηkBTδijδ(t − t′), where kB is the Boltzmann constant. Once the temperature has been reduced to zero, we apply a driving force FD = CFD∧x with C=+1 for easy direction driving and C=−1 for hard direction driving. For each value of FD we measure the average particle velocity 〈V〉 = Np−1∑Npiv i·∧x. We first consider driving in the absence of thermal fluctuations, but later we discuss the effect of finite temperature on the clogging dynamics.
Fig2.png
Figure 2: (a) The depinning force Fc vs filling fraction Nc for driving in the easy or +x direction with C=+1. (b) Representative 〈V〉 vs FD curves for driving in the easy direction at Nc = 3.0 (black), 4.0 (red), 4.5 (green) and 5.0 (blue). In each case there is a single depinning threshold Fc.
Results - Illustrations of representative FD = 0 particle configurations obtained through the annealing procedure appear in Fig. 1(b,c,d) for Nc = 3.0, 4.0, and 5.0, respectively. At integer values of Nc, each plaquette captures nc particles, while for noninteger values such as Nc=5.5, there is a mixture of plaquettes containing nc=5 or nc+1=6 particles. A more complete study of the particle ordering at the zero drive condition is provided in Ref. [30]. When a drive is applied in the positive x or easy flow direction, as in Ref. [30], the depinning force depends strongly on the commensuration ratio, as shown in Fig. 2(a). We plot four representative 〈V〉 versus FD curves in Fig. 2(b) for driving in the easy flow direction at Nc = 3.0, 4.0, 4.5, and 5.0. In each case, there is a single depinning threshold and 〈V〉 increases monotonically with increasing FD. Other fillings produce similar V−F curves.
Fig3.png
Figure 3: The average particle velocity |〈V〉| vs FD for driving in the hard or −x direction with C=−1 for Nc = 3.0 (black), 3.25 (red), 3.5 (green), and 3.75 (blue). There is a single depinning transition for Nc = 3.0 and Nc=3.5 as well as a region of reentrant pinning where |〈V〉| drops back to zero for Nc = 3.25. An example of negative differential conductivity appears for Nc = 3.75 in the form of a drop in |〈V〉| to a finite value near FD = 0.09.
Fig4.png
Figure 4: Instantaneous snapshots of the particle locations (circles) along with the particle trajectories (blue lines) obtained over 103 simulation time steps for driving in the hard or −x direction for the system in Fig. 3. The particles are moving from right to left in the images. (a) Nc = 3.0 at FD = 0.11 showing 1D channel flow. (b-f) Nc=3.25 for different FD. (b) An illustration of the transient depinning process at FD=Fc1 = 0.022 where particles fall away from the walls and flow along the center of the channel. (c) The first sliding phase at FD = 0.044. (d) The transient motion at the onset of the reentrant pinned phase at FD=Frp = 0.0625. (e) The reentrant pinned or clogged phase at Fd = 0.088, where particles have accumulated behind a blockage caused when particles trapped on the vertical funnel wall effectively decrease the width of the funnel aperture. (f) The sliding state at FD=0.11 above Fc2 where the particles are once again able to move through the funnel aperture.
In Fig. 3 we plot |〈V〉| versus FD for driving in the negative x or hard direction for Nc = 3.0, 3.25, 3.5, and 3.75. Here there are a variety of distinct dynamic behaviors and V−F curve characteristics that are very different from those found for driving in the easy direction. For the commensurate filling of Nc = 3.0, there is a single depinning transition at Fc = 0.02 followed by a linear increase in |〈V〉| with increasing FD. In Fig. 4(a) we show a snapshot of the particle positions along with the particle trajectories drawn over a period of 103 simulation time steps in the Nc=3.0 sample at FD=0.11. The particles are moving from right to left in the image. Here two particles are trapped at the corners of each plaquette, while the third particle in each plaquette is moving in the −x direction along a 1D channel at the center of the funnel array.
