Physical Review E Rapid Communications 95, 030902(R) (2017)

Clogging and Jamming Transitions in Periodic Obstacle Arrays

H.T. Nguyen1,2, C. Reichhardt1, and C. J. Olson Reichhardt1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
2Department of Physics, University of South Florida, Tampa, Florida 33620, USA

(Received 9 December 2016; published 29 March 2017)

We numerically examine clogging transitions for bidisperse disks flowing through a two dimensional periodic obstacle array. We show that clogging is a probabilistic event that occurs through a transition from a homogeneous flowing state to a heterogeneous or phase separated jammed state where the disks form dense connected clusters. The probability for clogging to occur during a fixed time increases with increasing particle packing and obstacle number. For driving at different angles with respect to the symmetry direction of the obstacle array, we show that certain directions have a higher clogging susceptibility. It is also possible to have a size-specific clogging transition in which one disk size becomes completely immobile while the other disk size continues to flow.
I. INTRODUCTION
II. MODEL AND METHOD
III. RESULTS
IV. DIRECTIONAL DEPENDENCE AND SIZE-DEPENDENT CLOGGING
V. CONCLUSION
REFERENCES

I.  INTRODUCTION

A loose collection of particles such as grains or bubbles can exhibit a transition from a flowing liquidlike state to a non-flowing or jammed state as a function of increasing density, where the density ϕj at which the system jams is referred to as Point J [1,2,3]. Jamming has been extensively studied in bidisperse two-dimensional (2D) packings of frictionless disks, for which ϕj ≈ 0.844, and where the system density is uniform at the jamming transition [1,2,4]. Related to jamming is the phenomenon of clogging, as observed in the flow of grains [5,6,7,8] or bubbles [9] through an aperture at the tip of a hopper. The clogging transition is a probabilistic process in which, for a fixed grain size, the probability of a clogging event occurring during a fixed time interval increases with decreasing aperture size. A general question is whether a system can exhibit features of both jamming and clogging. For example, in a system containing quenched disorder such as pinning or obstacles, jammed or clogged configurations can be created by a combination of particles that are directly immobilized in a pinning site as well as other particles that are indirectly immobilized through contact with obstacles or pinned particles. In many systems where pinning effects arise, such as for superconducting vortices or charged particles, the particle-particle interactions are long range, and there is no well defined areal coverage density at which the system can be said to jam [10], so a more ideal system to study is an assembly of hard disks with strictly short range particle-particle interactions. Previous studies have described the effect of a random pinning landscape on transport in a 2D sample of bidisperse hard disks [11], while in other work on the effect of obstacles, the density at which jamming occurs decreases when the number of pinning sites or obstacles increases [12,13].
Here we examine a 2D system of bidisperse frictionless disks flowing through a square periodic obstacle array with lattice constant a composed of immobile disks. The total disk density ϕt is defined as the area coverage of the mobile disks and the obstacles. We find that for ϕt far below the obstacle-free jamming density ϕj, the system can reach clogged configurations by forming a phase separated state consisting of a high density connected cluster surrounded by empty regions, and that the clogging probability Pc during a fixed time interval depends on both a and ϕt. There is also a strong dependence of Pc on the angle θ between the driving direction and the x-axis symmetry direction of the obstacle lattice, with an increase in Pc for certain incommensurate angles. Over a range of θ values we observe a novel size-dependent clogging effect in which the smaller disks become completely jammed while a portion of the larger disks continue to flow. This work is relevant for filtration processes [14,15,16], the flow of discrete particles in porous media [17,18], and the flow and separation of of colloids on periodic substrates [19,20,21,22].

