Gravitaxis of Asymmetric Self-Propelled Colloidal Particles

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Received 28 Oct 2013 | Accepted 9 Jul 2014 | Published 19 Sep 2014 DOI: 10.1038/ncomms5829 Gravitaxis of asymmetric self-propelled colloidal particles

Borge ten Hagen1, Felix Ku¨mmel2, Raphael Wittkowski3, Daisuke Takagi4, Hartmut Lo¨wen1 & Clemens Bechinger2,5

Many motile microorganisms adjust their swimming motion relative to the gravitational field and thus counteract sedimentation to the ground. This gravitactic behaviour is often the result of an inhomogeneous mass distribution, which aligns the microorganism similar to a buoy. However, it has been suggested that gravitaxis can also result from a geometric fore–rear asymmetry, typical for many self-propelling organisms. Despite several attempts, no conclusive evidence for such an asymmetry-induced gravitactic motion exists. Here, we study the motion of asymmetric self-propelled colloidal particles which have a homogeneous mass density and a well-defined shape. In experiments and by theoretical modelling, we demonstrate that a shape anisotropy alone is sufficient to induce gravitactic motion with either preferential upward or downward swimming. In addition, also trochoid-like trajectories transversal to the direction of gravity are observed.

1 Institut fu¨r Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universita¨tDu¨sseldorf, D-40225 Du¨sseldorf, Germany. 2 2. Physikalisches Institut, Universita¨t Stuttgart, D-70569 Stuttgart, Germany. 3 SUPA, School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK. 4 Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822, USA. 5 Max-Planck-Institut fu¨r Intelligente Systeme, D-70569 Stuttgart, Germany. Correspondence and requests for materials should be addressed to C.B. (email: [email protected]).

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ravitaxis (known historically as geotaxis) describes the response of motile microorganisms to an external Ggravitational field and has been studied for several decades. In particular, negative gravitaxis, that is, a swimming 0.2 motion opposed to a gravitational force FG, is frequently observed for flagellates and ciliates such as Chlamydomonas1 or 2 ) Paramecium . Since this ability enables microorganisms to ( counteract sedimentation, gravitaxis extends the range of their p 0.1 habitat and allows them to optimize their position in 2 y FG environments with spatial gradients . In case of photosynthetic α flagellates such as Euglena gracilis, gravitaxis (in addition to x phototaxis) contributes to their vertical motion in water, which enables them to adjust the amount of exposure to solar radiation3. 0 To achieve a gravitactic motion, in general, a stable orientation of –90 0 90 the microorganism relative to the gravitational field is required. (°) While in some organisms such an alignment is likely to be dominated by an active physiological mechanism4,5, its origin in 1,2,6–13 b other systems is still controversially discussed . It has been 100 suggested that gravitaxis can also result from purely passive 2 3 4 û effects like, for example, an inhomogeneous mass density within ⊥ F û the organism (bottom heaviness), which would lead to an G ll 14,15 CM alignment similar to that of a buoy in the ocean . (We use the a xCM term ‘gravitaxis’, which is not uniformly defined in the literature, m)

