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.

Strain-Stiffening Hydrogels with Dynamic, Secondary Cross-Linking.

Hydrogels are water-swollen, typically soft networks useful as biomaterials and in other fields of biotechnology. Hydrogel networks capable of sensing and responding to external perturbations, such as light, temperature, pH, or force, are useful across a wide range of applications requiring on-demand cross-linking or dynamic changes. Thus far, although mechanophores have been described as strain-sensitive reactive groups, embedding this type of force-responsiveness into hydrogels is unproven. Here, we synthesized multifunctional polymers that combine a hydrophilic zwitterion with permanently cross-linking alkenes, and dynamically cross-linking disulfides. From these polymers, we created hydrogels that contain irreversible and strong thiol-ene cross-links and reversible disulfide cross-links, and they stiffened in response to strain, increasing hundreds of kPa in modulus under compression. We examined variations in polymer composition and used a constitutive model to determine how to balance the number of thiol-ene vs disulfide cross-links to create maximally force-responsive networks. These strain-stiffening hydrogels represent potential biomaterials that benefit from the mechanoresponsive behavior needed for emerging applications in areas such as tissue engineering.

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.

Buckling-induced interaction between circular inclusions in an infinite thin plate.

Design of slender artificial materials and morphogenesis of thin biological tissues typically involve stimulation of isolated regions (inclusions) in the growing body. These inclusions apply internal stresses on their surrounding areas that are ultimately relaxed by out-of-plane deformation (buckling). We utilize the Föppl-von Kármán model to analyze the interaction between two circular inclusions in an infinite plate that their centers are separated a distance of 2ℓ. In particular, we investigate a region in phase space where buckling occurs at a narrow transition layer of length ℓ_{D} around the radius of the inclusion, R (ℓ_{D}≪R). We show that the latter length scale defines two regions within the system, the close separation region, ℓ-R∼ℓ_{D}, where the transition layers of the two inclusions approximately coalesce, and the far separation region, ℓ-R≫ℓ_{D}. While the interaction energy decays exponentially in the latter region, E_{int}∝e^{-(ℓ-R)/ℓ_{D}}, it presents nonmonotonic behavior in the former region. While this exponential decay is predicted by our analytical analysis and agrees with the numerical observations, the close separation region is treated only numerically. In particular, we utilize the numerical investigation to explore two different scenarios within the final configuration: The first where the two inclusions buckle in the same direction (up-up solution) and the second where the two inclusions buckle in opposite directions (up-down solution). We show that the up-down solution is always energetically favorable over the up-up solution. In addition, we point to a curious symmetry breaking within the up-down scenario; we show that this solution becomes asymmetric in the close separation region.

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.

Modeling the behavior of inclusions in circular plates undergoing shape changes from two to three dimensions.

Growth of biological tissues and shape changes of thin synthetic sheets are commonly induced by stimulation of isolated regions (inclusions) in the system. These inclusions apply internal forces on their surroundings that, in turn, promote 2D layers to acquire complex 3D configurations. We focus on a fundamental building block of these systems, and consider a circular plate that contains an inclusion with dilative strains. Based on the Föppl-von Kármán (FvK) theory, we derive an analytical model that predicts the 2D-to-3D shape transitions in the system. Our findings are summarized in a phase diagram that reveals two distinct configurations in the post-buckling region. One is an extensive profile that holds close to the threshold of the instability, and the second is a localized profile, which preempts the extensive solution beyond the buckling threshold. While the former solution is derived as a perturbation around the flat configuration, assuming infinitesimal amplitudes, the latter solution is derived around a buckled state that is highly localized. We show that up to vanishingly small corrections that scale with the thickness, this localized configuration is equivalent to that expected for ultra-thin sheets, which completely relax compressive stresses. Our findings agree quantitatively with direct numerical minimization of the FvK energy. Furthermore, we extend the theory to describe shape transitions in polymeric gels, and compare the results with numerical simulations that account for the complete elastodynamic behavior of the gels. The agreement between the theory and these simulations indicates that our results are observable experimentally. Notably, our findings can provide guidelines to the analysis of more complicated systems that encompass interaction between several buckled inclusions.

Modeling the formation of double rolls from heterogeneously patterned gels.

Both stimuli-responsive gels and growing biological tissue can undergo pronounced morphological transitions from two-dimensional (2D) layers into 3D geometries. We derive an analytical model that allows us to quantitatively predict the features of 2D-to-3D shape changes in polymer gels that encompasses different degrees of swelling within the sample. We analyze a particular configuration that emerges from a flat rectangular gel that is divided into two strips (bistrips), where each strip is swollen to a different extent in solution. The final configuration yields double rolls that display a narrow transition layer between two cylinders of constant radii. To characterize the rolls' shapes, we modify the theory of thin incompatible elastic sheets to account for the Flory-Huggins interaction between the gel and the solvent. This modification allows us to derive analytical expressions for the radii, the amplitudes, and the length of the transition layer within a given roll. Our predictions agree quantitatively with available experimental data. In addition, we carry out numerical simulations that account for the complete nonlinear behavior of the gel and show good agreement between the analytical predictions and the numerical results. Our solution sheds light on a stress focusing pattern that forms at the border between two dissimilar soft materials. Moreover, models that provide quantitative predictions on the final morphology in such heterogeneously swelling hydrogels are useful for understanding growth patterns in biology as well as accurately tailoring the structure of gels for various technological applications.

