Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution

Significance

Using computational modeling, we designed a self-oscillating materials system that is driven by a nonperiodic chemical reaction to undergo both periodic shape changes and motion. Catalytic reactions in a fluid-filled microchamber drive the movement of the fluid and immersed flexible sheets. The fluid affects the sheets' shape, and the sheets affect the fluid flow. This feedback enables remarkably rich and controllable oscillatory behavior: a single sheet fishtails periodically across the chamber or circulates continuously within a narrow region. Two sheets form coupled oscillators displaying not only synchronized temporal behavior, but also unique, coordinated morphological reconfigurations. These oscillators enable development of soft robots that operate through an inherent coupling of chemistry and motion, permitting novel autonomous and self-regulating behavior.

Abstract

The synchronization of self-oscillating systems is vital to various biological functions, from the coordinated contraction of heart muscle to the self-organization of slime molds. Through modeling, we design bioinspired materials systems that spontaneously form shape-changing self-oscillators, which communicate to synchronize both their temporal and spatial behavior. Here, catalytic reactions at the bottom of a fluid-filled chamber and on mobile, flexible sheets generate the energy to “pump” the surrounding fluid, which also transports the immersed sheets. The sheets exert a force on the fluid that modifies the flow, which in turn affects the shape and movement of the flexible sheets. This feedback enables a single coated (active) and even an uncoated (passive) sheet to undergo self-oscillation, displaying different oscillatory modes with increases in the catalytic reaction rate. Two sheets (active or passive) introduce excluded volume, steric interactions. This distinctive combination of the hydrodynamic, fluid–structure, and steric interactions causes the sheets to form coupled oscillators, whose motion is synchronized in time and space. We develop a heuristic model that rationalizes this behavior. These coupled self-oscillators exhibit rich and tunable phase dynamics, which depends on the sheets’ initial placement, coverage by catalyst and relative size. Moreover, through variations in the reactant concentration, the system can switch between the different oscillatory modes. This breadth of dynamic behavior expands the functionality of the coupled oscillators, enabling soft robots to display a variety of self-sustained, self-regulating moves.

Self-oscillating chemical reactions transduce a constant, nonperiodic input of energy into sustained periodic motion. Such self-oscillating chemistry is resplendent in biology, enabling the firing of neurons, the beating of the heart, and the cyclic behavior of predator–prey relationships (1). The development of self-oscillating, shape-changing materials would hasten the development of soft robots that autonomously perform self-sustained work and controllable movement (2, 3). With few exceptions (4–6), however, the creation of synthetic self-oscillating materials remains a significant challenge. Most candidate soft materials only exhibited periodic behavior when exposed to variations in the constant energy input (e.g., from changes in illumination, heat, humidity, or pH) (7–12). The rare exceptions include soft materials that incorporate one of three intrinsically self-oscillatory chemical reactions, e.g., self-oscillating gels driven by the Belouzov–Zhabotinsky reaction (13–17). Herein, we use computational modeling to design chemically driven, flexible micro- to millimeter-sized sheets powered by nonoscillatory chemical reactions that form self-oscillating, shape-changing systems in solution. A single two-dimensional sheet spontaneously morphs into a three-dimensional structure that moves periodically in time. Two sheets form coupled oscillators that communicate to synchronize both their motion and morphology. While biological and synthetic systems can exhibit self-organized motion and synchronization, there are few systems that exhibit coordinated spatial movement, structural change, and temporal synchronization (18). These responsive, active sheets encompass a level of autonomous spatiotemporal activity that extends the limited repertoire of self-oscillating soft materials and facilitates the fabrication of autonomous, self-regulating soft robots.

The mechanism driving the oscillatory, shape-changing behavior involves a distinctive combination of chemomechanical transduction, the confinement of the host fluid, and a feedback loop. Fig. 1A shows the key components in the system: a fluid-filled chamber that contains a surface-anchored catalytic patch and a deformable sheet (Fig. 1A). A fixed concentration of reactants is added to the solution to initiate the catalytic reaction at the central patch. The energy released from this reaction is converted into the mechanical motion (flow) of the surrounding fluid. This constitutes the chemomechanical transduction vital to the observed behavior. Since the system is symmetric about the central patch, the fluid flow occurs about each side of this patch (Fig. 1A). These symmetric streams eventually hit the confining walls, driving the fluid to circulate and form two convective rolls (as detailed in the results section). Hence, the confinement of the fluid is another critical component for these self-oscillations.

Fig. 1.