μ b ( 0 5 M also in the context of passive alignment effects.) Under such y y conditions, the organism’s alignment should be the same b CM regardless whether it sediments downward or upward in a hypo- or hyperdensity medium, respectively, as indeed observed 1 for pluteus larvae12. However, corresponding experiments with F 6 immobilized Paramecium and gastrula larvae showed an –100 b b alignment reversal for opposite sedimentation directions12. This 0 100 suggests another mechanical alignment mechanism that is caused x (μm) by the organism’s fore–rear asymmetry1,2,11,12. To explore the role of shape asymmetry on the gravitactic Figure 1 | Characterization of the experimental setup. (a) Measured behaviour and to avoid the presence of additional physiological probability distribution p(f) of the orientation f of a passive L-shaped mechanisms, in our experiments, we study the motion of colloidal particle during sedimentation for an inclination angle a ¼ 10.67° (see micron-sized swimmers with asymmetric shapes in a gravita- sketch in b). (c) Experimental trajectories for the same inclination angle and increasing illumination intensity: (1) I ¼ 0, (2–5) tional field. The self-propulsion mechanism is based on À 2 À 2 À 2 diffusiophoresis, where a local chemical gradient is induced in 0.6 mW mm oIo4.8 mW mm , (6) I44.8 mW mm . All trajectories the solvent around the microswimmer16. It has been shown that start at the origin of the graph (red bullet). The particle positions after 1 min the corresponding type of motion is similar to that of biological each are marked by yellow diamonds (passive straight downward microorganisms17–23. On the basis of our experimental and trajectory), orange squares (straight upward trajectories), green bullets theoretical results, we demonstrate that a shape anisotropy alone (active tilted straight downward trajectory) and blue triangles (trochoid-like is sufficient to induce gravitactic motion with either upward or trajectory). (d) Geometrical sketch of an ideal L-shaped particle (the Au downward swimming. In addition to straight trajectories, also coating is indicated by the yellow line) with dimensions a ¼ 9 mm and more complex swimming patterns are observed, where the b ¼ 3 mm and coordinates xCM ¼À2.25 mm and yCM ¼ 3.75 mm of the swimmers perform a trochoid-like (that is, a generalized centre of mass (CM) (with the origin of coordinates in the bottom right cycloid-like24) motion transversal to the direction of gravity. corner of the particle and when the Au coating is neglected). The propulsion mechanism characterized by the effective force F and the effective torque M induces a rotation of the particle that depends on the c ^ ^ Results length of the effective lever arm . u jj ðfÞ and u?ðfÞ are particle-fixed unit Experiments. For our experiments we use asymmetric L-shaped vectors that denote the orientation of the particle (see Methods for details). microswimmers with arm lengths of 9 and 6 mm, respectively, and 3 mm thickness that are obtained by soft lithography25,26 (see Methods for details). To induce a self-diffusiophoretic motion, Passive sedimentation. Figure 1a shows the measured orienta- the particles are covered with a thin Au coating on the front side tional probability distribution p(f) of a sedimenting passive of the short arm, which leads to local heating on laser L-shaped particle (I ¼ 0) in a thin sample cell that was tilted by illumination with intensity I (see Fig. 1d). When such particles a ¼ 10.67° relative to the horizontal plane. The data show a clear are suspended in a binary mixture of water and 2,6-lutidine at maximum at the orientation angle f ¼À34°, that is, the critical composition, this heating causes a local demixing of the swimmer aligns slightly turned relative to the direction of gravity solvent that results in an intensity-dependent phoretic propulsion as schematically illustrated in Fig. 1b. A typical trajectory for such in the direction normal to the plane of the metal cap26–28.To a sedimenting particle is plotted in Fig. 1c,1. To estimate the effect restrict the particle’s motion to two spatial dimensions, we use a of the Au layer on the particle orientation, we also performed sample cell with a height of 7 mm. Further details are provided in sedimentation experiments for non-coated particles and did not Methods. Variation of the gravitational force is achieved by find measurable deviations. This suggests that the alignment mounting the sample cell on a microscope, which can be inclined cannot be attributed to an inhomogeneous mass distribution. by an angle a relative to the horizontal plane (see Fig. 1b). The observed alignment with the shorter arm at the bottom

