Pull-in instability of carbon nanotube-reinforced nano-switches considering scale,surface and thermal effects

In this paper, an analytical method is presented to investigate the effect of surface characteristic and temperature change on the pull-in instability of electrically actuated nano-switches reinforced by carbon nanotubes (CNTs) based on Eringen's nonlocal beam theory. An extremely nonlinear fourth-order governing equation for the doubly clamped nano-switches made of CNTs/Si composites nanobeam is derived and solved by using the principle of virtual work, where Van der Waals force as atomic interactions and Casimir force as macro effects of quantum field fluctuation of vacuum are combined as an electrostatic force with fringing field effects. The results show that both the pull-in voltage and pull-in deflection of CNTs/Si composite nanobeam increase with the increase of CNTs volume ratio but decrease with the increase of temperature change. The coupling influences of small scale parameter, geometric behavior, surface characteristic and thermal effect on the pull-in instability of electrostatically actuated CNTs/Si nanobeam are detailedly discussed.     

A coupled bending-torsion model for electrostatically actuated torsional nano/micro-actuators with considering influence of van der Waals force

In the current paper, a coupled two degree of freedom model which considers both bending and torsion of the supporting torsion beams is presented for electrostatically actuated torsional nano/micro-actuators under the effect of van der Waals (vdW) force. Newton’s second law is utilized for finding the normalized equations governing the static behavior of the actuator. The implict function theorem is then utilized for finding the equations governing the pull-in state of the actuator. The related results show that torsion model considerably overestimates the pull-in parameters of the nano/micro-actuator. The concept of the instability mode is introduced, and it is shown that when the ratio of the bending stiffness to the torsion stiffness of the supporting torsion beams is relatively low, the dominant instability mode of the actuator would be the bending mode and otherwise the dominant instability mode would be the torsion mode. It is also observed that the presence of the vdW force can significantly reduce the pull-in angle and pull-in deflection of the nano/micro-actuator. The presented results also show that the vdW force can lead to considerable reduction in the pull-in voltage of the actuator. The equilibrium behavior of the actuator is studied, and it is observed that the vdW force and also bending of the supporting torsion beams greatly reduce the maximum allowable voltage which can be applied to the actuator. Results of this paper can be used for successful design of electrostatically actuated torsional nano/micro-actuators where the size of the actuator is sufficiently small, and as a result, the vdW force plays a major role in the system.     

Closed-form approximation and numerical validation of the influence of van der Waals force on electrostatic cantilevers at nano-scale separations

In this paper the two-point boundary value problem (BVP) of the cantilever deflection at nano-scale separations subjected to van der Waals and electrostatic forces is investigated using analytical and numerical methods to obtain the instability point of the beam. In the analytical treatment of the BVP, the nonlinear differential equation of the model is transformed into the integral form by using the Green's function of the cantilever beam. Then, closed-form solutions are obtained by assuming an appropriate shape function for the beam deflection to evaluate the integrals. In the numerical method, the BVP is solved with the MATLAB BVP solver, which implements a collocation method for obtaining the solution of the BVP. The large deformation theory is applied in numerical simulations to study the effect of the finite kinematics on the pull-in parameters of cantilevers. The centerline of the beam under the effect of electrostatic and van der Waals forces at small deflections and at the point of instability is obtained numerically. In computing the centerline of the beam, the axial displacement due to the transverse deformation of the beam is taken into account, using the inextensibility condition. The pull-in parameters of the beam are computed analytically and numerically under the effects of electrostatic and/or van der Waals forces. The detachment length and the minimum initial gap of freestanding cantilevers, which are the basic design parameters, are determined. The results of the analytical study are compared with the numerical solutions of the BVP. The proposed methods are validated by the results published in the literature.     

Nonlinear nonlocal pull-in instability of boron nitride nanoswitches

In the present study, nonlinear pull-in instability of boron nitride nanoswitches (BNNSs) subjected to electrostatic and van der Waals (vdW) forces is investigated. Based on Euler–Bernoulli beam theory, von Kármán geometric nonlinearity, nonlocal piezoelasticity theory and the principle of virtual work, the governing equations are obtained. The differential quadrature method is employed to discretize the nonlinear governing equations, which are then solved by a direct iterative method to obtain the nonlinear pull-in and pull-out voltages for cantilever and fixed–fixed boundary conditions. A detailed parametric study is conducted to elucidate the influences of nonlocal parameter, vdW force, fringing field, beam length and gap distance on the behavior of the pull-in instability voltage. Numerical results indicate that the magnitude of the pull-in voltage increases with increase in the gap distance. Furthermore, as the effective gap distance increases, the pull-out voltage tends toward the electrostatically pull-out voltage.     

