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Soft Matter
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Hydrodynamic si
aTheoretical So-Matter and Biophysi
Forschungszentrum Jülich, 52425 Jülich,
[email protected] National Laboratory for
Condensed
Matter Physics, Institute of Physics, Chine
China
Cite this: Soft Matter, 2014, 10, 6208
Received 21st March 2014Accepted 15th June 2014
DOI: 10.1039/c4sm00621f
www.rsc.org/softmatter
6208 | Soft Matter, 2014, 10, 6208–621
mulations of self-phoreticmicroswimmers
Mingcheng Yang,*ab Adam Wysockia and Marisol Ripolla
A mesoscopic hydrodynamic model to simulate synthetic
self-propelled Janus particles which is
thermophoretically or diffusiophoretically driven is here
developed. We first propose a model for a
passive colloidal sphere which reproduces the correct rotational
dynamics together with strong phoretic
effect. This colloid solution model employs a multiparticle
collision dynamics description of the solvent,
and combines stick boundary conditions with colloid–solvent
potential interactions. Asymmetric and
specific colloidal surface is introduced to produce the
properties of self-phoretic Janus particles. A
comparative study of Janus and microdimer phoretic swimmers is
performed in terms of their swimming
velocities and induced flow behavior. Self-phoretic microdimers
display long range hydrodynamic
interactions with a decay of 1/r2, which is similar to the decay
of gradient fields generated by self-
phoretic particle, and can be characterized as pullers or
pushers. In contrast, Janus particles are
characterized by short range hydrodynamic interactions with a
decay of 1/r3 and behave as neutral
swimmers.
I. Introduction
Synthetic microswimmers have recently stimulated consider-able
research interest from experimental1–6 and
theoreticalviewpoints.7–9 This is due to their potential practical
applica-tions in lab-on-a-chip devices or drug delivery, and
fundamentaltheoretical signicance in non-equilibrium statistical
physicsand transport processes. Self-phoretic effects have shown to
bean effective and promising strategy to design such
articialmicroswimmers,3–5,7,10–12 where the microswimmers are
drivenby gradient elds locally produced by swimmers themselves
inthe surrounding solvent. In particular, the collective behavior
ofa suspension of self-diffusiophoretic swimmers has recentlybeen
studied in experiments.13–16
Phoresis refers to the directed dri motion that
suspendedparticles experience in the presence of a gradient
eld.10
Important examples are thermophoresis (induced by gradientsof
temperature), diffusiophoresis (gradients of concentration),or
electrophoresis (gradients in the electric potential).
Self-phoretic swimmers are typically composed of two parts:
afunctional part which modies the surrounding solvent prop-erties
creating a local gradient eld, and a non-functional partwhich is
exposed then to the local gradient eld. Most existingexperimental
investigations of the self-phoretic microswimmers
cs, Institute of Complex Systems,
Germany. E-mail: [email protected];
Matter Physics and Key Laboratory of So
se Academy of Sciences, Beijing 100190,
8
consider Janus particles, which can be quite easily
synthesizedusing partial metal coating on colloidal spheres.3,5 In
dif-fusiophoretic microswimmers, the metal coated part catalyzes
achemical reaction to induce a local concentration gradient.
Inthermophoretic microswimmers, the metal coated part is ableto
effectively absorb heat from e.g. an external laser, whichcreates a
local temperature gradient. The investigations per-formed by
computer simulations have mostly considered dimerstructures
composed of two connected beads instead of Janusparticles.17–20
This is motivated by the simplicity of the structurewhich can be
approached by a two beads model. Janus particleshave been recently
simulated by employing a many beadsmodel,21,22 which has provided
an interesting but computa-tionally costly approach. The
fundamental differences on thehydrodynamic behavior of Janus and
dimer swimmers, as wellas the interest in the investigation of
collective phenomena ofthese systems strongly motivates the
development of simpleand effective models to simulate the
self-phoretic Janusparticles.
A single-bead model of the self-phoretic Janus particle
insolution is here proposed, together with a detailed
comparativestudy of the hydrodynamic properties of dilute solutions
of aself-phoretic Janus particle and a self-phoretic
microdimer.While the solvent is explicitly described by a
mesoscopicapproach known asmultiparticle collision dynamics (MPC),
it isnecessary to develop a description of a colloidal particle
able toproduce strong phoretic effect, and reproduce the correct
rota-tional dynamics. The proposed colloid model combines into
asingle bead, potential interactions with the solvent and
stickhydrodynamic boundary conditions. The properties of
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self-phoretic Janus particles are introduced then with
asym-metric and specic particle surface. The validity of the model
isshown by implementing the simulations of both the
self-dif-fusiophoretic and self-thermophoretic microswimmers. Theow
eld induced by the self-phoretic Janus particle ismeasured and
compared with that around the self-phoreticdimer and their
analytical predictions. The efficiency of themodel and the
consistency of the results puts this methodforward as a reliable
and powerful tool to investigate thecollective behavior of
self-phoretic microswimmers.
Fig. 1 Schematic diagram of the three regions of the
interactions as afunction of r the distance between the solvent
particle and the colloidcenter. The inset is a sketch of a Janus
particle.
II. Simulation of a Janusmicroswimmer in solution
The typical sizes and time scales of a Janus colloidal
particleand the surrounding solvent particles are separated by
severalorders of magnitude which are impossible to cover with
amicroscopic description. Over the last decades various meso-scopic
simulation methods have been developed to bridge suchan enormous
gap. Here, we employ an especially convenienthybrid scheme that
describes the solvent by MPC which is acoarse-grained
particle-based method,23–28 while the interac-tions of the Janus
particle with the solvent are simulated bystandard molecular
dynamics (MD).
MPC consists of alternating streaming and collision steps. Inthe
streaming step, the solvent particles of mass m moveballistically
for a time h. In the collision step, particles aresorted into a
cubic lattice with cells of size a, and their velocitiesrelative to
the center-of-mass velocity of each cell are rotatedaround a random
axis by an angle a. In each collision, mass,momentum, and energy
are locally conserved. This allows thealgorithm to properly capture
hydrodynamic interactions,thermal uctuations, to account for heat
transport and tomaintain temperature inhomogeneities.29,30
Simulation unitsare chosen to be m ¼ 1, a ¼ 1 and kB�T ¼ 1, where
kB is theBoltzmann constant and �T the average system
temperature.Time and velocity are consequently scaled with
(ma2/kB�T)
1/2 and(kB�T/m)
1/2 respectively. The solvent transport properties aredetermined
by the MPC parameters.31,32 Here, we employ thestandard MPC
parameters a ¼ 120�, h ¼ 0.1, and the meannumber of solvent
particles per cell r ¼ 10, which correspondsto a solvent with a
Schmidt number Sc ¼ 13. The simulationsystem is a cubic box of size
L ¼ 30a with periodic boundaryconditions.
