Using light to control the interactions between self-rotating assemblies of active gels

Polymer 55 (2014) 5924e5932

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Using light to control the interactions between self-rotating assemblies of active gels Debabrata Deb, Olga Kuksenok, Anna C. Balazs* Chemical Engineering Department, University of Pittsburgh, Pittsburgh, PA 15261, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 March 2014 Received in revised form 16 June 2014 Accepted 17 June 2014 Available online 25 June 2014

Polymer gels undergoing the oscillating Belousov-Zhabotinsky (BZ) reaction exhibit an autonomous, periodic swelling and deswelling, where the mechanical oscillations are driven by the chemical reaction within the polymer network. Using computer simulations, we show that these BZ gels can undergo a form of auto-chemotaxis, enabling the gels to spontaneously move in response to self-generated chemical gradients. Focusing on four millimeter-sized pieces of these BZ gels, we show that the pieces can organize into self-rotating clusters that resemble a moving pinwheel or gear. By analyzing the factors that promote the formation of a single self-rotating cluster, we attempt to design systems of multiple, interacting gears. We show that light, which suppresses the oscillations of the gels, can be harnessed to promote the formation of two self-rotating clusters. These studies point to a novel form of photo-chemomechanical transduction, where light is utilized to control the conversion of chemical and mechanical energy in the system. Moreover, the interaction between the BZ gel gears reveals a new form of entrainment between these moving units. Namely, their coordinated motion is achieved through chemical coupling or communication, rather than a mechanical coupling. These findings can lead to the formation of chemically “communicating” devices that can be programmed to perform autonomous work through the use of light. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Computer modeling Self-oscillating gels Photo-sensitive gels

1. Introduction Shape morphing polymer systems can refer not only to a single piece of material that undergoes distinct morphological changes, but also to multiple pieces that self-organize into various structures. Here, we examine a system that exhibits that latter behavior. In particular, through computational modeling, we investigate how multiple pieces of self-propelled polymer gels aggregate into larger structures, which can exhibit remarkable dynamic behavior. Namely, certain structures undergo spontaneous rotation, so that they resemble a rotating pinwheel or a moving gear. The polymer gels that enable this behavior are referred to as “BZ gels” [1e3] because the network is undergoing the oscillating Belousov-Zhabotinsky (BZ) reaction [4,5]. In solution, the periodic chemical oscillations produced by the BZ reaction can be seen by cyclic changes in the color of the fluid [2]. These color changes correspond to the repeated reduction and oxidation of a metal catalyst that is dissolved in the solution. Yoshida et al. [1e3] introduced the BZ reaction into a polymer gel by anchoring a ruthenium catalyst onto

* Corresponding author. E-mail address: [email protected] (A.C. Balazs). http://dx.doi.org/10.1016/j.polymer.2014.06.051 0032-3861/© 2014 Elsevier Ltd. All rights reserved.

the chains within the network. In this system, the reduction of catalyst to the Ruþ2 state created a less hydrating environment for the polymers and hence, the gel underwent a pronounced shrinking; on the other hand, the oxidation of catalyst to the Ruþ3 state yielded a more hydrating solution for the polymer and the gel underwent a distinct expansion. The cyclic redox reaction could now be visualized by the rhythmic deswelling and swelling of the gel, so that sample resembled a beating heart [1e3]. Notably, these volume changes occur autonomously, with the mechanical oscillations of the gel being driven by the chemical oscillations in the reaction. The autonomous changes in the volume of the BZ gels can be harnessed to perform mechanical work and thus, these gels can ultimately be used to design micro-scale soft robots that could perform useful tasks. One of the challenges in designing such autonomously functioning soft robots is determining optimal means of effectively “programming” the material to carry out a particular function. The BZ reaction is photo-sensitive [6] with light leading to a suppression of the oscillations in the BZ gels [7]. Thus, the judicious application of light could provide a means of programming the system both remotely and non-invasively. In the studies described below, we exploit the unique properties of photo-responsive BZ gels to design self-rotating assemblies,

D. Deb et al. / Polymer 55 (2014) 5924e5932

which can potentially function as gears. We first illustrate how separated pieces of the gel can undergo self-propelled motion and aggregate into clusters [8]. The latter behavior mimics the properties of chemotaxing amoebas, which both emit and sense chemical signals and thus, organize into larger structures. We then pinpoint conditions where four BZ gels, which interact through this chemical communication, can form a single rotating pinwheel. Here, we extend calculations from earlier studies [9] and thereby pinpoint the role that the local concentration of chemical species in the surrounding solution play in promoting the rotation of the four-gel assembly. Next, we examine the interactions between two distinct four-gel clusters and determine how light can be utilized to promote the robust formation of two rotating pinwheels. Through the latter studies, we show a novel form of photo-chemo-mechanical communication within the system, and furthermore, show how non-uniform illumination can be used to effectively program the dynamic behavior of the material. These BZ gels transduce the chemical energy from the reaction to sustain their motion. This motion can lead to the autonomous formation of gears, which ultimately can interact to form more complex machines. In essence, the BZ gels could spontaneously form both the components and the larger devices that can be harnessed to perform work. Below, we first describe the computational model we derived to capture the dynamic properties of these materials. 2. Model To describe the dynamic behavior of the BZ gels, we use a modified version [10] of the two-variable Oregonator model [11, 12] for the BZ reaction in solution. The modified equations explicitly take into account the polymer volume fraction, f in the system. The governing equations for this system are [10]


 vf ¼ V, fvðpÞ ; vt

(1)

additional flux of bromide ions due to the light that leads to the suppression of the oscillations. This parameter is assumed to be proportional to the light intensity [6]. Experiments have shown that illuminating the BZ gels with light of wavelength 436 nm and intensity of 6.45 mW/cm2 suppressed the oscillations; removing the light resulted in the reoccurrence of these oscillations [7]. This two-variable Oregonator model allows us to reproduce the experimentally observed suppression of oscillations within the BZ gels by visible light [7]. With F ¼ 0 in Eq. (5), we recover the model for BZ gels in the absence of light [10,13]. The constitutive equation for the gel is given by Refs. [10,14]: _

