Elastic Membrane Theory Research Articles

Elastometry of Complex Fluid Pendant Capsules.

Oil/water interfaces are ubiquitous in nature. Opposing polarities at these interfaces attract surface-active molecules, which can seed complex viscoelastic or even solid interfacial structure. Biorelevant proteins such as hydrophobin, polymers such as PNIPAM, and the asphaltenes in crude oil (CRO) are examples of some systems where such layers can occur. When a pendant drop of CRO is aged in brine, it can form an interfacial elastic membrane of asphaltenes so stiff that it wrinkles and crumples upon retraction. Most of the work studying CRO/brine interfaces focuses on the viscoelastic liquid regime, leaving a wide range of fully solidified, elastic interfaces largely unexplored. In this work, we quantitatively measure elasticity in all phases of drop retraction. In early retraction, the interface shows a fluid viscoelasticity measurable using a Gibbs isotherm or dilatational rheology. Further retraction causes a phase transition to a 2D elastic solid with nonisotropic, nonhomogeneous surface stresses. In this regime, we use new techniques in the elastic membrane theory to fit for the elasticities of these solid capsules. These elastic measurements can help us develop a deeper understanding not only of CRO interfaces but also of the myriad fluid systems with solid interfacial layers.

Bubble-Induced Strong Thermal Contraction for Graphene

Abstract The thermal expansion coefficient (TEC) of van der Waals (vdW) layered structures is of crucial importance in governing the dependability and longevity of advanced optoelectronic equipment. Bubbles are commonly found within layered structures, yet their impact on the thermal expansion coefficient of the structure is still not well understood. We derive an analytical formula that describes the influence of bubbles on the variation in length, utilizing both the elastic membrane theory and the improved van der Waals gas state equation. The results demonstrate that as the temperature increases, bubbles can induce significant thermal contraction in few-layer graphene structures. The analytical predictions are validated through molecular dynamic (MD) simulations, confirming the accuracy and reliability of the proposed model. These findings hold significant value in enhancing our understanding of the dependability and durability of functional equipment that relies on van der Waals layered structures.

The bubble effect on the thermal expansion coefficient of few-layer MoS2

The thermal expansion coefficient of van der Waals layered structures plays an important role in governing the thermal stability of high-performance electronic devices. Meanwhile, bubbles are frequently observed in the van der Waals layered structures due to the weak inter-layer interaction and the extremely small bending stiffness of the layered structure, but the effect of bubbles on the thermal expansion coefficient for layered structures remains unclear. We derive an analytic formula for the bubble effect on the length variation of the MoS2 layer. In the analytic derivation, the MoS2 layer is described by three mechanics models, including the elastic membrane theory, the nonlinear plate theory, and the linear plate theory. The gas inside the bubble is described by two types of state equations, including the ideal gas law and the generalized van der Waals equation. It is found that the nonlinear plate theory with the generalized van der Waals equation gives the most accurate description for the bubble effect, since MoS2 has a moderate bending stiffness. The analytic formula shows that the bubble can cause strong thermal contraction for few-layer MoS2 with increasing temperature. The analytic predictions are verified by comprehensive molecular dynamic simulations for few-layer MoS2 with different layer numbers and different gas molecule numbers. These results shall be valuable for understanding the thermal stability of functional devices based on the van der Waals layered structure.

Nanobubble-induced significant reduction of the interfacial thermal conductance for few-layer graphene.

The heat transport properties of van der Waals layered structures are crucial for ensuring the reliability and longevity of high-performance optoelectronic equipment. Owing to the two-dimensional nature of atomic layers, the presence of bubbles is commonly observed within these structures. Nevertheless, the effect of bubbles on the interfacial thermal conductance remains unclear. Based on the elastic membrane theory and the improved van der Waals gas state equation, we develop an analytical formula to describe the influence of bubble shape on the interfacial thermal conductance. It shows that the presence of bubbles has a considerable impact on reducing the interfacial thermal conductance across graphene/graphene interfaces. More specifically, the presence of nanobubbles can result in a reduction of up to 53% in the interfacial thermal conductance. The validity of the analytical predictions is confirmed through molecular dynamic simulations. These results offer valuable insights into the thermal management of van der Waals layered structures in the application of next-generation electronic nanodevices.

