Multistability of segmented rings by programming natural curvature.

Multistable structures have widespread applications in the design of deployable aerospace systems, mechanical metamaterials, flexible electronics, and multimodal soft robotics due to their capability of shape reconfiguration between multiple stable states. Recently, the snap-folding of rings, often in the form of circles or polygons, has shown the capability of inducing diverse stable configurations. The natural curvature of the rod segment (curvature in its stress-free state) plays an important role in the elastic stability of these rings, determining the number and form of their stable configurations during folding. Here, we develop a general theoretical framework for the elastic stability analysis of segmented rings (e.g., polygons) based on an energy variational approach. Combining this framework with finite element simulations, we map out all planar stable configurations of various segmented rings and determine the natural curvature ranges of their multistable states. The theoretical and numerical results are validated through experiments, which demonstrate that a segmented ring with a rectangular cross-section can show up to six distinct planar stable states. The results also reveal that, by rationally designing the segment number and natural curvature of the segmented ring, its one- or multiloop configuration can store more strain energy than a circular ring of the same total length. We envision that the proposed strategy for achieving multistability in the current work will aid in the design of multifunctional, reconfigurable, and deployable structures.

One-dimensional spatial patterning along mitotic chromosomes: A mechanical basis for macroscopic morphogenesis.

Spatial patterns are ubiquitous in both physical and biological systems. We have recently discovered that mitotic chromosomes sequentially acquire two interesting morphological patterns along their structural axes [L. Chu et al., Mol. Cell, 10.1016/j.molcel.2020.07.002 (2020)]. First, axes of closely conjoined sister chromosomes acquire regular undulations comprising nearly planar arrays of sequential half-helices of similar size and alternating handedness, accompanied by periodic kinks. This pattern, which persists through all later stages, provides a case of the geometric form known as a "perversion." Next, as sister chromosomes become distinct parallel units, their individual axes become linked by bridges, which are themselves miniature axes. These bridges are dramatically evenly spaced. Together, these effects comprise a unique instance of spatial patterning in a subcellular biological system. We present evidence that axis undulations and bridge arrays arise by a single continuous mechanically promoted progression, driven by stress within the chromosome axes. We further suggest that, after sister individualization, this same stress also promotes chromosome compaction by rendering the axes susceptible to the requisite molecular remodeling. Thus, by this scenario, the continuous presence of mechanical stress within the chromosome axes could potentially underlie the entire morphogenetic chromosomal program. Direct analogies with meiotic chromosomes suggest that the same effects could underlie interactions between homologous chromosomes as required for gametogenesis. Possible mechanical bases for generation of axis stress and resultant deformations are discussed. Together, these findings provide a perspective on the macroscopic changes of organized chromosomes.

Bending Instability of Rod-Shaped Bacteria.

A thin-walled tube, e.g., a drinking straw, manifests an instability when bent by localizing the curvature change in a small region. This instability has been extensively studied since the seminal work of Brazier nearly a century ago. However, the scenario of pressurized tubes has received much less attention. Motivated by rod-shaped bacteria such as E. coli, whose cell walls are much thinner than their radius and are subject to a substantial internal pressure, we study, theoretically, how this instability is affected by this internal pressure. In the parameter range relevant to the bacteria, we find that the internal pressure significantly postpones the onset of the instability, while the bending stiffness of the cell wall has almost no influence. This study suggests a new method to infer turgor pressure in rod-shaped bacteria from bending experiments.

Nanoscale ripples on polymers created by a focused ion beam.

We show that focused ion beam irradiation results in the creation of peculiar one- and two-dimensional nanoscale features on the surface of polyimide-a common polymer in electronics, large scale structures, and the automobile industry, as well as in biomedical applications. The role of ion beam incident angle, acceleration voltage, and fluence on the morphology of the structural features is systematically investigated, and insights into the mechanisms of formation of these nanoscale features are provided. Moreover, by using the maskless patterning method of the focused ion beam system, we have developed a robust technique for controlled modification of the polymeric surface. The technique, which is analogous to using a gray glass with varying darkness to control the radiation from the sun, but at a much smaller scale, enables the ion intensity and angle to be controlled at each surface point of the polymer, giving rise to structural surface features with desired shape and morphology.

Nonlinear dynamics of spherical shells buckling under step pressure.

Dynamic buckling is addressed for complete elastic spherical shells subject to a rapidly applied step in external pressure. Insights from the perspective of nonlinear dynamics reveal essential mathematical features of the buckling phenomena. To capture the strong buckling imperfection-sensitivity, initial geometric imperfections in the form of an axisymmetric dimple at each pole are introduced. Dynamic buckling under the step pressure is related to the quasi-static buckling pressure. Both loadings produce catastrophic collapse of the shell for conditions in which the pressure is prescribed. Damping plays an important role in dynamic buckling because of the time-dependent nonlinear interaction among modes, particularly the interaction between the spherically symmetric 'breathing' mode and the buckling mode. In general, there is not a unique step pressure threshold separating responses associated with buckling from those that do not buckle. Instead, there exists a cascade of buckling thresholds, dependent on the damping and level of imperfection, separating pressures for which buckling occurs from those for which it does not occur. For shells with small and moderately small imperfections, the dynamic step buckling pressure can be substantially below the quasi-static buckling pressure.