By construction, a Janus particle has a well-dened orienta-tion
with a corresponding well-dened rotation, and surfaceproperties are
different in the two colloid hemispheres. Inprevious studies of
colloid phoresis with MPC,18,19,33–35 a centraltype of interaction
such as the Lennard Jones potential has beenemployed, which does
not result in a rotational motion. Aprevious study of rotational
colloidal dynamics36 has alreadyemployed MPC with stick boundary
conditions. This was per-formed by drawing the relative
post-collisional solvent velocityfrom a Maxwell–Boltzmann
distribution with the temperatureas a control parameter. This means
that the solvent was effec-tively thermalized at the whole colloid
surface, which articially
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perturbs the temperature and concentration elds. In order
toproperly consider the effect of temperature and
concentrationnon-homogeneous elds, the development of an
alternativeapproach is necessary. In this work, we rst modify
existingtechniques to construct a specic model which allows us
tosimulate a colloid with stick boundary conditions together
withpotential interactions with the solvent. These boundary
condi-tions locally conserve not only mass and momentum, but
alsoenergy. Then, in order to reproduce the properties of a
Janusparticle, the spherical colloid is divided in two
hemispherescharacterized by different interactions with the
surroundingsolvent. One of these hemispheres (with a polar angle q
# p/2with respect to a dened colloid axis n) is considered to be
thefunctional part, while the other one is the non-functional
part.The functional part of the Janus particle is where the
materialhas special properties like enabling a chemical reaction
(cata-lytic) or carrying a high temperature due to a larger
heatadsorption. The special behavior of the functional part
origi-nates local gradients (as of concentration or temperature)
whichwill induce a phoretic force applied to the Janus colloid. In
thefollowing sections we introduce rst the model for a colloidwith
stick boundary conditions and a well-dened orientation,and then
consecutively the thermophoretic and diffusiopho-retic Janus
particles.
A. Passive colloid with stick boundary: simulation model
A colloidal particle with stick boundary conditions will
rotaterandomly. This is caused by the stochastic torque exerted on
theparticle due to collisions with the solvent. On a coarse
grainedlevel, stick boundary conditions can be modeled by the
bounceback (BB) collision rule,37,38 this is by reversing the
direction ofmotion of the solvent particle with respect to the
colloidalsurface. However, the bounce back rule does not
inducesignicant phoretic effects, such that it is necessary to
combineit with a so potential. Practically, we realize this by
deningthree interaction regions, as shown in Fig. 1. For solvent
parti-cles at distances to the center of a colloid, r, larger than
thecutoff radius, r > rc, there is no interaction. For rb > r
> rc just theso central potential is considered. And for r <
rb, both the so
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Fig. 2 Time auto-correlation function of the orientation vector
of
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potential and the bounce back collision are taken into
account.The value of the bounce-back radius rb should be large
enough toensure that a certain amount of solvent particles
participate inthe bounce-back collision such that a signicant
rotationalfriction is induced. On the other hand, the value of rb
shouldalso be small enough such that the colloid–solvent
potentialeffectively contribute to the phoretic force.
The interaction potential employed in this work is
ofLennard-Jones (LJ) type,39 with the general form
UðrÞ ¼ 43"�
s
r
�2k��s
r
�k#þ C; r# rc: (1)
The positive integer k controls the stiffness of the
potential,and rc is the potential cutoff radius. The potential
intensity ischosen as one of the system units 3 ¼ kB�T ¼ 1, and the
inter-action length parameter as s¼ 2.5a. In this work we choose
rb¼s which is also a good estimation for the colloid
radius.Attractive interactions are obtained with C ¼ 0, and rc ¼
2.5sand repulsive with C¼ 3 and rc¼ 21/ks. Themass of the
colloidalparticle is set to M ¼ 4ps3mr/3 ¼ 650m, such that the
colloid isneutrally buoyant. Between two MPC collision steps,
Nmdmolecular dynamics steps are employed. The equations ofmotion
are integrated by the velocity-Verlet algorithm with atime step Dt
¼ h/Nmd, where we use Nmd ¼ 50.
In most cases the bounce-back collision is applied to
theinteraction between solvent particles and immobile planarwalls,
where the particle velocity is simply reversed. Here incontrast, an
elastic collision is performed when a point-likesolvent particle
with velocity v is moving towards the sphericalcolloid and is
closer to it than rb, this is r < rb. The colloidalparticle has
a linear velocity V, an angular velocity u, and amoment of inertia
I ¼ cMs2, with c ¼ 2/5 the gyration ratio.Since the collision is
now performed with a moving object, therelevant quantity for the
collision is ~v, namely, the solventparticle velocity relative to
the colloid at the colliding point,
~v ¼ v � V � u � s, (2)
where s¼ r� R, with r and R, the position of the solvent
particleand of the center of the colloid, respectively. In the
following, werefer to s as the contact vector and ~v as the contact
velocity. Theconservation of linear and angular momentum imposes
thefollowing explicit expressions for the post-collision
velocities
v0 ¼ v � p/m,
V0 ¼ V + p/M, (3)
u0 ¼ u + (s � p)/I.
6210 | Soft Matter, 2014, 10, 6208–6218
The precise form of the momentum exchange p can becalculated in
terms of the normal and tangential components ofthe contact
velocity ~vn¼ ŝ(̂s$~v), and ~vt¼ ~v � ~vn, with ŝ¼ s/|s| theunit
contact vector. Imposing the conservation of kinetic energyand
stick boundary condition (see calculation details in theAppendix A)
leads to ~v 0n ¼ �~vn and ~vt 0t ¼ �~v, which determines
p ¼ pn þ pt ¼ 2m~vn þ2mcM
cM þ m~vt; (4)
where m¼mM/(m +M) is the reducedmass. This collision rule
issimilar to the one used in rough hard sphere systems,40,41
although in the present case the colliding pair is composed of
apoint particle and a rough hard sphere.42 This collision does
notchange the positions of the particles and, consequently,
thepotential energy does not vary discontinuously.
B. Passive colloid with stick boundary: simulation results
In order to test the correct rotational dynamics of the
proposedmodel, we rst verify the exponential decay of the
orientationaltime-correlation function. This is expected to
be,43
hn(t)$n(0)i ¼ exp(�2Drt), (5)
with n the body-xed orientation vector, and Dr the
rotationaldiffusion constant. A repulsive potential with k¼ 24 in
eqn (1) ischosen for the colloid–solvent interactions. Although
otherchoices would have been possible, the short range of
thispotential is convenient since the hydrodynamic radius
isexpected to be closer to the colloidal radius s, which
makeseasier the comparison with analytical predictions. A t of
eqn(5) to our data (shown in Fig. 2) yields Dr ¼ 0.0015 in units
of(kB�T/ma
2)1/2. In order to provide an analytical estimation of
thiscoefficient, it should be taken into account that within the
cut-off-radius, the number density of the solvent particles obeys
r(r)¼ re�U(r)/kBT due to the ideal gas equation of state of the
MPCsolvent. This results into a position-dependent viscosity. In
thefollowing, we refer to the local number density at the
colloidsurface as rs ¼ r(s) ¼ re�1. The corresponding dynamic
andkinematic viscosity at the particle surface are obtained using
thedependence of h on r from the kinetic theory.32 For the MPC
passive colloidal sphere. Symbols refer to simulation results,
and theline to eqn (5).
