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Modeling Chemoresponsive Polymer Gels Olga Kuksenok,1 Debabrata Deb,1 Pratyush Dayal,1,2 and Anna C. Balazs1 1

Department of Chemical Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261; email: [email protected]

2

Present address: Department of Chemical Engineering, Indian Institute of Technology, Gandhinagar 382424, India

Annu. Rev. Chem. Biomol. Eng. 2014. 5:35–54

Keywords

The Annual Review of Chemical and Biomolecular Engineering is online at chembioeng.annualreviews.org

spirobenzopyran-functionalized gels, self-oscillating gels, gel lattice spring model

This article’s doi: 10.1146/annurev-chembioeng-060713-035949 c 2014 by Annual Reviews. Copyright  All rights reserved

Abstract Stimuli-responsive gels are vital components in the next generation of smart devices, which can sense and dynamically respond to changes in the local environment and thereby exhibit more autonomous functionality. We describe recently developed computational methods for simulating the properties of such stimuli-responsive gels in the presence of optical, chemical, and thermal gradients. Using these models, we determine how to harness light to drive shape-changes and directed motion in spirobenzopyran-containing gels. Focusing on oscillating gels undergoing the Belousov-Zhabotinksy reaction, we demonstrate that these materials can spontaneously form self-rotating assemblies, or pinwheels. Finally, we model temperature-sensitive gels that encompass chemically reactive filaments to optimize the performance of this system as a homeostatic device for regulating temperature. These studies could facilitate the development of soft robots that autonomously interconvert chemical and mechanical energy and thus perform vital functions without the continuous need of external power sources.

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INTRODUCTION

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Recent advances in stimuli-responsive gels are paving the way for the creation of soft robotic devices that can perform useful functions in response to a range of environmental cues (1, 2). Namely, when responsive gels are exposed to variations in the thermal, optical, or chemical properties of the surrounding medium, these materials undergo distinct changes in volume or shape, which in turn can be harnessed to produce mechanical work. For example, the rhythmic expansions and contractions exhibited by thermoresponsive (3) or chemoresponsive (4, 5) gels can serve as a synthetic muscle that drives small-scale devices (3–5). An intriguing scientific challenge is determining how to regulate the dynamic behavior of these gels by modulating the local temperature, illumination, or chemistry and thereby enable the systems to perform more complex, multistep functions. Computational models can provide an ideal tool for addressing this challenge because they constitute an effective means of exploring a broad parameter space in reasonable timescales. Hence, models that can accurately capture the properties of stimuli-responsive gels could greatly accelerate the development of the next generation of smart robotic systems. In this review, we describe recent progress in theoretical and computational approaches that has led to a new, robust method for simulating the behavior of responsive gels in the presence of various stimuli (e.g., heat, light, and chemical gradients). We first provide a general description of the theoretical model (6–9) that forms the underpinnings of the computational approach. We then detail how this model can be tailored to describe specific chemistries, focusing on gels functionalized with spirobenzopyran chromophores (10, 11), as well as polymer networks encompassing grafted ruthenium catalysts and undergoing the oscillating Belousov-Zhabotinksy (BZ) reaction (4, 5, 12). As we further show below, both of these systems provide distinct means of driving the gels to undergo directed motion, which could be exploited in creating millimeter-sized robots. We then focus on a composite system in which elastic, catalyst-bearing posts are embedded into thermoresponsive gels (3). This composite structure enables a cyclic interconversion of chemical and mechanical energy that regulates the temperature of the system. This design provides a blueprint for creating various homeostatic devices that help maintain and regulate the functionality of the system. Finally, we examine multiple chemoresponsive gels undergoing the BZ reaction. We show that this system exhibits a distinct form of chemotaxis, where individual gel pieces respond to self-generated chemical gradients, which drive them to move and cooperatively self-organize into a variety of dynamic structures (13). Notably, these findings can lead to design rules for creating assemblies of soft robots that effectively communicate (via the self-generated gradients) to perform a concerted function. Ultimately, these different design rules could be integrated to aid in the development of autonomously powered robotic systems, which interconvert chemical and mechanical energy and emit self-sustained signals and, thus, could perform tasks for finite periods of time without the need for external power sources.

THEORETICAL MODELING OF CHEMORESPONSIVE GELS The approach for simulating the behavior of stimuli-responsive gels is based on the theoretical model described in this section. The total free energy of the chemoresponsive gels is taken as the sum of the elastic free energy associated with the deformation of the gel, U el , and the energy of the polymer-solvent interaction, U F H . The latter term has the following Flory-Huggins form (14):    1. U F H = I3 (1 − φ) ln(1 − φ) + χ F H (φ, T )φ(1 − φ) + fint . The first two terms in Equation 1 describe the mixing free energy of the gel, and the last term accounts for the interactions between the polymer network and chemical species that are anchored 36

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to this network (e.g., the spirobenzopyran chromophores or ruthenium catalyst). Here, χ F H (φ, T ) is the polymer-solvent interaction parameter, which depends on the polymer volume fraction, φ, ˆ is an invariant of the left Cauchy-Green (Finger) strain tensor, and temperature, T (15); I3 = det B ˆB, which characterizes the volumetric changes in the deformed gel (16). The local volume fractions −1/2 of polymer in the deformed state, φ, and undeformed state, φ0 , are related as (7) φ = φ0 I3 . √ Note that the factor I3 appears in Equation 1 because the energy density is defined with respect to a unit volume in the undeformed state (7). The elastic energy contribution, U el , describes the rubber elasticity of the crosslinked network (14, 17) and is proportional to the crosslink density in the undeformed state, c0 :  c 0 v0  1/2 I1 − 3 − ln I3 , U el = 2. 2 ˆ (16). Both energy contributions (Equawhere v0 is the volume of a monomeric unit and I1 = trB tions 1 and 2) are dimensionless and are measured in the energy units of kT , where k is the Boltzmann constant and T is temperature. Equations 1 and 2 yield the following constitutive equation for the chemoresponsive gels (6, 7): φ ˆ σˆ = −P(φ, c int , T )Iˆ + c 0 v0 B, φ0

