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The Casimir forces induced by fluctuations of electromagnetic field have been measured with a high precision at distances above $60$ nm. Although at shorter separations the magnitude of the force increases, it is more complicated to measure the forces at distances below $50$ nm. We review the problems that appear in this range of distances and their possible solutions. The first problem is related to the pull-in instability that occurs when two surfaces get too close to each other. As a particular manifestation of this problem, the spontaneous formation of the capillary bridges at distances $\sim 10$ nm is discussed. As an alternative, we discuss the method of adhered cantilever, which does not suffer from the pull-in instability. The second problem is related to the roughness of interacting surfaces that gives a significant deviation of the force from the expected scaling with the distance. We explain how the deviation can be related to the contribution of high asperities to the force. Characterization of the deposited rough films is also covered with a special emphasis on the excessive number of high asperities for the films deposited under nonequilibrium conditions. The third problem is related to the poor precision in the determination of the absolute distance between the bodies that results in a large total error in the force. We discuss the methods to determine the distance upon contact and cover a proposition to improve this precision in the method of adhered cantilever.

Beam-type nano-electromechanical system (NEMS) is one of the most components in constructing nano-devices. Herein, a distributed parameter model is used to study the influence of van der Waals (vdW) attraction on the pull-in performance of the cantilever NEMS. Homotopy perturbation method (HPM) is applied to solve the nonlinear governing equation of the actuators in the form of convergent series. Moreover, analytical results are compared with those of numerical method as well as a lumped parameter model. The pull-in voltage and critical cantilever tip deflection of NEMS are determined. Results depict that vdW attraction decreases the pull-in deflection and voltage of the NEMS. On the other hand, the fringing field increases the pull-in deflection while decreases the pull-in voltage of the system. The analytical results have good agreement with numerical results and those available in the literature.

This paper studies the effect of Casimir force on the pull-in instability of electrostatically actuated torsional nano/micro actuators. Dependence of the actuator's pull-in angle and pull-in voltage on several design parameters are investigated and it is found that Casimir force can considerably reduce the stability limits of the torsional actuators. Nonlinear equilibrium equation is solved numerically and analytically using straight forward perturbation expansion method. It is observed that a fourth-order perturbation approximation can precisely model the behavior of a torsional actuator. The results of this paper can be used for safe and stable design of torsional nano/micro actuators.

Experiments reveal that mechanical behavior of nanostructures is size-dependent. Herein, the size dependent pull-in instability of torsional nano-mirror is investigated using strain gradient nonclassic continuum theory. The governing equation of the mirror is derived taking the effect of electrostatic Coulomb and molecular van der Waals (vdW) forces into account. Variation of the rotation angle of the mirror as a function of the applied voltage is obtained and the instability parameters i.e., pull-in voltage and pull-in angle are determined. Nano-mirrors with square and circular cross-sectional beams are investigated as case studies. It is found that when the thickness of the torsional nano-beam is comparable with the intrinsic material length scales, size effect can substantially increase the instability parameters of the rotational mirror. Moreover, the effect of vdW forces on the size-dependent pull-in instability of the system is discussed. The proposed model is able to predict the experimental results more accurately than the previous classic models and reduce the gap between experiment and previous theories.

The Casimir force can strongly interfere with the pull-in performance of ultra-small structures. The strength of the Casimir force is significantly affected by the geometries of interacting bodies. Previous investigators have exclusively studied the effect of the Casimir force on the electromechanical instability of nanostructures with planar geometries. However no work has yet considered this effect on the pull-in instability of systems with cylindrical geometries such as nanotweezers fabricated from nanotube/nanowires. In our present work, the influence of the Casimir attraction on the electrostatic response and pull-in instability of nanotweezers fabricated from cylindrical conductive nanowires/nanotubes is theoretically investigated. An asymptotic solution, based on scattering theory, is applied to consider the effect of vacuum fluctuations in the theoretical model. The Euler–Bernoulli beam model is employed, in conjunction with the size-dependent modified couple stress continuum theory, to derive the governing equation of the nanotweezers. The governing nonlinear equations are solved by two different approaches, i.e., the modified Adomian–Padé method (MAD–Padé) and a numerical solution. Various aspects of the problem, i.e., the variation of pull-in parameters, effect of geometry, coupling between the Casimir force and size dependency effects and comparison with the van der Waals force regime are discussed.

Electromechanical nanocantilevers are promising for using as sensors/detectors in centrifugal-fluidic systems. For this application, the presence of angular speed and electrolyte environment should be considered in the theoretical analysis. Herein, the pull-in instability of the nanocantilever incorporating the effects of angular velocity and liquid media is investigated using a size-dependent continuum theory. Using d’Alembert principle, the angular speed is transformed into an equivalent centrifugal force. The electrochemical and dispersion forces are incorporated considering the corrections due to the presence of electrolyte media. Two different approaches, i.e., the Rayleigh–Ritz method (RRM) and proposing a lumped parameter model (LPM), were applied to analyze the system. The models are validated with the results presented in literature. Impacts of the angular velocity, electrolyte media, dispersion forces, and size effect on the instability characteristics of the nanocantilever are discussed.