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solvent employed parameters, we obtain a global shear viscosityh
¼ 7.93, a local shear hs ¼ 2.47, and a local kinematic viscosityns
¼ 0.67, corresponding to the solvent density at a distance sfrom
the center of the colloid. The Stokes–Einstein equation forthe
rotational diffusion provides the dependence Dr ¼ kBT/zH,with the
hydrodynamic rotational friction zH ¼ 8phss3. Withthis
approximation, we obtain Dr ¼ 0.001, which underesti-mates, but is
still consistent with the simulation result.
The rotation dynamics can be further analyzed by measuringthe
angular velocity autocorrelation function of the colloidalparticle.
For short times, Enskog kinetic theory36,44 predicts thatthe
autocorrelation function follows a exponential decay,
limt/0
huðtÞ$uð0Þi ¼ �u2�expð�zEt=IÞ; (6)with hu2i ¼ 3kBT/I, as
obtained from energy equipartitiontheorem, and zE the Enskog
rotational friction coefficient of asphere suspended in bath of
point-like particles,42
zE ¼8
3
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2pkBTm
prss
4 cM
mþ cM : (7)
For long times, the relaxation of the correlation function
ispredicted by hydrodynamic mode-coupling theory36,45 to
decayalgebraically,
limt/N
huðtÞ$uð0Þi ¼ 3pkBTmrsð4pnstÞ5=2
: (8)
The angular velocity autocorrelation function obtained
fromsimulations is displayed in Fig. 3. It agrees very well with
thetheoretical predictions at short and long time regimes
respec-tively in eqn (6) and (8), where no adjustable parameter
isemployed. On the other hand, the rotational diffusion
coeffi-cient can be understood to be determined by the total
friction z,with 1/z ¼ 1/zH + 1/zE. Considering both terms, the
analyticalprediction is Dr ¼ kBT/z ¼ 0.002, which overestimates
then themeasured value of Dr. Improvements to this estimation
shouldfollow different simultaneous routes. A more accurate
treat-ment of the density and viscosity inhomogeneity is to solve
the
Fig. 3 Time decay of the angular velocity autocorrelation
function ofpassive colloidal sphere. Symbols correspond to
simulation results, thedashed line to short-time Enskog prediction
in eqn (6) and solid line tothe long-time hydrodynamic prediction
in eqn (8).
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Stokes equation with a inhomogeneous viscosity prole.46
TheEnskog and the hydrodynamic times scales are not enoughseparated
in this case to consider the previous additive depen-dence as
accurate. In conclusion, these results ensure that
thecoarse-graining model introduced here describes
physicallycorrect rotational dynamics where no surface
thermalizationhas been employed. This is the basic colloidmodel on
which theJanus structure can be further introduced.
C. Self-thermophoretic Janus colloid
In the presence of a temperature gradient a suspended
colloidexperiences a directed force as a result of the unbalance of
thesolvent–colloid interactions, this translates into the
particledri most frequently towards cold areas, but eventually
alsotowards warm areas. This effect is know as
thermophoresis.47–49
A Janus particle partially made/coated with a material of
highheat absorption and heated, for example with a laser,
developsaround it an asymmetric temperature distribution.5 The
localtemperature asymmetry therefore induces a signicant
ther-mophoretic force on the Janus particle. Depending on thenature
(thermophilic/thermophobic) of the colloid–solventinteractions, the
thrust will be exerted towards or against thetemperature
gradient.
The simulation model combines now the rotating colloidintroduced
in the preceding section, with elements of thepreviously
investigated self-thermophoretic dimer.19 In partic-ular, we impose
a temperature Th (higher than the bulktemperature �T) in a small
layer (z0.08s) around the heatedhemisphere. The temperature Th is
achieved by rescaling thethermal energy of the solvent particles
within this layer. In thiswork we have restricted ourselves to Th ¼
1.25�T , although alarge range of possible values is accessible.
The inserted energyis drained from the system by thermalizing the
mean temper-ature of the system to a xed value �T. In experiments,
thethermalization is performed at the system boundaries.Although
these two thermalizations are intrinsically different,
Fig. 4 Temperature distribution induced by a
self-thermophoreticJanus particle. Here, the Janus particle has a
repulsive LJ potential withk ¼ 3. The right (h) and the left (n)
hemisphere correspond to theheated and the non-heated parts,
respectively. Because of axis-symmetry, only the distribution in a
section across the axis is displayed.
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Fig. 6 MSD of the center-of-mass of the Janus particle along
thepolar axis as a function of time. Simulation parameters are
those ofFig. 5. Lines correspond to fits with eqn (9).
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the differences are expected to be negligible, when the system
islarge enough, and especially when considering the neighbor-hood
of the Janus particle. A typical temperature distributiongenerated
by the Janus particle is plotted in Fig. 4.
Two different colloid–solvent potentials of LJ-type eqn (1)
areemployed in the simulations provided here, a so
repulsivepotential with k¼ 3, and a short-range attractive
potential with k¼ 24. The particular shape of the colloid–solvent
potential hasalready shown19,34 to inuence the magnitude of the
thermo-phoretic force, and interestingly also its direction. The
repulsiveLJ potential is expected to produce a thrust pointing to
theheated hemisphere; while the attractive potential will lead to
adriving force in the opposite direction. Two procedures
areemployed to quantify the self-propelled velocity vp. A
directcharacterization can be performed by projecting the
center-of-mass velocity of the Janus particle on its polar axis, vp
¼ hV$ni.Fig. 5 shows how direct measurements of vp are well-dened
fordifferent interaction potentials as a function of time, which
isemployed as an averaging parameter. Indirect determination ofvp
is obtained by measuring the mean square displacement(MSD) of the
Janus particle along its polar axis. In this direction,the motion
of the Janus particle can be divided into a purediffusion and a
pure dri, and it is related to the self-propelledvelocity via
h(Dxp)2i ¼ 2Dpt + vp2t2. (9)
Here Dp is the translational diffusion coefficient of the
Janusparticle along its axis. The mean square displacement in
thepolar direction is shown in Fig. 6 as a function of time for
Janusparticles with different colloid–solvent interactions. At
verysmall times, an initial inertial regime with a quadratic
timedependence is observed. For times larger than the Browniantime,
the diffusive behavior coexists with the presence of
theself-propelled velocity as indicated in eqn (9). A t to the
dataallows us to determine both Dp and vp with good accuracy.
