Geometry, mechanics and actuation of intrinsically curved folds.

We combine theory and experiments to explore the kinematics and actuation of intrinsically curved folds (ICFs) in otherwise developable shells. Unlike origami folds, ICFs are not bending isometries of flat sheets, but arise via non-isometric processes (growth/moulding) or by joining sheets along curved boundaries. Experimentally, we implement both, first making joined ICFs from paper, then fabricating flat liquid crystal elastomer (LCE) sheets that morph into ICFs upon heating/swelling via programmed metric changes. Theoretically, an ICF's intrinsic geometry is defined by the geodesic curvatures on either side, κgi. Given these, and a target 3D fold-line, one can construct the entire surface isometrically, and compute the bending energy. This construction shows ICFs are bending mechanisms, with a continuous family of isometries trading fold angle against fold-line curvature. In ICFs with symmetric κgi, straightening the fold-line culminates in a fully-folded flat state that is deployable but weak, while asymmetric ICFs ultimately lock with a mechanically strong finite-angle. When unloaded, freely-hinged ICFs simply adopt the (thickness t independent) isometry that minimizes the bend energy. In contrast, in LCE ICFs a competition between flank and ridge selects a ridge curvature that, unusually, scales as t-1/7. Finally, we demonstrate how multiple ICFs can be combined in one LCE sheet, to create a versatile intrinsically curved gripper that lifts a heavy weight.

Surface Instability in a Nematic Elastomer.

Liquid crystal elastomers (LCEs) are soft phase-changing solids that exhibit large reversible contractions upon heating, Goldstone-like soft modes, and resultant microstructural instabilities. We heat a planar LCE slab to isotropic, clamp the lower surface, then cool back to nematic. Clamping prevents macroscopic elongation, producing compression and microstructure. We see that the free surface destabilizes, adopting topography with amplitude and wavelength similar to thickness. To understand the instability, we numerically compute the microstructural relaxation of a "nonideal" LCE energy. Linear stability reveals a Biot-like scale-free instability, but with oblique wave vector. However, simulation and experiment show that, unlike classic elastic creasing, instability culminates in a crosshatch without cusps or hysteresis, and is constructed entirely from low-stress soft modes.

Lifting, Loading, and Buckling in Conical Shells.

Liquid crystal elastomer films that morph into cones are strikingly capable lifters. Thus motivated, we combine theory, numerics, and experiments to reexamine the load-bearing capacity of conical shells. We show that a cone squashed between frictionless surfaces buckles at a smaller load, even in scaling, than the classical Seide-Koiter result. Such buckling begins in a region of greatly amplified azimuthal compression generated in an outer boundary layer with oscillatory bend. Experimentally and numerically, buckling then grows subcritically over the full cone. We derive a new thin-limit formula for the critical load, ∝t^{5/2}, and validate it numerically. We also investigate deep postbuckling, finding further instabilities producing intricate states with multiple Pogorelov-type curved ridges arranged in concentric circles or Archimedean spirals. Finally, we investigate the forces exerted by such states, which limit lifting performance in active cones.

Mechanics of mouse blastocyst hatching revealed by a hydrogel-based microdeformation assay.

Mammalian embryos are surrounded by an acellular shell, the zona pellucida. Hatching out of the zona is crucial for implantation and continued development of the embryo. Clinically, problems in hatching can contribute to failure in assisted reproductive intervention. Although hatching is fundamentally a mechanical process, due to limitations in methodology most studies focus on its biochemical properties. To understand the role of mechanical forces in hatching, we developed a hydrogel deformation-based method and analytical approach for measuring pressure in cyst-like tissues. Using this approach, we found that, in cultured blastocysts, pressure increased linearly, with intermittent falls. Inhibition of Na/K-ATPase led to a dosage-dependent reduction in blastocyst cavity pressure, consistent with its requirement for cavity formation. Reducing blastocyst pressure reduced the probability of hatching, highlighting the importance of mechanical forces in hatching. These measurements allowed us to infer details of microphysiology such as osmolarity, ion and water transport kinetics across the trophectoderm, and zona stiffness, allowing us to model the embryo as a thin-shell pressure vessel. We applied this technique to test whether cryopreservation, a process commonly used in assisted reproductive technology (ART), leads to alteration of the embryo and found that thawed embryos generated significantly lower pressure than fresh embryos, a previously unknown effect of cryopreservation. We show that reduced pressure is linked to delayed hatching. Our approach can be used to optimize in vitro fertilization (IVF) using precise measurement of embryo microphysiology. It is also applicable to other biological systems involving cavity formation, providing an approach for measuring forces in diverse contexts.

