Soft Matter
PAPER
Diffusion of active dimers in a Couette flow
Cite this: Soft Matter, 2017,
13, 2793
Tanwi Debnath,a Pulak K. Ghosh,bc Franco Nori,c Yunyun Li,
Fabio Marchesonidf and Baowen Lig
*de
We study the 3D dynamics of an elastic dimer consisting of an active swimmer bound to a passive
cargo, both suspended in a Couette flow. Using numerical simulations, we determine the diffusivity of
Received 20th February 2017,
Accepted 16th March 2017
such an active dimer in the presence of long-range hydrodynamic interactions for different values of its
DOI: 10.1039/c7sm00356k
greatly enhanced under the condition that self-propulsion is strong enough to contrast the shear flow.
rsc.li/soft-matter-journal
The magnitude of the effect grows with the size of the dimer’s constituents relative to their distance,
which makes it appreciable under experimental conditions.
self-propulsion speed and the Couette flow. We observe that the effect of hydrodynamic interactions is
1 Introduction
Artificial microswimmers are active particles capable of autonomous
propulsion.1–4 A common class of such micromotors is the so-called
Janus particles (JPs), mostly spherical objects with two differentlycoated hemispheres, or ‘‘faces’’.5,6 Due to the different functionalization of their faces, JPs harvest kinetic energy from their environment,
by generating local (electric,7 thermal,8 or chemical9) gradients in the
suspension medium (self-phoresis).
Among the most promising technological applications of
artificial microswimmers is their usage as motors, whereby they
couple to a cargo, represented by a passive particle (PP), and
tow it from a docking station to an end station, often along an
assigned track engraved on a 2D microfluidic chip.2,3 In such a
configuration, the tower and the cargo form a dimer with one
active head, the JP, and a swerving tail, the PP.10 The diffusion
of such a dimer is fueled by the self-phoretic ‘‘force’’11 acting
upon the JP, while both monomers are subjected to a viscous
drag. The self-phoretic mechanism activated by the JP generates a
short-range hydrodynamic backflow in the viscous suspension
fluid,12 which may result in an additional pair interaction
between the monomers.13,14 However, at low Reynolds numbers,
a
Department of Chemistry, University of Calcutta, Kolkata 700009, India
Department of Chemistry, Presidency University, Kolkata 700073, India
c
CEMS-RIKEN, Saitama 351-0198, Japan
d
Center for Phononics and Thermal Energy Science, School of Physics Science and
Engineering, Tongji University, Shanghai 200092, People’s Republic of China.
E-mail: [email protected]; Fax: +86 (21)6598 6745; Tel: +86 (21)6598 1360
e
Shanghai Key Laboratory of Special Artificial Microstructure Materials and
Technology, School of Physics Science and Engineering, Tongji University,
Shanghai 200092, China
f
Dipartimento di Fisica, Università di Camerino, I-62032 Camerino, Italy
g
Department of Mechanical Engineering, University of Colorado, Boulder,
Colorado 80309, USA
b
This journal is © The Royal Society of Chemistry 2017
as the active swimmer tows its cargo, it generates a laminar flow
that tends to align the dimer’s axis parallel to the propulsion
force, with the PP trailing the JP. It is known12,15,16 that the
dipolar hydrodynamic interactions associated with the swimmer’s
propulsion mechanism decay faster with the distance than the
perturbation due to the laminar flow caused by its steady translation.
Therefore, the average tower–cargo distance can be conveniently
chosen, so as to ignore the short-range hydrodynamic interaction
between them with respect to the long-range hydrodynamic effects
on their viscous drag.
