## Introduction

Active or self-propelled colloidal-particle systems are currently a subject of great interest in soft condensed matter science, owing to their ability to mimic the collective behaviour of complex living systems, but also to serve as model systems to study intrinsically out-of-equilibrium systems1,2,3,4,5,6. Self-propelled particles can exhibit rich collective behaviour, such as clustering, segregation, and anomalous density fluctuations, by consuming internal energy or extracting energy from their local environment in order to generate their own motion1,2,3,4,5,7. In recent years, several experimental self-propelled particle systems have been developed, namely Janus spherical particles2,8,9,10,11, Janus rods12, bimetallic rods13, granular rods14, gear-shaped15,16, hetero-dimers17, and L-shaped particles18, based on different self-propulsion mechanisms. However, most of the experimental studies have been reported on Janus spherical19 and rod-like particle systems12. Moreover, the majority of self-propelled particles move actively by converting the input energy directly into translational motion on a single particle level. As a result, the self-propelled particles exhibit ballistic behaviour at time scales much smaller than the rotational diffusion time and diffusive behaviour at longer time scales, with an effective diffusion coefficient that is many times larger than the equilibrium diffusion coefficient. In this work we ask ourselves whether, and if so how, self-propelled Janus spheres can assemble into bead chains that exhibit rotational motion, possibly coupled with translational motion. Some biological systems display active rotational motion, e.g., certain bacteria confined to two dimensions20, and dancing algae21. To the best of our knowledge, only a few synthetic systems show active rotation, e.g., a granular gear-shaped particle in an active bacterial bath15,16, multi-layered bimetallic rods22,23, L-shaped particles24, and random clusters of active Janus particles25,26. However, the swimming behaviour of these systems cannot be tuned by controlling the internal structure. Here, we develop a new type of internally driven colloidal-particle system, namely self-propelled colloidal bead chains with tunable stiffness. This system is suitable to study dynamics on a single particle level in concentrated systems, and exploits directed self-assembly of simple self-propelled building blocks. Moreover, we show that introducing (semi-)flexibility between the beads in a chain is sufficient to break the time-reversibility constraint to realize flagellum-like or helical-like motion that is powered by the fuel. To the best of our knowledge, flagellum-like motion has not been reported before in internally powered (self-propelled) colloidal swimmers. Only externally powered propellers such as helix-shaped27,28,29,30 particles using rotating magnetic fields, and DNA-linked assemblies of magnetic particles tethered to a red blood cell31 in biaxial magnetic fields have shown this motion. However, these systems are not suitable for collective behaviour studies because the long-ranged magnetic interactions between individual units are difficult to minimize. Additionally, they do not satisfy the force-free and torque-free condition28 which is an essential condition for studying internally powered colloidal systems. Moreover, these systems are difficult to fabricate.

Several simulations and theoretical studies have predicted that particle shape and swimming direction can affect the macroscopic behaviour of self-propelled particle systems1,2,6,32, but the experimental realization of shape-anisotropic particles in combination with varying propulsion directions scarcely exists. Several synthesis routes have been reported for synthesizing ‘passive’ colloidal molecules, complex-shaped particles33,34,35,36,37,38, chains of particles using different starting building blocks, e.g., isotropic spherical particles using various linking mechanisms34,39,40,41,42, Janus particles43, and flattened particles44. However, making these model systems with self-propelling capabilities remains a challenge. Here, we take an inspiration from the design principles of synthesizing passive colloidal molecules, where the colloidal spherical particles are treated as atoms and the interactions between them are tuned such that they self-assemble into well-defined complex clusters33,34,36. We show that self-assembly of the Janus particles into polymer-like chains with tunable stiffness provides a powerful route towards the fabrication of complex self-propelled particle systems.