3.

where Iˆ is the unit tensor, σˆ is the dimensionless stress tensor measured in units of v0−1 kT , and c int denotes the dimensionless concentration of chemical species that defines the mechanical response of the gel. The isotropic pressure in Equation 3 is   4. P(φ, c int , T ) = − φ + ln(1 − φ) + χ (φ, T )φ 2 + c 0 v0 φ(2φ0 )−1 + πint . Here, the parameter χ (φ) is related to the Flory-Huggins interaction parameter χ F H in Equation 1 as χ (φ, T ) = χ F H (φ, T )−(1−φ)∂χ F H (φ, T )/∂φ. We chose χ (φ, T ) = χ0 (T )+χ1 φ, where χ0 (T ) = [δh − T δs ] /kT , with δh and δs being the respective changes in the enthalpy and entropy per monomeric unit of the gel (15). In Equation 4, πint describes the contribution to the isotropic pressure from the last term in Equation 1; we define the specific form of this term in the following sections for each type of chemoresponsive gel considered here. We describe the dynamics of a chemoresponsive gel within the framework of the two-fluid model (17–19). Both the respective polymer and solvent velocities, v( p) and v(s ) , contribute to the total velocity as v = φv( p) + (1 − φ)v(s ) , where φ and 1 − φ are the respective volume fractions of the polymer and solvent. The system is incompressible (i.e., ∇ · v = 0). We further assume there is no collective motion and that only the polymer-solvent interdiffusion contributes to the gel dynamics (6, 7, 20, 21); therefore, we neglect the total velocity of the polymer-solvent system and set v = 0. The volume fraction of the polymer, φ, obeys the following continuity equation: ∂φ = −∇ · (φv( p) ). ∂t

5.

The dynamics of the polymer network is assumed to be purely relaxational (18), so that the forces acting on the deformed gel are balanced by the frictional drag owing to the motion of the solvent. Therefore, the velocity of the polymer network, v( p) , can be written as (7)  ˆ 6. v( p) = 0 (φ φ0 )−3/2 (1 − φ) ∇ · σ. Here, 0 is the dimensionless kinetic coefficient (7), and the polymer-solvent friction coefficient was chosen as ξ (φ) ∼ φ 3/2 (18). Using this basic formulation, we can model the two distinct systems described below. www.annualreviews.org • Chemoresponsive Polymer Gels

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Modeling Spirobenzopyran-Functionalized Gels We focus on modeling photosensitive polymer gels containing spirobenzopyran chromophores, which are anchored onto the polymer network (10, 11). In the absence of light and in acidic aqueous solutions, the spirobenzopyran chromophores are primarily in the open ring form (the protonated merocyanine form, or McH) and are hydrophilic (10, 11, 22). Illumination with blue light causes the isomerization of these chromophores into the closed ring conformation [the spiro (or SP) form], which is hydrophobic (10, 11). The SP form is unstable in the absence of light, and hence, in the dark it undergoes spontaneous conversion back to the stable, hydrophilic McH form: kL McH ⇔ SP, kD

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where kL and k D are the reaction rate constants for the forward and backward reaction, respectively. Although spontaneous conversion back to the McH form is significantly slower than the photoinduced isomerization to the SP form, the conversion rate constants vary significantly, depending on the specific spirobenzopyran derivatives used (11, 22). The interconversion reaction in Equation 7 is described by the following reaction kinetics: ∂c SP = kL (I (r)) (1 − c SP ) − k D cSP , ∂t

8.

where c SP is the concentration of chromophores in the spiro form normalized by the total concentration of chromophores. The reaction rate constant kL is proportional to the light intensity at a given point, I (r). We consider only thin samples (with a thickness of approximately 60 μm); thus, we neglect attenuation of the light across the thickness of the sample and assume that kL depends only on the relative location of the illuminated region within the sample (23). The photostationary concentration of chromophores in the SP form is defined by the k D /kL ratio as c˜ SP = (1 + k D /kL (I ))−1 . For this chemoresponsive SP-functionalized gel, we specify the last term in Equation 1, which describes the energy of the polymer-solvent interaction in the following form (23): fint (φ, c SP ) = α(1 − φ)cSP .

9.

This term captures an experimentally observed (10, 11), photo-induced decrease in the hydration of the polymer network; here, we find c SP by solving Equation 8. Correspondingly, the contribution from this term to the isotropic pressure (Equation 4) is πint = −αc SP φ. We emphasize that the physical origin of the photo-induced volume change of the SPfunctionalized gels is distinctly different from the gel collapse that results from a direct lightinduced heating (24, 25). Experiments have shown that the temperature of the SP-functionalized gels remains constant during the illumination (10) and that the light-induced change in the conformation of the chromophores produces the decreases in the hydration of the gels. As we show below, it is the latter effect that can be exploited to achieve programmable reconfiguration and motion of gels.

