Letter | Published:

# Targeted assembly and synchronization of self-spinning microgears

## Abstract

Self-assembly is the autonomous organization of components into patterns or structures: an essential ingredient of biology and a desired route to complex organization1. At equilibrium, the structure is encoded through specific interactions2,3,4,5,6,7,8, at an unfavourable entropic cost for the system. An alternative approach, widely used by nature, uses energy input to bypass the entropy bottleneck and develop features otherwise impossible at equilibrium9. Dissipative building blocks that inject energy locally were made available by recent advances in colloidal science10,11 but have not been used to control self-assembly. Here we show the targeted formation of self-powered microgears from active particles and their autonomous synchronization into dynamical superstructures. We use a photoactive component that consumes fuel, haematite, to devise phototactic microswimmers that form self-spinning microgears following spatiotemporal light patterns. The gears are coupled via their chemical clouds by diffusiophoresis12 and constitute the elementary bricks of synchronized superstructures, which autonomously regulate their dynamics. The results are quantitatively rationalized on the basis of a stochastic description of diffusio-phoretic oscillators dynamically coupled by chemical gradients. Our findings harness non-equilibrium phoretic phenomena to program interactions and direct self-assembly with fidelity and specificity. It lays the groundwork for the autonomous construction of dynamical architectures and functional micro-machinery.

## Main

The self-assembly of complex structures which emulate the fidelity and tunability of their biological counterparts is a fundamental goal of materials science and engineering. In colloidal science, the paradigm for equilibrium self-assembly encodes information through specific interactions between building blocks that, in turn, promote assembly, so that the resulting (static) structure is the thermodynamic ground state. Emergent properties in active matter13, in which self-driven units convert an energy source into useful motion and work, have been the focus of intensive experimental and theoretical studies in recent years and primarily focused on meso-scale or macroscopic fluxes14, flocks15,16 or flows17,18 from local mechanical forcing. Their use to design nonequilibrium interactions and control self-assembly remains, however, largely unexplored19,20,21,22. We introduce a sequential approach as a means to architect structures, the dynamics of which arise from chemical gradients through diffusiophoretic interactions. The process is dissipative and makes use of a photocatalytic material, haematite, to harvest energy from a hydrogen peroxide fuel23 and form hierarchical superstructures through non-equilibrium pathways (Fig. 1). Instrumental to our work is the development of phototactic swimmers (Fig. 1a), which direct along light gradients and assemble into self-spinning microgears or rotors (Fig. 1b). We focus in this Letter on the self-organization of moderate numbers of rotors with manually sorted spins. They organize into stable patterns, whose dynamics originates from the interplay between the phase and the space coordinates of the components. For example, sets of seven rotors form hexagonal patterns (Supplementary Video 1, Fig. 1c), and exhibit an edge-current travelling at Ω ~ 0.1 rad s−1 for co-rotating rotors (Supplementary Video 2) or remain static for structures with alternating spins (Supplementary Video 3). Switching off the light, the system swiftly returns to equilibrium, the interactions vanish and the order is destroyed by thermal noise (Fig. 1c,inset).

A haematite cube (α–Fe2O3) decomposes hydrogen peroxide, at a rate ν = βI/(I + I0), dependent on the light intensity I (ref. 23), with β and I0 given by the photocatalytic properties of haematite, and constant parameters in the experiment. It travels towards the low intensity in a gradient as a result of the asymmetric fuel consumption and diffusiophoresis12, the migration of a colloidal particle in a concentration gradient c. The measured velocity shows good agreement with the diffusiophoretic velocity, Vcν(I), predicted from the experimental intensity profile (Fig. 2a). The migration is slow, V ~ 2 μm s−1, as the gradient on the particle scale is moderate. In the following, we devise swimmers with a fore–aft asymmetry, consisting of the extrusion of the haematite cube from a chemically inert polymer bead24 (Supplementary Information). In uniform light, they exhibit a persistent random walk as previously reported for Janus microswimmers11,25. The polymer bead heads and the velocity, up to 30 μm s−1, is controlled by the light intensity. In a light gradient, the haematite component is directed towards the low intensity, exerting a torque on the composite particle. The swimmer reorients and migrates towards the high intensity light (Fig. 2b and Supplementary Information). We use a custom optical set-up combining light sources to deliver spatio-temporal patterns with a resolution of a few micrometres and 0.01 s accuracy and guide the self-assembly (Supplementary Information).

