Circular motion of asymmetric self-propelling particles

Felix Kummel,1 Borge ten Hagen,2 Raphael Wittkowski,3 Ivo Buttinoni,1

Ralf Eichhorn,4 Giovanni Volpe,1, 5 Hartmut Lowen,2 and Clemens Bechinger1, 6

12. Physikalisches Institut, Universitat Stuttgart, D-70569 Stuttgart, Germany2Institut fur Theoretische Physik II, Weiche Materie,

Heinrich-Heine-Universitat Dusseldorf, D-40225 Dusseldorf, Germany3School of Physics and Astronomy, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom4Nordita, Royal Institute of Technology, and Stockholm University, SE-10691 Stockholm, Sweden5Present address: Department of Physics, Bilkent University, Cankaya, Ankara 06800, Turkey

6Max-Planck-Institut fur Intelligente Systeme, D-70569 Stuttgart, Germany(Dated: August 26, 2018)

Micron-sized self-propelled (active) particles can be considered as model systems for characterizingmore complex biological organisms like swimming bacteria or motile cells. We produce asymmetricmicroswimmers by soft lithography and study their circular motion on a substrate and near channelboundaries. Our experimental observations are in full agreement with a theory of Brownian dynamicsfor asymmetric self-propelled particles, which couples their translational and orientational motion.

PACS numbers: 82.70.Dd, 05.40.Jc

Micron-sized particles undergoing active Brownian mo-tion [1] currently receive considerable attention from ex-perimentalists and theoreticians because their locomo-tion behavior resembles the trajectories of motile mi-croorganisms [2–5]. Therefore, such systems allow in-teresting insights into how active matter [6] organizesinto complex dynamical structures. During the lastdecade, different experimental realizations of microswim-mers have been investigated, where, e.g., artificial flag-ella [7] or thermophoretic [8] and diffusiophoretic [9]driving forces lead to active motion of micron-sized ob-jects. So far, most studies have concentrated on spher-ical or rod-like microswimmers whose dynamics is welldescribed by a persistent random walk with a transitionfrom a short-time ballistic to a long-time diffusive behav-ior [10]. Such simple rotationally symmetric shapes, how-ever, usually provide only a crude approximation for self-propelling microorganisms, which are often asymmetricaround their propulsion axis. Then, generically, a torqueis induced that significantly perturbs the swimming pathand results in a characteristic circular motion.

In this Letter, we experimentally and theoreticallystudy the motion of asymmetric self-propelled particlesin a viscous liquid. We observe a pronounced circular mo-tion whose curvature radius is independent of the propul-sion strength but only depends on the shape of the swim-mer. Based on the shape-dependent particle mobilitymatrix, we propose two coupled Langevin equations forthe translational and rotational motion of the particlesunder an intrinsic force, which dictates the swimming ve-locity. The anisotropic particle shape then generates anadditional velocity-dependent torque, in agreement withour measurements. Furthermore, we also investigate themotion of asymmetric particles in lateral confinement.In agreement with theoretical predictions we find eithera stable sliding along the wall or a reflection, depending

on the contact angle.

Asymmetric L-shaped swimmers with arm lengths of9 and 6 µm were fabricated from photoresist SU-8 byphotolithography [11]. In short, a 2.5 µm thick layerof SU-8 is spin coated onto a silicon wafer, soft-bakedfor 80 s at 95 ◦C and then exposed to ultraviolet lightthrough a photo mask. After a post-exposure bake at95 ◦C for 140 s the entire wafer with the attached par-ticles is coated with a 20 nm thick Au layer by thermalevaporation. When the wafer is tilted to approximately90◦ relative to the evaporation source, the Au is selec-tively deposited at the front side of the short arms asschematically shown in Figs. 1(a),(b). Finally, the coatedparticles are released from the wafer by an ultrasonicbath treatment. A small amount of L-shaped particles issuspended in a homogeneous mixture of water and 2,6-lutidine at critical concentration (28.6 mass percent oflutidine), which is kept several degrees below its lowercritical point (TC = 34.1 ◦C) [12]. To confine the par-ticle’s motion to two dimensions, the suspension is con-tained in a sealed sample cell with 7 µm height. Theparticles are localized above the lower wall at an averageheight of about 100 nm due to the presence of electro-static and gravitational forces. Under these conditions,they cannot rotate between the two configurations shownin Figs. 1(a),(b), which will be denoted as L+ (left) andL– (right) in the following. When the sample cell is illu-minated by light (λ = 532 nm) with intensities rangingon the order of several µW/µm2, the metal cap becomesslightly heated above the critical point and thus inducesa local demixing of the solvent [13, 14]. This leads toa self-phoretic particle motion similar to what has beenobserved in other systems [15–17].

