Buckling and metastability in membranes with dilation arrays

Universal moiré buckling of freestanding 2D bilayers

The physics of membranes, a classic subject, acquires new momentum from two-dimensional (2D) materials multilayers. This work reports the surprising results emerged during a theoretical study of equilibrium geometry of bilayers as freestanding membranes. While ordinary membranes are prone to buckle around compressive impurities, we predict that all 2D material freestanding bilayers universally undergo, even if impurity-free, a spontaneous out-of-plane buckling. The moiré network nodes here play the role of internal impurities, the dislocations that join them giving rise to a stress pattern, purely shear in homobilayers and mixed compressive/shear in heterobilayers. That intrinsic stress is, theory and simulations show, generally capable to cause all freestanding 2D bilayers to undergo distortive bucklings with large amplitudes and a rich predicted phase transition scenario. Realistic simulations predict quantitative parameters expected for these phenomena as expected in heterobilayers such as graphene/hBN, [Formula: see text] heterobilayers, and for twisted homobilayers such as graphene, hBN, [Formula: see text]. Buckling then entails a variety of predicted consequences. Mechanically, a critical drop of bending stiffness is expected at all buckling transitions. Thermally, the average buckling corrugation decreases with temperature, with buckling-unbuckling phase transitions expected in some cases, and the buckled state often persisting even above room temperature. Buckling will be suppressed by deposition on hard attractive substrates, and survives in reduced form on soft ones. Frictional, electronic, and other associated phenomena are also highlighted. The universality and richness of these predicted phenomena strongly encourages an experimental search, which is possible but still missing.

Chirality effects in molecular chainmail

Motivated by the observation of positive Gaussian curvature in kinetoplast DNA networks, we consider the effect of linking chirality in square lattice molecular chainmail networks using Langevin dynamics simulations and constrained gradient optimization. Linking chirality here refers to ordering of over-under versus under-over linkages between a loop and its neighbors. We consider fully alternating linking, maximally non-alternating, and partially non-alternating linking chiralities. We find that in simulations of polymer chainmail networks, the linking chirality dictates the sign of the Gaussian curvature of the final state of the chainmail membranes. Alternating networks have positive Gaussian curvature, similar to what is observed in kinetoplast DNA networks. Maximally non-alternating networks form isotropic membranes with negative Gaussian curvature. Partially non-alternating networks form flat diamond-shaped sheets which undergo a thermal folding transition when sufficiently large, similar to the crumpling transition in tethered membranes. We further investigate this topology-curvature relationship on geometric grounds by considering the tightest possible configurations and the constraints that must be satisfied to achieve them.

Multistable sheets with rewritable patterns for switchable shape-morphing

Flat sheets patterned with folds, cuts or swelling regions can deform into complex three-dimensional shapes under external stimuli1-24. However, current strategies require prepatterning and lack intrinsic shape selection5-24. Moreover, they either rely on permanent deformations6,12-14,17,18, preventing corrections or erasure of a shape, or sustained stimulation5,7-11,25, thus yielding shapes that are unstable. Here we show that shape-morphing strategies based on mechanical multistability can overcome these limitations. We focus on undulating metasheets that store memories of mechanical stimuli in patterns of self-stabilizing scars. After removing external stimuli, scars persist and force the sheet to switch to sharply selected curved, curled and twisted shapes. These stable shapes can be erased by appropriate forcing, allowing rewritable patterns and repeated and robust actuation. Our strategy is material agnostic, extendable to other undulation patterns and instabilities, and scale-free, allowing applications from miniature to architectural scales.

Computational model of twisted elastic ribbons

We develop an irregular lattice mass-spring model to simulate and study the deformation modes of a thin elastic ribbon as a function of applied end-to-end twist and tension. Our simulations reproduce all reported experimentally observed modes, including transitions from helicoids to longitudinal wrinkles, creased helicoids and loops with self-contact, and transverse wrinkles to accordion self-folds. Our simulations also show that the twist angles at which the primary longitudinal and transverse wrinkles appear are well described by various analyses of the Föppl-von Kármán equations, but the characteristic wavelength of the longitudinal wrinkles has a more complex relationship to applied tension than previously estimated. The clamped edges are shown to suppress longitudinal wrinkling over a distance set by the applied tension and the ribbon width, but otherwise have no apparent effect on measured wavelength. Further, by analyzing the stress profile, we find that longitudinal wrinkling does not completely alleviate compression, but caps the magnitude of the compression. Nonetheless, the width over which wrinkles form is observed to be wider than the near-threshold analysis predictions: the width is more consistent with the predictions of far-from-threshold analysis. However, the end-to-end contraction of the ribbon as a function of twist is found to more closely follow the corresponding near-threshold prediction as tension in the ribbon is increased, in contrast to the expectations of far-from-threshold analysis. These results point to the need for further theoretical analysis of this rich thin elastic system, guided by our physically robust and intuitive simulation model.

