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He received his bachelor's degree in Chemical Engineering from Indian Institute of ... Her current research interests include theory and computer simu...
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Directing the Behavior of Active, Self-Oscillating Gels with Light Pratyush Dayal,† Olga Kuksenok,‡ and Anna C. Balazs*,‡ †

Chemical Engineering Department, Indian Institute of Technology, Gandhinagar, India Chemical Engineering Department, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States

‡

ABSTRACT: In the biological realm, light can act as a powerful stimulus, promoting both positive and negative phototaxis. Using computational modeling, we attempt to design systems that display analogous biomimetic behavior by exhibiting directed, autonomous motion in response to light. We specifically focus on polymer gels that undergo the oscillating Belousov−Zhabotinsky (BZ) reaction and thus manifest periodic chemomechanical pulsations, which can be modulated with light. Reviewing our recent computational studies, we describe how long, rectangular samples of BZ gels, or “worms”, can perform self-sustained movement and via a distinct form of negative phototaxis migrate along complex paths under nonuniform illumination. When the ends of multiple rectangular BZ gels are anchored to a surface, the dynamic behavior of the cilialike layer can be tuned by light to resemble the motion of a keyboard. With BZ gel pieces that move freely on a surface, we show that these gels exhibit autochemotaxis and, thereby, can self-organize in response to self-generated chemical signals. These examples illustrate that BZ gels constitute optimal materials for creating millimeter-sized soft robots whose self-sustained movement can be regulated through the use of light.

I. INTRODUCTION

isolating ideal polymeric materials exhibiting autonomous motion that can be regulated by light. Below, we review recent work on polymer gels undergoing the Belousov−Zhabotinsky (BZ) reaction,11 or as commonly referred to as BZ gels,12−14 which constitute an ideal material for the above task. The BZ gels present two important attributes for creating responsive, soft robots: they can undergo autonomous motion9,10 and they are light-sensitive.15,16 These distinctive characteristics are due to the BZ chemical reaction occurring within the polymer network. Discovered serendipitously in the 1950s, the BZ reaction now forms a cornerstone of the field of nonlinear dynamics. The BZ is an oscillating reaction that involves a metal catalyst, commonly ruthenium (Ru), which undergoes a periodic reduction and oxidation. Hence, the reagents in the BZ solution drive the Ru to cycle periodically between the Ru2+ and Ru3+ states. In the late 1990s, Ryo Yoshida anchored the ruthenium tris(2,2′-bipyridine) (Ru(bpy)3) catalyst onto cross-linked polymer chains of poly(N-isopropylacrylamide) (NIPAAm) and then placed the gel into an aqueous solution of the BZ reagents: NaBrO3, HNO3, and malonic acid.12,13,17 Remarkably, the gel oscillated autonomously, beating like a heart. Moreover, BZ gels that are millimeters in size can self-oscillate on the order of hours,18,19

Motion due to light is an ubiquitous phenomenon in biological systems. The most obvious example is the ability of plants to track the motion of sunlight. On the other hand, a number of organisms, such as Caenorhabditis elegans1 and the slime mold Physarum polycephalum,2 flee from light. Hence, in the biological realm, light provides a powerful stimulus, leading to both positive and negative phototactic motion. Recently, there has been a surge of activity aimed at designing biomimetic materials that display directed movement in response to external stimuli.3 This work is motivated in part by the growing interest in creating responsive robotic systems4−6 that move in a controllable, autonomous manner.7 Notably, researchers are turning to polymeric materials to create such robots since the flexibility of polymers allows these devices to both twist and bend. Moreover, polymers can be formulated to be stimuliresponsive, thereby providing a means of regulating the movement of these systems. Formed from responsive, polymeric materials, robots could ultimately be directed to perform a range of useful tasks for human-centric activities. As in the biological realm, light could provide a particularly effective stimulus for controlling the motion of soft, polymeric robots.8−10 Namely, light can be applied from a distance and in a noninvasive manner, i.e., without the need for physical contact. Moreover, the effects of this stimulus can easily be removed by simply switching off the light source. Hence, a critical issue for the development of maneuverable “soft-bots” is © 2014 American Chemical Society

Received: November 26, 2013 Revised: April 7, 2014 Published: April 14, 2014 3231