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(see Fig. 1b) is characteristic for sedimenting objects with a self-propelled swimmer is force-free and torque-free (see homogeneous mass distribution and a fore–rear asymmetry11. Methods). This can easily be understood by considering an asymmetric The self-propulsion strength P* is obtained from the experi- dumbbell formed by two spheres with identical mass density but ments by measuring the particle velocity (see Methods for different radii R1 and R24R1. The sedimentation speed of a single details). The rotational motion of the microswimmer depends on sphere with radius R due to gravitational and viscous forces is the detailed asymmetry of the particle shape and is characterized upR2. Therefore, if hydrodynamic interactions between the by the effective lever arm c relative to the centre-of-mass position spheres are ignored, the dumbbell experiences a viscous torque as the reference point (see Fig. 1d). Note that the choice of the resulting in an alignment where the bigger sphere is below the reference point also changes the translational and coupling smaller one11. elements of the diffusion tensor. If the reference point does not In Fig. 1c, we show the particle’s centre-of-mass motion in the coincide with the centre of mass of the particle, in the presence of x–y plane as a consequence of the self-propulsion acting normally a gravitational force an additional torque has to be considered in to the Au coating (see Fig. 1d). When the self-propulsion is equation (4). In line with our experiments (see Fig. 1c), the noise- sufficiently high to overcome sedimentation, the particle performs free asymptotic solutions of equations (3) and (4) are either a rather rectilinear motion in upward direction, that is, against straight (upward or downward) trajectories or periodic swimming gravity (negative gravitaxis11, see Fig. 1c,2–4). With increasing paths. Up to a threshold value of P*, the effective torque self-propulsion, that is, light intensity, the angle between the originating from the self-propulsion (first term on the right-hand trajectory and the y axis increases until it exceeds 90°. For even side of equation (4)) can be compensated by the gravitational higher intensities, interestingly, the particle performs an effective torque (second term bDC?FG on the right-hand side of downward motion again (see Fig. 1c,5). It should be mentioned equation (4)) so that there is no net rotation and the trajectories that in case of the above straight trajectories, the particle’s are straight. In this case, after a transient regime the particle orientation f remains stable (apart from slight fluctuations) and orientation converges to shows a monotonic dependence on the illumination intensity ! 0 1 (as will be discussed in more detail further below). Finally, for ? ? ? DC @ P qDffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC þ ‘DR A strong self-propulsion corresponding to high light intensity, the f1 ¼Àarctan jj þ arcsin À ð5Þ D bbFG jj 2 ?2 microswimmer performs a trochoid-like motion (see Fig. 1c,6). C DC þ DC To obtain a theoretical understanding of gravitaxis of asymmetric microswimmers, we first study the sedimentation of with the magnitude of the gravitational force FG ¼ |FG|. 29 passive particles in a viscous solvent based on the Langevin Obviously, fN is a superposition of the passive case (first term equations on the right-hand side, cf. equation (2)) with a correction due to self-propulsion (second term on the right-hand side). The r_ ¼ bD F þ f ; T G r ð1Þ theoretical prediction given by equation (5) is visualized in f_ b z ; ¼ DC Á FG þ f Fig. 2 by the solid line, which is fitted to the experimental data jj for the time-dependent centre-of-mass position r(t) ¼ (x(t), y(t)) (symbols) by using DC as fit parameter. and orientation f(t) of a particle. Here, the gravitational force The restoring torque caused by gravity depends on the FG, the translational short-time diffusion tensor DT, the orientation of the particle and becomes maximal at the critical jj ? translational–rotational coupling vector DC, the inverse effective angle fcrit ¼ arctanðDC =DC ÞÀp. According to equation (5), this thermal energy b ¼ 1/(kBT) and the Brownian noise terms fr and zf are involved (see Methods for details). Neglecting the stochastic contributions in equation (1), one obtains the stable long-time particle orientation angle ! –40 ? DC f1 fðt !1Þ¼Àarctan jj ; ð2Þ DC –60 jj ? which only depends on the two coupling coefficients DC and DC determined by the particle’s geometry. (°)

∞ –80 Swimming patterns under gravity. Extending equation (1) to also account for the active motion of an asymmetric microswimmer yields (see Methods for a hydrodynamic derivation) –100 ? r_ ¼ðP =bÞðDTu^? þ ‘DCÞþbDTFG þ fr; ð3Þ _ ? –120 f ¼ðP =bÞð‘DR þ DC Á u^?ÞþbDC Á FG þ zf; ð4Þ where the dimensionless number P* is the strength of the 0 100 200 300 self-propulsion, b is a characteristic length of the L-shaped P* particle (see Fig. 1d), c denotes an effective lever arm (see Figure 2 | Measured particle orientations. Long-time orientation fN of an Fig. 1d), and DR is the rotational diffusion coefﬁcient of the particle. As shown in Methods, one can view F ¼ |F| ¼ P*/(bb) L-shaped particle in the regime of straight motion as a function of the strength P* of the self-propulsion for a ¼ 10.67°. The symbols with error and M ¼ |M| ¼ Fc as an effective force (in u^?-direction) and an effective torque (perpendicular to the L-shaped particle), bars representing the s.d. show the experimental data with the self- respectively, describing the self-propulsion of the particle (see propulsion strength P* determined using equation (29) and orange squares Fig. 1d). This concept is in line with other theoretical work that and green bullets corresponding to upward and downward motion, has been presented recently30–35. The above equations of motion respectively (see Fig. 1c). The solid curve is the theoretical prediction based jj are fully compatible with the fact that, apart from gravity, on equation (5) with DC as ﬁt parameter (see Methods for details).

