Biomimetic growth in polymer gels.

By modeling gels growing in confined environments, we uncover a biomimetic feedback mechanism between the evolving gel and confining walls that enables significant control over the properties of the grown gel. Our new model describes the monomer adsorption, polymerization and cross-linking involved in forming new networks and the resultant morphology and mechanical behavior of the grown gel. Confined between two hard walls, a thin, flat "parent" gel undergoes buckling; removal of the walls returns the gel to the flat structure. Polymerization and cross-linking in the confined parent generates the next stage of growth, forming a random copolymer network (RCN). When the walls are removed, the RCN remains in the buckled state, simultaneously "locking in" these patterns and increasing the Young's modulus by two orders of magnitude. Confinement of thicker gels between harder or softer 3D walls leads to controllable mechanical heterogeneities, where the Young's modulus between specific domains can differ by three orders of magnitude. These systems effectively replicate the feedback between mechanics and morphology in biological growth, where mechanical forces guide the structure formation throughout stages of growth. The findings provide new guidelines for shaping "growing materials" and introducing new approaches to matching form and function in synthetic systems.

Dynamic behavior of chemically tunable mechano-responsive hydrogels.

Using theory and simulation, we model the mechanical behavior of gels that encompass loops and dangling chain ends. If the loops remain folded and dangling ends are chemically inert, then these topological features just serve as defects. If, however, the loops unfold to expose the hidden ("cryptic") binding sites and the ends of the dangling chains are reactive, these moieties can form bonds that improve the gel's mechanical properties. For gels with a lower critical solubility temperature (LCST), we systematically switch on the possible unfolding and binding events. To quantify the resulting effects, we derive equations for the gel's equilibrium and dynamic elastic moduli. We also use a finite element approach to simulate the gel's response to deformation and validate the analytic calculations. Herein, we show that the equilibrium moduli are highly sensitive to the presence of unfolding and binding transitions. The dynamical moduli are sensitive not only to these structural changes, but also to the frequency of deformation. For example, when reactive ends bind to exposed cryptic sites at T = 29 °C and relatively high frequency, the storage shear modulus is 119% greater than the corresponding equilibrium value, while the storage Young's modulus is 109% greater than at equilibrium. These findings provide guidelines for tuning the chemical reactivity of loops and dangling ends and the frequency of deformation to tailor the mechano-responsive behavior of polymer networks.

Controllable growth of interpenetrating or random copolymer networks.

Interpenetrating and random copolymer networks are vital in a number of industrial applications, including the fabrication of automotive parts, damping materials, and tissue engineering scaffolds. We develop a theoretical model for a process that enables the controlled growth of interpenetrating network (IPNs), or a random copolymer network (RCN) of specified size and mechanical properties. In this process, a primary gel "seed" is immersed into a solution containing the secondary monomer and crosslinkers. After the latter species are absorbed into the primary network, the absorbed monomers are polymerized to form the secondary polymer chains, which then can undergo further crosslinking to form an IPN, or undergo inter-chain exchange with the existing network to form a RCN. The swelling and elastic properties of the IPN and RCN networks can be tailored by modifying the monomer and crosslinker concentrations in the surrounding solution, or by tuning the enthalpic interactions between the primary polymer, secondary monomer and solvent through a proper choice of chemistry. This process can be used repeatedly to fabricate gels with a range of mechanical properties from stiff, rigid materials to soft, flexible networks, allowing the method to meet the materials requirements of a variety of applications.

Effects of an Imposed Flow on Chemical Oscillations Generated by Enzymatic Reactions.

Using analytical and computational models, we determine how externally imposed flows affect chemical oscillations that are generated by two enzyme-coated patches within a fluid-filled millimeter sized channel. The fluid flow affects the advective contribution to the flux of chemicals in the channel and, thereby, modifies the chemical reactions. Here, we show that changes in the flow velocity permit control over the chemical oscillations by broadening the range of parameters that give rise to oscillatory behavior, increasing the frequency of oscillations, or suppressing the oscillations all together. Notably, simply accelerating the flow along the channel transforms time-independent distributions of reagents into pronounced chemical oscillations. These findings can facilitate the development of artificial biochemical networks that act as chemical clocks.

Harnessing biomimetic cryptic bonds to form self-reinforcing gels.