Numerical simulation of the head-on collision of two equal-sized drops with van der Waals forces

The head-on collision of two equal-sized drops in a hyperbolic flow is investigated numerically. An axisymmetric volume-of-fluid (VOF) method is used to simulate the motion of each drop toward a symmetry plane where it interacts and possibly coalesces with its mirror image. The volume-fraction boundary condition on the symmetry plane is manipulated to numerically control coalescence. Two new numerical methods have been developed to incorporate the van der Waals forces in the Navier–Stokes equations. One method employs a body force computed as the negative gradient of the van der Waals potential. The second method employs the van der Waals forces in terms of a disjoining pressure in the film depending on the film thickness. Results are compared to theory of thin-film rupture. Comparisons of the results obtained by the two methods at various values of the Hamaker constant show that the van der Waals forces calculated from the two methods have qualitatively similar effects on coalescence. A study of the influence of the van der Waals forces on the evolution and rupture of the film separating the drops reveals that the film thins faster under stronger van der Waals forces. Strong van der Waals forces lead to nose rupture, and small van der Waals forces lead to rim rupture. Increasing the Reynolds number causes a greater drop deformation and faster film drainage. Increasing the viscosity ratio slows film drainage, although the effect is small for small viscosity ratio.     

A size-dependent model for instability analysis of paddle-type and double-sided NEMS measurement sensors in the presence of centrifugal force

Paddle-type and double-sided nanostructures have much potential for measuring the angular speed in rotary systems and aerospace applications. While most investigations in the literature have concentrated on the electromechanical performance of conventional beam-type nanostructures, few researchers have addressed the performance of these systems. The pull-in instability of the cantilever paddle-type and double-sided sensors in the presence of the centrifugal force have been investigated. The nonlinear governing equations are solved and the obtained results are compared with the numerical solution. The influences of the van der Waals force, geometric parameters, angular speed, and size phenomenon on the instability performance have been demonstrated.     

Stochastic analysis of dynamic characteristics and pull-in instability of FGM micro-switches with uncertain parameters in thermal environments

In this paper, the stochastic vibration characteristics of a functionally graded material micro-switch with random material properties near the pull-in instability are investigated. The uncertainties of the material properties and randomness of the composition of constituents due to the fabrication process are considered in this study. The micro-switch is modeled as a micro-beam and is under the effects of electrostatic and Casimir forces. The properties of the constituent materials are temperature dependent, and the system undergoes a change in temperature. The governing equations of motion of the Euler–Bernoulli micro-beam are derived based on modified couple stress theory and are solved using the state space form finite difference method. The statistics of the dynamic characteristics are obtained based on Monte Carlo simulation method. The effects of Casimir force, the length scale parameter, temperature dependencies of the material properties, the temperature change, the applied voltage and the volume fraction index on stochastic properties of the first natural frequency, the second natural frequency, the pull-in gap, and the pull-in voltage are studied in detail.     

Surface fluctuation spectroscopy by surface-light-scattering spectroscopy

In recent years surface-light-scattering spectroscopy has been transformed from a complex optical experiment requiring substantial effort to operate effectively to a simpler instrument for which an accurate theory of operation has been developed. The accuracy and precision are sufficiently enhanced that refinement of the theory of spectral band shapes is justified to include more subtle effects such as bending moduli and thin-film forces associated with the van der Waals and the Casimir effects. We show how to develop extensions of the theory of interfacial fluctuations through the mass and the momentum balances of interfacial transport. We also show that free-energy functionals can be used to express curvature effects crisply. The results are detailed formulas that can be used to fit experimental spectra.     

Three-dimensional analysis of the spontaneous instability for soft thin viscoelastic films

Based on the continuum mechanics and the bifurcation theory, a three-dimensional theoretical model for a soft thin viscoelastic film bonded to a rigid substrate is investigated. Considering the competition among van der Waals interaction potential energy, strain energy, and surface energy, three-dimensional governing equations of the spontaneous instability are derived, and the analytical results of time-dependent critical conditions are obtained. Furthermore, the phase diagram of instability due to van der Waals interaction and variation of the dimensionless characteristic wavenumber and the critical stiffness of interaction with the critical time are discussed.     

Electro-thermo-torsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model

Electro-thermo-torsional buckling response of a double-walled boron nitride nanotube (DWBNNT) has been investigated based on nonlocal elasticity and piezoelasticity theories. The effects of surrounding elastic medium such as the spring constant of the Winkler-type and the shear constant of the Pasternak-type are taken into account. The van der Waals (vdW) forces are considered between inner and outer layers of nanotube. According to the relationship between the piezoelectric coefficient of armchair boron nitride nanotubes (BNNTs) and stresses, the first order shear deformation theory (FSDT) is used. Energy method and Hamilton’s principle are employed to obtain coupled differential equations containing displacements, rotations and electric potential terms. The detailed parameter study is conducted to investigate the effects of nonlocal parameter, elastic foundation modulus, temperature change, piezoelectric and dielectric constants on the critical torsional buckling load. Results indicate that the critical buckling load decreases when piezoelectric effect is considered.     