Self-oscillations of a passive sheet. (A) Schematic of the fluidic chamber containing a chemical pump (marked by red rectangular region) and a passive elastic sheet (in blue). (Inset) The network of nodes (marked by blue dots) that form the sheet and the flexible bonds between nodes (white lines). Stretching and bending moduli of the sheet are ${\kappa }_s=60\text{pN}$ and ${\kappa }_b=7.2\text{pN}{\text{mm}}^2$, respectively. The sheet is $1.95\text{mm}\times \text{0}.6\text{mm}\times \text{0}.26\text{mm}$ in size. (B) Low reaction rate: $r_m^{\text{patch}}=52\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$(Movie S1). Black arrows indicate the direction and magnitude of the fluid flow; the arrows on the left vertical wall reveal the flows within a vertical plane passing through the center of the chamber. Arrows on the right sidewall indicate the flow in the orthogonal vertical plane. Color bar indicates the concentration of ${\text{H}}_2{\text{O}}_2$ in the solution. (C) Height ($z_s$) of the center of the sheet. Color bar indicates time. (D and E) High reaction rate: $r_m^{\text{patch}}=72\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$. (F) Center of the sheet oscillates back and forth across the catalytic pump. (G and H) At the highest reaction rate, $r_m^{\text{patch}}=96\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$, the sheet simply circulates within the right half of the domain. (I) Corresponding temporal motion of the center of the sheet. Times for the sheet configurations are marked on the bottom of the rightmost column.

The third critical component is the feedback loop that arises from the fluid–structure interactions between the circulating fluid and the compliant sheet (whether the sheet is passive or chemically active). The fluid transports the sheet within the chamber, but this sheet also exerts a force on the fluid that modifies the flow. The modified flow in turn affects the movement of the sheet. This feedback mechanism contributes to the oscillatory behavior; the magnitude of the effect depends on the geometry and flexibility of the sheet. The examples below illustrate the complex dynamics that emerges due to interactions of these three critical components and provide new systems for uncovering factors that regulate nonequilibrium behavior.

Computational Model

The surface-anchored catalytic patch (Fig. 1A) works as a “chemical pump” that drives the fluid flows along the bottom surface toward or away from the patch, depending on the nature of the chemical reaction. This flow occurs due to solutal buoyancy (19, 20). Specifically, the enzymes on the catalytic patch decompose the chemical reactants into products, which can be denser or less dense than the reactants and thus alter the local density of the fluid. The density variation in the solution gives rise to a buoyancy force per unit volume, ${\mathbf{F}}^b=\mathbf{g}{\rho }_0\sum {\beta }_i{C}_i$, where ${\rho }_0$ is the solvent density, $\mathbf{g}$ is gravitational acceleration, $C_i$ is the concentration of chemical species i, and ${\beta }_i=\frac{1}{{\rho }_0}\frac{\partial \rho }{\partial C_i}$ are the corresponding solutal expansion coefficients. ${\mathbf{F}}^{\mathbf{b}}$ drives the spontaneous fluid motion.

In previous studies, we combined modeling and experiments to show that the solutal buoyancy forces are primarily responsible for the observed pumping velocities in a number of systems involving catalytic patches on an immobile surface and mobile, spherical particles (chemical motors) (20–22). For example, we predicted and confirmed experimentally that catalytic reactions on the bottom wall of a chamber generate solutal buoyance forces, which deliver microparticle cargo to specified regions in microchambers and consequently, the cargo is deposited around a specific position on the surface (22). We also predicted and experimentally validated that the coordination among three enzymatic pumps leads to the formation of circuit containing a bifurcating stream, which simultaneously delivers microparticles to two different locations (23). In the case of chemical motors, we showed that microparticles uniformly coated with enzymes in solution could undergo self-sustained motion in the presence of the appropriate reactant (24). These studies were also verified by subsequent experimental studies (25). While there have yet to be experiments on chemically active sheets, we have developed an analytical theory that showed quantitative agreement with our computational model for the folding of a single sheet in solution (26).

In the case of the sheets, the fluid flow alters not only the position but also the sheet’s shape. The sheet is modeled as a single-layer network of nodes, with positions ${\mathbf{r}}_k$, that are interconnected by elastic rods (Fig. 1A, Inset and SI Appendix, Section I). The nodes of active sheets are uniformly coated with a catalyst and generate buoyancy forces. The passive sheets are uncoated and hence, in this case, the buoyancy forces originate solely from the surface-anchored chemical pump. The sheet nodes experience body forces, ${\mathbf{F}}^{\text{sheet}}={\mathbf{F}}^e+{\mathbf{F}}^s+{\mathbf{F}}^g$, which are the respective elastic forces, steric forces, and gravity. The elastic forces ${\mathbf{F}}^e$ are characterized by the stretching (${\kappa }_s$) and bending (${\kappa }_b$) moduli and are governed by the linear constitutive relations for a Kirchhoff rod (27). The steric forces on the sheet, ${\mathbf{F}}^s$, are the sum of the “node–node” (nn) repulsion between two nodes, ${\mathbf{F}}^{nn}$, and repulsion between nodes and any of the six confining walls, ${\mathbf{F}}^{nw}$. These steric forces are calculated as the gradient of Morse potential (SI Appendix). Gravity acting on the sheet is described by ${\mathbf{F}}^g=V({\rho }_s-{\rho }_0)\mathbf{g}$, where $V$ is the effective volume of each node. The density of the sheet (${\rho }_s$) is assumed to be greater than the density of the solvent $\left({\rho }_0\right)$. In the absence of reactants, gravity drives the sheet to sediment to the bottom of the chamber.

