Investigation of size-dependent quasistatic response of electrically actuated nonlinear viscoelastic microcantilevers and microbridges

This study investigates the size-dependent quasistatic response of a nonlinear viscoelastic microelectromechanical system (MEMS) under an electric actuation. To have this problem in view, the deformable electrode of the MEMS is modelled using cantilever and doubly-clamped viscoelastic microbeams. The modified couple stress theory in conjunction with Bernoulli–Euler beam theory are used for mathematical modeling of the size-dependent instability of microsystems in the framework of linear viscoelastic theory. Simultaneous effect of electrostatic actuation including fringing field, residual stress, mid-plane stretching and Casimir and van der Waals intermolecular forces are considered in the theoretical model. A single element of the standard linear solid element is used to simulate the viscoelastic behavior. Based on the extended Hamilton’s variational principle, the nonlinear governing integro-differential equation and boundary conditions are derived. Thereafter, a new generalized differential-integral quadrature solution for the nonlinear quasistatic response of electrically actuated viscoelastic micro/nanobeams under two different boundary conditions; doubly-clamped microbridge and clamped-free microcantilever. The developed model is verified and a good agreement is obtained. Finally, a comprehensive study is conducted to investigate the effects of various parameters such as material relaxation time, durable modulus, material length scale parameter, Casimir force, van der Waals force, initial gap and beam length on the pull-in response of viscoelastic microbridges and microcantilevers in the framework of viscoelasticity.     

Towards non-linear modeling of molecular interactions arising from adsorbed biological species on the microcantilever surface

Microcantilever-based biosensors are rapidly becoming an enabling sensing technology for a variety of label-free biological applications due to their extreme applicability, versatility and low cost. These sensors operate through the adsorption of species on the functionalized surface of microcantilevers. This paper presents a general framework towards modeling resonance frequency changes induced due to the surface stress arising from the adsorption of biological species on the surface of the microcantilever. Very few works have dealt with the effect of surface stress on the resonance frequency shifts of microcantilevers and have mainly assumed a simple model for the vibrating microcantilever beam. In the proposed modeling framework, the non-linear terms due to the beam's flexural rigidity from macro- to microscale as well as varying nature of the adsorption induced surface stress are considered. It is first shown that the non-linearity of the system originates from two different sources, namely, microcantilever flexural rigidity and adsorption induced surface stress. Through simulation results, it is demonstrated that the non-linearity due to the surface stress does not have a considerable effect on the resonance frequency change of the microcantilever. However, non-linearity due to flexural rigidity (which is directly attributed to beam's dimensions) plays an important role in the resonance frequency shift, and hence, in the resultant molecular recognition capability. It is then shown that the adsorption induced resonance frequency shift highly depends on the size of the microcantilever. It is also demonstrated that the effect of surface stress and molecular arrangement modeling presented in this paper is more sensitive to the change of the microcantilever length.     

Growth limit of carbon onions – A continuum mechanical study

The growth of carbon onions is simulated using continuum mechanical shell models. With this models it is shown that, if a carbon onion has grown to a critical size, the formation of an additional layer leads to the occurrence of a structural instability. This instability inhibits further growth of carbon onions and, thus, can be a reason for the limited size of such particles. The loss of stability is mainly evoked by van der Waals interactions between misfitting neighboring layers leading to self-equilibrating stress states in the layers due to mutual accommodation. The influence of the curvature induced surface energy and its consequential stress state is investigated and found to be rather negligible. Furthermore, it is shown that the nonlinear character of the van der Waals interactions has to be considered to obtain maximum layer numbers comparable to experimental observations. The proposed model gives insight into mechanisms which are assumed to limit the size of carbon onions and can serve as basis for further investigations, e.g., of the formation of nanodiamonds in the center of carbon onions.     