_

_

s ¼ Pðf; vÞ I þ c0 v0 fðf0 Þ1 B ;

b is the dimensionless stress tensor, bI is the unit tensor, B is where s the local strain tensor, and the isotropic pressure P(f,v) is:

h i Pðf; vÞ ¼  f þ lnð1  fÞ þ cðfÞf2  c* vf þ c0 v0 fð2f0 Þ1 ; (7) The osmotic term (in the square brackets) depends on cðfÞ ¼ c0 þ c1 f, which is derived from the Flory-Huggins parameter for the polymer-solvent interactions [10]. The parameter c* > 0 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 side of eq. (7) describes the pressure from the elasticity of the network; c0 represents the crosslink density of the gel, v0 is the volume of a monomeric unit, and f0 is the polymer volume fraction in the undeformed state. We assume that the total velocity of the gel/solvent system is v≡fvðpÞ þ ð1  fÞvðsÞ ¼ 0 [10,15] and we further assume [15] that dynamics of the gel is purely relaxational, so the forces that act on the deformed gel are balanced by the frictional drag due to the motion of the solvent. Consequently, the polymer velocity is written as [10]: _

(2)


 vu ¼ V, uvðsÞ  V,jðuÞ þ Fðu; v; fÞ: vt

(3)

The system of Equations (1)e(3) is written in dimensionless units of time and space; these units are normalized by T0 and L0, respectively, where T0 depends on the concentration of the sodium bromate and the characteristic reaction rate coefficient, and L0 is related to T0 through the diffusion coefficient of the dissolved reagent [10]. The estimates of the values of T0 and L0 based on the available experimental data are given at the end of this section. The variables v and u in Eqs. (1)e(3) are the respective dimensionless concentrations of the oxidized catalyst and the activator for the reaction, and v(p) and v(s) are the respective velocities of the polymer network and solvent. The dimensionless diffusive flux of the solvent j(u) through the gel is calculated as [10]: jðuÞ ¼ ð1  fÞVðuð1  fÞ1 Þ. The terms Gðu; v; fÞ and Fðu; v; fÞ, which describe the BZ reaction within the gel, are:

Gðu; v; fÞ ¼ ð1  fÞ2 u  ð1  fÞv;

(4) u  qð1  fÞ2 u þ qð1  fÞ2

:

(5)

The parameters q, f and ε have the same meaning as in the Oregonator model [11]. The parameter F characterizes the

(6) _

vðpÞ ¼ L0 ð1  fÞðf=f0 Þ3=2 V,s;


 vv ¼ V, vvðpÞ þ εGðu; v; fÞ; vt

F ðu; v; fÞ ¼ ð1  fÞ2 u  u2  ð1  fÞð f v þ FÞ

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(8)

where L0 is the dimensionless kinetic coefficient [10]. Given that we know the stress tensor (Eq. (6)), we can compute the polymer velocity (Eq. (8)) and hence, numerically integrate the dynamic equations (Eqs. (1)e(3)). We perform these calculations via our recently developed three-dimensional gel lattice spring model (gLSM) [13], which combines both finite element and finite difference techniques. The gLSM has proven to be a powerful approach for predicting the behavior of these self-oscillating gels [8,14,16e20]. Of particular relevance here, we predicted that rectangular-shaped BZ gels on a surface would undergo selfpropelled, directed motion in the presence of light [21, 22]; these predictions were recently confirmed experimentally [23]. We have augmented the gLSM [24] by incorporating the evolution of the activator concentration, u, outside the gels and within the external fluid:

vu ¼ V2 u  u 2 vt

(9)

The last term on the right hand side represents the decay of activator due to the dis-proportionation reaction [25]. We account for the diffusive exchange of reagents between the gels and fluid, capturing reactionediffusion processes occurring in the solution [24]. By incorporating a short range, repulsive potential with the cut-off distance fixed at rc ¼ 1.5 units between gel pieces [26], we introduce excluded volume interactions and thereby, prevent the motile samples from colliding and penetrating into each other [26].