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Table 1 | Experimentally determined values of the 600 parameters used for the comparison with our theory. 200 SUS

0 Translational diffusion coefﬁcients TLM m)

μ SDS+ À 3 2 À 1 ( D|| 7.2 Â 10 mm s y À 3 2 À 1 400 –200 D? 8.1 Â 10 mm s SDS+ ? 2 1 TLM À SDS– SDS+ D jj 0 mm s

P* 0 200 400 600 Coupling coefﬁcients SUS x (μm) jj À 4 À 1 DC 5.7 Â 10 mms 200 ? À 4 À 1 DC 3.8 Â 10 mms SDS+ Rotational diffusion coefﬁcient SDS– À 4 À 1 DR 6.2 Â 10 s 0 0 20406080 Effective lever arm α (°) c À 0.75 mm

Buoyant mass = 10.67° m 2.5 Â 10 À 14 kg SDSSUS SDS+ TLM +,– corresponds to a critical self-propulsion strength qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 200 400 600 P* D jj 2 þ D?2 P? ¼ bbmg sin a C C ð6Þ crit D? þ ‘D Figure 3 | State diagram for moderate effective lever arm. (a) State C R diagram of the motion of an active L-shaped particle with an effective lever with the particle’s buoyant mass m and the gravity acceleration of arm c ¼À0.75 mm (see sketch in Fig. 1d) under gravity. The types of earth g ¼ 9.81 m s À 2. When P* exceeds this critical value for a motion are straight downward swimming (SDS), straight upward swimming given inclination angle of the setup, the effective torque (SUS) and trochoid-like motion (TLM). Straight downward swimming is originating from the non-central drive can no longer be usually accompanied by a drift in negative (SDS À ) or positive (SDS þ ) x compensated by the restoring torque due to gravity (see direction. The transition from straight to circling motion is marked by a thick equation (4)) so that f_ 6¼ 0 and a periodic motion occurs. black line and determined analytically by equation (6). Theoretical noise- To apply our theory to the experiments, we use the values of free example trajectories for the various states are shown in the inset. All the various parameters as shown in Table 1. All data are obtained trajectories start at the origin and the symbols (diamonds: SDS À , circles: from our measurements as described in Methods. SDS þ , squares: SUS, triangles: TLM) indicate particle positions after 5 min each. (b) Experimentally observed types of motion for a ¼ 10.67°. The different symbols correspond to the various states as defined in a and are Dynamical state diagram. Depending on the self-propulsion shifted in vertical direction for clarity. The error bars represent the s.d. strength P* and the substrate inclination angle a, different types of motion occur (see Fig. 3a). (For clarity, we neglected the noise in Fig. 3, but we checked that noise changes the trajectories only prediction experimentally, we tilted the silicon wafer with the ° marginally.) For very small values of P*, the particle performs L-particles about 25 relative to the Au source during the straight downward swimming. The theoretical calculations reveal evaporation process. As a result, the Au coating slightly extends two regimes where on top of the downward motion either a small over the front face of the L-particles to one of the lateral planes which results in a shift of the effective propulsion force and a drift to the left or to the right is superimposed. For zero self- c propulsion, this additional lateral drift originates from the dif- change in the lever arm. The value of was experimentally c m ference D À D between the translational diffusion coefficients, determined to ¼À1.65 m from the mean radius of the jj ? a ° which is characteristic for non-spherical particles. Further circular particle motion which is observed for ¼ 0 (see ref. 26). increasing P* results in negative gravitaxis, that is, straight The gravitactic behaviour of such particles is shown for different upward swimming. Here, the vertical component of the self- self-propulsion strengths in the inset of Fig. 4. Interestingly, propulsion counteracting gravity exceeds the strength of the under such conditions, we did not find evidence for negative gravitational force. For even higher P*, the velocity-dependent gravitaxis. This is in good agreement with the corresponding torque exerted on the particle further increases and leads to theoretical state diagram shown in Fig. 4 and suggests that the trajectories that are tilted more and more until a re-entrance to a occurrence of negative gravitaxis does not only depend on the straight downward motion with a drift to the right is observed. strength of self-propulsion but also on the position where the For the highest values of P*, the particle performs a trochoid-like effective force accounting for the self-propulsion mechanism acts ? on the body of the swimmer. motion. The critical self-propulsion Pcrit for the transition from straight to trochoid-like motion as obtained analytically from equation (6) is indicated by a thick black line in Fig. 3a. Discussion Indeed, the experimentally observed types of motion taken Our model allows to derive general criteria for the existence of from Fig. 1c correspond to those in the theoretical state diagram. negative gravitaxis, which are applicable to arbitrary particle This is shown in Fig. 3b where we plotted the experimental data shapes. For a continuous upward motion, first, the self- for a ¼ 10.67° as a function of the self-propulsion strength P*. propulsion P* trivially has to be strong enough to overcome Apart from small deviations due to thermal fluctuations, quanti- gravity. Secondly, the net rotation of the particle must vanish, ? ? tative agreement between experiment and theory is obtained. which is the case for P Pcrit (see equation (6)). Otherwise, for According to equations (3) and (4), the state diagram should example, trochoid-like trajectories are observed. As shown in the strongly depend on the effective lever arm c. To test this state diagrams in Figs 3 and 4, the motional behaviour depends

