5038 | Soft Matter, 2017, 13, 5038--5043 This journal is©The Royal Society of Chemistry 2017

Cite this: SoftMatter, 2017,

13, 5038

Geometry-driven collective ordering of bacterialvortices†

Kazusa Beppu,a Ziane Izri,a Jun Gohya,a Kanta Eto,b Masatoshi Ichikawab andYusuke T. Maeda *a

Controlling the phases of matter is a challenge that spans from condensed materials to biological

systems. Here, by imposing a geometric boundary condition, we study the controlled collective motion

of Escherichia coli bacteria. A circular microwell isolates a rectified vortex from disordered vortices

masked in the bulk. For a doublet of microwells, two vortices emerge but their spinning directions show

transition from parallel to anti-parallel. A Vicsek-like model for confined self-propelled particles gives the

point where the two spinning patterns occur in equal probability and one geometric quantity governs

the transition as seen in experiments. This mechanism shapes rich patterns including chiral

configurations in a quadruplet of microwells, thus revealing a design principle of active vortices.

Introduction

In nature, collective rotational motion organized by motileelements is ubiquitous across scales, from motor proteins,1–3

and flagellated sperm,4 to the development of social amoebacells.5 Understanding the mechanism by which their motionsare organized into ordered patterns from flocking and propagatingbands to a lattice of vortices is a central subject in the emergingfield of active matter physics.6–11 In particular, the method ofcontrolling the patterns has attracted considerable interest due toits potential in exploiting the underlying mechanism as a universalfeature. As for bacteria in a quasi two-dimensional plane, densebacterial suspensions show mesoscale collective motions in whichthe jets and vortices result in a turbulent-like state.12 Moreover,hidden but weakly synchronized rotation appears at a higherdensity of bacteria,13 implying that rotational motion maycommonly be present. When swimming bacteria are confinedin a circular space, a rotational mode of vortex arises from theguiding interaction between the bacteria and the wall.14–16 With theaccumulated knowledge about the confinement-induced vortex ofactive matter, from vibrated rigid bodies17,18 to colloidal rollers,19 itis now apparent that a promising means of controlling their orderedphases lies in the setting of the boundaries.20,21

The vortex lattice is a conceptual basis for the descriptionof phases of matter from magnets to superfluids and super-conductors.22–24 This concept can impact active matter,because the correlation or frustration with defined interactions

can be constructed even for bacteria. It has been reported thatthe ferromagnetic lattice of bacterial vortices (uniform rotationaldirections) or the anti-ferromagnetic one (alternate clockwise andcounter-clockwise rotations) was constructed in a chamber, inwhich the vortices interact with their neighbors via advection ofbacteria through channels.25 However, the method of couplingthe vortices is actually not limited to indirect advection: vortices canbe directly collided by imposing a designed boundary based ongeometric quantities. Hence, the geometry-based approach, by whichone can control the exclusive interaction between bacterial vortices,is needed to elucidate the ordered phases and their transitions.

In this paper, we investigate the ordered phases of bacterialvortices inside microwells with designed geometries. We foundthat a single vortex and a doublet of vortices could be formed asorganized patterns under confinement. The pairing of vortices isclassified into two distinct phases: the first one is ferromagneticvortices (FMV) in which both vortices rotate in the same direction;the second one, anti-ferromagnetic vortices (AFMV), has thevortices rotating in opposite directions. The transition from FMVto AFMV occurs when the geometry of the boundary satisfies acertain condition. A theoretical model for self-propelled particleswith polar interaction in merging vortices is considered to explainthe observed transition, and the predicted transition point isexactly consistent with the experiments. This new approach high-lights a design principle of active vortices that could be relevant toa broad class of active matter systems.