Modeling Chemoresponsive Gels Undergoing the Belousov-Zhabotinsky Reaction Gels undergoing the BZ reaction, or so-called BZ gels, are unique because they can expand and contract autonomously (4, 5, 12, 26–40) (until the reagents driving the reaction are consumed). Discovered in the 1950s (41, 42), the original BZ reaction took place in a fluid. In the late 1990s, Yoshida et al. (12) fabricated the first BZ gel by covalently bonding a ruthenium catalyst into 38

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the polymer matrix. The BZ gels typically involve a poly(N-isopropylacrylamide) (PNIPAAm) polymer network, which is swollen in an aqueous solution of NaBrO3 , HNO3 , and malonic acid. The BZ reaction within this medium generates a periodic oxidation and reduction of the anchored ruthenium catalyst; the hydrating effect of the oxidized catalyst induces an expansion of the gel, which then contracts when the catalyst is in the reduced state (4, 5, 12, 26–40). To date, the PNIPAAm-based BZ gels are the most studied and well-understood self-oscillating gels; there have, however, been a number of recent advances in using different polymer networks to design these active materials. For example, recently Vaia et al. (43, 44) fabricated BZ gels using a post-functionalization technique, where Ru catalyst is printed into bioderived polypeptide gelatin gels (43) or polyacrylamide gels (44). Furthermore, while the PNIPAAm-based gels swell in response to the oxidation of the metal ion catalyst, there are examples of BZ gels that on the contrary deswell during the catalyst oxidation. Such behavior was observed by Konotop et al. (45, 46) in composite gels based on polyacrylamide (PAAm) and silica gel, and gels of poly(acrylamideco-acrylate), as well as in PAAm-based gels most recently fabricated by Nuzzo et al. (47). The observed changes in volume were attributed to the formation of additional reversible crosslinks when the polymer is in the oxidized state (45–47). While these newly synthesized BZ gel systems hold great promise, in this review, we focus on the more studied PNIPAAm-based BZ gels. The dynamics of the gels undergoing the BZ reaction is described by a modified version (6, 7) of the two-variable Oregonator model (48, 49) that explicitly accounts for the polymer volume fraction, φ. Hence, in addition to Equation 5, the governing equations (6, 7) are ∂v = −∇ · (v v( p) ) + εG(u, v, φ) ∂t

10.

and

∂u = −∇ · (u v(s ) ) − ∇ · j(u) + F (u, v, φ). 11. ∂t Here, v is the dimensionless concentration of the oxidized catalyst, and u is the dimensionless concentration of the activator. The velocity of the polymer network, v( p) , is found by solving Equation 6, and v(s ) = −φ/(1 − φ)v( p) . The dimensionless diffusive flux of the solvent j(u) through the gel in Equation 11 is calculated (7) as j(u) = −(1−φ)∇(u(1−φ)−1 ). The reactive terms G(u, v, φ) and F (u, v, φ) in Equations 10 and 11 are written as G(u, v, φ) = (1 − φ)2 u − (1 − φ)v

12.

and F (u, v, φ) = (1 − φ)2 u − u 2 − (1 − φ) [ f v + ]

u − q (1 − φ)2

. 13. u + q (1 − φ)2 The dimensionless parameters q, f, and ε in the above equations have the same meaning as in the original Oregonator model (48). The dimensionless variable  in Equation 13 accounts for the effect of light on the reaction kinetics (50). For the BZ occurring in a fluid, this two-variable, photosensitive Oregonator model has been used to successfully explain the observed experimental phenomena in several studies (51–54). Within this model,  specifically accounts for the additional production of bromide ions that are due to illumination by light of a particular wavelength, and  is assumed to be proportional to the light intensity (50). In our simulations, the above approach allowed us to reproduce (55, 56) the experimentally observed suppression of oscillations within BZ gels by visible light (32). Finally, the last term in Equation 1 is written as (6, 7) fint (φ, v) = −χ ∗ v(1 − φ),

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where χ ∗ > 0 is a fitting parameter in the model that describes the hydrating effect of the ruthenium catalyst and captures the coupling between the gel dynamics and BZ reaction. This term results in a contribution to the isotropic pressure in Equation 4 of the form πint = χ ∗ vφ. To model the behavior of the systems described above, we must now solve Equations 5, 6, and 8 in the case of the SP-containing gels and Equations 5, 6, 12, and 13 in the case of the BZ gels. Below, we describe a general approach that allows us to numerically integrate these governing equations and thereby simulate these different chemoresponsive gels.

COMPUTATIONAL MODELING OF CHEMORESPONSIVE GELS AND GEL-BASED COMPOSITES: THE GEL LATTICE SPRING MODEL

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Modeling Homogenous Gels in Three Dimensions The above equations can be integrated numerically using the recently developed gel lattice spring model, or gLSM (7–9), which combines a finite element approach for the spatial discretization of the elastodynamic equations and a finite difference approximation for the reaction and diffusion terms. Below, we briefly describe the 3D formulation of the gLSM; more details and a validation of the approach are found in Reference 9. (The 2D formulations of the gLSM are detailed in References 7 and 57.) Within the framework of the model, a 3D deformable gel is represented by a set of general linear hexahedral elements (58, 59) (see Figure 1a). Initially, the sample is undeformed and consists of (L x − 1) × (L y − 1) × (L z − 1) identical cubic elements (see inset in Figure 1a); here Li is the