## Fabrication of permanent bead chains

We first present our method to create catalytically powered colloidal rotators or active rigid linear bead chains by connecting the individual self-propelled spherical swimmers in such a way that neighbouring swimmers in the chain are always propelling in opposite directions. Our fabrication method consisted of two steps (Fig. 1a): (i) aligning Janus particles into linear chains using a high frequency external AC electric field45, and (ii) making these structures permanent using a combination of van der Waals attractions and linkers (i.e., inter-diffusion of polymer chains that are made of particles, and polyelectrolytes) to ensure that the chains remained intact as a chain even after removing the field. Suspensions consisting of 0.02 vol% half-side Pt coated polystyrene (PS) particles in deionized water were introduced to a rectangular electric cell (0.1 × 1.0 mm2). Upon application of a high frequency external AC electric field (E rms  = 0.01 Vμm−1, f = 800 kHz where E rms is the root-mean-square electric-field strength and f is the frequency), the particles assembled into staggered or zig-zag linear chains in the direction of the applied field (Fig. S1b). The difference in the polarizability of both sides (half-dielectric and half-metallic) of the Janus particles caused the particles to align into staggered chains as shown in Fig. S1b. At low particle concentrations and at low field strengths, the stable structure is a string fluid phase that consists of staggered chains of particles in the field direction and a liquid-like order in the

perpendicular to the field direction (Fig. S1b)45. However, these linear structures rely on the presence of external electric fields, i.e., the chains lost their identity (Fig. S1c) when the field was switched off 37,45. To circumvent this, we used a combination of a strong electric field and a heating step. At high field strengths (E rms  = 0.04–0.05 Vμm−1, f = 800 kHz), the induced dipolar attractions between the particles are strong enough to push the particles together into distances where the van der Waals attractions take over and bind the particles irreversibly. In particular, the van der Waals attractions between metallic halves of the particles are strong in comparison to those between the polymeric halves of the particles46. Due to some irregularities in the Pt coating a small fraction of the polymeric side of a particle can also touch the polymeric side of neighbouring particles. These polymer-polymer connections are the key to making permanent bonds between neighbouring Janus spheres through a heating step that was developed in an earlier study by Vutukuri et al.34,39. In the heating step, the sample cell was heated to 60–65 °C, which is well below the glass transition temperature47 (T g ≈ 107 °C) of polystyrene as shown in Fig. S2, for about 2–3 minutes using a hot-air stream that was much wider than the sample cell34,39. After 2–3 minutes, the field was slowly switched off and we found that the particles were permanently attached to each other in a staggered fashion and acted as a single rigid body. Fig. S3 shows the length distribution of the resulting permanent bead chains. It is important to mention here that the length of the permanent chains can be controlled by varying the distance between the two electrodes as has been developed in our previous study by Vutukuri et al.34. This procedure could thus significantly increase the monodispersity in length, if necessary, but was not used in the present study.

## Results and Discussion

### Dynamics of self-propelled rotators or active rigid bead chains

After creating the permanent structures, we studied their dynamic behaviour in the presence of H2O2. We note that it has been reported that in the case of individual half Pt-coated Janus spheres, the platinum-catalyzed decomposition reaction of H2O2 generates a concentration gradient of ions across the surface of the particles, inducing self-propulsion in the direction of the non-coated surface8,9,10,48. Although the underlying mechanism responsible for the propulsion is still under debate49, the dominant propulsion mechanism is probably a combination of diffusiophoresis and self-electrophoresis8,9,10,48. When we transferred the permanent staggered chains into a 1.0 vol% H2O2 solution, the bead chains showed autonomous sustained rotating behaviour on the surface of the bottom wall (Fig. 1b). The rotating movement can be attributed to the fact that the propulsion force on each Janus particle is acting in opposite directions in alternative fashion along the length of the chain (see the inset of Fig. 1b), resulting in a constant torque acting on the chain, which induces a rotational motion of the chain. The time-lapsed bright field images show the rotational motion of a 6-bead chain in 1.0 vol% of H2O2 solution as shown in Fig. 1b (see supplementary Movie S1). The inset of Fig. 2b shows the typical trajectory of the centre-of-mass of the bead chain. In order to characterize the trajectories, we first extracted the centroids of each bead (Janus spheres) in a chain using a particle-tracking algorithm50, and then calculated the orientation angle based on the coordinates of the first and the last bead as a function of time (see Materials and Methods section). From this, we calculated the mean-squared angular displacement (MSAD), and the translational mean-squared displacement (MSD) of the centre of mass of the chain as shown in Fig. 2. Due to the persistent angular propulsion, the bead chains are actively rotating, as shown by the cumulative rotational angle θ(t) continuously increasing with time (inset of the Fig. 2a). We obtained the average angular velocity (ω) and the passive (equilibrium) rotational diffusion coefficient (Dr,eq) of the rotator by fitting the MSAD curves with the equation $$\langle {\theta }^{2}\rangle =\langle {[\theta (t)-\theta (0)]}^{2}\rangle ={\omega }^{2}{t}^{2}+2{D}_{r}t$$ (see more details in Materials and Methods section). The rotator or active bead chain was propelling with an angular velocity ω = 0.48 ± 0.03 rad/s. The obtained equilibrium rotational constant,

