Collective behavior of strongly confined suspensions of squirmers

J.-B. Delfau∗1, J. Molina†2, and M. Sano‡3

1Institute for Cross-Disciplinary Physics and Complex Systems, University of the BalearicIslands, Palma de Mallorca E-07122 , Spain

2Department of Chemical Engineering, Kyoto University, Kyoto 615-8510, Japan3Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo

113-0033, Japan

May 7, 2021

Abstract

We run numerical simulations of strongly confined suspensions of model spherical swimmers called“squirmers”. Because of the confinement, the Stokeslet dipoles generated by the particles are quicklyscreened and the far-field flow is dominated by the source dipole for all the different kinds of squirmers.However, we show that the collective behavior of the suspension still depends on the self-propelling mech-anism of the swimmers as polar states can only be observed for neutral squirmers. We demonstrate thatthe near-field hydrodynamic interactions play a crucial role in the alignment of the orientation vectorsof spherical particles. Moreover, we point out that the enstrophy and the fluid fluctuations of an activesuspension also depend on the nature of the squirmers.

Understanding the physics of “active matter” hasbecome a major preoccupation in statistical physics.Self-propelling objects being fundamentally out-of-equilibrium, they can exhibit various interesting col-lective behaviors: experimental works pointed outswarming in systems of bacteria[10, 13, 31, 30, 35, 9]or self-propelled droplets[34], dynamic clustering[33,27] and phase separation of Janus particles[8] or di-rectional motion in suspensions of rotating colloids[6],vibrated rods[19] and asymmetric hard-disks[11, 12].The onset of directional motion in suspensions ofswimmers has been intensively studied numericallyand theoretically to try to determine in what kindof systems this collective behavior may be observed.Many numerical simulations focused on a simple andwell-known model of swimmers called “squirmers” in-troduced for the first time by Lighthill[22] and refinedfurther by Blake[5]. It was pointed out that the self-propelling mechanism of these squirmers has a stronginfluence on the collective dynamics and that onlysome specific kinds of active particles can exhibit po-lar states[16, 14, 37]. These results were explainedby demonstrating that if the motion of the swim-mers generates Stokeslet dipoles, these singularities

will dominate the far-field hydrodynamic flow anddestabilize the polar state[3, 28, 29, 4]. Yet whenit comes to strongly confined suspensions, the sit-uation is quite different: in that case, these singu-larities are screened exponentially[23] and hydrody-namics interactions become independent of the de-tails of the self-propelling mechanism, at least inthe far-field approximation[29, 4]. It was thus pre-dicted that the usual distinction between the differentkinds of squirmers (pushers, pullers, neutral squirm-ers) should become irrelevant for strongly confinedsuspensions[7]. However, this statement only holdsif we assume that the collective dynamics of a sus-pension is mainly determined by the far-field hydro-dynamic interactions. In particular, we can wonderif the near-field interactions do not play a significantrole in the alignment of the orientation vectors of theswimmers, which is essential in the formation of polarstates. To our knowledge, this question has not beenclearly answered yet for squirmers.

In this paper, we run periodic simulations of self-propelled spherical particles with 2D orientations ata constant density. The system is confined verti-cally by two walls with no-slip boundary conditions.

1

arX

iv:1

602.

0464

1v2

[ph

ysic

s.fl

u-dy

n] 1

8 M

ay 2

016

Figure 1: Velocity of the flow field u generated bythree different kinds of squirmers with respect to thedistance r/a. a) No confinement. b) Two no-slipwalls separated by h = 5.2 a. The dashed and dottedblack lines correspond to r−2 and r−3 respectively.