Cryptic sites, which lay hidden in folded biomolecules, become exposed by applied force and form new bonds that reinforce the biomaterial. While these binding interactions effectively inhibit mechanical deformation, there are few synthetic materials that harness mechano-responsive cryptic sites to forestall damage. Here, we develop a computational model to design polymer gels encompassing cryptic sites and a lower critical solution temperature (LCST). LCST gels swell with a decrease in temperature, thereby generating internal stresses within the sample. The gels also encompass loops held together by the cryptic sites, as well as dangling chains with chemically reactive ends. A decrease in temperature or an applied force causes the loops to unfold and expose the cryptic sites, which then bind to the dangling chains. We show that these binding interactions act as "struts" that reinforce the network, as indicated by a significant decrease in the volume of the gel (from 44% to 80%) and shifts in the volume phase transition temperature. Once the temperature is increased or the deformation is removed, the latter "cryptic bonds" are broken, allowing the loops to refold and the gel to return to its original state. These findings provide guidelines for designing polymer networks with reversible, mechano-responsive bonds, which allow gels to undergo a self-stiffening behavior in response to a temperature-induced internal stress or external force. When applied as a coating, these gels can prevent the underlying materials from undergoing damage and thus, extend the lifetime of the system.

Opto-chemo-mechanical transduction in photoresponsive gels elicits switchable self-trapped beams with remote interactions.

Next-generation photonics envisions circuitry-free, rapidly reconfigurable systems powered by solitonic beams of self-trapped light and their particlelike interactions. Progress, however, has been limited by the need for reversibly responsive materials that host such nonlinear optical waves. We find that repeatedly switchable self-trapped visible laser beams, which exhibit strong pairwise interactions, can be generated in a photoresponsive hydrogel. Through comprehensive experiments and simulations, we show that the unique nonlinear conditions arise when photoisomerization of spiropyran substituents in pH-responsive poly(acrylamide-co-acrylic acid) hydrogel transduces optical energy into mechanical deformation of the 3D cross-linked hydrogel matrix. A Gaussian beam self-traps when localized isomerization-induced contraction of the hydrogel and expulsion of water generates a transient waveguide, which entraps the optical field and suppresses divergence. The waveguide is erased and reformed within seconds when the optical field is sequentially removed and reintroduced, allowing the self-trapped beam to be rapidly and repeatedly switched on and off at remarkably low powers in the milliwatt regime. Furthermore, this opto-chemo-mechanical transduction of energy mediated by the 3D cross-linked hydrogel network facilitates pairwise interactions between self-trapped beams both in the short range where there is significant overlap of their optical fields, and even in the long range--over separation distances of up to 10 times the beam width--where such overlap is negligible.

Tuning the synchronization of a network of weakly coupled self-oscillating gels via capacitors.

We consider a network of coupled oscillating units, where each unit comprises a self-oscillating polymer gel undergoing the Belousov-Zhabotinsky (BZ) reaction and an overlaying piezoelectric (PZ) cantilever. Through chemo-mechano-electrical coupling, the oscillations of the networked BZ-PZ units achieve in-phase or anti-phase synchronization, enabling, for example, the storage of information within the system. Herein, we develop numerical and computational models to show that the introduction of capacitors into the BZ-PZ system enhances the dynamical behavior of the oscillating network by yielding additional stable synchronization modes. We specifically show that the capacitors lead to a redistribution of charge in the system and alteration of the force that the PZ cantilevers apply to the underlying gel. Hence, the capacitors modify the strength of the coupling between the oscillators in the network. We utilize a linear stability analysis to determine the phase behavior of BZ-PZ networks encompassing different capacitances, force polarities, and number of units and then verify our findings with numerical simulations. Thus, through analytical calculations and numerical simulations, we determine the impact of the capacitors on the existence of the synchronization modes, their stability, and the rate of synchronization within these complex dynamical systems. The findings from our study can be used to design robotic materials that harness the materials' intrinsic, responsive properties to perform such functions as sensing, actuation, and information storage.

"Patterning with loops" to dynamically reconfigure polymer gels.

The structural and mechanical properties of gels can be controlled by promoting the unfolding (and refolding) of loops (stored lengths) embedded within the networks. As a loop unfolds, the released chain length can increase the extensibility and reconfigurability of the gel. Here, we develop a theoretical model that couples the elasticity of the gel to the dynamic transitions occurring in loops that lie between the crosslinks. Using this model, we show that a thermally-induced swelling of the gel generates an internal strain, which unfolds the loops and thereby further increases the degree of gel swelling. We exploit this cooperative behavior to reconfigure the gel by patterning the location of the loops within the sample. Through this approach, we convert flat, two-dimensional layers into three-dimensional forms and introduce architectural features into uniform 3D slabs. At a fixed temperature, an applied force produces analogous structural transformations. The shape-changes are reversible: the systems return to their original structure when the temperature is reset or the force is removed. The findings provide guidelines for creating materials that interconvert thermal, chemical and mechanical energy to perform work. Such systems could be useful for designing soft robotic materials that convert environmental stimuli into useful functionality.