Effects of the van der Waals force,squeeze-film damping,and contact bounce on the dynamics of electrostatic microcantilevers before and after pull-in

The operational range of microcantilever beams under electrostatic force can be extended beyond pull-in in the presence of an intermediate dielectric layer. In this paper, a systematic method for deriving dynamic equation of microcantilevers under electrostatic force is presented. This model covers the behavior of the microcantilevers before and after the pull-in including the effects of van der Waals force, squeeze-film damping, and contact bounce. First, a polynomial approximate shape function with a time-dependent variable for each configuration is defined. Using Hamilton’s principle, dynamic equations of microcantilever in all configurations have been derived. Comparison between modeling results and previous experimental data that have been used for validation of the model shows a good agreement.     

Effect of the mass-transfer intensity on the dynamic relations on a phase interface

The dynamic relations on the phase interface are derived in the isothermal approximation for liquids in which the dependence of the surface energy on the mass flux across the phase interface must be taken into account. The criterion of stability of subsonic phase-transition fronts with plane waves with respect to small perturbations is determined. The dependence of the surface tension on the mass flux is calculated for the well-known exact solution to the problem of the structure of isothermal phase transition within the framework of the generalized van der Waals model.     

Bifurcation and pull-in voltages of primary resonance of electrostatically actuated SWCNT cantilevers to include van der Waals effect

In this work the voltage response of primary resonance of electrostatically actuated single wall carbon nano tubes (SWCNT) cantilevers over a parallel ground plate is investigated. Three forces act on the SWCNT cantilever, namely electrostatic, van der Waals and damping. While the damping is linear, the electrostatic and van der Waals forces are nonlinear. Moreover, the electrostatic force is also parametric since it is given by AC voltage. Under these forces the dynamics of the SWCNT is nonlinear parametric. The van der Waals force is significant for values less than 50 nm of the gap between the SWCNT and the ground substrate. Reduced order model method (ROM) is used to investigate the system under soft excitation and weak nonlinearities. The voltage-amplitude response and influences of parameters are reported for primary resonance (AC near half natural frequency).     

Bifurcations and chaos in vibrating microcantilevers of tapping mode atomic force microscopy

We study the dynamics of a microcantilever in tapping mode atomic force microscopy when it is close to the sample surface and the van der Waals force has an important influence. Utilizing the averaging method, the extended version of the subharmonic Melnikov method and the homoclinic Melnikov method, we show that abundant bifurcation behavior and chaotic motions occur in vibrations of the microcantilever. In particular, in the subharmonic Melnikov analyses, a degenerate resonance is treated appropriately. Necessary computations for the subharmonic and homoclinic Melnikov methods are performed numerically. Numerical bifurcation analyses and numerical simulations are also given to demonstrate the theoretical results.     

A cohesive law for carbon nanotube/polymer interfaces based on the van der Waals force

We have established the cohesive law for interfaces between a carbon nanotube (CNT) and polymer that are not well bonded and are characterized by the van der Waals force. The tensile cohesive strength and cohesive energy are given in terms of the area density of carbon nanotube and volume density of polymer, as well as the parameters in the van der Waals force. For a CNT in an infinite polymer, the shear cohesive stress vanishes, and the tensile cohesive stress depends only on the opening displacement. For a CNT in a finite polymer matrix, the tensile cohesive stress remains the same, but the shear cohesive stress depends on both opening and sliding displacements, i.e., the tension/shear coupling. The simple, analytical expressions of the cohesive law are useful to study the interaction between CNT and polymer, such as in CNT-reinforced composites. The effect of polymer surface roughness on the cohesive law is also studied.     

Interfacial slippage effect on the spontaneous instability of 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 effect of interfacial slippage and the competition among van der Waals interaction potential energy, strain energy, and surface energy, a set of three-dimensional governing equations of the spontaneous instability is 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 the variation of the dimensionless characteristic wavenumber and the critical interaction stiffness with the critical time are discussed by considering the effect of interfacial slippage.     

