N. Uchida and R. Golestanian, "Synchronization and Collective Dynamics in a Carpet of Microfluidic Rotors," Phys. Rev. Lett. 104 , 178103 ().

N. Uchida and R. Golestanian, "Synchronization and Collective Dynamics in a Carpet of Microfluidic Rotors," Phys. Rev. Lett. 104 , 178103 ().
PHYSICAL REVIEW LETTERS
PRL 104, 178103 (2010)
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Synchronization and Collective Dynamics in a Carpet of Microfluidic Rotors
Nariya Uchida1,* and Ramin Golestanian2,†
1
2
Department of Physics, Tohoku University, Sendai, 980-8578, Japan
Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
(Received 12 November 2009; published 26 April 2010)
We study synchronization of an array of rotors on a substrate that are coupled by hydrodynamic
interaction. Each rotor, which is modeled by an effective rigid body, is driven by an internal torque and
exerts an active force on the surrounding fluid. The long-ranged nature of the hydrodynamic interaction
between the rotors causes a rich pattern of dynamical behaviors including phase ordering and selfproliferating spiral waves. Our results suggest strategies for designing controllable microfluidic mixers
using the emergent behavior of hydrodynamically coupled active components.
DOI: 10.1103/PhysRevLett.104.178103
PACS numbers: 87.19.rh, 07.10.Cm, 47.61.Ne, 87.80.Fe
Introduction.—Microorganisms and the mechanical
components of the cell motility machinery such as cilia
and flagella operate in low Reynolds number conditions
where hydrodynamics is dominated by viscous forces [1].
The medium thus induces a long-ranged hydrodynamic
interaction between these active objects, which could
lead to emergent many-body behaviors. Examples of
such cooperative dynamical effects include sperms beating
in harmony [2], metachronal waves in cilia [3–5], formation of bound states between rotating microorganisms [6],
and flocking behavior of red blood cells moving in a
capillary [7]. For a collection of free swimmers, such as
microorganisms [8], hydrodynamic interactions have been
shown to lead to instabilities [9,10] that can result in
complex dynamical behaviors [10,11]. In the context of
simple microswimmer models where hydrodynamic interactions coupled to internal degrees of freedom can be
studied with minimal complexity, it has been shown that
the coupling could result in complex dynamical behaviors
such as oscillatory bound states between two swimmers
[12] and collective many-body swimming phases [13,14].
A particularly interesting aspect of such hydrodynamic
coupling is the possibility of synchronization between
different objects with cyclic motions [4,5,15–21]. This
effect has mostly been studied in simple systems such as
two interacting objects or linear arrays and very little is
known about possible many-body emergent behaviors of a
large number of active objects with hydrodynamic coupling. For example, in a recent experiment [22], Darnton
et al. observed chaotic flow patterns with complex vortices
above a carpet of bacteria with their heads attached to a
substrate and their flagella free to interact with the fluid
(see also [23]). On the other hand, recent advances from
micron-scale magnetically actuated tails [24] to synthetic
molecular rotors [25] now allow fabrication of arrays of
active tails that can stir up the fluid. It is therefore very
important to explore the possible complexity of the phase
behavior of such an actively stirred microfluidic system.
0031-9007=10=104(17)=178103(4)
Here, we consider a simple generic model of rotors [26]
positioned on a regular 2D array on a substrate and study
their collective dynamics numerically. We find that the
long-ranged hydrodynamic interactions could either enhance or destroy ordering, depending on the degree of a
built-in geometric frustration that originates from the interaction of the rotors with the fluid. More specifically, our
model adopts a fully synchronized state when the frustration is weak, and a randomly disordered state when it is
maximally frustrated. Moreover, the dynamics of the system leads to self-proliferating spiral waves between the
above two limiting behaviors. We also take into account
thermal fluctuations of the rotors and map out the phase
diagram of the system as a function of temperature and the
degree of frustration.
Model and dynamical equations.—We consider an array
of rotors that are assumed to be spherical beads of radius a
moving on circular trajectories of radius b, which are
positioned on a rectangular lattice of base length d and at
a height h above a substrate (see Fig. 1). The ith rotor is
anchored at ri0 to the surface of the substrate, which we
take to be the xy plane. The instantaneous position of the
rotating bead is ri ¼ ri0 þ bni þ hez , where the unit vector ni ðtÞ ¼ ð cosi ðtÞ; sini ðtÞ; 0Þ gives the orientation of
the arm of the rotor. Because of the constraint that the bead
is only allowed to move on the circular orbit of radius b, the
FIG. 1 (color online). Schematic representation of the array of
rotors. Inset: an immobilized bacterium with active flagella as a
possible realization of a rotor that can exert both a tangential
drag and an active radial force on the fluid.