Figures 1(a),(b) show trajectories of L+ and L– swim-mers obtained by digital video microscopy for an illu-mination intensity of 7.5 µW/µm2, which corresponds

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Figure 1. (Color online) (a),(b) Trajectories of an (a) L+ and(b) L– swimmer for an illumination intensity of 7.5 µW/µm2.(Red) bullets and (blue) square symbols correspond to ini-tial particle positions and those after 1 min each, respectively.The insets show microscope images of two different swimmerswith the Au coating (not visible in the brightfield image) indi-cated by an arrow. (c),(d),(e) Probability distributions p(α)of the angle α (see inset in (c)) between the normal vector u⊥of the metal coating and the displacement vector ∆r of an L+particle in time intervals of 12 s each for illumination intensi-ties (c) I = 0 µW/µm2, (d) 5 µW/µm2, and (e) 7.5 µW/µm2.

to a mean propulsion speed of 1.25 µm/s. In contrastto spherical swimmers, here a pronounced circular mo-tion with clockwise (L+) and counter-clockwise (L–) di-rection of rotation is observed. For the characterizationof trajectories we determined the center-of-mass positionr(t) = (x(t), y(t)) and the normalized orientation vectoru⊥ of the particles (see inset of Fig. 1(c)). From this, wederived the angle α between the displacement vector ∆rand the particle’s body orientation u⊥. Figures 1(c)-(e)show how the normalized probability distribution p(α)changes with increasing illumination intensity I. In caseof pure Brownian motion (see Fig. 1(c)) p(α) ≈ const.since the orientational and translational degrees of free-dom are decoupled when only random forces are actingon the particle. In presence of a propulsion force whichis constant in the body frame of the particle, however,the translational and rotational motion of an asymmet-ric particle are coupled. This leads to a peaked behaviorof p(α) as shown in Figs. 1(d),(e). The peak’s halfwidthdecreases with increasing illumination intensity since thecontribution of the Brownian motion is more and moredominated by the propulsive part. In addition, the peaksare shifted to positive (negative) values for a particleswimming in (counter-)clockwise direction. The posi-tion of the peak is given by α = π∆t/τ , where τ is theintensity-dependent cycle duration of the circle swimmer(cf., Fig. 2(b)) and ∆t is the considered time interval.This estimate (see arrows in Figs. 1(d) (τ = 60 s) and1(e) (τ = 40 s)) is in good agreement with the experi-mental data. The shift of the maximum of p(α) docu-ments a torque responsible for the observed circular mo-tion of such asymmetric swimmers. In contrast to an

externally applied constant torque [18], here it is due toviscous forces acting on the self-propelling particle. Thisis supported by the experimental observation that theparticle’s angular velocity ω(t) = dα/dt increases lin-early with its total translational velocity v(t) (see Fig.2(a)). As a direct result of the linear relationship be-tween ω and v, the radius R of the circular trajectoriesbecomes independent of the propulsion speed, which isset by the illumination intensity (see Fig. 2(b)).

