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.