"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.

Living Additive Manufacturing: Transformation of Parent Gels into Diversely Functionalized Daughter Gels Made Possible by Visible Light Photoredox Catalysis.

Light-initiated additive manufacturing techniques typically rely on layer-by-layer addition or continuous extraction of polymers formed via nonliving, free radical polymerization methods that render the final materials "dead" toward further monomer insertion; the polymer chains within the materials cannot be reactivated to induce chain extension. An alternative "living additive manufacturing" strategy would involve the use of photocontrolled living radical polymerization to spatiotemporally insert monomers into dormant "parent" materials to generate more complex and diversely functionalized "daughter" materials. Here, we demonstrate a proof-of-concept study of living additive manufacturing using end-linked polymer gels embedded with trithiocarbonate iniferters that can be activated by photoinduced single-electron transfer from an organic photoredox catalyst in solution. This system enables the synthesis of a wide range of chemically and mechanically differentiated daughter gels from a single type of parent gel via light-controlled modification of the parent's average composition, strand length, and/or cross-linking density. Daughter gels that are softer than their parent, stiffer than their parent, larger but with the same modulus as their parent, thermally responsive, polarity responsive, healable, and weldable are all realized.

Combining ATRP and FRP Gels: Soft Gluing of Polymeric Materials for the Fabrication of Stackable Gels.

Stackable gels comprised of layers of dissimilar polymers were synthesized by combining conventional free radical polymerization (FRP) and atom transfer radical polymerization (ATRP) using two approaches: (i) polymerization of a pre-gel solution containing a monomer and cross-linker introduced on top of a previously prepared gel, and (ii) simultaneous polymerization of two immiscible pre-gel solutions remaining in contact. All permutations of FRP and ATRP yielded single-piece, connected, amphiphilic gels regardless of the order of polymerization. Furthermore, multi-layer ATRP gels combining different polymers were synthesized with the FRP layer as a gluing agent. A 10-layer amphiphilic stackable gel combining n-butyl methacrylate (BMA) and 2-(dimethylamino)ethyl methacrylate (DMAEMA), and a 10-layer stackable gel combining BMA, DMAEMA and di(ethylene glycol) methyl ether methacrylate (PEO2MA) were synthesized. This patching method, combining conventional FRP gels with ATRP ones, offers an efficient path to the formation of complex stackable gel architectures.

Micromechanics of emergent patterns in plastic flows.

Crystalline solids undergo plastic deformation and subsequently flow when subjected to stresses beyond their elastic limit. In nature most crystalline solids exist in polycrystalline form. Simulating plastic flows in polycrystalline solids has wide ranging applications, from material processing to understanding intermittency of earthquake dynamics. Using phase field crystal (PFC) model we show that in sheared polycrystalline solids the atomic displacement field shows spatio-temporal heterogeneity spanning over several orders of length and time scales, similar to that in amorphous solids. The displacement field also exhibits localized quadrupolar patterns, characteristic of two dislocations of the opposite sign approaching each other. This is a signature of crystallinity at microscopic scale. Polycrystals being halfway between single crystals and amorphous solids, in terms of the degree of structural order, descriptions of solid mechanics at two widely different scales, namely continuum plastic flow and discrete dislocation dynamics turns out to be necessary here.

Coarsening in polycrystalline material using quaternions.

We develop a phase field model to study the phenomenon of recrystallization and grain coarsening in polycrystalline material. A unique feature of our model is that it can time-evolve the actual orientation field of a material, expressed in terms of quaternions, a four-dimensional non-conserved vector field. The quaternions evolve in time following a Langevin dynamics. The free energy that drives the evolution contains bulk energy for various preferred grain types and anisotropic grain boundary energy. As a proof of principle for the new formalism we show that the average grain size (L) follows the usual L ∼t(1/2) scaling law when the grain boundary energy is independent of the misorientation angle between neighboring grains, whereas the scaling exponent is less (∼0.42) when the grain boundary energy follows the misorientation-dependent, phenomenological Read-Shockley formula.

Predicting the coherence resonance curve using a semianalytical treatment.

Emergence of noise-induced regularity or coherence resonance in nonlinear excitable systems is well known. We explain theoretically why the normalized variance (V(N)) of interspike time intervals, which is a measure of regularity in such systems, has a unimodal profile. Our semianalytic treatment of the associated spiking process produces a general yet simple formula for V(N) , which we show is in very good agreement with numerics in two test cases, namely, the FitzHugh-Nagumo model and the chemical oscillator model.