For Nc = 3.25 in Fig. 3, we find a multi-step depinning process with an initial depinning at Fc1 = 0.022. Above Fc1, |〈V〉| increases with increasing FD until a drop to |〈V〉| = 0 occurs over the interval of 0.0625 < FD < 0.094. This reentrant pinned phase is followed by a second depinning transition at Fc2 = 0.094. In Fig. 4(b) we illustrate the particle positions and trajectories during the transient depinning process that occurs at FD=Fc1=0.11 in the Nc=3.25 sample, while Fig. 4(c) shows the particle motion in the first flowing region at FD = 0.044 where a 1D channel of flow forms. The transition into the reentrant pinned state at FD=0.0625 appears in Fig. 4(d). The 1D channel flow is abruptly halted when two particles that had been pinned against the wall fall simultaneously toward the center of the channel but have their motion interrupted by the vertical wall of the funnel. These particles become trapped on the vertical wall by the combination of the driving force and their own repulsive interactions with each other. The repulsive force of the freshly trapped particles effectively reduces the width of the funnel aperture so that particles in the 1D flowing channel can no longer pass through and instead pile up inside the plaquette containing the blockage. This configuration remains static as the drive is further increased, as illustrated in Fig. 4(e) for FD=0.088. The clog is characterized by the accumulation of more than nc+1 particles in a single plaquette. In this case, a similar clogging configuration appears in roughly every fourth plaquette, so the reentrant pinning can be viewed as a clogging event dominated by a small number of plaquettes. As FD is further increased, there is a second depinning transition at Fc2 to the second moving state, where particles in the 1D channel can once again pass through the funnel aperture, as illustrated in Fig. 4(f) for FD = 0.11. Additional upward jumps in |〈V〉| can occur at higher FD when other particles that are trapped along the walls become dislodged and join the flow.
At commensurate fillings such as Nc=3.0 in Fig. 4(a), the particle density is homogeneous and at any given time each plaquette contains nc particles. At incommensurate fillings with noninteger Nc, the initial pinned state is mostly homogeneous, although due to the discreteness of the particles some plaquettes contain nc particles and others contain nc+1 particles. The two types of plaquettes are distributed evenly across the sample, so that for Nc=3.25, every fourth plaquette contains four particles. The clogged states that form under an applied drive are much more heterogeneous, with some plaquettes containing less than nc particles and others containing more (often significantly more) than nc+1 particles. When flow resumes after a clogged phase has formed, the flow remains somewhat heterogeneous since a portion of the clogged structure is generally still present, but as the drive increases, this structure gradually disintegrates and the flow becomes more homogeneous. We note that since we are working in the hard wall limit, it is possible for some particles that have become trapped in the corners of the plaquette to remain permanently trapped even at high drives. We consider a sample with Npl=16 plaquettes since we are working with quarter-filling intervals of f=0, 0.25, 0.5, and 0.75. The behavior of the system is robust for larger or smaller samples provided that the number of plaquettes is a multiple of four. If Npl were changed to a value that is not divisible by four, additional incommensuration effects could arise since the extra particles in initial pinned states with nonzero f can no longer be evenly distributed across the plaquettes.
Fig5.png
Figure 5: Instantaneous snapshots of the particle locations (circles) along with the particle trajectories (blue lines) obtained over 103 simulation time steps for driving in the hard or −x direction for the system in Fig. 3 at Nc = 3.75 where there is a region of negative differential conductivity. The particles are moving from right to left in the images. (a) The flowing phase at FD = 0.066. (b) An illustration of the transient shift in particle positions from the slanted walls to the funnel aperture at the drop in |〈V〉| at FD=0.09. (c) At FD = 0.12, above the drop in |〈V〉|, a partial clog forms when the particles next to the funnel aperture slow down the flow. (d) At FD = 0.26, the size of the partial clog decreases.
There is only a single pinned state for Nc=3.5 in Fig. 3 with a much higher depinning threshold of Fc1=0.175, but not all of the particles depin simultaneously, as shown by the step features that appear as FD increases above depinning. At Nc = 3.75 in Fig. 3, there is a single depinning threshold at Fc = 0.0275; however, a drop in |〈V〉| occurs near FD = 0.9 indicating negative differential mobility with d〈V〉/dFD < 0. This drop in |〈V〉| is associated with the formation of a partially clogged state rather than a completely clogged state. In Fig. 5(a) we plot the 1D channel flow pattern in this system at FD = 0.066 prior to the drop in |〈V〉|. As FD increases, some trapped particles suddenly shift from sitting along the slanted walls to sitting next to the funnel aperture. This transient motion at FD=0.09 is illustrated in Fig. 5(b), where it appears in the rightmost plaquette. The repositioned particles constrict but do not halt the channel flow, resulting in a local accumulation of particles at the constriction in the flowing state, as illustrated in Fig. 5(c) at FD = 0.012. As FD is further increased, the flow gradually increases and the size of the partial clog reduces, as shown in Fig. 5(d) at FD = 0.26.
