Reconfigurable assemblies of active, autochemotactic gels

Abstract

Using computational modeling, we show that self-oscillating Belousov–Zhabotinsky (BZ) gels can both emit and sense a chemical signal and thus drive neighboring gel pieces to spontaneously self-aggregate, so that the system exhibits autochemotaxis. To the best of our knowledge, this is the closest system to the ultimate self-recombining material, which can be divided into separated parts and the parts move autonomously to assemble into a structure resembling the original, uncut sample. We also show that the gels’ coordinated motion can be controlled by light, allowing us to achieve selective self-aggregation and control over the shape of the gel aggregates. By exposing the BZ gels to specific patterns of light and dark, we design a BZ gel “train” that leads the movement of its “cargo.” Our findings pave the way for creating reconfigurable materials from self-propelled elements, which autonomously communicate with neighboring units and thereby actively participate in constructing the final structure.

Species ranging from single-cell organisms to social insects can undergo autochemotaxis, where the entities move toward a chemo-attractant that they themselves emit. This mode of signaling allows the organisms to form large-scale structures, with amoebas (1) and Escherichia coli (2) self-organizing into extensive multicellular clusters and termites constructing macroscopic mounds (3). Notably there are few equivalents of such autochemotactic-driven assembly in the synthetic world. Although researchers have devised a range of nano- and microscopic self-propelled particles (4), hardly any exhibit autochemotaxis (5) that leads to the formation of extended structures (6). Although recent theoretical models provide insight into the autochemotaxis of a self-propelled walker (7) and active Brownian particles (8), these studies provide few guidelines for synthesizing specific materials that display autochemotactic self-organization. The latter materials would open new routes for dynamic, reconfigurable self-assembly, where self-propelled elements communicate with neighboring units and thereby actively participate in constructing the final structure. Herein, we use computational modeling to show that millimeter-sized polymer gels can display such self-sustained, autochemotatic behavior. In particular, we demonstrate that gels undergoing the self-oscillating Belousov–Zhabotinsky (BZ) reaction (9) not only respond to a chemical signal from the surrounding solution, but also emit this signal and thus, multiple neighboring gel pieces can spontaneously self-aggregate into macroscopic objects. These findings indicate that BZ gels can undergo a form of “self-recombining”: if a BZ gel is cut into distinct pieces and the pieces are moved relatively far apart, then their autochemotactic behavior drives the parts to move autonomously and recombine into a structure resembling the original, uncut sample (Fig. 1). We also show that the gels’ coordinated motion can be regulated by light, allowing us to achieve selective self-aggregation and control over the shape of the gel aggregates, as well as reconfiguration of the entire structure.

Fig. 1.

Autochemotaxis of four cubic BZ gels, which autonomously self-aggregate in response to self-generated distribution of in solution. (A) Initial conditions for case 1: gels’ centers form square of length 22.5 units (E); distance between inner gel surfaces: 12 units; gels’ distance from the fluid boundaries: 5.75 units. Movie S1 corresponds to the early time behavior of the system in A. (B) Late-time behavior of system. Movie S2 corresponds to the late time dynamics of this system. (C) Distribution of u in solution. Color bar also indicates local concentration of in gel, with red corresponding to highest and highest degree of gel swelling; blue represents lowest concentration and contracted gel. (D) Initial conditions for case 2: relative displacements of gels’ centers from case 1: for gel 1, for gel 2, for gel 3, and for gel 4. Initial conditions for case 3: relative displacements of gels’ centers from case 1: for gel 1, for gel 2, for gel 3, and for gel 4. Early- and late-time dynamics corresponding to case 2 are shown, respectively, in Movies S3 and S4. Early- and late-time dynamics corresponding to case 3 are shown, respectively, in Movies S5 and S6. (E) Area occupied by self-aggregated cubes.