Our numerical simulations are based on the “Smooth-profile” method (SPM), which has been described indetails in previous publications[26]. The key conceptof this method is to consider that the interface be-tween the particles and the fluid is not discrete butcontinuous, with a given thickness. We define a con-tinuous density function ψ(xr, t) such as ψ = 1 insidethe particles, 0 < ψ < 1 inside the interface andψ = 0 in the fluid. If xup(xr, t) is the velocity ofthe particles, then the function ψxup(xr, t) is a con-tinuous velocity field for the particles over the wholesystem. The following modified Navier–Stokes equa-tion is then solved assuming momentum conservationbetween the fluid and the particles

∂txv + (xv · ∇)xv = −∇pρ + η∇2xv + (ψxfp + xfsq)

∇ · xv = 0(1)

where xv is the total fluid velocity given byxv = ψxup + (1 − xu) (xu being the fluid velocity),p is the pressure, η the kinematic viscosity and ρthe density of the solvant. ψxfp and xfsq are theforces necessary to enforce the particles rigidityand the required surface slip velocities xur|r=a

and xuθ|r=a (given in the following paragraph).Finally, equation 1 is coupled to the underdampedNewton-Euler equations for the center of massand the angular velocity of the rigid particles.The self-propelling force thus originates from theinteractions between the particles and the fluid,when we enforce the required slip velocities andassuming momentum conservation. Our simulationsare nondimensionalized so that ρ = η = ∆ = 1, ∆being the size of the integration grid. Rotational andtranslational thermal fluctuations can be added tothese equations[18] even though most of the resultsshown here were obtained for T = 0. Also, we willfocus here on rather dilute regimes (with packingfractions φ generally close to 0.3, far from the jam-ming threshold) so that the continuous nature of theinterface particle-fluid does not raise any controversy.

As for the particles, we are considering squirmerswhich means that we enforce the following “squirm-ing” slip velocity conditions at the interface betweenthe particles and the fluid:

xur|r=a =∑∞n=1AnPn (cos θ)xr

xuθ|r=a =∑∞n=1

2n(n+1)Bn sin θP ′n (cos θ)xθ

(2)

with xr and xθ the radial and tangential unit vectors,θ the angle of the orientation vector xe, Pn the n-th order Legendre polynomial and a the radius ofthe particles. In our case, we will consider the mostsimple squirmer with An = 0 for ∀n and Bn = 0 for∀n > 2. The slip-velocity is thus purely tangential:

xuθ|r=a = B1 sin θ (1 + α cos θ)xθ (3)

where α = B2/B1 is the ratio between the am-plitude of the first two squirming modes. It wasshown analytically[17, 36] and numerically[24] thatthe swimming velocity of a squirmer is entirely de-termined by B1 in Newtonian fluids. α on the otherhand characterizes the self-propelling mechanism ofthe squirmer: if α > 0, the particle is a “puller”,if α < 0 it is a “pusher” and if α = 0, it is a“neutral squirmer”. The velocity field generated bysuch swimmers can be calculated analytically[5] andthe agreement between the numerical streamlines andthe analytical ones is very good, thus validating ourmethod[2]. In the bulk, the far-field velocity fieldgenerated by pushers and pullers is dominated bythe Stokeslet dipole singularity and decays like r−2

whereas it is dominated by the source dipole singu-larity for neutral squirmers and decays like r−3 (seefig. 1 a)).

2

Figure 2: Simulations of 60 squirmers starting fromrandom positions and orientations at T = 0. Thepacking fraction is φ ≈ 0.29 and the Reynolds number8 × 10−3. a) and b) pushers with α = −5, c) andd) neutral squirmers, e) and f) pullers with α = 5.Left column: snapshots of the suspensions after 2millions of time steps[1]. The arrows correspond tothe orientation vectors of the particles and the colorsto their angle θ. Right column: probability densityfunction P (θi, θf ) where θi and θf are the relativeangles between two squirmers before and after theyinteract through near-field interactions respectively.We measured θi and θf when the distance betweenthe two particles crosses a given threshold ‖xrij‖ =2.6a. The colors correspond to the probability of thepair (θi, θf ) such that

∫∫P (θi, θf ) dθidθf = 1. The

angles were measured during the entire simulation(starting t = 0).

The orientation vector of the squirmers xe waskept two-dimensional here, with xez(t) = 0 for ∀t.