Fig. 5 Self-propelled velocity as a function of time as
averagingparameter. Triangles refer to self-thermophoretic Janus
particles withrepulsive and attractive LJ-type potentials. Circles
refer to self-dif-fusiophoretic Janus particles. Solid symbols
refer to forward motion,namely along the polar axis towards the
functional part. Open symbolsrefer to backwards motion. For
reference, squares denote the velocitymeasured for a purely passive
colloid.
6212 | Soft Matter, 2014, 10, 6208–6218
Direct and indirect determination of vp agree very well
withinthe statistical accuracy, as can be seen in Table 1. Note
that bymeasuring the MSD along the polar axis, instead of in
thelaboratory frame, we suppress the contribution due to
trans-lation–rotation coupling, such that direct comparison with
theanalytical prediction by Golestanian50 is not appropriate.
The quantitative values of the propelled velocities
aredetermined by the nature of the thermophoretic forces. As inthe
case of thermophoretic microdimers,19 these forces arerelated to
the temperature gradients VT, and the thermaldiffusion factor aT
which characterizes the particularities of thecolloid–solvent
interactions.34,49,51 The self-propelled velocity isthen vp ¼
�aTVkBT/gp, with gp the particle translational fric-tional
coefficient and Dp ¼ kBT/gp. The hydrodynamic trans-lational
frictional coefficient is gHp ¼ Bhs with B being anumerical factor
given by the boundary conditions. Colloidswith stick boundary
conditions have B¼ 6p, while colloids withslip boundary conditions
have B ¼ 4p. The here proposedmodel provides stick boundary
conditions for colloids at r x rb¼ swith the surface viscosity hs,
and slip for r > rb, whichmeansthat the overall colloid behavior
will be effectively partial slip.The stick boundary approach
predicts kBT/(6phs) x 0.0027,such that the slightly larger
simulation results in Table 1 areconsistent with the partial slip
prediction. In principle thesevalues should still be corrected by
considering the Enskogcontribution and nite size effects. However,
the precise formand validity of these corrections is still under
debate for colloidssimulated with MPC.26,42 It can be observed that
the values for
Table 1 Summary of the self-propelled velocities, and the
diffusioncoefficient of the thermophoretic and diffusiophoretic
Janus particlesobtained from the simulations with the direct and
indirect methods.For comparison the same quantities are displayed
for our prior resultson the thermophoretic microdimers19
Therm-att Therm-rep Diff-A / B Diff-B / A
vp (direct) �0.0131 0.0030 �0.0065 0.0059vp (indirect) �0.0133
0.0035 �0.0070 0.0059Dp (indirect) 0.0029 0.0035 0.0032 0.0032vp
(dimer) �0.0068 0.0047Dp (dimer) 0.0028 0.0034
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the thermophoretic attractive potential are smaller than
thosefor the repulsive one, which reects the larger viscosity
hsprovided by the attractive surface interactions. Interestingly,
thevalues for Dp of the thermophoretic dimers are very similar
tothose of the Janus particles. This can be understood as theresult
of two canceling effects. On the one hand the microdimerhas larger
size than the Janus particle, which decreases thetranslational
diffusion. On the other hand, the microdimer issimulated with slip
boundary conditions, which reduces thefriction in comparison with
the stick, or partial slip boundaryconditions employed for the
Janus particle. Given that Dp is notsignicantly changing for the
results in Table 1, the variation ofnumerical values of vp can be
related to the differences in VTand aT. The actual value of VT
varies along the particle surface,and it is not the same for both
particle geometries. The deter-mination of aT is given by the size,
the geometry, and thespecic interactions between the colloid and
the solvent. Thecomparison of the measured vp for the dimer and the
Janusparticles is therefore non-trivial and deserves a more
in-depthinvestigation. Furthermore, the bounce-back surface
consid-ered in the Janus particle model produces an additional
ther-mophobic thrust, which could explain the enhanced value ofthe
Janus particle with attractive interactions.
In the presence of a temperature gradient, the transport ofheat
is a relevant process which in experimental systems occursin a much
faster time scale than the particle thermopho-resis.5,47,52 For
thermal energy propagation the characteristictime is sk � a2/k with
k the thermal diffusivity, and the timescale of particle motion is
related to the self-propelled velocityby sm � a/vp. Using k
estimated from kinetic theory32 and themeasured vp, we have sk/sm �
10�1 for our simulation param-eters. This means that both times are
also well-separated in thesimulations, and that temperature prole
around the swimmeris almost time-independent.
Fig. 7 Potential interactions between the two solvent species
and theJanus particle. UA(r) is defined in eqn (10) and UB(r) in
eqn (1). Inset:schematic representation of the catalytic and
non-catalytic hemi-spheres of the Janus particle and the
interaction of the A and B specieswith each hemisphere, for the A /
B reaction.
D. Self-diffusiophoretic Janus colloid
A colloidal particle with a well-dened part of its surface
withcatalytic properties can display self-propelled
motion.1,3,13,14,53
Such functional or catalytic part of the Janus particle
catalyzes achemical reaction, which creates a surrounding
concentrationgradient of the solvent components involved in the
reaction,which typically have different interactions with the
colloid. Thisgradient in turn induces a mechanical driving force
(dif-fusiophoretic force) on the Janus particle and hence
propulsion.The direction of the self-propelled motion will be
related to theinteraction of each solvent component with the
colloid.Chemical reactions are generally accompanied by an
adsorptionor emission of energy. A catalytic Janus particle could
thereforegenerate a local temperature gradient which would induce
anadditional thermophoretic thrust. However, existing experi-ments
of Pt–Au micro-rods1 have shown the contribution of thiseffect to
be negligible.
The effect of irreversible chemical reactions has already
beenincluded in a MPC simulation study of chemically
powerednanodimers by Rückner and Kapral.17 Similar to that work,
wehere consider a solvent with two species A and B, together
with
This journal is © The Royal Society of Chemistry 2014
the model of the stick boundary colloid previously
introduced.The reaction A/ B is performed with a probability pR
wheneveran A-solvent particle is closer than a distance r1 to the
catalytichemisphere of the Janus particle (see inset of Fig. 7).
