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# Bubble casting soft robotics

## Abstract

Inspired by living organisms, soft robots are developed from intrinsically compliant materials, enabling continuous motions that mimic animal and vegetal movement1. In soft robots, the canonical hinges and bolts are replaced by elastomers assembled into actuators programmed to change shape following the application of stimuli, for example pneumatic inflation2,3,4,5. The morphing information is typically directly embedded within the shape of these actuators, whose assembly is facilitated by recent advances in rapid prototyping techniques6,7,8,9,10,11. Yet, these manufacturing processes have limitations in scalability, design flexibility and robustness. Here we demonstrate a new all-in-one methodology for the fabrication and the programming of soft machines. Instead of relying on the assembly of individual parts, our approach harnesses interfacial flows in elastomers that progressively cure to robustly produce monolithic pneumatic actuators whose shape can easily be tailored to suit applications ranging from artificial muscles to grippers. We rationalize the fluid mechanics at play in the assembly of our actuators and model their subsequent morphing. We leverage this quantitative knowledge to program these soft machines and produce complex functionalities, for example sequential motion obtained from a monotonic stimulus. We expect that the flexibility, robustness and predictive nature of our methodology will accelerate the proliferation of soft robotics by enabling the assembly of complex actuators, for example long, tortuous or vascular structures, thereby paving the way towards new functionalities stemming from geometric and material nonlinearities.

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## Data availability

The data used in this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

## Code availability

The codes that support the findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

This work was supported by NSF CAREER award (CBET 2042930) and through the Princeton University Materials Research Science and Engineering Center (NSF Grant DMR-1420541) and the Princeton Yang Family and David T. Wilkinson Innovation Funds.

## Author information

Authors

### Contributions

T.J.J., J.M. and P.-T.B. conceived the project. T.J.J., E.J.-P. and P.-T.B. conducted the experiments and analysed the data. T.J.J. performed the Kirchhoff rod simulations. E.J.-P. performed the finite element simulations. All authors wrote the manuscript.

### Corresponding author

Correspondence to P.-T. Brun.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Detlef Lohse, Orlin Velev and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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

## Extended data figures and tables

### Extended Data Fig. 1 Characterization of elastomers.

a, Oscillatory shear rheology measurements for VPS-08, 16 and 32 (strain 1%, frequencies {1, 2.5, 2.5} Hz). The cross indicates τc. b, Rescaled time varying viscosity for t < τc. The dashed lines are fits of equation (5). c, Dogbone uniaxial tensile tests data for multiple samples of VPS-32, VPS-16 and VPS-08. The black curves are fits of the various constitutive models. d, Table showing rheological and elastic material constants for VPS

### Extended Data Fig. 2 Cross-section shape.

a, Schematic for the computation of the shape of the liquid bath. b, Meniscus shape profiles obtained from solving equation (9) for various values of $$R/{{\mathscr{l}}}_{{\rm{c}}}$$ and hi/R. c, Model shape profiles obtained from matching of the bath solution (equation (9)) to the film thickness solution (equation (2)) for R = 1.6 mm. The solid lines represent the membrane and the dashed lines represent the bath. The color codes the average membrane thickness hf. d, e, Void fraction ϕ =  (1 − hi/R)2 (d) and average membrane thickness hf (e) as a function of the axial position for a metre-long actuator (sample 1). Error bars represent propagated measurement uncertainty. f, Void fraction for three different samples. Uncertainty is shown as box (middle 50%) and whisker (full range) plots overlaid on the markers. g, Membrane thickness for three different samples. The upward (respectively downward) facing triangles represent the maximum thickness $$h(0)$$ (minimum thickness $$h({\psi }_{0})$$), while the diamonds indicates the average thickness hf. Uncertainty is shown as box (middle 50%) and whisker (full range) plots overlaid on the markers

### Extended Data Fig. 3 Finite element simulations.

a, Deformation of the simulated actuator shown in b for $$P\approx 23$$ kPa (experimental shape, G = 0.36 MPa, $${J}_{m}=14$$). Inset: view of the deformed cross-section. The color codes the von Mises stress while the black lines show the edges of the undeformed configuration. b, Curvature κ as a function of the inflation pressure P. Circles represent experimental data (VPS-32, R = 1.57 mm, $$h(0)=52\,{\rm{\mu }}{\rm{m}}$$, hi/R = 0.34), the solid curve is the finite element simulation using the experimental cross-sectional shape extracted by image analysis, the dashed curve is the simulation using the modeled cross-section using the experimental value of $$h(0)$$. Inset: comparison between the experimental and model shapes. c, Curvature κ as a function of the applied pressure P for our model shapes as we vary hf and R (see legend in d). d, Same data as c rescaled according to equation (17), the dashed line is a power law fit