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Finally, due to the slow dynamics of the gels and low ratio of viscous to elastic forces, we neglect hydrodynamic effects [24]. Equations (1)e(3) are the governing equations for the polymersolvent 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 [24]. At every time step, the flux of activator from the fluid to the gel is interpolated from the Eulerian grid, while the flux of activator from the gel to the fluid is interpolated from the Lagrangian grid. For the surfaces of the Lagrangian grid that are attached to the wall, we impose no-flux boundary conditions for the activator. These boundary conditions allow for the exchange of activator between the fluid and gel only across the mobile gelefluid interfaces. We use no-flux, Dirichlet or periodic boundary conditions for the external boundary on the fluid grid. Where possible, our model parameters were taken from known experimental data. In particular, for the BZ reaction parameters, we set [10,27] ε ¼ 0.12, f ¼ 0.9 and q ¼ 9.52  105 and for the parameters characterizing the properties of the gel, we set f0 ¼ 0.139, and c0 ¼ 1.3  103 (based on experimental data provided in Ref. [28]). For the gelesolvent interaction parameters, we used c0 ¼ 0.338 and c1 ¼ 0.518, which correspond to the NIPAAm gelesolvent interaction parameters at 20  C for a gel with the above values of f0 and c0 [29]. We also set c* ¼ 0.105; c* is an adjustable parameter of the model and is chosen to have the same value as in Refs. [10,21,22,24,30]. For the initial conditions within the gels, we chose the concentrations of the oxidized catalyst and activator to be randomly distributed around their stationary solutions for the given reaction parameters, [13], vst ¼ 0.089 and ust ¼ 0.093. Specifically, a random noise is initially added to these values of vst and ust; the amplitude of the noise is 20% of the latter values. In addition, we set the initial degree of swelling l to its stationary value lst ¼ 1.48. Finally, to determine the characteristic length and time scales of the system, we assume that the diffusion coefficient of the activator, Du ¼ 2  109 m2/s, remains the same in the gel and surrounding fluid [31]. Using the above values, we find the correspondence between our dimensionless units of time and length and the respective physical values as T0 z 0.31 s and L0 ¼ DuT0 z 25 mm [26]. Hence, the initial dimensionless length of a cubic sample, l ¼ 8.9, corresponds to ~0.22 mm. 3. Results and discussion We consider multiple, equally-sized BZ gel cubes that can slide freely on the bottom wall of the simulation box. We apply no-flux boundary conditions at the top and bottom walls of this simulation box, and set u ¼ 0 at all the side walls. Each cube has an initial length of l ¼ 8.9 dimensionless units, which corresponds to ~0.22 mm. The BZ reaction within the gels gives rise to the production of the activator u, which diffuses into the surrounding solution. In previous studies [26], we showed that the BZ gels not only emit u, but also “sense” the local concentration of u in the solution. Furthermore, these BZ gels autonomously move towards the highest concentration of the self-generated u and hence, the gels show a remarkable form of auto-chemotaxis. Below, we show that this auto-chemotactic behavior can lead to the formation of “communicating”, self-rotating assemblies. We also show that externally applied light can be harnessed to control the communication between the different assemblies. 3.1. Communication among multiple BZ gels placed in a row Fig. 1 shows eight cubic BZ gels that are placed in a single row within a simulation box of size Lx  Ly  Lz ¼ 214  39  13 units.

The initial distance between the gels' outer surfaces and edges of the simulation box in both the x and y-directions is 15 units. Fig. 1a and b show the respective gel positions and distribution of activator u in the central plane (z ¼ 5) at early times (t ¼ 100). As the reaction proceeds, u produced within the gels diffuses into the surrounding solution. Given the initial arrangement of the cubes and the u ¼ 0 constraint at the side walls, the highest accumulation of u eventually (after a few oscillation cycles) is localized between neighboring gels, as can be seen in Fig. 1b (where red and blue correspond to the respective maximum and minimum of u). The non-uniform distribution of the activator gives rise to the chemotactic behavior noted above, where the gels move towards the highest concentration of u. In order to explain this behavior, we first note that the nodes on the gel's surface that are exposed to a higher activator concentration exhibit a higher intrinsic oscillation frequency than other surface nodes [24]. In a system with multiple oscillators, the one with the highest intrinsic frequency sets the directionality of the traveling chemical wave [32,33]. Due to the inter-diffusion of the solvent and gel, if a traveling wave of solvent moves in one direction, then the gel will move in the opposite direction. (The latter behavior was observed in our simulations for a number of distinct cases [17,21,22,24,26].) In other words, the chemical wave originates and moves outward from the regions boarding the highest concentration of u; consequently, the gels move towards this region of high u concentration [26]. The above explanation helps rationalize the behavior seen in Fig. 1aee. The two gels on either end of the row are exposed to an asymmetric distribution of u, which is lower near the side walls (due to imposed boundary conditions) and higher nearer the neighboring gel (see Fig. 1b). Consequently, these end cubes move towards their neighbors, i.e., towards the center of the box. In the time interval from Fig. 1c to .d, the gels come closer together and self-aggregate into clusters, driven by the higher concentration of u in the intervening regions. As can be seen in Fig. 1e, at late times all the gels move to the central region of the box and form a single large cluster. In this cluster, the average distance between the gels' centers is approximately 12 units; this value is determined by the average lateral size of the oscillating gels and the cut-off distance (rc ¼ 1.5 units) of the excluded volume interactions [26], which prevents the gels from coming closer together. We note that the effect of clustering is robust; the specific gel positions and clusters distributions. We note that the effect of clustering is robust. The specific gel positions and cluster distributions for the gels initially aligned along a single row do, however, depend on the initial random seed. As we show below, significantly more robust behavior is observed in the case where the same eight gels are aligned within two rows. Furthermore, in this arrangement, we could isolate a scenario where the gels' dynamics do not depend on initial fluctuations in the system (see below). Before introducing this system, however, we first illustrate the dramatic difference in the communication between the pieces when we introduce a second row of gels (Fig. 1f) in the system in Fig. 1a. The size of the simulation box size in Fig. 1f is now Lx  Ly  Lz ¼ 204  54  13 units, and the initial distance between the gels' outer surfaces and the edges of the simulation box in the x and y-directions is set to 10 units. All other parameters and boundary conditions are the same as in Fig. 1a. The snapshots in Fig. 1hej show the evolution of system. At early times, the dynamics of the gels is similar to the case of eight gels in a row; as the reaction takes place, the activator, u, diffuses out of the gels and into the surrounding solution. The distribution of u within the z ¼ 5 plane after the first few chemical oscillations is shown in Fig. 1g. This distribution resembles the one in Fig. 1b, with the highest accumulation of u lying between the gels. Hence, the gels also move towards each other due to self-generated gradients of u. Unlike the