Fig6.png
Figure 6: Instantaneous snapshots of the particle locations (circles) along with the particle trajectories (blue lines) obtained over 103 simulation time steps for driving in the hard or −x direction for the system in Fig. 3 at Nc = 3.5. The particles are moving from right to left in the images. (a) At FD = 0.15, the system is in a clogged state with a buildup of particles in the leftmost plaquette. (b) The partially clogged flow along the first velocity step at FD = 0.2. (c) Multiple transient rearrangements of the particles occur at the second step in |〈V〉| centered at FD = 0.26. (d) The flow above the second step in |〈V 〉| at FD = 0.3 where the partial clog is smaller.
For Nc = 3.5, the reentrant pinning is lost and Fc1 increases to Fc1=0.175 while the depinning occurs in a series of steps. The initial flow phase disappears due to the development of an instability within the pinned phase that causes particles to shift from the slanted walls to positions near the funnel aperture prior to the onset of 1D channel flow, producing a direct transition from the pinned state to a clogged state without an intermediate flowing phase. In Fig. 6(a) at FD = 0.15, the system is already in a clogged state even though no steady state particle flow has occurred. The number of particles in each plaquette varies from 2 to 5. Just above Fc, a rearrangement of the clogged state occurs into a partially clogged state where a small number of particles are able to push their way through the clog and begin to flow, as illustrated in Fig. 6(b) for FD = 0.2. As the drive increases, the partially clogged configuration begins to break apart, as shown in Fig. 6(c) for FD=0.26 at the second jump in |〈V〉|. Above this jump, the number of particles involved in the clog is reduced, as shown in Fig. 5(d) for FD=0.3. The size of the clog continues to decrease with further increases in FD as additional particles join the flow, leading to additional steps in |〈V 〉|.
Fig7.png
Figure 7: (a) |〈V〉| vs FD for driving in the hard direction at Nc = 3.25 (black), 4.25 (red), 4.75 (green), and 5.25 (blue). Each curve contains a reentrant pinning phase. (b) |〈V〉| vs FD for Nc = 5.5 (black), 5.75 (red), 6.0 (green), and 8.0 (blue), also showing reentrant pinning.
In Fig. 7(a) we plot |〈V〉| versus FD for representative fillings of Nc = 3.25, 4.25, 4.75, and 5.25, where in each case there is an interval of FD over which reentrant pinning occurs. The clogging effects become stronger at the higher fillings, where Fc2 ≈ 0.25. Figure 7(b) shows |〈V〉| versus FD for Nc = 5.5, 5.75, 6.0, and 8.0, which again exhibit reentrant pinning effects. The onset Frp of the repinned state shows some variations but generally shifts to higher values of FD for Nc = 5.75 and 6.0.
Fig8.png
Figure 8: |〈V〉| vs FD for driving in the hard direction at Nc = 8.25 (black), 8.5 (red), 8.75 (green), and 10.5 (blue). There is a single depinning transition for Nc = 8.5 and Nc=10.5, and at these fillings there is a direct transition from a pinned to a clogged state without an intermediate flowing state. (b) |〈V〉| vs FD for Nc = 9.25 (black), 10.75 (red), 11.0 (green), and 12.0 (blue). The point marked A indicates the value of FD at which there is a transition from the pinned to the clogged state for the Nc = 9.25 system, and the point marked B indicates the same transition for the Nc = 10.75 system.
In Fig. 8(a) we plot |〈V〉| versus FD for Nc = 8.25, 8.5, 8.75, and 10.5. Reentrant pinning occurs for Nc = 8.25 and Nc=8.75, while at Nc = 8.5 and Nc=10.5, there is only a single depinning transition but there is a flow-free transition from a pinned uniform state directly to a clogged inhomogeneous state. At higher values of Nc there are generally more steps and jumps in the sliding phases due to the increased number of particles participating in the partially clogged arrangements, which generates a larger number of partially clogged stages of flow.
