Experiments

For our experiments we use asymmetric L-shaped microswimmers with arm lengths of 9 and 6 μm, respectively, and 3 μm thickness that are obtained by soft lithography25,26 (see Methods for details). To induce a self-diffusiophoretic motion, the particles are covered with a thin Au coating on the front side of the short arm, which leads to local heating on laser illumination with intensity I (see Fig. 1d). When such particles are suspended in a binary mixture of water and 2,6-lutidine at critical composition, this heating causes a local demixing of the solvent that results in an intensity-dependent phoretic propulsion in the direction normal to the plane of the metal cap26,27,28. To restrict the particle’s motion to two spatial dimensions, we use a sample cell with a height of 7 μm. Further details are provided in Methods. Variation of the gravitational force is achieved by mounting the sample cell on a microscope, which can be inclined by an angle α relative to the horizontal plane (see Fig. 1b).

Figure 1: Characterization of the experimental setup. (a) Measured probability distribution p(φ) of the orientation φ of a passive L-shaped particle during sedimentation for an inclination angle α=10.67° (see sketch in b). (c) Experimental trajectories for the same inclination angle and increasing illumination intensity: (1) I=0, (2–5) 0.6 μW μm−2<I<4.8 μW μm−2, (6) I>4.8 μW μm−2. All trajectories start at the origin of the graph (red bullet). The particle positions after 1 min each are marked by yellow diamonds (passive straight downward trajectory), orange squares (straight upward trajectories), green bullets (active tilted straight downward trajectory) and blue triangles (trochoid-like trajectory). (d) Geometrical sketch of an ideal L-shaped particle (the Au coating is indicated by the yellow line) with dimensions a=9 μm and b=3 μm and coordinates x CM =−2.25 μm and y CM =3.75 μm of the centre of mass (CM) (with the origin of coordinates in the bottom right 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 length of the effective lever arm ℓ. and are particle-fixed unit vectors that denote the orientation of the particle (see Methods for details). Full size image

Passive sedimentation

Figure 1a shows the measured orientational probability distribution p(φ) of a sedimenting passive L-shaped particle (I=0) in a thin sample cell that was tilted by α=10.67° relative to the horizontal plane. The data show a clear maximum at the orientation angle φ=−34°, that is, the swimmer aligns slightly turned relative to the direction of gravity as schematically illustrated in Fig. 1b. A typical trajectory for such a sedimenting particle is plotted in Fig. 1c,1. To estimate the effect of the Au layer on the particle orientation, we also performed sedimentation experiments for non-coated particles and did not find measurable deviations. This suggests that the alignment cannot be attributed to an inhomogeneous mass distribution. The observed alignment with the shorter arm at the bottom (see Fig. 1b) is characteristic for sedimenting objects with homogeneous mass distribution and a fore–rear asymmetry11. This can easily be understood by considering an asymmetric dumbbell formed by two spheres with identical mass density but different radii R 1 and R 2 >R 1 . The sedimentation speed of a single sphere with radius R due to gravitational and viscous forces is υ∝R2. Therefore, if hydrodynamic interactions between the spheres are ignored, the dumbbell experiences a viscous torque resulting in an alignment where the bigger sphere is below the smaller one11.

In Fig. 1c, we show the particle’s centre-of-mass motion in the x–y plane as a consequence of the self-propulsion acting normally to the Au coating (see Fig. 1d). When the self-propulsion is sufficiently high to overcome sedimentation, the particle performs a rather rectilinear motion in upward direction, that is, against gravity (negative gravitaxis11, see Fig. 1c,2–4). With increasing self-propulsion, that is, light intensity, the angle between the trajectory and the y axis increases until it exceeds 90°. For even higher intensities, interestingly, the particle performs an effective downward motion again (see Fig. 1c,5). It should be mentioned that in case of the above straight trajectories, the particle’s orientation φ remains stable (apart from slight fluctuations) and shows a monotonic dependence on the illumination intensity (as will be discussed in more detail further below). Finally, for strong self-propulsion corresponding to high light intensity, the microswimmer performs a trochoid-like motion (see Fig. 1c,6).

To obtain a theoretical understanding of gravitaxis of asymmetric microswimmers, we first study the sedimentation of passive particles in a viscous solvent29 based on the Langevin equations

for the time-dependent centre-of-mass position r(t)=(x(t), y(t)) and orientation φ(t) of a particle. Here, the gravitational force F G , the translational short-time diffusion tensor , the translational–rotational coupling vector D C , the inverse effective thermal energy β=1/(k B T) and the Brownian noise terms ζ r and ζ φ are involved (see Methods for details).

Neglecting the stochastic contributions in equation (1), one obtains the stable long-time particle orientation angle

which only depends on the two coupling coefficients and determined by the particle’s geometry.

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)

where the dimensionless number P* is the strength of the self-propulsion, b is a characteristic length of the L-shaped particle (see Fig. 1d), ℓ denotes an effective lever arm (see Fig. 1d), and D R is the rotational diffusion coefficient of the particle. As shown in Methods, one can view F=|F|=P*/(bβ) and M=|M|=Fℓ as an effective force (in -direction) and an effective torque (perpendicular to the L-shaped particle), respectively, describing the self-propulsion of the particle (see Fig. 1d). This concept is in line with other theoretical work that has been presented recently30,31,32,33,34,35. The above equations of motion are fully compatible with the fact that, apart from gravity, a self-propelled swimmer is force-free and torque-free (see Methods).