Besidesthis reaction, A and B solvent particles interact simply via
theMPC collision. Another important element to induce
self-propelled motion is that the interaction of each componentwith
the colloid surface should be different.17 We thereforeconsider
that solvent species A and B interact with the Janusparticle with
different potentials UA(r) and UB(r), but with thesame bounce-back
rule. A change of potential energy at thepoint where the A / B
reaction occurs could be numericallyunstable, and would lead to a
local heating or cooling of thesurrounding solvent. In order to
model here a purely dif-fusiophoretic swimmer, we choose smoothly
varying potentialsUA(r) and UB(r) which completely overlap for r #
r1, ensuring areaction without an energy jump. We consider UB(r) as
therepulsive LJ-type potential in eqn (1) with k ¼ 12. UA(r) in
Fig. 7is constructed in four intervals by a cubic spline
interpolation,which yields to
UAðrÞ ¼
UBðrÞ ðr# r1Þa0 þ a1rþ a2r2 þ a3r3 ðr1 # r# r2Þb0 þ b1rþ b2r2 þ
b3r3 ðr2 # r# r3Þ0 ðr3 # rÞ
8>>><>>>:
(10)
where the coefficients and the distances to determine therelated
intervals are specied in the Table 2.
Simulations are initiated with a solvent composed only of A-type
particles. The considered chemical reaction A / B in thecatalytic
part of the Janus particle is irreversible, such that A-type
solvent particles are gradually consumed. Chemicallyreacting
systems with different spatial distributions can inprinciple be
implemented.54 We choose a simple scheme tokeep a stationary
concentration gradient. The reaction proba-bility is xed to pR ¼
0.1, and whenever a B-type particleis beyond a distance d from the
Janus particle (we considerd ¼ 5s), it automatically converts into
A. This allows the system
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Table 2 Coefficients employed in the simulations for the
potentialfunction UA in eqn (10)
a0 ¼ 844.6 a1 ¼ �849.7 a2 ¼ 280.7 a3 ¼ �30.3b0 ¼ 3283 b1 ¼ �3610
b2 ¼ 1322 b3 ¼ �161.4rb ¼ s r1 ¼ 1.0132s r2 ¼ 1.06s r3 ¼ 1.12s
Fig. 8 Local number density distribution of B-type particles
inducedby a self-diffusiophoretic Janus particle with the A / B
reaction. Theright hemisphere corresponds to the catalytic
part.
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to reach an steady-state concentration distribution of B
mole-cules around the swimmer. Fig. 8 shows rB, the number of
B-type particles per unit cell. It can be seen that on the
catalytichemisphere there are mostly B-type particles, while on the
non-catalytic hemisphere the situation is reversed and there
aremostly A-particles. For comparison, the reaction B / A is
alsoconsidered by reversing the roles of A and B. The
self-propelledvelocity is quantied by using the direct and indirect
methodsas already described for the self-thermophoretic Janus
particles.The results are displayed separately in Fig. 5 and 6, and
thenumerical values are summarized in Table 1 where the
niceagreement between the methods can be observed. The
diffusioncoefficients for both diffusiophoretic Janus particles are
thesame, which is related to the fact that at the surface
bothpotentials are the same. The value of the
self-propulsionvelocity, vp, is determined by the choice of the
colloid–solventpotentials, the reaction probability and the
boundary condi-tions. For the considered A / B reaction with UB(r)
repulsive,and UA(r) attractive, the concentration gradient pushes
theJanus particle against the direction of the polar axis
n.Conversely, the reaction B / A pushes the Janus particle alongn
as can be veried in Fig. 5 and Table 1. It should be noted thatthe
values of the velocities in both simulations are not
exactlyreversed, since the reciprocal choice of potentials does
notcorrespond to a perfectly reverse distribution of the
speciesconcentrations. A comparison of the velocities for the
dif-fusiophoretic Janus particle in this work, and the existing
datafor microdimers and Janus particles17,22 is not really
straight-forward since the employed parameters and potentials
aredifferent. The systems are though not so different, and
thevalues of vp range from similar values to approximately
fourtimes smaller.
The time scale of the particle motion of a
self-diffusiopho-retic swimmer sm needs to be compared with the
time scale of
6214 | Soft Matter, 2014, 10, 6208–6218
solvent molecule diffusion ss, which is a much faster process
inexperimental systems. The solvent diffusion coefficient
Dsdetermines ss¼ a2/Ds. For the employed simulation parameters,Ds
from the kinetic theory, and the measured vp determine
theseparation of both time scales to be ss/sm � 10�1.
III. Flow field around phoreticswimmers
In the previous section, an efficient model to simulate
thebehavior of self-phoretic Janus particles has been
introduced,and the obtained velocities have been related with the
employedsystem parameters. Another fundamental aspect in the
inves-tigation of microswimmers is the effect of
hydrodynamicinteractions,55 and how do these compare with the
effect ofconcentration or temperature gradients. In the case of
self-phoretic particles, the temperature or concentration
distribu-tions decay with 1/r around the particle, such that their
gradi-ents decay as 1/r2. Furthermore, the hydrodynamic
interactionshave shown to be fundamentally different for swimmers
ofvarious geometries and propulsion mechanisms, yielding
tophenomenologically different behaviors classied in threetypes:
pullers, pushers, and neutral swimmers.55 In thefollowing, we
investigate the solvent velocity elds generated bythe self-phoretic
Janus particles, as well as those generated byself-phoretic
microdimers, and in both cases the analyticalpredictions are
compared with simulation results. The velocityeld around a
self-propelled particle can be analytically calcu-lated from the
Navier–Stokes equation. Here, we solve theStokes equation, which
neglects the effect of inertia due to verysmall Reynolds number,
and consider the incompressible uidcondition.10,35 Note that
although MPC has the equation of stateof an ideal gas, the
compressibility effects of the associated owelds have shown to be
very small in the case of thermophoreticparticles.35 We also
implicitly assume that the standardboundary layer approximation is
valid, this is that the particle–solvent interactions are
short-ranged. Moreover, we haveassumed that the viscosity of
solvent is constant along theparticle surface, which neglects the
temperature or concentra-tion dependence of the viscosity. Finally,
in order to solve theStokes equation, three hydrodynamic boundary
conditionsneed to be determined. In the particle reference frame
thenormal component of the ow eld at the particle surfacevanishes.
Considering sufficiently large systems, it is reasonableto assume
vanishing velocity eld at innity. Finally, the inte-gral of the
stress tensor over the particle surface has to beidentied in each
geometry.
A. Self-phoretic Janus particle
For a self-phoretic Janus particle, the propulsion force
balanceswith the friction force due to the particle motion, such
that theintegral of stress tensor over the particle surface
vanishes. Theseconditions are the same as for a thermophoretic
particlemoving in an external temperature gradient, case
alreadyinvestigated in our previous work.35 The velocity eld
resultingfrom solving the Stokes equation reads,
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Fig. 10 Rescaled flow velocity, v$n, as a function of the
distance to thecenter of the Janus particle (positive direction
towards the functionalpart). Symbols refer to the simulation
results, and lines to the predic-tions in eqn (11). (a) Velocity
along the axis n, va. (b) Velocity along theaxis perpendicular to
n, vb. The insets correspond to the same data inlogarithm
representation.