Nonlinear buckling behaviour of spherical shells: barriers and symmetry-breaking dimples.

The nonlinear axisymmetric post-buckling behaviour of perfect, thin, elastic spherical shells subject to external pressure and their asymmetric bifurcations are characterized, providing results for a structure/loading combination with an exceptionally nonlinear buckling response. Immediately after the onset of buckling, the buckling mode localizes into a dimple at the poles. The relations among the pressure, the dimple amplitude and the change in volume of the shell are determined over a large range of pole deflections. These results allow accurate evaluation of criteria such as the Maxwell condition for which the energies in the unbuckled and buckled states are the same and evaluation of the influences of pressure versus volume-controlled loadings. Non-axisymmetric bifurcation from the axisymmetric state, which occurs deep into the post-buckling regime in the form of multi-lobed dimples, is also established and discussed.This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications.'

Period-doubling and period-tripling in growing bilayered systems.

Growing layers on elastic substrates are capable of creating a wide variety of surface morphologies. Moderate growth generates a regular pattern of sinusoidal wrinkles with a homogeneous energy distribution. While the critical conditions for periodic wrinkling have been extensively studied, the rich pattern formation beyond this first instability point remains poorly understood. Here we show that upon continuing growth, the energy progressively localizes and new complex morphologies emerge. Previous studies have often overlooked these secondary bifurcations; they have focused on large stiffness ratios between layer and substrate, where primary instabilities occur early, long before secondary instabilities emerge. We demonstrate that secondary bifurcations are particularly critical in the low stiffness ratio regime, where the critical conditions for primary and secondary instabilities move closer together. Amongst all possible secondary bifurcations, the mode of period-doubling plays a central role - it is energetically favorable over all other modes. Yet, we can numerically suppress period-doubling, by choosing boundary conditions, which favor alternative higher order modes. Our results suggest that in the low stiffness regime, pattern formation is highly sensitive to small imperfections: surface morphologies emerge rapidly, change spontaneously, and quickly become immensely complex. This is a common paradigm in developmental biology. Our results have significantly applications in the morphogenesis of living systems where growth is progressive and stiffness ratios are low.

pH-responsive scaffolds generate a pro-healing response.

A principal challenge in wound healing is a lack of cell recruitment, cell infiltration, and vascularization, which occurs in the absence of temporal and spatial cues. We hypothesized that a scaffold that expands due to local changes in pH may alter oxygen and nutrient transport and the local cell density, leading to enhanced cell deposition and survival. In this study, we present a pH-responsive scaffold that increases oxygen transport, as confirmed by our finite element model analysis, and cell proliferation relative to a non-responsive scaffold. In vivo, responsive scaffolds induce a pro-healing gene expression profile indicative of enhanced angiogenesis, granulation tissue formation, and tissue remodeling. Scaffolds that stretch in response to their environment may be a hallmark for tissue regeneration.

Ridge localizations and networks in thin films compressed by the incremental release of a large equi-biaxial pre-stretch in the substrate.

Two-dimensional ridge structures are induced by equi-biaxial compression with large equi-biaxial pre-stretch in a thin film by using a micro-fluidics technique. Whereas wrinkles tend to be uniformly distributed, ridges are localized. The wrinkle-to-ridge transition is unstable (subcritical), resulting in large amplitude changes. The nature and morphology of the ridges is studied and quantified by experiments and numerical simulations.

The role of nonlinear substrate elasticity in the wrinkling of thin films.

The role of substrate nonlinearity in the stability of wrinkling of thin films bonded to compliant substrates is investigated within the initial post-bifurcation range when wrinkling first emerges. A fully nonlinear neo-Hookean bilayer composed of a thin film on a deep substrate is analysed for a wide range of the film-substrate stiffness ratio, from films that are very stiff compared with the substrate to those only slightly stiffer. Substrate pre-stretch prior to film attachment is shown to have a significant effect on the nonlinearity relevant to wrinkling. Two dimensionless parameters are identified that control the stability and mode shape evolution of the bilayer: one specifying arbitrary uniform substrate pre-stretch and the other a stretch-modified modulus ratio. For systems with film stiffness greater than about five times that of the substrate the wrinkling bifurcation is stable, whereas for systems with smaller relative film stiffness bifurcation can be unstable, especially if substrate pre-stretch is not tensile.

Nanopore fabrication in amorphous Si: Viscous flow model and comparison to experiment.