Fibers on the surface of thermo-responsive gels induce 3D shape changes.

Using computational modeling, we pattern the outer layers of thin, thermo-responsive gels with elastic fibers and thereby design stable structures that could not be achieved with the fibers in the interior of the network. To perform these studies, we simulate the fiber-decorated gels with the gel lattice spring model (gLSM) and determine how the temperature, arrangement and number of the fibers control the bending and curvature of the thin films. We establish arrangements of the fiber on square- and rectangular-shaped gels that yield reversible, pronounced contraction and extension of the layer with changes in temperature. Hence, these composites provide useful actuation. Finally, we cut the gel into a cross-shaped sample to show that the modes of actuation and functionality of the material can be significantly expanded by tailoring not only the fiber placement, but also the initial geometry of the gel.

Convective Self-Sustained Motion in Mixtures of Chemically Active and Passive Particles.

We develop a model to describe the behavior of a system of active and passive particles in solution that can undergo spontaneous self-organization and self-sustained motion. The active particles are uniformly coated with a catalyst that decomposes the reagent in the surrounding fluid. The resulting variations in the fluid density give rise to a convective flow around the active particles. The generated fluid flow, in turn, drives the self-organization of both the active and passive particles into clusters that undergo self-sustained propulsion along the bottom wall of a microchamber. This propulsion continues until the reagents in the solution are consumed. Depending on the number of active and passive particles and the structure of the self-organized cluster, these assemblies can translate, spin, or remain stationary. We also illustrate a scenario in which the geometry of the container is harnessed to direct the motion of a self-organized, self-propelled cluster. The findings provide guidelines for creating autonomously moving active particles, or chemical "motors" that can transport passive cargo in microfluidic devices.

Designing self-powered materials systems that perform pattern recognition.

Inspired by the advances in both materials and computer science, we describe efforts to design "materials that compute" where the material and the computer are the same entity. Using theory and simulation, we devise systems that integrate the behavior of self-oscillating gels and fundamental concepts from oscillator-based computing. We specifically focus on gels that undergo the Belousov-Zhabotinsky (BZ) reaction and thus exhibit self-sustained oscillations. In our models, we couple the BZ gel to an overlaying piezoelectric (PZ) cantilever to create a BZ-PZ unit. By connecting the BZ-PZ units by electrical wires, we design networks that autonomously transduce chemical, mechanical, and electrical energy to propagate a signal across the device and achieve synchronization of the oscillating gels in the network. This synchronization allows the device to perform pattern recognition in a self-organized manner, without the need for external electrical power sources. In particular, we imposed a collection of input patterns onto different BZ-PZ networks, where each network encompassed a distinct stored pattern. The network encompassing the stored pattern closest to the input pattern exhibited the fastest convergence time to the stable synchronization behavior, and could be identified as the "winner". In this way, the networks of coupled BZ-PZ oscillators achieved pattern recognition. We demonstrated that the convergence time to the stable synchronization provides a robust measure of the degree of match between the input and stored patterns. Through these studies, we laid out fundamental and experimentally realizable design rules for creating "materials that compute".

Using Torsion for Controllable Reconfiguration of Binary Nanoparticle Networks.

Mechanical deformation can potentially provide an effective means of controlling the nanoscale morphology in hybrid materials. The challenge, however, is establishing optimal couplings of the deformation and mechano-responsive components in the material to achieve nanoscopic structural reorganization without causing catastrophic damage. Through computational modeling, we investigate how torsion can be utilized to induce controllable structural changes in networks formed from binary mixtures (A and B) of polymer-grafted nanoparticles (PGNs). The nanoparticles' rigid cores are decorated with a corona of grafted polymers, which contain reactive functional groups at the chain ends. With the overlap of the neighboring coronas, these reactive groups form labile bonds, which can reform after breakage. The labile bond energy between similar PGNs (UAA, UBB) is different than the energy between dissimilar species (UAB). By tailoring the relative values of these bond energies and the boundary conditions acting on the system, the application of a torsional deformation can result in a controllable reconfiguration of the network, leading to intertwining helical structures, or homogeneously mixed nanocomposites. In effect, our mechano-mutable system resembles a "Rubik's cube" material, whose nanostructure, and hence global properties, can be tailored by mechanically twisting the sample.

Pattern recognition with "materials that compute".

