Cooperative Behavior and Pattern Formation in Mixtures of
Driven and Nondriven Colloidal Assemblies
C. Reichhardt and C.J. Olson Reichhardt
Center for Nonlinear Studies and Theoretical Division,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
(Received 5 February 2006; revised manuscript received
18 May 2006; published 24 July 2006)
We simulate a disordered assembly of
particles interacting through a repulsive
Yukawa potential with a small fraction of the particles
coupled to an external drive.
Distortions in the arrangement of the
nondriven particles produce a dynamically induced effective
attraction between the driven particles,
giving rise to intermittent one-dimensional stringlike structures.
The velocity of a moving string increases with the number of
driven particles in the string.
We identify the average stable string length as a function of
driving force, background particle density, and particle charge.
This model represents a new type of collective transport system
composed of
interacting particles moving through deformable disorder.
The collective transport of interacting particles
driven through disordered backgrounds has been studied extensively in
systems such as moving
vortex lattices in the presence of random pinning [1],
driven charge density waves [2], and sliding friction [3].
A variety
of nonequilibrium behaviors occur in these systems, including
fractal flow patterns, avalanches,
nonlinear velocity-force responses, and dynamic reordering transitions.
Here, the disorder in the
substrate is quenched and does not change with time, but
substrates which
distort and respond to
the driven particles are also possible,
such as
in the case of
a single colloidal particle
driven through a disordered background of other particles
which are not coupled to the external drive [4,5].
Both simulations [5] and experiments [4]
show that such a system can exhibit a nonzero threshold force
for motion as well as
a nonlinear velocity vs applied force response. The driven
particle moves in an
intermittent manner
and causes local rearrangements and distortions in the surrounding
bath of colloids.
An open question is what happens if there are multiple particles
driven through a background of nondriven particles, rather than only
a single driven particle.
In particular, it is possible that the ability of the
surrounding media to react to the motion
of driven particles could induce
an emergent effective driven particle-particle interaction,
leading to new types
of moving structures or to pattern formation that is not observed in
driven systems with fixed or quenched background disorder.
Recent studies have focused on a related system
in which two sets of identical repulsive
particles are driven in opposite directions
[6,7,8,9].
Here, a laning phenomenon occurs, and
the species segregate into multiple streams flowing past one another.
This model has been studied
in
terms of pedestrian dynamics, where the particles
represent people moving in opposite directions
[8],
as well as in the context of
charged colloids in the
presence of a driving field [7].
The colloidal systems are particularly attractive
for further study since
binary mixtures of oppositely charged colloids which can
undergo laning transitions in the presence of a
driving field have been produced recently in experiment
[9]. This type of
system opens a wealth of new experimental possibilities.
To our knowledge, the case of a small fraction of particles driven
through a responsive but undriven background
has not been studied previously.
By using a smaller ratio of driven particles compared to previous work
[7],
we identify and characterize individual
strings of moving particles,
which has
not been done in previous studies.
Here we show that when multiple particles move
through a background of nondriven particles, an effective emergent attraction
arises between the driven particles even though all of the
pairwise interactions between the particles are repulsive.
We find that intermittent moving strings form
which have an average stable length that is a function of the
driving force, system density, and
particle charge.
Strings that are longer than average are very short lived.
The string velocity increases monotonically with the string length.
We also study the power spectra of the velocity fluctuations, which has
never been considered before in these types of systems.
We show that the power spectrum can be very valuable in
distinguishing between the cases of single driven
particles, small fluctuating strings, and states with
fully formed lanes of moving particles.
We consider a two-dimensional system with periodic boundary conditions
in the x and y directions
containing Nc particles interacting via a screened Coulomb
or Yukawa interaction.
Only
a small number ND of the particles couple to an external drive.
The particles move in an overdamped background, and we neglect
hydrodynamic effects, which is reasonable for the low volume fraction
limit considered here.
We use molecular dynamics (MD) to
numerically integrate the overdamped equations of motion
for the particles, given for particle i by
ηdri/dt = Ficc+FiD + FTi.
Here the damping coefficient η = 1.
The colloid-colloid interaction
force term
Ficc = −qiA∑j ≠ iNc∇iV(rij)
contains the Yukawa interaction potential
V(rij) = (qj/|ri − rj|)exp(−κ|ri − rj|), where
ri(j) is the position of
particle i(j),
qi(j) is the charge of colloid i(j), and all colloids have
the same sign of charge so that all interaction forces are repulsive.
