Ratchet Cellular Automata for Colloids in Dynamic Traps
C.J. Olson Reichhardt and C. Reichhardt
Center for Nonlinear Studies and Theoretical Division,
Los Alamos National Laboratory Los Alamos, NM 87545, USA
received 13 February 2006; accepted in final form 13 April 2006
published online 3 May 2006
PACS. 05.40.-a. - Fluctuation phenomena, random processes, noise, and Brownian motion
PACS. 05.40.Ca. - Noise.
PACS. 82.70.Dd. - Colloids Abstract. -
We numerically investigate the transport of kinks in
a ratchet cellular automata geometry for colloids interacting with
dynamical traps. We find that thermal effects can enhance the transport
efficiency in agreement with recent experiments. At high temperatures
we observe the creation and annihilation of thermally induced kinks
that degrade the signal transmission. We consider both the
deterministic and stochastic cases and show how the trap geometry can
be adjusted to switch between these two cases.
The operation of the dynamical trap geometry can be achieved
with the adjustment of fewer parameters than
ratchet cellular automata constructed using static traps.
Simulation References
Recently, a ratchet mechanism was proposed
for propagating logic states in a clocked manner
through a system of vortices in nanostructured type-II superconductors
[1]. Since the operation of the device depends on the
discrete positions of the vortices,
the system was termed a ratchet cellular automata (RCA).
It has been demonstrated that a complete logic architecture
can be constructed using the RCA, so that variations of RCAs
constructed in different systems might offer a promising
alternative to the current microelectronic
logic architectures based on silicon MOSFETs [1,2].
The use of discrete particles to store logic states or perform
logic operations has been studied previously in various forms
including the quantum dot cellular automata [3]
and magnetic dot cellular automata [4].
In contrast to these systems, which work in the adiabatic limit,
an RCA operates when the system is
far from equilibrium.
The original RCA geometry was proposed for
vortices in a type-II superconductor with
nanostructured pinning sites. The vortices act like repulsive
particles and adopt one of two possible configurations in the static
pinning sites.
In order to propagate a logic signal through the device,
an alternating external driving force must be applied, such as
by inducing an oscillating Lorentz force on the vortices by means of
an alternating current.
The basic concept of the RCA
should be applicable to any system of particles which have a repulsive
interaction with each other,
such as ions in optical traps, classical electrons,
vortices in Bose-Einstein condensates with
optical arrays of traps [5], or colloids
interacting with arrangements of optical traps [7].
In many of these systems, such as for the colloids,
it is already experimentally possible
to create dynamical traps; in this case, an external applied drive may not
be necessary.
In the original RCA geometry, the basic structural unit consists of three
elongated static traps, each containing a single vortex.
The three traps have different widths and biases in order to break the spatial
symmetry of the system. In each trap, the vortex sits either at the top
or the bottom of the trap and represents either state 0 or 1.
A three-stage alternating external drive is then applied which shifts
the vortices to the left and right inside the traps. The trap designs are
chosen such that the result of these shifts is to alter the distance
between vortices in neighboring traps
so that two out of every three vortices
are close to
one neighboring vortex and far from the other. The resulting
asymmetry permits propagation of the logic signal, which would otherwise
not occur in this overdamped system.
With this geometry, a logic signal in the form of a kink corresponding to
two adjacent up or down sites can
be propagated along a chain of traps.
The bare RCA functions at finite temperatures in a stochastic mode since
a small barrier remains at the center of each trap which must be overcome
thermally. If an additional potential is superimposed on the wells to
counteract this barrier, the RCA can operate in a completely deterministic
mode and can also run at T=0.
Recently, an experimental version of RCA has been realized for colloids
confined to two dimensions and
interacting with
optical traps [6,7]. The colloids are
micron-sized particles with
repulsive Yukawa or screened Coulomb interactions.
In this case, a series of optical traps are prepared and the wells are
labeled as sites A, B, and C in a pattern that repeats across all of the wells.