Driven by advances in materials and computer science, researchers are attempting to design systems where the computer and material are one and the same entity. Using theoretical and computational modeling, we design a hybrid material system that can autonomously transduce chemical, mechanical, and electrical energy to perform a computational task in a self-organized manner, without the need for external electrical power sources. Each unit in this system integrates a self-oscillating gel, which undergoes the Belousov-Zhabotinsky (BZ) reaction, with an overlaying piezoelectric (PZ) cantilever. The chemomechanical oscillations of the BZ gels deflect the PZ layer, which consequently generates a voltage across the material. When these BZ-PZ units are connected in series by electrical wires, the oscillations of these units become synchronized across the network, where the mode of synchronization depends on the polarity of the PZ. We show that the network of coupled, synchronizing BZ-PZ oscillators can perform pattern recognition. The "stored" patterns are set of polarities of the individual BZ-PZ units, and the "input" patterns are coded through the initial phase of the oscillations imposed on these units. The results of the modeling show that the input pattern closest to the stored pattern exhibits the fastest convergence time to stable synchronization behavior. In this way, networks of coupled BZ-PZ oscillators achieve pattern recognition. Further, we show that the convergence time to stable synchronization provides a robust measure of the degree of match between the input and stored patterns. Through these studies, we establish experimentally realizable design rules for creating "materials that compute."

Designing Synthetic Microcapsules That Undergo Biomimetic Communication and Autonomous Motion.

Inspired by the collective behavior of slime molds and amoebas, we designed synthetic cell-like objects that move and self-organize in response to self-generated chemical gradients, thereby exhibiting autochemotaxis. Using computational modeling, we specifically focused on microcapsules that encompass a permeable shell and are localized on an adhesive surface in solution. Lacking any internal machinery, these spherical, fluid-filled shells might resemble the earliest protocells. Our microcapsules do, however, encase particles that can diffuse through the outer shell and into the surrounding fluid. The released particles play two important, physically realizable roles: (1) they affect the permeability of neighboring capsules and (2) they generate adhesion gradients on the underlying surface. Due to feedback mechanisms provided by the released particles, the self-generated adhesion gradients, and hydrodynamic interactions, the capsules undergo collective, self-sustained motion and even exhibit antlike tracking behavior. With the introduction of a chemically patterned stripe on the surface, a triad of capsules can be driven to pick up four-capsule cargo, transport this cargo, and drop off this payload at a designated site. We also modeled a system where the released particles give rise to a particular cycle of negative feedback loops (mimicking the "repressilator" network), which regulates the production of chemicals within the capsules and hence their release into the solution. By altering the system parameters, three capsules could be controllably driven to self-organize into a stable, close-packed triad that would either translate as a group or remain stationary. Moreover, the stationary triads could be made to switch off after assembly and thus produce minimal quantities of chemicals. Taken together, our models allow us to design a rich variety of self-propelled structures that achieve complex, cooperative behavior through fundamental physicochemical phenomena. The studies can also shed light on factors that enable individual protocells to communicate and self-assemble into larger communities.

Achieving synchronization with active hybrid materials: Coupling self-oscillating gels and piezoelectric films.

Lightweight, deformable materials that can sense and respond to human touch and motion can be the basis of future wearable computers, where the material itself will be capable of performing computations. To facilitate the creation of "materials that compute", we draw from two emerging modalities for computation: chemical computing, which relies on reaction-diffusion mechanisms to perform operations, and oscillatory computing, which performs pattern recognition through synchronization of coupled oscillators. Chemical computing systems, however, suffer from the fact that the reacting species are coupled only locally; the coupling is limited by diffusion as the chemical waves propagate throughout the system. Additionally, oscillatory computing systems have not utilized a potentially wearable material. To address both these limitations, we develop the first model for coupling self-oscillating polymer gels to a piezoelectric (PZ) micro-electro-mechanical system (MEMS). The resulting transduction between chemo-mechanical and electrical energy creates signals that can be propagated quickly over long distances and thus, permits remote, non-diffusively coupled oscillators to communicate and synchronize. Moreover, the oscillators can be organized into arbitrary topologies because the electrical connections lift the limitations of diffusive coupling. Using our model, we predict the synchronization behavior that can be used for computational tasks, ultimately enabling "materials that compute".

Modeling the entrainment of self-oscillating gels to periodic mechanical deformation.