The dynamic interactions between the elastic sheet and flowing fluid are described by the following coupled equations: the continuity and Navier–Stokes [in the Boussinesq approximation (28)] equations for the dynamics of an incompressible flow; the equation for the advection, diffusion, and reaction of the dissolved chemical species $C_i$; and the equation for the velocity of the nodes of the elastic sheet. The respective equations are

$$\nabla \cdot \mathbf{u}=0,$$ [1]

$$\frac{\partial \hspace{0.17em}\mathbf{u}}{\partial \hspace{0.17em}t}+\left(\mathbf{u}\cdot \nabla \right)\hspace{0.17em}\mathbf{u}=-\frac{1}{{\rho }_0}\nabla \hspace{0.17em}p+\nu \hspace{0.17em}{\nabla }^2\mathbf{u}+\frac{1}{{\rho }_0}\mathbf{F},$$ [2]

$$\mathrm{where}\hspace{0.17em}\mathbf{F}=\underset{\text{solutalbuoyancy}}{\underbrace{\mathbf{g}{\rho }_0\sum {\beta }_i{C}_i}}+\frac{1}{V}\underset{\text{sheet}}{\underbrace{\left({\mathbf{F}}^e+\sum {\mathbf{F}}^{nw}+\sum {\mathbf{F}}^{nn}\right)}}+\left({\rho }_s-{\rho }_0\right)\mathbf{g},$$

$$\frac{\partial \hspace{0.17em}{C}_i}{\partial \hspace{0.17em}t}+(\mathbf{u}\cdot \nabla )\hspace{0.17em}{C}_i=D_i\hspace{0.17em}{\nabla }^2\hspace{0.17em}{C}_i\pm \underset{\text{patch}}{\underbrace{S^{\text{patch}}{K}_d^{\text{patch}}}}\pm \underset{\text{sheet}}{\underbrace{s{K}_d^{\text{sheet}}{\displaystyle \sum _{k=1}^{N_s}\delta ({\mathbf{r}}_k-\mathbf{r})}}},$$ [3]

$$\frac{\partial \hspace{0.17em}{\mathbf{r}}_k}{\partial \hspace{0.17em}t}=\mathbf{u}.$$ [4]

Here, $\mathbf{u}$ and $p$ (in Eq. 2) are the respective local fluid velocity and pressure, $\nu$ is the kinematic viscosity of the fluid, $\nabla$ is the spatial gradient operator. The body force (per unit volume) $\mathbf{F}$ arises from solutal buoyancy and the force exerted by the sheet nodes on the fluid (the fluid–structure interactions). The immersed boundary method (27) is used to treat the latter interactions. The ith reagent of concentration $C_i$ diffuses with the diffusion constant $D_i$.

The chemical is consumed or produced at the chemical pump with a reaction rate $S^{\text{patch}}{K}_d^{\text{patch}}$, where $S^{\text{patch}}$ is the surface of area of the patch. For active sheets, chemical reactions occur at the position of catalytic node ${\mathbf{r}}_k$ with a reaction rate given by $s{K}_d^{\text{patch}}$, where $s$ is the surface area per node. The catalytic reactions are modeled using Michaelis–Menten reaction rates (29) and the reaction rates are

$$K_d^{\text{patch}}=\frac{r_m^{\text{patch}}{C}_i}{K_M+C_i},\hspace{0.17em}\hspace{0.17em}{K}_d^{\text{sheet}}=\frac{r_m^{\text{sheet}}{C}_i}{K_M+C_i}.$$ [5]

$K_M$ (in units of molarity, M) is the Michaelis–Menten constant. $r_m^{\text{patch}}$ and $r_m^{\text{sheet}}$ are the maximal reaction rates at the chemical pump and active sheets, respectively, and are the product of the reaction rate per molecule of enzyme $k_e$ ${(\text{s}}^{-1})$ and the areal concentration of enzyme, $\left[E\right]$(${\text{mol m}}^{-2}$).

