Dynamic Phases, Clustering, and Chain Formation for Driven Disk Systems
in the Presence of Quenched Disorder
Y. Yang1,2, D. McDermott1,2,
C. J. Olson Reichhardt1, and C. Reichhardt1
1Theoretical Division, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545, USA 2Department of Physics, Wabash College, Crafordsville, Indiana
47933, USA
(Received 15 August 2016; published 10 April 2017)
We numerically examine the
dynamic phases and pattern formation of
two-dimensional monodisperse repulsive disks
driven over random quenched disorder.
We show that there is a series of distinct dynamic regimes as a function of increasing
drive, including a clogged or pile-up phase
near depinning, a homogeneous disordered flow state, and
a dynamically phase separated
regime consisting of
high density crystalline regions surrounded by a low density of
disordered disks. At the highest drives
the disks arrange into one-dimensional moving chains.
The phase separated regime has parallels with the phase separation observed in
active matter systems, but arises from a distinct mechanism consisting of the
combination of nonequilibrium fluctuations with density-dependent mobility.
We discuss the pronounced differences between this system and
previous studies of driven particles with longer range repulsive interactions
moving over random substrates,
such as superconducting vortices or electron crystals,
where dynamical phase separation and distinct
one-dimensional moving chains are not observed.
Our results should be generic to a broad class of systems in which the
particle-particle interactions are short ranged, such
as sterically interacting colloids or Yukawa particles with strong
screening driven over random pinning arrays,
superconducting vortices in the limit of small penetration depths,
or quasi-two-dimensional granular matter flowing over rough landscapes.
I. INTRODUCTION II. SIMULATION III. VARIED DISK DENSITY A. Intermediate disk densities B. Low disk density C. Transverse diffusion and topological order D. Dynamic phase diagram IV. VARIED PINNING DENSITY V. DISCUSSION VI. SUMMARY REFERENCES
A wide range of systems can be
effectively modeled as a collection of
repulsively interacting particles
that are coupled to a substrate that serves as quenched disorder,
and these systems
typically exhibit a transition from a pinned to a sliding state
under an applied external driving force [1].
Examples of such systems include vortices in
type-II superconductors [2,3,4,5,6], driven electron
or Wigner crystals [7,8,9],
skyrmions in chiral magnets [10,11], charge stabilized colloids [12,13,14],
and magnetically interacting colloidal systems [15,16].
The depinning transition
can either be elastic, where the particles keep their same neighbors, or plastic,
where the particles exchange neighbors and break apart [1,3].
In systems with intermediate or long range
repulsive particle-particle interactions, the ground state
is usually a defect-free triangular lattice.
When plastic depinning occurs, pinned and mobile particles coexist,
leading to
a proliferation of topological defects in the
lattice and producing highly disordered particle configurations during plastic flow
[1,2,3].
At higher drives
there can be a transition from the plastic flow state to
a moving anisotropic crystal [3,17,18] or
moving smectic state [19,20,21,22].
This transition is
associated with an increase in the ordering of the system and
produces a distinct change in the structure factor [20,21,22] and
the density of topological defects [20,22]
as well as cusps or dips in the transport curves and changes
in the fluctuation spectra [22,23,24].
Depending on the dimensionality and
anisotropy of the system, these dynamical transitions
can have continuous or first order characteristics [1,3,25].
In most of the systems where depinning and sliding dynamics
have been studied, the repulsive particle-particle interactions
are modeled as a smooth potential that is either long range, as in the case of
Coulomb or logarithmic interactions, or screened long range,
such as a Bessel function interaction for superconducting vortices or
a Yukawa interaction for colloidal systems.
There are many systems where the repulsive particle-particle interactions
are short range with sharp cutoffs,
such as sterically interacting colloids [26,27], emulsions [28],
micelles [29],
binary fluids [30], bubble rafts [31,32,33], granular matter
[34,35],
charged colloids under strong screening [36,37], and solid state systems
under certain conditions.
For most of these systems it should be possible to flow the particles over some type
of rough surface or landscape.
Systems with sharp repulsive interaction cutoffs, such as hard disks,
can exhibit very different behavior
from systems with long
range repulsion, such as a strong density dependence
of the response near a crystallization or
jamming transition [35,38].
Two-dimensional (2D) systems
with long
range repulsive interactions form an
ordered solid down to very low densities
since the particles are always within interaction range of each other,
whereas
hard disk systems form a crystalline solid only for
the density at which the particles can just touch each other,
which corresponds to a packing density or area coverage of ϕ = 0.9 for
2D monodisperse nonfrictional disk packings [35].
For densities below the crystallization density, the
hard disk system forms a disordered or
liquidlike state.
It is not clear whether a hard disk assembly driven
over random disorder would exhibit the same
types of dynamical transitions observed in
systems with longer range interactions such
as superconducting vortices, Wigner crystals, skyrmions, and charged colloids,
or whether it would simply form a moving disordered state at high drives.
Previous studies addressed
how pinning and obstacles affect the onset of the jamming transition in bidisperse
disk packs [39,40]; however, the driven dynamics for nonzero loading
above the jammed state have not been studied.
Although it may seem that hard disks driven
over quenched disorder would
simply exhibit the same general dynamics, such as dynamical reordering at high drives,
as repulsive particle systems with longer range interactions,
the question has surprisingly
not previously been addressed.
Here we examine an assembly of monodisperse
harmonically interacting repulsive disks
driven over a random array of pinning sites.
We focus on disk densities ϕ < 0.9,
below jamming or crystallization.
Despite the apparent simplicity of the model, we
find that this system exhibits dynamical phases distinct from those
observed in studies of
longer range repulsive particles driven over
random disorder.
When the number of pinning sites is
smaller than the number of disks,
the pinned phase is associated with a pile up or clogging phenomenon
in which the system breaks up into clumps or clusters,
with unpinned disks prevented from moving by interactions with
disks trapped at pinning sites.
As the drive is increased beyond depinning,
the system enters either a fluctuating uniform disordered state or
a phase separated cluster state consisting of
a low density gas of disks coexisting with high density clusters.
Within the clusters, the disks form a predominantly triangular lattice.
The phase separated states generally appear when the driving
force is close to the value of the maximum pinning force.
For even higher drives, the system can transition
into a collection of one-dimensional (1D) moving chains,
and the structure factor exhibits a strong smectic ordering signature.
We characterize the different phases and the transitions
between them using velocity-force curves,
the transverse root mean square displacements,
the structure factor, and the density of non-sixfold coordinated particles.
Dynamical phase separation does not normally occur
in systems with
longer range interactions
since the coexistence of a high density and a low density phase would have a
prohibitively large energy cost
due to the close spacing of the particles in the dense phase.
For the disk system, the energy cost of the
particle-particle interactions is zero until the disks
come into contact, which occurs only at the highest densities.
Similarly, strong 1D chain formation occurs when the disks can approach each
other very closely in the direction of the applied drive without overlapping.