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400 s=a 0 ^ F u⊥ CM G m)

300 μ –40 ( 123 y 2 F –80 s=–2b ^ 050100 150 u ll 200 μ P* x ( m) CM TLM 3 Vsl r a 100 2 SDS+ CM

SDS– 2b 1 0 020406080 α (°) Figure 5 | Sketch for the slender-body model. Illustration of a rigid Figure 4 | State diagram for large effective lever arm. The same as in L-shaped particle with arm lengths a and 2b. The position r(t) of the Fig. 3a but now for an active L-shaped particle with effective lever arm centre of mass (CM) and the orientation angle f(t) evolve over time c ¼À1.65 mm (see left inset). Notice that the regime of upward swimming due to the gravitational force FG and fluid slip velocity Vsl on the particle is missing here. Only straight downward swimming (with a drift in negative surface. The unit vectors u^ jj and u^? indicate the particle-fixed coordinate (SDS ) or positive (SDS ) x direction) and trochoid-like motion (TLM) À þ system. are observed. The right inset shows three experimental trajectories whose corresponding positions in the state diagram are indicated together with error bars representing the s.d. coated onto a silicon wafer, soft baked for 80 s at 95 °C and then exposed to ultraviolet light through a photomask. After a postexposure bake at 95 °C for 140 s, the entire wafer with the attached particles is covered by a several-nm-thick Au sensitively on several parameters such as the strength of the self- coating. During this process, the substrate is aligned by a specific angle relative to propulsion and the length of the effective lever arm. This may the evaporation source. Depending on the chosen angle, the Au coating can also provide an explanation why gravitaxis is only observed for selectively be applied to specific regions of the front sides of the short arms of the particles. Afterwards, the particles are released from the substrate by an ultrasonic particular microorganisms as, for example, Chlamydomonas treatment and suspended in a binary mixture of water and 2,6-lutidine at critical reinhardtii and why the gravitactic behaviour is subjected to composition (28.6 mass percent of lutidine) that is kept several degrees below its 7 large variations within a single population . lower critical point TC ¼ 34.1 °C. Time-dependent particle positions and orienta- In our experiments, the particles had an almost homogeneous tions were acquired by video microscopy at a frame rate of 7.5 f.p.s. and stored for mass distribution to reveal pure shape-induced gravitaxis. For real further analysis. biological systems, gravitaxis is a combination of the shape- induced mechanism studied here and bottom heaviness resulting Langevin equations for a sedimenting passive particle. The Langevin equations for the centre-of-mass position r(t) and orientation f(t) of an arbitrarily shaped from an inhomogeneous mass distribution36. In particular for passive particle that sediments under the gravitational force FG ¼ (0, À mg sin a) eukaryotic cells, small mass inhomogeneities due to the nucleus are given by equation (1). The key quantities are the translational short-time ? or organella are rather likely. Bottom heaviness can straight- diffusion tensor DT ¼ D jj u^ jj u^ jj þ D jj ðu^ jj u^? þ u^? u^ jj ÞþD?u^? u^? with denoting the dyadic product and the translational–rotational coupling forwardly be included in our modelling by an additional torque jj ? 