The advent of mesoscopic self-propelled objects that self-organize into macroscopic materials, which furthermore could be dynamically reconfigured, would have a dramatic impact in both manufacturing and sustainability (10). Namely, the same units could be used to form structurally different assemblies (potentially enabling different functionalities); this ability to reconfigure a material into a new shape, without building a new part, would contribute significantly to the recyclability of the system. Polymer networks undergoing the Belousov–Zhabotinsky (BZ) reaction (or so-called “BZ gels”) (11, 12), constitute ideal materials for meeting these challenging tasks. As we show below via computational modeling, these gels undergo self-propelled motion and biomimetic communication to organize into macroscopic assemblies, whose shape can be controlled by light. The BZ gels are unique materials because they can transduce chemical energy into mechanical oscillations in the absence of external stimuli (11–18). This self-oscillatory behavior is due to a ruthenium catalyst, which is covalently bonded to the polymers (11–15). The BZ reaction generates a periodic oxidation and reduction of the anchored metal ion; the hydrating effect of the oxidized metal catalyst induces an expansion of the gel, which then contracts when the catalyst is in the reduced state. Thus, the chemical energy from the BZ reaction fuels the mechanical oscillations of these active gels.

To describe the coupled elastodynamics and reaction–diffusion processes occurring within these BZ gels, we use our recently developed 3D gel lattice spring model (gLSM) (19). Within this model, the governing equations are written in terms of , v, and u, which are the respective volume fraction of polymer and the dimensionless concentrations of the oxidized catalyst and the activator of the reaction (Methodology). Via the gLSM, we obtained qualitative agreement between our computational findings and various experimental results (19). For example, in agreement with the experiments, we observed the in-phase synchronization of the chemical and mechanical oscillations for relatively small sized samples (19, 20) and the decrease of the oscillation period with an increase in the concentration of malonic acid (12, 19). Additionally, the shape- and size-dependent pattern formation observed in our gLSM simulations showed qualitative agreement with experimental studies (21). Our simulations were also useful in explaining how gradients in cross-link density could drive long, thin BZ gels to both oscillate and bend, and thereby undergo concerted motion (22). Hence, the gLSM (16, 18) has proven to be a powerful approach for predicting the behavior of these self-oscillating gels.

We combine this gLSM with a finite-difference approach that accounts for the diffusive exchange of reagents between the gels and fluid, and captures reaction–diffusion processes occurring in the solution. Hence, our system of multiple, self-oscillating gels can “communicate” via the diffusion of reagents in the surrounding fluid. Herein, we present an important modification of this hybrid model that prevents the motile gel pieces from colliding and penetrating into each other (details in SI Methodology, Introducing Interactions Between Gels and Fig. S1). By introducing these excluded volume interactions between mesoscopic pieces, we can readily capture the late-time behavior of the system, where the gels self-aggregate into larger structures with the individual samples oscillating in close proximity to each other.

We first consider the scenario where four equally sized BZ gel cubes are initially placed in a symmetric square arrangement on the bottom surface of the simulation box, as shown in Fig. 1A. The cubes can slide freely on this substrate. The side of each cube has a length of = 10.5 units, which corresponds to roughly 0.26 mm (SI Methodology, Model Parameters and Relationship Between Dimensionless Simulation and Experimental Values, which relates the simulation parameters to experimental values). Such cubes might be obtained by cutting a sample that is 2 × 2 × in size into four equal pieces. The distance between the separated cube centers is = 22.5 units and hence the cubes lie relatively far apart (i.e., the distance between the inner faces of the cubes is equal to 12 units and exceeds the linear dimension of the gel). We impose no-flux boundary conditions for the activator u at the top and bottom walls and set at all edges of the simulation box, which is 45 × 45 × 15 units in size.

