Session T07: Buckling Instabilities of Thin Materials II

Dynamics of strained nanoribbons at finite temperature

The mechanics and the equilibrium critical behavior (including scale-dependent elastic constants) of thermally fluctuating two-dimensional crystalline membranes, such as graphene, have been studied extensively over the last two decades through field theoretic approaches, numerical simulations and experiments. However, investigations of dynamics, such as the characteristic oscillation times and damping times, remain limited. Here, we use molecular dynamics simulations to study the time trajectory of the midpoint (the height center-of-mass) of doubly clamped nanoribbons under various strain conditions. By treating the nanoribbon midpoint as a Brownian particle confined to a double-well potential, we formulate an effective theory describing the ribbon's tunneling rate across the two wells and its oscillation inside the wells. We find that, for nanoribbons compressed above the Euler buckling point and thermalized above the temperature at which the non-linear effects become significant, the energy barrier increases linearly with temperature. The cancelation between the energy barrier and the thermal energy results in escape time that depends only on the geometry, which is quite different from the usual Arrhenius behavior. Similarly, the natural oscillation time of nanoribbons under tension also becomes temperature dependent due to bending stiffening. Our findings suggest a simple connection between the dynamical critical exponent describing the collective motion characterizing by the midpoint time history, and the static critical exponent near the buckling transition.   

Geometrical Control of the Tilt Transition in Single-Clamped Thermalized Elastic Sheets

We study the dynamics of the tilt transition of thermalized thin elastic sheets clamped at one end only in the manner of a cantilever. Beyond a critical strain, such a sheet undergoes a tilt transition with a finite energy barrier separating the spontaneously chosen tilt plane (up) from its oppositely oriented state (down). The finite barrier implies that, over long time scales, the elastic sheet may transition between the two states, residing in each for a finite time. Naively, temperature might be assumed as the primary driver for these transitions, but we find that geometric characteristics, in particular the aspect ratio of the sheet, is the key controller of the transition probability. Using a combination of an effective mean field elastic theory and transition-state theory, we derive an expression for the rate of transition between the two tilted states. We show that, at scales larger than a thermal length scale, renormalization of the elastic constants weakens the temperature dependence and allows for geometrical factors to dominate.

    

Active buckling of pressurized spherical shells

We study the buckling of pressurized spherical shells by Monte Carlo simulations in which the detailed balance is explicitly broken -- thereby driving the shell active, out of thermal equilibrium. Such a shell typically has either higher (active) or lower (quiescent) fluctuations compared to one in thermal equilibrium depending on how the detailed balance is broken. We show that for the same set of elastic parameters, a shell that is not buckled in thermal equilibrium can be buckled if turned active. Similarly, a shell that is buckled in thermal equilibrium can unbuckle if turned quiescent. Based on this result, we suggest that it is possible to experimentally design microscopic elastic shells whose buckling can be optically controlled   

Rate-controlled buckling of elastic thin films bonded to viscous substrates: Slow compression gives localized ridges; fast compression gives wrinkles

We examine the buckling of a thin elastic film bonded to a much thicker viscous substrate undergoing compression at a fixed rate. Experiments show two distinct buckling modes. At high rate of compression or at large liquid thickness, the buckles take on the form of approximately-sinusoidal wrinkles. At low rate of compression or at small liquid thickness, the buckles are highly localized into tall ridges separated by nearly flat regions. Such ridge formation is entirely distinct from other examples of curvature localization such as fold localization of films on liquids in static equilibrium or ridge formation of films bonded to hyperelastic substrates. We quantify how the two lengthscales that arise from the buckling process, the wrinkle amplitude and the interridge distance, depend on compression rate and liquid thickness. We also quantify the large lateral motion of the film in the near-ridge region that accompanies ridge growth; this lateral motion implies severe shear flow in the liquid near the ridges. Simulations show that ridge localization appears due to a competition between two effects: a buckle mode with a few well-spaced ridges offers a lower energy state than uniform wrinkles, but wrinkles can develop faster because they require the viscous fluid to move over shorter distances.   

Ridge Localization Driven by Natural Imperfections

While buckling is a time independent phenomena for filaments or films bonded to soft elastic substrates, time evolution plays an important role when the substrate is a viscous fluid. Here we show that buckling instabilities in fluid-structure interactions can be reduced to the analysis of a growth function that amplifies the initial noise characterizing experimental or numerical error. The convolution between a specific growth function and noise leads to natural imperfections that emerge in the form of wave packets with a large scale modulation that could transform into localized structures depending on nonlinear effects. Specifically, we provide an experimental example where these wave packets are amplified into ridges for sufficiently low compression rates or are diluted into wrinkles for high compression rates.   

