Accepted: February 08, 2023 | Published Online: February 10, 2023

Citation: Bianchi G, Radi E (2023) Lower and Upper Bound for the Pull-in Parameters of a Micro- or Nanocantilever Beam Immersed in Liquid Electrolytes. Int J Nanoparticles Nanotech 8:040.

Copyright: © 2023 Bianchi G, et al. This This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Abstract

An analytical method is proposed to accurately estimate the pull-in parameters of a micro- or nanocantilever beam immersed in liquid electrolytes with a flexible support at one end. The system is actuated by electrochemical force, namely the sum of electric and osmotic forces, and is subject to Casimir or van der Waals forces according to the spacing between the two electrodes. The deflection of the beam is described by a fourth-order nonlinear boundary value problem that can be formulated by an equivalent nonlinear integral equation. At first, a priori upper and lower analytical estimates on the beam deflection are derived and then very accurate lower and upper bounds for the pull-in voltage and tip deflection are obtained. The analytical predictions are in excellent agreement with the numerical results provided by the shooting method. Finally, a simple closed-form relation is proposed for the pull-in voltage under the effect of bulk ion concentration.

Keywords

MEMS, NEMS, Pull-in instability, Liquid electrolytes

Introduction

Micro- and nanoelectromechanical systems, MEMS and NEMS, have the potential to innovate many application fields: From medical and automobile to aerospace and information areas [1]. To this aim innovative NEMS devices include high precision sensors, nanoactuators and nanotweezers for miniaturized robotics and memory devices [2-5]. In addition to industrial and consumer uses, BioMEMS and BioNEMS are increasingly applied for chemical and biochemical analyses in medical diagnostics as DNA, cells, blood pressure and toxin identifications, tissue engineering, implantable pharmaceutical drug delivery, and minimal invasive surgery [6-8]. In these applications the components are immersed in liquid electrolytes which are typically 0.2 M ionic solutions, mainly NaCl or KCl [9]. Moreover, ionic liquid NEMS actuators find application in many other devices, such as fuel cells, batteries, supercapacitors, filters, electro-osmotic pumps, storage of hydrogen and electroactive polymer actuators [10].

These devices typically enclose a large number of nanoswitches, which are formed by a mobile electrode and a fixed one. The mobile part is modelled by a cantilever or clamped-clamped beam suspended above a conductive substrate and actuated by a voltage difference [11,12]. The application of electrostatic force then switches the mobile electrode between two stable positions. Indeed, as the voltage is increased the mobile electrode deflects towards the fixed electrode until the former suddenly collapses on the latter under a critical voltage, defined as pull-in voltage, thus closing the electric circuit. Investigation of this kind of instability, called pull-in instability, through analytical and accurate prediction of pull-in parameters is fundamental for MEMS and NEMS design in order to determine the operation voltage and power dissipation correctly [1,13,14]. Moreover, the pull-in voltage strongly limits the behaviour of the system. Indeed, to reduce the power consumption and the amount of energy stored in the system the design should aim at obtaining devices actuated by small pull-in voltage [15]. Conversely, increasing the pull-in voltage yields more stable devices. For these reasons, the best design represents also the best compromise between power consumption and performance of the system. In particular, a decrease in the spacing between the electrodes at the nanoscale implies a drastic reduction in the actuation voltage, since the intermolecular forces, van der Waals (vdW) or Casimir, significantly influence the pull-in instability at this scale [13,16].

The nature of the intermolecular attractions depends on the separation distance between the two electrodes. If the gap is smaller than the retardation length, namely below 20 nm for metals, then the dominant attraction is the vdW force. If the gap is much larger instead, typically above 20 nm, then the Casimir force becomes dominant [17,18]. When the switch is immersed in a liquid electrolyte, the osmotic (chemical) force appears because of ion concentration difference and this new term strongly affects the pull-in voltage. A model for investigating the pull-in instability under electrochemical force, namely the sum of electric and osmotic forces, has been introduced by Boyd, et al. [19,20], Noghrehabadi, et al. [9,21] and Ghalambaz, et al. [22]. These authors observed that the combined effect of electrochemical force and intermolecular interactions makes the problem governing the pull-in instability strongly nonlinear and, to author's knowledge, only numerical and approximate methods have been used in the Literature for solving this boundary value problem (BVP). In particular, 1D lumped models [19], modified Adomian decomposition method [9,21,22], Rayleigh-Ritz and numerical methods [23] and finite element analysis [20] have been employed for investigating the pull-in instability of NEMS devices immersed in ionic solutions. Moreover, the stability and pull-in voltage of electrostatic parallel plates in liquid solutions were experimentally investigated in [24]. The present work aims at providing an accurate analytical method, useful to estimate the pull-in parameters of electrochemically actuated nanobeam, and also to validate the various numerical approaches so far used for solving the BVP.

