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# 3D-printed machines that manipulate microscopic objects using capillary forces

## Abstract

Objects that deform a liquid interface are subject to capillary forces, which can be harnessed to assemble the objects1,2,3,4. Once assembled, such structures are generally static. Here we dynamically modulate these forces to move objects in programmable two-dimensional patterns. We 3D-print devices containing channels that trap floating objects using repulsive capillary forces5,6, then move these devices vertically in a water bath. Because the channel cross-sections vary with height, the trapped objects can be steered in two dimensions. The device and interface therefore constitute a simple machine that converts vertical to lateral motion. We design machines that translate, rotate and separate multiple floating objects and that do work on submerged objects through cyclic vertical motion. We combine these elementary machines to make centimetre-scale compound machines that braid micrometre-scale filaments into prescribed topologies, including non-repeating braids. Capillary machines are distinct from mechanical, optical or fluidic micromanipulators in that a meniscus links the object to the machine. Therefore, the channel shapes need only be controlled on the scale of the capillary length (a few millimetres), even when the objects are microscopic. Consequently, such machines can be built quickly and inexpensively. This approach could be used to manipulate micrometre-scale particles or to braid microwires for high-frequency electronics.

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## Data availability

Raw videos and experimental data are available on the Harvard Dataverse (https://doi.org/10.7910/DVN/9AHDUL).

## Code availability

The code for tracking floats from videos, processing surface profilometry data, and processing restoring force measurement data can be found at https://github.com/Faaborg/float_tracker and Zenodo (https://doi.org/10.5281/ZENODO.6916546). The code for numerical calculations of float motion can be found at https://github.com/falkma/capillarymachines-numerics and Zenodo (https://doi.org/10.5281/ZENODO.6816029). CAD files for 3D printing the machines in this paper are available at https://github.com/manoharan-lab/capillary-stl and Zenodo (https://doi.org/10.5281/ZENODO.6909015).

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## Acknowledgements

We thank A. Duwel, D. J. Carter, K. J. Russell, R. Gordon, A. Aydin, C. Chang, R. Garmann and Z. Rozynek for helpful discussions and D. Clarke and J. Aizenberg for use of equipment. This work was supported by Defense Advanced Research Projects Agency (DARPA) contract FA8650-15-C-7543 to the Charles Stark Draper Laboratory. It was supported in part by NSF through the Harvard University Materials Research Science and Engineering Center, grant DMR-2011754. Additional support was provided by NSF through grant ECCS-1541959, the Office of Naval Research through grant number N00014-17-1-3029, and the Simons Foundation.

## Author information

Authors

### Contributions

Y.B.-S. conceived the capillary tweezer. C.Z. and M.W.F. conceived capillary machines that use the tweezer to control the paths of multiple floats. C.Z. designed and built the motorized stage, controller and tank. C.Z., M.W.F. and A.S. designed, built and tested all the capillary machines and did all the experiments on the machines. C.Z., M.W.F., A.S., M.J.F. and R.H. developed the hysteresis-based mechanism for swapping. M.W.F. and V.N.M. conceived the switch and arbitrary braiding machine, and C.Z., M.W.F. and A.S. designed and built these machines. C.Z. did the SEM imaging. M.X. measured the contact angles, surface tension and meniscus profile. K.H. measured the capillary forces. M.J.F., R.H., Y.B.-S. and M.P.B. developed the theory and numerical approach and did all the simulations. M.J.F. developed the perturbative theory and did the design-space calculations. M.W.F. analysed the float trajectories. M.W.F., A.S. and V.N.M. wrote the main text and made the figures, based on initial drafts by C.Z., M.W.F. and V.N.M. and incorporating input from all authors. All authors wrote the Supplementary Information. A.S. created the videos. V.N.M. oversaw all experimental work and preparation of the paper. M.P.B. oversaw all theoretical and numerical work. V.N.M. and M.P.B. obtained funding for the work.

### Corresponding author

Correspondence to Vinothan N. Manoharan.

## Ethics declarations

### Competing interests

A US patent application (application serial number 17/639,088) on the design of capillary machines that manipulate and assemble micro- and nanoscopic objects has been filed by C.Z., M.W.F., A.S., Y.B.-S., M.P.B. and V.N.M.