We use no-slip boundary conditions for the fluid flow ($\mathbf{u}=0$) and no-flux conditions for chemical $C_i$ at the confining walls of the chamber. The numerical methods for solving the governing equations (Eqs. 1–4) with the specific boundary conditions are described in SI Appendix. The parameters relevant to chemical reactions are also given in SI Appendix, Tables S1 and S2.

Results: Chemical Pump and Elastic Sheets as a Self-Oscillatory System

Self-Oscillation of a Passive Elastic Sheet.

Fig. 1A shows the simplest system considered here: a centrally located chemical pump coated with the enzyme catalase and a passive elastic sheet, which lies orthogonal to the pump, with its left end crossing two units into the width of the patch. Initially, hydrogen peroxide (${\text{H}}_2{\text{O}}_2$) is added to the confined aqueous solution. Catalase decomposes hydrogen peroxide into the lighter products, water (${\text{H}}_2\text{O}$) and oxygen (${\text{O}}_2$),

$${\text{H}}_2{\text{O}}_2\stackrel{\text{Catalase}}{\to }{\text{H}}_2\text{O}+\frac{1}{2}{\text{O}}_2.$$ [6]

The solutal expansion coefficient for oxygen is approximately an order of magnitude smaller (22) than that for hydrogen peroxide, and hence, we neglect its contribution to the density variation in the solution. The reaction rates are sufficiently low that the formation of ${\text{O}}_2$ bubbles can be ignored. Since the temperature increase due to the above chemical reaction is relatively small and the thermal expansion coefficients are significantly smaller than the solutal expansion coefficients, we neglect the density variation due to temperature changes and treat the system as isothermal (22).

Since the products of the reaction are less dense than ${\text{H}}_2{\text{O}}_2$, the product-rich fluid rises upward. Due to the continuity of the confined fluid, and the symmetry of the physical setup, the flow near the top of the chamber splits into two equal streams. Each stream flows downward along the closest sidewall and back toward the center of the patch, forming “inward” flow at the bottom surface of the chamber.

Fig. 1B shows the steady-state conformation of the passive sheet for the lowest reaction rate considered here ($r_m^{\text{patch}}=52\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$). Driven by the underlying catalysis, the fluid velocity is highest above the center of the catalyst-coated patch. This flow drags the sheet toward to the left, i.e., closer to the patch’s center. At early times, the rising fluid (forming the inward flow) drags the sheet upward and increases the sheet height ($z_s$) as the sheet crosses the patch (Fig. 1 C, Top). The restoring force from the elastic sheet (i.e., the fluid–structure interaction) and the fluid drag resist this motion. Within a relatively short time, the system attains a steady state, where the competing forces are equilibrated. This rapid equilibration suggests that forces arising from the elasticity of the sheet (determined by ${\kappa }_s$ and ${\kappa }_b$) and hence, the material properties, play a significant role in establishing the final structure (as shown in Fig. 2).

Fig. 2.

State diagram of the stationary and oscillatory states of a passive sheet as a function of the bending stiffness of the sheet and catalytic reaction rate. Top panels (from left) show the typical trajectory of the center of the sheet for a stationary bridge state (S), a side-to-side oscillatory state (${\text{O}}_2$), and a state in which a sheet oscillates in one half of the domain (${\text{O}}_1$). The S, ${\text{O}}_2$, and ${\text{O}}_1$ states are represented by circle, triangle, and square, respectively. The colors provide a guide to the eye. Here, the stretching moduli of the sheet is ${\kappa }_s=60\text{pN}$.

At the steady state, the sheet forms a stable spanning bridge (Fig. 1B) centered about the patch width. The bridge height attains a maximum value relatively quickly and remains constant in time (Fig. 1C). Importantly, the flow is necessary for initiating the dynamic behavior and raising the height of the bridge. The flow profiles (Fig. 1B) at the steady state are seen to be symmetric about the center of the chamber. The letter “S” indicates this steady state in the state diagram in Fig. 2.

An increase in the reaction rate ($r_m^{\text{patch}}=72\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$) corresponds to an increase in the velocity of the generated flow (as evident through Eqs. 2 and 3). Now, the forces originating from the flow and the elastic sheet are comparable in value. This is evident from the “tug of war” between these competing forces that causes the sheet to oscillate, moving back and forth across the patch (Fig. 1 D and E). In each cycle, forces exerted by the rectangular sheet on the fluid compress a significant portion of the nearest convective roll (SI Appendix, Fig. S2); in turn, the momentum transfer from the fluid to the sheet drives the sheet into the adjacent compartment (Movie S1). The trajectory of this motion exhibits periodic behavior with time (Fig. 1F); this state is labeled “${\text{O}}_2$” in Fig. 2.