As reported by Radi, et al. [25], the current limitations of manufacturing techniques for micro- nanoelements make inaccurate the hypothesis of perfect clamped support at one end of the real micro- nanoswitch. This involves that the stiffness of the support plays a significant role in the behaviour of the device and, in many situations, modelling a flexible boundary condition is a prerequisite for obtaining a reliable pull-in response of the system. To this scope, some electro-mechanical experiments tested the influence of flexible supports [26]. As a result, they confirm that the effective rotational stiffness of the support should be included in conditions of the problem to obtain an accurate pull-in model for devices at micro- nanoscale.

In this work, the analytical approach proposed for the assessment of accurate lower and upper bounds for the pull-in parameters of a micro- or nanocantilever electrostatically actuated by Radi, et al. [25,27-29] is extended by taking into consideration a device immersed in liquid electrolytes with the contribution of the osmotic (chemical) force. The effects of intermolecular forces and support flexibility are also considered. The non-linear model governing the beam deflection is introduced in Section 2. An integral equation equivalent to the original BVP is derived by exploiting the Green's function of micro- nanocantilever beam in ionic solution. First, a priori bounds on the beam deflection are derived in Section 3 by exploiting positivity, monotonic behaviour and convexity of the beam deflection. These bounds are then used in Section 4 to obtain accurate lower and upper estimates of pull-in parameters, thus avoiding the difficult task of solving the strongly nonlinear BVP. Results are discussed in Section 5 where the proposed analytical technique is validated by comparing the computed estimates and the numerical results provided by the shooting method. Moreover, in Section 6, an approximated formula is provided for a fast estimation of pull-in voltage under the effect of osmotic force, thus simplifying the design of micro- nano-actuators.

Mathematical Model

Figure 1 shows a schematic view of an elastic cantilever beam of length l, where the axial position is 0 ≤ z ≤ l. The flexible support at the end z = 0 is modelled by a rotational spring of stiffness K. The deflection v(z) of the beam subjected to electrochemical actuation and intermolecular surface forces can be described by a fourth-order, non-linear ODE written in terms of the nondimensional variables u = v/d, x = z/l, where d is the initial gap between the electrodes, and φ = z_ceΨ/(k_B T), corresponds to the value of the non-dimensional applied voltage, as described in [9,20]. Moreover, z_c corresponds to the absolute value of the valence, e = 1.602•10^-19 C is the electronic charge, Ψ1 and Ψ2 are the applied electric potential to the two electrodes, k_B = 1.38054•10^-23 J K^-1 is the Boltzmann constant and T is the absolute temperature. By following the approach presented in [9,19,20], the governing nonlinear differential equation of the nanobeam in liquid electrolytes is

$u^{\text{IV}}\left(x\right)\text{ = }\frac{\beta }{{\sinh }^2\left({\xi }_0\left[1-u\left(x\right)\right]\right)}\left[\frac{1}{2}\left(1+{\left(\frac{{\varphi }_2}{{\varphi }_1}\right)}^2\right)-\frac{{\varphi }_2}{{\varphi }_1}\cosh \left({\xi }_0\left(1-u\left(x\right)\right)\right)\right]+\frac{{\alpha }_W}{{\left[1-u\left(x\right)\right]}^3}+\frac{{\alpha }_C}{{\left[1-u\left(x\right)\right]}^4},$ (1)

for x ε [0, 1], completed by the boundary conditions

$$u\left(0\right)\text{ = 0 }{u}^{\text{'}}\left(0\right)\text{ = }{u}^{\text{'}\text{'}}\left(0\right)/k,u^{\text{'}\text{'}}\left(1\right)\text{ = }{u}^{\text{'}\text{'}\text{'}}\left(1\right)\text{ = 0}\text{. (2)}$$

The nondimensional positive parameters β, α_W and α_C, appearing in Eq. (1), are proportional to the electrochemical, van der Waals and Casimir forces, respectively. Moreover, φ₂/φ₁ is the non-dimensional voltage ratio and ξ₀ represents the non-dimensional ionic concentration.