As we propose to utilize active microswimmers to transport
cargos in a controllable manner, it becomes important to study
the diffusive dynamics of an active dimer in the laminar flow
maintained, say, in a microfluidic device.17 The hydrodynamics
of active systems, including both single and clustering microswimmers, possibly in confined geometries, is an extensive
topic.18,19 In this work, we limit ourselves to considering the
case of an elastic dimer suspended in a highly viscous fluid
confined between two parallel plates. The two plates may slide
in opposite directions with equal speed, thus generating the
simplest and more tractable example of laminar shear flow in
the fluid (Couette flow).17 To avoid unnecessary complications,
we assume that the tower and the cargo have equal masses and
viscous constants. Therefore, here the two monomers can be
regarded as identical, except that the JP is subject to a fluctuating
force of constant modulus, which represents the pull from the selfpropulsion mechanism. When moving, each particle generates
a laminar flow that affects the diffusion of its partner and,
eventually, the diffusivity of the entire dimer.20,21
We perform extensive Brownian dynamics simulations of
such a bound system in 3D and focused on the hydrodynamic
corrections to the dimer’s diffusion constant. As discussed in
the following, these corrections are much more conspicuous than
reported for passive dimers.20 In a Couette flow, corrections due
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to long-range hydrodynamic interactions become prominent in
the dynamical regime when the speeds of the self-propelling
dimer and the sliding plates are comparable.
The content of this paper is organized as follows. In Section 2
we formulate the Brownian dynamics of the dimer, which we
implement in our numerical simulation code. We also introduce
an overdamped rigid-dimer approximation of the model to help
interpret the simulation results. New results are reported in
Sections 3 and 4. In Section 3 we analyze the hydrodynamic
corrections to the diffusion constant of a passive dimer made of
two identical PP’s, both in the bulk and in a Couette cell. These
increase with the particle radius to the dimer length, as expected,20
but are suppressed in a shear flow. Finally, in Section 4 we
investigate the diffusion of an active dimer in different dynamical
regimes. The main conclusion of this section has been anticipated
above: hydrodynamic effects are magnified by an appropriate
choice of the shear flow in the Couette cell. In Section 5, we
present some concluding remarks about possible improvements
of our approach and the relevance of our results to the technology
of active colloids.
2 The model
The overdamped Brownian dynamics of a pair of identical
interacting particles of mass m, radius a, and viscous constant g,
suspended in a fluid at rest with temperature T, has been
numerically simulated by Ermak and McCammon.20 Upon
expressing time in units of g1 and all lengths in units of
pffiffiffiffiffiffiffiffiffiffiffiffiffi
kT=m=g, their integration algorithm reads (in Einstein
summation convention)
@
Dij
ri ðt þ DtÞ ¼ ri ðtÞ þ vi Dt þ Dij Dt þ
fj Dt þ Zi ðDtÞ;
D0
@rj
(1)
where the repeated indices i and j run over the dimer coordinates,
1 r i, j r 6 (1 to 3 and 4 to 6, respectively, for the two
monomers), Dt = 106 or 105 is our integration step, and Zi
are zero-mean valued, delta-correlated Gaussian noises with
(co)variance hZi(Dt)Zj (Dt)i = 2DijDt. The drag velocities, vi, are
the components of the unperturbed shear flow in the Couette
cell at the point occupied by the center of the particles. Recall
that the velocity of a sphere may lag behind the local fluid
velocity by an amount proportional to (a/L)2, where L is the size
of the cell.22 Throughout the present work a { L, so that the
flow lag is considered negligible. Finally, we also neglect the
polarization effects a shear flow may have on the JP, owing to its
vorticity23 and different surface properties of the JP faces.24,25
The magnitude of such an effect is rather small and quite
sensitive to the actual fabrication details of the JP.
The force fj is composed of two terms: (1) An internal force,
(e)
f (e)
j = qU /qrij, due to the elastic pair potential,
U ðeÞ ¼
2
k ~
rij l :
2
(2)
-
Here, l is the average dimer length in vacuo, rij is the vector
pointing from the center of particle i to the center of particle j,
and k is the relevant restoration rate of the dimer’s internal
2794 | Soft Matter, 2017, 13, 2793--2799
degrees of freedom, i.e., a measure of its stiffness. (2) The selfpropulsion, p, acting on the JP. Its modulus, p, is fixed, while its
orientation fluctuates in time with the law,19
^_ ¼ ^
p
p nðtÞ;
(3)
where p̂ is the relevant unit vector, and the three Cartesian
components of the Gaussian noise n(t) are independent, zeromean valued, and delta-correlated with variance hni(Dt)2i =
2DrDt. The quantity Dr controls the time decay of the p auto2
correlation function, h pi(t)pi(0)i = ( p /3)exp[2Dr|t|], and Dr is
referred to as the JP rotational constant.