Notably, a sphere of a comparable size (diameter= 0.6 mm) does not generate fluid–structure interaction forces that span across the patch and generated convective roll (see SI Appendix, Section IV); this compression requires the long arm of the sheet. Consequently, the sphere does not oscillate across the patch, but simply circulates in one half of the chamber.

For a further increase of the reaction rate ($r_m^{\text{patch}}=96\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$), the fluid velocities generated at the pump are sufficiently large that they dominate the system’s behavior. The passive sheet now simply oscillates about the center of a half domain (Fig. 1 G and H, SI Appendix, Fig. S3, and Movie S1). Fig. 1I shows the elliptic trajectory of the center of the sheet as it undergoes this sustained periodic motion; this state is labeled “${\text{O}}_1$” in Fig. 2.

The state diagram (Fig. 2) is plotted as a function of the bending modulus of the sheet and catalytic reaction rate at the patch, and delineates the regions where the different oscillatory behavior is most pronounced. For all tested values of the bending modulus, systematic increases in the reaction rate drive the sheet from the steady-state bridge configuration (S), to oscillating across the patch (${\text{O}}_2$), and ultimately to oscillating in one half domain (${\text{O}}_1$). The observed oscillations remain stable for a few hours (until the reactant is consumed).

Notably, the frequency of the oscillations is constant for a fixed $r_m^{\text{patch}}$. It can, however, be tuned by altering this reaction rate at the patch (e.g., by varying the areal concentration of enzyme) and size of the sheet (SI Appendix, Fig. S4). Moreover at fixed $r_m^{\text{patch}}$, the oscillatory behavior can be altered by varying ${\kappa }_b$ or ${\kappa }_s$ since the deformations induced by the fluid will depend on the stiffness of the sheets (SI Appendix, Fig. S4).

In the above examples, we used the enzyme catalase to generate the activity. Similar behavior could be produced with other enzymes, such as acid phosphatase and glucose oxidase. Due to the patterns of the fluid flow created by the latter enzymes (26, 30, 31), they would also prompt the oscillatory motion of the sheet.

Self-Oscillation of a Half-Coated, Active Sheet.

The oscillatory behavior of a single sheet can be altered by coating a portion of it with catalase (Fig. 3A and Movie S2). Now the catalytic reaction (Eq. 6) occurs on both the patch and the coated area on the sheet. Hence, the state diagram (Fig. 3B) depends on both reaction rates, $r_m^{\text{patch}}$ and $r_m^{\text{sheet}}$ (at fixed values of ${\kappa }_b$ and ${\kappa }_s$). Fig. 3 A and B show the scenario where a half-coated sheet (marked in red) is placed on one side of the catalytic patch. The active half of the sheet also generates flow and thus, the $r_m^{\text{patch}}$ necessary to induce side-to-side oscillations (${\text{O}}_2$) decreases with an increase of $r_m^{\text{sheet}}$. In contrast to the passive sheet, the oscillations in the ${\text{O}}_2$ state of the half-coated, active sheet displays different oscillation amplitudes at each side of the patch, one corresponding to the active half being above the patch and another when the passive portion is above the patch. This patterning also affects the trajectory of the side-to-side oscillations; the center of the trajectory is now shifted toward right of the patch (SI Appendix, Fig. S5), indicating that the flow generated through the interactions between the patch and active part of the sheet pulls the passive sheet to the right. The similar trend is observed for the transition from the ${\text{O}}_2$ to ${\text{O}}_1$ state. These trends are relatively insensitive to the initial location of the sheet (SI Appendix, Fig. S6). As shown in SI Appendix, Fig. S7, the state diagram can be tuned by coating larger portions of the sheet or by changing the bending and stretching moduli.

Fig. 3.

Self-oscillation of a partially coated active sheet. (A) Schematic view of a fluid chamber containing a catalytic pump and an active elastic sheet. The red region of the sheet is made chemically active by coating it with catalase. (B) State diagram of the active sheet as a function catalytic reaction rate of the patch, $r_m^{\text{patch}}$ and catalytic reaction rate of the sheet, $r_m^{\text{sheet}}$. Symbols represent the following: stationary state (circle), ${\text{O}}_2$ state (triangle), and ${\text{O}}_1$ state (square). Here, the bending and stretching moduli of the sheet are ${\kappa }_b=7.2\text{pN}{\text{mm}}^2$ and ${\kappa }_s=60\text{pN}$, respectively.

Two Passive Sheets Act as Autonomously Coupled Oscillations.