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Fig. 1. Self-assembly of self-oscillating BZ gels initially arranged in one (left panel) or two (right panel) rows. Left panel: (a), (c)e(e) Snapshots of eight gels self-aggregating under the influence of self-generated distribution of u in solution at times t ¼ 100(a), 30,000(c), 55,000(d) and 100,000(e). (b) The distribution of activator concentration, u, in the z ¼ 5 plane at t ¼ 100. Right panel: (f), (h)e(j) Snapshots of sixteen gels self-aggregating under the influence of self-generated distribution of u in solution at times t ¼ 120(f), 20,000(h), 40,000(i) and 60,000(j). (g) The distribution of activator concentration, u, in the z ¼ 5 plane at t ¼ 120. The color blue (red) represents the minimum (maximum) value of the concentration of oxidized catalyst, v, in the gels according to the color bar in the bottom of the left panel. The same color bar is used to represent the concentration of u in (b) and (g). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

case of only eight gels in a single row, where the gels form a single cluster at late times and keep oscillating about the central region of the box (Fig. 1e), we now observe four separate clusters that are formed at late times (Fig. 1i). Interestingly, some of the clusters spontaneously rotate, forming a self-organized “pinwheel”; the other clusters self-aggregate and oscillate as a single cluster but do not undergo rotation. In this specific example, the system encompasses two such self-rotating pinwheels and the direction of rotation is indicated by the arrows over the respective clusters (Fig. 1iej). The translational behavior of the rotating and non-rotating clusters is distinctly different. While the center of the selfrotating pinwheel remains fixed in position, the non-rotating cluster moves towards the pinwheel as can be seen from Fig. 1j. The latter figure also shows that both non-rotating clusters have moved considerably closer towards the self-rotating clusters (compared to their positions in Fig. 1i). To understand the reason for this motion, we note that the distribution of u is higher around the pinwheel than around the non-rotating cluster [9]. As discussed above, the gels move toward the highest concentration of u and hence, the gels migrate towards the pinwheels. It is worth noting that the specific number of self-rotating and non-rotating clusters in the system in Fig. 1 depends on the initial random perturbation. Fig. 1 shows that multiple gels arranged in two parallel rows self-organize into pinwheels and non-rotating clusters. Below, we describe the mechanism of forming a pinwheel by focusing on a cluster of four gels, and contrasting the dynamic behavior of the rotating and non-rotating assemblies. We then detail the interactions between the two clusters and discuss means of controlling these interactions by externally imposed light.

3.2. Mechanism for forming a pinwheel We consider a system consisting of four BZ gels placed in a box of size Lx  Ly  Lz ¼ 49  49  13 units with initial gels' center-tocenter distance equal to 20 units and the distance between the gels' outer surfaces and the edges of the simulation box equal to 10 units in both the x and y-directions. Fig. 2a and e show the gel positions and distribution of activator in the central plane (z ¼ 5), respectively, at early times (t ¼ 460). Similar to the scenario in Fig. 1, owing to the symmetric position of the gels and u ¼ 0 condition set at the boundaries, the highest accumulation of u occurs in the center of the box after a few oscillation cycles (Fig. 2e). As discussed above, due to this distribution of u, chemical waves propagate from the inner to the outer faces of the gels and the gels move in the opposite direction, i.e. towards the center of the box [26]. As the gels move towards each other, at time t ¼ 4000 (Fig. 2b), the oscillations in the gels become phase locked [9], exhibiting a constant phase difference between the oscillations in neighboring gels; hence, the chemical waves travel successively from one gel to the next in a clockwise direction. As the cubes continue to approach each other, the excluded volume interactions begin to affect their motion and exert a torque on each gel. This torque results in the rotation of the individual gels around their centers of mass [9]; we refer to the time at which this rotation occurs as the moment of pinwheel formation (Fig. 2c). As the system evolves further, the pinwheel keeps rotating about the center of the box as shown in Fig. 2d, where the direction of rotation of the pinwheel is indicated by the arrow above the gels. We note that this example represents the outcome of one simulation; for another independent simulation (i.e., using a different random