Coupled mechanism and pull-in instability of several probe-membrane assemblies subjected to electrostatic force

The mathematical model of a coupled probe-membrane system subjected to electrostatic force is constructed. It is different than the conventional nano/micro actuator, which is constructed by two independent fixed/mobile conducting electrodes. The formula of pull-in voltage for different membranes is presented. The effects of several parameters on the pull-in instability are studied. The analytical method for the coupled vibration is proposed. The effects of the probe tips and several differential boundary conditions of membranes on the coupled characteristics are investigated. The coupled characteristic mechanism is clearly described.     

Pulsating fluid induced dynamic instability of visco-double-walled carbon nano-tubes based on sinusoidal strain gradient theory using DQM and Bolotin method

This research deals with the dynamic instability analysis of double-walled carbon nanotubes (DWCNTs) conveying pulsating fluid under 2D magnetic fields based on the sinusoidal shear deformation beam theory (SSDBT). In order to present a realistic model, the material properties of DWCNTs are assumed viscoelastic using Kelvin–Voigt model. Considering the strain gradient theory for small scale effects, a new formulation of the SSDBT is developed through the Gurtin–Murdoch elasticity theory in which the effects of surface stress are incorporated. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The van der Waals interactions between the adjacent walls of the nanotubes are taken into account. The size dependent motion equations and corresponding boundary conditions are derived based on the Hamilton’s principle. The differential quadrature method in conjunction with Bolotin method is applied for obtaining the dynamic instability region. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface stress and fluid velocity on the dynamic instability of DWCNTs. The results depict that the surface stress effects on the dynamic instability of visco-DWCNTs are very significant. Numerical results of the present study are compared with available exact solutions in the literature. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems in advanced biomechanics applications with magnetic field as a parametric controller.     

On Axisymmetric Longitudinal Wave Propagation in Double-Walled Carbon Nanotubes

An attempt is made into the investigation of longitudinal axisymmetric wave propagation in the DWCNT with the use of the exact equations of motion of the linear theory of elastodynamics. The DWCNT is modeled as concentricallynested two circular hollow cylinders between which there is free space. The difference in the radial displacements of these cylinders is coupled with the van der Waals forces and it is assumed that full slipping conditions occur on the inner surface of the outer tube and on the outer surface of the inner tube. Numerical results on the influence of the problem parameters such as the thickness/radius ratio, the distance between the tubes of the DWCNT and the van der Waals forces on the dispersion curves are presented and discussed. In particular, it is established that new types of modes arise under propagation of axisymmetric longitudinal waves as a result of the van der Waals interaction between the tubes of the DWCNT. The limit values of the wave propagation velocity are also analyzed.     

Vibration analysis of two orthogonal slender single-walled carbon nanotubes with a new insight into continuum-based modeling of van der Waals forces

Transverse vibrations of doubly orthogonal slender single-walled carbon nanotubes (SWCNTs) at the vicinity of each other are of interest. The van der Waals (vdW) forces play an important role in dynamic interactions between two adjacent nanotubes. Using Lennard-Jones potential function, such a phenomenon is appropriately modeled by a newly introduced vdW force density function. By employing Hamilton’s principle, the equations of motion are obtained based on the nonlocal Rayleigh beam theory. In fact, these are integro-partial differential equations and seeking an exact or even analytical solution to them is a very difficult job. Therefore, an efficient numerical solution is proposed. The effects of the intertube distance, slenderness ratio, small-scale parameter, aspect ratio, and elastic properties of the surrounding medium on the free vibration of the nanosystem are addressed. The obtained results could be regarded as a pivotal step for better realizing of dynamic behaviors of more complex systems consist of multiple orthogonal networks of nanotubes.     

Influence of surface effects on the pull-in instability of NEMS electrostatic switches

The influence of surface effects, including residual surface stress and surface elasticity, on the pull-in instability of electrostatic switches in nanoelectromechanical systems (NEMS) is studied using an Euler-Bernoulli beam model. This model is inherently nonlinear due to the driving electrostatic force and Casimir force which become dominant at the nanoscale. Since no exact solutions are available for the resulting nonlinear differential equation, He's homotopy perturbation method (HPM) is used to get the approximate analytical solutions to the static bending of NEMS switches, which are validated by numerical solutions of the finite difference method (FDM). The results demonstrate that surface effects play a significant role in the selection of basic design parameters of NEMS switches, such as static deflection, pull-in voltage and detachment length. Surface effects on low-voltage actuation windows are also characterized for these switches. The present study is envisaged to provide useful insights for the design of NEMS switches.     