The single-sheet studies provide insight for understanding self-oscillatory systems involving two or more sheets, which respond not only to the flow generated by the pump, but also to the presence of additional sheets. In the simplest case, two passive sheets are placed symmetrically on either side of the catalytic patch (Fig. 4A). In each half domain, the chemically generated, inward flow drags a sheet upward. The two sheets are coupled not only through the hydrodynamic interactions in the intervening fluid, but also through the mutual excluded volume interactions. For the low reaction rate, $r_m^{\text{patch}}=52\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$, the system reaches a dynamic steady state, forming a stationary towerlike structure (Fig. 4B), which is stable for relatively long times (Fig. 4C). The fluid dynamics primarily acts to initiate the motion of the sheet; the steric interactions play the dominant role. Steric interactions are also dominant at $r_m^{\text{patch}}=72\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$; they prevent the back and forth motion evident for the single sheet. Rather, each sheet circulates in the respective half domain.

Fig. 4.

Autonomous coupled oscillations of two passive sheets. (A) Two passive sheets (marked in blue) are placed on either side of the chemical pump (marked in red) in a fluid-filled chamber. (B) At low reaction rate at the catalytic pump, $r_m^{\text{patch}}=52\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$, the sheets form a stable towerlike structure (Movie S3). (C) The height ($z_s$) of the center of the sheets remains constant for a long time. The left and right color bars indicate the time elapsed during the motion of left and right sheets, respectively. (D and E) For a higher reaction rate, $r_m^{\text{patch}}=96\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$, both sheets oscillate about the center of each half domain. (F) The trajectory of the center of the sheets lies either side of the catalytic path (Top). The heights of the center of the sheets as a function of time show are identical and thus indicate that the sheets are synchronized in-phase. The stretching and bending moduli of the sheets are taken as ${\kappa }_s=60\text{pN}$ and ${\kappa }_b=2.2\text{pN}{\text{mm}}^2$, respectively.

The latter circulating behavior also occurs at the highest reaction rate, $r_m^{\text{patch}}=96\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$, where hydrodynamic interactions dominate the behavior of a single sheet (Fig. 1 G–I). At this reaction rate, a comparison of the oscillation frequency and amplitude for one sheet versus two reveals that for the two-sheet case, the oscillation frequency is lower and the amplitude is higher than for one sheet (SI Appendix, Fig. S8). This difference is due to a reduction in the flow velocity; the fluid exerts more drag and must perform greater mechanical work on two than on one single sheet. The flow velocity is also reduced due to the fluid–structure instructions from the additional sheet.

This difference in the one and two passive sheet cases also indicates that the sheets affect their mutual dynamic behavior. Indeed, Fig. 4F shows that the two act as coupled oscillators displaying in-phase synchronization (Movie S3). When two sheets are placed at different distances from the catalytic patch, then after an initial period of synchronization, the sheets oscillate in synchrony but with a finite phase difference (SI Appendix, Fig. S9). We verified this behavior by perturbing the initial position of the sheet with respect to the patch. Any initial placement led to the sheets’ out-of-phase synchronization with the same phase difference, indicating that out-of-phase oscillations represent a stable dynamical state.

Autonomous Coupled Oscillations of Two Active Sheets.

To illustrate the complex oscillatory behavior exhibited by sheets coated with catalytic patches, we first consider sheets containing both active (red) and passive (blue) regions. The initial orientations of the active regions are symmetric (Fig. 5A) and antisymmetric (Fig. 5B) about the chemical pump. The symmetrically placed sheets oscillate in-phase; however, the oscillation amplitudes are relatively lower when the passive regions come face-to-face and higher when the active portions face each other (Fig. 5G and Movie S4). The higher amplitudes within the cycle (Fig. 5G) are due to the increased fluid flow generated by the simultaneous action among three active regions, two on the sheets and the one on the pump. In the case of the asymmetric placement, the three active regions do not work in concert and thus generate nonuniform fluid flows about the pump. These flows impose different fluid drags on the sheets, causing the sheets to exhibit asynchronous oscillations (Fig. 5H and Movie S4).

Fig. 5.

Autonomous coupled oscillations of two active sheets. (A–C) Two partially coated (red) sheets are initially placed in symmetric (A) and antisymmetric configuration (B) whereas two fully coated sheets are placed in symmetric locations about the patch in C. The corresponding configurations of the sheets at 66 min after the start of the catalytic reaction are shown in D–F (Movie S4). Two partially coated sheets exhibit in-phase synchronized oscillations for symmetric initial placement (G) and display asynchronous oscillations for asymmetric initial placement (H). Two symmetrically placed, fully coated sheets oscillate in a phase-locked state with zero phase difference (I). The bending and stretching moduli of the sheets are ${\kappa }_b=2.2\text{pN}{\text{mm}}^2$ and ${\kappa }_s=60\text{pN}$, respectively. Reaction rates for the patch and sheets are $r_m^{\text{patch}}=90\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$ and $r_m^{\text{sheet}}=80\mu {\text{mol m}}^{-2}\cdot {\text{s}}^{-1}$.