Moreover, the confining action of the sliding plates in a
Couette cell has been mimicked21 by introducing an additional
force perpendicular to the plates, f (w). Denoted by L the distance
between the plates and by y the transverse coordinate of a given
particle, the corresponding confining force on that particle
was formulated as f (w) = qU (w)/qy, with the wall potential
U (w) = C( y 8 L/2)4, respectively, for y o L/2 (upper plate) and
y 4 L/2 (lower plate), and U (w) = 0 for L/2 o y o L/2. The choice
C = 100 ensures that the walls are stiff enough for all parameters of
the dimer and the Couette cell explored in our simulations.
The hydrodynamic coupling between monomers mediated
by the fluid enters only the diffusion tensor, Dij.22 The simulation
results reported in this paper have been obtained for the
approximate Oseen tensor,17,20–22
Dij = D0dij,
3 a ~ ~
rij~
rij
1þ 2 ;
Dij ¼ D0
4 rij
rij
(4)
respectively, for i and j on the same particle or on different
particles. Here 1 is the unit matrix and D0 is a measure of the
magnitude of the thermal fluctuations in the suspension fluid,
i.e., in the dimensional notation, D0 = kT/mg. The use of more
refined approximations for the diffusion tensor, like the Rotne–
Prager tensor,20 does not affect our conclusions. We remind
that the Oseen tensor handles well the hydrodynamic interactions between a pair of active particles, but only under the
condition that their distance allows the short-range active
stresses responsible for their propulsion to be neglected.12,26
Furthermore, we also assume that the Zi correlations corresponding
to eqn (4) are not appreciably affected by a laminar shear flow.27,28
To simplify our notation, we assumed a symmetric Couette
flow with plates sliding with opposite velocities, vs, in the x
direction. Accordingly, the shear flow across the cell assumes
the simple linear expression, v(y)/vs = 2y/L, while the remaining
z coordinate can be ignored. Due to symmetry considerations, the
net drift velocity of the dimer is identically zero. Its diffusivity is
quantified by the diffusion constant29 D lim xðtÞ2 2t, where
t!1
x is the coordinate of the JP and the stochastic average, h i, is
taken over at least 2 103 trajectories. Indeed, for exceedingly
long observation times, in our runs, t 4 109Dt or larger, the
dimer length becomes negligible with respect to the dimer
displacement, and so are the initial condition transients.30
For numerical purposes, we set k = 1, which is equivalent to
using the rescaled time variable t 0 = kt in eqn (1) and expressing
This journal is © The Royal Society of Chemistry 2017
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all speeds and diffusion constants in units of k, namely, vs 0 = vs/k,
p 0 = p/k, D0 0 = D0/k, Dr 0 = Dr/k, and finally, D 0 = D/k.
2.1
Rigid dimer model
To help interpret our numerical results, we now map the
dynamics of eqn (1) onto that of a single composite active
swimmer31,32 in 3D. We assume that the center of force, P, and
the center of mass, O, rest on the axis of a dumbbell of a fixed
length, l (rigid dimer), as illustrated in Fig. 1(a). The rigid dimer
approximation is certainly correct for large values of elastic
constant k, while for softer elastic bonds, one can expect that
the effective dimer length increases under the combined action
of self-propulsion and hydrodynamic interactions.20 In the
overdamped regime, the dimer’s instantaneous self-propulsion
velocity, v0, is oriented parallel to the force p, which fluctuates
with the law given by eqn (3). The (constant) modulus of selfpropulsion velocity is immediately related to the modulus of
force, v0 = p/2, the two monomers being identical. Furthermore,
due to the propulsion force applied in P, the swimmer tends
to rotate around its center of mass subject to the torque,
^r is oriented from the PP
tp ¼ ðl=2Þ~
r^ ~
p, where the unit vector ~
to the JP. The particular case of an active dumbbell self^r, has been
propelling along its axis, i.e., with p parallel to ~
investigated in ref. 33.