Defective nematogenesis: Gauss curvature in programmable shape-responsive sheets with topological defects.

Flat sheets encoded with patterns of contraction/elongation morph into curved surfaces. If the surfaces bear Gauss curvature, the resulting actuation can be strong and powerful. We deploy the Gauss-Bonnet theorem to deduce the Gauss curvature encoded in a pattern of uniform-magnitude contraction/elongation with spatially varying direction, as is commonly implemented in patterned liquid crystal elastomers. This approach reveals two fundamentally distinct contributions: a structural curvature which depends on the precise form of the pattern, and a topological curvature generated by defects in the contractile direction. These curvatures grow as different functions of the contraction/elongation magnitude, explaining the apparent contradiction between previous calculations for simple +1 defects, and smooth defect-free patterns. We verify these structural and topological contributions by conducting numerical shell calculations on sheets encoded with simple higher-order contractile defects to reveal their activated morphology. Finally we calculate the Gauss curvature generated by patterns with spatially varying magnitude and direction, which leads to additional magnitude gradient contributions to the structural term. We anticipate this form will be useful whenever magnitude and direction are natural variables, including in describing the contraction of a muscle along its patterned fiber direction, or a tissue growing by elongating its cells.

Peristaltic Elastic Instability in an Inflated Cylindrical Channel.

A long cylindrical cavity through a soft solid forms a soft microfluidic channel, or models a vascular capillary. We observe experimentally that, when such a channel bears a pressurized fluid, it first dilates homogeneously, but then becomes unstable to a peristaltic elastic instability. We combine theory and numerics to fully characterize the instability in a channel with initial radius a through an incompressible bulk neo-Hookean solid with shear modulus µ. We show instability occurs supercritically with wavelength 12.278…a when the cavity pressure exceeds 2.052…µ. In finite solids, the wavelength for peristalsis lengthens, with peristalsis ultimately being replaced by a long-wavelength bulging instability in thin-walled cylinders. Peristalsis persists in Gent strain-stiffening materials, provided the material can sustain extension by more than a factor of 6. Although naively a pressure driven failure mode of soft channels, the instability also offers a route to fabricate periodically undulating channels, producing, e.g., waveguides with photonic or phononic stop bands.

Pattern selection when a layer buckles on a soft substrate.

If a neo-Hookean elastic layer adhered to a neo-Hookean substrate grows equibiaxially, it will buckle into a topographic pattern. Here, we combine higher order perturbation theory and finite element numerics to predict the pattern formed just beyond the buckling threshold. More precisely, we construct a series of solutions corresponding to hexagonal, square and stripe patterns, and expand the elastic energy for each pattern as a Landau-like energy series in the topography amplitude. We see that, for square and stripe patterns, the elastic energy is invariant under topography inversion, making the instabilities supercritical. However, since patterns of hexagonal dents are physically different to patterns of hexagonal bumps, the hexagonal energy lacks this invariance. This lack introduces a cubic term which causes hexagonal patterns to be formed subcritically and are hence energetically favoured. Our analytic calculation of the cubic term allows us to determine that dents are favoured in incompressible systems, but bumps are favored in sufficiently compressible systems. Finally, we consider a stiff layer sandwiched between an identical substrate and superstrate pair. This system has topography inversion symmetry, so hexagons form supercritically, and square patterns are favoured. We use finite element calculations to verify our theoretical predictions for each pattern, and confirm which pattern is selected. Previous work has used a simplified elastic model (a plate & a linear elastic substrate) that possesses invariance under topography inversion, and hence incorrectly predicted square patterns. Our work demonstrates that large strain geometry is sufficient to break this symmetry and explain the hexagonal dent patterns observed in buckling experiments.