178103-1
Ó 2010 The American Physical Society
PHYSICAL REVIEW LETTERS
PRL 104, 178103 (2010)
di
i
velocity of the rotor can be written as vi ¼ b dn
dt ¼ b dt ti ,
where ti ¼ ez ni ¼ ð sini ; cosi ; 0Þ is the unit vector tangent to the trajectory.
We assume that the structure of the rotor is such that it
drags the fluid with it as it moves (tangentially) along the
circular trajectory, while it can also pump the fluid radially
due to some internal degrees of freedom. The inset of Fig. 1
shows a possible realization of such a system in the case of
bacteria whose heads are fixed on the substrate. In this
example, the spinning rotation of the flagella would produce the pumping effect, while the precession of the axis of
the flagella about the anchoring point would correspond to
the tangential motion of the bead. Therefore, in our simplified model, each rotor exerts a force, which can be
decomposed into the radial, tangential, and vertical components as Fi ¼ Fn ni þ Ft ti þ Fz ez . The velocity
Pfield of
the fluid created by the rotors is given by vðrÞ ¼ i Gðr ri Þ Fi where GðrÞ is the Blake-Oseen tensor [27], which
describes the hydrodynamic interaction near a flat surface
with the nonslip boundary condition. Assuming that the
arm length b and the height h are much smaller than the
characteristic distance d between the rotors, we can use the
3h2 r r
Oðh2 =d2 Þ approximation [4], G ðrÞ ¼ 2
for ,
jrj5
¼ x, y, and Gz ðrÞ ¼ Gz ðrÞ ¼ Gzz ðrÞ ¼ 0, for ¼ x,
y. Note that the z component of the force is not coupled to
flow and that the fluid velocity is lying in the xy plane.
To obtain the flow velocity at the position of the rotors,
we need to subtract the self-interaction, which involves the
Stokes drag coefficient ¼ 6a. This yields
3 X ti rij rij ð!n nj þ !t tj Þ
di
;
¼ !t þ
2 jÞi
jrij j5
dt
(1)
where rij ¼ ri rj , !t;n ¼ Ft;n =ðbÞ are the reduced
forces, and ¼ h2 = ¼ 6ah2 is the hydrodynamic
coupling constant. When the interaction is weak, we can
simplify the phase equation [Eq. (1)] following a standard
prescription [28]. To this end, we rewrite it in terms of the
slow variable i ¼ i !t t, and then integrate it over a
cycle under the approximation that i in the interaction
term is constant over the period 2=!t . We obtain
(a)
(b)
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3! X 1
di
sinði j Þ;
¼
4 jÞi jrij j3
dt
(2)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where ¼ tan1 ð!t =!n Þ and ! ¼ !2t þ !2n . This equation is correct to Oð=ðd3 sinÞÞ [28,29]. In this form, the
hydrodynamic coupling becomes isotropic, and our system
resembles existing models of nonlocally coupled oscillators with phase delay [30,31].
Simulation method.—The model is implemented on a
L L square lattice with the grid size d, and the phase
equation Eq. (1) is solved by Euler method with the time
step t. The system size used is L ¼ 256 for most of the
results below, while we have also used L ¼ 128 for obtaining some of the statistical data.
We have imposed periodic boundary condition and computed the velocity field at every time step by Fourier transformation. For > 0, we also solved the reduced phase
equation Eq. (2), to compare with the solution of Eq. (1),
and found a very good agreement. We have also incorporated thermal fluctuations, by adding an uncorrelated
Gaussian noise i ðtÞ to the right-hand side of Eq. (1).
The noise is assumed to have zero mean, and its fluctuations are controlled by the rotational diffusion constant of
the bead Dr ¼ kB T=ðb2 Þ as hi ðtÞj ðt0 Þi ¼ 2Dr ij ðt t0 Þ. We define the reduced (effective) temperature ¼
Dr d3
!
