Membrane Theory Research Articles

The Progressive Nature of Menière's Disease: Stress Projections and Lesion Analysis.

An engineering model of the labyrinth can provide a mechanism that accounts for the pattern of lesion development observed in Menière's disease. Membrane lesions in Menière's disease can occur in virtually every part of the membranous labyrinth. How these lesions are induced is unclear and their mode of distribution uncertain. Pressure induced stress in the membranous labyrinth may play a mechanistic role in lesion formation and distribution. An engineering model of the labyrinth was used to provide membrane stress formulations and projections for lesion induction in the several chambers using membrane theory. These were compared with an analysis of actual lesions observed in Menière's disease to evaluate the model's accuracy. The model projects that lesions in the membranous labyrinth will be induced progressively because of stress differentials among the chambers, with a chain of lesion pattern that follows the serial anatomic order and occurs with a frequency commensurate with chamber stress level. An analysis of lesions observed in actual cases of Menière's disease reveals a pattern of lesion development that is progressive, sequential, commensurate, and concordant with the model's stress projections. The concordance between the stress projections and the lesion analysis strengthens the hypothesis that Menière's is a progressive disease that follows a chain of lesions paradigm based on pressure-induced stress differentials in the variously configured chambers of the membranous labyrinth.

Energetics and Stability of Neck Formation in Yeast and Mammalian Endocytosis

The formation and constriction of membrane necks is a fundamental step that immediately precedes fission in membrane remodeling processes like endocytosis, viral egress, and cytokinesis. Even in endocytosis, the formations of invaginations and buds are different in yeast vs mammalian cells. In mammalian cells, structural and biophysical studies have identified that constriction of the neck by dynamin helices triggers membrane fission. However, yeast cells do not have dynamin. A detailed physical understanding of the force and energetic requirements for neck constriction in yeast vs mammalian cells is difficult to ascertain using experimental techniques due to the small length scales nearing self contact. In this work, we apply a modified Helfrich theory of lipid membranes to investigate membrane necking in yeast and mammalian endocytosis. Specifically, we seek to understand how applied forces and protein-induced spontaneous curvature contribute to induce necking and energy distribution on the membrane. Application of a radial constricting force, mimicking the action of dynamin, leads to a snapthrough instability in the shape of the membrane during budding. We observe a large energy barrier that can be overcome by force to achieve a narrow neck. This instability is eliminated by applying a region of spontaneous curvature, contracting the radius or widening the force. Application of a pulling force, essential to yeast endocytosis, also leads to snapthrough instability for a rigid protein coat. Here, the instability is eliminated by ensuring uniform rigidity of coat and membrane. We extend the analysis to images of clathrin coated buds from mammalian and yeast experiments and track the process of neck formation. To do this, we implement an image processing framework that was recently implemented by the authors. Our model sheds insight into the defining characteristics of membrane necking and related snapthrough instabilities.

Voltage-driven torsion of electroactive thick tubes reinforced with helical fibers

This work studies the voltage-driven torsional deformation of a thick-walled circular cylindrical tube composed of a dielectric elastomer (DE) reinforced with a family of helical fibers. The Helmholtz free energy of the DE composite tube involves separate DE matrix and fiber contributions. Upon the application of an electrical voltage difference on its inner and outer surfaces, the DE matrix contracts through the thickness and expands in cylindrical surfaces due to the Maxwell stress. Since the composite is stiffened along the fiber direction, it preferentially expands in the direction orthogonal to the fiber within the cylindrical surfaces. As a result, voltage-driven torsional deformation will occur in the helical fiber-reinforced DE tube. Although membrane theory was usually adopted to model fiber-reinforced thin DE membranes, the finite deformation theory is utilized in the present work to model relatively thick DE composite tubes. It is found that the bending effect plays an important role in the voltage-driven torsional deformation. The reorientation of the fibers and the distribution of the stress carried by the fibers in the tubes are discussed in detail so as to provide an interpretation to some new voltage-driven torsional behaviors. The theoretical model and physical insights provided in this work will aid the design and optimization of soft electroactive actuators and soft robotics.