A dilute suspension of the swimmers is fed to a capillary, residing near the bottom surface at a surface density Φs ~ 10−3 particles per μm2. They travel isotropically in uniform light, but gather and collide as we superimpose the bright spot of a focused laser (λ = 404 nm), assembling into a self-spinning microgear. The structure is composed of a core vertical swimmer surrounded by six peripheral close-packed particles (Fig. 1b). The rotor remains stable in uniform light after extinction of the laser spot and its lifetime is set by the duration of the experiment (~10–20 mins). We repeat the light sequence to form multiple rotors. The handedness is random and we measure 49.85% clockwise and 50.15% anticlockwise from 1,017 rotors. The formation is specific and high yield as added particles spontaneously detach from the microgears (Supplementary Video 4). It shows fidelity, forming self-powered microgears in the absence of a template26 and under limited feedback by an external operator, a key feature in self-assembly.

As haematite is partially absorbing at the wavelength of activation, it sets a gradient of light and reaction rate along the illumination axis. The effect is negligible in most situations but becomes significant as a swimmer crosses the brightest region of a laser spot: for a laser shone from below, the haematite is phoretically lifted, and the swimmer flips vertically. It swims downwards against the bottom wall, producing a hydrodynamic pumping27,28. It attracts the neighbouring swimmers, already converging to the area by phototaxis. The swimmers collide head-on with the central particle, collectively orienting as they pack at the periphery, forming the self-spinning structure (Figs.1d and 2b). In our experimental conditions, we observe neither decomposition of the hydrogen peroxide nor significant optical forces from the light. Shining the laser from the top does not allow the vertical flipping of a swimmer and inhibits the formation of the rotors. Transient rotors can be formed by focusing the laser on a sphere deposited on the substrate; however, with limited yield and lifetime, which stresses the importance of the hydrodynamic pumping for the cohesion of the structure. The angular speed ω is tuned by the light intensity (Fig. 2c), reflecting the translational velocity of the individual swimmers (Fig. 2c), and shows less than 10% variability amongst a population. At low speeds, ω < 3 rad s−1, a fluctuation can flip a peripheral swimmer, which induces the collective reorientation of the shell of the microgear and a reversal of the direction of rotation, although the magnitude of the rotation rate is unchanged (Fig. 2c,inset). At higher speeds, unidirectional motion persists over the time of the experiment with reduced fluctuations (Fig. 2c,inset). In uniform light, the centre of mass of the rotors displays a two-dimensional random walk with a diffusion coefficient DR = 0.4 ± 0.1 μm2 s−1, larger than a passive particle of comparable size (Supplementary Information). The rotors show slow migration in light gradients, which we harness to build superstructures.

We probe the influence of a rotor on its surrounding using fluorescent tracers (latex, 200 nm, diffusivity Dc), whose concentration ρ(r) is extracted by fluorescence microscopy and an azimuthal average (Fig. 2d,inset). We observe a radial repulsion from the rotor, constituting a sink of hydrogen peroxide, with c 1/r. The tracers migrate in the concentration gradient by diffusiophoresis, with velocity VDPc = α/r2 (with α > 0 for a repulsive interaction). At steady state, the flux of particles j = ρVDP − Dcrρ gives a Boltzmann distribution, $$\rho \propto \mathrm{e}^{ - \frac{\alpha }{{D_{\mathrm{c}}r}}}$$, in an effective repulsive potential, in agreement with the experiment for α = 48 ± 10 μm3 s−1 (Fig. 2d).