37 vector DC ¼ DC u^ jj þ DC u^? with the centre-of-mass position as reference point. contribution and would support negative gravitaxis. Even for u^ f; f u^ f; f 11 The orientation vectors jj ¼ðcos sin Þ and ? ¼ðÀsin cos Þ are fixed in microorganisms with axial symmetry but fore–rear asymmetry the body frame of the particle. (Notice that equation (1) could equivalently be like Paramecium2, both the associated shape-dependent hydro- written with friction coefficients instead of diffusion coefficients38. In this article, dynamic friction and bottom heaviness contribute to gravitaxis so we use the given representation since the diffusion coefficients can directly be that the particle shape matters also in this special case. obtained from our experiments.) Finally, fr(t)andzf(t) are Gaussian noise terms of zero mean and variances hfrðt1Þ frðt2Þi ¼ 2DTdðt1 À t2Þ; hfrðt1Þzfðt2Þi In conclusion, our theory and experiments demonstrate that ¼ 2DCdðt1 À t2Þ, and hzfðt1Þzfðt2Þi ¼ 2DRdðt1 À t2Þ. Within this formalism, ? the presence of a gravitational field leads to straight downward the specific particle shape enters via the translational diffusion coefficients D||, D jj , jj ? and upward motion (gravitaxis) and also to trochoid-like and D?, the coupling coefficients DC and DC , and the rotational diffusion trajectories of asymmetric self-propelled colloidal particles. On coefficient DR. the basis of a set of Langevin equations, one can predict from the shape of a microswimmer, its mass distribution, and the geometry Langevin equations for an active particle under gravity. In the following, we provide a hydrodynamic derivation of our equations of motion (3) and (4) to of the self-propulsion mechanism whether negative gravitaxis can justify their validity. Our derivation is based on slender-body theory for Stokes occur. Although our study does not rule out additional flow39. This approach has been applied successfully to model, for example, flagellar physiological mechanisms for gravitaxis4,5, our results suggest locomotion40,41. The theory is adapted to describe the movement of a rigid that a swimming motion of biological microswimmers opposed to L-shaped particle in three-dimensional Stokes flow as sketched in Fig. 5. A key E gravity can be entirely caused by a fore–rear asymmetry, that is, assumption is that the width 2 of the arms of the particle is much smaller than the 11 total arc length L ¼ a þ 2b of the particle, where a and 2b are its arm lengths. passive effects . Such passive alignment mechanisms may also be Though E ¼ 1.5 mm is only one order of magnitude smaller than L ¼ 15 mminthe useful in situations where directed motion of autonomous self- experiments, the slenderness approximation E L offers valuable fundamental propelled objects is required as, for example, in applications insight into the effects of self-propulsion and gravity on the particle’s motion, as examined below. where they serve as microshuttles for directed cargo delivery. The centreline position x(s) of the slender L-shaped particle is described by a Furthermore, their different response to a gravitational field could parameter s with À 2brsra: be utilized to sort microswimmers with respect to their shape and 33 su^ jj ; if À 2b s 0; activity . xðsÞ¼r À rCM þ ð7Þ su^?; if 0os a:

Methods Here, r is the centre-of-mass position of the particle, u^ jj and u^? are unit 2 2 Experimental details. Asymmetric L-shaped microswimmers were fabricated vectors deﬁning the particle’s frame of reference and rCM ¼ða u^? À 4b u^ jjÞ=ð2LÞ from photoresist SU-8 by photolithography. A 3-mm thick layer of SU-8 is spin is a vector from the point where the two arms meet at right angles to the