Indentation of a circular hyperelastic membrane by a rigid cylinder

In the paper we investigate the effect of friction on the stress–strain state of a hyperelastic membrane in contact with a massive solid body. We consider the axisymmetric indentation of a thin circular rubber membrane by a cylindrical indenter for different contact conditions: dry contact, liquid lubrication, and gel lubrication. We present experimental results and a mathematical model of the indentation. We use the non-linear theory of elastic membranes and the Coulomb friction model in the simulation. We describe different contact conditions with different friction coefficients. We compare the experimental and theoretical dependences between indentation force and indenter displacement.

Scaling behavior of crystalline membranes: An ε-expansion approach

We study the scaling behavior of two-dimensional (2D) crystalline membranes in the flat phase by a renormalization group (RG) method and an ε-expansion. Generalization of the problem to non-integer dimensions, necessary to control the ε-expansion, is achieved by dimensional continuation of a well-known effective theory describing out-of-plane fluctuations coupled to phonon-mediated interactions via a scalar composite field, equivalent for small deformations to the local Gaussian curvature. The effective theory, which will be referred to as Gaussian curvature interaction (GCI) model, is equivalent to theories of elastic D-dimensional manifolds fluctuating in a (D+dc)-dimensional embedding space in the physical case D=2 for arbitrary dc. For D≠2, instead, the GCI model is not equivalent to a direct dimensional continuation of elastic membrane theory and it defines an alternative generalization to generic internal dimension D. After decoupling interactions through a Hubbard-Stratonovich transformation, we study the GCI model by perturbative field-theoretic RG within the framework of an expansion in ε=(4−D). We calculate explicitly RG functions at two-loop order and determine the exponent η characterizing the long-wavelength scaling of correlation functions to order ε2. The value of η at this order is shown to be insensitive to Feynman diagrams involving vertex corrections. As a consequence, the self-consistent screening approximation for the GCI model is shown to be exact to O(ε2). In the physical case of a single out-of-plane displacement field, dc=1, the O(ε2) correction is suppressed by a small numerical prefactor. As a result, despite the large value of ε=2, extrapolation of the first and second order results to D=2 leads to very close numbers, η=0.8 and η≃0.795. The calculated exponent values are close to earlier reference results obtained by non-perturbative renormalization group, the self-consistent screening approximation and numerical simulations. These indications suggest that a perturbative analysis of the GCI model could provide an useful framework for accurate quantitative predictions of the scaling exponent even at D=2.

A Novel Full Boundary Element Formulation for Harmonic Analysis of Elastic Membranes Coupled to Acoustics Fluids

A novel full Boundary Element Formulation for the harmonic analysis of elastic membranes coupled to acoustics fluid is presented. The elastic membranes is modeled using the classical linear elastic pre-stretched membrane theory. The acoustic fluid is modeled using the acoustic-wave equation for homogeneous, isotropic, inviscid and irrotational fluids. Elastostatic fundamental solution is used in the boundary element formulation for the elastic membrane. The boundary element formulation for the acoustic fluid is based on the fundamental solution of three dimensional Poisson equation. Domain integrals related to inertial terms and those related with distributed pressure on membrane, were treated using the Dual Reciprocity Boundary Element Method. Fluid-structure coupling equations were established considering the continuity of the normal acceleration of the particles and dynamic pressure at fluid-structure interfaces. Results obtained shows the accuracy and efficiency of the proposed boundary element formulation.