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Fig. 2. Mechanism for forming a pinwheel. The left panel shows the positions of four self-oscillating BZ gels at times t ¼ 460(a), 4000(b), 9300(c) and 18015(d). The color represents the concentration of oxidized catalyst, v, in the gels. The red arrow in (d) marks the pinwheel's direction of rotation. The right panel shows the distribution of activator concentration, u, in the z ¼ 5 plane of the simulation box at times t ¼ 460(e) and 18015(f). The color scheme is same as in Fig. 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

seed), the system did not form a pinwheel, but simply exhibited auto-chemotaxis, with the gels coming together in a configuration similar to Fig. 2c. To gain insight into factors leading to the rotary motion, we plot the running average of the activator concentration around the gels, us, for both the rotating and non-rotating systems in Fig. 3a and b, respectively. To calculate us, we consider a cubic shell with a linear size of ds ¼ 20 around each gel and calculate the average value of u within this shell. In the plots of Fig. 3aeb, the colors of the curves correspond to the colors of the numbers 1e4 of the gels (see insets). Until t ¼ 4000, the dynamics of both the rotating and non-rotating

systems remains similar, and the curves for us in Fig. 3aeb follow the same path. At t ¼ 4000, however, the pinwheel forming system exhibits a phase locking between the neighboring gels (see Fig. 2b). Zooming in the portions of curves within the gray shaded region (at the time around t ¼ 4000, see insets in Fig. 3aeb), we find that for the pinwheel (Fig. 3a) us is distinctly different for gels 1 and 2 than for gels 3 and 4; this is in contrast to the non-rotating assembly, where all the curves are essentially indistinguishable from each other [34]. Hence, in the case of a pinwheel assembly, one of the gels (gel number 1 in Fig. 3a) on average is surrounded by the highest concentration of u and, correspondingly, has the highest

Fig. 3. The running average of activator concentration, us, within a cubic shell with the linear size of ds ¼ 20 encompassing each of the BZ gels in the case of a pinwheel in (a) and in the case of a non-rotating assembly in (b). The insets in (aeb) show the portions of the curves zoomed within the gray shaded region. The black vertical lines are at t ¼ 4000, when the chemical oscillations in the gels are phase locked for the case of pinwheel assembly (a).The color of each curve corresponds to that of the numbering of the gels as indicated in the snapshot given in the inset. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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intrinsic frequency of oscillations. Using the similar reasoning, gel number 2 exhibits the second higher intrinsic frequency. As discussed above, the oscillator with the highest intrinsic frequency acts like a pacemaker and leads the oscillations in the assembly [32,33], initiating a wave that travels from the gel with the highest frequency to the one with lowest frequency (in the direction 1/2/3/4/1 for the case in Fig. 3a). Hence, the particular direction of wave propagation is dictated by the local distribution of u around the gel, as characterized by the average value of us within a cubic shell encompassing each gel. As we showed earlier, the phase locking is a necessary condition for the formation of a pinwheel [9]. A comparison of the evolution of us around the gels for the cases of rotating and non-rotating assemblies indicates that: 1) an asymmetric distribution of u around different gels during the evolution of the system promotes the phase locking and thereby the pinwheel formation, and 2) the values of us around all the gels are higher when the pinwheel is formed and lower for the non-rotating assembly. The latter observation is consistent with our earlier finding [9]. In what follows, we rely on the measurements of us around the gels to understand the interactions between the gel clusters. Furthermore, we later show that understanding the main features of the behavior of rotating and non-rotating clusters allows us to control the communication and interactions between these assemblies through the application of light. 3.3. Interaction between two BZ gel pinwheels We now focus on eight BZ gels initially arranged in two rows with four gels in each (as in Fig. 4a). We show that with this initial arrangement, gels self-aggregate into two distinct clusters with

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four gels. Similar to the scenario with the sixteen gels in Fig. 1b, some of these clusters become pinwheels and some of the clusters form non-rotating assemblies. We find that the probability of pinwheel formation depends on the initial center-to-center distance between the gels. To quantify the behavior of this system, we classify the late-time states of these systems into the following three categories: 1) both clusters remain non-rotating (0 pinwheel), 2) one cluster is non-rotating while the other becomes pinwheel (1 pinwheel) and 3) both clusters become pinwheels (2 pinwheels). Fig. 4a shows the initial arrangement of eight cubic BZ gels that are placed in a simulation box of size 89  89  13 units. The initial center-to-center separation between neighboring gels is d ¼ 20 units and the distance between the gel's outer surface and the nearby edge of the simulation box is 10 units. The boundary conditions remain the same as in all the cases considered above. With time, the gels within each cluster move towards each other and at late times, we observe the formation of two self-aggregated clusters (see Fig. 4b). Namely, the cluster on the left forms a pinwheel undergoing rotary motion in the counterclockwise direction and the cluster on the right remains in the non-rotary state (Fig. 4b). (Note that the mechanism of the pinwheel formation follows the steps described above for the case of only four gels, see Fig. 2.) During the further evolution of this system, the center of the pinwheel on the left remains effectively pinned to its position, so that the gels continue to rotate in the same direction around the same central point. The non-rotating cluster on the right, however, moves considerably closer towards the pinwheel (see Fig. 4ced), under the influence of the higher average activator u produced within the pinwheel as compared to that around the non-rotating cluster, as we discussed in the previous section [9]. We note that this effect is robust; in a number of additional simulations with

Fig. 4. Interaction between pinwheel and non-rotating cluster. (a)e(d) Snapshots of eight BZ gels interacting in a box of size 89  49  13 units respectively at times t ¼ 140, 9000, 13,000 and 25,000. Initial center-to-center distance between neighboring gels is d ¼ 20 units. (e) Late time snapshot of a similar system of eight BZ gels interacting in a box of size 104  54  13 units and with initial gels' center-to-center distance d ¼ 25 units. (f) Distribution of cluster formation in the systems of eight gels initially placed in two rows with initial center-to-center distances d ¼ 20 and 25.