We now discuss the interactions between pairs of rotors. Rotors radially repel each other as previously observed for latex particles. The radial repulsion is isotropic and insensitive to the absolute or relative handedness of the rotors. We confine the rotors using light patterns: a uniform illumination in the central region solely actuates the rotors, while a sharp gradient at the edge confines them. Following this, we observe the existence of a short-range tangential interaction for r ~ 2.0R − 3.0R, where r is the centre-to-centre distance of rotors of radii R ~ 3 μm. Co-rotating pairs revolve around each other at Ω ~ 0.05 rad s−1, in the direction of their spin (Supplementary Video 5, Fig. 3a) but do not synchronize (Fig. 3b). Counter-rotating pairs are static (Fig. 3a, Supplementary Video 6) and phase-lock at π/6, as visible from the peak in the probability density function (PDF) of the phase lag $${\it{\Psi }} = \left| {\theta _1} \right| - \left| {\theta _2} \right|$$ (Fig. 3b). The PDF broadens as the rotors are further apart and becomes asymmetric for pairs with different angular speeds (Fig. 3b,inset). This behaviour is reminiscent of mechanical cogwheels, in the absence of any contact between the rotors. We rule out hydrodynamics as the main contribution to the tangential interaction, as a torque-free rotor generates a slip-flow opposite to its spin, which makes a co-rotating pair revolve along the common direction of spin, in contrast to the experiment. Besides, the confinement of the pairs hinders the hydrodynamic translation of contra-rotating rotors29.

In order to gain insight into the diffusiophoretic coupling, we study the concentration field surrounding a rotor and simulate the diffusion equation for a structure of seven impermeable and passive spheres, decorated by six chemically active sites (see Supplementary Information for details). It constitutes a sink of fuel, whose near-field concentration profile presents the six-fold symmetry of the rotor (Fig. 3c). We compute, from the simulated concentration field, the radial (respectively, azimuthal) phoretic velocities for a point particle $$\tilde v_{r,\theta } \propto \nabla _{r,\theta }c(r,\theta )$$ and obtain 1/r2 decay (respectively, 1/r8), with $$\left( {\tilde v_\theta {\mathrm{/}}\tilde v_r} \right) \left( {r{\mathrm{/}}R} \right)^6\sim 1$$ (Fig. 3d). The short range of the tangential interaction originates from the rapid decay of the azimuthal component. It arises from the superimposition of the monopolar concentration field generated by each active site and reflects the azimuthal dependence of the concentration surrounding a sphere with hexapolar chemical activity: c(r, θ) P6(cos θ)/r7, where P6 is the Legendre polynomial of 6th degree30. The result is qualitatively unchanged for a bead of finite size (Fig. 3d), whose velocity is obtained by integration of the slip on the surface12, or by addition of an impermeable wall delineating a semi-infinite space (see Supplementary Information).

As a result of the fast diffusion of the fuel, $$\varepsilon \sim 2\left( {\frac{R}{{r - R}}} \right)^8$$ ~ 103 μm2 s−1, the Péclet number of the rotation is low, Pe = R2ω/$$D_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}}$$ ~ 0.01, and the concentration field is steady in the rotating frame of the rotor. Neighbouring rotors interact through the modulation of the concentration field computed above, mirroring the effective energy landscape U(r, θ) c(r, θ) (ref. 20). The tangential interaction is short-ranged and we assume that the azimuthal coupling is set at distance r − R, separating the edge of a rotor from the centre of its neighbour. The phases θi (i = 1, 2) of the rotors in a pair, at fixed distance r, are described by coupled Langevin equations $${\mathrm{d}}\theta _i{\mathrm{/d}}t$$ = $$\omega _i + \varepsilon Q\left( {\theta _i,\theta _j} \right)$$ + $$\sqrt {2D_\theta } \zeta (t)$$, where i, j identifies the rotor, ωi their angular speed, εQ is the phoretic coupling between the rotors and ζ(t) a delta-correlated noise of amplitude Dθ. We independently estimate the amplitude of the phoretic coupling from the radial repulsion,

$$\varepsilon = \frac{\alpha }{{R^3}} \left( {\frac{R}{{r - R}}} \right)^8\sim 2\left( {\frac{R}{{r - R}}} \right)^8$$