II.  MODEL AND METHOD

We consider a 2D square system of size L ×L where L = 60 with periodic boundary conditions in the x and y-directions. The sample contains Nl disks of diameter σl = 1.4 and Ns=Nl disks of diameter σs = 1.0, giving a size ratio of 1:1.4. This same size ratio was studied in previous works examining jamming in bidisperse obstacle-free disk packings, where jamming occurs at ϕj = 0.844 and is associated with a contact number of Z = 4.0 [1,2,3,4]. We place Np obstacles, modeled as immobile disks of diameter σs=1.0, in a square lattice with lattice constant a. The disks interact through a repulsive short range harmonic force, Finij = k(σij − |rij|)Θ(σij − |rij|)∧rij where σij=(σi+σj)/2 is the sum of the radii of disks i and j, rij=ri−rj, ∧rij=rij/|rij|, and Θ is the Heaviside step function. The spring constant is set to k=300 which is large enough to ensure that the overlap between disks for the largest driving force we consider remains small. To prepare the initial disk configuration, we alternately place small and large disks in the sample at randomly chosen available spaces that are not already occupied by disks or obstacles. Then we apply a constant driving force Fd to the mobile disks which could arise from a gravity or fluid induced flow. The dynamics for a given disk i at position ri is obtained by integrating the following overdamped equation of motion, appropriate for colloidal particles or for disks on a frictional surface for which inertial effects can be neglected:
η d ri
dt
 = N
∑
i ≠ j 
 Finij + Fd  .
(1)
Here N=Ns+Nl+Np is the total number of disks and the damping constant η determining the proportionality between the disk velocity and the forces acting on the disk is set to unity. The external driving force is given by Fd = Fd(cos(θ)∧x + sin(θ)∧y), where θ is the angle of the driving direction with respect to the positive x axis. We take Fd=0.025 but, provided Fd is sufficiently small, our results are not sensitive to the choice of Fd. In the absence of obstacles, all the disks move in the driving direction at a speed of Fd/η. The total disk density ϕt is the area fraction covered by the free disks and obstacles, ϕt = [1/4] π(Nl σ2l + (Ns+Np)σ2s)/L2. To quantify the clogging transition, we monitor the average velocity of the mobile disks along the x and y directions, 〈Vx,y 〉 = ( Ns+Nl)−1∑i = 1Ns+Nl vi·(∧x, ∧y), where vi is the velocity of disk i. To ensure that the system has reached a steady state, we run all simulations for 3 ×108 simulation time steps and average the values of 〈Vx 〉 and 〈Vy 〉 over 105 simulation time steps. Generally we find that clogging occurs within 1 ×107 simulation time steps. We define Pc to be the probability that the system will reach a clogged state with 〈Vx〉 = 0.0 after a total of 3×108 simulation time steps, and perform 100 realizations for each value of ϕt and a.
Fig1.png
Figure 1: (a) The average disk velocities 〈Vx〉, (b) fraction of disks in a cluster Cl, and (c) average contact number Z versus time in simulation time steps for 2D bidisperse disks moving through a square periodic obstacle array with total disk density ϕt=0.54, lattice constant a = 3.0, and constant external drive Fd=0.025 applied in the positive x-direction. We illustrate three cases: steady state flow (blue), full clogging (red), and partial clogging (green). (d) Distribution P(ϕ) of local disk density ϕ in 2 ×2 spatial regions in the initial state (blue) and after reaching a clogged state (red), averaged over 40 clogged realizations.

