Self-Oscillating Membranes: Chemomechanical Sheets Show Autonomous Periodic Shape Transformation

Ido Levin, Robert Deegan, and Eran Sharon

Phys. Rev. Lett. 125, 178001 – Published 19 October 2020

Abstract

While living organisms have mastered the dynamic control of residual stresses within sheets to induce shape transformation and locomotion, man-made implementations are rudimentary. We present the first autonomously shape-shifting sheets made of a gel that shrinks and swells in response to the phase of an oscillatory chemical (Belousov-Zhabotinsky) reaction. Propagating reaction-diffusion fronts induce localized deformation of the gel. We show that these localized deformations prescribe a spatiotemporal pattern of Gaussian curvature, leading to time-periodic global shape changes. We present the computational tools and experimental protocols needed to control this system, principally the relationship between the Gaussian curvature and the reaction phase, and optical imprinting of the wave pattern. Together, our results demonstrate a route for developing fully autonomous soft machines mimicking some of the locomotive capabilities of living organisms.

Received 18 February 2020
Revised 24 June 2020
Accepted 22 September 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.178001

Physics Subject Headings (PhySH)

Biomimetic & bio-inspired materials Chemical waves Elasticity Functional materials Mechanical deformation Patterns in complex systems

Hydrogels Thin films

Condensed Matter, Materials & Applied PhysicsNonlinear DynamicsInterdisciplinary PhysicsPolymers & Soft Matter

Authors & Affiliations

Ido Levin ¹, Robert Deegan ², and Eran Sharon^1,*

¹Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
²Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA

^*Corresponding author. erans@mail.huji.ac.il

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Vol. 125, Iss. 17 — 23 October 2020

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Images

Figure 1
Experimental apparatus. (a) The redox state of ruthenium determines the volume of a BZ-gel disk. The area of our gels is typically 1.4 times larger in the oxidized than in the reduced state. (b) A BZ gel in a solution of the substrate reactants develops green oxidation traveling fronts that induce localized swelling and global shape changes. (c) Different spatiotemporal wave patterns emerge depending on the chemical concentrations and the geometry of the gel, leading to different mechanical responses (scale bar equal 1 cm). Disks small compared to the wavelength buckle globally (left); larger ones produce wrinkled structures (center and right). (d) BZ pattern can be controlled with blue-colored light, as shown by this before and after image where the blue curve indicates the illumination pattern.
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Figure 2
BZ front and the resultant plate bending propagation as a function of position. (a) Images at 80 sec intervals of a targetlike oxidization wavefront in a gel and (b) the corresponding phase of the reaction and (c) out-of-plane height along the colored lines in (a) with zero indicated by the dashed lines. These data show significant curvature changes, including sign changes (e.g., yellow versus teal), and a global geometric response despite the highly localized phase variation at the front. (d) Representative examples of gel shapes for the same wavefront: global types (top; 1 cm diameter), where the entire sheet buckles out of plane, and wrinkled types (bottom; 3 cm diameter.), displaying an emergent length scale. See also Videos S1 and S4, respectively [18].
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Figure 3
Local chemomechanical activity. (a) Average BZ phase (red) and area growth (blue) measured from the single patch indicated in (b). Both signals oscillate periodically, with a period of 450 sec, and in synchrony with a time delay $τ \approx 55 s$ (see Supplemental Material [18] for the full analysis). The magnitude of the areal swelling is roughly 40%. (b) Growth as a function of the BZ phase delayed by 55 s (blue dots). These data collapse to a single curve, indicating that the delay is uniform in time. The data from all patches are represented by the colored envelopes; 75% and 50% of all data fall within the blue and the purple envelopes, respectively. The red line is a linear fit to the data from all patches. This fit is used below to model the dependence of the growth on the BZ phase. Note that the fit to all data agrees well with the data from a single patch. Inset: the patch (teal) on the triangular grid from which the local data in (a) and (b) were collected.
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Figure 4
Analysis of the Gaussian curvature. (a) The time dependence of the reference (blue) and actual (red) Gaussian curvatures in a small patch. The large spikes in $\bar{K}$ are due to the sharp changes in the reference metric. The oscillations in $K$ display a much smaller amplitude. (b) Actual versus reference Gaussian curvature. These data show $K$ follows $\bar{K}$ for low values of $| \bar{K} |$ (the sloped black line is $\bar{K} \approx 5 K$ ) but saturates for $| \bar{K} | > 0.05 {mm}^{- 2}$ (c) Reference curvature versus the distance from the origin of the propagating front, obtained from numerical simulations for different thicknesses. The reference Gaussian curvature (black) estimated over the patch highlighted in (d) is asymmetric and has a much larger amplitude than the actual curvature (color). These data suggest that the observed relationship in (b) is due to a competition between bending and stretching (see Supplemental Material [18] for more details). Insets: the top inset illustrates the relation between the Gaussian curvature and the shape resulting from a circular swelling front. The sharp change in curvature corresponds to the sharp peaks in the black curve. The bottom inset demonstrates the spread of the Gaussian curvature due to finite thickness effects. The highlighted areas illustrate disks cut from the surfaces with diameters corresponding to the experiment and the color represents the Gaussian curvature (red, positive; blue, negative). (d) Actual versus reference Gaussian curvature from simulations. The actual curvature displays a similar cutoff for thicknesses of around 0.6 mm. Inset: the patch (teal) on the simulated triangular grid in which the local data in (c) and (d) were collected. The patch was chosen with correspondence to the patch in Fig. 3.
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