Decoration of plasmonic Mg nanoparticles by partial galvanic replacement.

Plasmonic structures have attracted much interest in science and engineering disciplines, exploring a myriad of potential applications owing to their strong light-matter interactions. Recently, the plasmonic concentration of energy in subwavelength volumes has been used to initiate chemical reactions, for instance by combining plasmonic materials with catalytic metals. In this work, we demonstrate that plasmonic nanoparticles of earth-abundant Mg can undergo galvanic replacement in a nonaqueous solvent to produce decorated structures. This method yields bimetallic architectures where partially oxidized 200-300 nm Mg nanoplates and nanorods support many smaller Au, Ag, Pd, or Fe nanoparticles, with potential for a stepwise process introducing multiple decoration compositions on a single Mg particle. We investigated this mechanism by electron-beam imaging and local composition mapping with energy-dispersive X-ray spectroscopy as well as, at the ensemble level, by inductively coupled plasma mass spectrometry. High-resolution scanning transmission electron microscopy further supported the bimetallic nature of the particles and provided details of the interface geometry, which includes a Mg oxide separation layer between Mg and the other metal. Depending on the composition of the metallic decorations, strong plasmonic optical signals characteristic of plasmon resonances were observed in the bulk with ultraviolet-visible spectrometry and at the single particle level with darkfield scattering. These novel bimetallic and multimetallic designs open up an exciting array of applications where one or multiple plasmonic structures could interact in the near-field of earth-abundant Mg and couple with catalytic nanoparticles for applications in sensing and plasmon-assisted catalysis.

Magnesium Nanoparticle Plasmonics.

Nanoparticles of some metals (Cu/Ag/Au) sustain oscillations of their electron cloud called localized surface plasmon resonances (LSPRs). These resonances can occur at optical frequencies and be driven by light, generating enhanced electric fields and spectacular photon scattering. However, current plasmonic metals are rare, expensive, and have a limited resonant frequency range. Recently, much attention has been focused on earth-abundant Al, but Al nanoparticles cannot resonate in the IR. The earth-abundant Mg nanoparticles reported here surmount this limitation. A colloidal synthesis forms hexagonal nanoplates, reflecting Mg's simple hexagonal lattice. The NPs form a thin self-limiting oxide layer that renders them stable suspended in 2-propanol solution for months and dry in air for at least two week. They sustain LSPRs observable in the far-field by optical scattering spectroscopy. Electron energy loss spectroscopy experiments and simulations reveal multiple size-dependent resonances with energies across the UV, visible, and IR. The symmetry of the modes and their interaction with the underlying substrate are studied using numerical methods. Colloidally synthesized Mg thus offers a route to inexpensive, stable nanoparticles with novel shapes and resonances spanning the entire UV-vis-NIR spectrum, making them a flexible addition to the nanoplasmonics toolbox.

Meniscus instabilities in thin elastic layers.

We consider meniscus instabilities in thin elastic layers perfectly adhered to, and confined between, much stiffer bodies. When the free boundary associated with the meniscus of the elastic layer recedes into the layer, for example by pulling the stiffer bodies apart or injecting air between them, then the meniscus will eventually undergo a purely elastic instability in which fingers of air invade the layer. Here we show that the form of this instability is identical in a range of different loading conditions, provided only that the thickness of the meniscus, a, is small compared to the in-plane dimensions and to two emergent in-plane length scales that arise if the substrate is soft or if the layer is compressible. In all such situations, we predict that the instability will occur when the meniscus has receded by approximately 1.27a, and that the instability will have wavelength λ ≈ 2.75a. We illustrate this by also calculating the threshold for fingering in a thin wedge of elastic material bonded to two rigid plates that are pried apart, and the threshold for fingering when a flexible plate is peeled from an elastic layer that glues the plate to a rigid substrate.