Each trap is composed of a double well potential created by two optical
tweezers, so the colloidal particles sit either in the up or down position
inside each trap. Unlike the original RCA, in the colloidal realization each
trap is identical in shape.
The ratchet effect is induced by dynamically relocating the positions
of the wells
periodically. The wells labeled B are moved
to the right and left, and separately the wells labeled C are also moved
to the right and left, so that the net effect is the same alteration of
spacing between neighboring colloids that was achieved by means of three
well shapes and a three-stage alternating drive in the vortex system
of Ref. [1].
For example, in Fig. 1(a), the colloid in well A1 has been switched from
the up to the down position. In Fig. 1(b), the wells in group B have
been shifted to the left, and the colloid in well B1 flips from the
down to the up position. In Fig. 1(c), the wells in group C are shifted
to the left which permits the colloid in well C1 to flip
to the down position.
Finally in Fig. 1(d), all of the wells in groups B and C are shifted
to the right, and as a result the colloid in well A2 flips to the
new state. This process repeats, with the signal moving over by one well
and following the "vacant" site, corresponding to a far spacing
between neighboring colloids, across the system.
The colloidal version of the RCA which has been experimentally realized
will be a useful system for studying alternative geometries and
further properties of the RCA.
The colloid version of RCA can also provide a valuable system with which
to understand
transport in noisy environments, which has connections to stochastic resonance.
Several important issues have not been studied directly
in the experiments, such as explicitly changing
the temperature or the clock frequency,
as well as understanding the role of thermally
induced kink and anti-kink creation in the signal propagation.
It would be valuable to probe
the effect of trap geometry on the transition
between the thermally dominated and deterministic or clocked regimes.
Here, we explore all of these possibilities through simulations.
Figure 1:
(a-d) Colloid positions (black dots)
inside the three sets of wells marked A, B and C during
propagation of the signal from well A1 to
well A2.
Wells A are stationary, while the wells in sets
B and C move back and forth to the dashed locations.
(e) Geometry of a lozenge shaped pin.
Simulation:
We consider a quasi one-dimensional geometry of Nc=144 traps
with open boundary conditions,
in analogy with the geometry considered
in the experiments of Ref. [7].
Each trap contains a single colloid which is
modeled as an overdamped particle
confined to two dimensions and interacting
with the other colloids via a Yukawa potential
U(rij) = q2A0exp(−κrij)/rij.
Here κ = 1 is the inverse screening length, q is
the colloid charge,
and A0=e2/(4πϵϵ0a0). Length is measured in
units of a0 and time in units of η0a02/A0.
The equation of motion for a colloid i
is
η0vi = Fi = Ficc + Ftrapi +FiT .
(1)
Here
η0 is the damping constant
from the surrounding fluid which is set to unity.
The colloid-colloid force
Ficc = −∑Ncj ≠ i∇iU(rij).
Since the colloid interaction force falls off exponentially for
large r we place a cutoff on the interaction
at r=4, further than the screening length, for computational
efficiency.
Taking a longer cutoff produces the same results.
The temperature is applied as random Langevin kicks
FT
with the
statistical properties 〈fT(t)〉 = 0
and 〈fT(t)fT(t′)〉 = 2η0kBTδ(t −t′).
The trap force
Ftrap is produced by lozenge shaped
pins, each of which is composed of
two half-parabolic traps separated by an elongated
region that confines only in the x direction, as shown in Fig. 1(e).
The aspect ratio of the pins is 3 to 1, with the
long direction running along the y axis perpendicular to the line
of pins. The pinning strength fp=11.0.
The lozenge shapes of the traps were chosen to model the
experiments closely.
The positions of the wells in group A is fixed in time, and the
distance between the wells in group A, which corresponds to the
lattice constant of the ratchet device, is 5.0.
The wells B and C are moved back and forth by a distance of 1.0
periodically in three stages, described earlier in Fig. 1(a-d).
The total length of time spent by the wells in one of
the three stages is reported as the clock period τ.
A kink in the form of a change in logic state is produced by
moving the leftmost colloid from the up to the down position
at time t=0. The kink appears in Fig. 1 as two adjacent wells with
colloids that are both up or both down.