Materials and methods

Bacteria Escherichia coli RP4979 strain, which was deficient oftumbling ability, was used in this study.26 The volume fraction

a Department of Physics, Kyushu University, Motooka 744, Fukuoka 812-0395,

Japan. E-mail: kazu.beppu@phys.kyushu-u.ac.jp, ymaeda@phys.kyushu-u.ac.jpb Department of Physics and Astronomy, Kyoto University, Kitashirakawa,

Kyoto 606-8502, Japan

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm00999b

Received 19th May 2017,Accepted 1st July 2017

DOI: 10.1039/c7sm00999b

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Soft Matter

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of the bacterial suspension was increased to 20% to inducecollective motion. To attain a controlled shape of the boundary,an SU-8 pattern on the flat surface of a Si wafer was fabricatedand then used as a mold for polydimethyl siloxane (PDMS)microwells. 0.5 mL of dense bacterial suspension was spotted ona surface treated glass slide and enclosed by PDMS microwellsfrom the top. The typical thickness of the microwells is 20 mm.We then recorded bacterial motion in an inverted microscopeat 30 frames per second using a CCD camera for 10 s. Thevelocity field of bacterial motion was acquired by PIV analysis.

Results

When bacteria RP4979 swarm on a two-dimensional plane,their collective motion is disordered (Fig. 1(a)). However, theenergy spectrum exhibits a bell-shaped distribution with a long-tail, thus showing a peak at wave number k*, such that disorderedvortices with a characteristic diameter, l* = p/k* are present,although they are apparently hidden in turbulent-like motion(Fig. 1(b)).26 To test whether a single vortex can be isolated byimposing a boundary condition, we construct microwells ofcircular boundary for various radius R, as the simplest model(Fig. 1(c)). Fig. 1(d) shows a typical example of a velocity fieldv(r,y) of bacterial motion in a circular microwell, in whicha single vortex is formed. Vorticity = � v exhibits positive

(or negative) values around the center of the microwell, whileclose to the wall at r = R, it shows opposite values because of thedecay of the velocity nearby the boundary.

What one needs to know in order to control collectivemotion is the upper limit for selecting a single vortex. We inturn examined the size-dependence of vortex formation incircular microwells with various R from 10 to 100 mm. Thevortex order parameter (VOP) of a single vortex is defined as

1

1� 2=p

Pi

vi � tij jPi

vik k� 2

p

0@

1A14,15 where i represents the index of

sites in the microwell and ti is the unit orthoradial vector atsite i. The VOP was employed to classify either a single vortex(VOP = 1) or disordered motion (VOP = 0). Fig. 1(e) (red) exhibitsthat the VOP is between 0.7 and 0.9 for R = 10 to 37 mm, whereasVOP drops to 0.2 for R Z 46 mm. This means that R = 37 mm isclose to the limit where rotational velocity correlation persistsover the confined space. This size is comparable to l* E 25 mm,implying that additional vortices arise in the same microwell asR becomes much larger than l*.

The correlation between R and l* was further examined byusing bacteria having a longer body. It is assumed that a longerrod-shaped body may enhance the local alignment of bacteriaand in turn alter the size of the vortices in mesoscale turbulence.10

For E. coli bacteria, a short exposure to a cell-division inhibitor(cephalexin, CEP) makes them become more elongated: theaverage length is 14.5 mm for elongated bacteria but the onefor untreated bacteria is 8.3 mm.26,27 The elongation of thebacteria shifted the peak in the energy spectrum to l* = 45.3 �0.1 mm (Fig. 1(b), blue). These bacteria also show a singleisolated vortex in circular microwells as we expected, and thensustained higher VOP Z 0.5 for R = 19 to 46 mm, which was, onceagain, comparable to l* (Fig. 1(e), blue). Thus, we demonstratethat the range of vortex formation is prolonged by altering thecorrelation length scale in velocity, indicating that the confine-ment by R t l* is required to host a single vortex. We found thatthe number of vortices with counter-clockwise rotation is 28 outof 48 samples, which consist of 14 for normal RP4979 bacteriaand 34 for elongated bacteria, selected by VOP Z 0.8. Theproportion of clockwise and counter-clockwise is 58%, meaningthat the rotational direction of single vortices is symmetric. Wealso note that the layer of counter-rotation against a confinedvortex (negative slip velocity15) is observed on occasion in 5 samplesout of 48 in total (vortices with large VOP Z 0.8). Most of thebacterial vortices observed here have either a no-slip or positivesmall slip boundary, implying that counter-rotating action appearsto be a secondary effect.