Energies are expressed in units of A=q2/4πϵϵ0 and
lengths in terms of a, the average colloid spacing.
The screening length 1/κ is
set to 1/2 in all our simulations.
We use a time step of ∆t=0.01 in dimensionless units, such that
a colloid moving with an average velocity of 1.5 requires roughly 100
time steps to translate by one lattice constant.
The background colloids are composed of a 50:50 mixture of two
charges with q1/q2 = 1/2, which produces a noncrystalline
background arrangement.
The average background charge q=(q1+q2)/2.
For the non-driven particles,
FDi = 0.
The colloids that couple to the external field have charge qD
and experience a constant drive FDi=FD∧x
applied after the system is equilibrated by simulated annealing.
The thermal force FiT has the properties
〈FiT 〉 = 0 and
〈FiT(t)FjT(t′) 〉 = 2ηkBTδijδ(t−t′).
For a system of length L the
colloid density is nc = Nc/L. We consider both the
constant nc case,
where L and Nc are increased simultaneously,
as well as the case
where nc is increased by fixing L and increasing Nc.
Figure 1:
Images of the system composed of a disordered
array of background colloids (small circles) and
particles that couple to
the external drive (large circles)
for qD/q = 6.67, nc = 0.94, FD = 3.0, and
ND/Nc = 0.01. (a) Initial configuration;
(b) the same system after 105 MD steps; (c) a typical steady state
snapshot after 106 MD steps; (d) snapshot of a string breaking
into shorter strings.
The driving force is from left to right.
In Fig. 1 we show four snapshots which highlight the
dynamically induced attraction of the driven particles, indicated as
large black dots,
and the formation of the string structures
for nc = 0.94,
qD/q = 6.67, and FD = 3.0.
Fig. 1(a) shows the initial state in which the
driven particles are well separated from each other. In Fig. 1(b),
after 105 MD time steps,
the driven particles begin
to form string like structures of length Ns=2 to 3 colloids
which are aligned in the direction of
the driving force (left to right in the figure).
In this nonequilibrium system, despite
the fact that all of the pair-wise interactions
are repulsive, an effective attraction
emerges between the driven particles.
What is difficult to convey through still images
is the fact that
as Ns
increases,
the average string velocity 〈Vxs〉 increases
so that strings move considerably
faster than isolated driven particles.
In Fig. 1(c) after 106 MD time steps,
the system has reached a nonequilibrium steady state.
Here the average stable length of a string 〈Ns〉 = 5 to 6.
Strings with Ns > 6 form occasionally but
have a very short lifetime.
In Fig. 1(c), strings with Ns=6, 5, 3, and 1 are present.
All of the strings have
an intermittent character, in that driven particles in the string
detach from the string, and new driven particles join the string.
Typically strings
add or shed one driven particle at a time; however,
long strings with Ns=7 or 8
tend to break up into separate
strings with Ns=3, 4, or 5, as illustrated in Fig. 1(d).
Figure 2:
(a) Time series Vx(t) for a single driven colloid
from the system in Fig. 1(a) with ND=20.
(b) The corresponding
Vy(t).
(c) Solid line: The normalized histogram of Vx
for the system in (a). Dotted line: Histogram of Vx for
ND=1.
(d) Solid line: The normalized histogram of Vy
for
the system in (b). Dotted line:
Histogram of Vy for
ND=1.
We note that if ND/Nc is increased while all other
parameters are held fixed,
then 〈Ns〉 remains unchanged at
〈Ns〉 = 5 to 6.
We find the formation of stable strings over an extensive range
of parameters, including all qD/q ≥ 1.0 (where we have checked up
to qD/q = 40),
for nc > 0.17,
and for arbitrarily large FD. The
value of 〈Ns〉
depends on the specific choice of parameters; however,
the phenomenon of string formation is very robust
provided that the driven particles
can distort the arrangement of the background particles.
In Fig. 1, ND/N ≈ 0.01,
so that the initial average distance between driven particles is many
times larger than the pair-interaction length. Thus, the attractive force
between driven particles is
mediated by the distortion field produced in the
background particles.