3
B Td
¼ 362ka
2 b2 h2 ! . For typical values of a b h 1 m, d 10 m, ! 102 Hz, with ¼ 1 103 Pa s and kB T ¼ 4 1021 J, we have 101 .
Note that the sharp dependence of on a, b, h, and d
makes it easy to control the reduced temperature by changing the size or density of rotors. In our simulations, we have
used the parameter values ¼ 0:1 and ! ¼ 0:1, with d ¼
1 and t ¼ 0:1. First, we turn off the thermal noise ( ¼ 0)
and vary the force angle to study the pattern dynamics.
Pumping-driven rotors.—When ¼ 0, each rotor
pumps the fluid radially and is driven by the fluid flow
generated by the other rotors. In this case, the initial
random perturbations develop into topological defects (singularities of the phase field) ðrÞ of winding numbers 1,
which coarsen by collision of þ1 and 1 defects and
finally disappear to establish global synchronization.
(c)
(d)
FIG. 2 (color online). Snapshots of coarsening defects (with the greyscale representing cosðrÞ) for (a) ¼ 0 and spiral waves for
(b) ¼ 45 and (c) ¼ 60 . These developed from random initial perturbations. (d) Spiral waves for ¼ 60 evolving from a defect
pair, with a schematic picture of the director field (red, solid arrows) and the velocity field (blue, dotted arrows) near a þ1 defect.
178103-2
Figure 2(a) shows a snapshot of the coarsening defects at
t ¼ 10 000. To characterize the phase ordering dynamics,
we define the correlation length ¼ h½rðrÞ2 i1=2
,
r
where r is the spatial gradient and h. . .ir means spatial
average. As shown in Fig. 3(a), we find that as a function
of time is well fitted by the power law / t , with ¼
0:75. The scaling of the hydrodynamic interaction GðrÞ jrj3 and Eq. (1) suggest that the characteristic time scale
of a pattern is proportional to its size, which would mean
¼ 1. The difference between the numerical and scaling
exponents suggests violation of dynamic scaling, which is
characteristic to coarsening of point defects in two dimensions [32].
Torque-driven rotors.—When ¼ 90 , each rotor is
driven by an active torque and exerts a force tangential to
its orbit. In this case, we find that the system reaches a
disordered state in which spatial correlation is almost
completely lost. The absence of orientational correlation
can be understood as follows. The average flow created by
a rotor is perpendicular to its arm, and a neighboring rotor
tends to align with the flow. Thus, the two rotors tend to be
perpendicular to each other on average. However, it is not
possible that every pair of rotors have their arms vertically
crossed (geometric frustration), and hence the system
evolves towards randomly oriented states.
Rotors driven by pumping and torque.—In the general
case, the rotor is driven by an active torque while it pumps
the fluid radially, and the total force exerted on the fluid has
an angle 0 < < 90 with respect to the arm of the rotor.
We varied the parameter and found two types of dynamical behavior. For 0 < 40 , the globally synchronized
state is still obtained as the final state. However, for 40 <
< 90 , we find that the dynamical steady state of the
system involves self-proliferating spiral waves as shown in
Figs. 2(b) and 2(c). Moreover, we find that the correlation
length ðtÞ converges to a finite value as t ! 1 as shown in
Fig. 3(a), and that the equilibrium correlation length decreases as is increased.
The flow pattern is locally correlated with the rotor’s
director nðrÞ ¼ ð cosðrÞ; sinðrÞÞ. The surface flow velocity vðrÞ makes the angle with nðrÞ except at the core of
(b)
δ = 0 deg
30 deg
40 deg
10
50 deg
60 deg
t^0.75
order parameter S
correlation length ξ
(a)
1
the defect (see the movies [33] for comparison of the two
fields). This observation leads us to an intuitive interpretation of the spiral waves. In the vicinity of a þ1 defect
from which the director emanates radially, the rotors create
an outgoing flow that has an anticlockwise slant with
respect to the radial direction, and hence form an anticlockwise spiral; see the inset of Fig. 2(d). The spiral is
tighter for a larger force angle . We confirmed this
scenario by choosing a defect pair as the initial configuration and following its evolution; see Fig. 2(d) and the
corresponding movie [33]. Initially, clockwise and anticlockwise spirals are formed around 1 and þ1 defects,
respectively. Then, the director is randomized on the thinning spiral arm, which collapses and proliferates a cascade
of new defects.
