Augmentation of the stable static travel range of electrostatically actuated slender nano-cantilevers by accounting for the influence of the van der Waals force

This paper investigates the pull-in instability of graphene sheets. The influence of geometry parameters such as chirality of graphene and length to gap ratio is studied using molecular dynamics (MD). For molecular interactions, the AIREBO potential is used. Furthermore, by applying the electrostatic and van der Waals (vdW) forces, pull-in voltages are calculated. Size effect is estimated, with adding the fringing field effect correction factor to the electrostatic force. In MD simulations, the graphene sheets on the armchair and zigzag structure have been investigated. The results show that the closer the moving electrode to the fixed electrode, the greater the effect of van der Waals force than the electrostatic force. The results also represent that the vdW force and fringing effect on the electrostatic load increases the pull-in deflection and decrease the pull-in voltage. The numerical results of the present study show good agreement with previous analytical and experimental researches.

On nanoscale, van der Waals (vdW) force has a very important influence on physical, chemical, electrical and mechanics properties of low-dimensional nanostructures. vdW force is an important factor affecting constitutive relation, boundary condition of low-dimensional nanostructures, and then affecting its natural frequency and nonlinear dynamics property. The nanostructures have wide applications in fabricating the nanoresonator and nanosensor, such as mass sensing, optical sensing, signal-processing, et al. In this paper, the latest research progress about the effect of vdW force on the vibrational properties of low-dimensional nanostructures is introduced. The main contents of this paper are as follows. The vdW force is introduced in the first part of this paper. The source of vdW force and the expression to describe vdW force are presented. The research progress on the basic development history of vdW force is discussed. In the second part, the effect of vdW force on the mechanical property of nanomaterials is discussed, including buckling, stretching, fracture, et al. vdW force is weak in resisting interlayer shear or sliding. However, it can affect the buckling and fracture behavior of nanomaterials, and can especially affect the deformation of nanomaterials. The effect of vdW force on the vibrational behavior of nanomaterials is presented in the third part. Different from the macro structure, vdW force has a significant influence on the vibrational behavior of low-dimensional nanoscale structures. The experimental method, molecular dynamic simulations method and continuum mechanics method are usually used for studying the influence of vdW force on the vibrational behavior of low-dimensional nanostructure. Due to the existence of vdW force between interlayer of low-dimensional nanostructures, the vibration of double-layered low-dimensional nanostructures appear anti-phase vibration mode and in-phase vibration mode. The frequency of anti-phase vibration mode is much higher than that of in-phase vibration mode, but the vdW force cannot affect the first order natural frequency. However, the existence of the vdW force between the interlayer of multi-layered low-dimensional nanostructure can cause the interlayer shear force. The vdW force can affect the vibrational behavior of low-dimensional nanostructure, e.g., the natural frequency is affected by vdW force. Some latest researches about the interlayer shear force caused by vdW force on the vibration property are discussed in this paper. The vdW force can also affect the boundary condition of the vibrational problems for low-dimensional nanostructure. For one-dimensional nanostructures, they are usually bridged on the substrate. But for two-dimensional nanostructures, they are usually suspended on the circular hole. The vdW force between the substrate and low-dimensional nanostructure is a very important factor that can affect the boundary conditions of vibrational property of these low-dimensional nanostructures. Some open problems and future researches in the dynamic behaviors of low-dimensional nanostructure with vdW force are presented. To study the effect of vdW force on the vibration characteristics of nanoscale structures by quantum mechanics method or semi quantum mechanics method is a prodigious challenge. The effect of nonlinear vdW force and long-range nonlocal vdW force on the vibration characteristics of nanoscale structures should be an interesting problem to be solved. The influence of vdW force on the vibration characteristics of nanostructures needs to be verified by experiments.