This would correspond to experimental systems likeJanus particles moving by self-induced electrophore-sis, which are subjected to a torque preventing xe togo out of the xy plane[32]. To prevent overlap, theparticles also interact through a soft-core repulsivecore potential (truncated Lennard-Jones). A lot ofactual experiments on swimmers involve monolayersof particles or bacteria moving in quasi-two dimen-sional systems. To take into account the influence ofthe fluid confinement, we added to the simulationstwo walls with no-slip boundary conditions such asu (z = ±h/2) = 0 with h the system height. In thispaper, h was kept constant at h = 5.2 a. Undersuch a strong confinement, the Stokeslet dipolesingularities are screened exponentially[23] and thesource dipole becomes dominant in the far-field forall squirmers. Thus, the far-field flow decays liker−2 independently of the self-propelling mechanismas can be seen on fig. 1 b) so that the hydrodynamicfar-field interactions become generic.

Figure 2 shows snapshots of suspensions of 60squirmers with α = −5, 0 and 5 after thousands oftime steps, initially starting from random positionsand orientations. One can immediately notice theonset of directional motion in the system of neutralsquirmers (fig. 2 b): over time, particles align theirorientation vectors, which is clearly shown on fig. 3by the regular increase of the system order parameter〈e(t)〉 = ‖

∑Ni xei(t)‖/N that eventually reaches 1.

On the other hand, the suspensions of strong pullersor pushers shown on fig. 3 never exhibit directionalmotion and their order parameter fluctuates around0 at all times for high values of |α|, even at zerotemperature. The existence of globally aligned statesin confined active suspensions has been predictedand observed numerically in simulations where theflow field is calculated using purely two-dimensionalfar-field approximations[7, 21]. In that context,it was claimed that the distinction between thedifferent types of squirmers should be irrelevant asthey all generate similar far-field flows, as shown onfigure 1 b). But in our simulations, the collectivebehavior of the suspension still clearly depends onthe value of α. We can draw two conclusions fromhere: first, the distinction between squirmers withdifferent self-propelling mechanisms does not system-atically become irrelevant under rigid confinement asonly the neutral squirmers exhibit polarized states.Second, it seems that the collective behavior of anactive suspension cannot always be predicted usingonly far-field approximation.

Our second conclusion suggests that the near-field

3

Figure 3: Time evolution of the order parameter 〈e〉in suspensions of 60 squirmers for different values ofα, temperatures and interaction forces. t is the num-ber of time steps. The squirmers start from randompositions and orientations, the Reynolds number is8 × 10−3and the packing fraction is φ ≈ 0.29. Thegray circles and red empty circles show the influenceof thermal fluctuations and of the additional repul-sive force given by eq. 4 on the collective behavior insystems of neutral squirmers.

hydrodynamic interactions also need to be taken intoaccount. In order to understand their role, we ransimulations with an additional repulsive short-rangeforce

xF (rij) =

{ [Uλ exp

(−rijλ

)+ c]xrij if rij ≤ 4a

x0 if rij > 4a(4)

with c a constant such that F (4a) = 0 and λ = 2a thecharacteristic length of the force. U and λ are tunedso that xF (rij) prevents the particles from beingtoo close but remains small enough so that thereis no jamming at this packing fraction. As xF (rij)has a cutoff equal to 4a, the far-field interactionsbetween squirmers remain purely hydrodynamic andare exactly identical as in our previous simulations.Moreover, it has no direct effect on the relativealignment of the squirmers as it does not induce anytorque. Figure 3 shows that with this additionalforce, the order parameter 〈e(t)〉 fluctuates aroundzero during the whole simulation. Therefore withoutnear-field hydrodynamic interactions, directionalmotion cannot be observed anymore which confirmsthat they play a crucial role in the collective behaviorof confined active suspensions.