The system operates stochastically due to the presence of a finite barrier
at the center of each well, generated by interparticle interactions,
and at T = 0
there is no transmission of kinks. At finite T the kink propagates
and the
system can exhibit either a clocked or
effectively deterministic behavior.
Figure 2:
(a) The kink position vs time
for a system with τ = 10000 at T = 0.4. (b)
The same system at T = 1.15. Dark dots indicate propagation of the
originally introduced kink, while light dots show the formation and
propagation of thermally activated kink-anti-kink pairs.
In Fig. 2(a) we plot the location of
a kink that was inserted at the edge of the sample
at t = 0 as a function of time for T = 0.4
in a system with q2=0.5 and τ = 10000.
At this temperature, the kink moves in a
clocked manner through the entire system of 144 dots, and
there are no thermally created kinks or anti-kinks.
The kink propagates at a constant speed,
as indicated by the linear
slope.
We observe a similar clocked
motion at lower temperatures until T < 0.1.
Below this temperature the kink becomes pinned near its entry point
and does not propagate across the system.
For T = 1.15 for the same system, shown in Fig. 2(b),
thermally
induced kinks can appear. The initial kink is marked as a thick black line,
and moves through the system
at the same speed as the kink in Fig. 2(a).
At later times, thermally activated kink-anti-kink pairs are
created, and the ratcheting mechanism
propagates both species of kinks in the same direction.
Kinks and anti-kinks collide and annihilate
when the leading kink takes a thermally induced step backwards
and the anti-kink is able to catch it.
In other cases the kinks and anti-kinks travel the length of
the system without annihilating.
Figure 3:
The same system as in Fig. 2 for (a) T = 1.3;
(b) T = 1.4.
Dark dots: the originally introduced kink; light
dots: thermally activated kink-anti-kink pairs.
In Fig. 3(a), we show the same system
at T = 1.3. In this case the
thermal fluctuations are sufficiently strong that the
induced kink does not move linearly but shows occasional steps backward
so the motion is no longer completely clocked.
A significant number of thermally induced kink anti-kink pairs form
and also show occasional steps backwards.
As the temperature increases the average number of thermally
created pairs
increases and the average lifetime of a given pair decreases.
Figure 3(b) illustrates the system at T = 1.4,
where thermally created pairs
proliferate rapidly and the initially introduced kink is
both hard to distinguish and short-lived.
For T > 1.4 the thermal fluctuations are so strong that
the colloids can hop out of the individual wells.
Figure 4:
(a) The efficiency η vs
T for q2=0.5 at clocking speeds
τ = 20000, 10000, 5000, 2500, 1500, and 1000 from top to bottom.
(b) η vs clock frequency ν
in inverse MD steps for fixed T = 0.5.
The kink motion can also be characterized by measuring
the transmission efficiency η, which is defined
in terms of the time τdet=τNc it would
take for a kink in a completely deterministic system to travel the
length of the system.
The actual travel time for a kink is given by τkink, so
the efficiency η = τdet/τkink.
If the kink travels at the clocked pace, η = 1. If the kink
takes steps backward, gets stalled, or annihilates with a
thermally activated anti-kink, η < 1.
In Fig. 4(a) we plot η vs T for
fixed q2 = 0.5, fp = 11.0,
and different frequencies or clocking speeds.
Each point has been averaged over five realizations of thermal
disorder.
In this system the kinks are
motionless for T < 0.1 since the particle-particle interactions induce a
barrier
at the center of the pinning sites that must be overcome by
a small amount of thermal activation.
For T ≥ 0.1, there are enough thermal fluctuations to overcome
the barrier and the kinks begin to propagate.
For
τ > 10000,
kinks can propagate effectively deterministically,
and η = 1 over a wide
range of temperatures.
The efficiency begins to drop when T > 0.8
since there are excessive
thermal fluctuations that cause the kinks to take occasional steps
backward rather than forward.