In this paper the two-point boundary value problem (BVP) of the cantilever deflection atnano-scale separations subjected to van der Waals and electrostatic forces is investigatedusing analytical and numerical methods to obtain the instability point of the beam.In the analytical treatment of the BVP, the nonlinear differential equation ofthe model is transformed into the integral form by using the Green’s functionof the cantilever beam. Then, closed-form solutions are obtained by assumingan appropriate shape function for the beam deflection to evaluate the integrals.In the numerical method, the BVP is solved with the MATLAB BVP solver, whichimplements a collocation method for obtaining the solution of the BVP. The largedeformation theory is applied in numerical simulations to study the effect of the finitekinematics on the pull-in parameters of cantilevers. The centerline of the beam under theeffect of electrostatic and van der Waals forces at small deflections and at the point ofinstability is obtained numerically. In computing the centerline of the beam, the axialdisplacement due to the transverse deformation of the beam is taken into account, usingthe inextensibility condition.The pull-in parameters of the beam are computed analytically and numerically under theeffects of electrostatic and/or van der Waals forces. The detachment length and theminimum initial gap of freestanding cantilevers, which are the basic design parameters, aredetermined.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.

The Journal of Physics: Condensed Matter special issue is dedicated to the memory of David C Langreth and his contribution to research in the field of Van der Waals (vdW) interactions in advanced materials. David C Langreth was an outstanding condensed matter theorist and a scholar who significantly influenced researchers through his particle-physics based insights into density functional theory (DFT), surface science, and related areas. His seminal works ranged from conserving formulations of interacting nonequilibrium transport and formal-scattering theory to an explicit formulation the exact DFT exchange-correlation energy in the adiabatic connection formula (ACF). David C Langreth's another significant contribution was in the area of vdW interactions in materials, as he played a key role in developing and formulating the vdW density functional (vdW-DF) method.

The quantum theory of the fluctuations of the van der Waals (vdW) force between macroscopic bodies is developed. Unlike the mean vdW force that is determined by all quantum states that contribute to the optical absorption, the energies of those excitations of the interacting bodies that contribute effectively to the vdW force noise are thermal energies. Contrary to the mean vdW force, the vdW force noise drops with decreasing temperature. Due to these differences the main mechanism of the mean vdW force and that of the vdW force noise may be different, e.g., the mean vdW force is determined by electronic excitations, the force noise by the random lattice or impurity dynamics. Since the vdW force is linear in the fields squared its spectral density depends not only on the frequency of its measurement ${\ensuremath{\omega}}_{S}$ but also on the integral over the second frequency. The dependence of its integrand on this frequency and temperature is quantumlike even at small ${\ensuremath{\omega}}_{S}⪡{k}_{B}T∕\ensuremath{\hbar}$, and such may be the behavior of the vdW noise. In the case of interacting metals the vdW force noise is enhanced by dc voltage applied across these metals. This additional noise is proportional to the voltage squared and to the spectral density of the random electric field at the frequency of noise measurement. The theory is in qualitative agreement with experiments.

Molecular charge asymmetrically distributed in a diffusing tagged particle causes a nonzero electrostatic force balanced by an opposing van der Waals (vdW) force. Fluctuations of electrostatic and vdW forces are highly correlated, and they destructively interfere in the force variance. This phenomenology is caused by the formation of a structurally frozen hydration layer for a particle diffusing in water and is responsible for a substantial speedup of translational diffusion compared to traditional theories of dielectric friction. Diffusion of proteins is insensitive to charge mutations, while smaller particles with asymmetric charge distribution can show a strong dependence of translational and rotational diffusion on molecular charge. Dielectric calculations of the electrostatic force require low values of ≃5 for the effective dielectric constant of interfacial water to be consistent with simulations.