Nanopores fabricated in free-standing amorphous silicon thin films were observed to close under 3 keV argon ion irradiation. The closing rate, measured in situ, exhibited a memory effect: at the same instantaneous radius, pores that started larger close more slowly. An ion-stimulated viscous flow model is developed and solved in both a simple analytical approximation for the small-deformation limit and in a finite element solution for large deformations. The finite-element solution exhibits surprising changes in cross-section morphology, which may be extremely valuable for single biomolecule detection, and are untested experimentally. The finite-element solution reproduces the shape of the measured nanopore radius versus fluence behavior and the sign and magnitude of the measured memory effect. We discuss aspects of the experimental data not reproduced by the model, and successes and failures of the competing adatom diffusion model.

Wrinkled hard skins on polymers created by focused ion beam.

A stiff skin forms on surface areas of a flat polydimethylsiloxane (PDMS) upon exposure to focused ion beam (FIB) leading to ordered surface wrinkles. By controlling the FIB fluence and area of exposure of the PDMS, one can create a variety of patterns in the wavelengths in the micrometer to submicrometer range, from simple one-dimensional wrinkles to peculiar and complex hierarchical nested wrinkles. Examination of the chemical composition of the exposed PDMS reveals that the stiff skin resembles amorphous silica. Moreover, upon formation, the stiff skin tends to expand in the direction perpendicular to the direction of ion beam irradiation. The consequent mismatch strain between the stiff skin and the PDMS substrate buckles the skin, forming the wrinkle patterns. The induced strains in the stiff skin are estimated by measuring the surface length in the buckled state. Estimates of the thickness and stiffness of the stiffened surface layer are estimated by using the theory for buckled films on compliant substrates. The method provides an effective and inexpensive technique to create wrinkled hard skin patterns on surfaces of polymers for various applications.

Highly selective directed assembly of functional actomyosin on Au surfaces.

The capability of assembling biomotors onto specific locations of solid substrates is a key for development of biomotor-based nanomechanical systems. We developed a method to direct the assembly of the heavy meromyosin fragment from rabbit skeletal muscle myosin onto specific locations of Au substrates utilizing surface molecular patterns. In this strategy, chemically directed patterns of streptavidin were achieved to direct highly specific assembly of biotinylated heavy meromyosin on the substrates--a strategy applicable for patterning a variety of biotinylated molecules--while BSA was utilized to avoid nonspecific adsorption. In vitro motility assays of filament sliding were used to confirm functionality of assembled actomyosin.

Self-assembled shells composed of colloidal particles: fabrication and characterization.

We construct shells with tunable morphology and mechanical response with colloidal particles that self-assemble at the interface of emulsion droplets. Particles self-assemble to minimize the total interfacial energy, spontaneously forming a particle layer that encapsulates the droplets. We stabilize these layers to form solid shells at the droplet interface by aggregating the particles, connecting the particles with adsorbed polymer, or fusing the particles. These techniques reproducibly yield shells with controllable properties such as elastic moduli and breaking forces. To enable diffusive exchange through the particle shells, we transfer them into solvents that are miscible with the encapsulant. We characterize the mechanical properties of the shells by measuring the response to deformation by calibrated microcantilevers.

Simulation of indentation fracture in crystalline materials using mesoscale self-assembly.

A new physical model based on mesoscale self-assembly is developed to simulate indentation fracture in crystalline materials. Millimeter-scale hexagonal objects exhibiting atom-like potential functions were designed and allowed to self-assemble into two-dimensional (2D) aggregates at the interface between water and perfluorodecalin. Indentation experiments were performed on these aggregates, and the stresses and strains involved in these processes were evaluated. The stress field in the aggregates was analyzed theoretically using the 2D elastic Hertz solution. Comparison of the experimental results with theoretical analysis revealed that fracture develops in regions subjected to high shear stress and some, albeit low, tensile stress. The potential for the broader application of the model is illustrated using indentation of assemblies with point defects and adatoms introduced at predetermined locations, and using a two-phase aggregate simulating a compliant film on a stiff substrate.

Self-assembled polymer membrane capsules inflated by osmotic pressure.

We fabricate and characterize capsules that are composite membranes, made of a polymer network stabilized by adsorption to colloids and inflated by osmotic pressure from internal free polyelectrolyte; here, poly-l-lysine forms the network and inflates the capsules. To assess these capsules' properties and structure, we deform capsules using microcantilevers and use finite element modeling to describe these deformations. Additional experimental tests confirm the model's validity. These capsules' resilient response to mechanical forces indicates that loading and shear should be good triggers for the release of contents via deformation. The osmotic pressure inflating these capsules has the potential to trigger release of contents via deflation in response to changes in the capsules' environment; we demonstrate addition of salt as a trigger for deflating capsules. Because these capsules have a variety of release triggers available and the technique used to fabricate them is very flexible and allows high encapsulation efficiency, these capsules have very high potential for application in many areas.