For a theoretical description of the motion of asymmet-ric swimmers, we consider an effective propulsion force F[19], which is constant in a body-fixed coordinate sys-tem that rotates with the active particle. With the unitvectors u⊥ = (− sinφ, cosφ) and u‖ = (cosφ, sinφ) (seeFigs. 1(c) and 3(a)), where – in case of L-shaped parti-cles – φ is the angle between the short arm and the xaxis, the propulsion force F is given by F = F uint withuint = (cu‖ + u⊥)/

√1 + c2 with the constant c depend-

ing on how the force is aligned relative to the particleshape. If the propulsion force is aligned along the longaxis u⊥, one obtains c = 0, i.e., uint = u⊥. In case ofan asymmetric particle, the propulsion force leads alsoto a velocity-dependent torque relative to the particle’scenter-of-mass. For c = 0 this torque is given by M = lFwith l the effective lever arm (see Fig. 3(a)). Our the-oretical model is valid for arbitrary particle shapes andvalues of c and l. However, for the sake of clarity, weset c = 0 as this applies for the L-shaped particles con-sidered here. Accordingly, we obtain the following cou-pled Langevin equations, which describe the motion ofan asymmetric microswimmer

r = βF(DTu⊥ + lDC

)+ ζr ,

φ = βF(lDR + DC·u⊥

)+ ζφ .

(1)

Here, β = 1/(kBT ) is the inverse effective thermal en-ergy of the system. These Langevin equations containthe translational short-time diffusion tensor DT(φ) =D‖u‖⊗u‖+D⊥‖ (u‖⊗u⊥+u⊥⊗u‖)+D⊥u⊥⊗u⊥ with thedyadic product ⊗ and the translation-rotation coupling

vector DC(φ) = D‖Cu‖ + D⊥C u⊥ [20]. The translational

diffusion coefficients D‖, D⊥‖ , and D⊥, the coupling coef-

ficients D‖C and D⊥C , and the rotational diffusion constant

DR are determined by the specific shape of the particle.Finally, ζr(t) and ζφ(t) are Gaussian noise terms of zeromean and variances 〈ζr(t1) ⊗ ζr(t2)〉 = 2DT δ(t1 − t2),〈ζr(t1) ζφ(t2)〉 = 2DC δ(t1 − t2), and 〈ζφ(t1) ζφ(t2)〉 =2DRδ(t1 − t2) [21].

In case of vanishing noise, Eq. (1) immediately leadsto a circular trajectory with radius

R =

√√√√ (D⊥‖ + lD‖C)2 + (D⊥ + lD⊥C )2

(D⊥C + lDR)2. (2)

In agreement with the experimental observation (seeFig. 2(b)) the radius does not depend on the parti-

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Figure 2. (Color online) (a) Angular velocity ω and (b) ra-dius R of the circular motion of an L+ swimmer plotted asfunctions of the linear velocity v = |v| and the illuminationintensity I ∼ v. The dashed lines correspond to a linear fitwith nonzero and zero slope, respectively.

cle velocity set by the propulsion force. Rather, thevalue of R is only determined by the particle’s geom-etry, which defines its diffusional properties. More-over, the translational and angular particle velocities

are v = βF√

(D⊥‖ + lD‖C)2 + (D⊥ + lD⊥C )2 and ω =

βF (D⊥C + lDR). Both quantities are proportional to theinternal force F and ensure R = v/|ω| in perfect agree-ment with the experimental results shown in Fig. 2(a).

For a quantitative comparison with the experimentaldata, most importantly, the diffusion and coupling co-efficients for the particles under study have to be de-termined. They constitute the components of the gen-eralized diffusion matrix and are, in principle, obtainedfrom solving the Stokes equation that describes the lowReynolds number flow field around a particle close to thesubstrate [22]. This procedure can be approximated byusing a bead model [23], where the L-shaped particle isassembled from a large number of rigidly connected smallspheres. Exploiting the linearity of the Stokes equation,the hydrodynamic interactions between any pair of thosebeads can be superimposed to calculate the generalizedmobility tensor of the L-shaped particle and from thatits diffusion and coupling coefficients; details of the cal-culation are outlined in Ref. [23]. This method is well es-tablished for arbitrarily shaped particles in bulk solution[23, 24]. We take into account the presence of the sub-strate by using the Stokeslet close to a no-slip boundary[25] to model the hydrodynamic interactions between thecomponent beads in the bead model. For the L-shapedparticles considered here, we find that the value of D⊥exceeds the terms D⊥‖ , lD