If the fraction is increased as much as an order of magnitude up to
ND/Nc=0.1, the preferred length of 〈Ns〉 = 5 to 6
persists. We note that if ND/Nc
is too high, finite size effects due to the boundary conditions
become important.
In order to illustrate
the intermittent behavior and the velocity increase as the number of
colloids in a string increases, in Fig. 2(a) we plot a typical
time series of the velocity in the direction of drive
Vx for one of the driven
colloids from the system in Fig. 1(a).
Each
point of the velocity time series is obtained by averaging the instantaneous
velocity over 50 time steps.
Here Vx(t) shows a series of
well defined increasing or decreasing jumps,
with a roughly constant velocity
between the jumps.
The lower value that Vx takes is around 0.39
corresponding to the average velocity
of a driven particle that is not in a string.
There are several places where Vx = 1.45 which
corresponds to the average velocity
of a string with Ns=5. Additionally, there are some plateaus
near a value of Vx = 0.7 which corresponds to Ns=3.
Fig. 2(a) also shows that for short times,
Vx > 1.5, which
corresponds to strings with Ns > 6.
This figure shows
that the driven colloid is exiting and joining strings of different lengths.
Figure 3: 〈Vxs〉 vs
Ns.
(a) The same system as in Fig. 1(a)
with qD/q=6.67 for FD = 3.0 (squares),
4.0 (triangles), and 5.0 (circles).
(b) The same system as in (a) for qD/q = 1.33 and
FD = 0.5 (squares), 0.75 (triangles), and 1.0 (circles).
Fig. 2(a) is only a portion of a much larger time series
which covers 107 MD steps.
In Fig. 2(c) we plot the histogram of
this entire time series along with
the histogram of Vx (dotted line) for
a case where
there is only one driven particle in the system so that strings
do not form.
The ND=1 curve shows a single peak
in P(Vx) near Vx = 0.35,
which falls near but slightly below the first peak in P(Vx) for
the multiple driven particle system. Thus, the lowest value
of Vx in Fig. 2(a) corresponds to time periods when the driven colloid is
moving individually and is not part of a string.
Additionally, there is a small peak in P(Vx)
for the ND=20 system near Vx=0.8,
which corresponds to the formation of strings with Ns=2.
There is a large peak in P(Vx) around Vx=1.45
corresponding to Ns=5 and 6.
For Vx > 1.5
there is a rapid decrease in P(Vx).
This result shows that there is a preferred string
length of 〈Ns〉 ≈ 5 for the ND=20 system.
The average velocity of a driven particle
in the ND=20 case is
much larger than the average velocity in the ND=1 case, even through
the fraction of driven particles is only ND/Nc=0.01.
Driven particles traveling in
strings show a more pronounced transverse wandering
motion than individual driven particles.
In Fig. 2(b), we plot the time series for the velocity transverse to the
driving force, Vy, for the same particle as in Fig. 2(a).
Here 〈Vy〉 = 0,
and there are numerous spikelike events
in the positive and negative directions. In
Fig. 2(d) we plot the histogram of the entire time series of Vy
(solid line) along with the same histogram
for a system containing only a single driven particle
(dashed line). The ND=1 curve
shows a well defined Gaussian distribution
of P(Vy) centered at zero. For ND=20,
the spread in Vy is much larger and the fluctuations
are non-Gaussian, as indicated by the presence of large tails
in the distribution.
There are an excess of events at larger values of |Vy|
which result from the fact that the particles in a chain move
in a correlated fashion. The non-Gaussian statistics also
indicate that the strings have transverse superdiffusive behavior.
Figure 4:
(a)
〈Ns〉 vs FD for the system in Fig 1.
(b) 〈Ns〉 vs qD for the same system as in (a) for
FD = 1.0.
(c) 〈Ns〉 vs nc for qD = 6.67 and FD = 2.0.
Inset: 〈Vxs〉 vs FD for the system in (a).
(d) 〈Vxs〉 vs system size L for the system in Fig. 1 for
FD= 1.5 (triangles), 3.0 (squares), 4.5 (diamonds), and 6.0 (circles).
Figure 5:
The power spectra S(ν) obtained from the velocity fluctuations
in the x-direction Vx(t) for a system
with the same parameters as in Fig. 2.
