Controlled pathways and sequential information processing in serially coupled mechanical hysterons.

The complex sequential response of frustrated materials results from the interactions between material bits called hysterons. Hence, a central challenge is to understand and control these interactions, so that materials with targeted pathways and functionalities can be realized. Here, we show that hysterons in serial configurations experience geometrically controllable antiferromagnetic-like interactions. We create hysteron-based metamaterials that leverage these interactions to realize targeted pathways, including those that break the return point memory property, characteristic of independent or weakly interacting hysterons. We uncover that the complex response to sequential driving of such strongly interacting hysteron-based materials can be described by finite state machines. We realize information processing operations such as string parsing in materia, and outline a general framework to uncover and characterize the FSMs for a given physical system. Our work provides a general strategy to understand and control hysteron interactions, and opens a broad avenue toward material-based information processing.

Patterning Complex Line Motifs in Thin Films Using Immersion-Controlled Reaction-Diffusion.

The discovery of self-organization principles that enable scalable routes toward complex functional materials has proven to be a persistent challenge. Here, reaction-diffusion driven, immersion-controlled patterning (R-DIP) is introduced, a self-organization strategy using immersion-controlled reaction-diffusion for targeted line patterning in thin films. By modulating immersion speeds, the movement of a reaction-diffusion front over gel films is controlled, which induces precipitation of highly uniform lines at the reaction front. A balance between the immersion speed and diffusion provides both hands-on tunability of the line spacing ( d = 10 - 300 µ m $d = 10-300 \; \umu \text{m}$ ) as well as error-correction against defects. This immersion-driven patterning strategy is widely applicable, which is demonstrated by producing line patterns of silver/silver oxide nanoparticles, silver chromate, silver dichromate, and lead carbonate. Through combinatorial stacking of different line patterns, hybrid materials with multi-dimensional patterns such as square-, diamond-, rectangle-, and triangle-shaped motifs are fabricated. The functionality potential and scalability is demonstrated by producing both wafer-scale diffraction gratings with user-defined features as well as an opto-mechanical sensor based on Moiré patterning.

Counting and Sequential Information Processing in Mechanical Metamaterials.

Materials with an irreversible response to cyclic driving exhibit an evolving internal state which, in principle, encodes information on the driving history. Here we realize irreversible metamaterials that count mechanical driving cycles and store the result into easily interpretable internal states. We extend these designs to aperiodic metamaterials that are sensitive to the order of different driving magnitudes, and realize "lock and key" metamaterials that only reach a specific state for a given target driving sequence. Our metamaterials are robust, scalable, and extendable, give insight into the transient memories of complex media, and open new routes towards smart sensing, soft robotics, and mechanical information processing.

Emergent nonlocal combinatorial design rules for multimodal metamaterials.

Combinatorial mechanical metamaterials feature spatially textured soft modes that yield exotic and useful mechanical properties. While a single soft mode often can be rationally designed by following a set of tiling rules for the building blocks of the metamaterial, it is an open question what design rules are required to realize multiple soft modes. Multimodal metamaterials would allow for advanced mechanical functionalities that can be selected on the fly. Here we introduce a transfer matrix-like framework to design multiple soft modes in combinatorial metamaterials composed of aperiodic tilings of building blocks. We use this framework to derive rules for multimodal designs for a specific family of building blocks. We show that such designs require a large number of degeneracies between constraints, and find precise rules on the real space configuration that allow such degeneracies. These rules are significantly more complex than the simple tiling rules that emerge for single-mode metamaterials. For the specific example studied here, they can be expressed as local rules for tiles composed of pairs of building blocks in combination with a nonlocal rule in the form of a global constraint on the type of tiles that are allowed to appear together anywhere in the configuration. This nonlocal rule is exclusive to multimodal metamaterials and exemplifies the complexity of rational design of multimode metamaterials. Our framework is a first step towards a systematic design strategy of multimodal metamaterials with spatially textured soft modes.

Machine Learning of Implicit Combinatorial Rules in Mechanical Metamaterials.

Combinatorial problems arising in puzzles, origami, and (meta)material design have rare sets of solutions, which define complex and sharply delineated boundaries in configuration space. These boundaries are difficult to capture with conventional statistical and numerical methods. Here we show that convolutional neural networks can learn to recognize these boundaries for combinatorial mechanical metamaterials, down to finest detail, despite using heavily undersampled training sets, and can successfully generalize. This suggests that the network infers the underlying combinatorial rules from the sparse training set, opening up new possibilities for complex design of (meta)materials.

Rapid formation of uniformly layered materials by coupling reaction-diffusion processes with mechanical responsiveness.