In this paper, the authors propose the neural modeling of optimized design for pull-in instability reduction of micromachined capacitive shunt radio frequency (RF) micro electro mechanical system (MEMS) switch. Prediction of effective dielectric constant and hence the critical collapse voltage that represents the bridge position instability for two typical bridge geometrics have been derived using artificial neural network (ANN). The effects of residual stress, length of center conductor and the gap between the bridge and center conductor of switch in lowering the driving voltage have been studied. Based on the neural model results, we have observed the reduction of 3 V in critical collapse voltage for an increase of 10 μm in length of center conductor. We have also noted the strong variation in voltage (reduction of 0.8 V for 1 MPa residual stress reduction) with respect to residual stress change. We achieved the reduction of 1.5 V in collapse voltage by reducing the gap between the bridge and the center conductor by 0.1 μm. Among the two structures considered, the structure with lower width of the center conductor proved as an optimum in achieving low critical collapse voltage. Further, the performance of trained neural network with the training datasets derived from MATLAB simulation has been evaluated in terms of convergence speed and mean square error.

Pull-in instability, as an inherent nonlinear problem, continues to become an increasingly important and interesting topic in the design of electrostatic Nano/Micro-electromechanical systems (N/MEMS) devices. Generally, the pull-in instability was studied in a continuous space, but when the electronic devices work in a porous medium, they need to be analyzed in a fractal partner. In this paper, we establish a fractal model for N/MEMS, and find a pull-in stability plateau, which can be controlled by the porous structure, and the pull-in instability can be finally converted to a stable condition. As a result, the pull-in instability can be completely eliminated, realizing the transformation of pull-in instability into pull-in stability.

The influence of Casimir and van der Waals forces on the instability of vibratory micro and nano-bridge gyroscopes with proof mass attached to its midpoint is studied. The gyroscope subjected to the base rotation, Casimir and van der Waals attractions is actuated and detected by electrostatic methods. The system has two coupled bending motions actuated by the electrostatic and Coriolis forces. First a system of nonlinear equations for the flexural-flexural deflection of beam gyroscopes is derived using the extended Hamilton's principle. In modeling, the nonlinearities due to mid-plane stretching, electrostatic forces, including fringing field, Casimir and van der Waals attractions, are considered. The method of homotopy perturbation is used to solve the equations of equilibrium, with the solution validated by numerical methods. In addition, the effect of nondimensional parameters on the instability and deflection of the gyroscope is investigated. The data presented can be used in the design of vibratory micro/nano gyroscopes.

This paper applies the homotopy perturbation method to the simulation of the static response of nano-switches to electrostatic actuation and intermolecular surface forces. The model accounts for the electric force nonlinearity of the excitation and for the fringing field effect. Using a mode approximation in the Galerkin projection method, the nonlinear boundary value differential equation describing the statical behavior of nano-switch is reduced to a nonlinear algebraic equation which is solved using the homotopy perturbation method. The number of included terms in the perturbation expansion for achieving a reasonable response has been investigated. Three cases have been specifically studied. These cases correspond to when the effective external force is the electrostatic force, the combined electrostatic and Casimir force and the combined electrostatic and van der Waals force. In all three cases the pull-in characteristics has been investigated thoroughly. Results have been compared with numerical results and also analytical results available in the literature. It was found that HPM modifies the overestimation of N/MEMS instability limits reported in the literature and can be used as an effective and accurate design tool in the analysis of N/MEMS.

In this paper, the generalized differential quadrature (GDQ) method is employed to investigate the static and dynamic pull-in instability of nano-switches with small width/height ratio. For nano-switches with slender narrow-width beam elements, the corrections due to surface layer, external forces and size effect are crucial. So the couple-stress theory combined with the Gurtin–Murdoch surface elasticity is employed to incorporate the coupled effects of size phenomena and surface energy. In addition, the model accounts for the external force corrections by incorporating the impact of narrow width on the distribution of Casimir attraction, van der Waals force and the fringing field. The effects of the above-mentioned corrections on the static and dynamic pull-in parameters are demonstrated. The results from the modified model are compared with those from conventional parallel beam-plate models in the literature. It is shown that the corrections significantly affect the pull-in parameters of the narrow-width switches and hence, cannot be ignored in theoretical models. For wide switches, the results of the proposed model approach to those of the conventional models.