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multiple clusters, we consistently observed that the centers of the rotating assemblies remained pinned to their locations and these pinwheels effectively attracted neighboring non-rotating assemblies (all the gels within these assemblies move towards the higher concentrations of u generated by the pinwheels). Fig. 4aed show one possible outcome of a simulation involving this initial arrangement of the gels. With small fluctuations in the initial conditions, in addition to yield one rotating and one nonrotating cluster, these simulations can yield either two pinwheels or two non-rotating clusters. In the latter case, the clusters move towards each other (and towards the center of the box); again, this behavior is due to the higher concentration of u in the center of the box [26]. This range of scenarios can also be observed when the initial center-to-center gel separation is increased to d ¼ 25 units. Fig. 4e shows a snapshot of the late-time behavior for this system, showing that the assembly ultimately forms two pinwheels. In Fig. 4f, we summarize the late-time results for systems with initial separations of d ¼ 20 and 25; the numbers in the plot were obtained from 16 independent simulation runs. The plot clearly illustrates that the probability of forming two pinwheels at late times depends on the initial separation between the gels and is significantly higher for the larger initial separation of d ¼ 25. For d ¼ 20, in the majority of cases, we observe either one pinwheel and one non-rotating cluster (as in the example in Fig. 4aed) or two non-rotating clusters, while for the case of d ¼ 25, we observe the formation of the two pinwheels in the majority of the cases and none of the cases yielded two non-rotating clusters. We attribute the observed increase in the probability of forming two pinwheels in the system with the larger initial separation to the following effects. With the larger d, the average distance between the two clusters becomes larger, resulting in a higher asymmetry in the distribution of u in the system. This asymmetry increases the

probability for one of the gels within a cluster to be surrounded by a locally higher activator concentration, i.e., experience a higher value of us. As noted above, this asymmetry in distribution of us leads to the phase locking of the chemo-mechanical oscillations among the neighboring gels (see Fig. 3a) and, correspondingly, to the pinwheel formation. Furthermore, when the initial separation between the gels, d, is small, the gels move towards each other more rapidly because the u produced within the neighboring gels has a shorter distance to diffuse to influence the clustering. We found that this faster clustering of the gels effectively disfavors the phase locking of oscillations among the neighboring gels. In Fig. 5, we compare the evolution of two systems with d ¼ 20 and 25; in both cases, one non-rotating assembly and one pinwheel have formed at late times, as shown in the snapshots in Fig. 5a and d. Via the procedure used to generate Fig. 3, we now measure us for all the gels in Fig. 5, and determine the time of the phase locking of the chemo-mechanical oscillations among the gels forming the pinwheels. The phase locking occurs at t ¼ 4000 when d ¼ 20 and at t ¼ 19,000 when d ¼ 25 (as indicated in Fig. 5c and f by the vertical black lines). Hence, the phase locking occurs at an earlier time in the system with the smaller d. At the time of the phase locking, the activator around the gels, us, is the highest for gel number 2, and the second highest for the gel 3 (see insets in Fig. 5c and f). This behavior is similar to that observed in Fig. 3 for the case of a single pinwheel. This asymmetric distribution of us results in a chemical wave that is initiated at gel 2 and travels to gel 3 (i.e., moving in the counterclockwise direction). The chemical wave then leads to the motion of the entire assembly in the opposite direction (clockwise direction). For the non-rotating assemblies (in both Fig. 5a and d), the values of us for pairs of gels are essentially undistinguishable. As one might anticipate, these values are lower for gels 7 and 6 (outer

Fig. 5. Average activator concentration around gels in system of eight BZ gels. (a,d) Late time snapshots (top view) of a system with one self-rotating pinwheel and one non-rotating self-assembly. The initial gel separation is d ¼ 20 units in (a) and d ¼ 25 units in (d). (b,c,e,f) Time evolution of a running average of activator concentration, us, within a cubic shell of a size ds ¼ 20 encompassing each of the BZ gels. The color of each curve corresponds to that of the numbering of the gels as indicated in the snapshots (a) and (d). The insets in (b), (c), (e) and (f) show portions of the curves within the gray shaded regions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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gels) and higher for gels 5 and 8; again we note that this nonuniform distribution of u drives the non-rotating assembly to migrate towards the pinwheel. To summarize, our simulations show that in the system with the larger d, the distribution of u around the gels is asymmetric in the majority of cases; this asymmetry increases the probability of phase-locking and hence, the formation of pinwheels. The results of these studies give us a clue on how to control the formation of rotating assemblies. In this context, it is important to recall that blue light suppresses the oscillations in the BZ gels [7]. As we show below, we can use non-uniform illumination to robustly create two communicating pinwheels. 3.4. Controlling interaction between pinwheels using light We now begin with the system shown in Fig. 6a, where the separation between the gels is d ¼ 25 and apply light with a fixed intensity F (see eq. (5)) over just the right half of the simulation box (as indicated by the blue patch in Fig. 6aeb) for a fixed period of time during the first stage of the simulation and then remove the light during the second stage (Fig. 6ced). We show that this twostage treatment of the system not only increases the probability of the formation of two pinwheels, but also provides a means to control and tune the interactions between the two clusters of gels by externally perturbing the system with light of the appropriate intensity. The snapshots in Fig. 6aeb show that during the first stage (t < 30,000) when the right half of the simulation box is illuminated with a light of intensity F ¼ 0.002, oscillations within the four right gels are suppressed and, correspondingly, these gels remain at their initial positions. This value of F is significantly higher than the critical value required to suppress the oscillations in an isolated gel sample [21]. Such a high value is necessary to counteract the influx of the activator from the neighboring, non-illuminated gels and undergoing the BZ reaction [24]. During the first stage, the gels on the left in Fig. 6a eventually form a pinwheel (Fig. 6b) via the mechanism described above (see Section 3.2). Importantly, this pinwheel formation is robust: we observed the same outcome as in Fig. 6aeb in 32 independent runs (16 runs with F ¼ 0.002 and 16

