Crossover from Jamming to Clogging Behaviours in Heterogeneous Environments

Jamming describes a transition from a flowing or liquid state to a solid or rigid state in a loose assembly of particles such as grains or bubbles. In contrast, clogging describes the ceasing of the flow of particulate matter through a bottleneck. It is not clear how to distinguish jamming from clogging, nor is it known whether they are distinct phenomena or fundamentally the same. We examine an assembly of disks moving through a random obstacle array and identify a transition from clogging to jamming behavior as the disk density increases. The clogging transition has characteristics of an absorbing phase transition, with the disks evolving into a heterogeneous phase-separated clogged state after a critical diverging transient time. In contrast, jamming is a rapid process in which the disks form a homogeneous motionless packing, with a rigidity length scale that diverges as the jamming density is approached.

The concept of jamming is used in loose assemblies of particles such as grains or bubbles to describe the transition from an easily flowing fluidlike state to a rigid jammed or solidlike state [1][2][3][4] . Liu and Nagel proposed a generalized jamming phase diagram combining temperature, load, and density, where a particularly important point is the density φ j at which jamming occurs 1 . Jamming has been extensively studied in a variety of systems [3][4][5] , and there is evidence that in certain cases, the jamming transition has the properties of a critical point, such as a correlation length that diverges as the jamming density is approached [2][3][4][5][6][7][8][9] . A related phenomenon is the clogging that occurs for particles flowing through a hopper, where as a function of time there is a probability for arch structures to form that block the flow [10][11][12][13] . Clogging is associated with the motion of particulate matter past a physical constraint such as wells, barriers, obstacles, or bottlenecks [14][15][16][17][18][19] ; however, it has not been established whether jamming and clogging are two forms of the same phenomenon or whether there are key features that distinguish jamming from clogging.
Here we show for frictionless disks moving through a random obstacle array that jamming and clogging are distinct phenomena and that a transition from clogging to jamming occurs as a function of increasing disk density. We identify the number of obstacles required to stop the flow and the transient times needed to reach a stationary clogged or jammed state. There are two critical obstacle densities, φ c j for the jammed state and φ c c for the clogged state. In the jamming regime, the obstacle density φ c j at which flow ceases drops to lower obstacle densities with increasing disk density, and the system forms a homogeneous jammed state when the rigidity correlation length associated with φ j becomes larger than the average distance between obstacles. In contrast, during clogging the system organizes over time into a heterogeneous or phase-separated state, and the transient time diverges at a critical obstacle density φ c c that is independent of the disk density. The phase-separated state consists of regions with a density near φ j coexisting with low density regions.