### Extended Data Fig. 4 Pneumatic bending of actuator.

a, Schematic for the pneumatic inflation of a bubble cast actuator cross-section. b, Schematic for the bending of a bubble cast actuator. c, Actuator curvature $$\kappa$$ as a function of applied pressure P for varied length L (i), thickness hf (ii), radius R (iii), and shear modulus G (iv; see legend in Fig. 3a, b). d, Three-dimensional plot of the rescaled curvature $$\kappa R$$ as a function of the rescaled pressure $$P/G$$ and the rescaled membrane thickness R/hf. Markers show the experimental data while the blue surface is our model (equation (16)) using a Gent constitutive law with $${\psi }_{0}={\rm{\pi }}/4,\,\varphi =0.4225,\,\chi =0.45,\,{J}_{m}=14$$

### Extended Data Fig. 5 Blocking force elastica.

a, Series of images (i) and calculated elastic curve (ii) for a bubble cast actuator inflated with one end clamped and the other end blocked by a wall (see Extended Data Fig. 7c; scale bar, 1 cm). b, Blocking force F as a function of the inflation pressure P for various actuators (R, G, hf and L have been varied)

### Extended Data Fig. 6 Fabrication with a twist.

a, c, Two tubes are kept tightly together with a connector (not represented here) that allows the rotation of one tube with respect to the other. Applying the rotation at the very end of the drainage step, that is, around the gelation point where the polymer is still deformable, yet would not flow significantly anymore under the action of gravity results in the actuator shown in b and d. Colours are a guide to eye; only one polymer is used. Rotating the cylinder (as shown in a (respectively c)) rotates the thin membrane and thus the direction of actuation under pressure. In this example, the 90° (180°) rotation produces two curvatures equal in magnitude but in orthogonal planes (resp. with opposite signs).

### Extended Data Fig. 7 Experimental set-ups.

a, Experimental set-up for Bretherton-like flow during fabrication. The camera captures the bubble front velocity U to determine the Capillary number Ca. b, Experimental set-up for inflation-bending experiments. The pressure sensor records the internal–external pressure difference P while the camera records the shape of the actuator. c, Experimental set-up for blocking force–pressure experiments. The pressure sensor records the internal–external pressure difference P while the load cell measures the force F. d, Experimental set-up for force-elongation experiments. The pressure sensor records the internal–external pressure difference P while the Instron measures the force F and displacement $${\ell }$$.

## Supplementary information

### Supplementary Information

Supplementary text, Supplementary equations and Supplementary references

### Supplementary Video 1

Bubble casting fabrication method. A tortuous channel in an acrylic mould is filled with a curing VPS melt. An elongated bubble is then injected through the channel. The VPS film is then given time to cure as gravity causes drainage. The resulting soft actuator bends when inflated, demonstrating the ability to grasp a blackberry.

### Supplementary Video 2

High aspect ratio pneumatic muscles. Comparison between an inflated actuator at pressures P and a simulation of a Kirchhoff rod with changing natural curvature κ according to equation (3) of the main text. A human-scale actuator demonstrates coiling behaviour when inflated with air. An artificial limb mimics a human arm with a contractile force. Actuators with diameters of 12.8 mm and 1.0 mm are inflated to lift a water bottle and paper clip. A high-aspect-ratio curved actuator wraps itself around a raspberry and lifts it without any damage.

### Supplementary Video 3

Sequential actuation of digits. The programming logic is depicted via schematic representation. A bubble is injected through the channel digits (1,2,3,4) in a sequential manner at various waiting times τw,i. The evolving viscosity increases as the polymer melt cross-links and diverges at the curing time τc as the melt cures into an elastic solid. Injecting air into the resulting actuator causes a sequential bending of the actuator digits.

### Supplementary Video 4

Morphing to complex shapes. An actuator with a tortuous path is inflated in water. The actuator bends out of plane to a curve that lives on the surface of a sphere. A spiral-shaped actuator is attached to a circular membrane, transforming local curvature change of the rod to a global bending of the membrane. An actuator is attached to the edge of a membrane to perform self-folding of a five-sided box. Two branched actuators are bonded to either side of a membrane in the shape of a caudal fin. The fin actuators are alternatively inflated in a water bath with tracer particles. During fabrication a tubular mould is rotated 180 degrees at the gelation point. The resulting actuator bends in two different directions. During fabrication a three sectioned tubular mould is rotated at 180 degrees and 90 degrees at the gelation point. The resulting actuator bends in three directions and in two planes.

### Supplementary Video 5

A soft machine to fetch in a constrained space. A modular soft machine made of two actuators is programmed to grip and then retrieve a ball from a cylindrical vessel using a single pressure source.

### Supplementary Video 6

Programming curvature variation. A rectilinear actuator, built by bubble casting, is programmed to have three sections of varying curvatures (curvature ratio 4:2:1). The coding is accomplished by injecting the bubble through the tube in a step wise sequence such that the final film thickness is predicted. The actuator is inflated on a water bath forming a spiral with appropriate curvatures.

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Jones, T.J., Jambon-Puillet, E., Marthelot, J. et al. Bubble casting soft robotics. Nature 599, 229–233 (2021). https://doi.org/10.1038/s41586-021-04029-6

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• DOI: https://doi.org/10.1038/s41586-021-04029-6