The noises independently applied to the two interacting
particles of eqn (1) impact the translational and rotational
fluctuations of the dimer. To begin with, we ignore the hydrodynamic effects due to the laminar flow around the two spheres,
i.e., we set a = 0. One sees immediately that the six independent
translational noise components Zi(t) act on the rigid dimer
pffiffiffi
through a translational noise,29 Zt ðtÞ 2, and a fluctuating torque
r^ Z ðtÞ. Here, the three components of Zt(t) and Zt(t)
tZ ¼ ðl=2Þ~
t
have the same statistics as the Zi(t) of eqn (1). [Recall that the
random forces, Zi(t), acting on a single monomer, cause a
translational velocity Zi(t)/2 of the entire dimer; moreover, adding
two independent Zi(t) oriented along the same axis is statistically
equivalent to replacing them with another noise of identical
statistics, but twice their variance29].
In conclusion, to study the dynamics of a freely diffusing
rigid dimer it is useful to separate the translational motion of
its center of mass, O, from the rotational motion around O.33
The random motion of the center of mass, denoted by the
vector rO, then obeys a 3D Langevin equation,
pffiffiffiffiffiffi
~
v0 þ Dd xðtÞ;
(5)
r_O ¼ ~
-
-
where v0 = p/2 fluctuates according to eqn (3) and the Cartesian
components of the Gaussian noise x(t) have mean hxi(t)i = 0 and
correlation functions hxi(t)xj (0)i = 2dijd(t). As discussed above,
for pointlike monomers with a = 0, Dd = D0/2. The rotational
motion of the dumbbell is driven by the two random torques, tp
and tZ, introduced above.
When incorporating the hydrodynamic effects, we have to
take into account the spatial dependence of the diffusion tensor
in eqn (4). In leading order20 this is equivalent to rescaling
D0 - D0(1 + a/l).
(6)
Moreover, in the integration algorithm implemented in our
simulation code, eqn (1), the force term, too, was multiplied
by Dij, which implies that, in the same order of a/l, the selfpropulsion speed must also be rescaled as
v0 - v0(1 + a/l).
(7)
The diffusion properties of the composite active swimmer
described by the Langevin equation, eqn (5), are detailed in
ref. 32. For the sake of this presentation, it will suffice to recall
here that the noise strength Dr plays the role of an orientational
diffusion constant, whose inverse, tr, quantifies the temporal
persistency of the isotropic Brownian motion of the dimer’s
center of mass. Indeed, for long observation times t, with t c tr,
the asymptotic law, lim r2 ðtÞ ¼ 6Dt, defines the effective active
t!1
dimer diffusion constant,32
D = Dd + Ds
(8)
Fig. 1 Active dimer self-propulsion mechanism: (a) an active elastic dimer model. O and P are, respectively, the center of mass and the center of force of
the dimer formed by an active and a passive particle of equal shape, bound together by an elastic string. p represents the instantaneous self-propulsion
~
~
force and l ¼ l ^
r the dimer’s length vector oriented from the passive to the active particle. (b) A Couette cell. A shear flow is generated between two
concentric rings of radii Rint and Rext, with Rext Rint = L, rotating with an opposite angular frequency O.
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The term Dd is due to environmental thermal fluctuations,
while Ds = v02/6Dr is the (typically) much larger self-propulsion
diffusion, which depends on the activation properties of the
dimer in the suspension fluid.