### Self-propelled or active semiflexible bead chains

In the presence of 1.0 vol% H2O2, the chain showed both rotational and translational motion. The active rotational velocity around the axis of the active 6-bead semiflexible chain (Fig. 5) is 3.5 rad/sec and the chain is propelling along the long axis with a velocity of 0.9 μm/s. The chain of 8 beads propelled with a translational velocity of 0.62 μm/s and the rotational velocity around the axis is 3.8 rad/sec in 1.0 vol% H2O2 solution. Due to the large persistence length (l p ≈ 20 σ) of the semiflexible chains, the shorter chains (≤4σ) behaved like rigid chains and showed rotator behaviour.

In addition, we will demonstrate now that our method can also be used to make more complex self-propelled chains composed of both semiflexible and rigid parts as shown in Fig. 6. These complex chains were fabricated by mixing rigid and semiflexible chains together and repeating the same protocol as was used in making the constituent bead chains. In the presence of fuel (1.0 vol% H2O2), the propelling behaviour (see supplementary Movie S3) of a half-rigid and half-semiflexible 8-bead chain resembles the swimming behaviour of microorganisms with a flexible tail and a rigid head, such as e.g., E. coli. The rotational velocity of the long axis of the half-rigid and half-semiflexible 8-bead chain is 1.9 rad/sec and the propelling velocity is 1.2 μm/s. We believe that our method opens up new possibilities to realize more complex swimmers, for instance, attaching a semiflexible chain to a ‘passive’ single big sphere or a cargo.

### Bead-shell minimal model and equations of motion

In order to better understand the experimentally observed propulsion behavior of bead chains, we complemented our study with a minimal model that includes full hydrodynamic interactions with the fluid (see more details in Materials and Methods section). Instead of modeling the catalytically powered self-propulsion in terms of a slip velocity on the surface of the Janus particles, we model the propulsion mechanism by an effective propulsion force of a fixed magnitude that acts on each of the Janus spheres. For a rigid chain of spheres, these propulsion forces sum up to a force and torque acting on this rigid configuration. If the 6 × 6 hydrodynamic resistance tensor of this rigid body is known, the equations of motion of this object can be solved after imposing the effective force and torque. Alternatively, if the motion of the body is known, these equations of motions may be inverted to calculate the effective force and torque. We apply this to the rigid rotating chains, for which the observed angular velocity is used to calculate, given the ‘zig-zag’ arrangement of the propulsion forces, the magnitude of the effective propulsion force (more details in Materials and Methods section). To check the validity of this procedure, we feed the obtained effective force magnitude back into the equation of motion, from which we calculate the angular velocity for rigid bead chains of different lengths. We found our calculations to be consistent with the experimentally measured angular velocities (Fig. 3) in all three cases of bead chains with an even number of beads, as shown in Fig. 7.

As shown experimentally, the active rigid bead chains did not exhibit any translational motion (Figs 1b and 2), whereas the semiflexible chains showed a net translational motion (Fig. 5). Due to the over-damped nature of the dynamics at low Reynolds numbers, we expect that the internal motions within a semi-flexible bead chain will quickly decay towards an equilibrium configuration, rather than showing oscillatory motion as in the case of a driven oscillator with inertia. Below we show that a small anisotropy in both the shape and the force distribution of the bead chain, induced by the semiflexibility, can lead to a translational motion of the bead chain, even when the net effective propulsion force on the chain remains zero. To this end, we analyzed small anisotropies in the shape by considering bead chains with the centers of the Janus spheres positioned in a C-shape and on a helical centerline, as parameterized by a helical radius r and helical pitch p (see more details in Materials and Methods section). In addition, we studied anisotropic arrangements of these propulsion forces that deviate from the ‘zig-zag’ arrangement.