Wrinkling of soft composite sheets

We examine the buckling shape and critical compression of a confined composite sheet floating on a liquid. The sheet is inhomogeneous due to the presence of liquid inclusions, such that its elastic properties vary in the direction of uniaxial compression. From the Föppl–von Kármán model of Hookean elasticity, we formulate a fourth order eigenvalue problem where the eigenvalue is the compressive load imposed on the sheet and the eigenfunction is the midplane displacement. Assuming that the volume fraction of the liquid inclusions is small, we can solve the eigenvalue problem perturbatively. To leading order the sheet is homogeneous and we find that the buckling shape has a symmetric and an antisymmetric buckling mode. The mode associated with the minimum compressive load depends on the natural length of the compressed sheet. The symmetry changes are realized at the next order of the analysis, where the liquid inclusions are influential. Hence, we treat the inhomogeneities using a Galerkin method with enforced P-T symmetry. In the case of a large sheet, we show that the wrinkles are localized and are treated using a WKB method.   

Viscoelastic creasing: Free-surface instability of a viscoelastic liquid under compression

It has been known for many decades that when the free surface of an elastic material such as a block of rubber experiences severe compression, its surface develops sharp cusp-like creases. We show that a viscoelastic liquid undergoes a similar, but rate-dependent, surface-creasing instability. Experiments on a well-entangled molten polymer compressed at a controlled rate show that the strain required for creasing increases as rate decreases. A model is developed wherein the creasing criterion known previously for neo-Hookean elastic solids is applied to the elastic portion of the deformation of a viscoelastic liquid. Using the upper-convected Maxwell model, we derive an analytical criterion for viscoelastic creasing which is in good agreement with experiments. It predicts that the strain for creasing increases with decreasing Weissenberg number, and creasing is not possible below a critical Weissenberg number.

Full article: Guan et al, Extreme Mechanics Letters, 2022 https://doi.org/10.1016/j.eml.2022.101784   

Pseudomomentum balance in thin structures, deferred

(This will essentially be the talk I was supposed to give in March 2020).  The balance of pseudomomentum is associated with material symmetry in continua.  We will employ it in the context of thin, flexible structures, to explain conserved quantities in rotating conical membranes and propulsive forces on confined elastic rods.

    

Consequences of constitutive choices in soft and thin materials

Biopolymer gels and soft tissues are known to present rich nonlinear elastic responses, including strain-stiffening and a surprising tendency to develop a tensile stress perpendicular to the shearing direction (reverse Poynting effect). However, popular strain energies adopted for constitutive modeling fail to capture these responses, lead to universal relations that conflict with experimental data, and do not provide a satisfactory link with the mesoscale. For soft materials governed by quadratic energies, the choice of strain measure in which we expand the energy can be key to unlocking some of these behaviors. I will discuss that by adopting the Biot/Bell strain, instead of the usual difference in metrics, diverse nonlinear deformations can be captured, including a novel transition in the Poynting effect. Other overlooked consequences of the strain choice appear when deriving energies for thin structures. In fact, most strains lead to undesirable behaviors, such as materials that stretch under a pure moment. I will explain how to avoid this contamination between bending and stretching, which is of importance in the study of buckling and defects on thin sheets. This talk contains contributions from J. A. Hanna and C. O. Horgan.   

Elasticity and Morphology of Flexible Solid Domains in Lipid Vesicles

In biological systems, lipid rafts play important roles in controlling structure and function in membranes.  While analogous phenomena are well-studied in the context of fluid-fluid domain separation in multi-component system, far less is understood about the interplay between elasticity and shape in multi-phase vesicles, where one of those phases is a 2D solid.  Motivated to understand the rich arrange of emergent shapes and morphologies in fluid-solid composite vesicles, we study the elasticity of the simplest case, a single solid domain embedded in a fluid vesicle with a spherical topology.  As nanometrically thin materials, both solid and fluid regions are highly-flexible to bending deformations. Yet, the resistance to in-plane shear imbues solid domains (i.e. elastic sheets) with a geometrically-nonlinear resistance to changes of Gaussian curvature, which frustrates the preference of bending energy for globally spherical shapes.  We first describe the analytical results for axisymmetric shapes under the assumption that the solid domain remains strictly planar, for which we show that elastic energy of the composite vesicle to be a non-monotonic function of internal volume, with an optimal degree of inflation dependent on the size of the solid domain.  Next, we describe the more complex shapes that emerge when the vesicle relaxes from axisymmetric shape based on Surface Evolver computations.  We find that, even in the limit of infinitely large solid stretching to bending costs (otherwise the limit of large vesicle radius), there exists a rich pattern of nearly-isometric deformations of the solid domain that lead to pronounced relaxation of the elastic cost of vesicle inflation.  We explore what controls the geometric pattern of nearly-isometric “folding” of the solid domain, as well as the transition to non-isometric shape equilibria when the ratio of solid stretching to bending modulus (or radius) decreases.  Motivated by observations of highly symmetry-broken solid domains in experiments, we describe preliminary results for the elastic energy of non-axisymmetric solid domain shapes on vesicles, motivated by recent observations of solidification process in multi-component vesicles exhibiting non-convex, flower-like solid domains.

    

Why do drying colloidal suspension drops bend out of plane?

