Hydrodynamic simulations of self-phoretic...

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


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.

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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).


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.

Soft Matter, 2014, 10, 6208–6218 | 6211


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


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

Soft Matter, 2014, 10, 6208–6218 | 6213


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,



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.


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)

Soft Matter, 2014, 10, 6208–6218 | 6215


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

6216 | Soft Matter, 2014, 10, 6208–6218 This journal is © The Royal Society of Chemistry 2014


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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)


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.

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Hydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmers

Hydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmers

Hydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmersHydrodynamic simulations of self-phoretic microswimmers

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