It is known that 2D granular systems that undergo
inelastic collisions can
exhibit cluster instabilities [41,42]; however, in our
system there are no frictional contacts between the disks.
The density phase separated regime has parallels with
an active matter clustering effect, and arises
when the combination of disk-disk collisions and pinning produce
nonequilibrium transverse fluctuations of the disks
as well as a density-dependent mobility.
Studies of active matter systems with short range particle-particle repulsion
and density-dependent mobility show similar clustering behavior [43,44,45,46].
At higher drives for the disk system, we find that a uniform moving state
forms when the transverse diffusion is lost.
We also find that
at the higher drives, the disks align in nearly 1D chains in which
the disk spacing is nearly zero
in the longitudinal direction but is larger in the transverse direction.
Such
strong chaining does not occur in systems with longer range
repulsive interactions since the high particle density along the 1D chains would
impose a prohibitively high energy cost. Due to the short range interactions
in the disk system, the disks incur no energy penalty when they form 1D chains.
Our work suggests that dynamical phase
separation and chain formation
are general features of driven systems with short range
or hard disk particle-particle interactions moving over random disorder.
A specific system of this type that
could be realized experimentally is sterically interacting colloidal
assemblies moving over random disorder.
There are already several experiments examining colloidal
particles interacting with random pinning [13,14]
and periodic pinning [36,37,47,48],
and similar studies could be performed for sterically interacting colloids.
Other realizations could be achieved using
flowing bubble rafts [31,32,33] where steric interactions come into play
or flowing microemulsions [28,29] where again only short range interactions
arise.
Further examples include
magnetic bubble systems with weak dipolar interactions or skyrmion systems [11],
where at high densities the short range repulsive core interactions could dominate
over the longer range repulsive interactions.
In bulk superconducting vortex systems, the vortex-vortex
interactions have a Bessel function form [1,4,6] which decays exponentially for
length scales longer than the London penetration depth,
so that in certain limits such as at low magnetic fields in samples with very small penetration
depths, the vortices could exhibit dynamics similar to those we observe,
including the phase separated states.
Additionally, there are numerous
multiband superconductors in which the vortex interactions
are modified and the vortex dynamics is dominated
by only short range repulsive forces [49,50,51].
There are a wealth of studies of particle-like soft matter systems such
as micelles, binary fluids, soft solids, and active matter
systems which can be described as having short range steric interactions.
Another
class of such systems is assemblies of quasi-2D
granular matter flowing over random disorder; however,
in these systems additional effects such
as friction or inertia can also play a role.
The paper is organized as follows.
We provide a description of the model and the numerical simulations in Section II.
In Section III, we describe
the different dynamic phases
that arise for a fixed amount of quenched disorder
when the disk density is varied.
At intermediate disk densities,
subsection III A shows that there are three dynamic phases
with distinct structure factor signatures
that appear as the applied
driving force increases: pinned disordered flow, phase separated flow,
and a moving chain state.
In subsection III B
we discuss the low disk density limit
where quasi-1D chaining effects
are particularly pronounced.
We show in subsection III C
how to categorize the dynamic phases based on
the amount of transverse diffusion
and topological order.
A dynamic phase diagram as a function of
driving force and disk density appears
in subsection III D, and
we explain how the basic features of the phase diagram
can be understood
in terms of a drive-dependent dynamic shaking temperature
that induces clustering reminiscent of that
observed in active matter systems.
In Section IV we show the evolution of the dynamic phases
as a function of increasing disorder strength by first
fixing the number of pinning sites while increasing the pinning
force, and then fixing the pinning force while increasing the
number of pinning sites.
In Section V
we discuss our results in the context of
other systems that exhibit depinning phenomena,
and in Section VI we summarize our work.
We consider a 2D system with
an area of L2 with periodic boundary conditions in the x and y directions.
The sample contains Nd harmonically repulsive disks of radius Rd
as well as Np pinning sites that
are modeled as non-overlapping parabolic
potential traps which can exert a maximum pinning force of Fp on a disk.
The disk dynamics are governed by the following overdamped equation of motion:
η
dRi
dt
= Fdd + Fp + FD .
(1)
Here η is the damping constant
and Ri is the location of disk i.
The
disk-disk interaction force is
Fdd = ∑i ≠ jk(2Rd − |rij|)Θ(2Rd − |rij|) ∧rij,
where rij = Ri − Rj, ∧rij = rij/|rij|, the disk radius Rd = 0.5, and
the spring constant k = 50.
Distances are measured in simulation units
l0 and forces are measured in simulation units f0 so
that k is in units of f0/l0 and the unit of simulation time is τ = ηl0/f0.
The pinning force Fp
is modeled as arising from randomly placed parabolic attractive
wells with a pinning radius of rp = 0.5,
such that only a single disk can be trapped in a given pinning site at a time.
Fp is the maximum force exerted by the pinning site at the edge of the well.
The driving force FD=FD∧x is applied along the x direction,
and
for each driving force we allow
at least 1 ×106 simulation time steps
to elapse before taking measurements
to ensure that the flow has reached a steady state.
At each value of FD we measure
the average disk velocity
〈Vx〉 = Nd−1∑Ndi=1vi·∧x ,
where vi is the
instantaneous velocity of disk i.
The density ϕ of the system is characterized by the
packing fraction or the area covered by the disks,
ϕ = NdπR2d/L2, where L = 60
in dimensionless simulation length units.
In the absence of disorder, the disks form a polycrystalline state near
ϕ ≈ 0.85
and a triangular solid at ϕ ≈ 0.9.
A variation of this model was previously used to study the depinning and jamming
of bidisperse disks driven over random pinning; in that work, with a disk radii ratio
of 1:1.4, the jamming density in a pin free sample was
ϕj ≈ 0.845 [39].
The main scale determining our
choice of parameters is the ratio FD/Fp of the driving force to the pinning force.
When FD/Fp ≥ 1.0, all the disks are moving.
All of the general features of the dynamic phases
we observe are robust for varied parameters,
and changing Fp simply introduces a linear shift of the phase boundaries.
The disk-disk repulsion in our model
is harmonic in form,
and we choose a large spring constant k = 50.
For larger values of k, the results are unchanged;
however, we must use smaller simulation time steps in order to maintain the
numerical stability of our algorithm.
The harmonic disk interaction we consider has been used in
numerous previous studies
to mimic hard disks, particularly for jamming systems [35,38,39,40].
We note that the model we use
is for strictly overdamped systems, whereas in
granular matter the role of inertial effects and frictional contacts
between grains can be important. We have
tested various system sizes and find that our general results are robust.
One example of a soft matter system with a controllable substrate is
colloids interacting with optical traps,
so a possible experimental realization of our system consists of
sterically interacting colloids in the presence of an optical trap array
subjected to an external drive.