Let us now consider two confined vortices interacting withone another, and how the geometry drives their behavior. Weconstruct a doublet of circular microwells (Dcm) with twoidentical and overlapping circles (Fig. 2(a)). Such a geometryis chosen because it can be simply drawn with two geometricquantities: the radius of the circles R and the distance betweentheir centers D. In fact, these quantities are also linked bythe angle of elevation f with regard to the horizontal axis, ascosf = D/(2R). Hence, a doublet shape offers explicit definition

Fig. 1 (a) Vorticity map of the disordered pattern of RP4979 bacteria inthe free boundary. (b) Energy spectrum of disordered mesoscale-turbulence.The characteristic length scale l* is 25.1 mm for bacteria without cephalexin(CEP) treatment (red) while it is 45.2 mm for bacteria treated with CEP (blue).(c) Circular microwells filled with bacteria (left) and the schematic design(right). Scale bar: 50 mm. (d) Vorticity map superposed with velocity field forCEP treated bacteria in a circular microwell of R = 37 mm. (e) Vortex orderparameter (VOP) for a circular shape as a function of R. We plotted twocurves of VOP calculated from normal bacteria without CEP (red) andelongated bacteria treated with CEP (blue).

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of the boundary to analyze geometry-induced phenomena. Fig. 2shows that bacterial motion was organized into two vorticeswhose rotational directions are either identical or opposite.When the two circles have a large overlapping area, i.e. smallD, a single vortex is no longer sustained but instead two vorticeswith the same direction of rotation appear (Fig. 2(b) and (c)). Wename this pattern as ferromagnetic vortices (FMV) after thespinnings in parallel. However, as D is increased, the spinningof the vortices becomes opposite, with a pair of clockwise andcounter-clockwise rotations (Fig. 2(d) and (e)). This configurationof vortices is named as anti-ferromagnetic vortices (AFMV) afterthe spinnings in anti-parallel. Both FMV and AFMV patterns arestable for 10 s during experimental observations and spontaneoustransition from one to the other is not observed. Changing D istherefore a key parameter to determine one favorable vortexpairing out of two possible patterns.

The transition from FMV to AFMV occurs at D = Dc but Dc

presents a dependence on the circle radius R: the onset of theAFMV pattern is at Dc = 26 mm for R = 19 mm but it is shifted toDc = 53 mm for larger R = 37 mm. What geometric rule associatedwith D and R characterizes the transition? Interestingly, whenwe display the occurrence of FMV and AFMV in the ratio of D/R,which is linked to cosf, the transition point between these twopatterns collapses to a single horizontal line, i.e. Dc/R E 1.4,even for R varying from 19 to 37 mm (Fig. 3(a)).

To resolve the relationship shown above, the order parameterof the AFMV pattern, F, is defined as

F = |h pi�uii| (1)

where pi is the orientational map measured experimentally atsite i in a Dcm and ui is the expected orientation of velocity ofAFMV calculated numerically at the corresponding site.26 Wetake ensemble average h�i over all possible sites i inside a Dcm.F reaches 1 for AFMV while it goes down to 0 for FMV due tothe opposite sign of the product with one counter-rotating

vortex. As a common feature among all sizes of microwell,Fig. 3(b) shows that F sharply increases from 0 to nearly 0.9 ataround Dc/R = 1.4, yet it is independent of R. This findingimplies that the ratio D/R is a suitable parameter to controlvortex pairing.

To elucidate the mechanism of the transition from FMV toAFMV, we decide to analyze a Vicsek-like model based onconfined self-propelled particles.6 Point-like particles at positionxm move at speed v0 along their heading ym(t), i.e.

:xm = v0e(ym) where

e(ym) = (cosym,sinym). The headings of particles evolve following the

relationship _ym ¼ �g@U

@ymþ ZmðtÞ where U is the potential

describing the polar alignment interaction with neighboringparticles and is defined as U xm; ymð Þ ¼ �

Pxm�xnj jo e

cos ym � ynð Þ.

e determines the effective radius of particle interaction. Z(t) isGaussian white noise hZm(t)Zn(t0)i = 2Ddmnd(t � t0) where dmn andd(t� t0) are Dirac delta functions. Moreover, we take the effect ofthe boundary as a nematic interaction with motile particles andset a Dcm boundary condition with a geometric quantity f.Assuming a homogeneous spatial distribution of the particles,necessary for mean-field approximation,28 the Fokker–Planckequation expressing the probability distribution P(y) of a particlehaving a heading y in the middle of the Dcm is