## Peer review

### Peer review information

Nature thanks Camille Duprat, Shigeki Saito and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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## Extended data figures and tables

### Extended Data Fig. 1 A braiding machine using contact-angle hysteresis.

a, A schematic of a machine that uses hysteresis-based ratchets to braid fibres, without any asymmetric junctions. b, A diagram of the machine with cross-sections shown on the right. c, Photograph of the three-strand braiding machine shown in Fig. 3 (left) and the hysteresis-based braiding machine (right). Both machines make the same braid, with the same number of fibres, but the hysteresis-based machine is much more compact. d, Top-view photographs of the machine and floats. As the machine moves up, the three floats execute two swaps to make a $${\sigma }_{1}{\sigma }_{2}^{-1}$$ braid. As the machine moves back down, the floats do not swap positions. The resulting braid is the same as that shown in Fig. 3.

### Extended Data Fig. 2 Machines to make arbitrary braid words.

a, A schematic showing how a single rotator, connected to four reversal-activated switches, enables a pair of floats to be swapped in two ways. Two switches are above the rotator and two below, as shown in the left diagram. If the motion of the machine is reversed in the operational zone of the bottom switches, the floats complete a $${\sigma }^{-1}$$ swap, as shown by the blue arrows in the centre diagram. If the motion of the machine is reversed in the operational zone of the top switches and then reversed again, the floats complete a $$\sigma$$ swap, as shown by the red arrows in the right diagram. b, Diagram of a four-strand braiding machine consisting of three vertically staggered copies of the machine shown in a. Black paths represent rotators, and grey bars represent heights at which the machine can be reversed to swap different pairs of floats. The reversal of the machine can therefore be used to select any $${\sigma }_{i}$$ or any $${\sigma }_{i}^{-1}$$. c, Diagram of a four-strand arbitrary braiding machine that has been modified (‘folded’) to reduce its horizontal extent and that has been divided into eight horizontal slices that can slide into each other. The modified design is equivalent to a ‘flat’ four-strand arbitrary braiding machine (right). d, Photographs of a four-strand braiding machine as separate parts (left) and combined to form a single machine (right).

### Extended Data Fig. 3 The design space of a capillary tweezer.

a, Diagram of a capillary tweezer showing the contact angle at the wall (θ), float radius (Rf), normal force (F), and channel radius (Ro). b, Plot of nondimensionalized tweezer stiffness (k/γ) as a function of Rf/Ro for a contact angle of 50°, and a negative F that scales with $${R}_{{\rm{f}}}^{2}$$. Circles show numerical calculation and solid line shows calculation from perturbative theory. The perturbative theory assumes that Ro, while all numerical calculations are performed with Ro = . c, Plot of nondimensionalized trap stiffness as a function of contact angle for a float with Rf = 0.1Ro and a normal force F = −0.2γRo. At very small or very large contact angles the trap stiffness becomes negative (dotted line), meaning that the float does not stably follow the centre of the channel. d, Plot of nondimensionalized trap stiffness as a function of nondimensionalized normal force for a contact angle of 50° and a float with Rf = 0.1Ro. e, Heatmap of trap stiffness as a function of float radius and contact angle from perturbative theory. White areas correspond to where the tweezer is unstable. See Supplementary Information for further details.

### Extended Data Fig. 4 Rotating rafts of microscopic particles with a capillary machine.

a, Diagram of 50-μm-diameter silica particles assembling into a raft under the influence of the mutual capillary attraction between them. The raft bends the interface downwards, allowing it to be trapped in the centre of a hydrophilic channel. b, An optical micrograph of a typical raft made in an oval channel taken from above. c, Diagram of a centimetre-scale rotator used to rotate the raft. d, Photographs of the machine taken from above show the raft (circled in red) rotating as the height of the machine changes. See also Supplementary Video 10.

### Extended Data Fig. 5 Manipulating microscopic particles with a capillary machine.

a, Diagram of meniscus formed when water fills a hydrophilic channel with contact angle θ. A polystyrene sphere with a diameter of 20 μm and that is denser than water bends the interface downwards (inset). b, A diagram of a centimetre-scale machine with a sloping channel used to translate a microscopic particle. c, Photographs of the machine taken from above show the 20-μm-diameter particle (circled in red) moving as the height of the machine changes. See also Supplementary Video 10. d, Measured particle displacement as a function of machine height, alongside a diagram of the vertical cross-section of the machine taken along its centre (inset).

## Supplementary information

### Supplementary Information

This file contains a description of experimental and numerical methods, as well as additional discussion of the results shown in the main text. This file also contains thirteen figures that serve to clarify the methods discussed in the Supplementary Methods section.