Sterically stabilized colloids with a hard disk
radius in the range of 2 to 5 μm can be captured by
optical traps of radius 2 to 5 μm
with an optical trapping force of 2.5 to 5.0 pN per trap, and
experimentally arrays of up to 700 such traps can be produced with
an intertrap spacing of 5 to 10 μm.
Such a system can be mapped to our simulation by taking l0=10 μm
and f0=5 pN.
This gives k=0.5 μN, a value consistent with what has been measured
experimentally [52].
The driving force can be produced by a fluid flow, but this could
induce additional hydrodynamic effects that are not included in our model.
The sample can be tilted in order to produce a gravitational driving force;
alternatively,
in many optical trap systems the traps themselves can be moved,
so an effective driving can be produced by translating the traps in order to induce
different dynamical phases.
Charged colloids with strong screening can be driven by an electric field.
Figure 1:
(a) The average disk velocity 〈Vx〉 vs driving force
FD/Fp for a system of harmonically interacting repulsive
disks in a sample with Fp = 1.0
and Np = 1440 at disk densities of
ϕ = 0.85 (red circles), 0.71 (orange squares), 0.61 (yellow diamonds),
0.55 (light green up triangles), 0.43 (medium green left triangles),
0.3 (dark green down triangles), 0.25 (blue right triangles), and 0.15 (purple stars).
(b) The corresponding d〈Vx〉/dFD vs FD/Fp
curves showing a peak near
FD/Fp=1.0.
Inset: The depinning threshold Fc vs ϕ, where
ϕ ≈ 0.3 corresponds to a 1:1 ratio of disks to pinning sites.
(c) The corresponding cluster size CL vs FD/Fp.
We first consider a fixed number of pinning sites
Np=1440 with Fp = 1.0 as we vary the disk density from
ϕ = 0.05 to ϕ = 0.85, giving a ratio of
pinning sites to disks ranging from
Np/Nd=6.159 to Np/Nd=0.37.
With these parameters, a disk density of ϕ = 0.31 corresponds
to a ratio of Np/Nd=1.0.
Figure 1(a) shows 〈Vx〉 versus
FD/Fp for different values of ϕ and
Fig. 1(b) shows the corresponding d〈Vx〉/dFD curves.
In the inset of Fig. 1(b) we plot
the depinning force Fc vs ϕ indicating that
Fc has a constant value of Fc ≈ Fp at low disk densities
Np/Nd > 5.0.
Here Fc is the value of FD at which disk motion first begins to occur.
In this density range,
almost every disk can be pinned directly by a pinning site,
so collective interactions between the disks do not play an important role in
the depinning process; instead, depinning occurs in the single particle limit and
the depinning threshold is determined only by the value of
Fp.
For Np/Nd < 1.0, some of the disks are
not trapped by pinning sites, and these untrapped disks exert a force on the
pinned disks
which lowers the depinning threshold, as shown in the inset of Fig. 1(a).
In Fig. 1(b),
for ϕ ≤ 0.55
there is a pronounced peak in
d〈Vx〉/dFD
near FD/Fp = 1.0. This corresponds to the maximum pinning force
from the substrate, so that for FD/Fp > 1.0 all the disks are moving.
For
Np/Nd > 0.8
or ϕ < 0.4, a large fraction of the disks
are located at pinning sites
and the collision rate is low, so that most of the disks do not become mobile
until FD/Fp > 1.0, producing the jump in 〈Vx〉
at depinning at the lower fillings.
For Np/Nd < 1.0, there are excess disks that cannot be trapped
directly by the pinning sites, and in principle these disks would be mobile
for arbitrarily low FD;
however, they can still be indirectly pinned
or blocked by disks that are located at the pinning sites, creating a local pile up
or clogging configuration [39].
Since these interstitial disks exert forces on the disks located at the
pinning sites, their presence reduces the depinning threshold
by more than a factor of 2.
For fillings
Np/Nd=1.0 to 0.571,
corresponding to 0.3 ≤ ϕ ≤ 0.55, some disks remain
pinned until FD ≥ Fp, producing
a weak peak in the d〈Vx〉/dFD curves
at FD/Fp=1.0. When ϕ is large enough, most of the disks are already
moving for FD/Fp < 1.0, and the peak feature is lost.
In Fig. 1(c) we plot the average value Cl
of the size of the largest cluster normalized by the number of disks
in the system as a function of FD/Fp.
To determine Cl, we use the cluster counting
algorithm of Luding and Herrmann [53].
For ϕ < 0.43, Cl is low and the largest clusters contain 10 or fewer disks.
For ϕ ≥ 0.43, there is an increase in the cluster size at low drives
due to a pile up effect in which unpinned disks
accumulate behind pinned disks.
For ϕ = 0.85, the system forms a large cluster and Cl = 1.0 for all FD.
At ϕ = 0.55, 0.61, and 0.71, there is a drop off in Cl
for FD/Fp > 1.05, 1.33, and 1.4, respectively,
indicating a decrease in the cluster size.
There is also a local maximum in Cl near FD/Fp = 1.0
at ϕ = 0.61.
Figure 2: (a) The disk positions (circles) for the system in
Fig. 1 at ϕ = 0.61 for FD/Fp = 0.3, showing
a clustering or pile up effect.
(b) The corresponding structure factor S(k) has a ringlike signature.
(c) The driven homogeneous phase
in the same system at FD/Fp = 0.7.
(d) The corresponding S(k) plot from (c).
In Fig. 1(c), for ϕ = 0.61 there is an initial increase in
Cl up to
Cl=0.95 at small but finite FD/Fp due to the pile up effect.
This is followed by a decrease in Cl to a local minimum near FD/Fp = 0.85,
and then by another increase to a
local maximum in the range 0.85 < FD/Fp < 1.4,
indicating a growth in the size of the largest cluster
near FD/Fp = 1.0.
In Fig. 2(a) we plot the disk configurations for the ϕ = 0.61 system at
FD/Fp = 0.3 where Cl = 0.95 showing large scale clustering.
An illustration of disk motion in the cluster state appears in [54].
Similar configurations appear at
FD/Fp = 0.3 for 0.43 < ϕ < 0.85.
In Fig. 2(b), the corresponding structure factor
S(k)=Nd−1|∑iNdexp(−ik ·ri)|2
of the disk configuration has
a ringlike feature indicative of a disordered system.
As the drive is increased beyond
the depinning transition, the clusters break apart and the
disk density becomes homogeneous,
as shown in Fig. 2(c) for FD/Fp = 0.7, where
a reduction in Cl has occurred.
The corresponding structure factor in Fig. 2(d)
still contains a ringlike feature but has excess weight in two peaks along kx = 0,
indicating the formation of some chainlike structures due to the
x-direction driving.
Figure 3: The disk positions (circles) for the system in Fig. 1
at ϕ = 0.61 for FD/Fp = 1.05, corresponding to the
local maximum in Cl in Fig. 1(c).
Here the system forms a density phase separated state.