Fig. 1B shows the location of the cubes at late times and clearly indicates that the cubes have moved dramatically closer together, occupying an area that is close to that of an uncut sample (Fig. 1E). This autonomous aggregation can be explained as follows. The self-oscillating BZ gels undergo a cyclic swelling and deswelling. During the deswelling phase, diffuses from the gel into the solution. As the gel swells, however, some fraction of this is reabsorbed. Multiple cycles of such pulsations produce a buildup of in the solution. Given the symmetric arrangement of the cubes and the constraint at the box edges, the highest accumulation of occurs in the region between the four cubes, as shown in Fig. 1C. Importantly, the oscillation frequency of such BZ gels increases with the concentration of in the outer solution (23). Because the concentration of is higher for the “inner” surfaces of the cubes than the outer surfaces, the inner surfaces have a higher intrinsic frequency.* It is known that in a system containing multiple oscillators, the region with the highest frequency determines the ultimate direction of wave propagation (24, 25). In our BZ cubes, each element effectively acts as an oscillator; hence, the propagation of the chemical wave through each sample originates from the region with the highest oscillation frequency (23). Thus, the waves propagate in the direction marked by arrows in Fig.1A. These traveling waves effectively “push” the solvent to the outer portions of the box, and due to the interdiffusion of the polymer and solvent (Methodology), all of the gel cubes are driven to the inner, central region (26). In other words, the gel undergoes a net displacement that is opposite in direction to the propagation of the traveling waves (27, 28), and consequently the cubes migrate toward the center of the surface.

Because the gel cubes not only release (through diffusive exchange with the solution) but also migrate to the highest concentration of , this system exhibits a distinct form of autochemotaxis. This behavior is robust and the cubes essentially self-aggregate identically even if we vary the size of the simulation box. In particular, in additional simulations (SI Methodology, Effect of Varying Box Size on the Self-Assembly of Four Cubic Gels and Fig. S2) we kept the relative initial positions of the cubes the same as those in Fig. 1A, but significantly increased the distance from the outer faces of these cubes to the edges of the simulation box (by correspondingly increasing the size of the box). We find that the self-aggregation dynamics is not affected by this increase in box size for the examples considered here (SI Methodology, Effect of Varying Box Size on the Self-Assembly of Four Cubic Gels).

Moreover, the observed behavior remains robust even when we shift the initial positions of the cubes (as illustrated in Fig. 1D). Here, the pieces still self-aggregate, moving toward the highest concentration of in the central region. For Case 3, where the relative location of the cubes has been shifted by the greatest amount, the final square-shaped aggregate is rotated relative to the structure in Fig. 1B. To quantitatively characterize the self-aggregation of these cubes, we calculate the time evolution of the effective area occupied by the entire structure; i.e., at each moment of time we calculate the area between the centers of all four cubes, as shown schematically in the inset in Fig. 1E. We note that the initial area A_t of the uncut sample calculated in this manner would be equal to 110, whereas the areas of the self-assembled structures in the examples in Fig. 1E are larger (∼135–175) due to the excluded volume interactions between the cubes (Methodology). Nonetheless, Fig. 1E clearly shows that the temporal evolution of the self-aggregation and the final areas occupied by the cubes are similar for all of the cases considered here.

One could equally well divide a 2 × 2 × sample into two cubes and one rectangular element. Fig. 2 reveals that these separated pieces also autonomously self-organize into a structure resembling the original square if the pieces are initially arranged in a symmetric manner. Namely, the configuration in Fig. 2A results in the highest accumulation of being at the box center, and consequently leads to the mutual attraction and migration toward this area.

Fig. 2.

Autochemotaxis of cubic and rectangular gels yields square sample (resembling structure in Fig. 1B). Gel 1: units. Other initial distances same as in Fig. 1A. (A) Initial distance between line joining centers of cubic gels and center of rectangular gel: 22.5 units. (B) Late time behavior of the system. Movies S7 and S8 correspond to A and B, respectively.

Rectangular pieces, such as in Fig. 2, can be harnessed to introduce additional complexity and gain further control over the observed autochemotactic behavior. Specifically, one can control the spatiotemporal pattern of in the system by altering the gel’s shape. Furthermore, the longer samples can sustain a unidirectional wave, which travels along the length of the sample. (The particular direction of wave propagation in these samples depends on small random perturbations; in an isolated sample, the wave is equally likely to propagate to the left or to the right.) We exploit these effects in the scenarios depicted in Figs. 3 and 4. Moreover, we achieve control over the chemotaxis of these samples by exploiting the photosensitivity of the BZ gels.