A Novel Full Boundary Element Formulation for Transient Analysis of Elastic Membranes Coupled to Acoustics Fluids

A full time-direct Boundary Element Formulation for the dynamic analysis of elastic membranes coupled to acoustics fluid is presented. The elastic membranes is modeled using the classical linear elastic pre-stretched membrane theory. The acoustic fluid is modeled using the acoustic-wave equation for homogeneous, isotropic, inviscid and irrotational fluids. Elastostatic fundamental solution is used in the boundary element formulation for the elastic membrane. The boundary element formulation for the acoustic fluid is based on the fundamental solution of three dimensional Poisson equation. Domain integrals related to inertial terms and those related with distributed pressure on membrane, were treated using the Dual Reciprocity Boundary Element Method. Fluid-structure coupling equations were established considering the continuity of the normal acceleration of the particles and dynamic pressure at fluid-structure interfaces. The time integration is carried out using the Newmark method. Results obtained shows the accuracy and efficiency of the proposed boundary element formulation.

In situ nanomechanical properties of natural oil bodies studied using atomic force microscopy

Natural oil bodies (OBs) from plant organs represent an important category of functional ingredients and materials in a variety of industrial sectors. Their applications are closely related to the membrane mechanical properties on a single droplet level, which remain difficult to determine. In this research, the mechanical properties of the membranes of OBs from soybean, sesame, and peanut were investigated in-situ by atomic force microscopy (AFM). Different regions of the force-deformation curves obtained during compression were analyzed to extract the stiffness Kb or Young’s modulus of the OB membranes using Hooke’s law, Reissner theory, and the elastic membrane theory. At higher strains (ε = 0.15–0.20), the elastic membrane theory breaks down. We propose an extension of the theory that includes a contribution to the force from interfacial tension based on the Gibbs energy, allowing effective determination of Young’s modulus and interfacial tension of the OB membranes in the water environment simultaneously. The mechanical properties of the OBs of different sizes and species, as well as a comparison with other phospholipid membrane materials, are discussed and related to their membrane compositions and structures. It was found that the natural OBs are soft droplets but do not rupture and can fully recover following compressive strains as large as 0.3. The OBs with higher protein/oil ratio, have smaller size and stronger mechanical properties, and thus are more stable. The low interfacial tension due to the existence of phospholipid-protein membrane also contributes to the stability of the OBs. This is the first report measuring the mechanical properties of OB membranes in-situ directly.

Model of graphene nanobubble: Combining classical density functional and elasticity theories.

A graphene nanobubble consists of a graphene sheet, an atomically flat substrate, and a substance enclosed between them. Unlike conventional confinement with rigid walls and a fixed volume, the graphene nanobubble has one stretchable wall, which is the graphene sheet, and its volume can be adjusted by changing the shape. In this study, we developed a model of a graphene nanobubble based on classical density functional theory and the elastic theory of membranes. The proposed model takes into account the inhomogeneity of the enclosed substance, the nonrigidity of the wall, and the alternating volume. As an example application, we utilize the developed model to investigate fluid argon inside graphene nanobubbles at room temperature. We observed a constant height-to-radius ratio over the whole range of radii considered, which is in agreement with the results from experiments and molecular dynamics simulations. The developed model provides a theoretical tool to study both the inner structure of the confined substance and the shape of the graphene nanobubble. The model can be easily extended to other types of nonrigid confinement.

Bending of inflated curved hyperelastic tubes

AbstractWe discuss large flexure of an inflated curved thin‐walled tube within the framework of the nonlinear theory of elastic membranes. Wrinkling of the membrane is taken into account by using a relaxed strain energy density derived from a strain energy function of incompressible hyperelastic material. We consider curved tubes with elliptical cross‐section and analyze the influence of a cross‐section shape and initial curvature of the tube at its deformation. Dependence of the bending moment on the tube curvature is obtained and discussed. In particular, the comparison with the pure bending of a cylindrical tube same material is made. We study the fact that the curving of the curved tube under in‐plane bending might be realized in two directions. Depending on the flexure direction the behavior of a curved tube can be similar or significantly different from the behavior of a cylindrical tube.