For two fully coated sheets, the generated fluid flows are symmetric about the pump (Fig. 5F). Consequently, the sheets are pulled over the patch at the same time and thus oscillate in-phase (Fig. 5I). While appearing qualitatively similar, the amplitude and frequency of the in-phase oscillations are higher for the two fully coated sheets than the two passive sheets. The phase differences between the two oscillating active sheets can be altered by changing the relative size of the sheet (SI Appendix, Fig. S10). Moreover, a difference in oscillation frequencies can be tuned by changing the rate of reaction on one of the sheets (SI Appendix, Fig. S11).

Discussion

To rationalize the synchronization of the two-sheet oscillators, in SI Appendix we use a heuristic phase model to demonstrate that attractive hydrodynamic interactions can synchronize rotations of rigid spheres. On the other hand, when the aspect ratio of the particle is increased (to an oblate spheroid) the relative increase in the drag forces can give rise to repulsive interactions, which can induce out-of-phase motion and more complex dynamics. These results can be applied to the sheets carried by convective flow in the closed domains. In particular, for a range of parameters, each sheet rotates in its respective half domain, with a trajectory that resembles the periodic movement of a sphere. Hence, like the spheres, the rotating sheets can also exhibit synchronization of their mutual motion. The deformation of the rotating sheets leads to changes in the interactions between oscillators and changes in their respective trajectories. As revealed for the case of oblate spheroids, this asymmetry can lead to asynchronous behavior.

The phase behavior of the sheets, however, is more complex than that of the rigid spheres and spheroids. The rich phase behavior is attributable to the sheet’s flexibility. Namely, the sheets can dynamically adjust their configuration in response to the hydrodynamic forces imposed by the fluid and, in turn, exert forces back on the fluid. The coupling ensures the exchange between the kinetic energy of the moving fluid and elastic energy stored in the deformed configurations of the elastic sheets. As observed herein, this coupling mechanism, realized through complex hydrodynamic fluid–structure interactions, yields a range of remarkable phenomena ranging from the feedback that drives single-sheet oscillations, to synchronization–desynchronization dynamics of coupled oscillating sheets.

In summary, we devised a model of chemomechanical energy transduction that gives rise to a new class of self-oscillating materials system. Given a nonoscillating input, the system organizes into mobile oscillators that not only show temporal synchronization, but also exhibit coordinated shape changes. The results yield guidelines for realizing yet other examples of self-oscillating active sheets. Namely, changes in the shape, the patterning of catalysts on the sheet, the size and shape of the chemical pump, and the geometry of the chamber can all influence the system’s dynamics. Clearly, computational models are necessary to probe the rich design space. The studies presented here, however, pinpoint necessary conditions for achieving the observed spatiotemporal coordination: an input of chemical reactants that initiate the catalytic reaction at the pump, the confinement of the host fluid, and sufficiently flexible materials that both respond to and perturb the surrounding fluid (the feedback loop).

These coupled oscillators can serve as “chemical clocks,” which constitute valuable tools in analytical chemistry (32, 33). The oscillators can also be useful for creating autonomously self-regulating soft robots (34), where the material itself provides the control. For example, as illustrated in Fig. 1, the material (whether an internal component or the entire robot) can dynamically adapt its motion to variations in the chemical environment. Alternatively, the material’s motion and transition between different movements can be controlled by varying the reactant concentration (SI Appendix, Fig. S12). Additionally, a specified phase difference in oscillatory behavior between coupled oscillators, and hence, parts of the soft robot, can be achieved through the relative placement of the sheets (as in Fig. 5 A and B). In these ways, the soft robot’s performance is directed through inherent coupling of chemistry and motion, enabling an enhanced degree of autonomous behavior.

To guide the experimental realization of these systems, we provide order-of-magnitude estimates for the dimensionless numbers that characterize the system. The magnitude of the fluid flow is characterized by the ratio of the solutal to viscous forces, expressed by the dimensionless Grashof number, $\text{Gr}=\frac{g\beta \mathrm{\Delta }C{L}^3}{{\nu }^2}$, where $\beta$ is the coefficient of solutal expansion, and $\mathrm{\Delta }C$ is the characteristic chemical variation across the domain (see SI Appendix for more details). The typical Grashof number corresponding to the fluid velocity in Fig. 1G is ∼2 × 10². The elastic properties of the sheet are characterized by the dimensionless flexural rigidity, which is a ratio of the restoring elastic force to the solutal buoyancy force, $\stackrel{\sim }{\text{B}}=\frac{B^{\text{'}}/ds}{{\rho }_{0}g\beta \mathrm{\Delta }C{L}^3}$, where $B\text{'}$ is the dimensional flexural rigidity of the sheet and ds is the distance between two neighboring nodes of the sheet. The typical dimensionless flexural rigidity of the sheet considered in this study is $\sim 5\times {10}^{-5}$.