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and the system can be “resuscitated” by replenishing the solution with reagents that were consumed in the reaction.20 This rhythmic “beating” of the gel is driven by the periodic reduction and oxidation of the metal catalyst. When the catalyst is in the reduced state, the host fluid provides a less hydrating solvent and the gel contracts; when the catalyst is in the oxidized state, the fluid acts as a more hydrating solvent and the gel expands.12−14 In other words, the chemical energy from BZ reaction is transduced into the mechanical oscillations of the gels. To date, the NIPAAm-based BZ gels are the most analyzed and understood self-oscillating gels;12−21 there have, however, been a number of recent advances in utilizing different polymer networks to design these active materials. For example, recently Vaia et al.22,23 fabricated BZ gels using a postfunctionalization technique where Ru catalyst is printed into bio-derived polypeptide gelatin gels22 or polyacrylamide gels;23 such a reactive printing approach allows a variety of heterogeneous patterns to be fabricated. It is worth noting that in contrast to the NIPAAm-based gels, hybrid gels based on polyacrylamide (PAAm) and silica gel, and gels formed from poly(acrylamide-co-acrylate),24,25 as well as the PAAm-based gels recently fabricated by Nuzzo et al.,26 actually shrink when the catalyst is in the oxidized state. The observed volumetric changes in the latter systems were attributed24−26 to the formation of additional reversible crosslinks when the polymer is in the oxidized state.24−26 While these newly synthesized BZ gel systems hold great promise, in this review, we focus on the more studied NIPAAm-based BZ gels. Herein, we describe recent computational studies on the influence of light on BZ gels. Using these models, researchers have designed a polymeric “worm” that undergoes selfsustained motion and migrates along tortuous paths when exposed to nonuniform illumination. These phototaxing BZ worms could be utilized as effective, soft robots within microfluidic devices, controllably transporting and delivering payloads to particular regions within the devices. We also highlight recent experimental results validating predictions that these BZ gel worms move in controllable ways in response to light. We then show that the photosensitivity of the BZ gels leads to novel behavior even when the worm is attached to a wall by one end, so that the system resembles biological cilia. Finally, we discuss how the interaction between multiple, freely moving BZ gel pieces can be controlled via light, leading to guidelines for coordinating the actions of a group of soft, microscopic robots. Below, we briefly summarize the model used to simulate the behavior of these chemoresponsive gels. A more detailed description of this method is given in refs 14, 27, and 28. We then discuss how this model was used to determine the dynamic behavior of single or multiple BZ gel samples when the system was exposed to nonuniform illumination.

to capture the effect of the polymer gel on the evolution of the system. With ϕ being the polymer volume fraction, then the governing equations become27,28 ∂φ = −∇·(ϕ v(p)) ∂t

(1)

∂v = −∇·(v v(p)) + εG(u , v , ϕ) ∂t

(2)

∂u = −∇·(u v(s)) − ∇·j(u) + F(u , v , ϕ) ∂t

(3)

The variables v and u are the respective dimensionless concentrations of the oxidized catalyst and the activator for the reaction; the activator represents the chemical species that promote the oscillations in the system. v(p) and v(s) are the respective velocities of the polymer network and solvent. In eq 3, j(u)is the dimensionless diffusive flux of the solvent through the gel and is calculated as27 j(u) = −(1 − ϕ)∇(u(1 − ϕ)−1). The terms G(u,v,ϕ) and F(u,v,ϕ), which describe the BZ reaction within the gel, are G(u , v , ϕ) = (1 − ϕ)2 u − (1 − ϕ)v

(4)

F(u , v , ϕ) = (1 − ϕ)2 u − u 2 − (1 − ϕ)[fv + Φ] u − q(1 − ϕ)2 u + q(1 − ϕ)2

(5)

The parameters q, f, and ε in the above equations have the same meaning as in the Oregonator model.29,31 The dimensionless variable Φ in eq 5 accounts for the influence of light on the BZ reaction kinetics by capturing the effect of the additional flux of bromide ions that arise due to the illumination. It is assumed that the value of Φ is proportional to the light intensity. Importantly, the oscillations in the BZ gel become suppressed at a sufficiently high value of Φ, i.e., the critical value, Φc,9,32 which depends on the reaction parameters and the physical properties of the gel. For BZ solutions that do not encompass the gels, this two-variable, photosensitive Oregonator model has reproduced and helped rationalize a range of experimentally observed phenomena.33−36 For systems encompassing the polymer networks, this two-variable model allows us to reproduce the experimentally observed suppression of oscillations within BZ gels by visible light.15 By setting Φ = 0 in eq 5, we recover our model for BZ gels27,28,37 in the absence of light. Finally, we obtain the following the constitutive equation for the gels:14,27,28 σ̂ = −P(ϕ,v)I ̂ + c0v0ϕ(ϕ0)−1B̂ , where I ̂ is the unit tensor and σ̂ is the dimensionless stress tensor. The isotropic pressure P(ϕ,v) is P(ϕ , v) = −[ϕ + ln(1 − ϕ) + χ (ϕ)ϕ2 − χ * vϕ] + c0v0φ(2ϕ0)−1

II. METHODOLOGY In modeling the BZ gels, we take advantage of the experimentally observed photosensitivity of these materials.15,16 In particular, when the sample is illuminated by a specific wavelength of light, the reaction produces a higher concentration of bromide ions, which in turn inhibit the reaction and, thereby, suppress the periodic expand and contraction of the gels.15,16 To describe the kinetics of the gels undergoing the BZ reaction, we modified27 the two-variable Oregonator model29,30

(6)

Here, we set χ(ϕ) = χ0 + χ1ϕ, which is derived from the Flory− Huggins parameter for the polymer−solvent interactions.27,28 The parameter χ* > 0 accounts for the chemomechanical coupling in these systems; c0 represents the cross-link density, ϕ0 is the polymer volume fraction in the undeformed state of the gel, and v0 is the volume of a monomeric unit. We assume the dynamics of the polymer network to be purely relaxational.38 Furthermore, we neglect the total velocity of the solvent−polymer system, thereby assuming that only the 3232