Atomistic study of the solid state inside graphene nanobubbles

A two-dimensional (2D) material placed on an atomically flat substrate can lead to the formation of surface nanobubbles trapping different types of substances. In this paper graphene nanobubbles of the radius of 7–34 nm with argon atoms inside are studied using molecular dynamics (MD). All modeled graphene nanobubbles except for the smallest ones exhibit an universal shape, i.e., a constant ratio of a bubble height to its footprint radius, which is in an agreement with experimental studies and their interpretation using the elastic theory of membranes. MD simulations reveal that argon does exist in a solid close-packed phase, although the internal pressure in the nanobubble is not sufficiently high for the ordinary crystallization that would occur in a bulk system. The smallest graphene bubbles with a radius of 7 nm exhibit an unusual “pancake” shape. Previously, nanobubbles with a similar pancake shape were experimentally observed in completely different systems at the interface between water and a hydrophobic surface.

The influence of strain on the elastic constants of graphene

Recent advances in the understanding of graphene elasticity have shown that suspended graphene does not behave as a conventional thin plate but has a more complex behavior where flexural modes play a significant role. Among other effects, out-of-plane thermal fluctuations modify the in-plane elastic properties. Here we report indentation experiments on graphene subjected to strain. This strain is achieved by applying pressure difference across suspended graphene drumheads. Our indentation curves show an increased mechanical response at strains larger than 0.3%. Finite element simulations of the indentation curves on pressurized membranes show that this observation can be ascribed to a twofold increase of the in-plane elastic modulus and the Poisson ratio. Based in the thermodynamic theory of elastic membranes, this increase is attributed to the suppression of out-of-plane fluctuations by strain. This result reinforces the idea that suspended graphene behaves as a fluctuating membrane and points out that a careful analysis should be done when analyzing indentation curves on atomic thick materials.

Dynamic and Static Pull-in instability of electrostatically actuated nano/micro membranes under the effects of Casimir force and squeezed film damping

In the current study, the effects of Casimir force and squeeze film damping on pull-in instability and dynamic behavior of electrostatically actuated nano and micro electromechanical systems are investigated separately. Linear elastic membrane theory is used to model the static and dynamic behavior of the system for strip, annular and disk geometries. Squeeze film damping is modeled using nonlinear Reynolds equation. Both equation of motion and nonlinear Reynolds equation are first nondimensionalized, and then discretized and solved by means of finite element method. Static pullin analysis is performed and validated by previous researches, and then dynamic pull-in values are investigated and compared with static pull-in parameters. In the next step, the effect of squeeze film damping, ambient pressure and Casimir force on the system dynamics is studied. Results show significant effect of Casimir force and squeeze film damping on the system behavior which is considerable for fabrication and design

The two-ball bounce problem

A popular classroom demonstration is revisited in which a light ball and a much larger heavier ball are vertically aligned and dropped together onto a hard surface. Careful experimental data obtained using a high-speed camera is compared to a lumped-mass Newtonian restitution model. Good macroscopic agreement is found, provided there is sufficient separation between the two balls as they are dropped. An alternative continuum model based on elastic membrane theory is developed to explain the limit in which the balls are initially touching. The model assumes the lower ball deforms to a truncated sphere upon its impact with the floor, exciting an elastic wave which subsequently launches the upper ball like a particle on a trampoline, before the lower ball leaves the ground. A favourable comparison with experimental data is found for the case of negligible initial separation between the balls.

Trapping and deformation of microbubbles in a dual-beam fibre-optic trap

We present results of numerical calculations to evaluate the performance of a dual-beam fibre-optic trap for low refractive index particles such as ultrasound contrast agent microbubbles. Using a geometrical optics approach, we determine the range of parameters of microbubble size and beam dimensions over which the optical trap is stable and evaluate the trapping forces and spring constants. Additionally, we calculate the optically induced stress profile over the surface of the microbubble and evaluate the resulting deformation of the microbubble using elastic membrane theory. Our results suggest that such an experiment could be a useful tool for quantifying the mechanical properties (elastic modulus) of the shell material of an ultrasound contrast agent microbubble.

