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.

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.

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.

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.

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.

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.

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.

Metric mechanics with nontrivial topology: Actuating irises, cylinders, and evertors

Liquid crystal elastomers contract along their director on heating and recover on cooling, offering great potential as actuators and artificial muscles. If a flat sheet is programed with a spatially varying director pattern, then it will actuate into a curved surface, allowing the material to act as a strong machine such as a grabber or lifter. Here we study the actuation of programed annular sheets which, owing to their central hole, can sidestep constraints on area and orientation. We systematically catalog the set of developable surfaces encodable via axisymmetric director patterns and uncover several qualitatively new modes of actuation, including cylinders, irises, and everted surfaces in which the inner boundary becomes the outer boundary after actuation. We confirm our designs with a combination of experiments and numerics. Many of our actuators can reattain their initial inner or outer radius upon completing actuation, making them particularly promising, as they can avoid potentially problematic stresses in their activated state even when fixed onto a frame or pipe.

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.

On the identification of twinning in body-centred cubic nanoparticles

Many metals and alloys, including Fe and W, adopt body-centred cubic (BCC) crystal structures and nanoparticles of these metals are gaining significant scientific and industrial relevance. Twinning has a marked effect on catalytic activity, yet there is little evidence for or against the presence of twinning in BCC nanoparticles. Here, we explore the potential shapes of twinned BCC nanoparticles, and predict their electron microscopy and diffraction signatures. BCC single crystal and twinned shapes often appear similar and diffraction patterns along common, low-index zone axes are often indistinguishable, casting doubt on many claims of single crystallinity. We conclude by outlining how nanoparticles can be characterized to conclusively prove the presence or absence of twinning.

Ballooning, bulging, and necking: An exact solution for longitudinal phase separation in elastic systems near a critical point

Prominent examples of longitudinal phase separation in elastic systems include elastic necking, the propagation of a bulge in a cylindrical party balloon, and the beading of a gel fiber subject to surface tension. Here we demonstrate that if the parameters of such a system are tuned near a critical point (where the difference between the two phases vanishes), then the behavior of all systems is given by the minimization of a simple and universal elastic energy familiar from Ginzburg-Landau theory in an external field. We minimize this energy analytically, which yields not only the well known interfacial tanh solution, but also the complete set of stable and unstable solutions in both finite and infinite length systems, unveiling the elastic system's full shape evolution and hysteresis. Correspondingly, we also find analytic results for the the delay of onset, changes in criticality, and ultimate suppression of instability with diminishing system length, demonstrating that our simple near-critical theory captures much of the complexity and choreography of far-from-critical systems. Finally, we find critical points for the three prominent examples of phase separation given above, and demonstrate how each system then follows the universal set of solutions.

Evolving, complex topography from combining centers of Gaussian curvature

Liquid crystal elastomers and glasses can have significant shape change determined by their director patterns. Cones deformed from circular director patterns have nontrivial Gaussian curvature localized at tips, curved interfaces, and intersections of interfaces. We employ a generalized metric compatibility condition to characterize two families of interfaces between circular director patterns, hyperbolic and elliptical interfaces, and find that the deformed interfaces are geometrically compatible. We focus on hyperbolic interfaces to design complex topographies and nonisometric origami, including n-fold intersections, symmetric and irregular tilings. The large design space of threefold and fourfold tiling is utilized to quantitatively inverse design an array of pixels to display target images. Taken together, our findings provide comprehensive design principles for the design of actuators, displays, and soft robotics in liquid crystal elastomers and glasses.

Tents, Chairs, Tacos, Kites, and Rods: Shapes and Plasmonic Properties of Singly Twinned Magnesium Nanoparticles

Nanostructures of some metals can sustain light-driven electron oscillations called localized surface plasmon resonances, or LSPRs, that give rise to absorption, scattering, and local electric field enhancement. Their resonant frequency is dictated by the nanoparticle (NP) shape and size, fueling much research geared toward discovery and control of new structures. LSPR properties also depend on composition; traditional, rare, and expensive noble metals (Ag, Au) are increasingly eclipsed by earth-abundant alternatives, with Mg being an exciting candidate capable of sustaining resonances across the ultraviolet, visible, and near-infrared spectral ranges. Here, we report numerical predictions and experimental verifications of a set of shapes based on Mg NPs displaying various twinning patterns including (101̅1), (101̅2), (101̅3), and (112̅1), that create tent-, chair-, taco-, and kite-shaped NPs, respectively. These are strikingly different from what is obtained for typical plasmonic metals because Mg crystallizes in a hexagonal close packed structure, as opposed to the cubic Al, Cu, Ag, and Au. A numerical survey of the optical response of the various structures, as well as the effect of size and aspect ratio, reveals their rich array of resonances, which are supported by single-particle optical scattering experiments. Further, corresponding numerical and experimental studies of the near-field plasmon distribution via scanning transmission electron microscopy electron-energy loss spectroscopy unravels a mode nature and distribution that are unlike those of either hexagonal plates or cylindrical rods. These NPs, made from earth-abundant Mg, provide interesting ways to control light at the nanoscale across the ultraviolet, visible, and near-infrared spectral ranges.