Straightforward manufacturing pathways toward large-scale, uniformly layered composites may enable the next generation of materials with advanced optical, thermal, and mechanical properties. Reaction-diffusion systems are attractive candidates to this aim, but while layered composites theoretically could spontaneously arise from reaction-diffusion, in practice randomly oriented patches separated by defects form, yielding nonuniformly patterned materials. A propagating reaction front can prevent such nonuniform patterning, as is the case for Liesegang processes, in which diffusion drives a reaction front to produce layered precipitation patterns. However, while diffusion is crucial to control patterning, it slows down transport of reactants to the front and results in a steady increase of the band spacing as the front advances. Here, we circumvent these diffusive limitations by embedding the Liesegang process in mechanically responsive hydrogels. The coupling between a moving reaction front and hydrogel contraction induces the formation of a self-regulated transport channel that ballistically carries reactants toward the area where patterning occurs. This ensures rapid and uniform patterning. Specifically, large-scale ([Formula: see text]5-cm) uniform banding patterns are produced with tunable band distance (d = 60 to 160 µm) of silver dichromate crystals inside responsive gelatin-alginate hydrogels. The generality and applicability of our mechanoreaction-diffusion strategy are demonstrated by forming patterns of precipitates in significantly smaller microscopic banding patterns (d = 10 to 30 µm) that act as self-organized diffraction gratings. By circumventing the inherent limitations of diffusion, our strategy unlocks the potential of reaction-diffusion processes for the manufacturing of uniformly layered materials.

Sequential snapping and pathways in a mechanical metamaterial.

Materials that feature bistable elements, hysterons, exhibit memory effects. Often, these hysterons are difficult to observe or control directly. Here, we introduce a mechanical metamaterial in which slender elements, interacting with pushers, act as mechanical hysterons. We show how we can tune the hysteron properties and pathways under cyclic compression by the geometric design of these elements and how we can tune the pathways of a given sample by tilting one of the boundaries. Furthermore, we investigate the effect of the coupling of a global shear mode to the hysterons as an example of the interactions between hysteron and non-hysteron degrees of freedom. We hope our work will inspire further studies on designer matter with targeted pathways.

Contraction and Expansion of Nanocomposites during Ion Exchange Reactions.

The next generation of advanced functional materials can greatly benefit from methods for realizing the right chemical composition at the right place. Nanocomposites of amorphous silica and metal carbonate nanocrystals (BaCO3/SiO2) form an attractive starting point as they can straightforwardly be assembled in different controllable three-dimensional (3D) shapes, while the chemical composition of the nanocrystals can be completely converted via ion exchange. Nevertheless, it is still unknown-let alone predictable-how nanoscopic changes in the lattice volume of the nanocrystals translate to changes in the microscopic dimensions of 3D BaCO3/SiO2 structures during ion exchange. Here, we demonstrate that the microscopic shape adapts to contraction and expansion of the atomic spacing of nanocrystals. Starting from BaCO3/SiO2, we systematically decrease and increase lattice volumes by converting the BaCO3 nanocrystals into a range of chalcogenides and perovskites. Based on geometrical analysis, we obtain a precise prediction for how the microscopic nanocomposite volume follows the change in nanoscopic crystal volume. The silica matrix facilitates mechanical flexibility to adapt to nanoscopic volume changes, while preserving the 3D morphology and fine details of the original composite with high fidelity. The versatility and predictability of shape-preserving conversion reactions open up exciting opportunities for using nanocomposites as functional components.

Conformal elasticity of mechanism-based metamaterials.

Deformations of conventional solids are described via elasticity, a classical field theory whose form is constrained by translational and rotational symmetries. However, flexible metamaterials often contain an additional approximate symmetry due to the presence of a designer soft strain pathway. Here we show that low energy deformations of designer dilational metamaterials will be governed by a scalar field theory, conformal elasticity, in which the nonuniform, nonlinear deformations observed under generic loads correspond with the well-studied-conformal-maps. We validate this approach using experiments and finite element simulations and further show that such systems obey a holographic bulk-boundary principle, which enables an analytic method to predict and control nonuniform, nonlinear deformations. This work both presents a unique method of precise deformation control and demonstrates a general principle in which mechanisms can generate special classes of soft deformations.

Complex pathways and memory in compressed corrugated sheets.