Nanomechanical Properties of Phospholipid Microbubbles

This study uses atomic force microscopy (AFM) force-deformation (F-Δ) curves to investigate for the first time the Young's modulus of a phospholipid microbubble (MB) ultrasound contrast agent. The stiffness of the MBs was calculated from the gradient of the F-Δ curves, and the Young's modulus of the MB shell was calculated by employing two different mechanical models based on the Reissner and elastic membrane theories. We found that the relatively soft phospholipid-based MBs behave inherently differently to stiffer, polymer-based MBs [Glynos, E.; Koutsos, V.; McDicken, W. N.; Moran, C. M.; Pye, S. D.; Ross, J. A.; Sboros, V. Langmuir2009, 25 (13), 7514-7522] and that elastic membrane theory is the most appropriate of the models tested for evaluating the Young's modulus of the phospholipid shell, agreeing with values available for living cell membranes, supported lipid bilayers, and synthetic phospholipid vesicles. Furthermore, we show that AFM F-Δ curves in combination with a suitable mechanical model can assess the shell properties of phospholipid MBs. The "effective" Young's modulus of the whole bubble was also calculated by analysis using Hertz theory. This analysis yielded values which are in agreement with results from studies which used Hertz theory to analyze similar systems such as cells.

Dynamics of Sedimentation and Deformability of GUVs in the Presence of an Electrostatic Surface Repulsion

POPC Giant unilamellar vesicles (GUVs) with 1 mol% NBD-PE were prepared in different sucrose solutions by electroformation. These vesicles were then resuspended in glucose solutions with the same sugar concentrations to obtain isotonic conditions. We studied sedimentation velocities and vesicle deformation through SPIM and confocal microscopy. The results obtained show a relationship between area strain and surface tension that is in agreement with the elastic theory of membranes. The presence of an overall negative charge density in the surface of the glass and a net negative charge in the vesicle membrane leads to an electrostatic repulsive interaction. This is modeled using the linear Deryaguin approximation for an electrolyte solution. We find that the separation distances between GUVs and the glass surface, predicted by the theory, are in the same range than those measured by confocal imaging of sedimented vesicles. These distances range between 1.5-5 μm and influence the hydrodynamic interactions of vesicles moving close to surfaces.Boundary element simulations were performed to model vesicle deformation during sedimentation within a viscous fluid (Stokes flow) as a function of gravitational pull. A force field is implemented that takes into account local tension, local curvature force, gravitational pull, and the electrostatic interaction with the surface. Vesicle sedimentation, deformation and membrane tension were evaluated as a function of g0, a dimensionless factor relating gravitational and curvature energies. Results from the simulations are in agreement with the experimental results and gave us additional information of the deformability and sedimentation dynamics.

Dynamic deformation of red blood cell in Dual-trap Optical Tweezers

Three-dimensional dynamic deformation of a red blood cell in a dual-trap optical tweezers is computed with the elastic membrane theory and is compared with the experimental results. When a soft particle is trapped by a laser beam, the particle is deformed depending on the radiation stress distribution whereas the stress distribution on the particle in turn depends on the deformation of its morphological shape. We compute the stress re-distribution on the deformed cell and its subsequent deformations recursively until a final equilibrium state solution is achieved. The experiment is done with the red blood cells in suspension swollen to spherical shape. The cell membrane elasticity coefficient is obtained by fitting the theoretical prediction with the experimental data. This approach allows us to evaluate up to 20% deformation of cell's shape.

Topological instabilities of spherical vesicles

Within the framework of the Helfrich elastic theory of membranes and of differential geometry, we study the relative stability of spherical vesicles and double bubbles. We find that not only temperature but also magnetic fields can induce topological transformations between spherical vesicles and double bubbles and provide a phase diagram for the equilibrium shapes.