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.

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.

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.

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.

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.

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.

Mechanics of invagination and folding: Hybridized instabilities when one soft tissue grows on another

We address the folding induced by differential growth in soft layered solids via an elementary model that consists of a soft growing neo-Hookean elastic layer adhered to a deep elastic substrate. As the layer-to-substrate modulus ratio is varied from above unity toward zero, we find a first transition from supercritical smooth folding followed by cusping of the valleys to direct subcritical cusped folding, then another to supercritical cusped folding. Beyond threshold, the high-amplitude fold spacing converges to about four layer thicknesses for many modulus ratios. In three dimensions, the instability gives rise to a wide variety of morphologies, including almost degenerate zigzag and triple-junction patterns that can coexist when the layer and substrate are of comparable softness. Our study unifies these results providing understanding for the complex and diverse fold morphologies found in biology, including the zigzag precursors to intestinal villi, and disordered zigzags and triple junctions in mammalian cortex.

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.

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.

Surface sulci in squeezed soft solids

The squeezing of soft solids, the constrained growth of biological tissues, and the swelling of soft elastic solids such as gels can generate large compressive stresses at their surfaces. This causes the otherwise smooth surface of such a solid to become unstable when its stress exceeds a critical value. Previous analyses of the surface instability have assumed two-dimensional plane-strain conditions, but in experiments isotropic stresses often lead to complex three-dimensional sulcification patterns. Here we show how such diverse morphologies arise by numerically modeling the lateral compression of a rigidly clamped elastic layer. For incompressible solids, close to the instability threshold, sulci appear as I-shaped lines aligned orthogonally with their neighbors; at higher compressions they are Y-shaped and prefer a hexagonal arrangement. In contrast, highly compressible solids when squeezed show only one sulcified phase characterized by a hexagonal sulcus network.

Supersoft elasticity in polydomain nematic elastomers

We consider the equilibrium stress-strain behavior of polydomain liquid crystal elastomers (PLCEs). We show that there is a fundamental difference between PLCEs cross-linked in the high temperature isotropic and low temperature aligned states. PLCEs cross-linked in the isotropic state then cooled to an aligned state will exhibit extremely soft elasticity (confirmed by recent experiments) and ordered director patterns characteristic of textured deformations. PLCEs cross-linked in the aligned state will be mechanically much harder and characterized by disclination textures.

Characterization of soft stripe-domain deformations in Sm-C and Sm-C* liquid-crystal elastomers

The neoclassical model of Sm-C (and Sm-C*) elastomers developed by Warner and Adams predicts a class of "soft" (zero energy) deformations. We find and describe the full set of stripe domains-laminate structures in which the laminates alternate between two different deformations-that can form between pairs of these soft deformations. All the stripe domains fall into two classes, one in which the smectic layers are not bent at the interfaces, but for which--in the Sm-C* case--the interfaces are charged, and one in which the smectic layers are bent but the interfaces are never charged. Striped deformations significantly enhance the softness of the macroscopic elastic response.

Semisoft elastic response of nematic elastomers to complex deformations

We consider a relaxed semisoft elastomer with its director oriented along the z axis that is first subjected to a large stretch in the x direction then to a slight x-z shear. We give a general argument that in any theory including director rotation, at the onset and end of the director rotation induced by these large stretches, there will be kinks in the stress-large strain curve (forming a stress-strain plateau) and zeros in the x-z shear modulus (C5) associated with small shears imposed on top of the stretches. We then find the analytical forms of the C5 -strain curves for a particular model of semisoftness (arising from compositional fluctuations) and show that it, together with the known stress-strain curve, provides the basis for a strong test of this theory. Finally, we consider the scope for other semisoft models and show that the compositional fluctuations model in fact yielded a generic form, that is, it is the most general quadratic free energy that does not explicitly include a final state direction other than the director. By introducing such additional directions, a large range of alternative models could be developed.