Ishikawa and Pedley[17] found analytical expres-sions for the force and torque between squirmers inthe near-field approximation (in the absence of con-

finement but here we assume the confinement will notmodify the interactions at very small distances). Thestructure of the torque map appears quite differentbetween pullers or pushers and neutral squirmers[2].Moreover, the amplitude of the torque increases with|α|. In order to see how this can influence the align-ment between squirmers, we plotted the relative an-gle between the orientation vectors of two particles,before and after they interact at near-field distances(fig. 2 right column). This plot can be viewed as somekind of “collision-rules”[15], even if in our case a col-lision is not strictly defined and we have to choose anarbitrary distance to delimit this event. In the caseof neutral squirmers, we can see that a lot of datapoints collapse on the straight line of slope 1 whichmeans that many near-field interactions are ineffec-tive with respect to the relative angle. However, onealso has to note the presence of a red rectangle in themiddle of the plot showing that for small θi, we getalmost all the time |θf | ≤ |θi|. This means that aneffective alignment is induced by the near-field inter-actions at small angles. For pushers or pullers on theother hand, the near-field interactions have a muchmore complicated effect on the relative orientationsand the same initial θi can lead to very different θf ,even at T = 0. In particular, it is frequent to get|θf | ≥ |θi| even for low values of θi, which corre-sponds to dis-aligning interactions. Note that this isparticularly visible in the case of the pullers. Oneway to quantify the overall alignment induced by thenear-field interactions is to compute the alignmentintegral defined as[15]

〈∆A〉 = | cos (θi/2)| − | cos (θf/2)| (5)

where the bracket 〈. . .〉 is an average over all the pos-sible collisions. Using the data of figure 2 b), d)and f), we found 〈∆A〉 ≈ 0.017, 0.028 and 0.005 forα = −5, 0 and 5 respectively. We thus have the con-firmation that the effective alignment decreases when|α| increases.

Another notable difference between the squirmersis the strength of the far-field flow they generate.As we can see on fig. 1 b), even if the far-field flowdecays in the same way for all swimmers, |xu(r)|still increases with |α|. In order to quantify thedisorder in hydrodynamic interactions for severalsuspensions, we calculated the averaged enstrophy〈(∇ × xu)2〉 for several suspensions. Figure 4 a)shows that the enstrophy clearly increases with |α|.It is then logical to assume that the fluid fluctuationsshould also increase with respect to this parameter,which is confirmed by the numerical measurements ofthe effective diffusion coefficient of the particles[24]shown on fig. 4 b). As a consequence, increasing

4

Figure 4: a) Enstrophy and b) effective diffusion co-efficient with respect to α in suspensions of squirm-ers at T=0 for two different packing fractions. Thevalues of DT are obtained by calculating the mean-square displacement of the particles. At long times,we have 〈∆r2〉 ∝ DT t so that we can extract theeffective diffusion coefficients. The two plots havebeen normalized by |∇×u|2(α = 0) = 5.5×10−6 andDT (α = 0) = 8.1× 10−2 respectively.

|α| is equivalent to increasing the strength of theorientational noise which is known to prevent theonset of directional motion. Note that a systematicasymmetry between pullers and pushers can onceagain be observed.

Recent numerical and theoretical studies haveproved that the onset of directional motion stronglydepends on how much particles align their ori-entations during binary collisions. Hanke andcollaborators showed that collision rules generatinga weak relative alignment were enough to inducepolarized state for similar packing fractions atlow noise[15]. Simulations of active hard-disks

highlighted that the details of the scattering mapdo not matter as long as an effective alignmenttakes place for low angular separation collisions[20].Figure 2 d) clearly shows that there is such a relativealignment at small angles for neutral swimmers.When |α| increases however, the alignment becomesweaker as shown by the decrease of the alignmentintegral 〈∆A〉 and the strength of the orientationalnoise increases simultaneously as shown by theincrease of the effective diffusion coefficient. Webelieve that these two phenomena explain whydirectional motion cannot be observed in suspensionsof strong pushers or pullers. Even if we reporthere simulations of small systems, we believe theeventual finite size effects will not change quali-tatively these conclusions as the suspensions arebig enough to see the influence of the confinementon the far-field hydrodynamic flow field (see fig. 1 b)).