Towards understanding the roles of position and geometry on cell fate decisions during preimplantation development.

The first lineage segregation event in mouse embryos produces two separate cell populations: inner cell mass and trophectoderm. This is understood to be brought about by cells sensing their position within the embryo and differentiating accordingly. The cellular and molecular underpinnings of this process remain under investigation and have variously been considered to be completely stochastic or alternately, subject to some predisposition set up at fertilisation or before. Here, we consider these views in light of recent publications, discuss the possible role of cell geometry and mechanical forces in this process and describe how modelling could contribute in addressing this issue.

Limited predictive value of blastomere angle of division in trophectoderm and inner cell mass specification.

The formation of trophectoderm (TE) and pluripotent inner cell mass (ICM) is one of the earliest events during mammalian embryogenesis. It is believed that the orientation of division of polarised blastomeres in the 8- and 16-cell stage embryo determines the fate of daughter cells, based on how asymmetrically distributed lineage determinants are segregated. To investigate the relationship between angle of division and subsequent fate in unperturbed embryos, we constructed cellular resolution digital representations of the development of mouse embryos from the morula to early blastocyst stage, based on 4D confocal image volumes. We find that at the 16-cell stage, very few inside cells are initially produced as a result of cell division, but that the number increases due to cell movement. Contrary to expectations, outside cells at the 16-cell stage represent a heterogeneous population, with some fated to contributing exclusively to the TE and others capable of contributing to both the TE and ICM. Our data support the view that factors other than the angle of division, such as the position of a blastomere, play a major role in the specification of TE and ICM.

Gyrification from constrained cortical expansion.

The exterior of the mammalian brain--the cerebral cortex--has a conserved layered structure whose thickness varies little across species. However, selection pressures over evolutionary time scales have led to cortices that have a large surface area to volume ratio in some organisms, with the result that the brain is strongly convoluted into sulci and gyri. Here we show that the gyrification can arise as a nonlinear consequence of a simple mechanical instability driven by tangential expansion of the gray matter constrained by the white matter. A physical mimic of the process using a layered swelling gel captures the essence of the mechanism, and numerical simulations of the brain treated as a soft solid lead to the formation of cusped sulci and smooth gyri similar to those in the brain. The resulting gyrification patterns are a function of relative cortical expansion and relative thickness (compared with brain size), and are consistent with observations of a wide range of brains, ranging from smooth to highly convoluted. Furthermore, this dependence on two simple geometric parameters that characterize the brain also allows us to qualitatively explain how variations in these parameters lead to anatomical anomalies in such situations as polymicrogyria, pachygyria, and lissencephalia.

Digital instability of a confined elastic meniscus.

Thin soft elastic layers serving as joints between relatively rigid bodies may function as sealants, thermal, electrical, or mechanical insulators, bearings, or adhesives. When such a joint is stressed, even though perfect adhesion is maintained, the exposed free meniscus in the thin elastic layer becomes unstable, leading to the formation of spatially periodic digits of air that invade the elastic layer, reminiscent of viscous fingering in a thin fluid layer. However, the elastic instability is reversible and rate-independent, disappearing when the joint is unstressed. We use theory, experiments, and numerical simulations to show that the transition to the digital state is sudden (first-order), the wavelength and amplitude of the fingers are proportional to the thickness of the elastic layer, and the required separation to trigger the instability is inversely proportional to the in-plane dimension of the layer. Our study reveals the energetic origin of this instability and has implications for the strength of polymeric adhesives; it also suggests a method for patterning thin films reversibly with any arrangement of localized fingers in a digital elastic memory, which we confirm experimentally.

Far-field, near-field and photothermal response of plasmonic twinned magnesium nanostructures.