3 Passive dimer diffusion
The diffusion of a symmetric passive dimer, or dumbbell, has
already been investigated under operating conditions close to
those detailed in Section 2, namely assuming a linear shear-flow
and explicitly accounting for hydrodynamic effects, see, e.g.,
ref. 34. However, in addition to those earlier reports, we derive
here a phenomenological expression for the spatial diffusion of
the dumbbell parallel to a confined shear flow.
We start our discussion by analyzing simulation data for the
simple system composed of two identical passive particles of
radius a, bound by the elastic potential U(e) of eqn (2). According
to the authors of ref. 20, see eqn (6), in the limit k - N, one
expects that a passive rigid dimer diffuses with constant
D = Dd(1 + a/l),
(9)
where l is its length. Lowering k, the average distance between
monomers increases due to rotation. A simple perturbation
expansion yields20
l(k) = l [1 + 2D0/(kl2) +. . .].
(10)
For the simulation results displayed in Fig. 2(a) such a correction
is not appreciable. However, when plotted versus a/l, our data
agree with the analytical prediction of eqn (9) up to rather large
a/l values. We anticipate in Fig. 2(a) that the a/l corrections to the
diffusion constant of an active dimer are comparatively larger
than those of a passive one (see also Section 4).
When placed in a Couette cell with a plate sliding speed vs,
the actual constant D increases, because the dimer now diffuses
between opposite fluid flows, as illustrated in Fig. 1(a). The
datasets plotted in the inset of Fig. 2(a) clearly show that in the
limit vs - 0 one recovers the corresponding D values reported in
Fig. 2(a), while at large vs the dependence of D on vs turns quadratic.
Such a quadratic law can be explained based on a simple
phenomenological argument. We postulate that the dimer diffuses
between the two sliding plates with a certain mean-first-passage
time t. As we look at the dimer’s motion along the x axis, we
assume that the shear flow is so strong that we can ignore the
thermal diffusion in that direction. In contrast, when taking
into account transverse thermal diffusion, the dimer appears to
randomly switch between two kinematic states with opposite
effective velocities, v, and lifetime t. The diffusion constant in
the flow direction for such a dichotomic process is well
established,29 namely D = v2t. The speed v is of the order of vs.
For a more precise estimate, we chose v2 = hv2(y)i = vs2/3, where
v(y) is the shear flux across the Couette cell and the average was
taken over the transverse coordinate y.
Finally, we calculate t by integrating over time the normalized stationary autocorrelation function h y(t)y(0)i/h y2i, with
h y2i = L2/12. After rewriting eqn (5.2.113) of ref. 29 in our
notation, we arrived at t = L2/10D, with D given by eqn (9). On
passing, we notice that in our calculations we neglected two
relevant features of the fluidic system at hand, namely, the
softness of the cell walls, modeled by the potential, U(w), and
the finite length of the diffusing objects. Both impact the
effective width of the cell and therefore the mean-first passage
time across it. This may cause a small discrepancy between the
actual exponential decay time of hy(t)y(0)i and the predicted
value of t [not detectable in the inset of Fig. 2(a)].
Making use of our estimates for v2 and t we now compute
the product v2t, and eventually adding the thermal diffusion
term, eqn (8), we obtain our best fitting law,
D ¼ Dd ð1 þ a=lÞ þ
vs2
L2
;
30 Dd ð1 þ a=lÞ
(11)
which reproduces quite closely the data in the inset of Fig. 2(b).
Our phenomenological law for D also accounts for the peculiar
Fig. 2 Passive dimer diffusion in a Couette flow. (a) D/D(0) vs. a/l for l = 2, L = 20, vs = 0, and different values of self-propulsion force, p. The dashed
curve is the analytical prediction of eqn (9). Inset: Normalized autocorrelation function of y(t) for a = 0, p = 0 and different widths, L, of the cell and plate
sliding speeds, vs. The decay times of the fitting exponential curves compare well with the estimated mean-first passage time, t = L2/5D0 (see text);
(b) D/D(0) vs. vs for p = 0 and different a values. In the inset are the corresponding data for D. The dashed curve is the fitting law of eqn (11) for a = 0. D(0)
denotes the particle’s diffusion in the absence of hydrodynamic corrections, a = 0. Other simulation parameters are: l = 1, k = 1, and D0 = Dr = 0.03.
