1 Introduction

Beam-type electrostatic actuators are common components for the construction of micro-electromechanical systems (MEMSs) and nano-electromechanical systems (NEMSs)^[1]. A can-tilever MEMS/NEMS actuator is composed of a cantilever beam of the length L with a uniform rectangular cross section of the width w and the thickness h, which is suspended over a con-ductive substrate and separated by a dielectric spacer (see Fig. 1).

A voltage difference between the two electrodes causes the upper movable electrode to de-flect towards the fixed ground electrode. At a critical voltage, the movable electrode becomes unstable, and collapses onto the ground electrode. The voltage and deflection of the actuator at this critical state are used to designate the pull-in parameters^[2].

The pull-in behavior and parametric analyses of MEMS/NEMS actuators have been widely studied by many authors^{[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]}. For NEMS actuators, the effects of the intermolecular forces^{[19, 20, 21, 22]} such as the van der Waals force and the Casimir force are always considered.

The distributed parameter model of the deflection of nano-beams constitutes a boundary value problem for a non-linear fourth-order ordinary differential equation. Lin and Zhao^[16] and Guo and Zhao^[23] studied the dynamic behavior of nano-scale electrostatic actuators by consid-ering the effect of the van der Waals force. Guo and Zhao^[24] and Lin and Zhao^[25] investigated the influence of the van derWaals force and the Casimir force on the stability of the electrostatic torsional NEMS actuators. Abadyan et al.^[9], Koochi et al.^{[5, 13]}, Noghrehabadi et al.^[14], and Duan et al.^[26] used the Adomian decomposition method to treat similar models of non-linear boundary value problems. Ma et al.^[17] studied the instability of cantilever NEMS actuators by the homotopy perturbation method. Ramezani et al.^{[6, 15]} and Duan and Rach^[18] investigated the instability of nano-cantilevers with the assumption of a second-degree polynomial as the shape function of the beam during the deflection.

Surface energy effects are very important in modeling a nano-scale structure due to the large surface area/volume ratio of the structure. Surface effects play a crucial role in the pull-in performance of nano-actuators. Gurtin and Murdoch^[27] developed a continuum theory to model both the residual surface stress and the surface elasticity to investigate the surface effects on the elastic behavior of beam-type nano-structures^{[28, 29, 30]}, where the Casimir attraction and fringing field effects were neglected.

In this paper, we consider the distributed parameter model of the NEMS cantilever actuators, incorporating the effects of the surface energy, the Casimir force, and the fringing field, and investigate the pull-in parameters. A new shape function is used for the beam during the deflection. In Section 2, we establish the model as a boundary value problem for a non-linear fourth-order ordinary differential equation. Our main results are presented in Section 3, where a quartic shape function of the beam during the deflection is proposed, satisfying all of the four boundary values. The Gaussian quadrature rule is used to treat the involved integrations, retaining the designed parameters in the evaluated formulas. Section 4 summarizes our conclusions.

2 Distributed parameter model

The governing equation for the distributed parameter model based on the Euler-Bernoulli beam assumptions can be written as follows^{[5, 6]}:

where U is the transverse deflection of the beam as measured from its axis towards the surface of the fixed ground electrode, X is the position along the axis of the beam as measured from the clamped end, (EI)_eff is the effective bending rigidity of the beam incorporating the surface elasticity effect, and F_s, F_elec, and F_cas are the distributed transverse forces resulting from the surface effect, the electrostatic force, and the Casimir force per unit length of the beam, respectively.

Based on the composite beam theory and the assumption that the thickness of the surface layer is much smaller than the beam thickness h, the effective bending rigidity (EI)_eff for a beam with a rectangular cross section can be derived as follows:

where E is the corresponding elastic modulus of the material, I is the area moment of the inertia of the cross section of the beam, i.e., I = wh³/12, and E^s is the surface elastic modulus^{[28, 29, 31, 32]}. The distributed transverse force resulting from the surface effect is^{[28, 29, 32]} where τ₀ is the residual surface stress along the beam longitudinal direction. The first-order fringing field correction of the electrostatic force per unit length of the beam is^{[33, 34]} where ε₀ is the vacuum permittivity expressed by V is the applied voltage, and g is the original gap between the two electrodes when there is no deflection. The Casimir force per unit length of the beam is^{[5, 14, 21]} where ℏ is the reduced Planck’s constant, and v is the speed of light, which are

Substituting Eqs. (2), (3), and (4) into Eq. (1) and introducing the dimensionless variables u = U/g and x = X/L, we can transform the governing equation into the dimensionless form as follows:

where In the above equations, η, α, β, and γ are the surface energy parameter, the intermolecular Casimir force parameter, the electrostatic force parameter, and the fringing field parameter, respectively. They can characterize the magnitudes of the responding forces. These parameters are within their variation ranges^[8]. The surface energy parameter η can be negative. η = 0, α = 0, and β = 0 correspond to the cases of free-surface-energy nano-actuators, MEMS microactuators, and freestanding nano-actuators, respectively.