Fig. 3.

Regulating self-aggregation of gels via light; gel 2 can be controllably shuttled to the left or right. Simulation box: units. Width of blue illuminated region (): 11.375 units (half the size of gel 2). (A) Initial distance of gels 1 and 3 from the y-boundary: 15 units. Distance of gels from the fluid x-boundaries: 7.5 units. Light is applied to left side of gel 2. (B) Late time behavior of system in A. (C) Same initial position as in A. Light is applied to the right side of gel 2. (D) Late time behavior of system in C. Movies S9, S10, S11, and S12 correspond to A, B, C, and D, respectively.

Fig. 4.

BZ cubic cargo driven to oscillate and follow self-propelled BZ train. (A–D) Top-down view of 3D simulations at times specified in the plots (namely, the values of are provided on the images). (E) Trajectories of gels 1–4 as marked on the image during the timeframe from to . Here, the size of gel 1 is units and the sizes of gels 2–4 are units. Simulation box: units. Size of dark region (): units. Distance of cubic gels from closest fluid boundary: 7.0 units. Distance of gels 1 and 2 from left fluid boundary: 7.0 units. Distance between inner surfaces of gels 1 and 2: 3.5 units. Movies S13, S14, S15, and S16 correspond to A, B, C, and D, respectively.

Light of a particular wavelength and intensity suppresses the oscillations in BZ gels (29). In the following simulations, we now explicitly include the presence of light by setting the parameter Φ, which accounts for the light-induced production of bromide ions (30), sufficiently high that oscillations in fully illuminated samples are inhibited (Methodology). In Fig. 3A, however, only a fraction of the rectangular sample is illuminated (the region marked in blue). Because the left half of gel 2 is lit, the oscillations are initially suppressed in that region and the pulsations originate from the nonilluminated right half of the gel. The traveling wave then moves from right to left and, correspondingly, the gel moves to the right. Here, the light is harnessed to break the symmetry in the system and thereby direct the motion of gel 2 so that the gels achieve a selective self-aggregation, only joining with a specified partner. Notably, gel 2 can be shuttled to the left by illuminating the right portion of this gel in the initial setup (Fig. 3C). As shown in Fig. 3D, the rectangular sample is now successfully directed to the cube on the left. In effect, light permits control over the autochemotactic behavior and thus can be used to regulate the shape of the self-assembled cluster.

In the final example, we show how patterns of light and dark can be used to design a BZ train that not only picks up BZ cargo, but also “turns on” the oscillations within each transported gel. In Fig. 4, blue again marks the illuminated regions and the white stripe in the center indicates the nonilluminated area. Initially gels 2–4 are entirely illuminated and hence these cubic pieces do not oscillate. On the other hand, the right portion of the rectangular sample (gel 1) is located in the dark. Consequently, the oscillations in gel 1 originate in this nonexposed region, giving rise to a traveling wave that moves from right to left; hence, this rectangular train moves autonomously to the right, as shown in Fig. 4B. The latter figure also indicates that gels 2 and 3 are now in the oscillatory phase, undergoing a rhythmic expansion and contraction; this behavior is due to the diffusion of the activator from the moving BZ train. The local concentration becomes sufficiently high to overcome the effects of light and initiate the pulsations within each illuminated sample. Notably, here we exploited the rectangular shape of gel 1 to tailor the distribution of in the solution, i.e., to deliver sufficient to both gels 2 and 3.