The drying of sessile drops of concentrated silica nanoparticle suspensions creates fractured and bent dried deposits that resemble blooming flowers. As water evaporates from a freshly deposited drop, a solidification front propagates from the edge of the drop leaving behind a thin close-packed particle deposit that eventually covers the entire wetted area. Evaporation from the porous particle deposit and adhesion to the substrate lead to in-plane strains in the deposit that are released by the formation of regular radial cracks that define the 'petals' of the flower-like pattern. The deposit then progressively delaminates from the substrate and the petals bend upward. Using a poroelastic model combined with non-Euclidian plate simulations, we show that evaporation leads to non-uniform in-plane strains that cause the deposit to buckle out of plane. The balance between the in-plane stretching energy released by buckling and the out-of-plane bending energy cost sets the amplitude of the deformation. Our work provides a unified picture of crack formation and deformation as two ways to release strains in thin drying deposits.   

Adhesion-adjustable passive suction cup using an elasticity of hemispherical cap

The passive suction cup can produce adhesion without relying on a pneumatic device, which fundamentally solves the low energy efficiency, large volume, and weight problems of an active suction cup. It is especially attracting the attention of wall-climbing robots, where lightweight is essential. In addition, the wall-climbing robots need a moderately small adhesion when moving on the wall and, conversely, a high adhesive force to be fixed and work for a long time at a specific location. Thus, the adjustability of the adhesive force of the suction cup will expand the versatility of the robot. Here, we develop passive suction cups applying a mechanism that can produce high adhesive performance and can easily control the formed inner pressure. The suction cup consists of an elastic hemispherical cap and a flat brim that contacts with a wall. The cap largely deforms when subjected to indentation and creates a negative pressure returning to the hemispherical shape upon the disappearance of the indentation. The formed inner pressure can be changed by varying the indentation depth. We measure the generated pressure difference depending on indentation depth according to several conditions of the indenter. Furthermore, we also measure vertically and horizontally the pull-off force of the suction cup with various pulling speeds and the peel-off time with various additional loading.   

Polymer physics-based classification of neurons

Recognizing that diverse morphologies of neurons are reminiscent of structures of branched polymers, we put forward a principled and systematic way of classifying neurons that employs the ideas of polymer physics. In particular, we use 3D coordinates of individual neurons, which are accessible in recent neuron reconstruction datasets from electron microscope images. We numerically calculate the form factor, F(q), a Fourier transform of the distance distribution of particles comprising an object of interest, which is routinely measured in scattering experiments to quantitatively characterize the structure of materials. For a polymer-like object consisting of n monomers spanning over a length scale of r, F(q) scales with the wavenumber q(=2π/r) as F(q) ~ q^-D at an intermediate range of q, where D is the fractal dimension or the inverse scaling exponent (D=ν^-1) characterizing the geometrical feature (r ~ n^ν) of the object. F(q) can be used to describe a neuron morphology in terms of its size (R_n) and the extent of branching quantified by D. By defining the distance between F(q)s as a measure of similarity between two neuronal morphologies, we tackle the neuron classification problem. In comparison with other existing classification methods for neuronal morphologies, our F(q)-based classification rests solely on 3D coordinates of neurons with no prior knowledge of morphological features. When applied to publicly available neuron datasets from three different organisms, our method not only complements other methods but also offers a physical picture of how the dendritic and axonal branches of an individual neuron fill the space of dense neural networks inside the brain. Additionally, we compare the morphology-based and proximity-based classification using the Drosophila olfactory system, where the spatial organization of neurons is shown to be associated with functional and connectivity features suggestive of a degree of labeled-line strategy at work.   

Quantized Gravity and Cells at Planck Scale

A novel experiment tests Planck masses and a quantized gravity. The Planck mass 2.2 x 10^[-8] kg is an observable quantity similar to a flea's egg. We place two spherical sub-Planck masses on a level low-friction surface, grounded within a vacuum chamber, and observe for gravitational attraction. A negative result indicates that gravitational mass is quantized at the Planck scale. Quantum mechanics has applications for astrobiology and living cells, explaining why most cells are limited by the Planck mass. A living cell exchanges material within its walls through processes such as diffusion. In the process of mitosis, chromatids containing genetic material migrate to opposite ends of the cell. If the cell were larger than Planck mass, self-gravity would interfere with its internal processes. This continuing research connects cosmology of the large Universe with the microscopic world   

Submicron Ni/Au/Ge contacts to an AlGaAs/GaAs two-dimensional electron gas

We report on the results of a study to optimize submicron alloyed ohmic contacts to the two-dimensional electron gas (2DEG) in AlGaAs/GaAs heterostructures. We vary the geometry, annealing parameters, and metal orientation relative to the crystallographic axes of the host GaAs crystal to determine the optimal parameters for low resistance contacts and to study the mechanisms that limit the formation of Ohmic behavior for very small areas. We discuss both electrical and structural characterization data. Our study has generated contact resistance below 600 Ω for contact areas <1 um².

 

Work supported by the Quantum Science Center at Oak Ridge National laboratory under contract number DE-AC05-00OR22725.