$$\beta \text{ = }w\frac{2c_b{k}_{B}T{\varphi }_1^2{l}^4}{dEI},{\alpha }_W\text{ = }\frac{Aw{l}^4}{6\pi d^{4}EI},{\alpha }_C\text{ = }\frac{{\pi }^{2}hcw{l}^4}{240d^{5}EI},\frac{{\varphi }_2}{{\varphi }_1},{\xi }_0\text{ = }kd.\text{ (3)}$$

In Eq. (3) w is the cross-section width, c_b represents the bulk ion concentration, h = 1.055.10^-34 Js is the Planck's constant divided by 2π, c = 2.998.108 m/s is the speed of light, A is the Hamaker constant (typical value A = 0.4 ÷ 4.10-20 J), E is the Young's modulus of the beam material, I is the moment of inertia of the beam cross-section, κ² = 2 c_b (z_c e)²/εε₀ k_B T and 1/κ is the Debye length, finally ε₀ = 8.854.10^-12 C²N^-1m^-2 is the permittivity of vacuum and ε is the relative permittivity of the dielectric medium between the two electrodes. Moreover, the constant

$$k\text{ = }\frac{Kl}{EI},\text{ (4)}$$

denotes the nondimensional ratio between the rotational spring stiffness and beam bending stiffness. Note that the built-in end condition at x = 0 is recovered as k approaches infinity.

Following the analytical approach introduced by Radi, et al. [25,27-29] the BVP is equivalent to a nonlinear integral equation obtained exploiting the Green's function G(t) of the differential problem (1) and (2). In particular, imposing the boundary conditions of cantilever beam and the continuity conditions, the coefficients of the general solution to this problem can be univocally determined.

$$u\left(x\right)\text{ = }{\displaystyle \underset{0}{\overset{x}{\int }}\left[\frac{t^3\left(3x-t\right)}{6}+\frac{xt}{k}\right]}{f}_{EC}\left(u\right)dt+{\displaystyle \underset{x}{\overset{1}{\int }}\left[\frac{x^2\left(3t-x\right)}{6}+\frac{xt}{k}\right]}{f}_N\left(u\right)dt.\text{ (5)}$$

Where the non-dimensional electrochemical and intermolecular surface forces are

$$f_{EC}\left(u\right)\text{ = }\frac{\beta }{{\sinh }^2\left({\xi }_0\left(1-u\right)\right)}\left[\frac{1}{2}\left(1+{\left(\frac{{\varphi }_2}{{\varphi }_1}\right)}^2\right)-\frac{{\varphi }_2}{{\varphi }_1}\cosh \left({\xi }_0\left(1-u\right)\right)\right]\text{ (6)}$$

$f_N\left(u\right)\text{ = }\frac{{\alpha }_N}{{\left[1-u\right]}^m}$ with N = W for n = 3 or N = C for n = 4 (7)

Where, W and C mean van der Waals and Casimir forces, respectively. Note that the function fN is positive for 0 ≤ u ≤ 1, namely the effect of intermolecular forces is attractive between the two electrodes. Otherwise, function fEC is not defined in sign, namely the electrochemical force can produce an attractive or repulsive action between the electrodes according to the applied voltage and ionic concentration of the liquid solution. Clearly, the pull-in instability can appear only if the total force which act on the system is positive, namely attractive

$$f\left(u\right)\text{ = }{f}_{EC}\left(u\right)+f_N\left(u\right)\ge 0,\text{ for 0}\le \text{u}\le \text{1 (8)}$$

Inequality (8) allows to obtain useful a priori estimates on the solution of the BVP (1) and (2), following the procedure described in Section 3 of [25].

A Priori Estimates

Upper and lower bounds for the solution u(x) to the problem (1) and (2)

$$u\left(x\right)\le u\left(1\right)b\left(x\right),\text{ for }x\in \left[0,1\right],\text{ (9)}$$

$$u\left(x\right)\ge u\left(1\right)a_1\left(x\right)+{\beta }^{*}{a}_2\left(x\right),\text{ for }x\in \left[0,1\right],\text{ (10)}$$

are the same provided in [25] where the extended expression for b(x), a1(x) and a2(x) are reported for the elastically constrained cantilever beam. Otherwise, β* is different from value in [25] because the load functions of the two model are different. The value of β* for a device immersed in a liquid electrolyte is

$${\beta }^{\ast }\text{ = }\beta \left(\frac{1+{\left(\frac{{\varphi }_2}{{\varphi }_1}\right)}^2-2\frac{{\varphi }_2}{{\varphi }_1}\cosh \left({\xi }_0\right)}{2{\sinh }^2\left({\xi }_0\right)}\right)+{\alpha }_W+{\alpha }_C.\text{ (11)}$$