In the experiments of Ref. [7],
there was also a certain range of parameters over which the
system operated effectively deterministically and η = 1.
For shorter clock periods, τdet decreases and
the kink should in principle move through the chain at a faster rate.
Instead, as the clock frequency increases,
the colloids are no longer able to respond
and the efficiency decreases. Fig. 4(a) shows that
as the clock period decreases,
the temperature at which the system reaches an
effectively deterministic mode increases. For
τ < 10000
the system is never able to enter the effectively deterministic region.
There is an upper bound on the temperature that can be applied to this
system, since for T > 1.4 the colloids begin to jump completely out of the
wells by thermal activation and the ratchet device is destroyed.
In Fig. 4(b) we show the efficiency vs clock
frequency ν for fixed T = 0.5.
There is a plateau of η = 1
at low clock frequencies followed by an exponential decrease
of η with increasing ν
which is cut off at the highest frequencies.
Optimum speed of signal transmission can be achieved by choosing the
highest value of ν that still gives η = 1.
Figure 5:
(a) Efficiency vs q2 for constant T and fixed τ = 5000.
Diamonds: T=1.0. Open squares: T=0.75. Filled circles: T=0.5.
(b) Efficiency vs T for elliptical traps at τ = 5000.
Filled circles: q2=18. Open squares: q2=18.5. Filled diamonds:
q2=19.
In the experimental work of Ref. [7],
it was argued that increasing
the strength of the central trap in each pin has an effect that is
similar to decreasing the temperature.
This implies that if the central trap strength is
fixed, then there should be an optimal temperature range for
effectively deterministic
transport of logic signals.
The results in Fig. 4 support this conclusion.
The effect of temperature is more pronounced at the lower clock periods
τ and the temperature window in which
effectively deterministic behavior occurs
shrinks with
decreasing τ.
In the case of τ = 5000 in Fig. 4(a),
there is a peak value of η near
T = 0.9.
In our model, varying the trap strength fp does not affect the
ratchet efficiency. This is because it is the interaction
forces between the colloids that control both the strength of the
induced barrier
at the center of a pin as well as the magnitude of the force that leads
to propagation of the logic signal from pin to pin. We have tested both
larger and smaller values of fp and find that the
pinning can be made arbitrarily
strong without affecting our results. In the experimental system, there is
a practical limit on the amount of laser power that can be provided to the
sample in order to form the optical traps.
There is also a physical limit to the amount of
energy that can be absorbed by the colloids and the bath medium over a given
time period before damage occurs. For smaller values of fp, which can
be accessed readily in experiment
by reducing the laser power, we find that
the system is limited by the requirement
of keeping the colloids inside the pinning sites at all times.
The colloids may depin if the colloid-colloid interaction force
overcomes the pinning force, if thermal activation out of the pins
becomes possible, or if a combination of these two
effects occur. Once the colloids depin, the ratchet is destroyed.
We have considered the effect of varying the strength of the colloid-colloid
interaction, q2, as shown in Fig. 5(a) for different temperatures
at τ = 5000.
If q2 is reduced toward zero, there is insufficient coupling between
the colloids for the ratchet mechanism to function, and η drops
to zero. As q2 increases, the system enters the
effectively deterministic
regime with η = 1. At q2=0.5,
the same system with clock period τ = 5000 was shown in Fig. 4(a) never to
reach η = 1 even as T is increased.
Fig. 5(a) indicates that for this clock speed, the
system can enter the
effectively deterministic limit if q2 is increased to a value
of at least 1. If q2 is increased too much, however, the efficiency
drops again when the thermal fluctuations that are required
for operation of the ratchet are washed out by the very strong colloid-colloid
interaction forces. We show the decrease in η at higher values of
q2 in Fig. 5(a). At the lower temperature T=0.5,
η drops below 1 once q2 > 1.6, while for the higher temperature
T=1.0, thermal fluctuations are not washed out until q2 > 5.5.