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Figure 1 (a) Schematic of the 3D element. The entire sample consists of L x × L y × L z nodes. The set of indexes i = 1...L x , j = 1...L y , and k = 1...L z defines the respective position of the nodes in x, y, and z directions. Forces acting on node 1 (marked by the green circle) of element m = (i, j, k) are marked by the cyan and blue arrows. Namely, cyan arrows inside the element mark the springlike elastic forces acting between node 1 and the next-nearest and next-next-nearest neighbors within the same element m. The blue arrows outside the element mark contributions to nodal forces from the isotropic pressure within this element. (b) Schematic of the elastic filament (red ) introduced into the gel. Green circles mark nodes common to the gel and cilium. Arrows represent forces acting on the node j owing to its interaction with the gel (blue and cyan) and with the neighboring cilia nodes (black). The black circle marks a reactive tip, and the red plane marks the position of the layer above which the reaction takes place. 40

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number of nodes in the i-direction, i = x, y, z; the linear size of the elements in the undistorted state is set to = 1. Each element is labeled by the vector m = (i, j, k), and the element nodes are numbered by the index n = 1−8 (see Figure 1a) and characterized by the coordinates rn (m) (9). The volume fraction of the polymer, φ(m), is related to the volume of the element, V (m), as φ(m) = 3 φ0 /V (m). As noted above, the dynamics of the polymer network is assumed to be purely relaxational (18). Hence, the velocity of node n of the element m is proportional to the total force acting on this node (7, 9): d rn (m) = M n (m) [F1,n (m) + F2,n (m)] , 15. dt where M n (m) = 8 0 −3 (1− < φ(m) >n )(< φ(m) >n /.φ0 )−1/2 is the nodal mobility (7, 9). M n (m) depends on the volume fraction of polymer at this node, which is calculated by taking the average  value of φ(m ) over all elements m adjacent to it, as < φ(m) >n = 18 m φ(m ). The total force acting on node n of the element m consists of contributions from two types of forces (7, 9). The first contribution, F1,n (m), accounts for the neo-Hookean elasticity contribution to the energy of the system and is a combination of linear springlike forces (9): ⎞ ⎛  c 0 v0 ⎝  w(n , n)[rn (m ) − rn (m)] + [rn (m ) − rn (m)]⎠ . 16. F1,n (m) = 12   NN (m )



NNN (m )



Here, NN (m ) and NNN (m ) represent the respective summations over all the next-nearestneighbor and next-next-nearest-neighbor nodal pairs belonging to all the neighboring elements m adjacent to node n of the element m. Above, w(n , n) = 2 or w(n , n) = 1 if n and n belong to an internal face or to a boundary face (9). Note that there is no contribution from the interaction between nearest neighbors (9). The second contribution to the total force acting on node n can be written as (9) F2,n (m) =

1 P[φ(m )][n1 (m )S1 (m ) + n2 (m )S2 (m ) + n3 (m )S3 (m )], 4 

17.

m

where the summation is over all neighboring elements m that include node n of element m. The pressure within each element, P[φ(m )], is calculated according to Equation 4. The pressure within the element depends on the concentration of chemical species that modify the mechanical response of the polymer network, c int . The value of c int is represented either by the concentration of the chromophores in the spiro form, c SP , for the SP-functionalized gels or by the concentration of the oxidized catalyst, v, for the BZ gels. Hence, changes in the concentration of c int affect the pressure within the element and the corresponding force acting on the node (Equation 17) and thus give rise to the mechanical response (motion) of the gel. Finally, in Equation 17, the vector nl (m ) denotes the outward normal to the face l of element m , and Sl is the area of this face. The schematic in Figure 1a shows contributions from both types of forces that act on node n = 1 of element m and arise from within this element. The total force acting on node n of element m includes similar contributions from each of the neighboring elements containing this node. To couple the gel’s dynamics to specific reaction-diffusion processes occurring within a chemoresponsive gel, Equation 15 must be supplemented with the discretized reaction-diffusion equations for the concentrations of the relevant chemical reagents. A detailed derivation of these equations for the BZ gels is provided in Reference 9. The findings from computational studies using the gLSM approach are in good qualitative agreement with various experimental results. For example, for BZ gels, in agreement with experiments, simulations showed (9) the in-phase synchronization of the chemical and mechanical www.annualreviews.org • Chemoresponsive Polymer Gels

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oscillations for relatively small-sized samples (9, 29) and a decrease of the oscillation period with an increase in the concentration of malonic acid (9, 27). Additionally, the shape- and size-dependent pattern formation in BZ gels observed in these gLSM simulations showed qualitative agreement with experimental studies (60). Such simulations were also useful in explaining how gradients in crosslink density could drive long, thin BZ gels to both oscillate and bend, and thereby undergo concerted motion (61). Recently, a modified gLSM approach successfully captured the synchronization between circular BZ gel patches (57) that encompassed different concentrations of Ru catalyst. The gLSM model was further modified to capture the behavior of novel PAAm-based BZ gels in which the oxidation of the catalysts leads to a deswelling of the sample owing to the formation of additional reversible crosslinks (47). Hence, the gLSM (6–9) has proven to be a powerful approach for predicting the behavior of a variety of self-oscillating gels. Although the gLSM has been applied primarily to model BZ gels, this approach can be adapted in a straightforward manner to simulate other types of chemoresponsive gels. For example, as described below, we recently adapted the gLSM to model SP-functionalized gels, and our simulation results showed good agreement with the prior experiments (10). Additionally, we adapted the gLSM to model gels with embedded elastic filaments, which encompass catalyst-coated tips (3). This modification is described in the next section.

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Modeling Responsive Gels with Embedded Elastic Filaments We consider the heterogeneous samples shown in the inset in Figure 1b, where elastic filaments are embedded in the gel. We assume that the nodes comprising the filaments are attached to the gel nodes, i.e., the filaments can move only with the gel nodes (3). We define the filament energy as U f il =

 kb  ka (ri j − δ)2 + (cos(θi j k ) − cos(θ0 ))2 . 2 2

18.