III.  RESULTS

We first consider the θ = 0 case with the external drive applied along the x direction. For a=3.0 we find that the clogging probability Pc = 1.0 for ϕt > 0.62, Pc ≈ 0 for ϕt < 0.52, and Pc=0.31 at ϕt=0.54. We illustrate three representative realizations of the ϕt=0.54 sample in Fig. 1(a,b,c) showing steady state flow, complete clogging, and partial clogging in which at least three-quarters of the disks are no longer moving. The plot of 〈Vx〉 versus time in Fig. 1(a) indicates that in realizations that reach a clogged state, the system does not pass instantly from a flowing to a non-flowing state, but instead exhibits a series of steps in which a progressively larger number of disks become clogged, with 〈Vx〉 continuing to diminish until it reaches zero. This behavior is different from that typically observed in hopper flows, where a single event brings the flow to a sudden and complete halt. The red curve in Fig. 1(a) contains time intervals during which the number of flowing grains, which is directly proportional to the value of 〈Vx〉, temporarily increases prior to the system reaching a final clogged state with 〈Vx〉 = 0 after 2.5 ×107 simulation time steps. Since there are no thermal fluctuations or external vibrations, once the system is completely clogged, all of the dynamical fluctuations disappear and the system is permanently absorbed into a clogged state. There can also be a steady flowing state in which the disks no longer undergo any collisions and remain unclogged. When collisions produce nonequilibrium fluctuations, it is possible that if we were to consider a longer time interval, some of the flowing or partially clogged states could fully clog. In Fig. 1(b) we plot the fraction Cl of mobile disks that are in the largest connected cluster versus time, while in Fig. 1(c) we show the corresponding average disk contact number Z. For the realization that fully clogs, Cl gradually increases with time, indicating that there is a single growing cluster, while Z also increases. When 〈Vx〉 reaches zero, Cl = 0.98, indicating that almost all the disks have formed a single cluster, while Z = 3.25, which is well below the critical value Zc = 4.0 expected at the obstacle-free jamming transition. In contrast, for the system that remains flowing, 〈Vx〉 = 0.025, indicating that almost all of the mobile grains are freely flowing. At the same time, Cl is close to zero and Z = 2.0 since the disks tend to form effectively one-dimensional chains. Changing the system size changes the time required to reach a clogged state, but the nature of the clogged state remains the same.
Fig2.png
Figure 2: Images of the obstacle locations (green circles) and the mobile disks (large disks: blue; small disks: orange) for the samples shown in Fig. 1 with a=3.0 and ϕt = 0.54. (a) Initial configuration of the sample that clogs. (b) Final clogged configuration of the same sample. (c) Late time configuration of the flowing sample. (d) Late time configuration of the partially clogged sample.
In Fig. 2(a) we show an image of the initial uniform density disk configuration for the system in Fig. 1(a) which reaches a clogged state, while we show the final 〈Vx〉 = 0 clogged state in Fig. 2(b). The mobile disks phase separate into a high density connected cluster surrounded by empty regions. In contrast, Fig. 2(c) shows a late time image of the sample from Fig. 1(a) that remains flowing. Here the overall disk density is uniform and the motion is confined in one-dimensional (1D) channels that run between the rows of obstacles. For the partially clogged sample at late times, Fig. 1(b) indicates that the cluster fraction Cl=0.84 is lower than the value Cl=0.98 observed in the fully clogged state, and Fig. 2(d) shows that a large jammed cluster forms, while in the middle of the sample there is a region of uniform disk density through which the grains flow in 1D channels. The partially clogged state thus combines features of the clogged and flowing states in Fig. 2(b,c).
In Fig. 1(d) we plot the distribution P(ϕ) of the local packing density ϕ at initial and late times for a sample that reaches a clogged state. To measure ϕ, we divide the sample into squares of size 2 ×2 and find the area fraction of each square covered by free disks and obstacles. In the initial state, there is a peak in P(ϕ) centered at the total disk density of ϕt=0.54. In contrast, in the clogged state P(ϕ) has multiple peaks centered at ϕ = 0 from empty regions, ϕ = 0.2 from the obstacle density, and ϕ = 0.82 from clogged regions, which have a density close to the free disk jamming density.
Fig3.png
Figure 3: (a) The clogging probability Pc vs ϕt for varied obstacle lattice constant a=2.5 (dark blue circles), 2.609 (light blue squares), 2.727 (light green diamonds), 2.857 (dark green up triangles), 3.0 (orange left triangles), 3.158 (red down triangles), and 3.33 (magenta right triangles). (b) The average value of Z for realizations that clog vs a increases monotonically. (c) Angles θs (red squares) and θl (blue diamonds) at which x-direction channeling is lost for the small and large disks, respectively, vs a. For driving angles falling within the green shaded region, size-dependent clogging can occur.
In Fig. 3(a) we plot the clogging probability Pc versus ϕt for samples with obstacle lattice constant ranging from a = 2.5 to a=3.33 obtained from 100 realizations for each value of ϕt. When a = 3.33, Pc = 0 for ϕt < 0.79, and there is a sharp increase to Pc = 1.0 at ϕt = 0.8, indicating that when the spacing between obstacles is large, a high density of mobile particles must be introduced in order for the system to clog. We define the critical density ϕtc as the value of ϕt at which Pc passes through Pc=0.5. As a decreases, ϕtc also decreases, and at a=2.5, ϕtc=0.49. For some values of a there are particular combinations of disk configurations that can better fit in the constraint of a square obstacle lattice, so ϕct does not decrease strictly monotonically with a. The combination of our finite sample size and the square symmetry of our obstacle lattice constrains us to a discrete set of values for a. When we average the contact number Z over only realizations that clog, we find a monotonic increase in Z with a, as shown in Fig. 3(b), where Z increases from Z=2.9 at a = 2.5 to Z=3.6 at a = 3.33. In principle, Z will approach the value Z=4.0 for very large values of a or in the limit of a single obstacle when ϕt=ϕj ≈ 0.84; however, the time required to reach clogged states at large a increases well beyond our simulation time window. We can compare the ratios Rcl=a/σl and Rcs=a/σs to the critical pore size Rc for hopper flow clogging identified in Ref. [23]. At a=3.33 we have Rcl=3.33 and Rcs=2.37, giving an average value of Rc=2.85, close to the value Rc=2.5 to 3.5 for vibrated hoppers in Ref. [23] and to the value Rc=3.0 for hopper flow in Ref. [24].
Fig4.png
Figure 4: (a) Pc vs θ, the driving direction angle, in samples with ϕt = 0.5272 and a = 2.857. Pc is enhanced for θ > 25°. (b) 〈Vx〉 vs time in simulation time steps for the large disks only (red), the small disks only (blue), and all disks (purple) for a driving angle of θ = 20°. We find a size dependence, with only the smaller disks becoming clogged while the large disks continue to flow. (c) The disk configuration in the clogged state at θ = 320 from panel (a). (d) The disk configuration for the size-dependent clogged state from panel (b).