In the experiments of Ref. [7], the traps used to construct the
ratchet had a multi-well shape which we have represented in our model by
a lozenge-shaped pin. The same ratchet mechanism can also operate for
other types of wells. To test this, we have considered a much simpler model
for the traps consisting of elliptical pins. These are simply
parabolic traps with unequal aspect ratios in the x and y directions.
Unlike the lozenge-shaped pins, which have a central region that
is flat in the y direction and
confines only in the x direction, the elliptical pins have a minimum
in both the x and y directions
at the center of the pin. If the ratio of the pinning force to the
colloid-colloid interaction force is too small in the elliptical pin case,
the system loses its two logic states and all the colloids sit in the center
of the wells. On the other hand, when the colloid-colloid interaction force
is strong enough that the alternating up-down configuration appears in the
elliptical pin
system, the ratchet mechanism can operate in a fully deterministic mode
even when T=0. This is illustrated in Fig. 5(b) where we plot η as
a function of T for elliptical wells with fp=11.0,
τ = 5000, and q2=18,
18.5, and 19.
Here, we see that η = 1 all the way down to and
including T=0, in contrast to the case in Fig. 4(a) where η drops to
zero at T=0. The central minimum in the elliptical pins compensates for
the potential barrier at the center of the pin induced by the colloid-colloid
interaction forces. There is no benefit to adding temperature
to a system with elliptical pinning sites. For low T, η = 1, but as
T increases, thermally activated kinks and antikinks appear and interfere
with the signal transmission, causing η to drop. Unlike the case
shown in Fig. 3 for the lozenge-shaped pins, where thermally activated kinks
tended to ratchet at nearly the clock speed through the system, in the
case of elliptical pins the thermally activated kinks tend to diffuse and are
much more likely to travel backwards than the kinks in the lozenge-shaped
pins.
Fig. 5(b) also demonstrates that the transport efficiency of the
elliptical traps at elevated temperatures is strongly
sensitive to the value of q2.
We note that the RCA constructed using dynamical traps has several
advantages over the originally proposed RCA which involved static traps
in combination with an external drive.
The static trap RCA requires three different
trap geometries to be constructed in the same system.
In order to
achieve deterministic kink propagation
without temperature, an additional attractive
potential must be added to the center of each trap to compensate for
the barrier in the middle of the trap caused by particle-particle
interactions.
Finally, a three stage external drive must be applied. In
the dynamical trap RCA,
many of these extra parameters are eliminated, making it easier to adjust
the system into an effectively deterministic mode of operation.
In the simplest case of elliptical traps, the system
can function deterministically even without thermal fluctuations.
We also note that it may be possible to create
dynamical traps for vortices in type-II superconductors, the system in
which the RCA was first proposed.
It has been suggested recently that if artificial magnetic pinning sites
are created in a superconducting sample,
then the strength and shapes of the pinning can be changed dynamically
by applying time dependent magnetic
fields [8].
In conclusion, we have numerically investigated ratchet cellular automata
constructed for colloidal particles using dynamic traps.
Our results are in good
agreement with the recent experiments of Ref. [7].
We considered the effects of changing several parameters that have
not been explored experimentally,
including temperature, colloid-colloid interaction strength,
clock frequency, and the influence of
the trap geometry.
We find that
temperature can enhance the transport of kinks and can
permit the RCA to operate
in an effectively deterministic
limit even without the addition of an attractive potential to compensate
for the barrier created at the center of each pin by particle-particle
interactions. We also examined the proliferation of kink-antikink pairs
which are created at higher temperatures. These
pairs can be transported by the ratchet effect and
can recombine and annihilate.
At high clock frequencies, the efficiency of the ratchet transport is
degraded since the colloids can no longer fully respond to the
ratchet mechanism. We also identify the existence of
optimal temperature regimes and particle interaction
regimes for signal transport.
In the case of elliptical traps,
we show that the system can operate deterministically even in the
T = 0 limit.
Acknowledgments-We thank D. Babic, C. Bechinger, and M.B. Hastings
for useful discussions.
This work was supported by the US DoE under Contract No. W-7405-ENG-36.