The nonlinear response of driven complex materials-disordered magnets, amorphous media, and crumpled sheets-features intricate transition pathways where the system repeatedly hops between metastable states. Such pathways encode memory effects and may allow information processing, yet tools are lacking to experimentally observe and control these pathways, and their full breadth has not been explored. Here we introduce compression of corrugated elastic sheets to precisely observe and manipulate their full, multistep pathways, which are reproducible, robust, and controlled by geometry. We show how manipulation of the boundaries allows us to elicit multiple targeted pathways from a single sample. In all cases, each state in the pathway can be encoded by the binary state of material bits called hysterons, and the strength of their interactions plays a crucial role. In particular, as function of increasing interaction strength, we observe Preisach pathways, expected in systems of independently switching hysterons; scrambled pathways that evidence hitherto unexplored interactions between these material bits; and accumulator pathways which leverage these interactions to perform an elementary computation. Our work opens a route to probe, manipulate, and understand complex pathways, impacting future applications in soft robotics and information processing in materials.

Profusion of transition pathways for interacting hysterons.

The response, pathways, and memory effects of cyclically driven complex media can be captured by hysteretic elements called hysterons. Here we demonstrate the profound impact of hysteron interactions on pathways and memory. Specifically, while the Preisach model of independent hysterons features a restricted class of pathways which always satisfy return point memory, we show that three interacting hysterons generate more than 15 000 transition graphs, with most violating return point memory and having features completely distinct from the Preisach model. Exploring these opens a route to designer pathways and information processing in complex matter.

Design of Pseudo-Mechanisms and Multistable Units for Mechanical Metamaterials.

Mechanisms-collections of rigid elements coupled by perfect hinges which exhibit a zero-energy motion-motivate the design of a variety of mechanical metamaterials. We enlarge this design space by considering pseudo-mechanisms, collections of elastically coupled elements that exhibit motions with very low energy costs. We show that their geometric design generally is distinct from those of true mechanisms, thus opening up a large and virtually unexplored design space. We further extend this space by designing building blocks with bistable and tristable energy landscapes, realize these by 3D printing, and show how these form unit cells for multistable metamaterials.

Shape-Preserving Chemical Conversion of Architected Nanocomposites.

Forging customizable compounds into arbitrary shapes and structures has the potential to revolutionize functional materials, where independent control over shape and composition is essential. Current self-assembly strategies allow impressive levels of control over either shape or composition, but not both, as self-assembly inherently entangles shape and composition. Herein, independent control over shape and composition is achieved by chemical conversion reactions on nanocrystals, which are first self-assembled in nanocomposites with programmable microscopic shapes. The multiscale character of nanocomposites is crucial: nanocrystals (5-50 nm) offer enhanced chemical reactivity, while the composite layout accommodates volume changes of the nanocrystals (≈25%), which together leads to complete chemical conversion with full shape preservation. These reactions are surprisingly materials agnostic, allowing a large diversity of chemical pathways, and development of conversion pathways yielding a wide selection of shape-controlled transition metal chalcogenides (cadmium, manganese, iron, and nickel oxides and sulfides). Finally, the versatility and application potential of this strategy is demonstrated by assembling: 1) a scalable and highly reactive nickel catalyst for the dry reforming of butane, 2) an agile magnetic-controlled particle, and 3) an electron-beam-controlled reversible microactuator with sub-micrometer precision. Previously unimaginable customization of shape and composition is now achievable for assembling advanced functional components.

Non-Euclidean origami.

Traditional origami starts from flat surfaces, leading to crease patterns consisting of Euclidean vertices. However, Euclidean vertices are limited in their folding motions, are degenerate, and suffer from misfolding. Here we show how non-Euclidean 4-vertices overcome these limitations by lifting this degeneracy, and that when the elasticity of the hinges is taken into account, non-Euclidean 4-vertices permit higher order multistability. We harness these advantages to design an origami inverter that does not suffer from misfolding and to physically realize a tristable vertex.

Excess floppy modes and multibranched mechanisms in metamaterials with symmetries.

Floppy modes-deformations that cost zero energy-are central to the mechanics of a wide class of systems. For disordered systems, such as random networks and particle packings, it is well-understood how the number of floppy modes is controlled by the topology of the connections. Here we uncover that symmetric geometries, present in, e.g., mechanical metamaterials, can feature an unlimited number of excess floppy modes that are absent in generic geometries, and in addition can support floppy modes that are multibranched. We study the number Δ of excess floppy modes by comparing generic and symmetric geometries with identical topologies, and show that Δ is extensive, peaks at intermediate connection densities, and exhibits mean-field scaling. We then develop an approximate yet accurate cluster counting algorithm that captures these findings. Finally, we leverage our insights to design metamaterials with multiple folding mechanisms.

Multi-step self-guided pathways for shape-changing metamaterials.