Active processes make mixed lipid membranes either flat or crumpled

Whether live cell membranes show miscibility phase transitions (MPTs), and if so, how they fluctuate near the transitions remain outstanding unresolved issues in physics and biology alike. Motivated by these questions we construct a generic hydrodynamic theory for lipid membranes that are active, due for instance, to the molecular motors in the surrounding cytoskeleton, or active protein components in the membrane itself. We use this to uncover a direct correspondence between membrane fluctuations and MPTs. Several testable predictions are made: (i) generic active stiffening with orientational long range order (flat membrane) or softening with crumpling of the membrane, controlled by the active tension and (ii) for mixed lipid membranes, capturing the nature of putative MPTs by measuring the membrane conformation fluctuations. Possibilities of both first and second order MPTs in mixed active membranes are argued for. Near second order MPTs, active stiffening (softening) manifests as a super-stiff (super-soft) membrane. Our predictions are testable in a variety of in vitro systems, e.g. live cytoskeletal extracts deposited on liposomes and lipid membranes containing active proteins embedded in a passive fluid.

Asymptotics of Willmore Minimizers with Prescribed Small Isoperimetric Ratio

We consider surfaces in ${\mathbb R}^3$ of type ${\mathbb S}^2$ which minimize the Willmore functional with prescribed isoperimetric ratio. The existence of smooth minimizers was proved by Schygulla (Archive Rational Mechanics and Analysis, 2012). In the singular limit when the isoperimetric ratio converges to zero, he showed convergence to a double round sphere in the sense of varifolds. Here we give a full blowup analysis of this limit, showing that the two spheres are connected by a catenoidal neck. Besides its geometric interest, the problem was studied as a simplified model in the theory of cell membranes, see e.g. Berndl, Lipowsky, Seifert (Physical Review A, 1991).

Experimental characterization of graphene by electrostatic resonance frequency tuning

In the last decade, graphene membranes have drawn tremendous attention due to their potential application in Nano-Electro-Mechanical Systems. In this paper, we show that the frequency response curves of graphene resonators are powerful tools for their dynamic characterization and for extracting their equivalent Young's modulus. For this purpose, vibrations of an electrostatically actuated circular graphene membrane are studied both experimentally and numerically. The experiments reveal the dependency of the linear and nonlinear resonance frequency of the nano-resonator on the driving DC and AC voltages. A numerical model is proposed based on the nonlinear membrane theory, and by fitting the numerically calculated change in resonance frequency due to the DC voltage to those of the experimental observations, the Young's modulus is determined. It is shown that by using the obtained equivalent Young's modulus, the numerical model can accurately describe the nonlinear dynamics of the graphene membrane in other sets of measurements.

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.

Modelling of shell structures using the scaled boundary finite element method

AbstractIn view of recent trends towards lightweight structural designs, the numerical modelling of shell structures is a subject of high interest. Here, major challenges are associated with locking effects. This contribution presents a first approach to the numerical analysis of shell structures using the scaled boundary finite element method (SBFEM). First studies such as a simplified plane strain arch formulation to approximate an axisymmetric cylindrical shell are presented. This approximation already shows a high correlation with the membrane theory of shells. Furthermore, first results obtained for 3D shell formulations illustrate the potential of the proposed approach based on the SBFEM. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Unprecedented homotopy perturbation method for solving nonlinear equations in the enzymatic reaction of glucose in a spherical matrix.

The theory of glucose-responsive composite membranes for the planar diffusion and reaction process is extended to a microsphere membrane. The theoretical model of glucose oxidation and hydrogen peroxide production in the chitosan-aliginate microsphere has been discussed in this manuscript for the first time. We have successfully reported an analytical derived methodology utilizing homotopy perturbation to perform the numerical simulation. The influence and sensitive analysis of various parameters on the concentrations of gluconic acid and hydrogen peroxide are also discussed. The theoretical results enable to predict and optimize the performance of enzyme kinetics.

Nonlinear dynamic characterization of two-dimensional materials

Owing to their atomic-scale thickness, the resonances of two-dimensional (2D) material membranes show signatures of nonlinearities at forces of only a few picoNewtons. Although the linear dynamics of membranes is well understood, the exact relation between the nonlinear response and the resonator’s material properties has remained elusive. Here we show a method for determining the Young’s modulus of suspended 2D material membranes from their nonlinear dynamic response. To demonstrate the method, we perform measurements on graphene and MoS2 nanodrums electrostatically driven into the nonlinear regime at multiple driving forces. We show that a set of frequency response curves can be fitted using only the cubic spring constant as a fit parameter, which we then relate to the Young’s modulus of the material using membrane theory. The presented method is fast, contactless, and provides a platform for high-frequency characterization of the mechanical properties of 2D materials.