We summarized our numerical predictions in terms of ‘swimming’ behavior for the active helical bead chains with varying shape and propulsion force distributions in the Figure TI. We clearly found that anisotropy in both shape and force distribution is required to induce translational motion.

Hence, shape anisotropy in the form of a C-shaped or (moderately) helically shaped bead chain, combined with a heterogeneous distribution of propulsion forces, leads to a ‘swimming’ motion that qualitatively reproduces the experimentally observed propulsion behavior (Fig. 5). This can also be clearly seen in animation 6 found in the Supplementary information. We now take a close look at the influence of the force distribution on the swimming behavior. In Fig. 8, we compare the in-plane (projected) velocity v 2D and angular velocity around the long axis of the bead chain, ω ||, for a set of 100 randomly chosen configurations of propulsion forces for a fixed helically shaped bead chain, to the (angular) velocity for a range of different helical bead chains and the experimentally observed values. Here, we considered configurations with 6 randomly chosen orientations (blue), which are (almost) never force-free. In addition, we considered zig-zag configurations with a single randomly chosen orientation (purple), as well as configurations with three randomly chosen directions of propulsion forces, augmented with the three opposite vectors. For the latter two categories, the total effective propulsion force vanishes. From Fig. 8, we observed that these random configurations do not produce translational motion that qualitatively agrees with the observed experimental motion and therefore we conclude that the arrangement of the effective propulsion forces requires a certain correlation of the directions of the forces on the beads. Since the experimental system is semiflexible in nature (persistence length, l p ≈ 20σ; contour length, l c ≈ 6σ) and initially made from a linear zig-zag beadchain, there are always correlations between the force directions of neighboring beads in the chain.

## Materials and Methods

### Particle synthesis

We synthesized54 electrostatically stabilized and negatively charged polystyrene (PS) particles of 1.35 μm with a size polydispersity of 4%, and sterically stabilized (polyvinylpyrrolidone, PVP, Mw = 40 kg/mol) with a size polydispersity of 3%. These two types of particles were used in the fabrication of rigid bead chains. Whereas a longer molecular weight PVP (Mw = 360 kg/mol) PS particles were used in the fabrication of semiflexible bead chains. Sterically stabilized PS particles were synthesized using the method of Song et al.55. The size of the sterically stabilized PS particles was 1.40 μm with a size polydispersity of 5%. The particle size and polydispersity were determined using static light scattering, and scanning electron microscopy. We note that based on our camera frame rate, we deemed it easier to follow the dynamics of tracer particles in the case of rigid bead chains when we used bigger particles (2.1μm), because the chains are rotating faster. On the other hand, smaller particles (1.0 μm) are sufficient to capture the dynamics of tracer particles in the case of semiflexible case.

### Janus particles preparation

We first prepared a monolayer of spherical particles by slowly drying particles from a dilute suspension (0.05 wt%) on a clean microscope glass slides. A 15 nm thick layer of platinum (Pt) was then vertically deposited using a sputter coater (Cressington 208HR). Prior to particle detachment, slides were thoroughly washed with DI water, and subsequently particles were detached by sonicating the slides in DI water for 10 mins. Next, particles were washed a couple of times with DI water and then the suspension with the desired concentration was transferred to the electric cell.

### Electric-field setup

The electric cell consisted of a 0.1 mm × 1.0 mm cross section capillary with two 50 µm thickness nickel-alloy wires (Goodfellow) threaded along the side walls56,57. We used a function generator (Agilent, Model 3312 OA) and a wide band voltage amplifier (Krohn-Hite, Model 7602M) to generate the electric fields. We used a high frequency AC fields to minimize the polarization effects of the electric double layer around the particle. The field strength and the frequency were measured with an oscilloscope (Tektronix, Model TDS3052). We used a homemade DC filter to remove the DC component in the signal.