The first term in Equation 18 represents the elastic energy of the harmonic bond between the two neighboring nodes (i,j) within a filament (see Figure 1b). Here, ri j (t) is the length of the bond between these nodes at time t, δ is the equilibrium bond length, kb is the spring constant, and the summation is taken over all the bonds within the filament. The second term in Equation 18 represents the cosine harmonic angle potential (62); here, θi j k is the angle between the two neighboring bonds sharing a common node (as marked in Figure 1b), θ0 is the equilibrium value of this angle, the constant ka controls the stiffness of the filament, and the summation is taken over all the pairs of bonds sharing a common filament node. In the studies below, we set θ0 = π . The schematic in Figure 1b illustrates all the forces acting on the gel/filament node marked j. Four types of forces act on such a node. The first two contributions (cyan and blue) represent forces arising from interactions with the neighboring gel elements, and the second two contributions (black) are due to interactions with the neighboring filament nodes. Forces marked F1 represent springlike elastic forces (see Equation 16, above) acting between this node and all the next-nearest and next-next-nearest neighboring nodes [for simplicity, only three of these contributions (cyan) are shown in Figure 1b]. Forces F2 (blue) represent the contributions from the isotropic pressure (see Equation 17) within all the gel elements containing this node; this force includes both osmotic and elastic contributions (9). The forces F1 and F2 represent the same respective forces as F1,n (m, i) and F2,n (m, i) in Figure 1a. (In Figure 1a, the index i marks different contributions to the same type of forces, and the index n marks the nodal number; in Figure 1b, for clarity, we omit these details.) Finally, the contributions from the forces that account for the filament energy (Equation 18) are marked by black arrows. The springlike elastic forces acting between the nearest neighboring 42

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nodes comprising the elastic filament that account for the contribution from the first term in f il Equation 18 are marked as Fb in Figure 1b. These forces act to restore the length of the bond in the filament to its equilibrium length, δ (3). The force marked Faf il accounts for the contribution from the second term in Equation 18. This force ensures that the angle between each of the two neighboring bonds sharing a common node remains close to its equilibrium value and controls the stiffness (62) of the filament. The dynamics of the polymer network is again assumed to be purely relaxational (18); hence, the velocity of node n of the element m is proportional to the total force acting on this node (7, 9) as   d rn (m) f il f il = M n (m) F1,n (m) + F2,n (m) + Fa,n (m) + Fb,n (m) . 19. dt The last two terms in Equation 19 are defined only for the nodes that are common for the filament and the gel (3); the dynamics of the gel node in the absence of filament is described by Equation 15 (9). In the example further below, the embedded filaments are immersed into a two-layer fluid, and a chemical reaction occurs only when the tips are above the red plane shown in Figure 1b (3). We update the coordinates of the tip, rti p , based on the position of the two upper nodes of the filament and require that (a) the angle between two neighboring bonds sharing a common node k (a node on the gel’s top surface) always remains equal to its equilibrium value, θ0 , and (b) the filament length extending out of the gel remains constant (3).

RESULTS OF SIMULATING CHEMORESPONSIVE GELS AND GEL-BASED COMPOSITES Photo-Induced Shape Changes and Directed Motion of SP-Functionalized Gels Using the formalism described above, we focus on SP-functionalized gels and uncover two remarkable phenomena. Not only can the shapes of these gels be reconfigured on demand through the application of light (see Figure 2), but also, by repeatedly sweeping the light over the material, the sample can be moved in a specified direction (see Figure 3) (23). The model for the SP-functionalized gels (23) accounts for the experimentally observed, photoinduced decrease in the hydration of the polymer network (10, 11). In the context of the model, this photoresponsive gel can attain a steady state if the following conditions are satisfied: (a) The concentration of the chromophores in the spiro form has reached the photostationary value, c˜ SP , (see Equation 8 above) and (b) the elastic stresses are balanced by the osmotic pressure, i.e., σˆ = 0 (see Equation 3). Hence, a steady-state solution for the polymer volume fraction, φs t (c˜ SP , T ), at a given temperature T and value of c˜ SP can be found as (23)      φs t φs t 1/3 = − φs t + ln(1 − φs t ) + [χ0 (T ) + χ1 φs t ] φs2t − α c˜SP φs t . − 20. c 0 v0 φ0 2φ0 The left- and right-hand sides of Equation 20 represent the elastic stresses for the isotropic 3D deformations and the osmotic pressure, respectively. We first focus on the effect of light on the volume phase transition in SP-containing gels and compare simulation results with the corresponding experimental data. The details of the simulation parameters and their correspondence to the experimental values can be found in Reference 23; where possible, simulation values were chosen based on the experimental data (10, 11). The dimensionless units of time and length in the simulations correspond to the following respective values: T 0 ≈ 5s and L0 = 30μm (23). www.annualreviews.org • Chemoresponsive Polymer Gels

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Figure 2 (a) Volume phase transition in SP-functionalized gels in the dark (black curve) and upon illumination (red and blue curves), with the values of c˜ SP given in the legend. Inset shows sample in the dark at T = 25◦ C. (b) Relative light-induced shrinking, δλ. (c) Sample under nonuniform illumination through photomasks with different apertures; illuminated regions are collapsed (dark blue). The dimensionless time for which the sample is illuminated is t = 5.6 × 105 . The color shows the volume fraction of polymer; in the color bar, φmin = 0.06 and φmax = 0.2. The size of the sample is 50 × 50 × 3 nodes, which corresponds to 1.47mm × 1.47mm × 0.06mm at T = 25◦ C.