Guided by the numerical results, we approximate the coupling term, Q = sin(6Ψ), with the phase-lag $${\it{\Psi }} = \left| {\theta _1} \right| - \left| {\theta _2} \right|$$. We finally obtain by summation:

$$\frac{{{\rm{d}{\it{\Psi} }}}}{{{\mathrm{d}}t}} = - \delta \omega + 2\varepsilon {\kern 1pt} {\mathrm{sin}}(6{\it{\Psi }}) + \sqrt {4D_\theta } \zeta (t)$$

where δω is the relative speed of the rotors (see Supplementary Information for details). This is a classical Adler equation of synchronization with noise, also seen as the dynamics of an overdamped Brownian particle in a tilted washboard potential31,32, $$(1{\mathrm{/}}A){\mathrm{e}}^{ - {\cal V}({\it{\Psi }})/2D_\theta }{\int}_{\it{\Psi }}^{{\it{\Psi }} + \pi /3} {\kern 1pt} {\mathrm{e}}^{{\cal V}({\it{\Psi }}^\prime )/2D_\theta }\mathrm{d}{\it{\Psi }}^\prime$$ = δωΨ + (ε/3)cos(6Ψ). In the absence of noise, the phase is trapped in a potential well at high coupling, 3δω/ε < 1, and slides down the corrugated energy landscape at low coupling33. This qualitatively describes the phase-locking observed for contra-rotating rotors, for which δω ~ 0, and the slow revolution at Ω of co-rotating pairs, for which δω ~ 2ω. In the presence of noise, the PDF for Ψ follows a Fokker–Planck equation, whose stationary solution is:

$$P(\Psi) = (1{\mathrm{/}}A){\mathrm{e}}^{ - {\cal V}({\it{\Psi }})/2D_\theta }{\int}_{\it{\Psi }}^{{\it{\Psi }} + \pi /3} {\kern 1pt} {\mathrm{e}}^{{\cal V}({\it{\Psi }}^\prime )/2D_\theta }\mathrm{d}{\it{\Psi }}^\prime$$

where A is the normalization constant (see Supplementary Information for details). It shows an excellent agreement with our experimental measurements of the PDF for the phase lag (Fig. 3b), with the phoretic coupling ε as single fit parameter, as we independently extract the rotation speeds and noise amplitude, Dθ = 0.03 ± 0.01 rad2 s−1 from individual tracking of the rotation of the rotors (see Supplementary Information). The model remarkably captures the asymmetric PDF observed for pairs of contra-rotating rotors with a notable speed difference, up to Δω/ω ~ 20%, without additional parameters (Fig. 3b,inset). The coupling parameter ε decays with increasing distance and shows a good agreement with our prediction,

$$\varepsilon \sim 2\left( {\frac{R}{{r - R}}} \right)^8$$

obtained from the radial repulsion between rotors, without adjustable parameters (Fig. 3e). For co-rotating pairs, guided by the persistence of the phase lag, we neglect the noise and the difference of rotation speed of the rotors, and obtain dθ/dt = ω + ε sin(12θ), predicting a reduced rotation rate $$\omega \sqrt {1 - \left( {\varepsilon {\mathrm{/}}\omega } \right)^2}$$ (see Supplementary Information). The short range of the tangential interaction confines the slowing down to the nearest parts of rotors, effectively acting as partial friction. It qualitatively agrees with the experiment and rotors revolve in the direction of their spin, with

$${\tilde{\it \Omega }}\sim \omega - \omega \sqrt {1 - (\varepsilon {\mathrm{/}}\omega )^2} \sim \frac{{\varepsilon ^2}}{{2\omega }}\sim 0.005\;{\rm{rad}}\,{\rm{s}}^{-1}$$

Following our understanding of the pair interaction, we architect machineries whose collective dynamics arise from microgears with individually selected spins. First, we control the travel speed Ω of the edge-current in hexagonal patterns of seven co-rotating gears (Figs. 1c and 4a), increasing the azimuthal coupling ε through confinement (Fig. 4a, inset). Next, we form higher-order assemblies by the combination of superstructures: we initiate two contra-rotating sets of three co-rotating rotors, each collectively rotating along their common direction of spin (Fig. 4b). They are subsequently combined to constitute the synchronized gears of a micromachine (Supplementary Video 7, Fig. 4c), stressing the robustness and versatility of our findings.

Colloidal spinners provide a platform to explore active spinning matter and test theoretical predictions34,35,36,37,38,39, although the competition in our system of long-range diffusiophoretic interactions with hydrodynamics may significantly enrich the dynamics. The interplay between phase synchronization and spatial organization has the potential to achieve a new form of self-organization, without equilibrium counterparts, and not observed for collections of translational self-propelled particles40. Our bottom-up approach shows a desirable level of accuracy towards the targeted structure. Combining non-equilibrium interactions with spatiotemporal light sequences, we program dynamical superstructures that autonomously regulate. It makes us architects of matter and emulates biological organization into the material world.

### Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.

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## Acknowledgements

We thank L. Bocquet and P. Chaikin for enlightening discussions. This material is based on work supported by the National Science Foundation under Grant No. DMR-1554724. J.P. thanks the Sloan Foundation for support through grant FG-2017-9392. S.S. acknowledges support from the NSF CAREER award DMR-1653465.

## Author information

### Affiliations

1. #### Department of Physics, University of California, San Diego, CA, USA

• Antoine Aubret
•  & Jérémie Palacci
2. #### Department of Chemistry, New York University, New York, NY, USA

• Mena Youssef
•  & Stefano Sacanna

### Contributions

M.Y. and S.S. conceived and synthesized the colloidal particles. A. A. performed the experiment and analysed the experimental results. A.A. and J.P. conceived the project, designed the experiment, worked out the model and wrote the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Jérémie Palacci.

## Supplementary information

1. ### Supplementary Information

Supplementary Methods, Supplementary Figures 1–6, Supplementary References 1–12

2. ### Supplementary Video 1

Superstructure made of seven rotors. Four rotors rotate clockwise and three anticlockwise. Their relative arrangement is random. The structure is confined by an isotropic light pattern and the dynamics is sustained under illumination. When the light is turned off, the system returns to equilibrium and the order is destroyed as the dissipative interactions vanish. Real-time video, bright field transmission illumination.

3. ### Supplementary Video 2

Superstructure made of seven co-rotating rotors and collective rotation. We confine seven co-rotating rotors, which form a hexagonal pattern and show a collective rotation along the direction of the common spin (clockwise). The edge-current travels at Ω 0.1 rad s–1, and persists over the entire experiment (duration 10 min). The dynamics is a result of the anisotropic, short-range chemical interactions between the rotors. To enhance the collective rotation, the movie is momentarily accelerated 3× at time t = 10 s, before returning to real time at t = 36 s. The video is obtained by reflected illumination.

4. ### Supplementary Video 3

Assembly of seven rotors with alternating spins, constituted of three rotors rotating clockwise and four anticlockwise. The structure remains static without collective rotation. Alternatively, pairs of counter-rotating neighbouring rotors synchronize. The video is accelerated 3× at time t = 10 s and goes back to real time at t = 33 s. The movie is obtained by reflected illumination.

5. ### Supplementary Video 4

Sequential formation of rotors and superstructures. Phototactic swimmers are dispersed in the solution and exhibit persistent random walk under homogeneous illumination. We create multiple rotors by superimposing a focused laser spot to a uniform activation light. Laser ON, the swimmers gather, collide, and swiftly form rotors composed of seven particles. The laser is switched OFF to be displaced and switched ON again to form a new rotor. The sequence is repeated to obtain three rotors with high yield and control. Finally the rotors are confined by an isotropic light pattern with a flat central region to form a superstructure. As the light is switched off, the system returns to equilibrium and the rotors break. Video is accelerated 2×, bright field transmission illumination.

6. ### Supplementary Video 5

Rotational dynamics of a pair of co-rotating rotors. Co-rotating rotors revolve at Ω 0.1 rad s–1, in the direction of the spins of the rotors. Real-time video. Bright field transmission illumination.

7. ### Supplementary Video 6

Synchronized dynamics of a pair of counter-rotating rotors. The movie is initially in real time and switches to stroboscopic, at a frequency matching the rotation rate of a rotor, at t = 10 s. The phase locking and cogwheel-like behaviour is apparent. Bright field transmission illumination.

8. ### Supplementary Video 7

Higher level superstructure formed of two contra-rotating sets of three co-rotating rotors. We form two contra-rotating sets of three co-rotating rotors and combine them using an anisotropic light pattern. This leads to the formation of a colloidal machine with the synchronous motion of the two sets as gears. The video is accelerated 10× at time t = 13 s and goes back to real time at t = 49 s. The video is obtained by reflected illumination

### DOI

https://doi.org/10.1038/s41567-018-0227-4