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runs with F ¼ 0.003). Hence, applying the light to half of the box increases the probability of pinwheel formation within the assembly in the left by increasing the asymmetry in the system, as confirmed by calculations for the evolution of us around the gels within the assembly on the left (data are not shown). In addition, since during this stage, the four gels on the right are held in place, the possibility of one of the clusters forming a non-rotating pinwheel and moving towards the pinwheel (similar to the system in the dark in Fig. 4aed) is now excluded. During the second stage (t  30,000), we remove the light from the right half of the system, and the gels that were in the nonoscillatory state now begin to oscillate as shown in Fig. 6c at t ¼ 30,990. Thus, during the second stage, the pinwheel on the left continues to rotate, while the gels on the right move closer towards each other through the auto-chemotactic motion [26] and then finally form another pinwheel (see Fig. 6d) (following the same steps as described in Section 3.2). Our simulations show that this two-stage scheme of creating the two pinwheels is robust; two pinwheels were formed in the same manner as described in Fig. 6 in sixteen independent simulations. Moreover, one might anticipate that the outcome of these simulations should not be sensitive to the value of light intensity as long as this value is sufficiently high. To test this hypothesis, we again conducted sixteen independent simulations with different initial random seeds, but we set F ¼ 0.003 during the first stage over the right half of the box. Again, we found that in all sixteen simulations, the system formed two pinwheels (similar to the ones shown Fig. 6d). Note that in all the above simulations the direction of rotation is arbitrary and depends on initial fluctuations. Thus, we have devised a robust approach to controllably create two pinwheels out of eight self-oscillating BZ gels interacting in a box. As noted in the Introduction, the pinwheels also resemble simple gears, which could be used to perform mechanical work. Here, the gear-like structures are chemically entrained, i.e., the chemical interaction between the two gears (mediated by the selfgenerated concentration of u) promotes the mutual rotation of these assemblies. In effect, the two “mechano-chemical gears” are coupled through chemical interactions rather than though wellknown mechanical coupling in typical mechanical gears.

Fig. 6. Photo-controlled formation of two communicating pinwheels. (aeb) Stage 1: Light with intensity F ¼ 0.002 is applied over the right half of the simulation box as marked by the blue patch at 0 < t < 30,000. During this stage, chemo-mechanical oscillations are suppressed in the four gels under the light and they remain stationary while the four gels on the left form a self-rotating pinwheel as shown in (b). (ced) Stage 2: During this stage (t  30,000), light is removed (F ¼ 0) and four gels on the right begin to undergo chemomechanical oscillations as shown in (c). At later times, the four gels on the right form another self-rotating pinwheel (d). The arrows show the direction of rotation of the pinwheels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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4. Conclusions The autonomous pulsations of the BZ gels lead to the production of the activator, u, within the gels. Some fraction of this u diffuses from the gels into the surrounding solution. In turn, the local concentration of u in the solution affects the oscillations of the neighboring gels and consequently, the autonomous motion of the sample. In effect, the gels move in response to self-generated gradients and specifically, migrate toward the highest concentration of the u in the solution [26]. This behavior is particularly evident in Fig. 1aef and Fig. 2aec, which both illustrate the migration of the gels toward the highest u in the system. The above scenario takes an interesting twist when four cubic gels come sufficiently close to each other and, due to fluctuations in the system, the local concentration of activator around these gels, us, exhibits an asymmetry. In particular, if two neighboring gels subtend the highest and next higher values of us, then the system exhibits a distinct form of synchronization. Namely, the gels become phase locked so that they “beat” in a successive pattern, with a chemical wave traveling from the gel that is surrounded by the highest us to the gel with the next highest us. In the four-gel sample, this pattern then leads to a clockwise or counterclockwise rotation of the traveling wave and the rotation of the gels in the opposite direction [9]. Hence, it is this asymmetric distribution of us and the subsequent mode of synchronization that are key to forming the pinwheels. These studies on the single rotating cluster informed our studies on two four-gel clusters, which could potentially yield two rotators. Systems of multiple rotating clusters are interesting because the clusters could form chemically interacting gears, which in turn could act as useful devices. In the two-cluster system, we found that an increase in the separation between the clusters promoted the formation of two rotators. A close inspection of the distribution of us revealed that the clusters separated by a greater distance did in fact yield the desired asymmetric distribution of the activator. Hence, the conclusions about the role of us that emerged from our study of a single pinwheel also applied to systems involving two rotating clusters. From the distinctive behavior of the two clusters, we hypothesized that light, which suppresses the oscillations in the illuminated regions, could be used to regulate the interaction between the four-gel clusters and promote the robust formation of two gears. We did in fact find that with the aid of light, gels separated by d ¼ 25 formed two rotators in all 16 of the simulations examined. By illuminating one of the clusters for a fixed time, we suppress the release of u from that cluster. In effect, the non-illuminated, oscillating gel is now localized in a relatively asymmetric domain, which promotes an asymmetric distribution of us. This asymmetry in the local concentration of the activator enables the oscillating cluster to develop into a rotator. When the light is lifted from the first cluster, the asymmetric distribution of the activator generated by the oscillating gel promotes the necessary local conditions for forming a second rotator. The above scenario points to a novel form of photo-chemomechanical transduction, where light is harnessed to control the conversion of chemical and mechanical energy in the system. Moreover, as noted above, the interaction between the gears reveals a new form of entrainment between these moving units.