NATURE COMMUNICATIONS | 5:4829 | DOI: 10.1038/ncomms5829 | www.nature.com/naturecommunications 5 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5829 centre of mass (CM) (see Fig. 5). The fluid velocity on the particle surface is 2a3b3K D? ¼ ; ð18Þ approximated by x_ þ vslðsÞ, where vsl(s) is a prescribed slip velocity averaged over jj L2 the intersection line of the particle surface and the plane through the point x(s) adjacent to the centreline. Motivated by the slip flow generated near the Au coating a4b2 D? ¼ 2ða þ bÞA À K; ð19Þ in the experiments, we set vsl ¼ÀVslu^? along the shorter arm ( À 2brsr0) and L2 no slip (that is, vsl ¼ 0) along the other arm. According to the leading-order terms 39 2 in slender-body theory , the fluid velocity is related to the local force per unit jj a bða þ 4bÞK D ¼ ; ð20Þ length f(s) on the particle surface and given by C L x_ þ v ¼ cðI þ x0 x0Þf ð8Þ 4ab2ða þ bÞK sl D? ¼ ; ð21Þ C L with the constant c ¼ log(L/E)/(4pZ), the viscosity of the surrounding fluid Z, the 0 identity matrix I, the vector x ¼ qx/qs that is locally tangent to the centreline and DR ¼ 2ða þ bÞða þ 4bÞK ð22Þ the dyadic product . The force density f satisfies the integral constraints that the with the constant net force on the particle is the gravitational force, 2c Z K ¼ b ð2ða þ bÞða þ 4bÞA a ð23Þ 2 2 FG ¼ f ds; ð9Þ À a b ð4abL þ a3 þ 8b3ÞÞ À 1: À 2b L2 and that the net torque relative to the centre of mass of the particle vanishes, Defining 2 R ? 2bb Vsl ^ a 0 P ¼ ; ð24Þ ez Á À 2bðÀrCM þ sx ÞÂf ds c R R 0 a ¼ su^? Á f ds À su^ jj Á f ds ab À 2b 0 ð10Þ ‘ ¼À ð25Þ 1 2 2 L þ 2L ða u^ jj Á FG þ 4b u^? Á FGÞ one finally obtains ¼ 0; ? r_ ¼ðP =bÞðDTu^? þ ‘DCÞþbDTFG; ð26Þ where êz is the unit vector parallel to the z axis. Thus, in the absence of gravity, _ ? the self-propelled particle is force-free and torque-free as required. f ¼ðP =bÞð‘DR þ DC Á u^?ÞþbDC Á FG; ð27Þ We differentiate equation (7) with respect to time and insert it into where D is the translational short-time diffusion tensor, DC the translational– equation (8). This leads to T rotational coupling vector and DR the rotational diffusion coefficient of the particle. _ Up to the noise terms, which follow directly from the fluctuation–dissipation f 2 2 r_ þ vsl þ ða u^ jj þ 4b u^?Þ theorem and can easily be added, equations (26) and (27) are the same as our ( 2L equations of motion (3) and (4). This proves that our equations of motion are _ ð11Þ cðI þ u^ jj u^ jj Þf À sfu^?; if À 2b s 0; physically justified and that they provide an appropriate theoretical framework to ¼ understand our experimental results. cðI þ u^ u^ Þf þ sf_ u^ ; if 0os a: ? ? jj Finally, we justify the concept of effective forces and torques by comparing the