2796 | Soft Matter, 2017, 13, 2793--2799
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dependence of the ratio D/D(0) displayed in Fig. 2(b). D(0)
denotes here the numerically simulated diffusion constant for
the same parameters as D, except for a = 0. This quantity is thus
a measure of the hydrodynamic corrections to the diffusion
constant of a dimer under general shear and activation conditions.
As suggested by our fitting law, at small vs the ratio increases with
increasing a/l, whereas at large vs it decreases. The division between
pffiffiffiffiffiffiffiffiffiffi
such two regimes is located at around vcs ’ 15=2D0 ð1 þ a=lÞ=L,
where the two diffusion terms on the r.h.s. of eqn (9) become
comparable. Hydrodynamic interactions clearly have a distinct
impact on the dimer’s diffusion in the presence of a shear flow.
Most remarkably, the hydrodynamic corrections to D eventually
vanish for extremely large shear flows, as implied by the
asymptotic limit D/D(0) - 1, for vs/k - N. We explain this
property recalling that, during their roto-translational motion,
two bound monomers maintained at a finite distance are, in
general, subject to different local shear flows, which cause an
additional strain on the dimer’s bond. One thus expects the
ratio a/l to diminish with increasing vs, which leads to a
progressive suppression of the hydrodynamic effect.
Paper
typically much larger than the former. Hydrodynamic corrections
to these two diffusion terms must be treated separately. Moreover,
when propelling itself in a shear flow, an elastic dimer is subject to
hydrodynamic effects that result from the non-trivial interplay
of shear and activation. For this reason, in the forthcoming
subsection, we start discussing our numerical results for an
active dimer in a suspension fluid at rest. The more complicated case
of an active dimer in a Couette flow will be addressed in Section 4.2.
4.1
Zero flow
The datasets displayed in Fig. 3(a) clearly show a transition
between two diffusive regimes, which we agree to term thermal
and active, respectively. Indeed, at low p, the diffusion constant
is insensitive to p and dominated by thermal fluctuations, while
at large p it grows quadratically with p. The dependence of D on
the self-propulsion force is qualitatively reproduced well by
eqn (8) after the sequence of substitutions Dd - D0/2, Ds - v02/6Dr,
D0 - D0(1 + a/l), v0 - v0(1 + a/l), and v0 - p/2, introduced at the
bottom of Section 2.1. The final expression,
D ¼ Dd ð1 þ a=lÞ þ
4 Active dimer diffusion
The diffusive dynamics of active dimers has recently attracted
the attention of several authors. Observed experimentally,8 dimers
made of two coupled JPs with unaligned axes are known to spiral,
so that their diffusivity gets suppressed (chiral diffusion).35,36
Moreover, numerical simulation shows that in mixtures of JP–PP
pairs, the diffusivity of a tagged active dimer decreases due to
the coupling (direct or hydrodynamically mediated) with the
surrounding dimers.33,37,38 On the other hand, the hydrodynamics
of an isolated rigid JP–PP dimer has also been investigated in detail,
but limited to its impact on the dimer’s motility.13 To the best of our
knowledge the hydrodynamic effects on the diffusivity of a single
active dimer have not been explored yet.
The diffusion constant of an active swimmer, eqn (8), consists
of a thermal, Dd, and a self-propulsion term, Ds, the latter being
p2
ð1 þ a=l Þ2 ;
24Dr
(12)
for a = 0 is plotted in Fig. 3(a) for the sake of comparison. It
suggests a crossover between thermal and active diffusion at
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
around pc ’ 12D0 Dr in fair agreement with our numerical data.
In view of eqn (12), one expects that for p - N (active diffusion
regime), the ratio D/D(0) is expected to coincide asymptotically with
the square of the same ratio at p = 0 (thermal regime), namely,
D/D(0) = (1 + a/l) for p = 0 and D/D(0) - (1 + a/l)2 for p - N.