ν is given in units of inverse molecular dynamics steps.
Bottom curve: S(ν) for a single driven particle.
The remaining curves show S(ν) for Vx(t) for multiple
driven particles from Fig. 2(a).
Second lowest curve: ND/Nc = 0.01.
Second highest curve: ND/Nc = 0.1.
Top curve: ND/NC = 0.4. The lower dashed line is a power law
fit with α = 1.5 and the upper dashed line is a power law fit with
α = 0.25.
The curves have been offset in the y direction for presentation purposes.
We next examine the average velocity of the strings 〈Vxs〉
vs Ns, which we plot in
Fig. 3(a) for systems with the same parameters as in Fig. 1 at
FD = 3.0, 4.0, and 5.0.
The bottom curve, which
corresponds directly to the system in Figs. 1 and 2,
shows that for Ns = 1.0, 〈Vxs 〉 = 0.35,
while for Ns = 5 and 6, 〈Vxs 〉 ≈ 1.4.
The string velocity increases rapidly with Ns for Ns < 5 and
then increases much more slowly with Ns. The longer strings become
considerably more winding and experience additional frictional drag,
whereas the shorter strings move rigidly.
Due to the meandering of the longer strings, there
is a tendency for the driven particles at the back
of the string to be ripped away from the string very quickly.
For curves with higher FD,
there is still a
monotonic increase in the average velocity with Ns.
In order to show that this phenomenon is robust over a wide range
of parameters, in Fig. 3(b) we
plot 〈Vxs〉 vs Ns for a system with qD/q = 1.33 for
FD = 0.5, 0.75, and 1.0.
For these parameters we again observe the string formation
and an increase in 〈Vxs〉
with Ns. The best functional fit we find
is 〈Vxs〉 ∝ ln(Ns);
however, other forms can also be fit, such as
two linear fits for the system with qD=1.33.
In Fig. 4(a) we show the most probable number of particles
〈Ns〉 in a string for
the same system in Fig. 1 for varied
FD.
Here 〈Ns〉
goes through a maximum as a function of FD. As FD decreases
below FD < 0.2,
the distortion created by the driven particles in the surrounding
background particles decreases, and therefore there is less
attraction between the driven particles.
At high FD, the velocity fluctuations along the string
become important, which tends to limit 〈Ns〉.
In the inset of Fig. 4(c), the corresponding 〈Vxs〉
vs FD
shows a sharp increase in 〈Vxs〉 at FD = 2.0
when the longest strings form.
This is followed by a saturation of 〈Vxs〉
just above FD = 2.0, caused when the decreasing 〈Ns〉
counterbalances the
increasing FD.
In Fig. 4(b) we show that 〈Ns〉 for
a system with a fixed FD = 1.0 also goes through a maximum as a
function of qD/q. For small qD/q, the driven
particles produce very little distortion in the
surrounding medium so there is
little attraction between them. For
large qD/q, the repulsion between the driven particles becomes
more prominent and the average velocity
drops, which again reduces the distortion field.
In Fig. 4(c) we plot 〈Ns〉 vs
the particle density nc for a system with qD = 6.67 and
FD = 2.0. At low nc, all the particles are far apart and the
distortion in the bath particles is reduced or absent.
As nc increases,
the particle-particle interactions are enhanced and the string
can grow longer. Presumably for
very high nc, 〈Ns〉 will decrease again
as the average velocity will have to be reduced.
We are not able to access particle densities in this range.
In Fig. 4(d) we plot 〈Vxs〉 vs the
system size L for
FD=1.5 (lower curve), 3, 4.5, and 6.0 (upper curve).
In each case,
〈Vxs〉
initially decreases with L but reaches a saturation value by
L ≈ 40.
The change in 〈Vxs〉 at small L occurs because, in
small systems, the periodic boundary conditions allow driven colloids
to interact strongly with their own distortion trails.
For larger FD, larger system sizes must be used to avoid this effect.
In addition to the increase in 〈Vxs〉 in the small systems,
〈Ns〉
also increases for the smallest systems.
This emphasizes the fact
that sufficiently large systems must be considered in order
to avoid finite size effects.
We have used L=48 throughout this work,
which is large enough to avoid finite size
effects for the
quantities we have measured.
The attraction between the driven particles
is mediated by the
distortion in the surrounding bath of colloids.