Results
Time evolution to a jammed or clogged state. We numerically examine disks driven through a two-dimensional array of obstacles in the form of immobile disks. Simulation details appear in the Methods section. The total area density of the system is φ tot = φ m + φ obs , where φ m is the area density of the moving disks and φ obs is the area density of the obstacles. Starting from a uniformly dense sample, we apply a driving force and find that over time the system evolves either to a steady free flowing state or to a motionless clogged or jammed state. In Fig. 1a-c we illustrate the time evolution of a system with a disk density of φ m = 0.2186 and an obstacle density of φ obs = 0.175, beginning with the uniform density initial state in Fig. 1a. Upon application of a drive, we find a transient flowing state as shown in Fig. 1b which gradually evolves into the final motionless phase-separated or clustered clogged state in Fig. 1c. For a higher disk density of φ m = 0.436, Fig. 1d-f shows that the same evolution from uniform initial state to transient flowing state to static clogged state occurs, but the dense clusters in the clogged state are larger. At much higher disk densities of φ m = 0.785, we find jamming behavior when the obstacle density is larger than a critical value φ c j . Below φ c j , the system quickly settles into steady state flow, as shown in Fig. 2(a,b) for φ obs = 0.043 and in Fig. 2(c,d) for φ obs = 0.065. The magnitude of the flow decreases with increasing φ obs . Above φ c j the disks quickly form a disordered jammed state when driven, as illustrated in Fig. 2(e,f) for φ obs = 0.0872. In contrast to the density phase-separated clogged states that form at lower φ m , the jammed states are homogeneously dense. The randomly placed obstacles prevent the monodisperse disks from developing long-range crystalline order.
Velocity measurement of the transition from clogging to jamming. To characterize the system we perform a series of simulations with varied φ m and φ obs . We measure the final velocity V 0 of the mobile disks after a fixed time interval, and average over ten different realizations. In Fig. 3 we plot a velocity heat map as a function of φ obs versus φ m . We find a flowing regime at small φ obs , a clogged regime for φ m < 0.67, and a jammed regime for φ m > 0.67. The critical obstacle density φ c c above which the velocity V 0 drops to zero in the clogging regime remains roughly constant at φ ≈ . between obstacles, similar to the manner in which hopper clogging is controlled by the aperture size. In contrast, in the jamming regime the critical obstacle density φ c j separating flowing from jammed states decreases linearly with increasing φ m and reaches φ = 0 c j for φ m ≈ 0.9069, indicating that this transition is controlled by a growing correlation length ξ associated with the jamming point φ j 5 . We argue that the system jams when ξ = l obs . If we assume that near jamming in a clean system, the correlation length grows as ξ ∝ (φ j − φ m ) −ν , then the transition to the jammed state varies with obstacle den- Fig. 3, φ φ ∝ c j m , implying that ν = 1/2, consistent with the exponent ν = 1/2 proposed for jamming in refs. 20,21 , as well as with simulation measurements giving ν in the range 0.6 to 0.7 for two-dimensional bidisperse disks 6,8 . The exponent we find is also in agreement with that observed for the shift in the jamming point in bidisperse disks on random pinning arrays 22 . Studies of bidisperse disk jamming with dilute obstacles very near φ j also show that φ j decreases linearly with obstacle density, giving ν = 1/2 23 .
Previous simulations of bidisperse disks of radius R s = 0.5 and R l = 0.7 flowing through a periodic array of obstacles with radius R s = 0.5 showed that clogging is strongly enhanced when .  l 2 35 obs 24 . This is because in order for a pair of disks, one large and one small, to fit between two obstacles, the lattice constant a of the obstacle array must be at least large enough to accommodate the size of the obstacle itself plus the size of the two disks, a ≥ 2R s + 2R s + 2R l = 2.4. In our monodisperse disk system, the obstacles are placed randomly, but one can obtain an estimate of the l obs for the onset of clogging by considering the circular holes in the obstacle array 25 . If we construct a circle that just touches any three disks and/or obstacles, this circle is defined to be a hole when it does not overlap any disks. For a pair of disks to pass between two obstacles, the obstacle spacing must once again accommodate the size of the obstacle itself plus the size of the two disks, . This spacing can be achieved by placing the obstacles such that circular holes of size R d can form on all sides of the hole on average, giving an effective obstacle radius of 3R d and a critical obstacle density of φ π = =.     percolation 32 , and we find similar values of γ for φ m < 0.67 throughout the clogging regime. Directed percolation is often used to describe nonequilibrium absorbing phase transitions 32 , and in our case the steady state flow corresponds to a fluctuating state, while the clogged state is the non-fluctuating or absorbed state.
In the jamming regime the transient times are much shorter, as shown by the plot of V(t) in Fig. 4b for φ m = 0.785 at φ obs = 0.022 to 0.087. In Fig. 5c, τ versus φ obs in the range φ m = 0.785 to 0.872 has a value that is an average of 20 times smaller than in the clogging regime from Fig. 5a. The peak in τ shifts to lower φ obs with increasing φ m , reflecting the behavior of the critical jamming density φ c j . By fitting the curves in Fig. 5c to obs c j , as demonstrated in the inset of Fig. 5d for φ m = 0.872, we obtain γ ≈ −0.66, as shown in the plot of γ versus φ m in the main panel of Fig. 5d. This indicates that there is a pronounced difference in the dynamics of the jamming regime compared to the clogging regime.
In Fig. 6 we show a heat map of the transient time τ obtained by fitting The transient times become large near the crossover from flowing to clogging for φ m < 0.67, while in the jamming regime for φ m > 0.67, the transient times are strongly reduced. This provides further evidence that in the clogging regime it is necessary for the system to organize over time into a clogged state, gradually forming phase-separated regions of high and low density as illustrated in Fig. 1. In contrast, the jammed system has strong spatial correlations, and once the correlation length associated with φ j is larger than the distance l obs between defects, very few disk rearrangements are needed to bring the system into a stationary, nonflowing state.  In Fig. 7a we show the transition from the flowing to the clogged or jammed state as a function of φ obs versus φ m by identifying the points from Fig. 3 for which V = 0.01. Figure 7b shows the transient times τ along this transition line, and in Fig. 7c we plot the transient exponent γ. The dashed vertical line at φ m = 0.67 indicates a transition from clogging to jamming behavior, correlated with a change from γ ≈ −1.29 in the clogging regime to γ ≈ −0.66 in the jamming regime, as well as with a drop in φ obs and τ. The point φ m = 0.67 matches the density at which 2D continuum percolation of disks is expected to occur. We find a third value of γ for φ m < 0.07 in a density regime where the value of φ obs at which a clogged state appears decreases with decreasing φ m . This regime is dominated by the trapping of single disks rather than collective clogging dynamics.
Local disk densities in clogged and jammed states. The clogged and jammed systems can also be distinguished by examining the local disk density φ loc measured in areas 6R d × 6R d in size. In Fig. 8a we plot the local density distribution P(φ loc ) averaged over ten realizations of the final clogged state for a system with φ m = 0.5 and φ obs = 0.175. As shown in the inset of Fig. 8a, the disks phase separate into low density regions associated with the peak at φ loc = 0.1 and high density regions which produce a second peak at φ loc = 0.85. The local density of the dense regions is lower than the value of φ loc = 0.9069 for a dense ordered hexagonal disk arrangement due to the considerable disorder introduced in the packing by the randomly placed obstacles. In Fig. 8b, P(φ loc ) for a system with φ m = 0.8 and φ obs = 0.06 that reaches a jammed state has a single peak near φ loc = 0.9, reflecting the uniform disk density at jamming that is illustrated in the inset of Fig. 8(b).