The self-propulsion strength P* is obtained from the experiments by measuring the particle velocity (see Methods for details). The rotational motion of the microswimmer depends on the detailed asymmetry of the particle shape and is characterized by the effective lever arm ℓ relative to the centre-of-mass position as the reference point (see Fig. 1d). Note that the choice of the reference point also changes the translational and coupling elements of the diffusion tensor. If the reference point does not coincide with the centre of mass of the particle, in the presence of a gravitational force an additional torque has to be considered in equation (4). In line with our experiments (see Fig. 1c), the noise-free asymptotic solutions of equations (3) and (4) are either straight (upward or downward) trajectories or periodic swimming paths. Up to a threshold value of P*, the effective torque originating from the self-propulsion (first term on the right-hand side of equation (4)) can be compensated by the gravitational torque (second term β D C ˙F G on the right-hand side of equation (4)) so that there is no net rotation and the trajectories are straight. In this case, after a transient regime the particle orientation converges to

with the magnitude of the gravitational force F G =|F G |. Obviously, φ ∞ is a superposition of the passive case (first term on the right-hand side, cf. equation (2)) with a correction due to self-propulsion (second term on the right-hand side). The theoretical prediction given by equation (5) is visualized in Fig. 2 by the solid line, which is fitted to the experimental data (symbols) by using as fit parameter.

Figure 2: Measured particle orientations. Long-time orientation φ ∞ of an L-shaped particle in the regime of straight motion as a function of the strength P* of the self-propulsion for α=10.67°. The symbols with error bars representing the s.d. show the experimental data with the self-propulsion strength P* determined using equation (29) and orange squares and green bullets corresponding to upward and downward motion, respectively (see Fig. 1c). The solid curve is the theoretical prediction based on equation (5) with as fit parameter (see Methods for details). Full size image

The restoring torque caused by gravity depends on the orientation of the particle and becomes maximal at the critical angle . According to equation (5), this corresponds to a critical self-propulsion strength

with the particle’s buoyant mass m and the gravity acceleration of earth g=9.81 m s−2. When P* exceeds this critical value for a given inclination angle of the setup, the effective torque originating from the non-central drive can no longer be compensated by the restoring torque due to gravity (see equation (4)) so that and a periodic motion occurs.

To apply our theory to the experiments, we use the values of the various parameters as shown in Table 1. All data are obtained from our measurements as described in Methods.

Table 1 Experimentally determined values of the parameters used for the comparison with our theory. Full size table

Dynamical state diagram

Depending on the self-propulsion strength P* and the substrate inclination angle α, 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 marginally.) For very small values of P*, the particle performs straight downward swimming. The theoretical calculations reveal two regimes where on top of the downward motion either a small drift to the left or to the right is superimposed. For zero self-propulsion, this additional lateral drift originates from the difference between the translational diffusion coefficients, which is characteristic for non-spherical particles. Further increasing P* results in negative gravitaxis, that is, straight upward swimming. Here, the vertical component of the self-propulsion counteracting gravity exceeds the strength of the gravitational force. For even higher P*, the velocity-dependent torque exerted on the particle further increases and leads to trajectories that are tilted more and more until a re-entrance to a straight downward motion with a drift to the right is observed. For the highest values of P*, the particle performs a trochoid-like motion. The critical self-propulsion 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.

Figure 3: State diagram for moderate effective lever arm. (a) State diagram of the motion of an active L-shaped particle with an effective lever arm ℓ=−0.75 μm (see sketch in Fig. 1d) under gravity. The types of motion are straight downward swimming (SDS), straight upward swimming (SUS) and trochoid-like motion (TLM). Straight downward swimming is usually accompanied by a drift in negative (SDS−) or positive (SDS+) x direction. The transition from straight to circling motion is marked by a thick black line and determined analytically by equation (6). Theoretical noise-free example trajectories for the various states are shown in the inset. All trajectories start at the origin and the symbols (diamonds: SDS−, circles: SDS+, squares: SUS, triangles: TLM) indicate particle positions after 5 min each. (b) Experimentally observed types of motion for α=10.67°. The different symbols correspond to the various states as defined in a and are shifted in vertical direction for clarity. The error bars represent the s.d. Full size image

Indeed, the experimentally observed types of motion taken from Fig. 1c correspond to those in the theoretical state diagram. This is shown in Fig. 3b where we plotted the experimental data for α=10.67° as a function of the self-propulsion strength P*. Apart from small deviations due to thermal fluctuations, quantitative agreement between experiment and theory is obtained.

According to equations (3) and (4), the state diagram should strongly depend on the effective lever arm ℓ. To test this prediction experimentally, we tilted the silicon wafer with the L-particles about 25° relative to the Au source during the evaporation process. As a result, the Au coating slightly extends 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 change in the lever arm. The value of ℓ was experimentally determined to ℓ=−1.65 μm from the mean radius of the circular particle motion which is observed for α=0° (see ref. 26). The gravitactic behaviour of such particles is shown for different self-propulsion strengths in the inset of Fig. 4. Interestingly, under such conditions, we did not find evidence for negative gravitaxis. This is in good agreement with the corresponding theoretical state diagram shown in Fig. 4 and suggests that the occurrence of negative gravitaxis does not only depend on the strength of self-propulsion but also on the position where the effective force accounting for the self-propulsion mechanism acts on the body of the swimmer.