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vðrÞ ¼ s3
2r3
�3rr
r2� I�$vp; (11)
with I the unit tensor, r the distance to the colloid center,
and r¼ |r|. eqn (11) indicates that the velocity eld is a
sourcedipole, which decays fast with the distance as 1/r3. It
istherefore to be expected that in suspensions of the
self-pho-retic Janus particles, the hydrodynamic interactions are
negli-gible in comparison to the effects of concentration
ortemperature gradients.
Direct measurements of the ow eld around the micro-swimmers can
be performed in the simulations and allow aquantitative comparison
with the analytical expression. Sinceonly small differences are
expected between the two discussedtypes of phoretic swimmers, we
focus in the following on thethermophoretic microswimmers. Fig. 9
shows the velocity eldinduced by a self-thermophoretic Janus
particle with a ther-mophobic surface in a section across the
particle center. Themeasured velocity eld has a source-dipole type
pattern, inwhich propulsion and ow eld along the particle axis have
thesame direction as expected from the analytical prediction in
eqn(11). The position where the ow eld is measured correspondsto
the reference frame of the Janus particle. However, the owvelocity
itself is given in laboratory frame, namely uid velocityat the
surface of the Janus particle is nite and vanishes atinnity. The
quantitative values of the simulated velocity eldsare compared with
the analytical predictions in Fig. 10 for boththe
self-thermophoretic and the self-diffusiophoretic Janusparticle.
The ow eld component along the Janus particle axis,v$n, is
displayed along the Janus particle axis n in Fig. 10a
andperpendicular to it in Fig. 10b. Simulation results and
analyticalpredictions are in very good agreement without any
adjustableparameter, although on the axis perpendicular to n the
theoryslightly underestimates the velocity eld of the
self-thermo-phoretic Janus particle at short distances. The
underestimation
Fig. 9 Velocity field induced by a self-thermophoretic Janus
particlewith a thermophobic surface. The left (h) and the right (n)
hemispherecorresponds to the heated and the non-heated part,
respectively.Propulsion and flow field on the axis n point in the
same direction.Small arrows represents the flow velocity magnitude
and direction,and lines refer to the streamlines of the flow field.
The backgroundcolor code does not precisely correspond to the
temperature distri-bution, and should be taken as a guide to the
eye.
This journal is © The Royal Society of Chemistry 2014
probably arises from the sharp change of the solvent
propertiesat the border between the functional and
non-functionalhemispheres,56 which is disregarded in the present
analyticalcalculation. Interestingly this effect seems smaller for
thecatalytic Janus particle in the direction perpendicular to
thepropulsion axis. Further investigation and more accurate
datawill shed some light in this respect.
B. Self-phoretic microdimer
Besides the Janus particle, other particle geometries have
beenshown to be easy to construct phoretic swimmers. Such
analternative is the microdimer,17,19,57 composed of two
stronglyattached beads, in which one bead acts as the functional
end,and the other bead as the non-functional one. The
Stokesequation can be solved independently for each bead, and
thetotal velocity eld around the self-propelled microdimer can
beapproximated as a superposition of these two velocityelds. The
dimer is a typical force dipole such that integral ofstress tensor
over each bead is non-zero, although their sumvanishes. This is
fundamentally different from the case of theJanus particle.8,56 The
integral over the functional bead corre-sponds to the frictional
force, which is associated withthe propulsion velocity by �gvp,
with g the friction coefficient.The integral over the
non-functional bead corresponds to thedriving force which has the
same magnitude as thefriction force, but opposite direction; this
results in zero netforce on the dimer. By solving the Stokes
equation, thevelocity eld produced by the functional and
non-functionalbeads are
vfðrÞ ¼ s2r� rf
r� rf
�
r� rf
�r� rf 2 þ I
!$vp; (12)
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Fig. 11 Solvent velocity field and stream lines induced by
self-ther-mophoretic microdimers. (a) Pusher-type of swimmer for a
thermo-philic microdimer. (b) Puller-type of swimmer for a
thermophobicmicrodimer. The left bead (h) corresponds to the heated
bead, and theright (n) to the non-heated one, nh stands for
thermophilic bead, andnc for thermophobic.
Fig. 12 Rescaled flow velocity as a function of distance to the
dimercenter of mass. Symbols refer to the simulation results, and
discon-tinuous lines to the theoretical prediction in eqn (14).
Solid symbolsregard dimers with a thermophilic bead and a
pusher-like behavior.Open symbols regard dimers with a thermophobic
bead and a puller-like behavior. For comparison, thin solid lines
corresponds to the flowof the Janus particle in eqn (11). (a)
Velocity along the dimer axis, va.Triangles and circles correspond
to the velocities on the left and rightsides of the dimer center,
respectively. (b) Velocity perpendicular tothe dimer axis, vb, with
positive direction pointing to the dimer center.The insets
correspond to the same data in logarithm representation.
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and
vnfðrÞ ¼ � s2r� rnf
r� rnf
�
r� rnf
�r� rnf 2 þ I
!$vp
þ s3r� rnf 3
3
r� rnf
�
r� rnf
�r� rnf 2 � I
!$vp; (13)
respectively. Here, rf and rnf are the position coordinates of
thefunctional and non-functional beads, respectively. Note that
thesecond term on the right side of eqn (13) corresponds to asource
dipole, which arises from the excluded volume effect ofthe bead
(vanishing for point particle). Thus, the total velocityeld around
the self-propelled microdimer can be approxi-mated by
v(r) ¼ vf(r) + vnf(r) � 1/r2 (14)
where, 1/r2 refers to the far-eld behaviour of the
ow.Consequently, in suspensions composed of phoretic micro-
dimers the hydrodynamic interactions are comparable to
thecontributions coming from concentration or temperaturegradients.
Furthermore, the near-eld hydrodynamic behaviorsof the dimer also
differ remarkably from the Janus particle.