Mechanical interaction between single-walled carbon nanotubes during the formation of a bundle

This paper investigates the intertubular van der Waals interactions that produce the initial cross sectional distortion of single-walled carbon nanotubes during a bundle formation. By combining the analysis of molecular dynamics with the continuum mechanics, the distributions of the van der Waals forces were determined. The dependence of the load parameters, deformation variables and the lattice constant on the nanotube radius, was also investigated. It was found that the van der Waals forces are attractive and vary circumferentially in a harmonic manner. For the considered zigzag nanotubes of radius 7–14 Å, the intensity of the van der Waals forces is radius-dependent and can be as large as 6–11 GPa in the channels of the bundle and 1–5 GPa at the closest points between the single-walled nanotubes.     

Pull-in instability of geometrically nonlinear micro-switches under electrostatic and Casimir forces

This paper investigates the pull-in instability of micro-switches under the combined electrostatic and intermolecular forces and axial residual stress, accounting for the force nonlinearity and geometric nonlinearity which stems from mid-plane stretching. The micro-switch considered in the present study is made of either homogeneous material or non-homogeneous functionally graded material with two material phases. Theoretical formulations are based on Euler?CBernoulli beam theory and von Karman type nonlinear kinematics. The principle of virtual work is used to derive the nonlinear governing differential equation which is then solved using the differential quadrature method (DQM). Pull-in voltage and pull-in deflection are obtained for micro-switches with four different boundary conditions (i.e. clamped?Cclamped, clamped-simply supported, simply supported and clamped-free). The present solutions are validated through direct comparisons with experimental and other existing results reported in previous studies. A parametric study is conducted, focusing on the combined effects of geometric nonlinearity, gap ratio, slenderness ratio, Casimir force, axial residual stress and material composition on the pull-in instability.     

Self-organization of PbTe and SnTe nanostructures on the van der Walls GaSe(0001) surface

The formation of nanodimensional defects on the van der Waals (0001) surface of a gallium selenide (GaSe) layered crystal and the growth of lead telluride (PbTe) and tin telluride (SnTe) nanostructures by deposition on this surface from the vapor phase has been studied. It is established that linear defects of various shapes and nanodimensional cavities (nanocavities) are formed on the van der Waals surface of a GaSe crystal as a result of plastic deformation. The nucleation and growth of PbTe and SnTe nanostructures on the flat van der Waals surface and in the nanocavities proceeds according to different mechanisms. It is demonstrated that selective growth of lead and tin chalcogenide nanostructures is possible on the van der Waals surfaces with nanocavities formed as a result of self-organization in the uppermost (less than 1 nm thick) layer of a GaSe crystal.     

Nanoscale modeling of an embedded multi-shell fullerene and its application to vibrational analysis

In this paper, nanoscale modeling of a multi-shell fullerene embedded in an elastic medium and its application to vibrational analysis is investigated. The spherical layers of the multi-shell fullerene are concentrically nested, with carbon-carbon van der Waals interactions between them. Also, the whole multi-shell fullerene is influenced by polymer-carbon van der Waals forces from the surrounding elastic medium. The elasticity generated by the carbon-carbon bonds is assumed to be distributed isotropically over the fullerene surfaces. Following derivation of explicit equations for the motion of the multi-shell fullerene, vibrational behavior of a double-shell fullerene is analyzed and resonant frequencies and the associated mode shapes are determined.     

Unified nonlinear quasistatic and dynamic analysis of RF-MEMS switches

In this paper, the modified couple stress-based strain gradient theory is used to provide a unified nonlinear model of the quasistatic and dynamic behavior of an electrostatic microelectromechanical systems microbeam capacitive switch of the Euler–Bernoulli type. Our model not only accounts for the contact between the microbeam and the dielectric substrate using nonlinear springs and dampers, but also accounts for the system size by introducing an internal material length scale parameter. In view of the size of the microbeam and electrostatic gaps involved, Casimir and Van der Waals forces, damping force due to the squeeze membrane effect and electrostatic force with first-order fringing field effects were accounted for in our model. The resulting nonlinear system of PDEs was expanded into a coupled system using series expansion and integrated into ODEs using weighted residuals of the Galerkin type. To overcome the difficulties associated with the determination of the contact length, the Heaviside function for deflection was replaced with a Heaviside function for the contact length, and an iterative procedure was adopted to determine the contact length. To obtain the time variation of the microbeam, the dynamic system of equations was solved using Newmark’s integration scheme. The outcome of our work shows the dependence of the pull-in voltage upon the inertia force, slenderness ratios of the microbeam, the electrostatic gap and the initial boundary conditions of the switch. In addition, we were also able to provide the full history of the microbeam past the pull-in threshold.