This paper deals with the static behavior of an electrostatically actuated bilayered microswitch on the basis of the modified couple stress theory. The beam is modeled using Euler–Bernoulli beam theory and equivalent elastic modulus and length scale parameter are presented for the bilayer beam. Static deflection and pull-in voltage of the beam is calculated using numerical and analytical methods. The numerical method is based on an iterative approach while the homotopy perturbation method (HPM) is utilized for the analytical simulation. Results show that there is a very good agreement between these methods even in the vicinity of the pull-in instability. Moreover, the effects of different parameters such as thicknesses of layers and length scale parameter on the static deflection and instability of the microcantilever are studied. Results show that for the cases with the equivalent length scale parameter comparable to the thickness of beam, the size-dependency plays significant roles in the static behavior of the bilayer microcantilevers.

This paper introduces a new nonlinear model for microswitches based on the modified couple stress theory. The microswitch includes a microbeam which is connected to the clamped support from one side and attached to an electrostatically driven proof mass with a large gap from the other side. The microswitch operates in the pull-in instability with large deformation. The effects of fringing field and large curvature as well as size dependency are considered in the modeling. With regard to the size-dependent model, the equations of motion are obtained using Hamilton’s principle and solved by both numerical and analytical approaches. Consequently, dynamic pull-in instability is investigated based on the analytical and numerical solutions for dynamic conditions. The results depict that the dynamic deflection predicted by the modified couple stress theory is smaller than that obtained by the classical theory. The classical theory underestimates the pull-in instability voltage of the microswitches especially when the beam’s thickness is in the order of material length scale parameter. Furthermore, it is shown that neglecting nonlinearity due to large deflection leads to significant errors in the pull-in instability of the microstructures and these errors are calculated. The novelty of this paper is to provide a nonlinear size-dependent model for microswitches and to investigate the nonlinearity and instability of microswitches based on this model using the analytical and numerical methods.

In this paper, an integrated non-classical continuum model is developed to investigate the pull-in instability of electrostatically actuated functionally graded nanocantilevers. The model accounts for the simultaneous effects of local-microstructure, surface elasticity and surface residual in the presence of fringing field as well as Casimir and van der Waals forces. The modified couple stress and Gurtin–Murdoch surface elasticity theories are employed to conduct the scaling effects of microstructure and surface energy, respectively, in the context of Euler–Bernoulli beam hypothesis. Bulk and surface material properties are varied according to the power-law distribution through the beam thickness. The physical neutral axis position for mentioned FG nanobeams is considered. Hamilton principle is employed to derive the nonlinear size-dependent governing equations and the non-classical boundary conditions. The resulting nonlinear differential equations are solved utilizing the generalized differential quadrature method (GDQM). In addition, the non-classical boundary conditions of nanocantilever beams due to surface residual stress are exactly implemented. After validation of the obtained results by previously available data in the literature, the influences of different geometrical and material parameters on the pull-in instability of the FG nanocantilevers are examined in detail. It is concluded that the pull-in behavior of electrically actuated FG micro/nanocantilevers is significantly influenced by the material distribution, material length scale parameter, surface elasticity constant, surface residual stress, initial gap, slenderness ratio, Casimir, and van der Waals forces. The obtained results can be considered for modeling and analysis of electrically actuated FG nanocantilevers.

The Casimir forces induced by fluctuations of electromagnetic field have been measured with a high precision at distances above 60 nm. Although at shorter separations the magnitude of the force increases, it is more complicated to measure the forces at distances below 50 nm. We review the problems that appear in this range of distances and their possible solutions. The first problem is related to the pull-in instability that occurs when two surfaces get too close to each other. As a particular manifestation of this problem, the spontaneous formation of the capillary bridges at distances ∽10 nm is discussed. As an alternative, we discuss the method of adhered cantilever, which does not suffer from the pull-in instability. The second problem is related to the roughness of interacting surfaces that gives a significant deviation of the force from the expected scaling with the distance. We explain how the deviation can be related to the contribution of high asperities to the force. Characterization of the deposited rough films is also covered with a special emphasis on the excessive number of high asperities for the films deposited under nonequilibrium conditions. The third problem is related to the poor precision in the determination of the absolute distance between the bodies that results in a large total error in the force. We discuss the methods to determine the distance upon contact and cover a proposition to improve this precision in the method of adhered cantilever.

Pull-in instability of a cantilever nano-actuator was analyzed considering the fringing field and the Casimir force. The instability parameters of the actuator were determined analytically under the assumption of a second-degree shape function for the beam. The influence of the Casimir force and the fringing field effects on the pull-in parameters was investigated. The results demonstrate that the Casimir force decreases the pull-in deflection and voltage, the fringing field effects decrease the pull-in voltage, but the influence on the pull-in deflection depends on the Casimir force parameter.