Simulations of self-thermophoretic dimers allow us toperform
precise measurements of the induced ow eld. Thesimulation model is
the same one as employed in our previouswork19 where each bead has
a radius s ¼ 2.5a and the distancebetween the beads centers is d ¼
|rnf � rf| ¼ 5.5a. The inter-actions between the beads and the
solvent are of Lennard-Jonestype, cf. eqn (1). The heated bead
interacts with the solventthrough a repulsive potential (C ¼ 3 and
k ¼ 24), while for thephoretic bead two different interactions have
been chosen, anattractive (C ¼ 0 and k ¼ 48) and a repulsive
interaction (C ¼ 3and k ¼ 3). The solvent velocity eld is computed
around thedimer and displayed for dimers with both interaction
types inFig. 11. In spite of the opposite orientations and the
differencein intensity, the pattern of the two ow elds are very
similar.The velocity eld on the axis across the dimer center
andperpendicular to the symmetry axis is, for the microdimer
withthermophilic interactions (repulsive), oriented towards
thedimer center, while for the thermophobic dimer
(attractiveinteractions) is oriented against the dimer center. This
isconsistent with the well-known hydrodynamic character offorce
dipoles,55 and has further important consequences. Ifanother dimer
or particle is placed lateral and close to thedimer, the ow eld
will exert certain attraction in the case of athermophilic
microdimer and certain repulsion in the case of athermophobic
microdimer, which allows us to identify themrespectively as pushers
and pullers.
A quantitative comparison of the simulated velocity eldswith the
analytical prediction in eqn (14) is presented in Fig. 12for both a
pusher- and puller-type microdimer. The ow eldcomponent on the
microdimer axis analyzed along such axis isdisplayed in Fig. 12a
for the le and right branches. The oweld component perpendicular to
the microdimer axis analyzedalong such axis is displayed in Fig.
12b. The small observeddeviations of the simulations from the
analytical predictions are
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due to different factors. Besides the statistical errors,
thesuperposition approximation in eqn (14) is less precise in
thecase of nearby beads. In the direction perpendicular to the
dimeraxis, nite size effects play also an important role. Due to
thesymmetry of the ow lines and the presence of periodic images,the
condition of vanishing ow velocity occurs at the border ofthe
simulation box. This decreases the values of the ow velocitywith
respect to the analytic solution which considers vanishingow
velocity at innite distance. In spite of these considerations,Fig.
12 shows in all cases that the analytical solution of theStokes
equation agrees very nicely with the results from the
MPCsimulations without any adjustable parameter, which consti-tutes
a convincing validation for both the analytical approxi-mations and
the model employed in the simulations.
IV. Conclusions
A coarse grained model to simulate a self-phoretic Janusparticle
in which hydrodynamic interactions are consistentlyimplemented is
here proposed and analyzed. The Janus particleis provided with a
proper rotation dynamics through stickparticle boundary conditions.
These are modeled by bounce-back collisions which reverse the
direction of motion of thesolvent particle with respect to the
moving colloidal surface.The collisions are imposed to conserve
linear and angularmomentum, as well as kinetic energy. A strong
self-phoreticeffect is realized by using a so particle–solvent
potentialimplemented in a larger interaction distance than the
bounce-back collisions. With this model both the
self-thermophoreticand the self-diffusiophoretic Janus particles
are simulated in astraightforward manner, which further justies the
model val-idity. The model implementation details are most likely
alsoapplicable to other simulation methods like lattice
Boltzmann,or MD. Simulations to quantify the ow elds induced by
theself-phoretic Janus and dimer microswimmers are then
alsoperformed, and satisfactorily compared with
correspondinganalytical predictions. The ow eld around the
self-phoreticJanus particle shows to be short ranged, as it is
typical fromneutral swimmers. In contrast, self-phoretic
microdimersinduce a long-ranged ow eld. Dimers propelled towards
thefunctional bead, as thermophilic microdimers, show a
hydro-dynamic lateral attraction typical from pushers.
Conversely,dimers propelled against the functional bead, as
thermophobicmicrodimers, show a hydrodynamic lateral repulsion
typicalfrom pullers. These fundamental differences will result
insystems with very different collective properties, for which
oursimulation model is very adequately suited.
V. Appendix A: bounce-back with amoving spherical particle
Considering the contact velocity in eqn (2) and the
post-colli-sion quantities in eqn (3), the post-collision contact
velocity canbe calculated as
~v0 ¼ ~v� pmþ 1cM
ŝ ŝ$pð Þ � p½ �; (A1)
This journal is © The Royal Society of Chemistry 2014
where the relation of the vector triple product with the
scalarproduct has been employed. The difference between the
relativepre- and post-collision velocity, D~v ¼ ~v0 � ~v, can be
decomposedinto a normal and a tangential component as
D~vn ¼ � 1mŝ ŝ$pð Þ (A2)
D~vt ¼hŝ ŝ$pð Þ � p
i�1mþ 1cM
�; (A3)
with which p can expressed as
p ¼ m�D~vn þ cM
cM þ mD~vt�; (A4)
The difference in kinetic energy before and aer the collisioncan
be calculated from the pre- and post-collision velocities ineqn (3)
as
DE ¼ �2p$~vþ p2
mþ 1cM
p2 � ŝ$pð Þ2h i
; (A5)
where the circular shi property of the mixed product has
beenused. Employing the expression of D~vn in eqn (A2) and of p
andp2 which can be obtained from eqn (A4), the previous expres-sion
can be rewritten as
DE ¼ m2
2~vþ D~vn
�$D~vn þ 1
2
cM
cM þ m
2~vþ D~vt
�$D~vt: (A6)
To ensure a collision with energy conservation, it is neces-sary
that both components of the previous expression vanish,since the
prefactors are determined by the system under study.Using
orthogonality of normal and tangential velocity compo-nents the two
previous conditions translate into, ~vn
2 ¼ ~v 0n2 and~vt2 ¼ ~v 0t2. Two physical meaningful solutions
exist, both with ~vn
¼ �~v 0n. One is the specular reection of smooth hard spheres,
~vt¼ ~v 0t, which is well-known to imply slip-boundary
condition.Another solution is the bounce-back reection of rough
hardspheres, ~vt ¼ �~v 0t, which enforces a no-slip boundary
conditionbetween the solvent and the solute. With both conditions
it ispossible to express D~v and hence p in terms of the
componentsof the pre-collision contact velocity ~v, which is
specied in eqn(4) for the no-slip condition employed in this
work.
Acknowledgements
M.Y. acknowledges partial support from the “100 talent plan”
ofInstitute of Physics, Chinese Academy of Sciences, China.
A.W.acknowledges nancial support by the VW Foundation
(Volks-wagenStiung) within the program Computer Simulation
ofMolecular and Cellular Bio-Systems as well as Complex So
Matter.
References
1 W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen,S. K. S.
Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert andV. H. Crespi, J.
Am. Chem. Soc., 2004, 126, 13424.
Soft Matter, 2014, 10, 6208–6218 | 6217
http://dx.doi.org/10.1039/C4SM00621F
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Soft Matter Paper
Publ
ishe
d on
19
June
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4. D
ownl
oade
d by
Ins
titut
e of
Phy
sics
, CA
S on
05/
04/2
015
07:0
8:17
. View Article Online
2 R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A.
Stoneand J. Bibette, Nature, 2005, 437, 862.
3 J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough,R.
Vafabakhsh and R. Golestanian, Phys. Rev. Lett., 2007,99,
048102.