IV.  DIRECTIONAL DEPENDENCE AND SIZE-DEPENDENT CLOGGING

We next consider the effect of changing the direction θ of the drive relative to the x axis symmetry direction of the square obstacle array. In Fig. 4(a) we plot Pc versus θ in samples with ϕt = 0.527 and a = 2.857. For each value of θ, we perform 100 realizations. Here, Pc = 0 for θ = 0, consistent with Fig. 3(a). As θ increases, a local maximum in Pc with Pc=0.3 appears near θ = 10. This is followed by a drop to Pc=0 over the range 15° < θ < 25°, and an increase to Pc=0.98 for 25° ≤ θ < 40°, with a dip to Pc=0.72 occurring near θ = 45°. Due the symmetry of the obstacle lattice, the same features repeat over the range 45° < θ < 90°. The increase of Pc near θ = 10° occurs due to a breakdown of the 1D channeling that arises for the θ = 0° flow. Similarly, the dip in Pc near θ = 45° appears when the disks follow 1D flow channels along the diagonal direction. Angles such as θ = 0° and θ = 45° allow 1D channeling motion, whereas for 25° < θ < 40° there is no easy flow direction so the disks are forced to collide with the obstacles, producing an increase in Pc. In Fig. 4(c) we illustrate a clogged state that is aligned with the driving angle of θ = 32°.
For 20° ≤ θ ≤ 24° we observe a size-dependent clogging behavior in which the smaller disks become completely clogged while a portion of the larger disks continue to flow. In Fig. 4(b) we plot 〈Vx〉 for the large and small disks separately and for all disks combined for a driving angle of θ = 20°. After 2×107 simulation time steps, 〈Vx〉 = 0 for the small disks, which clog completely, while the larger disks saturate to a steady state flow. This result is counter-intuitive since it might be expected that the larger disks would clog first. In Fig. 4(d) we show a snapshot of the size-dependent clogged state from Fig. 4(b). All of the smaller disks are jammed in a cluster along with a portion of the larger disks, while in the lower density regions there are a number of larger disks undergoing channeling motion along the x-direction. The size-dependent clogging can be understood as a consequence of a directional locking effect [19,20,21,22,25] in which the flow of the larger disks remains locked to the θ = 0° direction of the obstacle lattice while the flow of the smaller disks follows the angle of the drive, which increases the chance for the smaller disks to become clogged. According to a simple geometric argument [26], the driving angle θm at which a disk of size σm ceases to channel along the x direction between obstacles and begins to move in the driving direction is given by the real root of tan−1[[((cos(θm)−sin(θm)))/(2a/(σm+σs)−(sin(θm)+cos(θm)))]]−θm=0. At a=2.857 the solutions are θs=20.49° and θl=24.84° for the small and large disks respectively, giving a window of size-dependent clogging that agrees with the numerical results. The variation of this window with a appears as a shaded region in Fig. 3(c). The directional locking effect, in which particles preferentially move along lattice symmetry directions, has been observed for colloids [19,20,21,22] and superconducting vortices [25] moving over periodic substrates. It can be used to perform particle separation by having one species lock to a symmetry direction while the other does not. In our case, the disk size that does not undergo directional locking ends up in a clogged state, suggesting that species separation by selective clogging could be a new method for particle separation.

V.  CONCLUSION

We have investigated the clogging transition for a bidisperse assembly of frictionless disks moving through a two-dimensional square obstacle array. We find that the probability of clogging during a fixed time interval increases with increasing total disk density ϕt and decreases with the obstacle spacing a. For disk densities well below the obstacle-free jamming density, the clogged states are phase separated and consist of a connected high density jammed cluster surrounded by a low density disk-free region. In the clogged state, the contact number Z increases monotonically with decreasing obstacle density. We also find that the clogging probability has a strong dependence on the angle between the driving direction and the symmetry axes of the square obstacle array. The clogging is enhanced for incommensurate angles at which the 1D channeling flow of the disks between the obstacles is suppressed. For a window of drive angles, a size-dependent clogging effect arises in which the smaller disks become completely clogged while a portion of the larger disks remain mobile. Here the motion of the larger disks remains locked along the x-axis of the obstacle array whereas the smaller disks move in the driving direction. This suggests that selective clogging could be used as a particle separation method.

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. H.N. gratefully acknowledges support from NSF Grant No. DMR-1555242.

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