After filling the cell with the Janus colloidal particles suspension, we sealed both ends of the capillary with UV-curing optical adhesive (Norland No. 68) and cured the glue with UV light (λ = 350 nm, UVGL-58 UV lamp). The dispersion was then exposed to an AC field (E rms  = 0.04–0.05 Vμm−1, f = 800 kHz), and the ramping time was about 2 mins. After 2–3 mins, all the particles were assembled into zig-zag chains of one particle wide in the field direction. The dispersion was subsequently heated to 60–65 °C, which is well below the glass transition temperature (T g ≈ 107 °C) of polystyrene47, for about 2–3 minutes using a hot-air stream that was much wider than the sample cell34,39,58. We further kept the field on for 2–3 mins while the sample was cooled down to room temperature. After the fixation step, we carefully opened both ends of the electric cell and the bead chains were subsequently transferred to a small eppendrof tube (0.5 ml) for further dilution. The dynamics of the chains were studied in Lab-Tek chambered cover-glass (Thermo Fisher Scientific).

### Internal structure of staggered or zig-zag chains

Our method yielded regular and staggered (zig-zag) linear chains independent of whether it is used in the fabrication of rigid or semiflexible chains. We note that in our fabrication method, we exploited the difference in the polarizability of both sides (half-dielectric and half-metallic) of the Janus particles in external AC electric fields, causing a zig-zag chain configuration, as is shown in the Fig. S1. In the case of semiflexible (persistence length, l p ≈ 20 σ; contour length, l c ≈ 6 σ), we expect that there remains always some correlation between the neighboring particles, stemming from the zig-zagged pattern at synthesis, while a combination of flexibility of the chain and hydrodynamic propulsion may generate an asymmetry in the self-propulsion direction of the beads.

### Particle tracking

We recorded the particle dynamics using an Olympus IX81 confocal microscope, equipped with an Andor iXon3 camera, Andor 400-series solid-state lasers, a Yokogawa CSU-X1 spinning disk. Bright-field optical micrographs and videos were recorded using a Nikon microscope equipped with a CCD camera (Olympus CKX41). We processed recorded images and extracted the centroids of each particle in a chain using the particle tracking programs of Rogers et al.50. We then calculated the orientation of the chain from the pixel coordinates of the first bead and the last bead in the chain. To uniquely identify the both ends of the chains, it is required that the rotation angle of the chain between successive frames is less than π/2.

### Mean squared displacement of rigid chains

The MSAD and MSDs can be determined by solving the equations of motion for a circular swimmer52. The overdamped equations can be written in 2D as

$$\frac{dx(t)}{dt}=\nu \,\cos \,\theta (t)+{\xi }_{1}(t)$$
(1)
$$\frac{dy(t)}{dt}=\nu \,\sin \,\theta (t)+{\xi }_{2}(t)$$
(2)
$$\frac{d\theta (t)}{dt}=\omega +\zeta (t)$$
(3)

where ω is the rotational velocity, x and y are the center of mass coordinates, $$\nu$$ is the propulsion velocity, and θ is the orientation of the bead chain. The Brownian noise terms, $$\xi$$, $$\zeta$$, are Gaussian random variables with zero mean whose magnitudes are taken from theoretical Brownian diffusivities. By solving the Eqs 13 we obtained

$$\mathrm{MSAD}:\langle {\rm{\Delta }}{\theta }^{2}\rangle =\langle {[\theta (t)-\theta (0)]}^{2}\rangle ={\omega }^{2}{t}^{2}+2\,{D}_{r}t$$
(4)
$$\begin{array}{c}{\rm{M}}{\rm{S}}{\rm{D}}:\quad \quad \langle {\rm{\Delta }}{L}^{2}\rangle =\frac{2\,{v}^{2}}{{({D}_{r}^{2}+{\omega }^{2})}^{2}}[({D}_{r}^{2}+{\omega }^{2})\,{D}_{r}t+({\omega }^{2}-{D}_{r}^{2}\,)\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \,\,\,\,\,+{e}^{-{D}_{r}t}([{D}_{r}^{2}-{\omega }^{2}]\cos \,(\omega t)-2\omega {D}_{r}\,\sin \,(\omega t)]+2({D}_{\parallel }+{D}_{\perp })t\end{array}$$
(5)

where D r is the rotational diffusion coefficient, $${D}_{\parallel }$$, $${D}_{\perp },$$ are translational diffusion coefficients along the major and minor axis, and $$v$$ is the propulsion velocity.