The oscillating cubes now also emit and the autochemotactic behavior draws the cubes closer to the moving train, i.e., toward the highest concentration. Fig. 4C clearly shows that gels 2 and 3 have moved from their original position and are following the train. (Due to the unidirectional wave within the rectangular piece, it moves faster than gels 2 and 3, which thus lag behind gel 1.) The figure also shows that emanating from the train has driven gel 4 to oscillate. Finally, Fig. 4D reveals that all of the cubes not only pulsate, but also move with gel 1. The trajectories of the different gel pieces shown in Fig. 4E further highlight the fact that the cubes follow the path of the rectangular gel train. (In SI Methodology, Interaction Between One Rectangular and Three Cubic BZ Gels: Highlighting the Effect Of Nonuniform Illumination and Fig. S3, we examine the same setup in the absence of these illuminated regions and show that the light is vital for realizing a BZ train that picks up BZ cargo and turns on the oscillations within each transported gel.)

The above examples demonstrate a primitive form of materials’ regeneration; whereas the severed pieces are not regrown de novo, the self-directed, self-sustained motion of the gels nonetheless leads to an effective recovery of the sample’s original shape. Notably, millimeter-sized pieces of the BZ gels can actually undergo self-oscillations for a few hours without replenishment of reagents (14, 15). Moreover, recent experiments demonstrated that upon depletion of reagents the system can be readily “refueled” by replenishing these solutes (21). The scenarios involving the rectangular samples also reveal that the self-generated distributions of activator can be tailored by simply altering the shape of the gel. Hence, by exposing different-shaped pieces to different patterns of illumination, one can drive the gels to configure and reconfigure into a vast array of structures. The process can be performed in one step, as illustrated above, or sequentially, where each piece of the assembly is added a step at a time. The latter process further expands the variety of morphologies that can be formed through this autochemotactic self-assembly. By introducing high concentrations of u at the outer boundaries of the system, the assembled structures can be “disassembled” because the pieces would migrate to the high u at these edges. These separated units could then be guided by light to self-organize into a completely different morphology. In this sense, the BZ gels resemble pieces of a construction toy that can be reused to build multiple structures and thus, provide a different route for creating dynamically reconfigurable materials.

Finally, we note that the findings presented herein constitute a vital step toward designing the next generation of self-healing materials. Currently, severed pieces of self-healing gels or networks can only form bonds when these pieces are brought together by an external agent (e.g., placed in contact via human intervention) (31–34). To the best of our knowledge, there is no example of synthetic materials that bring themselves together as the first step in the self-healing process. Notably, the sides of the isolated BZ gels can potentially be functionalized with reactive groups that would promote bond formation as the separate pieces come within a critical distance of each other. Hence, these autochemotactic materials could pave the way for designing severed materials that autonomously come together and then form strong bonds that bind the units and thereby, restore the mechanical properties of the system, to yield a truly self-healing material.

Methodology

The kinetics of the gels undergoing the photosensitive BZ reaction is described by a modified version (35) of the two-variable Oregonator model (36, 37) that explicitly takes into account the polymer volume fraction . The governing equations for this system are (19, 35)

Here, v and u are the respective dimensionless concentrations of the oxidized catalyst and activator; and are the respective velocities of the polymer network and solvent. The dimensionless diffusive flux of the solvent through the gel in Eq. 3 is calculated as (35) . The terms and , which describe the BZ reaction within the gel, are

The parameters q, f, and in the above equations have the same meaning as in the Oregonator model (36, 38). The two-variable version of the Oregonator model used above has been widely used to describe the spatiotemporal behavior of a variety of BZ reactive systems (39). The dimensionless variable (Eq. 5) was introduced into the Oregonator model to include the effect of light on the reaction kinetics (30). In the absence of the polymer gel (i.e., in the BZ solution), this two-variable, photosensitive Oregonator model has been used to successfully explain the observed experimental phenomena in a number of studies (40–43). Within this model, specifically accounts for the additional production of bromide ions that is due to illumination by light of a particular wavelength and is assumed to be proportional to the light intensity (30) [This approximation is valid in the limit of a constant concentration of bromomalonic acid (44)].^† We note that more complicated three- (45), four- (46), and five- (47) variable Oregonator-class models were formulated to account for a range of additional effects and more complicated reaction kinetics of photosensitive BZ reactions.^‡ Here, however, we assume that the simple two-variable model used above accounts for the most essential effect of light on BZ gels; namely, it allows us to reproduce the experimentally observed suppression of oscillations within BZ gels by visible light (29). Finally, we note that by setting and in Eq. 5, we recover our previous model for BZ gels (18, 19, 35) in the absence of light.