In this study, nonlinear dynamic behavior of a capacitive carbon nano-tube switch is investigated considering van der Waals (vdW) force. The carbon nano-tube is considered as a nano-beam and dynamic equation of motion for this switch is presented based on the Euler–Bernoulli beam model. In order to simplify the bifurcation analysis of the carbon nano-switch, dynamic equation based on the modified mass–damper–spring model has been extracted. The fixed points of the switch have been studied in the absence and presence of the electrostatic force. Global and local criterion for stability study of the nano switch is employed. In order to study the local stability of the fixed points, the associated eigenvalues were extracted and the stability of these points was identified. For studying global stability, motion trajectories of the switch with and without considering damping effect are provided and presented. Basin of attraction set and region of periodic set and dependency of them on the applied voltage, as well as the vdW force and damping effect are investigated. Critical values of the applied voltage and vdW parameter leading to qualitative changes in the nano switch behavior through a saddle node bifurcation are obtained.

Ubiquitous van der Waals (vdW) forces are very important for nanostructures. Although the vdW forces between two surfaces (or two layers) have been measured for several decades, a direct detection at the single-molecule level is still difficult. Herein, we report a novel method to solve this problem in high vacuum by means of AFM-based single-molecule force spectroscopy (SMFS). Solvent molecules and surface adsorbed water are removed thoroughly under high vacuum so that the situation is greatly simplified. A constant force plateau can be observed when a polymer chain is peeled off from a substrate in high vacuum. Accordingly, the vdW forces between one polymer repeating unit and the substrates can be obtained. The experimental results show that the vdW forces (typical range: 21–54 pN) are dependent on the species of substrates and the size of polymer repeating unit, which is in good accordance with the theoretical results. It is expected that this novel method can be applied to detect other non-covalent interactions (such as hydrogen bond and π-π stacking) at the single-molecule level in the future.

The pull-in instability is one of the most important phenomena which is usually associated with nanobeams when they are used in nanoelectromechanical systems (NEMS). This phenomenon may occur without electrical excitation and depends on different parameters. The aim of this paper is to investigate the nonlinear vibrations and pull-in instability of nanobeams in the presence of the van der Waals (vdW) force without electrical excitation. Utilizing Galerkin's method, the partial differential equation of motion is transferred to a nonlinear ordinary differential equation. Afterwards, the variational iteration method (VIM) is employed to obtain the nonlinear frequency and deflection of the nanobeam. The study is performed on doubly clamped, doubly simply supported and clamped-simply supported boundary conditions. The effects of boundary conditions, axial load, aspect ratio and the vdW force on nonlinear frequency and deflection as well as pull-in instability are discussed in details. In addition, three simple and useful equations are developed for predicting the critical values of the vdW force parameter in terms of axial load and aspect ratio parameters. These equations can be employed to estimate the dimensions of nanobeams before their fabrication and using them in the NEMS devices.

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.

The strong force that originates from breaking covalent bonds can be easily quantified through various testing platforms, while weak interfacial sliding resistance (ISR), originating from hydrogen bonding or van der Waals (vdW) forces, is very challenging to measure. Facilitated by an in-house nanomechanical testing system, we are able to precisely quantify and clearly distinguish the interfacial interactions between individual carbon fibers and several substrates governed by either hydrogen bonding or vdW forces. The specific ISR of the interface dominated by vdW forces is 3.55 ± 0.50 μN mm−1 and it surprisingly increases to 157.86 ± 44.18 μN mm−1 if the interface is bridged by hydrogen bonding. The ad hoc studies demonstrate that hydrogen bonding rather than vdW forces has great potential in sewing the interface if both surfaces are supportive of the formation of hydrogen bonds. The findings will enlighten the engineering of interfacial interactions and further mediate the entire mechanical performance of structures.

In this research, the static behavior of torsional nano-/micro-actuators under van der Waals (vdW) force is studied. First, the equilibrium equation governing the static behavior of torsional nano-/micro-actuators under vdW force is obtained. Then the energy method is utilized to investigate the statical stability of nano-/micro-actuator equilibrium points and a useful equation is suggested for the successful and stable design of nano-/micro-actuators under vdW force. Then, the equilibrium angle of nano-/micro-actuators is calculated both numerically and analytically using the homotopy perturbation method (HPM). It is observed that, with increasing instability number, defined in this paper, the rotation angle of the actuator is increased and pull-in suddenly occurs. Since analytical results are in good agreement with the numerical ones, the analytical method presented in this paper can be used as a fast, precise, and stable design tool in nano-/micro-actuators under vdW force.