The u^ jj and u^? components of equation (11) are equations of motion of a self-propelled particle with the equations of motion of a passive particle that is driven by an external force Fext, which is constant in the 2 _ 2cu^ jj Á f; if À 2b s 0; particle’s frame of reference, and an external torque M . To derive the equations ^ _ a f ext u jj Áðr þ vslÞþ 2L ¼ _ ð12Þ cu^ jj Á f þ sf; if 0os a; of motion of an externally driven passive particle, we assume no-slip conditions for the fluid on the entire particle surface. In analogy to the derivation for a self- and propelled particle (see further above) one obtains 0 1 0 10 1 2 2 _ ^ _ u^ jj Á r_ 2ða þ bÞ 0 À a b=L u^ jj Á r_ ^ _ b f cu? Á f À sf; if À 2b s 0; u? Áðr þ vslÞþ2 L ¼ ð13Þ @ ^ _ A 1 @ 2 A@ ^ _ A 2cu^? Á f; if 0os a: ZH u? Á r ¼ 2c 0 a þ 4b À 2ab =L u? Á r _ 2 2 _ f 0 À a b1=L 0À 2ab =LA1 f After integrating equations (12) and (13) over s from À 2b to 0 and separately from ð28Þ u^ jj Á FG u^ jj Á Fext 0toa, the unknown f can be eliminated using equations (9) and (10). This results @ A @ A ¼ u^? Á FG þ u^? Á Fext : in the system of ordinary differential equations 0 M 0 1 0 10 1 ext 2 u^ jj Á r_ 2ða þ bÞ 0 À a b=L u^ jj Á r_ We emphasize that the grand resistance matrix H in equation (28) for an externally B C B CB C B C 1 B 2 CB C driven passive particle is identical to that in equation (14) for a self-propelled ZH@ u^? Á r_ A¼ 2c @ 0 a þ 4b À 2ab =L A@ u^? Á r_ A particle. Formally, the two equations are exactly the same if u^ jj Á Fext ¼ 0, _ 2 2 _ 2 f À a b=L À 2ab =LA f u^? Á Fext ¼ 2bVsl=c, and Mext ¼À2ab Vsl/(cL). This shows that the motion of a 0 1 0 1 ð14Þ self-propelled particle with fluid slip v ¼ÀV u^ along the shorter arm is u^ Á F 0 sl sl ? B jj G C B C identical to the motion of a passive particle driven by a net external force B C B C ^ = ^ c ¼ @ u^? Á FG A þ @ 2bVsl=c A Fext ¼ Fu? ¼ð2bVsl cÞu? and torque Mext ¼À(ab/L)F ¼ F with the effective self-propulsion force F ¼ 2bV /c and the effective lever arm c ¼Àab/L. (Note that 2 sl 0 À 2ab Vsl=ðcLÞ while the equations of motion of a self-propelled particle and an externally driven passive particle are the same in this case, the flow and pressure fields generated by with the constant A ¼ ((8L2 À 6ab)(a3 þ 8b3) À 6L(a4 þ 16b4))/(12L2). It is the particles are different). important to note that the matrix H is identical to the grand resistance matrix (also Consequently, the concept of effective forces and torques, where formally called ‘hydrodynamic friction tensor’42) for a passive particle (see further below). external (‘effective’) forces and torques that move with the self-propelled particle Since equation (14) is given in the particle’s frame of reference, we next transform are used to model its self-propulsion, is justified and can be applied to facilitate the it to the laboratory frame by means of the rotation matrix 0 1 understanding of the dynamics of self-propelled particles. cos f À sin f 0 @ A MðfÞ¼ sin f cos f 0 : ð15Þ Analysis of the experimental trajectories. For the interpretation of the experi- 001 mental results in the context of the theoretical model, it is necessary to determine the self-propulsion strength P* for the various observed trajectories. This is As obtained from the hydrodynamic derivation, the generalized diffusion tensor 0 1 achieved by means of the relation D D? D jj u À bF ðD À D Þ sin f cos f B jj jj C C ? x G jj ? P ¼ b jj ; ð29Þ 1 À 1 B C ? D ¼ H ¼ B D? D D? C; ð16Þ ‘DC cos f ÀðD? þ ‘DC Þ sin f bZ @ jj ? C A jj ? which is obtained from equation (3). It provides P* as a function of the measured D D DR C C centre-of-mass velocity ux in x direction. On the other hand, the gravitational force FG ¼ mg sina is directly obtained from the easily adjustable inclination angle a of where b ¼ 1/(kBT) is the inverse effective thermal energy, is determined by the parameters the experimental setup. 2 4 4a b Determination of the experimental parameters D jj ¼ða þ 4bÞA À K; ð17Þ . For the comparison of the L2 theory with our measurements, specific values of a number of parameters have to

6 NATURE COMMUNICATIONS | 5:4829 | DOI: 10.1038/ncomms5829 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5829 ARTICLE

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