We assume the dynamics of the polymer network to be purely inertialess (relaxational) (48), so that the forces acting on the deformed gel are balanced by the frictional drag due to the motion of the solvent. Hence, the corresponding force balance equation can be written as (35)

Here, is the dimensionless stress tensor measured in units of , where is the volume of a monomeric unit and T is temperature in energy units; is the friction coefficient; D_u is the diffusion coefficient of the activator. We assume (48) that the total velocity of the gel/solvent system (35) and hence, the incompressibility condition is automatically satisfied. In other words, the net velocity of the polymer–solvent system is set to zero and it is solely the polymer–solvent interdiffusion that contributes to the gel dynamics (35, 48). Hence, by setting (18, 35), we neglect the hydrodynamic interactions within the gel. The hydrodynamic effects in the gels are indeed small, as shown below; in fact, neutral, nonresponsive polymer gels are often used as a medium for the BZ reaction to specifically suppress hydrodynamic effects (49).

Based on the above assumptions, we find the polymer velocity as (35)

where is the mobility coefficient of the polymer matrix (35).

The stress tensor in Eqs. 6 and 7 is derived from the free-energy density of the deformed gel, which consists of the elastic energy associated with the deformations of the polymer network and the polymer–solvent interaction energy. The constitutive equation for the gels (17, 19, 35) is given by , where is the unit tensor, is the dimensionless stress tensor, and the isotropic pressure is

The osmotic term (in the square brackets) depends on , which is derived from the Flory–Huggins parameter for the polymer–solvent interactions (19, 35). The parameter describes the hydrating effect of the oxidized catalyst and captures the coupling between the gel dynamics and the BZ reaction. The last term on the right-hand side of Eq. 8 describes the pressure from the elasticity of the network; represents the cross-link density of the gel; is the polymer volume fraction in the undeformed state.

The evolution of the activator concentration u outside the gels and within the external fluid is given by

The last term on the right-hand side represents the decay of activator due to the disproportionation reaction (50). We focus only on reaction–diffusion processes in the outer fluid and neglect the effects of hydrodynamics. In these simulations, the dimensionless velocity of the gel’s expansion/contraction is ∼0.1, which corresponds to v ∼ (using the scaling described in the SI Text). If hydrodynamic effects were taken into account, then it would be reasonable to assume that the characteristic velocities created by these oscillating gels within the fluid would be of the same order of magnitude as v, which would yield a low Reynolds numbers of Re = (v l/) (using the dynamic viscosity and density of water, and taking the characteristic length scale to be the linear dimension of the gel, ). Additionally, the hydrodynamic forces acting from the fluid on the gel’s surface would be negligibly small (23) due to the low ratio of the viscous to elastic forces; this ratio, which represents the capillary number (51), can be estimated as Ca = v/ l K, where the elastic modulus of the poly(N-isopropylacrylamide) (PNIPAAm) gel is taken to be (52). Hence, due to the slow dynamics of the gels and the low viscous forces, we can neglect hydrodynamic effects (23).

Eqs. 1–3 are the governing equations for the polymer–solvent system and are solved solely inside the gel, on a Lagrangian grid. The deformable hexahedral elements of this grid are defined by gel nodes. Eq. 9 is solved solely in the fluid, on a fixed, regular Eulerian grid. At every time step, the flux of activator from the fluid to the gel is interpolated from the Eulerian grid, whereas the flux of activator from the gel to the fluid is interpolated from the Lagrangian grid. These boundary conditions allow for the exchange of activator between the fluid and gel only across the mobile gel–fluid interfaces. For the surfaces of the Lagrangian grid that are attached to the wall, we impose no-flux boundary conditions for the activator. For the external boundary on the fluid grid, we impose no-flux boundary conditions for the activator u at the top and bottom walls and set at all of the side walls of the simulation box.