Buckling in a rotationally invariant spin-elastic model

Scanning tunneling microscopy experiments have revealed a spontaneous rippled-to-buckled transition in heated graphene sheets, in absence of any mechanical load. Several models relying on a simplified picture of the interaction between elastic and internal, electronic, degrees of freedom have been proposed to understand this phenomenon. Nevertheless, these models are not fully consistent with the classical theory of elasticity, since they do not preserve rotational invariance. Herein, we develop and analyze an alternative classical spin-elastic model that preserves rotational invariance while giving a qualitative account of the rippled-to-buckled transition. By integrating over the internal degrees of freedom, an effective free energy for the elastic modes is derived, which only depends on the curvature. Minimization of this free energy gives rise to the emergence of different mechanical phases, whose thermodynamic stability is thoroughly analyzed, both analytically and numerically. All phases are characterized by a spatially homogeneous curvature, which plays the role of the order parameter for the rippled-to-buckled transition, in both the one- and two-dimensional cases. In the latter, our focus is put on the honeycomb lattice, which is representative of actual graphene.

Memory from coupled instabilities in unfolded crumpled sheets

Crumpling an ordinary thin sheet transforms it into a structure with unusual mechanical behaviors, such as enhanced rigidity, emission of crackling noise, slow relaxations, and memory retention. A central challenge in explaining these behaviors lies in understanding the contribution of the complex geometry of the sheet. Here we combine cyclic driving protocols and three-dimensional (3D) imaging to correlate the global mechanical response and the underlying geometric transformations in unfolded crumpled sheets. We find that their response to cyclic strain is intermittent, hysteretic, and encodes a memory of the largest applied compression. Using 3D imaging we show that these behaviors emerge due to an interplay between localized and interacting geometric instabilities in the sheet. A simple model confirms that these minimal ingredients are sufficient to explain the observed behaviors. Finally, we show that after training, multiple memories can be encoded, a phenomenon known as return point memory. Our study lays the foundation for understanding the complex mechanics of crumpled sheets and presents an experimental and theoretical framework for the study of memory formation in systems of interacting instabilities.

Anomalous Thermal Expansion in Ising-like Puckered Sheets

Motivated by efforts to create thin nanoscale metamaterials and understand atomically thin binary monolayers, we study the finite temperature statistical mechanics of arrays of bistable buckled dilations embedded in free-standing two-dimensional crystalline membranes that are allowed to fluctuate in three dimensions. The buckled nodes behave like discrete, but highly compressible, Ising spins, leading to a phase transition at T_{c} with singularities in the staggered "magnetization," susceptibility, and specific heat, studied via molecular dynamics simulations. Unlike conventional Ising models, we observe a striking divergence and sign change of the coefficient of thermal expansion near T_{c} caused by the coupling of flexural phonons to the buckled spin texture. We argue that a phenomenological model coupling Ising degrees of freedom to the flexural phonons in a thin elastic sheet can explain this unusual response.

Shape multistability in flexible tubular crystals through interactions of mobile dislocations

Crystalline sheets rolled up into cylinders occur in diverse biological and synthetic systems, including carbon nanotubes, biofilaments of the cellular cytoskeleton, and packings of colloidal particles. In this work, we show, computationally, that such tubular crystals can be programmed with reconfigurable shapes, due to motions of defects that interrupt the periodicity of the crystalline lattice. By identifying and exploiting stable patterns of these defects, we cause tubular crystals to relax into desired target geometries, a design principle that could guide the creation of versatile colloidal analogues to nanotubes. Our results suggest routes to tunable and switchable material properties in ordered, soft materials on deformable surfaces.

We study avenues to shape multistability and shape morphing in flexible crystalline membranes of cylindrical topology, enabled by glide mobility of dislocations. Using computational modeling, we obtain states of mechanical equilibrium presenting a wide variety of tubular crystal deformation geometries, due to an interplay of effective defect interactions with out-of-tangent-plane deformations that reorient the tube axis. Importantly, this interplay often stabilizes defect configurations quite distinct from those predicted for a two-dimensional crystal confined to the surface of a rigid cylinder. We find that relative and absolute stability of competing states depend strongly on control parameters such as bending rigidity, applied stress, and spontaneous curvature. Using stable dislocation pair arrangements as building blocks, we demonstrate that targeted macroscopic three-dimensional conformations of thin crystalline tubes can be programmed by imposing certain sparse patterns of defects. Our findings reveal a broad design space for controllable and reconfigurable colloidal tube geometries, with potential relevance also to architected carbon nanotubes and microtubules.

Flatness and intrinsic curvature of linked-ring membranes

Recent experiments have elucidated the physical properties of kinetoplasts, which are chain-mail-like structures found in the mitochondria of trypanosome parasites formed from catenated DNA rings. Inspired by these studies, we use Monte Carlo simulations to examine the behavior of two-dimensional networks ("membranes") of linked rings. For simplicity, we consider only identical rings that are circular and rigid and that form networks with a regular linking structure. We find that the scaling of the eigenvalues of the shape tensor with membrane size are consistent with the behavior of the flat phase observed in self-avoiding covalent membranes. Increasing ring thickness tends to swell the membrane. Remarkably, unlike covalent membranes, the linked-ring membranes tend to form concave structures with an intrinsic curvature of entropic origin associated with local excluded-volume interactions. The degree of concavity increases with increasing ring thickness and is also affected by the type of linking network. The relevance of the properties of linked-ring model membranes to those observed in kinetoplasts is discussed.

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

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