In Fig. 8(b) we show |〈V〉| versus FD for Nc = 9.25, 10.75, 11.75, and 12.0. Each of these fillings exhibits only a single depinning threshold, but the transition from the uniform pinned state to the inhomogeneous clogged state can still be detected in the form of small jumps in |〈V〉| in the Nc = 9.25 and Nc=10.75 curves. These jumps, marked by the letters A and B, are generated by a transient rearrangement of the particles in the pinned phase to produce a clogged structure. For Nc ≥ 11 the system transitions directly from a pinned state to a flowing state without any kind of clogging. The particle density remains uniform and there are no jumps in |〈V〉|, as shown in the Nc = 11.0 and Nc=12.0 curves.
Fig9.png
Figure 9: Dynamic phase diagram as a function of FD vs Nc. In the pinned phase (black circles), |〈V〉| = 0 and the particle density is uniform. In the sliding phases, |〈V〉| > 0. Red squares indicate the initial sliding phase and blue triangles indicate a reentrant sliding phase. In the clogged or reentrant pinned phase (green diamonds), |〈V〉| = 0 and the particle density is heterogeneous.
In Fig. 9 we plot a dynamic phase diagram as a function of FD versus Nc created from a series of simulations that highlights the pinned phase, sliding phase, clogged phase, and reentrant sliding phase. In the pinned state, |〈V〉| = 0 and the particles are distributed evenly throughout the system. In general, the depinning force Fc marking the end of the pinned phase shifts to higher FD with increasing Nc. In the sliding states |〈V〉| > 0, while the clogged state has |〈V〉| = 0 with a heterogeneous distribution of particles in the funnel array. It is possible to identify additional dynamic regimes within the sliding state. For example, the moving particles are evenly spaced in the sliding states at low drives below Frp, while in the sliding states for FD > Fc2 the spacing between the moving particles is initially heterogeneous, but gradually become more homogeneous with increasing FD as the clogs break apart. For Nc > 6, the pinned state can transition directly into a clogged state with no intermediate flowing phase, while for Nc ≥ 11, the system passes directly from a uniform pinned state to a uniform moving state and the clogging behavior is lost. At low fillings 2.0 < Nc < 3.25, there is only a pinned phase and a flowing phase, while for 3.25 < Nc < 11.0 there are extended regions where clogging occurs. We note that due to the stochastic nature of the clogging, the value of FD at which clogging first appears can vary from realization to realization when the filling is high enough for multiple particles to accumulate in a small number of plaquettes.
Fig10.png
Figure 10: |〈V〉| vs FD at Nc = 4.75 for driving in the hard direction at zero temperature (blue) and at a finite temperature of FT = 0.1 (red). At this filling there is an extended reentrant pinned region for FT = 0. At finite temperature, the clogged region persists but is narrower, and it also contains a series of spikes in |〈V〉| that are produced by the thermal disintegration of a clog which then rapidly reforms.
One question is how robust the clogged states are to the introduction of a finite temperature, which could break apart the clogged structures. To address this we measure the velocity-force curves at a finite temperature of FT = 0.1, as shown in Fig. 10 for Nc = 4.75. At zero temperature, this filling exhibits an extended clogged region. Although the depinning transition at Fc1 is unaffected by the addition of temperature, the onset of the clogged state shifts to higher FD and the reentrant depinning transition Fc2 shifts to lower FD, giving a narrower clogged region. Thermal fluctuations produce numerous jumps in |〈V〉| as well as spikes of temporary flow within the clogged region due to the thermally assisted disintegration of clogged structures. Within the reentrant pinned region, the clogs rapidly reform, but in the flowing regimes, the clogs are permanently destroyed and the net transport through the system is larger than at zero temperature. In general we find that in the range of FD where the clogging susceptibility is high, clogging behavior still occurs for finite temperature but the clogging is interspersed with jumps or avalanche events, leading to a finite long-time average value of |〈V〉| which remains much lower than the average velocity in the flowing states.