@P

@t¼ D

@2P

@y2þ g

@

@y

ðp�psinðy� y0Þ �Pðy0; t;fÞdy0Pðy; t;fÞ

� �(2)

where %P(y0,t; f) is the probability of heading y0 from either the leftor right microwell in a doublet described by f.26 Becauseparticles start interacting at the tip in the middle of a Dcm,%P(y0; f) from the left microwell is given either by %P(y0; f) = d(y0 �p/2 + f) (counter-clockwise rotation) or %P(y0; f) = d(y0 + p/2 + f)(clockwise rotation) with low noise limit where bacteria movealong the boundary (Fig. 4(a)).26 Symmetrically, the probability ofhaving an orientation y0 for particles at the tip, coming from theright circle is %P(y0; f) = d(y0 � p/2 � f) (clockwise rotation) or%P(y0; f) = d(y0 + p/2 � f) (counter-clockwise rotation). As the

Fig. 2 Pattern formation of vortex pairing in a doublet of circular micro-wells (Dcm). (a) Representative pictures and the schematic description of aDcm of R = 28 mm. Its geometry can be given by two geometric quantitiesD and R. These quantities are also linked by the elevation angle f ascosf = D/(2R). Scale bar is 50 mm. (b) Velocity field merged with thevorticity map of a Dcm of R = 19 mm and D = 25 mm. (c) Orientation map ofthe velocity field corresponding to (b). (d) Velocity field and vorticity map ofa Dcm of R = 19 mm and D = 31 mm. (e) Orientation map of the velocity fieldcorresponding to (d).

Fig. 3 Transition of vortex configuration. (a) Phase diagram of the vortexpattern plotted on the D/R � R plane. Vortices switch from FMV (blue) toAFMV (red) across D E 1.4R. The dashed line is the transitionpoint theoretically predicted by eqn (5). (b) Order parameter F ofthe AFMV pattern is plotted in D/R for varying R = 19 mm (diamond),25 mm (square), 28 mm (triangle), and 37 mm (circle). F increases sharply atDc/R E 1.4.

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vortex pair forms respectively FMV and AFMV patterns, onederives the solution of eqn (2) at the steady state

PFMVðy;fÞ ¼exp

2gD

cos y sinf� �

2pI0gDsinf

� � (3)

PAFMVðy;fÞ ¼exp

2gD

sin y cosf� �

2pI0gDcosf

� � (4)

Hence, at the condition where sinfc = cosfc, i.e. fc = p/4, P FMV

(y = 0; f = p/4) = PAFMV(y = p/2; f = p/4). In other words, thegeometry of f o p/4 (or f 4 p/4) selects the AFMV pattern (orFMV pattern) (Fig. 4(b)). This symmetry allows us to obtain Dc as

Dc

2R¼ cos

p4

� �: (5)

Eqn (5) immediately leads to Dc=R ¼ffiffiffi2p� 1:41, which exactly

agrees with the experimental results. This fact indicates that thetip in the central area plays a crucial role for controlling thepreferred direction of the alignment and in turn determines onevortex pairing as either FMV or AFMV.

It is worth mentioning about the hydrodynamic effect in ourobservation. Bacteria are regarded as a force-dipole8 anddriven-fluid flow leads to disordered motion such as mesoscaleturbulence. In our study, this effect may have little amplitudebecause the radius of the circular microwell is smaller than l*so as to avoid loss of alignment correlation over distance. In

addition, it has been reported that swimming bacteria aretrapped at curved walls21,29 whereas bacteria uniformly distributewith less heterogeneity in our microwells (Fig. 2(a) and 3(a)),implying that hydrodynamic trapping may be suppressed due tothe high density of bacteria.

Finally, in order to test the predictability of our theoreticalmodel for a more complex boundary, we examined an orderedpattern of vortices in a quadruplet of circular microwells (Qcm)(Fig. 5(a)). On the one hand, we found that all four vortices

rotate in the same direction (FMV) when both Dx

�R �

ffiffiffi2p

and

Dy

�R �

ffiffiffi2p

were satisfied. On the other hand, the vortex

pairing becomes AFMV when both Dx

�R �

ffiffiffi2p

and

Dy

�R �

ffiffiffi2p

(Fig. 5(b)). By imposing not only a symmetricboundary Dx/R = Dy/R but also an asymmetric one such as

Dx

�R �

ffiffiffi2p

and Dy

�R �

ffiffiffi2p

, we find a mixed configuration ofFMV and AFMV with achiral symmetry (Fig. 5(b), green).