### Resistance tensor, bead-shell model and equations of motion

Often, the catalytically powered self-propulsion of (half-)platinum coated particles, such as the bead chains considered this work, is modelled by a slip-velocity on the active part of the particle surface. Moreover, since there are no external forces acting on the bead chains, the total hydrodynamic force and torque on the chain both vanish, such that the flow field around the bead chain is, to leading order, at most of dipolar character. However, as was proven by Ten Hagen et al.59, the motion of a self-propelled particle is equivalent to that of a passive particle (of identical shape), which is driven by an effective external force F eff and torque T eff. As described in ref.59, if both the distribution of slip-velocity on the active particle surface, and flow field solutions around the driven passive particle for arbitrary external force and torque are known, the motion of the active particle may be solved by taking suitable choices of the driving forces in the conjugate passive particle system. Subsequently, the effective force F eff and torque T eff are calculated from the passive case by imposing the velocity and angular obtained in the previous step. Due to the linearity of low-Reynolds number hydrodynamics, the force and torque relate to the particle velocity U and angular velocity ω in a linear fashion:

$${\mathscr{F}}=\eta \,{\mathscr{R}}\,{\mathscr{V}}$$
(6)

where the 6-vector $${\mathscr{F}}=({{\boldsymbol{F}}}^{{\boldsymbol{e}}{\boldsymbol{f}}{\boldsymbol{f}}},{{\boldsymbol{T}}}^{{\boldsymbol{e}}{\boldsymbol{f}}{\boldsymbol{f}}})$$ denotes the (effective) force and torque on the particle, the 6-vector $${\mathscr{V}}=({\boldsymbol{U}},{\boldsymbol{\omega }})$$ denotes the (angular) velocity with respect to a chosen reference frame and the 6 × 6 tensor $${\mathcal R}$$ denotes the hydrodynamic resistance tensor, which depends itself on the shape of the self-propelled particle under consideration. The linearity of this dependence is due to the linearity of the Stokes equation that governs the hydrodynamics at low Reynolds numbers.

Here, rather than calculating the effective force and torque from the slip-velocity on the active bead chains, and flow field solutions of the passive problem, we adopt the opposite strategy by directly determining the effective force and torque from bead chain systems for which both the motion and the geometry are accurately known (from the experiment). To this end, we numerically (using a bead-shell model, see p. 21) determine the tensor $${\mathcal R}$$ for the rigid bead chains, for which the (angular) velocity is measured in the experiment: U = 0, while $${\boldsymbol{\omega }}=\omega \hat{{\boldsymbol{z}}}$$, which leads to $${{\boldsymbol{F}}}^{eff}=0$$ and $${{\boldsymbol{T}}}^{eff}={T}^{eff}\hat{{\boldsymbol{z}}}$$, where $${T}^{eff}$$ is solved from the equation of motion $${\mathscr{F}}=\eta \,{\mathscr{R}}\,{\mathscr{V}},$$ after we determined the resistance tensor $${\mathcal R}$$.

To proceed, we decompose the effective force and torque into effective forces acting on each of the individual Janus sphere that constitute the chain, as

$${{\boldsymbol{F}}}^{{\boldsymbol{eff}}}=\sum _{i}{{\boldsymbol{F}}}_{i}\,{\rm{and}}\,{{\boldsymbol{T}}}^{{\boldsymbol{eff}}}=\sum _{i}{{\boldsymbol{F}}}_{i}\times ({{\boldsymbol{r}}}_{i}-{{\boldsymbol{r}}}_{cm})$$
(7)

where we assume that the force magnitude |F i | is equal for all i, while the direction is determined by the orientation of the catalytic hemispheres: pointing along the axis that connects the ‘south’ pole on the passive hemisphere to the ‘north pole’ of the active hemisphere, as is shown in e.g. Fig. 9 of the main text. This assumption is equivalent to assuming that the (chemical) interaction between catalytic surfaces is small, such that all F i are of equal magnitude, and that the symmetry of the active hemisphere is preserved such that (effective) contributions of individual torques on each Janus sphere may be neglected, as would be the case of a single Janus sphere. Since an alternating (‘zig-zag’) orientation of the active hemispheres is clearly observed in the experiment, the magnitude |F i | is the only unknown and may be solved by plugging Eq. (7) into Eq. (6) for the rigid rotating chains, as described in the next section.