Figure 2a shows the equilibrium degree of swelling, λeq (c˜ SP , T ), normalized by its value in the dark at T = 20◦ C. Specifically, we plot λ˜ eq = λeq /λeq (0, 20) as a function of temperature for two values of the photostationary concentration of chromophores in the SP form, c˜ SP (red and blue lines), and in the absence of light (black line). The solid lines in Figure 2a represent exact analytical solutions, where we first obtain φs t from Equation 20 and then calculate λeq = (φ0 /φs t )1/3 . The symbols in Figure 2a represent data points obtained from simulations of the phase transition of small samples (23). The excellent agreement between the analytical solution and simulation data indicates the accuracy of our simulation approach. Figure 2a shows that under illumination, λ˜ eq decreases and the phase transition is significantly smoother than that for the nonilluminated sample, similar to the behavior observed 44

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Figure 3 Repeated motion of the illuminated region over the tubular sample in the x-direction ( yellow arrow) results in the sample’s motion in the opposite direction (red arrow). The images (a–d ) correspond to the motion of the stripe of light during a single swipe; the collapsed regions indicate the influence of the light as the stripe passes over the sample. The image at the late time in section e corresponds to the position of the sample after multiple swipes and is taken at a time when the light is away from the gel. Here, we use the same color bar as given in Figure 2 to show the volume fraction of polymer. The size of the sample is 50 × 70 × 4 nodes, and the width of the stripe of light is w = 20. The rest of the parameters are the same as in Reference 23.

in experiments (10). Furthermore, the relative light-induced shrinking, δλ = (λeq (0, T ) − λeq (c˜ SP , T ))/λeq (0, 20), increases with increasing T until it reaches a peak value at the phase transition temperature (see Figure 2b). Further increases in T lead to a sharp decrease in δλ. This distinctive behavior of δλ is also in good agreement with the corresponding experimental findings (10). Such photo-induced shrinking allows us to dynamically reconfigure this material into a variety of shapes (23). For example, consider a sample that is flat in the absence of light (see inset in Figure 2a). By exposing this sample to different patterns of illumination, the material can be dynamically reconfigured into different shapes, as shown in Figure 2c, where the dark blue regions correspond to the illuminated areas. Here, the temperature is held at T = 25◦ C and kL (I )/k D = 10 within the illuminated regions. The nonuniform illumination can be readily obtained and altered by shining the light through different apertures in a photomask. The shapes of the SP-functional gels can be altered further by changing the light intensity or temperature (23). Because these light-induced shape changes are achieved by modifying solely the gel’s hydrophilicity and do not involve any local heating effects (10), this approach allows us to inscribe distinct features that are on the micrometer- to sub-micrometer-length scales. This resolution would not be feasible in photoresponsive gels where light causes the deswelling through local heating (24, 25) because the thermal diffusion is significantly faster (24, 63) than the collective diffusion of the polymer network. Beyond the ability to dynamically pattern the gels, our simulations showed that in the presence of spatially and temporally varying illumination, the photosensitive gel can be driven to undergo autonomous, directed motion, as illustrated in Figure 3. Starting with an initially nonilluminated www.annualreviews.org • Chemoresponsive Polymer Gels

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tubular sample, we repeatedly move a light source along the positive x direction, as marked by the yellow arrow in Figure 3a. Specifically, the sample is illuminated through a rectangular aperture in a photomask (the aperture of width w = 20 is indicated by the yellow rectangle in Figure 3a); both the light and aperture are rastered over the sample along the x direction with a speed of v L . With the scaling for the units of time and length provided above (see also Reference 23), this dimensionless velocity for the moving stripe of light corresponds to 0.6μ/s . When the aperture is located directly over a specific area, the light causes that region to shrink. With a shifting of the light along the sample, the region that is now in the dark swells, and the newly illuminated region shrinks. The continual swelling/shrinking of contiguous regions results in the movement of the sample (see Figure 3a–d) in the direction opposite to the motion of the light (23). Although the displacement is small during a single pass of the light over the sample (23), after multiple passes of the stripe of light, the gel undergoes a relatively large net displacement in the negative x direction (see Figure 3e). As discussed above, the gel moves owing to the interdiffusion of the solvent and polymer (6, 7, 20, 21). For example, as the light is moved from left to right, a region of contraction propagates along the sample that effectively pushes the solvent toward the right and thereby causes the polymer to move to the left (23). [In previous studies, we showed that the propagation of a traveling chemical wave in one direction resulted in the overall movement of the BZ gels in the opposite direction (7, 55, 56, 61, 64, 65); this motion was also caused by the polymer-solvent interdiffusion.] This light-induced motion is robust and is observed for a wide range of parameters; it can be controlled by altering the velocity of the light source, the reaction rate constant kL , or the temperature of the sample (23). The introduction of a temperature gradient provides a means of further controlling this autonomous movement (23). The results point to a robust method for controllably reconfiguring the morphology of polymer gels and driving the self-organization of multiple reconfigurable pieces into complex architectures.