When increasing p larger than pc, the numerical data seem
to support our expectations also for rather large ratios a/l
reported in Fig. 3(b). However, on further increasing p, this
asymptotic prediction fails. A more detailed analysis of our
numerical data clearly shows that all ratios D/D(0) eventually
decay to 1 like [D/D(0) 1] p a/p; this produces the bell-shaped
profile of the curves D/D(0) versus p. Such a property can be
qualitatively explained by recalling that the dimer’s length is
not fixed. The force, p, acting on one monomer only tends to
Fig. 3 Active dimer diffusion at zero shear flow: (a) D vs. p for different values of the monomer radius, a, and the angular diffusion, Dr. The dashed curves
are the analytical prediction of eqn (12) with a = 0; (b) D/D(0) vs. p for different a and Dr values. D(0) denotes the particle’s diffusion in the absence of
hydrodynamic corrections, a = 0. Other simulation parameters are: k = 1, l = 1, L = 10, vs = 0, and D0 = Dr = 0.03.
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Fig. 4 Active dimer diffusion in a Couette cell with different plate sliding speeds, vs: (a) D vs. p for a = 0 and 0.3. The dashed curve is the analytical
prediction of eqn (12) with a = 0; (b) D/D(0) vs. p for a = 0.3 and different vs. D(0) denotes the particle’s diffusion in the absence of hydrodynamic
corrections, a = 0. Other simulation parameters are: k = 1, l = 1, L = 10, and D0 = Dr = 0.03.
stretch the elastic bond modeled by the potential U(e) of eqn (2),
so that l grows with p. Indeed, upon neglecting the rotational
corrections in eqn (10) and assuming steady propulsion conditions,
a simple force balance calculation yields
l(p) = l + p/2k.
(13)
As a consequence, on increasing p the ratio a/l(p) tends to
vanish, no matter what a is, and so does the hydrodynamic effect
on the active dimer’s diffusion, D/D(0) - 1. More precisely, for
exceedingly large values of p/k, one easily derives the following
decay law in leading order of a/p, D/D(0) = [1 + a/l( p)]2 E 1 + 4ak/p.
This mechanism is appreciable when the strain of the dimer’s
bond, l(p) l, grows larger than its thermal fluctuations, which
at p = 0 have a variance Drij2 = D0/k.20 This occurs for p 4 pd,
pffiffiffiffiffiffiffiffiffiffiffi
where pd ¼ 2kD0 . Accordingly, pd roughly locates the broad
peaks of the curves plotted in Fig. 3(b).
4.2
Couette flow
Simulation data for the diffusion constant of an active dimer
self-propelling in a Couette cell with plate sliding speed vs are
plotted in Fig. 4. Note that for the small values of vs considered
here, the effect of the shear vorticity on the active dimer
diffusion is, indeed, negligible.23 In Fig. 4(a) we illustrate the
dependence of D on the modulus of the self-propulsion force, p,
for different values of vs. At low p, as expected, one recovers the
diffusion constant of the corresponding passive dimer, reported in
the inset of Fig. 2(b). At large p, instead, D grows quadratically with
p, as discussed in Section. 4.1 in the absence of shear flow, vs = 0.
The crossover between shear and active diffusion occurs for a value
of the force, ps, which can be estimated by comparing the relevant
(approximate) expressions for D at p = 0, eqn (11), and vs = 0,
eqn (12). For large shear flows, vs 4 vcs, one immediately sees that
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ps Dr =D0 vs L, in good agreement with the data in Fig. 4(a).
On closer inspection, one also notices that, for higher shear
flows, in Fig. 4(a) the low-p branch of the curves D(p) develops
a negative curvature. This is a consequence of the fact that
(for p = 0 and vs 4 vcs) the diffusion constant is approximately
proportional to the mean-first passage time, t, across the
2798 | Soft Matter, 2017, 13, 2793--2799
Couette cell, see the derivation of eqn (11) in Section 3. An explicit
calculation shows that t is quite sensitive to the active component
of the dimer’s Brownian motion,36 and, more precisely, decreases
with increasing p. This causes the small dips in the diffusion
curves observed for the highest vs plotted in Fig. 4(a).