(b) The corresponding S(k) plot contains
sixfold peaks due to the triangular ordering in the dense phase.
(c) The same system at FD/Fp = 2.0 where a moving chainlike state
forms. (d) The corresponding S(k) shows smectic ordering.
For 0.7 < FD/Fp < 1.4, the system forms a density phase separated state,
as illustrated in Fig. 3(a)
for FD/Fp = 1.05.
The motion of the disks in this state appears in [54].
Here
there is a high density region with ϕ ≈ 0.85 in which
the disks have triangular ordering coexisting with
a low density region where the disks are disordered.
The corresponding structure factor in Fig. 3(b) shows six
peaks due to the triangular ordering within the dense phase.
There is some smearing of the peaks along ky due to the tendency of the
crystallites in the dense phase to align with the driving direction.
For FD/Fp > 1.4, where Cl drops, the disks become more spread out and form
1D moving chains of the type shown in
Fig. 3(c) at FD/Fp = 2.0
and illustrated in a movie in [60].
The corresponding S(k) in Fig. 3(d) has strong smectic ordering.
In general,
for ϕ ≥ 0.43
we find a phase separation in the vicinity of FD/Fp ≈ 1 similar to that
shown in Fig. 3(a),
where the extent of the dense region grows with increasing
ϕ while the low density regions become smaller.
Figure 4:
(a) The disk positions (circles) for the system in Fig. 1 at ϕ = 0.3
for FD/Fp = 0.15, showing the formation of small clusters.
(b) The corresponding S(k) plot.
(c) The same system at FD/Fp = 0.6
in the moving phase where the disk density becomes homogeneous.
(d) The corresponding S(k) shows a diffuse or liquidlike pattern.
Figure 5:
(a) The disk positions (circles) for the system in
Fig. 1 at ϕ = 0.3 for FD/Fp = 1.05, where the disks form
chainlike patterns.
(b) The corresponding S(k) plot.
(c) The same system at FD/Fp = 2.0 in the moving phase
where the disks form a series of chains or stripes. (d) The corresponding S(k)
has smectic ordering.
(e) Blow up of disk positions from panel (c) showing formation of 1D chains.
For ϕ < 0.43, the clumps that form near depinning are small, as illustrated
in Fig. 4(a) at ϕ = 0.3 and FD/Fp = 0.15.
The clumps are anisotropic and
show some alignment along the y-direction,
while the corresponding structure factor in Fig. 4(b)
has a ringlike signature.
At higher drives above depinning when some of the disks are moving,
the disk density is more homogeneous, as
shown in Fig. 4(c) at FD/Fp = 0.6.
The corresponding S(k) plot
in Fig. 4(d) has a more diffuse structure.
Near FD/Fp = 1.0, most of the disks
are in motion and form chainlike structures, as illustrated in
Fig. 5(a,b) and in [59] for FD/Fp = 1.05.
The disk density is not uniform, with some chains closer together and others
further apart; however, the denser regions are still too sparse to form sections
of triangular lattice of the type that appear at ϕ = 0.61 in Fig. 3(a).
As FD increases for the ϕ = 0.3 sample, the
moving chains of disks become better defined, as shown
in Fig. 5(c) and in [60] at FD/Fp = 2.0.
The interchain spacing becomes small enough that the disks in neighboring
chains are almost touching,
and the corresponding
structure factor in Fig. 5(d) shows strong smearing along the ky direction.
These results indicate that even though ϕ is below the close-packed density of
ϕ = 0.9,
different dynamic phases can arise and there can be
transitions into states with smectic ordering,
similar to the smectic states observed for driven superconducting vortices
[9,20,21,22,24].
In general, the 1D channeling effect illustrated in Fig. 5(c)
is much more pronounced in the disk system
than in systems with longer range interactions.
The moving disks are unstable against the formation of chainlike structures due to
a velocity collapse phenomenon. If one moving disk slows down, the disk immediately
behind it can run into it and cause it to speed up again, but once the two disks move
beyond their steric interaction range, there are no particle-particle interactions to push them
further apart, so the disks tend to pile up behind each other in the longitudinal
direction.
For ϕ = 0.85, the system forms a dense cluster with polycrystalline
triangular ordering, and for FD/Fp > 1.0 the disks
form a single triangular domain that is aligned with the driving direction.
We find two specific phenomena that differ from what is observed in systems
of externally driven particles with
longer range interactions.
These are:
(1) a density phase separation,
where high and low density phases coexist as shown in Fig. 3(a), and
(2) the formation of 1D chains as illustrated in Fig. 3(c)
and Fig. 5(c).
The density phase separation generally occurs
in the range 0.9 < FD/Fp < 1.2, just above the drive at which all the disks
become mobile,
while the 1D chains appear for FD/Fp > 1.2 when all the disks are rapidly
flowing.
In Fig. 5(e) we show a blow up of moving 1D chains from the system in
Fig. 5(c,d) indicating that the chains form in the longitudinal direction,
and that in this direction the disks are almost touching
to give a density along the length of the chain close to ϕ = 0.9.
Although clustering has been observed in active matter systems [44,45,46],
such 1D chaining does not occur for active disk systems, and results from a combination
of the x direction driving force and the
highly anisotropic fluctuations of the moving disks.
Figure 6:
The transverse displacements 〈δy2〉
obtained after 4×106 simulation time steps (red squares)
and the diffusive exponent α (blue circles) vs FD/Fp
for the system in Fig. 1 at ϕ =
(a) 0.25, (b) 0.3, (c) 0.43, (d) 0.55, (e) 0.61, and (f) 0.71.
We can characterize the different phases by measuring the particle displacements in the
direction transverse to the applied drive,
〈δy2〉 = Nd−1∑i=1Nd(yi(t) − yi(t0))2,
for varied FD/Fp.
In general we find
〈δy2〉 ∝ tα at long times.
In the disordered homogeneous density regimes, α = 1.0,
indicative of diffusive behavior, while
α < 1.0 just above depinning and in the moving chain state.
In Fig. 6 we plot the value of 〈δy2〉 obtained at a
fixed time of 5×106 simulation time steps versus FD/Fp along with
the corresponding
value of α for the system in Fig. 1
at ϕ = 0.25, 0.3, 0.43, 0.55, 0.61, and 0.71.
For ϕ = 0.25 and ϕ = 0.3 in Fig. 6(a,b),
there is a peak in 〈δy2〉 near FD/Fp = 1.0, where
α ≈ 1.0, indicating diffusive behavior.
The maximum amount of transverse diffusion falls at the same
value of FD/Fp as the peak in
d〈Vx〉/dFD shown in Fig. 1(b).
At low drives where the system forms a clogged state,
the transverse diffusion is suppressed.
At higher drives where the disks form 1D channels,
the diffusion in the direction transverse to the drive is
strongly suppressed and α→ 0,
indicating that the 1D channels
are frozen in the transverse direction.