4 L. F. Valadares, Y. G. Tao, N. S. Zacharia, V. Kitaev,F.
Galembeck, R. Kapral and G. A. Ozin, Small, 2010, 6, 565.
5 H. R. Jiang, N. Yoshinaga and M. Sano, Phys. Rev. Lett.,
2010,105, 268302.
6 G. Volpe, I. Buttinoni, D. Vogt, H.-J. Kümmerer andC.
Bechinger, So Matter, 2011, 7, 8810.
7 R. Golestanian, T. B. Liverpool and A. Ajdari, Phys. Rev.
Lett.,2005, 94, 220801.
8 R. Golestanian, Phys. Rev. Lett., 2012, 108, 038303.9 B.
Sabass and U. Seifert, J. Chem. Phys., 2012, 136, 064508.10 J. L.
Anderson, Annu. Rev. Fluid Mech., 1989, 21, 61.11 R. Golestanian,
T. B. Liverpool and A. Ajdari, New J. Phys.,
2007, 9, 126.12 J. A. Cohen and R. Golestanian, Phys. Rev.
Lett., 2014, 112,
068302.13 J. Palacci, C. Cottin-Bizonne, C. Ybert and L.
Bocquet, Phys.
Rev. Lett., 2010, 105, 088304.14 I. Theurkauff, C.
Cottin-Bizonne, J. Palacci, C. Ybert and
L. Bocquet, Phys. Rev. Lett., 2012, 108, 268303.15 J. Palacci,
S. Sacanna, A. P. Steinberg, D. J. Pine and
P. M. Chaikin, Science, 2013, 339, 936.16 I. Buttinoni, J.
Bialké, F. Kümmel, H. Löwen, C. Bechinger
and T. Speck, Phys. Rev. Lett., 2013, 110, 238301.17 G. Rückner
and R. Kapral, Phys. Rev. Lett., 2007, 98, 150603.18 Y. G. Tao and
R. Kapral, So Matter, 2010, 6, 756.19 M. Yang and M. Ripoll, Phys.
Rev. E: Stat., Nonlinear, So
Matter Phys., 2011, 84, 061401.20 S. Thakur and R. Kapral, Phys.
Rev. E: Stat., Nonlinear, So
Matter Phys., 2012, 85, 026121.21 F. Lugli, E. Brini and F.
Zerbetto, J. Chem. Phys., 2012, 116,
592.22 P. de Buyl and R. Kapral, Nanoscale, 2013, 5, 1337.23 A.
Malevanets and R. Kapral, J. Chem. Phys., 1999, 110, 8605.24 A.
Malevanets and R. Kapral, J. Chem. Phys., 2000, 112, 7260.25 M.
Ripoll, K. Mussawisade, R. G. Winkler and G. Gompper,
Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2005,
72,016701.
26 J. T. Padding and A. A. Louis, Phys. Rev. E: Stat.,
Nonlinear,So Matter Phys., 2006, 93, 031402.
27 R. Kapral, Adv. Chem. Phys., 2008, 140, 89.28 G. Gompper, T.
Ihle, D. M. Kroll and R. G. Winkler, Adv.
Polym. Sci., 2009, 221, 1.29 D. Lüsebrink and M. Ripoll, J.
Chem. Phys., 2012, 136,
084106.
6218 | Soft Matter, 2014, 10, 6208–6218
30 M. Yang and M. Ripoll, So Matter, 2014, 10, 1006.31 E.
Tüzel, M. Strauss, T. Ihle and D. M. Kroll, Phys. Rev. E:
Stat., Nonlinear, So Matter Phys., 2003, 68, 036701.32 E.
Tüzel, T. Ihle and D. M. Kroll, Phys. Rev. E: Stat.,
Nonlinear,
So Matter Phys., 2006, 74, 056702.33 C. Echeveŕıa, K. Tucci and
R. Kapral, J. Phys.: Condens.
Matter, 2007, 19, 065146.34 D. Lüsebrink, M. Yang and M.
Ripoll, J. Phys.: Condens.
Matter, 2012, 24, 284132.35 M. Yang and M. Ripoll, So Matter,
2013, 9, 4661.36 J. T. Padding, A. Wysocki, H. Löwen and A. A.
Louis, J. Phys.:
Condens. Matter, 2005, 17, S3393.37 S. Chen and G. D. Doolen,
Annu. Rev. Fluid Mech., 1998, 30,
329.38 A. Lamura, G. Gompper, T. Ihle and D. M. Kroll,
Europhys.
Lett., 2001, 56, 319.39 G. A. Vliegenthart, J. F. M. Lodge and
H. N. W. Lekkerkerker,
Phys. A, 1999, 263, 378.40 M. P. Allen and D. J. Tildesley,
Computer Simulations in
Liquids, Clarendon, Oxford, 1987.41 B. J. Berne, J. Chem. Phys.,
1977, 66, 2821.42 J. K. Whitmer and E. Luijten, J. Phys.: Condens.
Matter, 2010,
22, 104106.43 M. Doi and S. F. Edwards, The Theory of Ploymer
Dynamics,
Oxford University Press, Oxford, 1986.44 G. Subramanian and H.
T. Davis, Phys. Rev. A, 1975, 11, 1430.45 M. H. Ernst, E. H. Hauge
and J. M. J. van Leeuwen, Phys. Rev.
Lett., 1970, 25, 1254.46 D. Rings, D. Chakraborty and K. Kroy,
New J. Phys., 2012, 14,
053012.47 S. Wiegand, J. Phys.: Condens. Matter, 2004, 16,
R357.48 R. Piazza and A. Parola, J. Phys.: Condens. Matter, 2008,
20,
153102.49 A. Würger, Rep. Prog. Phys., 2010, 73, 126601.50 R.
Golestanian, Phys. Rev. Lett., 2009, 102, 188305.51 M. Braibanti,
D. Vigolo and R. Piazza, Phys. Rev. Lett., 2008,
100, 108303.52 D. Rings, R. Schachoff, M. Selmke, F. Cichos and
K. Kroy,
Phys. Rev. Lett., 2010, 105, 090604.53 Y. Hong, N. M. K.
Blackman, N. D. Kopp, A. Sen and
D. Velegol, Phys. Rev. Lett., 2007, 99, 178103.54 K. Rohlf, S.
Fraser and R. Kapral, Comput. Phys. Commun.,
2008, 179, 132.55 E. Lauga and T. R. Powers, Rep. Prog. Phys.,
2009, 72, 096601.56 T. Bickel, A. Majee and A. Würger, Phys. Rev.
E: Stat.,
Nonlinear, So Matter Phys., 2013, 88, 012301.57 M. N. Popescu,
M. Tasinkevych and S. Dietrich, Europhys.
Lett., 2011, 95, 28004.
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