Morphological stability of epitaxial thin elastic films by van der Waals force

Summary  The morphological stability of epitaxial thin elastic films on a substrate by van der Waals force is discussed. It is found that only van der Waals force with negative Hamaker constant tends to stabilize the film, and the lower bound for the Hamaker constant is also obtained for the stability of thin film. The critical value of the undulation wavelength is found to be a function of both film thickness and external stress. The charateristic time-scale for surface mass diffusion scales to the fourth power to the wavelength of the perturbation. Received 4 December 2000; accepted for publication 31 July 2001     

范德华力对磁头/硬盘系统气体润滑特性的影响

通过数值方法分析范德华力对磁头承载力的影响以及随飞行姿态的变化规律.结果表明,范德华力主要作用在磁头滑块尾部的凸台上,当飞行高度低于10 nm后,范德华力使得滑块的承载能力明显降低,从而导致飞行高度下降.范德华力主要与飞行高度和俯仰角有关,随着飞行高度和俯仰角减小,范德华力使承载能力和俯仰转矩明显减小.范德华力对侧翻角的影响不敏感,随着侧翻角的增加,范德华力使滑块的承载能力和侧翻转矩略微减小.在磁头设计时,通过增加俯仰角和减小尾部凸台面积可以减小范德华力的影响.     

Spinodal surface instability of soft elastic thin films

When the thicknesses of thin films reduce to microns or even nanometers, surface energy and surface interaction often play a significant role in their deformation behavior and surface morphology. The spinodal surface instability induced by the van der Waals force in a soft elastic thin film perfectly bonded to a rigid substrate is investigated theoretically using the bifurcation theory of elastic structures. The analytical solution is derived for the critical condition of spinodal surface morphology instability by accounting for the competition of the van der Waals interaction energy, elastic strain energy and surface energy. Detailed examinations on the effect of surface energy, thickness and elastic properties of the film show that the characteristic wavelength of the deformation bifurcation mode depends on the film thickness via an exponential relation, with the power index in the range from 0.749 to 1.0. The theoretical solution has a good agreement with relevant experiment results.     

Parametric variational principle and quadratic programming method for van der Waals force simulation of parallel and cross nanotubes

A parametric variational principle for van der Waals force simulation between any two adjacent nonbonded atoms and the corresponding improved quadratic programming method for numerical simulation of mechanical behaviors of carbon nanotubes are developed. Carbon nanotubes are modeled and computed based on molecular structural mechanics model. van der Waals force is simulated by the network of bars (called bar network) with a special nonlinear mechanical constitutive law (called generalized parametric constitutive law) in the finite element analysis. Compared with conventional numerical methods, the proposed method does not depend on displacement and stress iteration, but on the base exchanges in the solution of a standard quadratic programming problem. Thus, the model and method developed present very good convergence behavior in computation and provide accurate predictions of the mechanical behaviors and displacement distributions in the nanotubes. Numerical results demonstrate the validity and the efficiency of the proposed method.     

STRENGTH, PLASTICITY, INTERLAYER INTERACTIONS AND PHASE TRANSITION OF LOW-DIMENSIONAL NANOMATERIALS UNDER MULTIPLE FIELDS

Atoms are hold together to form different materials and devices through short range interactions such as chemical bonds and long range interactions such as the van der Waals force and electromagnetic i...     

Axial buckling of multi-walled carbon nanotubes and nanopeapods

In this paper, we investigate both pre- and post-buckling behaviors of multi-walled carbon nanotubes and multi-walled carbon nanopeapods by incorporating into the applied forces of a prescribed beam equation both van der Waals interactions between the adjacent walls of the nanotubes and the interactions between the fullerenes and the inner wall of the nanotube. Two beam theories are employed. First, we utilize Donnell’s equilibrium equation to derive an axial stability condition for the multi-walled carbon nanotubes and multi-walled carbon nanopeapods. We then determine analytically the critical forces for single-walled and double-walled nanotubes and nanopeapods. Given the outer nanotube of a fixed radius, we observe that the critical force and strain derived from the axial buckling stability criterion decrease as a result of the molecular interactions between the adjacent layers of the nanotubes and the molecular interactions between the embedded fullerenes and the inner carbon nanotube, which is in agreement with existing literature. Next, we utilize an Euler–Bernoulli beam equation incorporating the curvature effect to obtain the post-buckled axial bending displacement for the multi-walled nanotubes and nanopeapods. We find that the interactions between molecules generate an inward force, which tends to resist any applied forces. While the inward force induced by the fullerenes to the inner wall of the nanotube vanishes as we increase the applied force, the inward force induced by the layers increases as the applied force increases. The main contribution of this paper is the incorporation of both van der Waals interactions and the curvature effect into prescribed beam theories to accurately measure the critical forces and the buckled displacements of multi-walled nanotubes and nanopeapods subject to a small external force. Our analysis is relevant to future nano devices, such as biological sensors and measuring devices for small forces arising from electrical charges or Casimir forces.     