For ϕ = 0.43, 0.55, and 0.61 in Fig. 6(c,d,e),
〈δy2〉 has a
double peak feature.
The first peak corresponds to the onset of the
homogeneous moving phase,
while the second peak occurs when
the system starts to undergo phase separation.
For ϕ = 0.61, where the strongest phase separation is observed,
there is even a region of drive for which
〈δy2〉 exhibits superdiffusive
behavior with α > 1.0.
At longer times the behavior transitions to regular diffusion.
For higher drives, both 〈δy2〉 and α decrease
with increasing drive
as the system forms a moving chain state.
For ϕ = 0.71 in Fig. 6(f), the double peak feature begins to
disappear.
Numerical studies of vortices in type-II superconductors [24,55]
show that the vortices exhibit strong transverse diffusion above the depinning
transition,
while at higher drives where a moving smectic state appears,
the transverse diffusion is strongly suppressed and the
system freezes in the transverse direction.
The vortex system typically has only
a single peak in 〈δy2〉
rather than the double peaks we observe here.
The regime
of superdiffusive behavior for ϕ = 0.61
arises due to collective transverse motion of the disks in the dense phase.
Another measure often used to characterize
interacting particles driven over disorder is the fraction
P6 of sixfold coordinated particles.
Here P6=Nd−1∑iNdδ(zi−6), where zi is the coordination
number of disk i obtained from a Voronoi tessellation.
In the case of superconducting vortices in the absence of
pinning, the ground state is a triangular lattice with P6 = 1.0,
while when
strong disorder is present, the pinned state
is disordered and contains numerous topological defects so that P6 < 1.0.
At high drives, where the effect of pinning is reduced, the
system can dynamically reorder into a moving triangular lattice
with P6 = 1.0 or into a moving smectic where
some topological defects persist that are aligned with the direction of drive,
giving P6 <~1
[3,4,17,20,21,22,24].
Figure 7: The fraction P6 of sixfold coordinated disks vs FD/Fp for the system in
Fig. 1 for ϕ = (a) 0.25, (b) 0.3, (c) 0.43, (d) 0.61,
(e) 0.71, and (f) 0.85.
For ϕ = 0.61 in panel (d), the local maximum in P6
near FD = 1.0 is correlated with the formation of the phase separated
state shown in Fig. 3(a).
In Fig. 7 we plot P6
versus FD/Fp
for the system in Fig. 1 at ϕ = 0.25, 0.3, 0.43, 0.61, 0.71, and 0.85.
Although there are several similarities to the behavior of P6 observed for
superconducting vortices,
there are a number of notable differences.
For ϕ = 0.25 and ϕ = 0.3 in Fig. 7(a,b), there is an
increase in P6 above FD/Fp=1.0 which corresponds to the formation of the
moving chain state illustrated in Fig. 5(a), followed by a saturation of
P6 at higher drives
to P6 = 0.55.
This is in marked contrast to the behavior observed in the vortex system,
where P6 saturates to a value much closer to P6=1.0
due to the longer range particle-particle repulsion
which favors the formation of a triangular vortex lattice
down to quite low vortex densities.
At ϕ = 0.43 in Fig. 7(c),
P6 shows a similar trend as in the systems with lower disk densities;
however, P6 saturates
to a higher value of P6=0.68.
In Fig. 7(d) at
ϕ = 0.61, there is a local maximum in P6 for
0.9 < FD/Fp < 1.4 that coincides with the density phase separated regime.
The disks in the dense phase have mostly triangular ordering,
as shown in Fig. 3(a,b).
For higher drives of FD/Fp > 1.4,
where the disks become more spread out, P6 drops again.
At ϕ = 0.71 in Fig. 7(e),
for low drives P6 ≈ 0.55, and then P6 gradually increases with increasing
drive up to a value of P6=0.9,
indicating that most of the sample has developed triangular ordering.
Finally, for ϕ = 0.85 in Fig. 7(f), at the lowest drives the system forms
a polycrystalline solid containing a small number of defects, so that the initial
value of P6 ≈ 0.81, while as
FD increases,
the polycrystal anneals into a single domain crystal that is aligned in the direction of drive,
with P6 = 0.99, indicating almost complete triangular ordering.
For 0.3 < ϕ < 0.85, the P6 curves in Fig. 7 show
a small peak
near FD/Fp = 0.2 due to the pile up or clustering effect.
Within the clusters the local density ϕloc
is ϕloc ≈ 0.85, producing increased sixfold ordering and a
corresponding increase in P6.
Once the drive is large enough to break apart these clusters,
there is a drop in P6 as the system enters the
homogeneous moving phase.
Figure 8: Schematic phase diagram as a function of FD/Fp vs
ϕ for the system in Fig. 1.
I: Pinned or clogged state.
II: Homogeneous plastic flow.
III: Density phase separated state.
IV: Moving smectic or moving chain state.
V: Moving polycrystalline state.
VI: Moving crystal state.
From the features in the velocity-force curves, P6, 〈δy2〉,
and the disk configurations,
we can construct a schematic phase diagram of the evolution of the different phases,
as shown in Fig. 8.
Phase I corresponds to the pinned or clogged state,
phase II is homogeneous disordered plastic flow,
phase III is the density phase separated state,
phase IV is the moving smectic or moving chain state,
phase V is the moving polycrystalline state, and
phase VI is the moving single domain crystal state.
Many of the features in the phase diagram can be understood with force balance arguments.
The depinning line separating phase I from phase IV
for ϕ < 0.2 falls at the constant value of FD/Fp=1.
At these low disk densities
the disks are pinned individually, so the depinning threshold is determined
only by the value of Fp.
For ϕ > 0.2, not all of the disks are directly captured by pinning sites; instead,
some disks are unable to find an empty pinning site and move through the system
as interstitials, pinned only through their interaction with directly pinned disks.
The interstitials can flow plastically, so phase II exists only when interstitials are
present.
Interstitials emerge once a percolating fraction
pf ∼ 0.67 of the pinning sites are filled, so if we write
ϕequiv=0.314 as the density of disks that we would have if every pinning site were
filled with exactly one disk, we expect the onset of phase II flow to occur
for ϕ >~pfϕequiv = 0.21.
For ϕ > 0.2,
due to the pairwise disk interactions
the depinning threshold gradually becomes dominated
by the driving force at which an unpinned
disk can depin a pinned disk with which it is in contact, reducing the
depinning threshold from Fc=Fp to Fc=Fp/2.
In Fig. 8, the
depinning line separating phases I and II gradually decreases from
FD/Fp=1 at ϕ = 0.2 to FD/Fp=0.5 at ϕ = 0.3.
As ϕ increases further,
three or more disks can come into contact and the depinning force falls off as
Fc=Fp/(Navg+1) where Navg is the average number of unpinned
disks in force contact with a pinned disk. Since Navg increases with disk
density we expect Fc ∼ Fp/ϕ, consistent with the decrease in the depinning
line marking the end of phase I for ϕ > 0.3 in Fig. 8.