Polymer gels undergoing the oscillatory Belousov-Zhabotinsky (BZ) reaction are one of the few synthetic materials that exhibit biomimetic mechano-chemical transduction, converting mechanical input into chemical energy. Here, we consider self-oscillating BZ gels that are subjected to periodic mechanical forcing, and model the entrainment of the oscillatory gel dynamics to this external stimulus. The gel size is assumed to be sufficiently small that the chemo-mechanical oscillations are spatially uniform. The behavior of the system is captured by equations describing the kinetics of the oscillatory BZ reaction in the gel coupled to equations for the variations in gel size due to the inherent reaction and imposed force. We employ the phase dynamics approach for analyzing the entrainment of the BZ gel to force- and strain-controlled compressive deformations. The phase response curves are obtained using Malkin's method, and time-averaging is applied to extract the slow phase dynamics caused by the periodic forcing. We demonstrate that the entrainment of the self-oscillating BZ gel is sensitive to the chemo-mechanical coupling in gel, the mode of deformation, and the level of static compression. Kuramoto's model of phase oscillators is shown to be applicable if the external forcing is purely harmonic.

Self-assembly of microcapsules regulated via the repressilator signaling network.

One of the intriguing challenges in designing active matter is devising systems that not only self-organize, but also exhibit self-regulation. Inspired by biological regulatory networks, we design a collection of self-organizing, self-regulating microcapsules that move in response to self-generated chemical signals. Three microcapsules act as localized sources of distinct chemicals that diffuse through surrounding fluid. Production rates are modulated by the "repressilator" regulatory network motif: each chemical species represses the production of the next in a cycle. Depending on the maximum production rates and capsule separation distances, we show that immobile capsules either exhibit steady or oscillatory chemical production. We then consider movement of the microcapsules over the substrate, induced by gradients in surface energy due to adsorbed chemicals. We numerically simulate this advection-diffusion-reaction system with solid-fluid interactions by combining lattice Boltzmann, immersed boundary and finite difference methods, and thereby, construct systems where the three capsules spontaneously assemble to form a close-packed triad. Chemical oscillations are shown to be critical to this assembly. By adjusting parameters, the triad can either remain stationary or translate as a cohesive group. Stationary triads can also be made to "turn off", producing chemicals at minimal rates after assembly. These findings provide design rules for creating synthetic material systems that encompass biomimetic feedback loops, which enable dynamic collective behavior.

Designing mechanomutable composites: reconfiguring the structure of nanoparticle networks through mechanical deformation.

Via a new dynamic, three-dimensional computer model, we simulate the tensile deformation of polymer-grafted nanoparticles that are cross-linked by labile bonds, which can readily rupture and reform. For a range of relatively high strains, the network does not fail, but rather restructures into a stable, ordered structure. Within this network, the reshuffling of the labile bonds enables the formation of this new morphology. The results provide guidelines for designing mechano-responsive hybrid materials that undergo controllable structural transitions through the application of applied forces.

Modeling polymer grafted nanoparticle networks reinforced by high-strength chains.

Using a multi-scale computational approach, we determine the effect of introducing a small fraction of high-strength connections between cross-linked nanoparticles. The nanoparticles' rigid cores are decorated with a corona of grafted polymers, which contain reactive functional groups at the chain ends. With the overlap of neighboring coronas, these reactive groups can form weak labile bonds, which can reform after breakage, or stronger bonds, which rupture irreversibly and thus, the nanoparticles are interconnected by dual cross-links. We show that this network can be reinforced by the addition of high-strength connections, which model polymer arms bound together by bonds with energies on the order of 100 kBT. We demonstrate that in the course of these simulations, these high-strength connections can be treated as unbreakable chains. By subjecting networks with a random distribution of the unbreakable chains to tensile deformation at a constant strain-rate, we determine the distribution of strain at break and toughness. With even a small amount of unbreakable chains, the nanoparticle networks can survive strains far greater than the networks without these connections. Furthermore, networks containing the high-strength connections tend to form long, thin threads, which enable a larger strain at break. The findings provide guidelines for creating polymer grafted nanoparticles networks that could show remarkable strength and ductility.

Chemo-responsive, self-oscillating gels that undergo biomimetic communication.

Species ranging from single-cell organisms to social insects can undergo auto-chemotaxis, where the entities move towards a chemo-attractant that they themselves emit. Polymer gels undergoing the self-oscillating Belousov-Zhabotinsky (BZ) reaction exhibit autonomous, periodic pulsations, which produce chemical species collectively referred to as the activator. The diffusion of this activator into the surrounding solution affects the dynamic behavior of neighboring BZ gels and hence, the BZ gels not only emit, but also respond to self-generated chemical gradients. This review describes recent experimental and computational studies that reveal how this biomimetic behavior effectively allows neighboring BZ gels to undergo cooperative, self-propelled motion. These distinctive properties of the BZ gels provide a route for creating reconfigurable materials that autonomously communicate with neighboring units and thereby actively participate in constructing the desired structures.