Membrane theory, revisited

Membrane theory, revisited

Simplified shell modeling for submerged structures

Faithful numerical models of the acoustical response of submerged structures are complicated by the necessity to account for the interaction of three fundamentally different systems: the outer pressure hull, the internal substructure, and the surrounding fluid. Standard finite element techniques are the only recourse for a complex substructure, and boundary element or finite element techniques usually are necessary for the fluid. This paper proposes an alternative approach for the pressure hull based on an observation regarding the relative importance of extensional and flexural effects in a curved shell. The formulation entails applying an energy-based correction to membrane theory in order to account for flexural deformation. In some cases doing so enables an analytical representation of the shell's behavior, and it also simplifies a finite element representation. Whereas the interaction laws between the shell and the fluid are enforced conventionally, coupling of the shell and the substructures may be implemented a procedure rooted in analytical dynamics. It imposes constraint equations on the displacement variables, which introduces Lagrange multipliers to the equations of motion. In this manner, explicit consideration of the forces at attachments is avoided.

Equilibrium analysis of masonry domes. on the analytical interpretation of the Eddy-Lévy graphical method

ABSTRACTThe 19th century graphic method of Eddy-Lévy is analyzed and proposed in an analytical form as an assessment tool for masonry domes. The method determines the neutral hoop that separates the upper part of the dome that behaves as a compressed membrane from and the bottom that behaves as independent arches, due to the presence of vertical cracks. This method allows determining a no-tension equilibrated solution, accounting for the natural behavior of masonry domes of fracturing along meridian planes.Six case studies were investigated, considering spherical and pointed dome, complete, with hole and with hole and lantern. A modified version has been studied to model the biaxial stress of upper part properly through the membrane theory, and compared to the solution obtained by a numeric discrete block model.Limit thickness-to-radius ratio was identified, and a parametric analysis was carried out to investigate the extension of meridian fracture varying dome shape and thickness.

Influence of Particle Size and Loading on Particle Accessibility in Electrospun Poly(ethylene oxide) and ZIF-8 Composite Fibers: Experiments and Theory.

Developing electrospun nanofiber/nanoparticle composites (ENNCs) is an emerging strategy for immobilizing functional particles for a variety of applications. The radial location of the particle along the fiber, either at the surface or in the bulk, has implication into the resulting properties. To explore particle location along fibers, ZIF-8 impregnated poly(ethylene oxide) (PEO) nanofibers are formed by electrospinning particle suspensions. Fibers impregnated with two different ZIF-8 particle sizes (200 nm and 12.5 μm) were electrospun and shown by nitrogen porosimetry to be nearly completely wrapped by PEO in each case at loadings near 10 wt %. This was favorably compared to developed theory of polymeric membrane encapsulated particles and extended to other electrospun fiber/particle composites through a brief literature review. ENNCs with varying loadings of nanosized ZIF-8 particles were then fabricated and tested with nitrogen porosimetry to find that the particles became available for adsorption at the surface of the fibers starting from 25 wt % (28 vol %) loading. This suggests that the particles are kinetically trapped at this loading level since, if allowed to exhibit random close-packing, the ZIF-8 would be expected to fully imbedded inside the fibers up to 56 vol % loading.

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.

Nonlinear Axisymmetric Free Vibration Analysis of Liquid-Filled Spherical Shell With Volume Constraint

Nonlinear axisymmetric free vibration analysis of liquid-filled spherical shells with volume constraint condition using membrane theory is presented in this paper. The energy functional of the shell and contained liquid can be expressed based on the principle of virtual work using surface fundamental form and is written in the appropriate forms. Natural frequencies and the corresponding mode shapes for specified axisymmetric vibration amplitude of liquid-filled spherical shells can be calculated by finite element method (FEM). A nonlinear numerical solution can be obtained by the modified direct iteration technique. The results indicate that the Lagrange multiplier is a parameter for adapting the internal pressure in order to sustain the shell in equilibrium state for each mode of vibration with the volume constraint condition. The axisymmetric mode shapes of the liquid-filled spherical shells under volume constraint condition were found to be in close agreement with those in existing literature for an empty spherical shell. Finally, the effects of support condition, thickness, initial internal pressure, bulk modulus of internal liquid, and elastic modulus on the nonlinear axisymmetric free vibration and change of pressure of the liquid-filled spherical shells with volume constraint condition were demonstrated. The parametric studies showed that the change of pressure has a major impact on the fundamental vibration mode when compared with the higher vibration modes.