Phase II in Fig. 8 consists of a combination of pinned and moving disks.
Since all of the disks depin for FD/Fp ≥ 1.0, the upper boundary
of phase II should be close to FD/Fp=1, as we observe. Similarly to the depinning
line, the upper boundary of phase II gradually decreases with increasing ϕ as
multiple disk interactions, which tend to depin the pinned disks, become more important.
For ϕ >~0.77 the system is so dense that
the depinning becomes elastic,
as observed in earlier studies of depinning for binary disk systems, so that phase
II disappears and is replaced by phase V, which again extends up to a maximum drive
of FD/Fp=1. The ϕ ≈ 0.77
line separating the high density phases V and VI from the lower
density phases is a type of random close packing (RCP), but it falls below the clean
system RCP value of ϕ = 0.82[56,57] due to the
presence of the quenched disorder.
Phase III, the phase separated state, in Fig. 8
occurs when two conditions are met:
(1) all the disks are moving,
and (2) the dynamical fluctuations are strong enough that
the moving disks have a component of their root mean square (RMS) motion in the direction
transverse to the drive that is ballistic over a sufficiently long time interval to permit
noticeable transverse grain motion to occur.
Phase III only occurs for ϕ > 0.2, in agreement with previous work [19],
and consistent with the observation in
active matter that cluster formation occurs only for sufficiently high density and activity.
For FD/Fp < 1.0 there are still pinned disks present that can interfere with
the phase separation and make the density more uniform,
so the lower boundary of phase III falls near
FD/Fp ≈ 1.0.
The transverse RMS motion of the disks decreases with increasing FD since
the magnitude of the fluctuations
δy induced by the pinning sites diminishes as the disks travel
faster, δy ∝ 1/FD, similar to the effective temperature found in
superconducting vortex systems.
Once these transverse fluctuations become small enough,
the clustering is lost.
This has similarities to the loss of clustering
in active matter systems as the run length is reduced [45,46]
or the active diffusion is reduced [58,59], but the origin of the
fluctuations in the disk and the active matter systems is quite different.
When the drive is high enough and the transverse displacements
〈δy2〉 are smaller than the longitudinal fluctuations, the system
enters the moving chain state marked phase IV in Fig. 8.
This phase is analogous to the moving smectic state found for vortices driven over random
disorder [19,20,21,22,24];
however, unlike the vortices, the disks
can form chains that have an almost close-packed density of ϕ = 0.9
along their length while still experiencing zero overlap energy
as long as the disks are not touching.
In contrast, for a system with longer range interactions
of 1/r, e−κr/r, or
Bessel function form,
due to the energy divergence at small r such
extreme chaining would be very energetically costly
and hence would be unstable.
As FD further increases, the longitudinal fluctuations δx also decrease
in magnitude; however, once the system
has entered the chain state, the chains can persist up to arbitrarily high drives.
At densities ϕ > 0.77, above the RCP transition to
elastic pinning, Fig. 8 indicates that for high drives the
system forms the moving solid state marked phase VI.
Figure 9:
(b) Schematic plot of
the effective shaking temperature or activity A vs FD/Fp
for a sample with ϕ = 0.3.
The vertical dashed black line at FD/Fp=0.5 separates the pinned
phase (I) from homogeneous plastic flow (II).
At FD/Fp = 1.0
all the particles begin to move, producing
an effective shaking temperature or activity
with an amplitude A that decreases
as 1/FD.
There is a critical shaking activity, Ac (horizontal red line),
above which clustering can begin to occur,
so that a phase separated state (III)
appears above FD/Fp = 1.0.
Phase III disappears above
the value of FD/Fp marked by a vertical solid green line, which is
determined by the point at which A drops below Ac.
At high drives where A < Ac, Phase IV, the moving smectic state,
appears as the externally applied driving force begins to dominate
the behavior of the system and the effective shaking temperature
becomes unimportant.
To highlight the role of the effective shaking temperature or activity in
inducing dynamic phase changes,
in Fig. 9
we plot a schematic of A,
the amplitude of the effective shaking temperature or the effective activity,
versus FD/Fp at ϕ = 0.3.
The I-II transition occurs
when the force FDnet acting on all n interstitial disks
in contact with a pinned disk as well as on the pinned disk itself exceeds the
pinning force Fp.
At ϕ = 0.3, as shown in
Fig. 9, there is an average of n=1 interstitial
disk in contact with each pinned disk, so FDnet=(n+1)FD=2FD
and depinning occurs at FD/Fp=0.5; in contrast,
for ϕ < 0.2, n=0 and depinning occurs at FD/Fp=1.0.
The activity A in the pinned phase I is A=0.
For 0.5 < FD/Fp < 1.0,
the system is in phase II and contains both pinned and moving disks.
In this case the shaking activity is bimodal and the
fluctuations are strongly non-Gaussian,
so A
is not well defined and we indicate
its value as A=0.
For Fd/Fp > 1.0 all the particles are moving,
so A is well defined and has its highest value
at FD/Fp = 1.0 before
decreasing according to A ∼ 1/FD.
We can define a disk density-dependent critical activity level, Ac,
needed for clustering to occur.
As long as A > Ac, the system remains in phase III,
but when A drops below Ac,
clustering is lost and the system transitions into phase IV.
Figure 10: (a) 〈Vx〉 vs FD/Fp at ϕ = 0.55
and Fp = 1.0 for
Np/Nd = 0.0, 0.072, 0.144, 0.216, 0.288, 0.36, 0.432, 0.504,
and 0.576, from top to bottom.
(b) The corresponding
d〈Vx〉/dFD vs FD/Fp curves showing peaks near
FD/Fp=0.5 and FD/Fp=1.0.
Figure 11: Cluster size Cl vs FD/Fp for the system in Fig. 10 at
Np/Nd = (a) 0.072, (b) 0.216, (c) 0.288, and (d) 0.432. The local peaks
in (b) and (c) correspond to the formation of a density phase separated state.
In panel (c) the lettering indicates the FD/Fp values represented in the
real space images in Fig. 12.
We next consider the case of a fixed disk density of ϕ = 0.55, corresponding to
Nd = 2500, and vary the number of pinning sites to give a ratio
of Np/Nd ranging from
Np/Nd = 0 to Np/Nd=0.576.
In Fig. 10(a,b) we show
〈Vx〉 and d〈Vx〉/dFD versus FD/Fp
for a sample with Fp = 1.0.
There is one peak in d〈Vx〉/dFD
near FD/Fp = 1.0, the drive above which all of the disks are moving,
and a second peak near FD/Fp = 0.5,
the drive at which the clogged state breaks apart.
We observe a similar set of dynamical phases
as those described in Section III, but find that
the density phase separated state is more prominent at lower pinning density,
as shown in the plots of Cl versus FD/Fp
in Fig. 11 for Np/Nd = 0.072, 0.216, 0.288, and 0.432.
