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# Coupled liquid crystalline oscillators in Huygens’ synchrony

## Abstract

In the flourishing field of soft robotics, strategies to embody communication and collective motion are scarce. Here we report the synchronized oscillations of thin plastic actuators by an approach reminiscent of the synchronized motion of pendula and metronomes. Two liquid crystalline network oscillators fuelled by light influence the movement of one another and display synchronized oscillations in-phase and anti-phase in a steady state. By observing entrainment between the asymmetric oscillators we demonstrate the existence of coupling between the two actuators. We qualitatively explain the origin of the synchronized motion using a theoretical model and numerical simulations, which suggest that the motion can be tuned by the mechanical properties of the coupling joint. We thus anticipate that the complex synchronization phenomena usually observed in rigid systems can also exist in soft polymeric materials. This enables the use of new stimuli, featuring an example of collective motion by photo-actuation.

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## Data availability

The data that support the findings of this study are available within the article and its Supplementary Information files, and from the corresponding authors upon reasonable request.

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## Acknowledgements

This work was financially supported by The Netherlands Organization for Scientific Research (NWO-TOP PUNT grant no. 10018944 and NWO-VENI grant no. 722.017.003), Dutch Ministry of Education, Culture and Science (Gravity Programme grant no. 024.001.035) and the European Research Council (Vibrate ERC grant no. 669991).

## Author information

Authors

### Contributions

G.V., E.W.M. and D.J.B. conceived and planned the project. G.V. and A.H.G. performed the experiments. L.C.M.E. performed the simulations. D.J.B., H.N., A.Y.P. and E.W.M. supervised the effort. G.V. wrote the manuscript. All authors discussed the results and commented on the manuscript.

### Corresponding authors

Correspondence to Ghislaine Vantomme or Dirk J. Broer.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Thermal oscillation.

a,b, Temperature of the coupled LCN films during in-phase oscillation (a) and anti-phase oscillation (b). c, Picture of the thermogram with the two points A and B where the temperatures are measured. The temperature at point A is the blue trace and at point B is the orange trace. The films geometry is 20 mm × 4 mm × 20 µm and are connectively separated by 2 mm.

### Extended Data Fig. 2 Thermal conductivity.

a, Scheme of the LCN film positioned on a hot plate at 63 °C. b, Temperature profile over the length of the film showing the poor thermal conductivity of the LCN.

### Extended Data Fig. 3 Modeling of the stiffness and damping parameters.

Tensile moduli of the LCN film as a function of temperature (red traces) and their modeled fits (black traces).

### Extended Data Fig. 4 Modeling of the actuation torque with temperature.

Displacement of the film tip over temperature (dotted line) and the linear fit obtained (black trace).

### Extended Data Fig. 5 Modeling of the temperature dynamics.

Temperature of the LCN film upon switching the LED on (left) and off (right). The experimental data are the red traces and the calculated fits are the black traces.

### Extended Data Fig. 6 Comparison of the experimental and calculated oscillations.

Experimental (green trace) and simulation (magenta trace) results of the oscillation over time.

### Extended Data Fig. 7 Modeling with a weak coupling showing changes in the final state depending on the starting conditions.

a, Example of non-periodic oscillations modeled with the coupling stiffness parameter kcoupling = 5 × 10−19 kg.m2.s−2 and the coupling damping parameter dcoupling. = 7.29 × 10−19 kg.m2.s−1, with the starting conditions $$\theta _1^0 = \theta _2^0$$ = 2.3 rad showing instability and b, with the starting conditions $$\theta _1^0 = 2.6\,{\mathrm{rad}}$$ and $$\theta _2^0 = 3.3\,{\mathrm{rad}}$$ showing anti-phase oscillation.

### Extended Data Fig. 8 Modeling with a weak coupling showing changes in the final state depending on the starting conditions.

a, Example of non-periodic oscillations modeled with the coupling stiffness parameter kcoupling = 5 × 10−19 kg.m2.s−2 and the coupling damping parameter dcouplin. = 7.29 × 10−19 kg.m2.s−1, with the starting conditions $$\theta _1^0 = 3.1\,{\mathrm{rad}}$$ and $$\theta _2^0 = 2.3\,{\mathrm{rad}}$$ showing non-periodic oscillations and b, with the starting conditions $$\theta _1^0 = 2.9\,{\mathrm{rad}}$$ and $$\theta _2^0 = 2.6\,{\mathrm{rad}}$$ showing in-phase oscillation.

### Extended Data Fig. 9 Complex synchronized motion of coupled LCN experimentally obtained.

a, Displacement of the tips of the two coupled oscillators over time in the z-direction. The displacement of one oscillator is represented with the orange trace and the other one with the blue trace. b, A different motion obtained stochastically after switching the LED on and off. The data reported in graphs a and b are plotted from two different sequences combined in Supplementary Video 7. The data plotted on graph a are from the first minute of Supplementary Video 7, and the data on the graph b are from the second sequence of Supplementary Video 7.

### Extended Data Fig. 10 Molecular structures.

Structures of the reactive mesogens RM82 and RM23, and structure of the chromophore Tinuvin 328.

## Supplementary information

### Supplementary Information

Supplementary Table 1 and captions for Supplementary Videos 1–7.

### Supplementary Video 1

Photo-induced in-phase synchrony of the coupled oscillators. The video shows two thin films joined by the same material oscillating in-phase at a frequency of 8.5 Hz (Fig. 1c).

### Supplementary Video 2

Photo-induced anti-phase synchrony of the coupled oscillators. The video shows two thin films joined by the same material oscillating in anti-phase at a frequency of 9.5 Hz (Fig. 1d).

### Supplementary Video 3

Control experiments showing situations with no synchronization of the oscillation for two films. This video presents five successive video clips showing absence of synchrony between two strips with (1) the light beam focused exclusively on one of the two strips, (2) one of the two strips without LCN alignment, (3) no joint between the two independent strips, (4) the joint between the two strips clamped directly on the hinge with a rigid pair of tweezers and (5) a long joint of 10 mm between the two strips.

### Supplementary Video 4

Photo-induced oscillation of a long strip of 18 mm length. The video shows a single strip of a length of 18 mm oscillating at a frequency of 5.5 Hz (Fig. 2c).

### Supplementary Video 5

Photo-induced oscillation of a single short strip of 12 mm length. The video shows a single strip of a length of 12 mm oscillating at a frequency of 10.1 Hz (Fig. 2c).

### Supplementary Video 6

Photo-induced in-phase oscillation of two asymmetric oscillators. The video shows joint strips of 12 mm and 18 mm oscillating at a frequency of 6.2 Hz. Although these two oscillators have different lengths and oscillate independently at different frequencies (Videos 4 and 5), they show here in-phase oscillation at the same frequency. This video demonstrates entrainment between asymmetric oscillators (Fig. 2d).

### Supplementary Video 7

Complex motion of the coupled oscillators. This video shows the occasional complex motion observed and as plotted in Extended Data Fig. 9.

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Vantomme, G., Elands, L.C.M., Gelebart, A.H. et al. Coupled liquid crystalline oscillators in Huygens’ synchrony. Nat. Mater. (2021). https://doi.org/10.1038/s41563-021-00931-6