These two distinct diffusion regimes are also clearly visible
in Fig. 4(b), where our data for the ratio D/D(0), at p = 0, start
with the corresponding point on the curve a = 0.3 of Fig. 2(b)
and for p - N, decay asymptotically to 1. More interesting is
the intermediate p range, where the hydrodynamic effect on the
dimer diffusion is magnified by the interplay between shear
and activation. At low vs, we recover the bell-shaped curves
already reported in Fig. 3(b) for zero shear. On increasing the
self-propulsion force for vs larger but close to vcs, the ratio D/D(0)
jumps from below 1 up to a maximum of the order of 2. The
relative correction to the constant D due to the hydrodynamic
interactions thus proves to be quite large. The maxima of the
curves D/D(0) versus p are not higher than the maximum
obtained in the absence of shear flow, i.e., for vs = 0, which
based on eqn (12), we know to be centered at p C ps and of the
order of (1 + a/l)2. Note, however, that the validity of eqn (12) is
restricted by the condition p c vs. Accordingly, as one increases
vs, the ratio D/D(0) becomes suppressed for p o vs and,
consequently, its maximum lowered and shifted to higher p.
5 Conclusions
We studied the 3D dynamics of an elastic active dimer consisting
of a Janus particle bound to a passive cargo freely diffusing in
a Couette flow. Extensive numerical simulations led to the
conclusion that corrections due to long-range hydrodynamic
interactions affect the dimer’s diffusive properties to an extent that
depends on three control parameters: the size of its constituents,
its self-propulsion speed, and the shear flow. We will show in a
forthcoming paper that similar results also apply to active dimers
diffusing in a Poiseuille flow, even if in that case the shear-flow
vorticity cannot be neglected.23 A direct demonstration of the
hydrodynamic effects investigated in the present work requires
This journal is © The Royal Society of Chemistry 2017
Soft Matter
only an affordable experimental setup of the kind sketched in
Fig. 1(b). More complicated setups are also conceivable.39
We conclude this paper with a final remark. We simulated
the dimer probability density across the Couette cell for different
values of control parameters and noticed that the dimers tend to
accumulate against the sliding plates, thus resulting in density peaks
near the cell walls. This phenomenon cannot be explained as a mere
manifestation of the long-range hydrodynamic interactions, even if
its magnitude actually increases with the radius of the dimer’s
constituents. As a matter of fact, a marked dimer accumulation
was also detected for active dimers with pointlike constituents, i.e.,
in the absence of hydrodynamic corrections, under the condition
that their self-propulsion speed is sufficiently high; such an effect
gets further enhanced in strong shear flows. Our observations point
to a deterministic mechanism, whereby, in the presence of strong
torques, either from the propulsion mechanism or the shear flow,
the rotation of the dimer is hampered by the walls, thus determining
its prolonged sojourn against them. A consistent analysis of all these
boundary effects requires an efficient encoding of the hydrodynamic
interactions between the dimer and the walls.10 We leave this task
for a forthcoming publication.
Acknowledgements
We thank RIKEN’s RICC for computational resources. Y. Li was
supported by the NSF China under grant No. 11505128 and the
Tongji University under grant No. 2013KJ025. P. K. G. was
supported by a SERB Start-up Research Grant (Young Scientist)
No. YSS/2014/000853 and the UGC-BSR Start-Up Grant No. F. 30-92/
2015. F. N. was partially supported by: the RIKEN iTHES Project, the
MURI Center for Dynamic Magneto-Optics via the AFOSR Award No.
FA9550-14-1-0040, the Japan Society for the Promotion of Science
(KAKENHI), the IMPACT program of JST, CREST, and a grant from
the John Templeton Foundation. T. D. thanks UGC, New Delhi,
India, for the award of a Junior Research Fellowship.
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