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Figure 1. Motion of BZ gel (of size 30 × 10 × 10 nodes) under nonuniform light intensity. The dark region (Φ = 0) is shaded in gray. (a) Initial position of the gel such that a one-third portion (on the left) is in the dark region. (b) Motion of the gel demonstrated by the time snapshots at early (t = 120) and late (t = 985) stages. (c) Evolution of the x-coordinate of the gel’s center shows locomotion in the negative x-direction. The color bar indicates the concentration of the oxidized catalyst, v, in the gel.

entire simulation box (fluid grid), we impose no-flux boundary conditions for u at the bottom and top walls and set u = 0 at all the side walls. By applying our gLSM approach to simulate the BZ gels, we have obtained qualitative agreement between the findings from our computational studies and various experimental results. For example, our simulations allowed us to reproduce the in-phase synchronization of the mechanical and chemical oscillations for relatively small-sized samples28,46 and the dependence of the period of oscillations on the concentration of malonic acid.17,28 Additionally, the pattern formation and dynamics of BZ gels observed in our simulations on uniform samples20 and on samples with gradients in cross-link density47 are in a good agreement with the corresponding experimental studies.20,47 Recently, the gLSM approach was modified to model synchronization between circular BZ gel patches48 embedded within a neutral, nonresponsive gel. Most recently, we further modified the gLSM to capture the behavior of novel PAAmbased self-oscillating BZ gels in which the oxidation of the catalysts leads to a deswelling of the sample due to the formation of additional reversible cross-links.26 Hence, the gLSM37,49 is a powerful approach for simulating the dynamic behavior of self-oscillating gels. Below, we consider the dynamic behavior of BZ gels when the samples are exposed to different patterns of illumination. Within a given sample, we set Φ = 0 for each element that is located within the dark, and we specify a nonzero value of Φ for each element that is illuminated. At each time step, the value of Φ is updated so that when an element moves away from the light, the value of Φ for that element is set to zero. At the onset of the simulation, the swelling of the gel is characterized by its stationary value λst (taken at Φ = 0), and the values of u and v

solvent−polymer interdiffusion contributes to the gel motion.27,38 Additional details on this theoretical model can be found in ref 27. Where possible, the systems parameters are taken from known experimental data. The values of these parameters are summarized in ref 27. To determine the characteristic length and time scales of the system, we assume that the diffusion coefficient of the activator,27 Du = 2 × 10−9 m2/s, remains the same in the gel and surrounding fluid.40 With the parameter values in ref 27, our dimensionless units of time and length correspond to the respective physical values of T0 = 0.31 s and L0 = (DuT0)1/2 = 25 μm. In the cases where we consider interactions between multiple BZ gel pieces, we also take into account the evolution of the activator concentration, u, outside the gels and within the external fluid. The evolution equation for u is given by ∂u = ∇2 u − u 2 ∂t

(7)

The last term on the right-hand side of eq 7 accounts for the consumption of the activator in the disproportionation reaction.41 We neglect hydrodynamic effects in these systems (see refs 42 and 43 for details). The above governing equations are numerically integrated as detailed in ref 43. Equations 1−3 are solved within the gels, using our three-dimensional gel lattice spring model (gLSM).28 Equation 7 is solved within the fluid on a fixed, regular grid using a finite difference approach. Our implementation of the coupling between the two grids is described in ref 43. Finally, we note that in all the simulations below we apply no-flux boundary conditions for u on the gel’s surfaces that are anchored to the bottom substrate. For the boundaries of the 3233

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directions, depending on the initial random fluctuations in the system. Notably, in order to achieve directed motion toward the dark region, the cross section of the sample must be approximately the same or smaller than the characteristic diffusion length in the system.14 The velocity of the gel’s directed motion can be tailored by altering χ* (see eq 6). Altering χ* does, however, change the stationary values ust, vst, and λst, and thus, samples with different values of χ* exhibit different initial degrees of swelling (and hence have different initial sizes). Hence, we characterize the directed motion of the different samples by calculating the change in the x-coordinate of the gel’s center, xc(t), with respect to its initial value, xc(0). Figure 2 shows the value of

have small random fluctuations around their respective stationary values, ust and vst (the latter values are determined by calculating the stationary solutions28 of eqs 1−3). In these calculations, the attenuation of light within the gel is neglected and the temperature is assumed to be constant.