For the cantilever NEMS, the boundary conditions are

We denote the maximum cantilever tip deflection as u_tip = u(1), and investigate the pull-in parameters u_tip^PI and β^PI. From the third equality in Eq. (6), the pull-in voltage V ^PI can be characterized in terms of the pull-in parameter β^PI as follows:

3 Quartic shape function and pull-in parameters

First, we convert Eqs. (5)-(7) into an equivalent non-linear integral equation^[35] so as to determine all of the integration constants in terms of the boundary conditions. Secondly, we rewrite Eq. (5) in its operator form as follows:

where

Introducing the definite integral operators L₀^-2(·) and L₁^-2(·) as follows:

where the iterated integrals can be rewritten as the respective simple integrals by exchanging the integration order, we have Substituting the boundary conditions in Eq. (7), we have Φ(x) = 0. Hence, applying the operator L₀^-2 L₁^-2(·) on both sides of Eq. (8) yields the equivalent non-linear Volterra integral equation as follows: in which all of the boundary conditions have been incorporated. Setting x = 1 and denoting u(1) = u_tip, we obtain

To analyze the deflection and pull-in parameters of the cantilever beam, some authors pro-posed the following second-degree function to approximate the beam deflection^{[6, 15, 18, 36, 37]}:

which satisfies the boundary conditions but does not satisfy the remaining boundary condition u′′(1) = 0. This means that the curvature at the free end should be zero.

Here, we present a new approximation of the beam by the quartic function

We observe that this new approximation satisfies all of the four boundary values in Eq. (7) by examining the first- and second-order derivatives, respectively, i.e., where 0 < x < 1. The quartic shape function in Eq. (12) is a strictly increasing monotone and convex function over the interval [0,1] just as the quadratic shape function in Eq. (14).

Substituting u(x) = 0 into the right-hand side of Eq. (12), we can obtain the approximation of the beam as follows:

It is in the form of Eq. (15). We use the new quartic shape function (15) to analyze the deflection and pull-in parameters of the beam-type cantilever actuators. Substituting Eq. (15) into Eq. (13) yields By exchanging the integration order in Eq. (16) and calculating the integration with respect to the variable s, we obtain

The integral on the right-hand side of Eq. (17) cannot be analytically calculated by the available computer algebra systems that utilize the Risch algorithm such as MATHEMATICA. We mention that this integral can be exactly and analytically calculated if the second-degree shape function (14) is used. Here, we rely on a numeric integral method. To acquire the superior accuracy of a numeric integral by use of the minor integral nodes, we use the Gaussian quadrature rule. For the relevant details of the Gaussian quadrature rule, see Appendix A^[38].

To perform the integration in Eq. (17) by the Gaussian quadrature rule, we use the trans-formation τ = (z + 1)/2 to obtain the canonical form as follows:

Applying the Gaussian quadrature rule, we have where n is the number of the integral nodes, z_i are the integral nodes, and ω_i are the weights, which can be readily computed for a specified positive integer n (see Appendix A).

From Eq. (19) and substituting the operator N into Eq. (9), we can obtain

where

For example, we take n = 5 and solve for β from Eq. (19). Then, we have

which retains all of the designed parameters.

We take α = 0.4, γ = 0.6, and η = 0 and 2, respectively, and plot the curves of β versus u_tip in Fig. 2.

The pull-in parameters u_tip^PI and β^PI are determined by

i.e., when the slope is zero. We note that, although we have the applied numeric integration with respect to x, all of the designed parameters are preserved, e.g., the designed parameters in Eq. (21). The result is an analytic expression of parameters, and is applicable to the parametric analyses.