In particular,
Np/Nd = 0.216 in Fig. 11(b) and
Np/Nd=0.288 in Fig. 11(c) exhibit strong
peak features associated with the density phase separated state.
The double peak feature in the d〈Vx〉/dFD curves
is generally absent for 2D studies of particles with longer range repulsion driven over
disorder,
where typically only one peak is observed, and is thus a unique feature of the 2D disk
system.
In addition, for
particles with longer range interactions,
measures of P6 and hence Cl generally show only monotonic
behavior above depinning, in contrast to the disk system,
which shows a clear nonmonotonic behavior with a second
peak near FD/Fp = 1.0.
Figure 12: The disk positions for the system in
Figs. 10 and 11 at
Np/Nd = 0.288 for drive values marked with letters in Fig. 11(c).
In panels (a) and (d),
red disks are part of clusters containing three or more disks, while blue disks
are isolated or in a cluster containing only two disks.
(a) The pinned cluster state at FD/Fp = 0.05.
(b) At FD/Fp = 0.3 the moving disks form more spread out clusters.
(c) At FD/Fp = 0.6, corresponding to the local minimum of Cl
in Fig. 11(c), a homogeneous disordered state forms.
(d) At FD/Fp = 1.05, corresponding to the peak in Cl in Fig. 11(c),
a density phase separated state forms.
At (e) FD/Fp = 1.5 and (f) FD/Fp = 2.0, the disks are in a moving chain state.
To show more clearly the evolution of the cluster state,
in Fig. 12 we illustrate the disk positions for the
system at Np/Nd = 0.288 for increasing FD.
The letters a through f in Fig. 11(c)
indicate the values of FD/Fp that match these images.
In Fig. 12(a) at FD/Fp = 0.05, where Cl = 0.85,
the system forms a clogged state.
Within the cluster regions, which are colored red,
the disk density is close to ϕ = 0.85,
and these clusters are separated by low density regions of disks.
As the drive increases, the
large cluster becomes more spread out,
as shown in Fig. 12(b) for FD/Fp = 0.3, where Cl drops to
Cl=0.78.
At FD/Fp = 0.6 in Fig. 12(c), which corresponds to a local
minimum in Cl in Fig. 11(c), the disks
are completely spread out and form a homogeneous disordered phase.
In Fig. 12(d) at FD/Fp = 1.05, which corresponds to a local
maximum in Cl in Fig. 11(c),
a density phase separated state appears.
Disks that are in a cluster containing at least three disks are colored red in order to more
clearly highlight the dense region, within which the disks have developed triangular
ordering.
As the drive is further increased, the disks
spread apart in the direction transverse to the drive to form
the moving chain state
illustrated in Fig. 12(e,f)
at FD/Fp = 1.5 and FD/Fp=2.0, respectively,
which also coincides with
a reduction of Cl in Fig. 11(c).
For Np/Nd = 0.55 and above, the density phase
separated state becomes less well defined,
as indicated in Fig. 11(d) at Np/Nd = 0.432.
Figure 13: The disk positions for a system with
ϕ = 0.55 at Np/Nd = 0.072, where there is no peak in
Cl in Fig. 11(a).
Red disks are part of clusters containing three or more disks, while blue disks
are isolated or in a cluster containing only two disks.
(a) A density phase separated state at FD/Fp = 0.3.
(b) A moving chain state forms at higher drives, shown here at
FD/Fp = 1.5.
Figure 14: (a) Schematic phase diagram
as a function of FD/Fp vs Np/Nd for the system in
Fig. 10 at fixed ϕ = 0.55.
I: Pinned or clogged state.
II: Homogeneous plastic flow.
III: Density phase separated state.
IV: Moving smectic or moving chain state.
(b) Phase diagram for the same system at ϕ = 0.55 and Np/Nd=0.288
as a function of FD vs Fp.
In Fig. 11(a) at Np/Nd=0.072,
although Cl does not show a peak near FD/Fp = 1.0,
there is still
a pronounced density phase separated state;
however, this phase has shifted to lower FD/Fp.
Since the low density clogged state transitions directly into the
flowing density phase separated state,
there is no dip in Cl.
The density phase separated state breaks apart at lower
values of FD/Fp compared to samples with higher values of Np/Nd.
In Fig. 13(a) we show the disk
configurations at Np/Nd = 0.072 and FD/Fp = 0.3 where
a density phase separated state appears,
while in Fig. 13(b) we illustrate the moving chain phase that forms
at FD/Fp=1.5 in the same system.
From the images we can construct a schematic phase diagram for the
ϕ = 0.55 sample as a function of
FD/Fp versus Np/Nd, as shown in Fig. 14(a), which
highlights the extents of regions I through IV.
Here, the widths of regions I and II
grow with increasing Np/Nd, while
region III reaches its largest extent near Np/Nd = 0.3.
We note that for Np/Nd = 0, the system forms a moving disordered state
for all FD > 0.
In the dynamic phase diagram of Fig. 14(a), the depinning transition
marking the upper bound of phase I increases linearly with increasing Np, a
behavior similar to that observed in other systems, such as superconducting
vortices, that exhibit plastic depinning [1].
The upper boundary of phase III varies non-monotonically with Np
due to the behavior of the fluctuations. Phase III arises due to the
transverse fluctuations produced by a combination of interactions with the pinning sites
and disk-disk collisions.
When Np is small, there are
not enough pinning sites to create strong nonequilibrium fluctuations, so the extent of
phase III decreases with decreasing Np.
At high Np, the situation is similar to that in the phase diagram of Fig. 8 at
high ϕ, where fluctuations in the disk motion are larger in the longitudinal or x
direction than in the transverse or y direction, and as a result chain-like structures
are destabilized and the width of phase III decreases with increasing Np.
Figure 15:
(a) The cluster size Cl vs FD for
samples with ϕ = 0.55 and Np/Nd = 0.288
at
Fp = 0.0 (blue circles), 0.2 (blue squares),
0.4 (green diamonds), 0.6 (orange triangles),
and 1.0 (red circles).
(b) 〈Vx〉 vs
FD − Fc for
the velocity-force curve obtained at ϕ = 0.55 and Np/ND = 0.576. The solid
line is a power law fit with an exponent of β = 1.6.
We have considered varying Fp while holding
ϕ and Np/Nd fixed,
and find that the same general phases appear.
In Fig. 14(b), the upper boundary of phase I increases
linearly with increasing Fp, as expected for a depinning transition.
The line separating phases II and III marks the point at which
all of the disks are moving, and this line also increases linearly with Fp.
The line separating phases III and IV appears at the point when the transverse fluctuations
become too small to permit density phase separation to occur, and since the
fluctuations are affected by the pinning strength, this line also increases linearly with
Fp.
In Fig. 15(a) we plot Cl versus
FD in a sample with ϕ = 0.55 and Np/Nd = 0.288 for
Fp = 0.0, 0.2, 0.4, 0.6, and 1.0 to show the evolution of the
second peak, which both increases in width and shifts to higher values
of FD as Fp increases.