### Supplementary Video 1 Mechanical stage and capillary machine that translates a float.

We move capillary machines by placing them on a motorized stage in a tank. The video (sped up by a factor of 5) first shows the stage moving a translator vertically. The video then shows a perspective rendering (left) of the translator (light grey) and float (blue), a top-view video of the actual machine and float (top right, sped up by a factor of 5), and a rendering of the float at the cross-section (orange) of the machine at the height of the float (bottom right). As the translator moves vertically, the float translates horizontally.

### Supplementary Video 2 A rotator.

We show the same perspective and top views as in Supplementary Video 1. Video is sped up by a factor of 5. As the rotator moves up with respect to the interface, repulsive capillary forces from the walls apply a torque to a pair of floats (red and blue circles).

### Supplementary Video 3 A separator.

We show the same perspective and top views as in Supplementary Video 1. Video is sped up by a factor of 2. As the separator moves up with respect to the interface, the repulsion from the walls overcomes the attraction between a pair of floats (red and blue circles) and separates them.

### Supplementary Video 4 An asymmetric junction.

A float in an asymmetric junction follows two different paths, depending on the direction of motion of the machine. As the machine moves up, a float (blue circle) that starts in the smaller channel moves along this channel and exits into the centre of the junction; as the machine moves back down, the float traverses the larger channel. We show the same perspective and top views as in Supplementary Video 1. Video is sped up by a factor of 5.

### Supplementary Video 5 A braiding machine.

We combine translators, rotators, separators, and asymmetric junctions in a single machine to move floats along the paths necessary to form a twist-free three-strand braid defined by the braid word $${({{\rm{\sigma }}}_{1}{{\rm{\sigma }}}_{2}^{-1})}^{m}$$. We show the same perspective and top views as in Supplementary Video 1. Video is sped up by a factor of 10. The second part of the video shows an annotated perspective rendering of the three-strand braiding machine as it completes several units of a $${{\rm{\sigma }}}_{1}{{\rm{\sigma }}}_{2}^{-1}$$ braid.

### Supplementary Video 6 The ratchet.

We take advantage of contact angle hysteresis to create a machine that can operate on a float in different ways, depending on its direction of motion. As the ratchet moves up, the rectangular float (white, with orientation shown by the red and blue circles) tends to align with the slot, resulting in no net rotation of the float. As the ratchet moves down, the rectangular float tends to align with the rectangular channel, resulting in a 180° rotation. We show the same perspective and top views as in Supplementary Video 1. Video is sped up by a factor of 2.

### Supplementary Video 7 A hysteresis-based braiding machine.

We can make a contact angle hysteresis-based braiding machine by using ratchets in place of rotators and asymmetric junctions. When the machine moves up with respect to the interface, it rotates floats, completing the braid word $${{\rm{\sigma }}}_{1}{{\rm{\sigma }}}_{2}^{-1}$$. When the machine moves down, it causes no net change in the position or orientation of the floats. We show a video of the machine taken from above during an experiment, showing one full cycle (video sped up by a factor of 5).

### Supplementary Video 8 A reversal-activated switch.

Moving the switch in a full cycle causes the float (blue circle) to move along a cyclical path, resulting in no net change in the position of the float. However, if we reverse the motion of the device within the operational zone of the switch, the float exits this cyclical path and enters a different channel. We show the same perspective and top views as in Supplementary Video 1. Video is sped up by a factor of 5.

### Supplementary Video 9 A machine that creates arbitrary braid words.

This movie shows a side rendering (left) of a machine that creates arbitrary braids of the paths of n = 4 floats (red, blue, yellow and green circles), a side-view video of the actual machine and floats (top right, sped up by a factor of 10), and a rendering of the floats at the cross-section (orange) of the machine at the height of the floats (bottom right). We reverse the machine at different heights to complete the seven available operations (the identity operation I, each of the three σn swaps, and each of the three $${{\rm{\sigma }}}_{n}^{-1}$$ swaps).

### Supplementary Video 10 Machines that translate and rotate microscopic particles.

This movie shows machines that can manipulate microscopic particles. The first part of the movie shows a machine that can be used to translate an individual 20-μm polystyrene sphere (circled in red). The second part of the movie shows a machine that can be used to rotate a raft of many 50-μm silica spheres (indicated by a red arrow). Video is sped up by a factor of 2.

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Zeng, C., Faaborg, M.W., Sherif, A. et al. 3D-printed machines that manipulate microscopic objects using capillary forces. Nature 611, 68–73 (2022). https://doi.org/10.1038/s41586-022-05234-7

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• DOI: https://doi.org/10.1038/s41586-022-05234-7