Equation (22) determines the pull-in parameter u_tip^PI . We next replace u_tip by u_tip^PI and rewrite Eq. (22) as follows:

From Eq. (20), we have from which we can solve for any of the designed parameters, including the Casimir force param-eter α, the fringing field parameter γ , and the surface energy parameter η. Using the pull-in parameter u_tip^PI, we can calculate the pull-in parameter βPI by virtue of Eq. (20) as follows:

From Eq. (23), we can analyze the effects of α, γ , and η on u_tip^PI. From Eq. (25), we can consider the relation between βPI and α, γ, or η.

To achieve higher accuracy, we take ten integral nodes, i.e., n = 10, to apply the Gaussian quadrature rule on Eq. (19) in the sequel. We remark that increasing the number of the integral nodes does not cause any technical difficulty, but only adds some additional calculation work.

For the specified values of α = 0.2 and γ = 0.65, α = 0.45 and γ = 0.65, and α = 0.7 and γ = 0.65, we plot the curves of u_tip^PI and βPI versus η by use of Eqs. (23) and (25) (see Fig. 3). It is shown that u_tip^PI decreases monotonically and βPI decreases when η increases. Moreover, u_tip^PI and βPI decrease when α increases, which is consistent with the results in Refs. ^[5] and ^[6].

Through Eqs. (23) and (25), we can obtain the numeric values of u_tip^PI and βPI for different η. The obtained results are listed in Tables 1-3. From the tables, we can see that βPI decreases non-linearly as η increases, which means that the curves in Fig. 3(b) are not straight, but are slightly concave.

Next, we consider two typical special cases.

Case 1 Electrostatic microactuators

Neglecting the intermolecular forces, i.e., setting α = 0, leads to the cantilever actuator model in the MEMS. In this case, from Eqs. (20) and (24), we can obtain the equations as follows:

from which we can obtain u_tip^PI and βPI. From Eqs. (26) and (27), we can conveniently consider the effects of γ on u_tip^PI and γ and η on βPI.

Especially, if we neglect the influence of the fringing field and the surface effect, i.e., setting γ = 0 and η = 0, we can obtain the pull-in parameters as follows:

Case 2 Freestanding nano-actuators

When the nano-actuator becomes freestanding, the voltage difference V between the can-tilever beam and the substrate shown in Fig. 1 vanishes. Therefore, the electrostatic force parameter satisfies β = 0. In this case, from Eq. (20), we have

In Fig. 4, we plot the curve of α versus u_tip for η = 0. The critical values of α and u_tip are determined by

From Eq. (29), we can calculate the critical value u_tip^*. Substituting u_tip^* into Eq. (28), we can obtain the critical value α^* for η = 0 as follows:

No solution exists when α is greater than the critical value α^*, since the cantilever beam collapses onto the substrate. Therefore, the critical value α^* is considered to be crucial in the cantilever design.

From the second formula in Eq. (6), we can write α ∝ L⁴/g⁵, where L is the length of the cantilever beam, and g is the gap between the two electrodes. The maximum length of the cantilever beam, which does not collapse onto the substrate due to the intermolecular forces, is called the detachment length^[16]. It is a basic design parameter for NEMSs. If the length of the beam is fixed, one can calculate the minimum gap gmin between the beam and the substrate to ensure that the beam does not adhere to the substrate due to the intermolecular forces. The detachment length and the minimum gap of the cantilever beam can be obtained from the critical value α^* and the second formula in Eq. (6).

Finally, we indicate that the calculation is readily amenable to the increase in the number of the integral nodes in the Gaussian quadrature rule, although the present results for 10 integral nodes are sufficiently accurate. We have checked that the results for 20 integral nodes are almost identical with the present results for 10 integral nodes. When α = 0.7, γ = 0.65, and η = 3, the difference between the obtained u_tip^PI for 10 nodes and 20 nodes is less than 1.117 77× 10^-12.

4 Conclusions

We have considered the distributed parameter model of the NEMS cantilever actuators incorporating the effects of the surface energy, the Casimir force, and the fringing field. A quartic polynomial is proposed as the shape function for the cantilever beam, and the analytic expressions for the tip deflection and pull-in parameters of the beam are derived by using the Gaussian quadrature rule. The new quartic shape function satisfies all of the four boundary values, but the quadratic shape function used in the previous investigations satisfies only three of the four boundary values. Although we have applied a numeric integral method, we derive the analytic expressions preserving the model parameters for the tip deflection and pull-in parameters of the beam. The MEMS cantilever actuators and the freestanding nano-actuators are included as two special cases in our parametric analyses. The results show that our analyses for the influence of the surface energy, the Casimir force, and the fringing field on the pull-in instability are convenient.