We also provide estimates of the dimensional, physical values for salient features of the system in SI Appendix, Section V. Notably, there exist both synthetic and biological thin films that exhibit a comparable or lower flexural rigidity than our model sheet and therefore, these materials could enable experimental studies. For example, ultrathin flexible sheets of functionalized nanoparticles (NPs) exhibit a flexural rigidity of $10K_{b}T\sim 4\times {10}^{-20}\text{Nm}$ at room temperature (35, 36). The NP composition of these sheets provides sites for catalyst attachment, or using the particles themselves as the catalyst [e.g., platinum NPs and other nanozymes (37)]. Photo-cross-linkable polymer films (38, 39) constitute another candidate material for corresponding experimental studies; the catalysts can be incorporated into these films by postmodification of reactive functional groups or direct inclusion of metal NPs (39). Alternatively, oleosin surfactant proteins can be utilized to form sheets (40) that are hundreds of micrometers long and wide, but nanoscale in thickness, mirroring the conditions in our models. Functional residues permit enzymes and other catalysts to be anchored to these sheets (19), providing the desired active coating. It is worth noting that the typical flexural rigidity of lipid bilayers is ∼10⁻¹⁸–10⁻¹⁹ Nm (41, 42) and hence, these materials also form suitable candidates for further studies.

Methods

A lattice Boltzmann method (LBM) with a single relaxation-time D3Q19 scheme is used to solve the continuity and Navier–Stokes equations (Eqs. 1 and 2) at each time step of the simulation (43). The equation for advection, diffusion, and reaction of the chemicals (Eq. 3) is solved using a finite-difference approach with a forward-time central-space scheme. The boundary condition for the fluid flow at the confining walls of the chamber satisfies the no-slip boundary condition ($\mathbf{u}=0$). For the concentration of chemical $C_i$, we use the no-flux boundary condition at the solid walls of the chamber, $\frac{\partial C_i}{\partial n}=0$, where, $\widehat{n}$ is surface normal pointing into the fluid domain. The immersed boundary (IB) approach is used to capture the fluid–structure interactions between the elastic sheet and fluid (27, 44). Each node of the elastic sheet is represented by a sphere with effective hydrodynamics radius $a$ that accounts for a fluid drag characterized by the mobility $\mu ={\left(6\pi \eta a\right)}^{-1}.$ The forces exerted by the nodes of the elastic sheet on the fluid, calculated using the IB method, provide zero fluid velocities at the discretized nodes of the elastic sheet. Therefore, the IB approach approximates no-slip conditions for the fluid velocities at the nodes, as well as no fluid permeation through the nodes constituting the sheet. Since the elastic sheet is composed of one layer of the nodes, the effective thickness of the elastic sheet is equal to the diameter $2a$ of a single node. We keep the thickness of the sheet constant and vary the elastic moduli to alter the mechanical properties of the sheet.

The velocity field $\mathbf{u}=\left(u_x,u_y,u_z\right)$ computed using the LBM is used to advect the chemical concentration (Eq. 3) and to update the position of nodes of the elastic sheet (Eq. 4). The updated concentration field is then used to determine the buoyancy forces in Eq. 2. The time-step size, $\mathrm{\Delta }t,$ in the simulation is $1.67\times {10}^{-3}$s. The size of the computational domain is $42\mathrm{\Delta }x\times 42\mathrm{\Delta }x\times 17\mathrm{\Delta }x,$ where the lattice Boltzmann unit $\mathrm{\Delta }x$ is $100\text{μm}.$ Thus, the physical dimension of the simulation box is $4\text{mm}\times 4\text{mm}\times 1.5\text{mm}.$ The hydrodynamic radius of the node, $a$ is taken as $1.3\mathrm{\Delta }x.$ In the discretization of the elastic sheet, the distance between two nearest-neighboring nodes is set to $1.5\mathrm{\Delta }x$. The lateral dimensions of the elastic sheet in Figs. 1–5 are $1.95\text{mm}\times \text{0}.6\text{mm}$ and $1.5\text{mm}\times \text{0}.6\text{mm}$, respectively, and $\text{0}.26\text{mm}$ in thickness. The dimension of the catalytic patch is $0.4\text{mm}\times 2.5\text{mm}.$

Data Availability

All study data are included in the article and/or supporting information.