‖C, and lD⊥C in the numerator

of Eq. (2) by more than one order of magnitude (giventhat l is in the range of 1 µm). On the other hand, thevalue of D⊥C is negligible compared to lDR. This finallyyields

R = |D⊥/(lDR)| (3)

as an approximate expression for the trajectory radiusand, correspondingly,

ω = βDRlF (4)

for the angular velocity.

Table I. Diffusion coefficients for the L-shaped particle in Fig.3(a) on a substrate: translational diffusion along the long(D⊥) and the short (D‖) axis of the L-shaped particle as wellas rotational diffusion constant DR.

experiment theory

D⊥ [10−3µm2s−1] 8.1± 0.6 8.3

D‖ [10−3µm2s−1] 7.2± 0.4 7.5

DR [10−4s−1] 6.2± 0.8 6.1

We determined the diffusion coefficients D⊥, D‖, andDR experimentally under equilibrium conditions (i.e., inthe absence of propulsion) from the short-time correla-tions of the translational and orientational components ofthe particle’s trajectories [26, 27] (see Tab. I). The exper-imental values are in good agreement with the theoreticalpredictions.

Inserting the experimentally determined values forthe diffusion coefficients and the mean trajectory ra-dius R = 7.91 µm into Eq. (3), we obtain the effectivelever arm l = −1.65 µm. This value is about a factorof two larger compared to an ideally shaped L-particle(see Fig. 3(a)) with its propulsion force perfectly centeredat the middle of the Au layer. This deviation suggeststhat the force is shifted by 0.94 µm in lateral direction,which is most likely caused by small inhomogeneities ofthe Au layer due to shadowing effects during the graz-ing incidence metal evaporation. Accordingly, from Eq.(4) we obtain the intensity-dependent propulsion forceF/I = 0.83× 10−13 Nµm2/µW.

To compare the trajectories obtained from theLangevin equations (1) with experimental data, we di-vided the measured trajectories into smaller segmentsand superimposed them such that the initial slopes andpositions of the segments overlap. After averaging thedata we obtained the noise-averaged mean swimmingpath, which is predicted to be a logarithmic spiral (spiramirabilis) [28] that is given in polar coordinates by

r(φ) = βF

√D2⊥

D2R + ω2

exp

(− DR

ω(φ− φ0)

). (5)

Qualitatively, such spirals can be understood as follows:in the absence of thermal noise, the average swimmingpath corresponds to a circle with radius R given byEq. (3). In the presence of thermal noise, however,single trajectory segments become increasingly differentas time proceeds. This leads to decreasing distancesdi between adjacent turns of the mean swimming path(di/di+1 = exp (2πDR/|ω|), see Fig. 3(d)) and, finally, tothe convergence in a single point for t→∞. Due to thealignment of the initial slope, this point is shifted relativeto the starting point depending on the alignment angleand the circulation direction of the particle.

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Figure 3. (Color online) (a) Geometrical sketch of an idealL+ swimmer as considered in our model. The dimensions area = 9 µm, b = 6 µm, xS = 2.29 µm, and yS = 3.55 µm (forhomogeneous mass density and an additional 20 nm thick Aulayer). The internal force F induces a torque M on the center-of-mass S depending on the lever arm l. (b),(c) Visualizationof the experimental trajectory (for an illumination intensity ofI = 7.5 µW/µm2) that is used for the quantitative analysis ofthe fluctuation-averaged trajectory in (d). The dashed curvein (d) is the experimental one, and the solid curve shows thetheoretical prediction with the starting point indicated by ared bullet. Inset: close-up of the framed area in the plot.