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Self-Oscillations in Thermoresponsive Gels with Embedded Elastic Filaments Here, we focus on a distinctly different type of chemoresponsive system: a thermoresponsive gel layer with embedded elastic filaments whose ends are chemically reactive. The gel and filaments are immersed in the fluid that lies below the red plane (Figure 4a). The catalyst-coated tips of the filaments can extend into the upper fluid, which lies above this plane and contains reagents. When the catalyst comes in contact with these reagents, it generates an exothermic chemical reaction (3). The swelling and deswelling of the gel in response to the ensuing changes in temperature induce the reversible actuation of the catalyst-bearing filaments into and out of the reactant-containing nutrient layer and serve as an on/off switch for chemical reactions (3). By using various exothermic catalytic reactions, researchers demonstrated different examples of autonomous, self-sustained homeostatic systems that were able to regulate the overall temperature (3). Notably, the gel layer in this system is not chemoresponsive; hence, we model the gel using the gLSM approach with fint = 0 in Equation 1. We introduce a row of elastic filaments at given locations within the gel (see Figure 4a). Initially, the temperature is set at T0 = 22◦ C and all the filaments are straight (along the z direction), with the catalyst-coated tips reaching into the upper reactive layer. We set the filament equilibrium bond length, δ, in Equation 18 to be equal to the equilibrium degree of swelling of the gel at T 0 ; without the catalyst, this configuration remains at equilibrium at T 0 . We assume that the position of the plane separating the two fluids remains constant (3). We account for the exothermic reaction and the temperature increase when the tips are located above this plane and for the heat dissipation throughout the whole system (3). Finally, we assume that the temperature changes instantaneously and uniformly across the whole sample. 46

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When the filaments extend into the reactive layer, they turn on the exothermic chemical reaction (3). Owing to the heat generated by this reaction, the thermoresponsive gel shrinks, causing the filaments to bend and the tips to dip below the reagent layer. Once the tips are removed from the upper layer, the heating stops and the system cools down, causing the gel to expand. The gel’s expansion drives the filaments to straighten so that the tips cross the plane and move into the reagent-filled layer, once again giving rise to the exothermic reaction. Figure 4b shows that the phase trajectory of the z coordinate of the tips, zti p (T ), develops into a limit cycle and stays in this cycle indefinitely, corresponding to robust, self-sustained oscillations; here, P1 indicates the tips’ initial height at 22◦ C, and P2 is where the tips first cross the interface. This oscillatory behavior is also evident in Figure 4c, which shows the time evolution of both zti p (t) and T (t). The time evolution during a single oscillation cycle (see Figure 4d ) clearly shows that the temperature of the system decreases when the tips are below the plane separating the two fluids and increases when they are above this plane, resulting in a phase shift between the oscillations in temperature and tip position. Furthermore, at any temperature T within the oscillation cycle, zti p can attain one of the two possible values, so that at the same T, the tips are higher c ool ∗ ∗ when the gel shrinks and lower when it swells, as marked by ztiheat p (T ) > zti p (T ) in Figure 4d. This feature is also evident in Figure 4b, where the upper/lower portion of the limit cycle corresponds to the tips being located above/below the reactive interface marked by the green line. www.annualreviews.org • Chemoresponsive Polymer Gels

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This bistability, as well as the negative feedback provided by the localized reaction, results in the oscillations seen in Figure 4. Bistability has been shown to play a key role in various self-oscillating gels (66–69). The dynamic behavior shown in Figure 4b–d is in good qualitative agreement with the experimental findings in Reference 3. Moreover, our model predicted that the oscillation amplitude and period could be controlled by varying the position of the fluid-fluid interface (red plane) and the heating rate; these trends were confirmed by corresponding experimental studies (3). Overall, the system displays a remarkable form of chemo-mechano-chemical transduction where chemical and mechanical energy are interconverted in a continuous, autonomous manner. Notably, this cyclic transduction of energy will stop when the reagents in the upper solution are consumed, but new reagents could be added to maintain the operation and functionality of the system. Below, we describe a gel system that undergoes a different form of chemomechanical transduction.

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Communication Between Gels Undergoing the Belousov-Zhabotinksy Reaction The BZ gels are unique materials because they can transduce chemical energy into mechanical oscillations in the absence of external stimuli (4, 5, 12, 26–40). In other words, the periodic chemical oscillations arising from the BZ reaction fuel the mechanical oscillations of these active gels. Hence, in a solution of BZ reagents, the gels swell and deswell autonomously, beating like a heart. Millimeter-sized pieces of the BZ gels can oscillate like this for hours (39, 40), and the system can be resuscitated by replenishing the solution with reagents that were consumed in the reaction (60). In this section, we focus on multiple chemoresponsive BZ gels immersed in solution and show that the individual gels can communicate by sending and receiving a chemical signal, which gives rise to large-scale collective behavior. To model this complex system, Dayal et al. (64) recently combined the gLSM technique with a finite-difference approach that accounts for the diffusive exchange of the activator for the BZ reaction, u, between the gels and outer solution, as well as the reaction-diffusion processes occurring within the solution. Using this approach, we can capture the dynamic interplay between the gels’ motion and local variations in the concentration of u. In the examples below, the gels can slide along the bottom surface of the simulation box. No-flux boundary conditions are applied at the top and bottom walls of this box, and u = 0 is imposed at the side walls. We first consider four gels that are arranged in a square array, as shown in Figure 5a. The system is observed to be bistable, with the gels exhibiting two distinct modes of behavior. In both cases, at relatively early times, the gels move closer together and toward the center of the simulation box (13, 65). At late times, however, the system forms a rotating pinwheel in one scenario (left panel in Figure 5), and in the other, the gels continue to oscillate as a nonrotating assembly (right panel in Figure 5). This complex dynamic behavior can be explained as follows. Multiple cycles of oscillations produce a buildup of u in the solution; given the initial symmetric arrangement of the cubes and the u = 0 constraint at the box edges, the highest accumulation of u occurs in the region between the four cubes (shown in red in Figure 5b). Importantly, the oscillation frequency, ω, of such BZ gels increases with the concentration of u in the outer solution (64). Because the concentration of u is higher for the inner surfaces of the cubes than for the outer surfaces, the inner surfaces have a higher intrinsic frequency1 (66). In a system containing multiple oscillators, the one with the highest frequency determines the ultimate direction of wave propagation (70–72). Hence, in

1 By intrinsic frequency of the small region of the gel (with a given value of u in the outer fluid), we are referring to the frequency of the oscillations that an isolated gel of the same size would exhibit if it were placed in a fluid with the same value of u (see Reference 64).