Thermal fluctuations.—We then introduce the thermal torque and study the phase behavior of the system.
In Fig. 3(b), we plot the equilibrium order parameter S ¼
jhcosij as a function of the effective temperature. For ¼
0, we find a critical temperature c at which S vanishes as
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
S / c to a good approximation [34]. We find the
critical temperature for this phase transition as c ¼ 0:76,
which is about 30% smaller than the mean-field value c ¼
1:08 by Guirao and Joanny [4].
As we increase the phase delay up to c ¼ 40 , the
critical temperature is lowered down to c ¼ 0:61. For >
c , the order parameter S is less than 1 even at ¼ 0 and is
smaller for a larger system size L, suggesting that S ¼ 0 in
the thermodynamic limit. However, the existence of local
order (spiral waves) is reflected in the finite-L data, using
which we can define the transition temperature c in the
same way as mentioned before [34]. The resulting phase
diagram is shown in Fig. 3(c). We distinguish three regions: (O) ordered, which is a distinct thermodynamic
phase characterized by global synchronization, and (S)
spiral waves and (D) disordered, which are inherently
the same phase but with different local ordering and dynamical structure. Also shown in Fig. 3(c) are the contours
of the correlation length . While the O-D and O-S transitions are sharp, the S-D transition is a crossover characterized by gradual decrease of . We also note that the
1
(c)
δ =0 deg
30 deg
40 deg
50 deg
60 deg
(τc -τ)1/2
0.8
0.6
0.4
0.2
1000
time t
10000
1
ξ =1,2,3,5,10
0.8
disordered
0.6
ordered
0.4
spiral
wave
0.2
0
0
100
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PHYSICAL REVIEW LETTERS
reduced temperature τ
PRL 104, 178103 (2010)
0
0.2
0.4
0.6
0.8
reduced temperature τ
1
0
10
20
30 40 50 60 70
phase delay δ (deg)
80
90
FIG. 3 (color online). (a) Correlation length as a function of time and for different values of . For ¼ 0 , it is well fitted by
/ t0:75 . For c ¼ 40 , the correlation length diverges after t ¼ 10 000 (not shown), while for > c , it remains finite.
(b) Equilibrium order parameter S as a function of temperature and for different values of . (c) Phase diagram and contour of the
correlation length . The correlation length is smaller for larger and higher temperature.
178103-3
PRL 104, 178103 (2010)
PHYSICAL REVIEW LETTERS
frustrated state for ¼ 90 , ¼ 0, and the thermally
disordered state for ¼ 0 , > c are qualitatively different, though they are not distinguished in the phase diagram.
Discussion.—The case of no active torque ( ¼ 0) could
be regarded as a simplified and idealized model of bacterial
carpets [22,23]. Our model reproduced the enhancement of
orientational ordering, while it predicts global ordering and
not the finite-size correlation as observed in the experiments. The experimental patterns might be explained by
some kind of frozen disorder in the flagellar configuration,
which can be readily incorporated in our model [29,35].
The case of ¼ 90 is realized by rigid spheres without
pumping. It is related to a recently studied model of two
rigid spheres making tilted elliptic orbits [17], which show
both in-phase and antiphase synchronization. Our results
suggest that the interaction between many of such rotors is
frustrated, and the system does not attain full synchronization. Spiral waves for finite phase delay have been
observed in previous models of 2D coupled oscillators
[30,31,36]. However, in our case, the pattern is intrinsically
turbulent and self-proliferating, in contrast to the case of
finite-range coupling, for which the phase is spatially
smooth except near the defect core [30,31,36].
In conclusion, we have introduced a generic model of
microfluidic rotors that shows a variety of dynamical patterns including global synchronization, fully disordered
states, and self-proliferating spiral waves. The patterns
are sensitively controlled by the angle of active force (the degree of frustration) and the temperature . Our
results suggest that arrays of active microfluidic components could be designed to induce a rich variety of dynamical behaviors in the vicinal fluid, and could be used to make
switchable microfluidic mixers.
N. U. thanks the hospitality at University of Sheffield
where this work was initiated, and financial support from
Grant-in-Aid for Scientific Research from MEXT. R. G.
acknowledges financial support from the EPSRC.
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*uchida@cmpt.phys.tohoku.ac.jp
†
r.golestanian@sheffield.ac.uk
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