基于气液相界面捕捉的统一气体动理学格式

气液相界面运动的研究无论是在科学还是工程领域都是非常重要的. 其中, 非平衡流动的计算尤其受到关注. 基于此, 我们构造了捕捉气液相界面的统一气体动理学格式. 由于统一气体动理学格式将自由输运和粒子碰撞耦合起来更新宏观物理量和微观分布函数, 故而可以求解非平衡流动. 具体思路是, 通过将范德瓦尔斯状态方程所表达的非理想气体效应引入统一气体动理学格式之中来捕捉气液相界面, 两相的分离与共存通过范德瓦尔斯状态方程描述. 由于流体在椭圆区域是不稳定的, 因此气液相界面可以通过蒸发和凝结过程自动捕捉. 如此, 一个锋锐的相界面便可以通过数值耗散和相变而得到. 利用该方法得到麦克斯韦等面积律(Maxwell construction)对应的数值解, 并与其相应的理论解相比较, 二者符合良好. 而后, 通过对范德瓦尔斯状态方程所描述的液滴表面张力进行数值计算, 验证了Laplace定理. 此外, 通过模拟两个液滴的碰撞融合过程, 进一步证明了该格式的有效性. 但是, 由于范德瓦尔斯状态方程的特性, 其所构造的格式仅适用于液/气两相密度比小于5的情况.      

SURFACE-ENHANCED CANTILEVER SENSORS WITH NANO-POROUS FILMS

Developing surface-enhanced microcantilevers with improved sensitivities is of longstanding interest. In this paper, the design of surface-enhanced cantilever sensors using nano- （micro-） porous films as surface layers is proposed. The static deformation and resonance frequencies of these surface-enhanced sensors with the simultaneous effects of the eigenstrain, the surface stress and the adsorption mass are analyzed. It is shown that the sensitivities of these novel cantilever sensors for the static deformation and resonance frequencies can be tuned by the porosity, the size of the pores and the structure of the porous films. For the three kinds of cantilever consisting of solid films, films with aligned cylindrical micro-scale pores, and those with nano-scale pores, the nano-porous one has the highest static and dynamic sensitivities, whereas the solid one has the lowest.     

Surface instability of a semi-infinite anisotropic elastic body under surface van der Waals forces

We investigate the surface instability of an anisotropic elastic half-plane subjected to surface van der Waals forces due to the influence of another rigid contactor by means of the Stroh formalism. It is observed that the surface of a generally anisotropic elastic half-plane subjected to van der Waals forces from another rigid flat is always unstable. The wave number of the surface wrinkling is only reliant on the positive M₂₂ component of the 3 × 3 surface admittance tensor M, the van der Waals interaction coefficient β and the surface energy γ of the elastic half-plane. The decay rate of surface perturbation along the direction normal to the surface of the anisotropic half-plane is different from the wave number, a phenomenon different from that observed for an isotropic half-plane.     

Prediction of the critical buckling load of multi-walled carbon nanotubes under axial compression

In this paper, we propose a new explicit analytical formula of the critical buckling load of double-walled carbon nanotubes (DWCNT) under axial compression. This formula takes into account van der Waals interactions between adjacent tubes and the effect of terms involving tube radii differences generally neglected in the derived expressions of the critical buckling load published in the literature. The elastic multiple Donnell shells continuum approach is employed for modelling the multi-walled carbon nanotubes. The validation of the proposed formula is made by comparison with a numerical solution. The influence of the neglected terms is also studied.