In this work, we calculate the rigid body resistance tensor of (active) bead chains using a bead-shell model60,61, in which the surface of a rigid body is homogeneously covered by a large number (M) of small spheres of radius a. When this cluster of M spheres is given a non-zero common velocity, a disturbance flow field is created, which in turn causes hydrodynamic interactions between the small spheres that are given by the Rotne-Prager mobility tensor62,63. The forces on the individual spheres can then be calculated by a 3 M × 3 M matrix inversion, from which the total force and torque on the rigid body follow as the sum of the individual forces and torques around a chosen reference point. Subsequently, these results are extrapolated to $$M\to \infty$$, while keeping Ma 2 finite, where typically $$M$$ = 1000–3000 spheres are used in this procedure. In this large-$$M$$ limit, we retrieve the boundary integral formulation of the Stokes equation, guaranteeing accurate results for the resistance tensor $${\mathcal R}$$.

We first validate our model for the active rigid bead chains by comparing the rotational diffusion coefficient for the 6-bead chain as obtained from the bead-shell model calculations to the experimental data. Assuming that the direction perpendicular to the plane of view of the microscope is the z-axis, the relevant rotational resistance coefficient is R 66, from which the rotational diffusion coefficient is expressed by the Stokes-Einstein relation as D r  = k B T/(η R 66). Here, using the viscosity of water and the experimental size of the Janus spheres, we find D r  = 0.0096 rad2/s, which is consistent with the experimentally measured value. We included the experimental results of rotational speed of the rigid chains in the presence of the fuel and the rotational diffusivity of chains in water, and the rotational and the translational speed of semiflexible chains in the fuel in supplementary information (see Table SII).

For completeness, we give the complete resistance tensor in the below paragraph. Next, we use the measurements of the rotational velocity of the rigid bead chains to estimate the value of the effective propulsion force acting on each individual Janus sphere. Note that in each case, we try to stay as closely as possible to the observed ‘zig-zag’ shape, as shown in Fig. S5, where we estimate from experimental snapshots that the angle between the vectors connecting the centers of adjacent Janus spheres is 0.15π on average. Also, the distribution of the propulsion forces is arranged in an alternating zig-zag fashion.

The validity of our model and consistency of our assumptions, Eq. (7), is verified from Fig. 7 above, where we compare the experimentally observed angular velocity ω with the angular velocity that we calculate from Eq. (6) for chains of different length, where the values of |F i | is averaged over the data of the rigid bead chains.

The complete 6 × 6 resistance tensor of a rigid linear ‘zigzag’ bead chain of 6 beads, aligned along the x-axis, is found to be:

$${\mathscr{R}}=(\begin{array}{cccccc}38.0\,\mu m & -0.35\,\mu m & -0.042\,\mu m & 0.097\,\mu {m}^{2} & -0.0041\,\mu {m}^{2} & -0.015\,\mu {m}^{2}\\ -0.35\,\mu m & 30.0\,\mu m & -0.056\,\mu m & -0.005\,\mu {m}^{2} & -0.01\,\mu {m}^{2} & 0.022\,\mu {m}^{2}\\ -0.042\,\mu m & -0.056\,\mu m & 38.0\,\mu m & 0.022\,\mu {m}^{2} & -0.0039\,\mu {m}^{2} & -0.09\,\mu {m}^{2}\\ 0.097\,\mu {m}^{2} & -0.005\,\mu {m}^{2} & 0.022\,\mu {m}^{2} & 430.0\,\mu {m}^{3} & -11.0\,\mu {m}^{3} & 1.1\,\mu {m}^{3}\\ -0.0041\,\mu {m}^{2} & -0.01\,\mu {m}^{2} & -0.0039\,\mu {m}^{2} & -11.0\,\mu {m}^{3} & 46.0\,\mu {m}^{3} & 0.15\,\mu {m}^{3}\\ -0.015\,\mu {m}^{2} & 0.022\,\mu {m}^{2} & -0.09\,\mu {m}^{2} & 1.1\,\mu {m}^{3} & 0.15\,\mu {m}^{3} & 430.0\,\mu {m}^{3}\end{array})$$