Magnesium nanoparticles offer an alternative plasmonic platform capable of resonances across the ultraviolet, visible and near-infrared. Crystalline magnesium nanoparticles display twinning on the (101̄1), (101̄2), (101̄3), and (112̄1) planes leading to concave folded shapes named tents, chairs, tacos, and kites, respectively. We use the Wulff-based Crystal Creator tool to expand the range of Mg crystal shapes with twinning over the known Mg twin planes, i.e., (101̄x), x = 1, 2, 3 and (112̄y), y = 1, 2, 3, 4, and study the effects of relative facet expression on the resulting shapes. These shapes include both concave and convex structures, some of which have been experimentally observed. The resonant modes, far-field, and near-field optical responses of these unusual plasmonic shapes as well as their photothermal behaviour are reported, revealing the effects of folding angle and in-filling of the concave region. Significant differences exist between shapes, in particular regarding the maximum and average electric field enhancement. A maximum field enhancement (|E|/|E0|) of 184, comparable to that calculated for Au and Ag nanoparticles, was found at the tips of the (112̄4) kite. The presence of a 5 nm MgO shell is found to decrease the near-field enhancement by 67% to 90% depending on the shape, while it can increase the plasmon-induced temperature rise by up to 42%. Tip rounding on the otherwise sharp nanoparticle corners also significantly affects the maximum field enhancement. These results provide guidance for the design of enhancing and photothermal substrates for a variety of plasmonic applications across a wide spectral range.

Machine learning for nanoplasmonics.

Plasmonic nanomaterials have outstanding optoelectronic properties potentially enabling the next generation of catalysts, sensors, lasers and photothermal devices. Owing to optical and electron techniques, modern nanoplasmonics research generates large datasets characterizing features across length scales. Furthermore, optimizing syntheses leading to specific nanostructures requires time-consuming multiparametric approaches. These complex datasets and trial-and-error practices make nanoplasmonics research ripe for the application of machine learning (ML) and advanced data processing methods. ML algorithms capture relationships between synthesis, structure and performance in a way that far exceeds conventional simulation and theory approaches, enabling effective performance optimization. For example, neural networks can tailor the nanostructure morphology to target desired properties, identify synthetic conditions and extract quantitative information from complex data. Here we discuss the nascent field of ML for nanoplasmonics, describe the opportunities and limitations of ML in nanoplasmonic research, and conclude that ML is potentially transformative, especially if the community curates and shares its big data.

Optoplasmonic Effects in Highly Curved Surfaces for Catalysis, Photothermal Heating, and SERS.

Surface curvature can be used to focus light and alter optical processes. Here, we show that curved surfaces (spheres, cylinders, and cones) with a radius of around 5 µm lead to maximal optoplasmonic properties including surface-enhanced Raman scattering (SERS), photocatalysis, and photothermal processes. Glass microspheres, microfibers, pulled fibers, and control flat substrates were functionalized with well-dispersed and dense arrays of 45 nm Au NP using polystyrene-block-poly-4-vinylpyridine (PS-b-P4VP) and chemically modified with 4-mercaptobenzoic acid (4-MBA, SERS reporter), 4-nitrobenzenethiol (4-NBT, reactive to plasmonic catalysis), or 4-fluorophenyl isocyanide (FPIC, photothermal reporter). The various curved substrates enhanced the plasmonic properties by focusing the light in a photonic nanojet and providing a directional antenna to increase the collection efficacy of SERS photons. The optoplasmonic effects led to an increase of up to 1 order of magnitude of the SERS response, up to 5 times the photocatalytic conversion of 4-NBT to 4,4'-dimercaptoazobenzene when the diameter of the curved surfaces was about 5 µm and a small increase in photothermal effects. Taken together, the results provide evidence that curvature enhances plasmonic properties and that its effect is maximal for spherical objects around a few micrometers in diameter, in agreement with a theoretical framework based on geometrical optics. These enhanced plasmonic effects and the stationary-phase-like plasmonic substrates pave the way to the next generation of sensors, plasmonic photocatalysts, and photothermal devices.