Effect of ripples on the finite temperature elastic properties of hexagonal boron nitride using strain-fluctuation method

This work intents to put forth the results of a classical molecular dynamics study to investigate the temperature dependent elastic constants of monolayer hexagonal boron nitride (h-BN) between 100 and 1000 K for the first time using strain fluctuation method. The temperature dependence of out-of-plane fluctuations (ripples) is quantified and is explained using continuum theory of membranes. At low temperatures, negative in-plane thermal expansion is observed and at high temperatures, a transition to positive thermal expansion has been observed due to the presence of thermally excited ripples. The decrease of Young's modulus, bulk modulus, shear modulus and Poisson's ratio with increase in temperature has been analyzed. The thermal rippling in h-BN leads to strong anharmonic behaviour that causes large deviation from the isotropic elasticity. A detailed study shows that the strong thermal rippling in large systems is also responsible for the softening of elastic constants in h-BN. From the determined values of elastic constants and elastic moduli, it has been elucidated that 2D h-BN sheets meet the Born's mechanical stability criterion in the investigated temperature range. The variation of longitudinal and shear velocities with temperature is also calculated from the computed values of elastic constants and elastic moduli.

Mathematical modeling of local perfusion in large distensible microvascular networks

Microvessels – blood vessels with diameter less than 200μm – form large intricate networks, functionally organized into arterioles, capillaries and venules. In these networks, the distribution of flow and pressure drop stems in a complex manner from single vessel behavior and mutual vessel interactions. In this paper, we propose a mathematical and computational model to study the behavior of large networks of compliant microvessels. The network geometry is simplified for computational purposes. The arteriolar and venular trees are represented by graphs of straight cylinders, each one corresponding to a vessel. The two trees are connected through a capillary bed. The blood flow and pressure drop across each vessel are related via a simplified fluid–structure interaction (FSI) model, represented by a generalized Ohm’s law featuring a conductivity parameter. The conductivity is a function of the vessel cross section area (shape and area), which, in turn, undergoes deformations due to luminal and external pressure loads. The membrane theory is used for the description of the deformation of vessel lumina, adapted to consider thick-walled arterioles and thin-walled venules. An original point of the present work is represented by the inclusion of a buckling model in the FSI problem for venules. As a matter of fact, venules can experience negative values of transmural pressure (difference between luminal and interstitial pressure) and may assume a deformed and even collapsed configuration due to their minimal cross-sectional bending stiffness. The complete model including flow in distensible arterioles, capillaries and venules represents a nonlinear coupled system of PDEs, which is approached numerically by finite element discretization and linearization techniques. As an example of application, we use the model to simulate flow in the microcirculation of the human eye retina, a terminal system with a single inlet and outlet. After a phase of validation against experimental measurements of the correctness of the blood flow and pressure fields in the network, we compute the network response to different conditions. Significant redistributions of the blood flow in the network are observed, highlighting the importance of considering the single vessel behavior along with its position and connectivity in the network.

Effect of double-fibre reinforcement on localized bulging of an inflated cylindrical tube of arbitrary thickness

We consider localized bulging of an inflated cylindrical hyperelastic tube of arbitrary thickness that is helically reinforced by two families of fibres. It is shown that localized bulging may become impossible, irrespective of the end conditions, when the tube wall becomes thick enough. This is in sharp contrast with an isotropic hyperelastic tube without fibre reinforcement for which localized bulging has previously been shown to be possible no matter how thick the tube wall is and for which the membrane theory provides a very good approximation for the ratio of wall-thickness/radius as large as 0.67. Our findings provide a feasible explanation on why aneurysms cannot occur in healthy arteries but become possible following pathological changes. They can also be used to guide the design of tubular structures where localized bulging should be prevented.

Confined chromonics and viral membranes

ABSTRACTRecent work at the University of Pennsylvania on two distinct topics will be reviewed: (1) chromonic liquid confined to spherical and cylindrical cavities, the former with degenerate planar boundary conditions and the latter with both degenerate planar and homeotropic boundary conditions, and (2) a theory for membranes formed by filamentous viruses in water solution with depletants.