For depinning in systems with longer range interactions,
such as superconducting vortices, colloidal particles, and electron crystals,
scaling near the depinning threshold is observed
in the velocity-force curves,
which have the form V ∝ (FD − Fc)−β.
In plastic depinning, where particles exchange neighbors as they move,
β > 1.0,
while for elastic depinning, in which the particles maintain the same neighbors as
they move, β < 1.0 [1].
In systems
with long range Coulomb interactions [8] and
screened Coulomb interactions [12,60], plastic depinning is
associated with exponents of
β ≈ 1.65 and β ≈ 2.0, respectively.
More recently, simulations of depinning of
superconducting vortices with a Bessel function
vortex-vortex interaction give β = 1.3 [61].
Due to the observed variations in these exponents,
it is not clear that existing simulations of plastic depinning
are large enough to accurately obtain the true scaling since
it is expected that
a critical phenomenon would be associated with a unique exponent.
It is interesting to ask whether
similar scaling of the velocity-force curves occurs in the disk system.
In Fig. 15(b) we plot 〈Vx〉 versus
FD − Fc on a log-log scale for
a sample with ϕ = 0.55 at Np/ND = 0.576.
The solid line indicates a scaling fit with
β = 1.6.
At higher drives, well above depinning, the slope of the velocity-force curve
becomes linear, as expected since the effectiveness of the pinning is lost in this regime.
In general, we find that for Np/Nd > 0.288,
the velocity-force curves can be fit to a power law with 1.4 < β < 1.7.
The variation in the exponents we obtain is
a result of the limited size of our simulation,
but our values are
within the range of those reported
for plastic depinning of systems
with longer range interactions [8,12,60].
It remains an open question
whether plastic depinning for systems with
short range interactions falls in the same universality
class as plastic depinning
for systems
with long range interactions.
For Np/Nd < 0.288, the depinning threshold Fc=0
since there are few enough pinning sites that some disks
can pass completely through the system without being trapped directly by pinning
or indirectly by becoming lodged behind pinned disks.
The dynamic density phase separation we find has not been
observed in studies of superconducting vortices or colloids driven over random disorder.
As noted previously, under certain conditions such as low flux density or very small
penetration length, superconducting vortices
could behave like a hard disk system and exhibit density phase separation or
the formation of 1D flowing chains.
Observation of such effects would require the use of
weak pinning samples that provide access to the flux flow regime at low fields.
There have been examples of clump-like vortex states observed at low fields in
certain materials; however, these clumps may be the result of
competing attractive and repulsive interactions between the vortices [51], rather
than from reaching an effective hard disk interaction limit.
There have been some
numerical studies of vortex avalanches in which the vortex-vortex
repulsion was modeled as as harmonic repulsion [62]; however,
these studies were performed in a 1D system, which is a very different limit from the
system we consider.
Numerical studies
of vortices moving through periodic substrate arrays showed that
under certain conditions the system can form vortex
density or soliton waves [63];
however, these studies are again in a very different regime from
that which we consider.
There has also been work showing that
phase separation into high
density regions as well as stripe ordering occurs for particles
driven over random disorder when the pairwise interactions between
particles include
both a repulsive and an attractive term [64,65];
however, in the disk system we consider here, the disk-disk interaction is
purely repulsive.
The phase separation we observe has similarities to
the active matter clustering
found in simulations
of hard disks undergoing active Brownian motion or run-and-tumble type dynamics.
In the active matter systems, when the
activity is high enough, the particles phase separate into
a dense solidlike region and a low density fluid [43,44,45,46]
due to a combination of the nonequilibrium nature of the fluctuations
and the fact that the mobility of the particles is dependent on the local
particle density [44].
In the driven disk system, velocity fluctuations transverse to the driving direction are
generally largest when there is a coexistence of disks being pinned or slowed down
by the pinning along with faster moving unpinned disks.
When the disks collide with each other,
they generate velocity fluctuations that have
a ballistic component in the transverse direction, similar to the
motion of active particles.
This also produces time intervals in the transverse diffusion that exhibit superdiffusive
behavior similar to that found in active matter systems [46].
Additionally, the disks have a reduced mobility
when the disk density increases.
When the drive is large enough,
both the speed differential of the disks and the velocity fluctuations transverse to the
drive are lost, and since these effects are necessary to produce the clustering and the
density phase separation, the clustering and density phase separation also disappear.
The same effects could arise in systems with longer range interactions;
however, the large energy cost of high density regions would
suppress the density phase separation we observe for the short
range repulsive disks.
Experimentally, the dynamic phase separation could be
observed using colloids, emulsions, or micelles that have only steric interactions
moving over random substrates.
Experiments with quasi-2D granular systems could include grains
flowing over a rough landscape under the influence of gravity or shaking; however, in
this case,
inertial and intergrain frictional effects would also need to be taken into account.
In our work we focus on the case of monodisperse disks,
so that the system forms triangular ordering in the dense phase; however, we have
also considered a case for bidisperse disks with a radius ratio of
1:1.4 and
find the same features,
where the phase separated state is shifted to a somewhat lower density,
suggesting that the dense phase separated regions are in fact jammed since
it is well known that jamming occurs for lower densities in bidisperse disks than
for monodisperse disks [38].
We have numerically examined the dynamical phases for monodisperse
repulsive disks driven
over random disorder. Despite the simplicity of this system,
we observe a rich variety of distinct dynamics,
many of which have significant differences from the dynamic phases
observed for other systems of
collectively interacting particles with longer range repulsion,
such as vortices in type-II superconductors and colloids with Yukawa interactions.
The phases we find include a heterogeneous clogged state
where the disks form local immobile clumps,
a homogeneous disordered plastic flow state, a moving density phase separated state
where the system forms a dense region with mostly triangular ordering coexisting with a
low density disordered phase,
and a stripe or chainlike state at higher drives.
The density phase separation occurs due to the density dependent mobility of the
disks
and the short range nature of their interaction with each other, which permits the
disks to pack closely together with little overlap energy.
In contrast,
in systems with longer range repulsion, density phase separated states are prevented
from
forming since
more homogeneous states have a much lower
particle-particle interaction energy.
The chain formation can occur in the disk system since the disks can
approach almost within a radius of each other without paying an overlap energy cost,
whereas in systems with longer range interactions,
such strongly anisotropic structures would have a very high energy cost.
From the features in the transverse diffusion, structure factor, and velocity-force curves, we
map the evolution of the different phases as a function of disk density,
pinning site density, and pinning force.
Our results suggest that the dynamic density phase separation and the chainlike state
should be general features in systems
with short range steric interactions driven over random disorder.
These effects could be observed experimentally using
sterically interacting colloids, emulsions, micelles and even
superconducting vortices at low fields
moving over random disorder.
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.
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