Strikingly, despite the absence of asymmetry in geometry,FMV and AFMV coexist in the region close to the transitionpoint, e.g. Dx/R = Dy/R = 1.33 (Fig. 5(c)). Eqn (3) and (4) accountfor this coexistence because the probabilities of FMV and AFMVbecome comparable as f is close to p/4. The exclusive inter-action among vortices no longer discriminates between FMVand AFMV, and the ordered patterns can also exhibit chiralconfigurations (Fig. 5(b), purple). Thus, we conclude that thetheoretical model constructed here can be applied even tomiscellaneous geometries so that it draws a design principlefor bacterial vortices pairing.

Discussion

We have studied collective ordering of bacterial vortices in aconfined space with defined geometries. For bacterial vortices,the probability of orientation of bacterial motion from the tipcan be controlled by geometry, e.g. an FMV pattern is permitted

when D=R �ffiffiffi2p

. Our data show that one geometric parameterD/R is sufficient to control the vortex pairing. This findingallows one to consider geometry as a powerful means to dictatespatial orderings of active vortices. For an original Vicsekmodel, the point-like particles with less fluctuation aredepleted from the center of the confined space and then resultin large density heterogeneity while, for our experiment, thedensity of bacteria is almost uniform inside the microwells.Eqn (3)–(5) derived from mean-field approximation explain thetransition point but further investigation, e.g. self-propelledparticles with excluded volume, remains as a future work tounderstand the pattern formation of FMV and AFMV. We alsofound that the typical size of the vortices increases withelongated bacteria and this empirical observation was employedin order to examine the transition for various sizes of confinement.The underlying mechanism of this size-dependence is left as asubject for future study. Thanks to its simplicity, our theoreticalmodel may provide a versatile protocol for not only bacteria butalso active cytoskeletons.30 In that case, an elucidated mechanismcan be in turn used to rationally design an autonomous

Fig. 4 Theoretical model for the transition of vortex pairing. (a) Schematicof particle motion at the vicinity of the tip between two circles. At the tip,the particles in the left circle go either in the direction of I or I*, while theones in the right circle go either in the direction of II or II*. Alignment atthe tip decides whether the two vortices make an FMV or AFMV pattern.(b) Probability distributions of PFMV (blue) and PAFMV (red) are plottedas functions of y at f = p/12, p/4, and 5p/12. Here we take 2g/D = 1.The dashed black line indicates the equal probability of PFMV and PAFMV atf = p/4.

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transporter of small objects over a long distance.31 To realize suchan isothermal engine, the extended analysis of frustrated vorticesin a triplet of circular microwells appears to be promising. Finally,the transition of vortex pairings may provide new insight into theuniversality for the ordered phases of matter; it has analogoussymmetry to the transition between type I and II in asuperconductor.24 In the future, a comprehensive exploration ofcollectively ordered vortices will be developed for uncoveringdesign principles that hold in wide classes of matter from colloidalrollers19 and cytoskeletons32 to quantum systems.24

Acknowledgements

We thank I. Kawagishi for providing the RP4979 strain. Thiswork was supported by PRESTO (no. 11103355, JPMJPR11A4)

from JST, JSPS KAKENHI (no. JP16H00805, JP25103012: ‘‘Synergyof Fluctuation and Structure’’, and no. JP26707020) from MEXT,and HFSP Research Grant (RGP0037/2015).

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Fig. 5 Design of complex patterns of vortices. (a) Quadruplet of microwells(Qcm) and its schematic design. Scale bar: 50 mm. (b) Phase diagram ofcollective vortices. Mixed states of FMV and AFMV with achiral symmetry occurin asymmetric quadruplets (green diamond). The mixed configurations withchiral symmetry are found only close to the transition point Dx;c

�R ¼ Dy;c

�R ¼ffiffiffi

2p

(purple star). (c) (left) Orientation map of vortices in a Qcm of R = 25 mm andDx/R = Dy/R = 1.33. (right) Line scan of normal velocity over the dotted diagonallines on the orientation map.

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