MEMS devices utilize electrostatics as preferred actuation method. The accurate determination of pull-in instability parameters (i.e., pull-in voltage and pull-in displacement) of such devices is critical for their correct design. It should be noted that similar to parallel plate capacitors, the electrostatic force between the surface of the deformable microbeam and stationary ground is non-linear in nature. Hence the analysis associated with MEMS devices is always inherently non-linear. In the literature, these devices have been majorly analyzed as Bernoulli-Euler microbeams with cantilever or clamped-clamped beam end conditions. However, Dileesh et al. (doi: 10.1115/ESDA2012-82536) have studied the static and dynamic pull-in instability behavior of slender cantilever microbeams by developing a six-nodded spectral finite element based on the Timoshenko beam theory (TBT-SFE). They have demonstrated the accuracy of the TBT-SFE by comparing their results with corresponding results of COMSOL-based three-dimensional finite element simulations. In addition, effects of shear deformation also start to play significant role as the beam thickness-to-length ratio increases. In this paper, authors have developed the TBT-SFE based on the work by Dileesh et al. for the case of statics. However, unlike Dileesh et al. where they have developed a six-nodded TBT-SFE, authors have investigated the best combination of number of nodes per element and total number of elements to carry out the study. For this purpose, authors have first calculated results of the maximum beam transverse displacement, for a shear deformable propped-cantilever microbeam under the action of uniformly distributed transverse load, obtained by utilizing the developed TBT-SFE with different combinations of number of nodes per element and total number of elements. These results are then compared with corresponding analytical results available in the literature to arrive at the best combination of number of nodes per element and total number of elements for the electrostatic-elastic analysis. In the second step, the finalized TBT-SFE is utilized to determine static pull-in instability parameters of narrow microbeams with various fixity conditions and beam thickness-to-length ratios. This study highlights the importance of transverse shear effects on pull-in instability parameters of Timoshenko microbeams.

The coupling effect between torsion and bending in nano/micromirrors under the combined effect of capillary force and van der Waals (vdW) force is presented in this paper. At the first, the dimensionless equations governing the statical behavior of the nano/micromirror are obtained using the minimum total potential energy principle. Then the equations governing the pull-in state of the mirror are obtained using the implicit function theorem. The related results show that neglecting bending effect can lead to considerable overestimation in predicting the pull-in limits of the nano/micromirror under combined vdW and capillary forces. It is observed that vdW force reduces the pull-in angle and pull-in deflection of the supporting torsion beams of the mirror. The static behavior of the nano/micromirror under capillary and vdW loading is also studied and the results reveal that the static behavior of the nano/micromirror under capillary and vdW forces highly depends on the bending of the torsion beams. The results of this paper can be used for a safe and stable design and fabrication of mirrors using the wet etching process, where the gap between the mirror and the underneath substrate is sufficiently small and as a results both capillary and vdW forces have significant role in the stability of the system.

In recent years, carbon nanotubes (CNTs) have attracted great attention in the fabrication of probe tips and actuators for scanning microscopes. Herein, the pull-in instability of CNT-based probe is investigated using a nanoscale continuum model. The Euler–Bernoulli beam theory is applied to model the elastic response of the CNT. The van der Waals attraction is computed from the simplified Lennard-Jones potential. Two analytical methods (i.e., Homotopy perturbation method and Adomian decomposition method) are applied to solve the nonlinear governing equation of the system. Furthermore, the obtained results are validated by comparing with experimental results in the literature as well as numerical solutions of the finite difference method. The pull-in parameters are determined and effect of van der Waals force and a geometrical parameter effect on the instability behavior of the CNT is discussed. Moreover, the detachment length and minimum initial gap of the freestanding CNT probe are determined.