Namely, their coordinated motion is achieved through chemical coupling or communication, rather than a mechanical coupling. These findings can lead to the formation of chemically “communicating” devices that can be programmed to perform autonomous work through the use of light. Acknowledgments ACB gratefully acknowledges financial support from AFOSR (for partial support of DD) and DOE (for partial support of OK). The authors gratefully acknowledge useful discussions with Dr. Pratyush Dayal. References [1] Yoshida R, Takahashi T, Yamaguchi T, Ichijo H. J Am Chem Soc 1996;118(21): 5134e5. [2] Yoshida R. Bull Chem Soc Jpn 2008;81(6):676e88. [3] Maeda S, Hara Y, Sakai T, Yoshida R, Hashimoto S. Adv Mater 2007;19(21): 3480e4. [4] Belousov BP. Collection of short papers on radiation medicine. Moskow: Medgiz; 1959. [5] Zaikin AN, Zhabotinsky AM. Nature 1970;225:535e7. [6] Krug HJ, Pohlmann L, Kuhnert L. J Phys Chem 1990;94(12):4862e6. [7] Shinohara S, Seki T, Sakai T, Yoshida R, Takeoka Y. Angew Chem Int Ed 2008;47(47):9039e43. [8] Kuksenok O, Dayal P, Bhattacharya A, Yashin VV, Deb D, Chen IC, et al. Chem Soc Rev 2013;42(17):7257e77. [9] Deb D, Kuksenok O, Dayal P, Balazs AC. Mater Horizons 2014;1(1):125e32. [10] Yashin VV, Balazs AC. J Chem Phys 2007;126(12):124707. [11] Tyson JJ, Fife PC. J Chem Phys 1980;73:2224e37. [12] Tyson JJ. A quantitative account of oscillations, bistability, and traveling waves in the Belousov-Zhabotinskii reaction. In: Field RJ, Burger M, editors. Oscillations and traveling waves in chemical systems. New York: Wiley; 1985. pp. 93e144. [13] Kuksenok O, Yashin VV, Balazs AC. Physical Review E 2008;78(4):041406. [14] Yashin VV, Kuksenok O, Dayal P, Balazs AC. Rep Prog Phys 2012;75(6): 066601. [15] Barriere B, Leibler L. J Polym Sci Part B-Polym Phys 2003;41(2):166e82. [16] Chen IC, Kuksenok O, Yashin VV, Moslin RM, Balazs AC, Van Vliet KJ. Soft Matter 2011;7:3141e6. [17] Dayal P, Kuksenok O, Balazs AC. Macromolecules 2014;47(10):3231e42. [18] Yuan P, Kuksenok O, Gross DE, Balazs AC, Moore JS, Nuzzo RG. Soft Matter 2013;9:1231e43. [19] Chen IC, Kuksenok O, Yashin VV, Balazs AC, Van Vliet KJ. Adv Funct Mater 2012;22(12):2535e41. [20] Yashin VV, Suzuki S, Yoshida R, Balazs AC. J Mater Chem 2012;22(27): 13625e36. [21] Dayal P, Kuksenok O, Balazs AC. Langmuir 2009;25(8):4298e301. [22] Dayal P, Kuksenok O, Balazs AC. Soft Matter 2010;6(4):768e73. [23] Lu X, Ren L, Gao Q, Zhao Y, Wang S, Yang J, et al. Chem Commun 2013;49(70): 7690e2. [24] Dayal P, Kuksenok O, Bhattacharya A, Balazs AC. J Mater Chem 2012;22(1): 241e50. [25] Steinbock O, Kettunen P, Showalter K. Science 1995;269:1857e60. [26] Dayal P, Kuksenok O, Balazs AC. Proc Natl Acad Sci 2013;110(2):431e6. [27] Kuksenok O, Yashin VV, Kinoshita M, Sakai T, Yoshida R, Balazs AC. J Mater Chem 2011;21(23):8360e71. [28] Sasaki S, Koga S, Yoshida R, Yamaguchi T. Langmuir 2003;19(14):5595e600. [29] Hirotsu S. J Chem Phys 1991;94:3949e57. [30] Yashin VV, Balazs AC. Science 2006;314(5800):798e801. [31] Yoshida R, Otoshi G, Yamaguchi T, Kokufuta E. J Phys Chem 2001;105(14): 3667e36723. [32] Mikhailov AS, Engel A. Phys Lett A 1986;117(5):257e60. [33] Kheowan OU, Mihaliuk E, Blasius B, Sendina-Nadal I, Showalter K. Phys Rev Lett 2007;98(7):074101. [34] Notably, while the difference in average values of us between the different gels is small (does not exceed 4% for the case in Fig. 3a), it is not negligible; this feature of the pinwheel dynamics is robust and is observed in all our simulations resulting in the formation of the pinwheels.