In summary, we have shown that in confined ge-ometries, the collective behavior of an active sus-pension still strongly depends on the self-propellingmechanism of the swimmers. Despite the fact thatpushers, neutral squirmers and pullers generate simi-lar far-field hydrodynamic fields in thin suspensions,stable polar states were only observed for small val-ues of α. Our results suggest that collective behaviorscannot be predicted only by using far-field approxi-mations of the flow field. In particular, one also hasto look at the scattering map induced by the com-bination of near-field hydrodynamic forces and stericinteractions. For squirmers, we demonstrated thatan effective alignment exist for collisions at small rel-ative angles. When the parameter |α| increases, thereduction of this effective alignment coupled to the in-crease of the amplitude of the fluid fluctuations makethe polar states unstable. We believe our results helpus understand why there are so few experimental ob-servations of directional motion in active suspensions:even if these ones are strongly confined, the formationof polar states is still strongly dependent on the na-ture of the mutual near-field interactions, which willbe function of the self-propelling mechanism of theswimmers. To our knowledge, many artificial swim-mers used in experiments like Janus particles can beconsidered as pushers or pullers, for which we haveseen that this effective alignment was quite weak.

Acknowledgements

The authors wish to thank Professor Ryoichi Ya-mamoto for helpful discussions. This work is sup-ported by the Japan Society for the Promotion of

5

Science Grants-in-Aid for Scientific Research KAK-ENHI.

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Supplementary material: collective behavior of stronglyconfined suspensions of squirmers

1 Comparison between numerical and analytical streamlines

One very big advantage of the model of squirmers is that it is possible to find analytical expressions for thevelocity and pressure fields induced by the motion of these particles. Solving the Stokes equation with theboundary conditions of squirmers given by equation (2), Blake found that[5]:

u(r) = a3

r3B1

[(e.r r − 1

3e)

+(a4

r4 −a2

r2

)αP2 (e.r) r + a4

r4αW2 (e.r) (e.r r − e)]

(S1)

where the function Wn is defined from the Legendre polynomial Pn as

Wn(e.r) =2

n(n+ 1)P ′n (e.r) (S2)

To validate our numerical method, we have compared the numerical streamlines induced by a single particlein the bulk to equation S1. As can be seen on figure 5, the agreement is excellent.

Figure 5: Comparison between the numerical and analytical streamlines induced by the three different kindsof squirmers. Top row: numerical streamlines. Bottom rows: analytical streamlines given by equation S1.Left column: pusher with α = −1. Middle column: neutral squirmer. Right column: puller with α = 1. Theblack arrow shows the orientation of the squirmer.

8

2 Hydrodynamic torque between squirmers

Using Blake’s results, Ishikawa and Pedley found analytical expressions for the torque induced by a squirmerin the near-field approximation[17]. Its z-component (the only one not equal to zero in our case) is given inthe first approximation by:

Tz (rij) = −µπa2e.xij B1 [W1 (−e.yij) + αW2 (−e.yij)] log(rij2) (S3)

with µ the viscosity of the fluid. Using this expression, we plotted the torque map for different values ofα (figure 6). Whereas it is dipolar for α = 0, it gets more and more quadrupolar as |α| increases, which wasconfirmed recently by numerical works [25]. Note that the intensity of the torque increases when |α| increases.The zones of strong torques correspond to the recirculation zones of the fluid. The near-field hydrodynamicinteractions between strong pullers or pushers and neutral squirmers being particularly different, it is notsurprising that their influence on the relative orientations of the particles depends on the value of α.

Figure 6: z-component Tz of the torque induced by a squirmer according to equation S3 in arbitrary units.From top-left to bottom right, α = −5, −3, −1, 0, 1, 3 and 5 respectively. The orange disk in the middle ofeach plot corresponds to the squirmer and the arrow to its orientation vector e. The black disk surroundingit comes from the fact that steric interactions prevents particles from overlapping, so that rij ≥ 2 a.

9