Multi-step pathways-which consist of a sequence of reconfigurations of a structure-are central to the functionality of various natural and artificial systems. Such pathways execute autonomously in self-guided processes such as protein folding1 and self-assembly2-5, but have previously required external control to execute in macroscale mechanical systems, provided by, for example, actuators in robotics6-9 or manual folding in origami8,10-12. Here we demonstrate shape-changing, macroscale mechanical metamaterials that undergo self-guided, multi-step reconfiguration in response to global uniform compression. We avoid the need for external control by using metamaterials that are made purely of passive components. The design of the metamaterials combines nonlinear mechanical elements with a multimodal architecture that enables a sequence of topological reconfigurations caused by the formation of internal self-contacts between the elements of the metamaterial. We realize the metamaterials by using computer-controlled water-jet cutting of flexible materials, and show that the multi-step pathway and final configuration can be controlled by rational design of the nonlinear mechanical elements. We also demonstrate that the self-contacts suppress errors in the pathway. Finally, we create hierarchical architectures to extend the number of distinct reconfiguration steps. Our work establishes general principles for designing mechanical pathways, opening up new avenues for self-folding media11,12, pluripotent materials9,13 and pliable devices14 in areas such as stretchable electronics and soft robotics15.

From Bouncing to Floating: The Leidenfrost Effect with Hydrogel Spheres.

The Leidenfrost effect occurs when a liquid or stiff sublimable solid near a hot surface creates enough vapor beneath it to lift itself up and float. In contrast, vaporizable soft solids, e.g., hydrogels, have been shown to exhibit persistent bouncing-the elastic Leidenfrost effect. By carefully lowering hydrogel spheres towards a hot surface, we discover that they are also capable of floating. The bounce-to-float transition is controlled by the approach velocity and temperature, analogously to the "dynamic Leidenfrost effect." For the floating regime, we measure power-law scalings for the gap geometry, which we explain with a model that couples the vaporization rate to the spherical shape. Our results reveal that hydrogels are a promising pathway for controlling floating Leidenfrost objects through shape.

Softening and yielding of soft glassy materials.

Solids deform and fluids flow, but soft glassy materials, such as emulsions, foams, suspensions, and pastes, exhibit an intricate mix of solid- and liquid-like behavior. While much progress has been made to understand their elastic (small strain) and flow (infinite strain) properties, such understanding is lacking for the softening and yielding phenomena that connect these asymptotic regimes. Here we present a comprehensive framework for softening and yielding of soft glassy materials, based on extensive numerical simulations of oscillatory rheological tests, and show that two distinct scenarios unfold depending on the material's packing density. For dense systems, there is a single, pressure-independent strain where the elastic modulus drops and the particle motion becomes diffusive. In contrast, for weakly jammed systems, a two-step process arises: at an intermediate softening strain, the elastic and loss moduli both drop down and then reach a new plateau value, whereas the particle motion becomes diffusive at the distinctly larger yield strain. We show that softening is associated with an extensive number of microscopic contact changes leading to a non-analytic rheological signature. Moreover, the scaling of the softening strain with pressure suggest the existence of a novel pressure scale above which softening and yielding coincide, and we verify the existence of this crossover scale numerically. Our findings thus evidence the existence of two distinct classes of soft glassy materials - jamming dominated and dense - and show how these can be distinguished by their rheological fingerprint.

Criticality in Vibrated Frictional Flows at a Finite Strain Rate.

We evidence critical fluctuations in the strain rate of granular flows that are weakly vibrated. Strikingly, the critical point arises at finite values of the mean strain rate and vibration strength, far from the yielding critical point at a zero flow rate. We show that the global rheology, as well as the amplitude and correlation time of the fluctuations, are consistent with a mean-field, Landau-like description, where the strain rate and the stress act as conjugated variables. We introduce a general model which captures the observed phenomenology and argue that this type of critical behavior generically arises when self-fluidization competes with friction.

Programmable mechanical metamaterials: the role of geometry.

We experimentally and numerically study the role of geometry for the mechanics of biholar metamaterials, which are quasi-2D slabs of rubber patterned by circular holes of two alternating sizes. We recently showed how the response to uniaxial compression of these metamaterials can be programmed by lateral confinement. In particular, there is a range of confining strains εx for which the resistance to compression becomes non-trivial-non-monotonic or hysteretic-in a range of compressive strains εy. Here we show how the dimensionless geometrical parameters t and χ, which characterize the wall thickness and size ratio of the holes that pattern these metamaterials, can significantly tune these ranges over a wide range. We study the behavior for the limiting cases where the wall thickness t and the size ratio χ become large, and discuss the new physics that arises there. Away from these extreme limits, the variation of the strain ranges of interest is smooth with porosity, but the variation with size ratio evidences a cross-over at low χ from biholar to monoholar (equal sized holes) behavior, related to the elastic instabilities in purely monoholar metamaterials. Our study provides precise guidelines for the rational design of programmable biholar metamaterials, tailored to specific applications, and indicates that the widest range of programmability arises for moderate values of both t and χ.