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these BZ cubes, the waves propagate from the inner to the outer surfaces (13, 73). These traveling waves effectively push the solvent to the outer portions of the box, and owing to the interdiffusion of the polymer and solvent, all the gel cubes are driven to the inner, central region (13, 73). In other words, the gel undergoes a net displacement that is opposite in direction to the propagation of the traveling waves (55, 56), and consequently, the cubes migrate toward the center of the box. In the cases where the system forms a pinwheel, the chemomechanical oscillations in the gels become phase locked at later times (65) (Figure 5c,d) so that the chemical wave propagates from one gel to another in a specific circular direction (clockwise in Figure 5d ). As the gels come closer to each other, they ultimately rotate in the direction opposite to that of the chemical wave (65) (shown in Figure 5d ). Because the system is bistable (65), a small perturbation in the initial conditions (i.e., a different initial random seed in the simulations) can drive the system in Figure 5a to form a nonrotating self-assembly (Figure 5e). However, by modifying the boundary conditions, one can controllably promote the formation of either the nonrotating assembly or a pinwheel and can even regulate the direction of the pinwheel’s rotation (65). Notably, we showed that the frequency of oscillations is higher for the case of pinwheel formation than for the case of the nonrotating assembly of four oscillating gels placed in the same simulation box (65). (With www.annualreviews.org • Chemoresponsive Polymer Gels

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the scaling for the dimensionless units of time provided in Reference 65, these frequencies can be estimated as 0.06 Hz and 0.03 Hz for the respective pinwheel and non-pinwheel assembly for the examples considered in that reference.) By placing eight identical BZ gels in a row (Figure 6a), we can highlight another intriguing form of dynamic behavior. As reaction progresses, the BZ gels move toward each other under the influence of the self-generated distributions of u and thus arrange themselves first in smaller clusters of two or three gels (see Figure 6b,c) and then finally in a single cluster around the central region of the box (Figure 6d,e). The cluster continues to oscillate with waves propagating 50

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throughout the array of gels (Figure 6d,e) (72). This structural rearrangement broadly resembles the auto-chemotactic behavior of the cellular slime mold, which aggregates into large-scale structures in response to chemical signals that they themselves emit. The colony of 16 BZ gels shown in Figure 6f also exhibits a form of auto-chemotaxis, as the gels migrate toward each other at early times. Given the initial square-like arrangement of these pieces, we might anticipate from the findings in Figure 5 that these gels would form clusters of BZ pinwheels at late times. The latter behavior is clearly seen in Figure 6g. However, because of the bistability in pinwheel formation noted above, we do find that out of eight independent simulations for this system, only one case displayed four pinwheels, whereas either two or three pinwheels were formed in the rest of the cases. Finally, we note that the BZ reaction is photosensitive (see section on Modeling Chemoresponsive Gels). Simulations have predicted that light can provide an effective means of controlling the motion of these gels (13, 55, 56, 64). Notably, some of these predictions (56) were recently confirmed experimentally (74). Hence, we anticipate that the motion of such active pinwheels could be tailored by light. Given the resemblance of the pinwheels to mechanical gears, these simulations can provide guidelines for creating simple, self-assembled tools that can perform autonomous work and could be regulated with light.

CONCLUSIONS Through computational modeling, we uncovered a range of biomimetic behavior in chemoresponsive gels. Specifically, we demonstrated that photosensitive SP-functionalized gels can undergo both dynamic reconfiguration and directed motion. The ability to alter the gel’s shape on demand can have significant consequences in manufacturing because the same material can be refashioned into multiple forms and thus be used (or reused) for various purposes, alleviating the need to build different parts for different functions. We also demonstrated a remarkable form of chemo-mechano-chemical transduction in a thermoresponsive gel that contained catalyst-coated filaments. In this system, chemical and mechanical energy are interconverted in a continuous, autonomous manner, resulting in another biomimetic function: homeostatic behavior. These findings can guide the development of smart switches for regulating chemical reactions in microfluidic devices or maintaining the temperature in these small-scale chambers. Finally, we showed that assemblies of the self-oscillating gels undergoing the BZ reaction are capable of auto-chemotaxis, moving in response to self-generated chemical gradients. We isolated cases where the gels self-organize into rotating assemblies. Hence, these materials could form simple self-propelled machines, such as gears, that perform autonomous work. Overall, the studies indicate that these computational models can facilitate the development of the next generation of smart devices or soft robotic systems that can sense and dynamically respond to environmental cues and thereby exhibit a range of biomimetic functionality.

DISCLOSURE STATEMENT The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS A.C.B. gratefully acknowledges the ARO (for partial support of O.K. for developing the codes and performing computer simulations and data analysis on spirobenzopyran-functionalized gels and www.annualreviews.org • Chemoresponsive Polymer Gels

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gel-based composites with elastic filaments), the National Science Foundation (for partial support of P.D. for developing the code for communicating BZ gels), and the AFOSR (for partial support of D.D. for performing the computer simulations and data analysis on communicating BZ gels).

LITERATURE CITED

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