III. AUTONOMOUS NEGATIVE PHOTOTAXIS OF BZ GELS A. Dynamics of BZ Gel in Nonuniform Illumination. We first focus on a long, regular sample that is 30 × 10 × 10 nodes in size and exposed to nonuniform illumination, which is characterized by a step function in the light intensity. Hence, one-third of the gel is initially located in the dark and two-thirds is illuminated with a constant intensity. The gel lies along the xdirection, and the gray shading in Figure 1a indicates the region that is in the dark (Φ = 0). Within the illuminated region, we set Φ = Φc. As determined from our simulations,9,10 the oscillations would be completely suppressed in the sample if it were uniformly illuminated with light of an intensity corresponding to this value of Φc.9,10 In the absence of light (at Φ = 0), the sample is in the oscillatory regime for the parameters selected here. Consequently, the nonilluminated region (the left end) generates a chemomechanical wave, which travels toward the right. Figure 1b displays the early and late time positions and morphologies of the BZ gel; the gel is drawn within a larger box to more clearly illustrate the path of its motion. Figure 1c shows the temporal evolution of the x-coordinate of the center of the gel, xc(t), and clearly indicates that xc systematically decreases with time. (The small scale oscillations are due to the periodic swelling and deswelling of the gel.) This systematic decrease in xc reveals that the gel has migrated along the negative xdirection and, hence, away from the light. In other words, the gel undergoes a form of negative phototaxis. In this sense, the BZ gel resembles the behavior of the C. elegans roundworms, which flee from light,1 and we thus refer to these samples as BZ “worms”. In this scenario, the spatially varying light breaks the symmetry in the system. In the absence of the applied light, traveling waves will still appear in this sample; there is, however, no preferred direction for the wave propagation. Consequently, the probability of the wave traveling to the left is equal to its probability of traveling to the right, and averaged over a large number of samples, the net displacement of the gel will be zero.9 In the case described above, however, the oscillations always originate from the nonilluminated region, and thus, as shown in Figure 1, the traveling waves continually propagate from left to right. As noted in the Methodology section, motion in this system is due to the interdiffusion of the polymer and solvent. With the traveling waves moving to the right, they effectively “push” the solvent to the right, and because of the polymer−solvent interdiffusion, the gel is thereby driven from right to left.9 In other words, the net displacement of the gel is opposite in direction to the propagation of the traveling waves; we also observed this behavior for two-dimensional samples of BZ gels for similar reaction parameters.27,50 While we focused on a relatively small sample in Figure 1, it is important to note that longer samples with the same cross section exhibit the same type of directed motion. For samples of size 100 × 10 × 10 nodes, the gel also bend in the perpendicular direction as they move along the negative xdirection.9 This bending can occur in any of the perpendicular

Figure 2. Deviation of the x coordinate of gel’s center from its initial position (xc(0) − xc(t)) as a function of time. The intensity profile is the same as Figure 1a. The vertical lines represent the error bars calculated over ten independent runs.

xc(0) − xc(t) as a function of time for different χ* and reveals that the slope of the curves, and hence the velocity increases with increasing χ*.9 We emphasize that the negative phototaxic motion of the gels demonstrated here depends critically on chemomechanical response of the polymer matrix; in order to observe distinct directed motion, the value of χ* has to be relatively high. Decreasing χ* below the lowest values used in simulations in Figure 2 causes the effect to become negligibly small and would correspond to the case of essentially nonchemoresponsive gels. The gel’s velocity can be obtained from the slope of the curve for the evolution of the x-coordinate at relatively early times (but after the oscillations were fully developed within the sample). This calculation gives the gel’s velocity for the reference value of χ* = 0.105 along the x-direction as approximately 0.03 dimensionless units. This value corresponds to a velocity of ∼1.2 μm s−1 at the characteristic length and time scales noted above.9 Importantly, recent experimental studies have confirmed the above predictions on the photocontrolled autonomous motion of BZ gels. In particular, Epstein et al.8 considered a freely sliding BZ gel confined in a capillary tube; when one end of the tube was illuminated with light of a higher intensity than the other illuminated end, the gel displayed negative phototaxis, moving along the length of the tube toward the darker region. (Notably, for the selected intensities, the frequency of oscillations was higher in the darker than in the brighter region.8) In these experiments, an increase in the light intensity resulted in a decrease in frequency of oscillations; we capture 3234

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Figure 3. Top view of the BZ gel making a 90° turn. (a) The position of the gel at early time (t = 297); the inset shows the initial position of the gel (green) with respect to the dark regions (gray) A and B. (b) Details of the dynamic behavior as the center of the gel crosses x = 0 threshold. The snapshots show the propagation of the two waves that originate in the dark region. The plot shows the trajectory of the gel’s center for a period of time encompassing a few oscillations; the direction of the net motion is indicated by the black arrow). (c) and (d) show the sequence of snapshots demonstrating subsequent bending and moving the gel into region A. (e) Trajectory of the gel’s center; black bars represent both position of the gels’ center and gels’ orientation at early and late times, respectively. The color bar indicates the concentration of oxidized catalyst, v, in the gel.

illumination, these macroscopic BZ gel worms can undertake various complicated maneuvers, including making 90° turns.10 To drive the sample to make a 90° right turn, we use the setup shown in the inset in Figure 3a; the two nonilluminated regions, marked A and B, are indicated in gray and are twice the height of the gel sample. Region A is offset in the y-direction so that its edge aligns with the center of region B (see Figure 3a). Region B extends from the left most edge of the gel and initially spans 20% of the gel’s length; the x-axis of the B region coincides with the long axis of the gel in its initial position. For the sake of clarity, Figure 3 shows only the top view of the three-dimensional system. To model the spatial variation in light intensity, we set Φ = 0 within the nonilluminated regions, and within the illuminated regions, we set Φ = 1.95 × 10−3, which is greater than Φc for this sample. As discussed above, the gel’s net displacement is opposite to the direction of the traveling chemical wave, and hence, the gel moves toward the dark. This behavior is evident in Figure 3; as the gel moves leftward through the dark B region, it “senses” the contiguous, nonilluminated A patch and hence enters this darkened area.10 The snapshots in Figure 3b allow us to obtain a better understanding of how the gel navigates the turn as it crosses the