We note that any type of particle flow that passes through an aperture is susceptible to forming a clogged state, and that asymmetry of the channel is not required to induce clogging. Thus, clogged states could form even in a symmetric channel consisting of a constricted rectangle. With such a geometry, clogging can occur for both directions of driving, whereas in our asymmetric sample, clogging occurs only for flow in the hard direction of driving. The transition from clogging to nonclogging flow in the symmetric constricted rectangle geometry would be a function both of the aperture size and of the difference in width of the two segments of the channel. Clogging behavior should decrease as the channel segments approach equal width, eliminating the constriction, and as the width of the smaller channel increases, permitting two or more particles to pass through the aperture simultaneously.
We next discuss how the clogging we consider could be observed experimentally. One candidate realization is superconducting vortices in nanostructured asymmetric funnel arrays [25]. A possible issue with this system is that the vortices are not true point particles, so that in the clogged state the vortices could merge and form multiquantum flux configurations, causing the particle picture to break down. Such a state could still exhibit clogging, and there is the interesting possibility of studying a transition from a multiquantum clogged vortex configuration to a flowing state, where the multiquantum vortices would dissociate back into individual moving flux quanta. Random point pinning would likely add a stochastic element to the behavior of the superconducting vortices; however, there should still be extended regions of clogging dynamics at intermediate fillings. An example of a specific superconducting realization would be a variant on the experiments performed with constricted geometries [25], where fillings of Nc = 2 to 6 would correspond to magnetic fields in the range 2 to 6 Oe, while the depinning current Fc1 would be in the range 0.5 to 3 mA. For systems such as magnetic skyrmions or bubbles, there is a similar issue in which the particle model can break down due to strong spatial distortions of the skyrmion or bubble. Such distortions could act to prevent clogging from occurring. Systems such as charged colloids or classical Wigner or ion crystals are also good candidates for observing the effects we describe.
We note that the constricted or blocked flow in our system resembles clogged granular states rather than jammed granular states. Granular jamming typically occurs when the density of granular particles reaches a unique jamming density known as point J [47]. In contrast, granular clogging is a cessation of granular flow due to a local buildup of high density regions, which can occur over a very wide range of global density [48]. Jammed systems are typically jammed for all directions of loading [47] while clogged systems have a memory of the loading direction and can become unclogged if this direction is changed. Our system exhibits clogging behavior, with strongly heterogeneous particle density in the clogged state and a strong sensitivity to the driving direction for the formation of the clogged state. The soft interactions between our particles mean that there is no well-defined unique jamming density of the type found for the hard sphere interactions of granular particles, and thus in the clogged state the local particle density producing the clog can vary much more than in a granular system. In granular or hard particle systems, reentrant clogging could occur; however, due to the strong contact interactions the system may remain in a reentrant pinned or clogged state even for very high values of FD, so that it could be difficult to observe the reentrant flowing state above Fc2.
Summary - In summary, we have investigated a system of repulsively interacting particles driven in the hard direction of an asymmetric funnel array for varied ratios of the number of particles to the number of funnel plaquettes. Previous work on this model focused on driving in the easy flow direction and showed only a single depinning threshold. In contrast, for driving in the hard direction we find that the system can exhibit a rich variety of dynamical phases including a reentrant pinned phase, where the system flows at lower drives but becomes pinned when the drive is increased. This reentrant pinning is the result of a clogging instability in which a buildup of particles in a small number of plaquettes block the flow of other particles. At higher drives, particles can force their way through the clog, leading to a second depinning transition. The clogging is a robust effect that occurs over the range of three to nearly eleven particles per plaquette. In addition to the reentrant pinning, we find that in some cases a partial clogging can produce negative differential conductivity, and we observe step-like features in the velocity-force curves due to the sequential breakup of a clog. We map the dynamic phases as a function of drive and filling fraction and identify a pinned phase where the particles are uniformly dense and do not move, a clogged phase where the particles do not move but are heterogeneously distributed with an accumulation of particles in certain plaquettes, and a moving phase. When a finite temperature is introduced, the clogging regime remains robust but exhibits intermittent bursts of flow when the clogs disintegrate under thermal fluctuations but then rapidly reform. Finally, we describe systems in which this clogging behavior could be observed, including superconducting vortices, colloids, Wigner crystals, or skyrmions in asymmetric channels.
Acknowledgments
This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.
ORCID iDs
C J O Reichhardt https://orcid.org/0000-0002-3487-5089

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