The concept of nonequilibrium
depletion forces, which are highly anisotropic and have both an
attractive and a repulsive portion,
was introduced recently for a
case where two large colloidal particles have
an additional bath of smaller particles flowing past them
[10].
The surrounding particle density is
higher in front of the large particle
and lower in back, which causes an anisotropic
attraction force between the large colloids that
aligns them with the direction of flow.
Although this
model was only studied for two particles, it seems to
capture several of the features that we find in our simulations, including
the alignment of the strings in the direction of drive.
This suggests that nonequilibrium depletion
force models could be applied for multiple large particles.
The longer strings move faster since the front particle has
additional particles pushing on it
from the back.
Since the chains are aligned, the cross section with the
background particles remains the
same as for a single driven particle; however, the string is
coupled Ns times more strongly to the external field.
As longer strings become
increasingly meandering, the cross section with the
background particles increases
and limits the speed that long strings can attain.
The role of the effective cross section of the chain in determining
the chain speed is also supported by work on a related system of chains
of ions moving through a dense jellium target. In Ref. [11],
the stopping power per ion for an ion chain moving parallel to its length
initially decreases with chain length for short chains before saturating
for longer chains. The stopping power, which would correspond to an
effective viscosity contribution in our context, is reduced for the
chain of ions compared to the value it would have for an equivalent number of
isolated ions due to dynamic interferences between the moving ions.
The power spectrum of the velocity fluctuations in the direction of drive
has not been examined in previous work on these types of systems.
This is defined as
S(ν) =
⎢ ⎢
⌠ ⌡
Vx(t) e−2πiνtdt
⎢ ⎢
2
.
(1)
If the velocity fluctuations are highly intermittent, we would expect
to observe a
1/fα noise signal.
In Fig. 5 we plot the power spectrum for the system in Fig. 2 for
a single particle (lowest curve), and for multiple driven particles
at ND/Nc = 0.01, ND/Ns = 0.1, and
ND/Ns = 0.4 (top curve).
The curves have been shifted from each other in the y direction
for clarity.
The lower dashed line is a 1/fα fit for
α = 1.5 and the upper dashed line is the same for α = 0.25.
Here the single driven
particle shows a white noise spectrum which reflects the lack
of correlations in the velocity fluctuations.
Additionally, the histogram of velocities for a single driven
particle from Fig. 2 shows a Gaussian distribution in this case.
If FD is much lower and close to the single driven particle
depinning threshold, the velocities become
non-Gaussian and a 1/f noise signal
is observed [12].
In the case where the intermittent strings form,
at ND/Nc = 0.01 and 0.1, the
velocity histograms are non-Gaussian as in Fig. 2 and
the velocity fluctuations are highly intermittent.
In these cases, the power spectrum has a 1/fα form
with α > 1.0.
When the density of driven particles becomes large,
stable lanes of driven particles form and the moving particles fluctuate
rapidly.
The velocity histograms again take a Gaussian form
and the power spectrum becomes white with
α ≈ 0. The noise spectrum of the velocity fluctuations
is thus another measure which can be used
to identify the different flow regimes.
In conclusion, we have investigated a system of repulsively interacting
Yukawa particles
which form a disordered background for a small fraction of Yukawa
particles that
couple to an external drive. In this system, despite
the fact that all of the pair-wise interactions are
repulsive, there is an emergent effective attraction between the
driven particles which results in the formation of
one-dimensional intermittent stringlike structures.
The velocity of a string increases with the
number of particles in the string.
The attraction between the driven particles
is mediated by the distortion of the
surrounding particles and is likely
a result of the recently proposed nonequilibrium depletion forces.
Longer strings move faster than shorter strings due to
the additional pushing force from the back particles in the string.
For a single driven particle at high drives the power spectrum
of the velocity fluctuations has a white
noise characteristic, while in the
intermittent string regime, the power spectrum is of the
form 1/fα with α > 1.0.
In the regime where close to half of the particles are driven,
the noise spectrum becomes white again.
This system represents a new class of collective transport
where multiple interacting
colloids are driven through a deformable
disordered background, which is distinct from the case where
the background disorder is quenched.
We thank E. Ben-Naim for useful discussions.
This work was supported by the U.S. DoE under Contract No.
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