the same behavior in our simulations. As in the above discussion, the wave in these experiments originated at the region with the higher intrinsic oscillation frequency (darker region in this example) and propagated toward the brigher end; due to the interdiffusion of solvent and polymer, the sample was shown to simultaneously move in the direction opposite to that of wave propagation.51 B. Navigation along Complex Paths. The above studies pave the way for designing “programmable” self-propelled, soft robots whose motion could be controlled via light. It would be particularly useful if such soft robots could navigate tortuous paths as well as dynamically alter their routes. For example, it might be advantageous to remotely (and noninvasively) direct a gel from one site to another site, and hence, the robot could visit multiple locations. Finally, it would be beneficial to control the time span that this object spends at each location before it is sent “on demand” to additional sites. Our recent simulations reveal that the BZ gels can be driven to perform these complex tasks when the material is exposed to the appropriate arrangement of the imposed light.10,14,42 The desired intensity profiles could be realized experimentally through the use of holographic techniques52−54 or masks. As described below, through exposure to the appropriate 3235

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Figure 4. Motion of BZ gel subjected to “T-shaped” configuration of regions A and B. (a) Early time position of the gel with respect to regions A and B. (b−e) Snapshots of the steps that the gel follows to align parallel to region A. (f) Trajectory of the gel’s center revealing back and forth movement of the gel; black bars represent both position of the gels’ center and gels’ orientation at early and late times, respectively. The color of the gel’s surface represents the value of v using the color scheme given in Figure 3.

x = 0 threshold and makes a move from B into A. To facilitate the discussion, we have drawn arrows in the snapshots to indicate the directionality of the traveling waves within the gel. As can be seen, the traveling waves originate in the two different dark regions and then propagate along the sample. As the opposing waves approach each other, they interact and, thereby, cause the gel sample to bend. The trajectory of the center of the gel’s motion during the relevant time interval (bottom plot in Figure 3b) shows a complex pattern, whereby large amplitude movements toward A alternate with smaller amplitude motions away from A. The snapshots in Figures 3c,d clearly demonstrate that the gel ultimately makes a 90° right turn and orients along the length of the A region, approximately parallel to y-direction. Once the gel is completely within the dark region, it appears to straighten out. (The driving force for the latter behavior is that the straighter structure has a lower elastic energy than the bent configuration.) The trajectory of the gel’s center in Figure 3e also shows that the gel becomes trapped within the dark

region. In this case, the cross sections of the A and B regions, as well as the length of the B region, were chosen to be smaller than the length of the gel; therefore, the only possible straight configuration that the gel can attain while still remaining within the dark regions is when it is oriented along the long axis of region A. Another of the BZ gel’s maneuvers can be seen in Figure 4; here, the dark A and B regions are arranged in a “T-shaped” configuration (Figure 4a). The total length (in the x-direction) of region B is 40% of the gel’s initial length, and the height and width of the B region are twice the gel’s initial cross section. The width (x-direction) of region A is 20% of the gel’s initial length. The gel is placed in region B so that only 30% of its length is in the dark (Figure 4a). Figures 4b−e highlight the gel’s movement and structural changes as it traverses the darkened path to arrive at its final configuration within A. The gel shuttles back and forth in B (see trajectory in Figure 4f) until it ultimately aligns perpendicular to the latter domain (Figure 4e). These back and forth movements correspond to 3236

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Figure 5. Dynamics of the five cilia system. (a−c) Late time snapshots in the absence of light (Φ = 0). (d−f) Late time snapshots when cilia 1, 2, and 3 are illuminated by light (Φ = 1.5 × 10−3). The size of the simulation box is 69 × 20 × 23 in x, y, and z directions. The distance between outer cilia surfaces and the fluid boundaries is 5.5 units; the distance between the inner surfaces of the cilia is 3.5 units. The cilium size is 6 × 6 × 30 nodes. The color bar indicates the concentrations of u in the fluid and v in the gel.

the images shown at t = 24 999 and t = 49 998 in Figures 4c,d. This behavior remains robust even when the long axis of the gel is initially not perfectly aligned along the long axis of the B region. (In particular, similar behavior occurs when we initially set a 5° angle between the long axis of the gel and that of the B region.) In samples with large cross sections (i.e., greater than the diffusion length), the system exhibited the generation of 3D spiral waves inherent to the BZ reaction.10 Consequently, the bending and reorientation of thicker BZ gels cannot be controlled effectively by light. Hence, in order to maneuver gel samples along complex paths using light, one has to choose samples with a cross section that is smaller than the characteristic length scale associated with the diffusion of the reagents.10,14 As mentioned above, these synthetic BZ worms exhibit striking biomimetic behavior in their ability to move away from the presence of light and thus exhibit negative phototaxis. Interestingly, the latter action mimics the adaptive behavior of the slime mold Physarum polycephalum, which also undergoes negative phototaxis, responding to the presence of light by moving toward the dark.2