The solid curve in Fig. 3(d) is the theoretical predic-tion (see Eq. (5)) with the measured values of D⊥, DR,and ω. On the other hand, the dashed curve in Fig.3(d) visualizes the noise-averaged trajectory determineddirectly from the experimental data (see Figs. 3(b),(c)).The agreement of the two curves constitutes a direct ver-ification of our theoretical model on a fundamental level.

Finally, we also address the motion of asymmetricswimmers under confinement, e.g., their interaction witha straight wall. This is shown in Fig. 4(a) exemplarily foran L+ swimmer which approaches the wall at an angleθ. Due to the internal torque associated with the activeparticle motion, it becomes stabilized at the wall and

(a)

(b)

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critθθ

slθ

(d)

(e)θ

θ

0 20 40 60 80 100 120 140 160 180

(c)

θcrit

x [µm]

θ [deg]

critθθ

Figure 4. (Color online) (a),(b) Trajectories of an L+ swim-mer approaching a straight wall at different angles (symbolscorrespond to positions after 1 min each). (c) Experimen-tally determined particle motion for different contact anglesθ. Bullets and open squares correspond to particle sliding andreflection. (d),(e) Visualization of the predicted types of mo-tion for an L+ swimmer with arrows indicating the directionof the propulsion force: (d) stable sliding and (e) reflection.The angles are defined in the text.

smoothly glides to the right along the interface. In con-trast, for a much larger initial contact angle the internaltorque rotates the front part of the particle away fromthe obstacle, the motion is unstable, and the swimmeris reflected by the wall (see Fig. 4(b)) [29]. Figure 4(c)shows the observed dependence of the motional behavioras a function of the approaching angle.

The experimental findings are in line with an instabil-ity analysis based on a torque balance condition of anL-shaped particle at wall contact as a function of its con-tact angle θ. For θcrit < θ < π (see Figs. 4(b),(e)) witha critical angle θcrit, the particle is reflected, while for0 < θ < θcrit (see Figs. 4(a),(d)) stable sliding with anangle θsl occurs. Both, θsl and θcrit are given as stableand unstable solutions, respectively, of the torque bal-ance condition

|l| = [(a− yS) cos θ − xS sin θ] sin θ . (6)

For l = −0.71 µm corresponding to an ideal L-shapedparticle with the propulsion force centered in the middleof the Au layer, we obtain θsl = 8.0◦ and θcrit = 59.2◦,which is in good agreement with the measured value ofabout θcrit = 60◦ (see Fig. 4(c)). The observed scatter inthe experimental data around the critical angle is due tothermal fluctuations that wash out the sharp transitionbetween the sliding and the reflection regime.

In conclusion, we have demonstrated that due toviscous forces of the surrounding liquid, asymmetricmicroswimmers are subjected to a velocity-dependenttorque. This leads to a circular motion, which is ob-served in experiments in agreement with a theoreticalmodel based on two coupled Langevin equations. In achannel geometry, this torque leads either to a reflectionor a stable sliding motion along the wall. An interesting

5

question for the future addresses how asymmetric swim-mers move through patterned media. In the presence ofa drift force, one may expect Shapiro steps in the particlecurrent similar to what has also been found in colloidalsystems driven by a circular drive [30]. Another appeal-ing outlook addresses the motion of chiral swimmers inthe presence of external fields such as gravity [31]. Incase of asymmetric particles, this leads to an orienta-tional alignment during their sedimentation, which mayresult in a preferential motion relative to gravity similarto the gravitactic behavior of asymmetric cells as, e.g.,Chlamydomonas [32, 33].

We thank M. Aristov for assistance in particle prepara-tion and M. Heinen for helpful discussions. This work wassupported by the DFG within SPP 1296 and SFB TR6-C3 as well as by the Marie Curie-Initial Training Net-work Comploids funded by the European Union SeventhFramework Program (FP7). R. W. gratefully acknowl-edges financial support from a Postdoctoral Research Fel-lowship (WI 4170/1-1) of the DFG.

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