Discussion
Our results suggest that clogging and jamming processes have different dynamics. Clogging in the presence of random obstacles has signatures of an absorbing transition falling in a directed percolation universality class, and its dynamics are controlled by the average spacing of the obstacles. In the jamming that occurs for higher φ tot , the dynamics are controlled by the growing correlation length associated with φ j , the jamming density of an obstacle-free system. These results show that jamming and clogging in obstacles are indeed different phenomena. Jamming is associated with an equilibrium critical point, the formation of a homogeneous rigid state, and short transient times to reach this state, while clogging is a nonequilibrium dynamical phenomenon in which the system evolves over an extended time into a strongly spatially heterogeneous state. Our results have implications for flow though heterogeneous media 33 , erosion 34 , depinning transitions in particle assemblies 35 , and active matter in disordered environments 36,37 . Experimentally our results could be tested using colloidal particles at low flow rates to reduce hydrodynamic effects. It would also be interesting to examine the effects of adding frictional contacts between the disks, since these can change the characteristics of the jamming transition 38,39 , or to replace the disks by elongated particles 40 or chains 41,42 .

Methods
Numerical simulation details. We conduct simulations of nonoverlapping disks and obstacles confined to a two-dimensional plane. The system size is L × L with L = 60, and we use periodic boundary conditions in both the x and y directions. We introduce N m mobile disks of radius R d = 0.5 along with N obs obstacles represented by disks of radius R d that are not allowed to move. The area coverage of the mobile disks is φ π = N R L /   . We apply a uniform driving force =F F x d d to all mobile disks, with F d = 0.5. Distances are measured in simulation units l 0 and forces are measured in simulation units f 0 so that k is in units of f 0 /l 0 and the unit of simulation time is t 0 = ηl 0 /f 0 . We initialize the system by placing N m + N obs disks of reduced radius in randomly chosen nonoverlapping positions, and then gradually expanding the radii to size R d while allowing all disks to move. This produces a randomized packing of homogeneous density with no internal tensions. We then randomly assign N obs of the disks to be obstacles, and apply an external driving force. After a fixed simulation time of 1 × 10 6 simulation time steps, we determine whether the system has reached a clogged or jammed state based on whether the average disk velocity V has dropped to zero.