the bottom surface, where no-flux boundary conditions are imposed. Hence, the diffusion of u in the surrounding fluid can enable a mode of communication among the BZ cilia, as shown below. The cilia (each of size 6 × 6 × 30 nodes) are arranged along the x-direction, at a distance of 5.5 from the side walls (xdirection) of the simulation box, and at a distance of 3.5 from each other. The size of the simulation box is 69 × 20 × 63 in the respective x, y, and z directions. The value of u is taken to be zero at the sides of the simulation box, whereas periodic and no-flux boundary conditions are imposed in the y and z directions, respectively.43 Figures 5a−c are snapshots of the system’s late-time behavior in the absence of light, while Figures 5d−f show late-time snapshots of the array under nonuniform illumination. Specifically, the ciliary arrangement and boundary conditions are the same in both systems, but cilia 1, 2, and 3 (as marked by arrows in Figure 5e) are illuminated in Figures 5d−f. Here, Φ = 1.5 × 10−3, which is greater than the critical value required to inhibit the production of u and suppress the oscillatory behavior.9,55 Hence, under these conditions, an isolated cilium would not exhibit chemomechanical oscillations. The oscillations in the five cilia array in Figures 5c,d would also be suppressed if the entire array were illuminated with uniform light corresponding to this value of Φ. In Figure 5e, however, the nonilluminated cilia (numbered 4 and 5) continue to produce the activator u, which diffuses through the fluid to cilia 1, 2, and 3. As the chemical waves travel from right to left (in the negative x direction), the local activator concentration increases sharply and thereby switches “on” each cilium. Once the concentration of u becomes sufficiently high within this system, the waves travel synchronously from the top to bottom of the cilium length (as marked by arrows in Figure 5e). Because of the light source, the leading wavefront shifts toward the nonilluminated side (compare Figures 5b and 5e).

IV. BZ GELS CILIA: COMMUNICATION IN THE PRESENCE OF LIGHT To investigate the potential for regulating the interactions among multiple BZ gels via light, we focus on five rectangular gels that are anchored to a substrate and immersed in fluid (see Figure 5). The long, anchored gels resemble biological cilia, hairlike structures that protrude from the surface of various cells. The fluid surrounding these synthetic cilia contains BZ reagents but is devoid of the Ru catalyst. Consequently, the activator, u, is produced solely inside each cilium. The boundary conditions on the ciliary surfaces allow for the diffusion of u, both to and from the surrounding fluid, except at 3237

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Figure 5 reveals particularly remarkable behavior in these ciliary arrays (in both the absence and presence of light) that is due to the local distribution of u. Without illumination, the highest concentration of u in the outer fluid occurs above cilium number 3 (Figure 5a); this is due to the symmetric arrangement of gels on either side of this central cilium. In Figure 5c, it is clear that the outer cilia are bent toward this central unit. In the presence of the nonuniform illumination indicated in Figure 5e, the highest concentration of u is shifted toward cilium number 4 (Figure 5d). As seen in Figure 5f, the cilia are all now tilted toward cilium 4. These results indicate that the cilia not only sense but also bend toward the highest values of u within the simulation box. This behavior is even more pronounced in systems involving the nonanchored, mobile gels, as discussed further below. The above phenomena can be understood from studies43 on the behavior of a small, isolated BZ gel cube as u was systematically increased in the outer solution, in both the presence and absence of light. In both cases, the oscillation frequency, ω, increases with the increase in u. (Notably, ω in the presence of light remains lower than that in the absence of light for a wide range of u.43) In a system of oscillators, the one beating with the highest frequency is the “pacemaker”, which sets the directionality of wave propagation.56,57 For each cilium, the regions closest to the highest concentrations of u can be viewed as the oscillators with the higher intrinsic frequency,43 and hence, they serve to initiate the traveling chemical waves in each of the gels. Since the waves move away from these “pacemakers”, the gels move in the opposite direction due to the interdiffusion of polymer and solvent,42,43 as discussed in the section above. In other words, the gels move toward the highest concentration of u in the outer solution; since the bottom surfaces of the cilia are attached to the substrate, this motion results in the “bunching” of the cilia observed in Figures 5e,f. As noted above, ω was higher for the nonilluminated cube; this explains the shift in the position of the leading front toward the cilia in the dark (see Figure 5e) as well as the asymmetric “bunching” of the cilia toward the nonilluminated region (Figure 5f). The dynamic variations in the behavior of the cilia at early times can be seen from a top-down view. In particular, the height of each cilium increases, reaches the maximum value when the catalyst is in the oxidized state, and then decreases when the catalyst is in the reduced state. We call this variation in height the “piano effect”43 since the cilia appear to move up and down like the keys of a piano. To illustrate this piano effect, in Figure 6 we plot the temporal behavior of the top cilial surfaces corresponding to the case in Figure 5e. Figure 6 shows the pronounced transient pattern that develops while the activator concentration in the fluid remains relatively low, i.e., until about t ∼ 103, which corresponds to more than a dozen oscillation cycles. Each cilium reaches its corresponding maximum height in a sequential manner. The fact that the light source causes the leading wavefront to shift toward the nonilluminated side is also reflected in the variations of the heights of the cilia. In summary, through these studies, we demonstrated that the BZ cilia exhibit a remarkable chemosensing capability and the ability to autonomously translate this chemosensitivity into a distinct mechanical response. These attributes could be exploited in a range of microfluidic applications, where, for example, externally applied light could be utilized to tailor the

Figure 6. The “piano” effect. Variation of the positions of the top cilial surfaces at early times when cilia 1, 2, and 3 (as shown in Figure 5) are illuminated by light (Φ = 1.5 × 10−3). Each cilium reaches its corresponding maximum height in a sequential manner and appears to move like the keys of a piano.

movement of a soft “conveyor belt”58−60 and thus regulate the transport of microscopic objects within the devices.