Curvature-driven instabilities in thin active shells.

Spontaneous material shape changes, such as swelling, growth or thermal expansion, can be used to trigger dramatic elastic instabilities in thin shells. These instabilities originate in geometric incompatibility between the preferred extrinsic and intrinsic curvature of the shell, which may be modified by active deformations through the thickness and in plane, respectively. Here, we solve the simplest possible model of such instabilities, which assumes the shells are shallow, thin enough to bend but not stretch, and subject to homogeneous preferred curvatures. We consider separately the cases of zero, positive and negative Gauss curvature. We identify two types of supercritical symmetry-breaking instability, in which the shell's principal curvature spontaneously breaks discrete up/down symmetry and continuous planar isotropy. These are then augmented by inversion instabilities, in which the shell jumps subcritically between up/down broken symmetry states and rotation instabilities, in which the curvatures rotate by 90° between states of broken isotropy without release of energy. Each instability has a thickness-independent threshold value for the preferred extrinsic curvature proportional to the square root of Gauss curvature. Finally, we show that the threshold for the isotropy-breaking instability is the same for deep spherical caps, in good agreement with recently published data.

Interfacial metric mechanics: stitching patterns of shape change in active sheets.

A flat sheet programmed with a planar pattern of spontaneous shape change will morph into a curved surface. Such metric mechanics is seen in growing biological sheets, and may be engineered in actuating soft matter sheets such as phase-changing liquid crystal elastomers (LCEs), swelling gels and inflating baromorphs. Here, we show how to combine multiple patterns in a sheet by stitching regions of different shape changes together piecewise along interfaces. This approach allows simple patterns to be used as building blocks, and enables the design of multi-material or active/passive sheets. We give a general condition for an interface to be geometrically compatible, and explore its consequences for LCE/LCE, gel/gel and active/passive interfaces. In contraction/elongation systems such as LCEs, we find an infinite set of compatible interfaces between any pair of patterns along which the metric is discontinuous, and a finite number across which the metric is continuous. As an example, we find all possible interfaces between pairs of LCE logarithmic spiral patterns. By contrast, in isotropic systems such as swelling gels, only a finite number of continuous interfaces are available, greatly limiting the potential of stitching. In both continuous and discontinuous cases, we find the stitched interfaces generically carry singular Gaussian curvature, leading to intrinsically curved folds in the actuated surface. We give a general expression for the distribution of this curvature, and a more specialized form for interfaces in LCE patterns. The interfaces thus also have rich geometric and mechanical properties in their own right.

Opportunities and Challenges for Alternative Nanoplasmonic Metals: Magnesium and Beyond.

Materials that sustain localized surface plasmon resonances have a broad technology potential as attractive platforms for surface-enhanced spectroscopies, chemical and biological sensing, light-driven catalysis, hyperthermal cancer therapy, waveguides, and so on. Most plasmonic nanoparticles studied to date are composed of either Ag or Au, for which a vast array of synthetic approaches are available, leading to controllable size and shape. However, recently, alternative materials capable of generating plasmonically enhanced light-matter interactions have gained prominence, notably Cu, Al, In, and Mg. In this Perspective, we give an overview of the attributes of plasmonic nanostructures that lead to their potential use and how their performance is dictated by the choice of plasmonic material, emphasizing the similarities and differences between traditional and emerging plasmonic compositions. First, we discuss the materials limitation encapsulated by the dielectric function. Then, we evaluate how size and shape maneuver localized surface plasmon resonance (LSPR) energy and field distribution and address how this impacts applications. Next, biocompatibility, reactivity, and cost, all key differences underlying the potential of non-noble metals, are highlighted. We find that metals beyond Ag and Au are of competitive plasmonic quality. We argue that by thinking outside of the box, i.e., by looking at nonconventional materials such as Mg, one can broaden the frequency range and, more importantly, combine the plasmonic response with other properties essential for the implementation of plasmonic technologies.