Analytical investigation and numerical verification of Casimir effect on electrostatic nano-cantilevers

Abstract

In this paper, the two-point boundary value problem (BVP) of the nano-cantilever deflection subjected to Casimir and electrostatic forces is investigated using analytical and numerical methods to obtain the instability point of the nano-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. The pull-in parameters of the beam are computed under the combined effects of electrostatic and Casimir forces. Electrostatic microactuators and freestanding nanoactuators are considered as special cases of our study. The detachment length and the minimum initial gap of freestanding nanocantilevers, which are the basic design parameters for NEMS switches, are determined. The results of the analytical study are verified by numerical solution of the BVP. The centerline of the beam under the effect of electrostatic and Casimir forces at small deflections and at the point of instability is obtained numerically to test the validity of the shape function assumed for the beam deflection in the analytical investigation. Finally, the large deformation theory is applied in numerical simulations to study the effect of the finite kinematics on the pull-in parameters of nano-canilevers.

Modeling the size dependent pull-in instability of beam-type NEMS using strain gradient theory

“…Now, by Considering the first order fringing field correctness effect the electrostatic force per unit length of the beam is written as the following (Ramezani et al, 2008):…”

“…In distance less than few micrometers, Casimir force significantly affects the nano-beams stability. Casimir force between two plates can be explained via electromagnetic quantum vacuum fluctuations and motion of virtual photons between two conductive surfaces (Lifshitz, 1956) .In recent years, various approaches such as experimental measurements (Buks et al, 2001a;2001b), using finite element methods (Moghimi Zand et al, 2010;Tadi Beni et al, 2013), experimental observations (Wilson and Beck, 1996;Sundararajan and Bhushan, 2002), applying classic continuum theory (Ramezani et al, 2008;2007;Farrokhabadi et al, 2013;, semi-analytical approaches Koochi and Abadyan 2011;Duan et al, 2013) are utilized to investigate the effect of Casimir force on performance of nano-systems. Another dispersion force that is dominant in nano-scales is the vdW force that affects the stability of nano-structures.…”

The size-dependent electromechanical instability of double-sided and paddle-type actuators in centrifugal and Casimir force fields

“…When double-sided actuator exhibits large displacements, the curvature of the de ection should be considered; therefore, governing changes into [54]:…”

Instability analysis of torsional MEMS/NEMS actuators under capillary force

“…And they always exist between two objects close to each other. The influence of vdW and Casimir forces on the stability of the MEMS and NEMS actuators with suspended structures has been extensively reported in many literatures [6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Chan et al [10] showed that Casimir force can be used to actuate the NEMS devices under certain condition.…”

“…Gusso and Delben [18] performed a theoretical analysis focusing on the effect of the Casimir force on the pull-in parameters of electrostatic torsional actuators made from silicon. Ramezani and coworkers [19] investigated the two-point boundary value problem (BVP) of the nano-cantilever deflection subjected to Casimir and electrostatic forces, and derived the instability point of the nanobeam by using analytical and numerical methods.…”