V. AUTOCHEMOTAXIS OF BZ GELS IN THE PRESENCE OF LIGHT In the previous section, we considered tethered BZ gels that were immersed in the fluid and demonstrated how light can be used to control the interaction between gel filaments. In this section, we again consider multiple BZ gels, but now the pieces can slide freely over the substrate. Given this mobility, multiple gels no longer bend but, rather, undergo directed motion toward the highest concentration of u.42,55 Thus, these BZ gels exhibit a distinct form of autochemotaxis, moving in response to a self-generated chemical gradient. Here, we demonstrate that this chemotactic behavior can be controlled by taking advantage of the photoresponsive behavior of the BZ gels. The system shown in Figure 7 consists of two identical gel cubes (of a linear size l = 6 nodes) placed symmetrically on either side of a rectangular gel (of a size 2l × l × l nodes). We impose no-flux boundary conditions for the activator u at the top and bottom walls and set u = 0 at all edges of the simulation box, which is 114 × 26 × 18 units in size. Here, we set Φ = 2 × 10−3 to account for the light-induced suppression of the oscillations in BZ gels (due to the production of bromide ions in the presence of light).15 As indicated in Figure 7a, we illuminate just the left half (indicated in blue) of the rectangular gel, “gel 2”, and thereby initially suppress the oscillations in this region. Consequently, the traveling wave originates from the nonilluminated right half of the gel and propagates from right to left. Consequently, the gel moves to the right, in the direction opposite to that of wave propagation (as discussed above). As in the case of the BZ gel “worm”, the application of light serves to break the symmetry in the system and, thereby, direct the motion of gel 2. In this manner, the gels in Figure 7 can be driven to undergo selective 3238

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Figure 7. Regulating self-aggregation of gels via light; gel 2 can be controllably shuttled to the left (in (a)) or to the right (in (b)). Simulation box size is 114 × 26 × 18 units. Width of the blue illuminated region (Φ = 2.0 × 10−3): 11.375 units (half the size of gel 2). (a) Initial distance of gels 1 and 3 from the boundary: 15 units. Distance of the gel from the x-boundary: 7.5 units. Left side of gel 2 is illuminated by light. (b) Late time behavior of system in (a). (c) Same initial positions of the gels as in (a). Right side of gel 2 is illuminated by light. (d) Late time behavior of system in (c).

redistribution of u due to the moving train, gel 4 is also driven to oscillate and ultimately move with gel 1. The trajectories of the moving gels are shown in Figure 8e; the plot clearly indicates that all the gels eventually migrate toward the moving train.

self-aggregation, only joining with a specified partner. Namely, in Figure 7a, gel 2 is directed toward the cube on the right. Similarly, gel 2 could be directed toward the cube on the left by illuminating just the right portion of this gel in the initial setup (Figure 7c); the preferential aggregation of gel 2 and the cube on the left is in fact seen in Figure 7d for this case. These examples show that light permits control over the autochemotactic behavior and thus can be utilized to regulate the selfassembly process. Finally, we show how nonuniform illumination can be used to design a BZ “train” that picks up BZ cargo and “turns on” the oscillations within each transported gel.42 The blue shading in Figure 8 again indicates the illuminated regions, and the white stripe in the center indicates the nonilluminated area. Initially, gels 2−4 are completely localized in the illuminated region (Figure 8a), and thus, these cubic pieces do not undergo oscillations. In the case of the rectangular gel 1, however, only the left portion is illuminated, and hence, the oscillations originate in this nonexposed region. The resulting traveling waves move from right to left, and this rectangular “train” moves autonomously to the right, as shown in Figure 8b. The pulsations and motion of this train gives rise to the diffusion of u in the vicinity of gels 2 and 3, which are then driven into the oscillatory state (see Figure 8b). In particular, the local concentration of u around these cubes is now sufficiently high that the influence of the activator overwhelms the effects of light and promotes the oscillations within each illuminated sample. Note that in this case, by choosing gel 1 to have a rectangular shape, we could effectively “tune” the distribution of u in the solution so that this “train” would deliver sufficient levels of the activator around gels 2 and 3. Since the oscillating cubes now also emit u, they are drawn closer to the moving train through autochemotaxis, which promotes the autonomous motion of the gels toward the highest concentration of u. The fact that gels 2 and 3 have migrated from their original positions and are following the train can be clearly seen in Figure 8c. (Because of the unidirectional wave within the rectangular piece, it moves faster than gels 2 and 3, which thus lag behind gel 1.) With the

VI. CONCLUSIONS The development of useful soft robots requires the effective utilization of compliant materials that can undergo controllable, self-sustained movement. In effect, the robots should perform their functions in a relatively autonomous manner that can be regulated by external cues. Ideally, the cues or “instructions” should be straightforward to apply, without the need for considerable peripheral connections to the material itself. The examples described above illustrate that BZ gels constitute optimal materials for creating millimeter-sized “soft-bots”. Importantly, the oscillating reaction inherent to the system provides the “fuel” for the transduction of chemical energy into mechanical motion. Namely, the reaction-induced periodic oxidation and reduction of the anchored metal catalysts drives the rhythmic expansion and contraction of the gel. As shown here, the latter morphological changes of the gel can be regulated with light, and thus, the system can exhibit controllable self-sustained movement. Light provides a particularly useful stimulus for tailoring the dynamic behavior of responsive polymer gels since it can be applied from a distance in a noninvasive manner. Furthermore, the light source can be readily turned on and off, yielding a means of tailoring the exposure of the gels to the stimulus. Hence, the effective combination of nonuniform illumination and the photoresponsive BZ gels can accelerate the creation of the next generation small-scale robotic systems. To provide useful design rules for creating such small-scale, responsive BZ robots, we first considered a single rectangular sample or BZ gel “worm” and elucidated guidelines for patterning the intensity profiles of the imposed light to produce the directed motion of the sample. Importantly, the gel’s path and the direction of motion can be dynamically, remotely, and noninvasively reconfigured. Hence, these 3239

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proving a means of controlling the motion of the particles (which could be bound to or are effectively “pushed” by these motile gels). We also considered multiple rectangular pieces of BZ gels that were anchored by one end to a substrate and thus resembled biological cilia. We demonstrated that BZ cilia exhibit a remarkable chemosensing capability, driving the tethered gels to bend toward the highest concentration of u.43 Since the gels generated the gradients of u that prompted their autonomous movement, the system can be said to exhibit autochemotaxis. We then showed how an external light source can be utilized to regulate the pulsations and “bunching” of these BZ cilia. Again, these attributes could be usefully exploited in a range of microfluidic applications. In particular, the gel layer could act as a conveyor belt,59,61 with the light directing the transverse movement of each cilium (as in Figure 6) and thereby managing the transport of microscopic objects along the surface. Finally, we modeled multiple BZ gels sliding freely on a surface and showed that the autochemotactic behavior of this assembly could be controlled by exposing the samples to nonuniform illumination.42 The light-controlled self-organization can be carried out either in one step, as illustrated above, or sequentially, where each piece of the assembly is added a step at a time. Consequently, the system can be driven to form a great variety of morphologies through this autochemotactic self-organization. We further note that the assembled structures can be “disassembled” by introducing high concentrations of u at the outer boundaries of the system (since the pieces would migrate to the high u at these edges).55 Through the use of light, these separated units could then be steered to selforganize into a completely different shape. In this sense, the BZ gels resemble pieces of a construction toy that can be reused to build multiple structures. Hence, the directed assembly and disassembly of these autonomously moving gels provides a new route for forming dynamically reconfigurable materials. More generally, the findings provide guidelines for fashioning soft active materials to create useful components in fluidic environments. On a fundamental level, it is important to recall that the BZ reaction is a classic example of nonlinear chemical dynamics, and hence, the results provide insight into factors that control the behavior of systems that are driven far from equilibrium.

Figure 8. BZ cubic cargo driven to oscillate and follow self-propelled BZ-train. (a−d) Top-down view of 3D simulations at times specified in plots. (e) Trajectories of gels during the time frame t = 0 and t = 106. The size of gel 1 is 10.5 × 22.75 × 10.5 units, and the sizes of gels 2−4 are 10.5 × 10.5 × 10.5 units. Simulation box: 56 × 116 × 18 units. The size of the dark region (Φ = 0) is 19 × 93 units. Distance of the cubic gels from the closest boundary: 7.0 units. Initial distance between inner surfaces of gels 1 and 2:3.5 units. The color of the gel’s surface indicates the value of v (using the color scheme in Figure 1).

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autonomously moving soft robots could be reprogrammed “on demand” to not only move to a specific target location but also remain at this location for a specified time interval. These worms could be particularly useful for conveying particulates through microfluidic devices, with the application of the light

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.C.B.). Notes

The authors declare no competing financial interest. 3240

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Biographies

She received her B.A. in physics at Bryn Mawr College and her Ph.D. in Materials Science at the Massachusetts Institute of Technology. Her research involves developing theoretical and computational models to capture the behavior of polymeric materials, nanocomposites, and multicomponent fluids in confined geometries.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from the NSF (for partial support of P.D.), the AFOSR (for partial support of O.K.), and ARO (for partial support of A.C.B.).

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REFERENCES

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Pratyush Dayal is an Assistant Professor in the Department of Chemical Engineering at the Indian Institute of Technology Gandhinagar. He received his bachelor’s degree in Chemical Engineering from Indian Institute of Technology−Banaras Hindu University (IIT-BHU) and his Ph.D. in Polymer Engineering from the University of Akron. His current research focuses on modeling and simulation studies on polymer synthesis, polymer blends, and smart biomimetic systems.

Olga Kuksenok is a Research Associate Professor in the Chemical Engineering Department at the University of Pittsburgh. She received her Ph.D. in Physics and Mathematics in 1997 at the Institute of Physics in Kiev, Ukraine. Her current research interests include theory and computer simulations of reactive multicomponent systems and chemoresponsive polymer gels as well as computational design of biomimetic materials.

Anna C. Balazs is the Distinguished Professor of Chemical Engineering and the Robert von der Luft Professor at the University of Pittsburgh. 3241

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