Bounds on the Pull-In Parameters

From Eq. (5) we obtain r = u(1) that denotes the normalized deflection of the cantilever tip. Then, by using r and the estimates (9) and (10) on the solution of the BVP (1)-(2), lower and upper bounds can be derived for the pull-in parameters βPI and rPI, as described in [25].

$$r\le \frac{1}{6}\left[\beta L\left(r\right)+F\left(r\right)\right],\text{ (12)}$$

$$r\ge \frac{1}{6}\left[\beta K\left(r,\beta \right)+H\left(r,\beta \right)\right],\text{ (13)}$$

Results

At first, we investigate the effects of the total force affecting the micro- nanocantilever when it is immersed in a liquid electrolyte. As remarked in Sec. 2, the electrochemical force is undefined in sign, or in a more practical way it can be attractive or repulsive depending of the non-dimensional ionic concentration ξ₀ and non-dimensional voltage ratio φ₂/φ₁. When the electrochemical force is attractive, namely fEC > 0, the total force f = fEC + fN is always attractive, so the pull-in instability can appear for the cantilever actuator bended toward the substrate. Otherwise, the total force of the system can close or separate the electrodes under an attractive or a repulsive contribution. In particular, the pull-in instability is possible only for a positive, namely attractive, total force, |fN| > | fEC|, so for a positive deflection of the mobile electrode toward the substrate. On the contrary, the pull-in instability cannot appear for a negative, namely repulsive total force, |fN| < |fEC|, which bends the cantilever beam away from the substrate. Then, useful qualitative graphs, which represent regions where the total force is positive under fixed sets of nondimensional parameters, can be obtained exploiting inequality (8). The region, where the total force is attractive, is shown in Figure 2a and Figure 2b for different values of ξ₀ and φ₂/φ₁. The coloured area provides positive attractive total force. We can observe that the region increases when the surface forces increase. In particular, for high value of surface forces, namely αN > 0.5, in Figure 2a there's no more combinations of the parameters ξ₀ and φ₂/φ₁ that produce a repulsive total force. However, in Figure 2b where higher electrochemical force β is considered, a white uncoloured region is present, so a repulsive force can appear. To reduce the white region a greater surface force, αN > 0.5, is necessary.

The relationships between the non-dimensional voltage ratio φ₂/φ₁ and tip deflection of the nanocantilever r = u(1), obtained from the shooting method (function NDSolve in Mathematica® [30]), are presented in Figure 3a and Figure 3b for constant values of electrochemical force β, vdW force α_W and ionic concentration ξ₀, and different values of the non-dimensional support stiffness k. In Figure 3a we can see that the cantilever tip deflection reaches zero for φ₂/φ₁ = 0.518 and φ₂/φ₁ = 7.006. Indeed, the total force at these voltage ratios is zero since the vdW and the electrochemical forces have the same intensity but opposite direction, and thus no deflection of the beam is observed at these values. Moreover, for the interval 0.518 < φ₂/φ₁ < 7.006 the total force is repulsive causing the separation of the mobile electrode from the substrate, namely a negative displacement. Considering different values for β and ξ₀, in Figure 3b, we obtain similar trends but diverse critical values of voltage ratio, namely φ₂/φ₁ = 0.633 and φ₂/φ₁ = 2.453. In particular, the total force is repulsive force for 0.633 < φ₂/φ₁ < 2.453 that are narrow limits respect to those shown in Figure 3a for upper values of β and ξ₀. Moreover, we can observe that the maximum normalized tip deflection u(1) in module depends on the support stiffness k and it can be negative under a repulsive force, see Figure 3a, or positive under an attractive force, see Figure 3b. Values of maximum tip deflection can be useful design parameter in various MEMS and NEMS applications.

The relationships between the electrochemical parameter β and tip deflection of the beam r are presented in Figure 4 for different values of ionic concentration ξ₀. The pull-in critical estimates predicted by using the proposed theoretical method are indicated by small (black) circles for lower bounds and by small black points for upper ones. It can be observed that the analytical lower and upper estimates of β and r are very close each other and perfectly bound the numerical pull-in parameters βPI and rPI, which correspond to the maximum of the curves reported in figures. In particular, observing Figure 4 we note that increasing the ionic concentration ξ₀ of the liquid solution, the critical electrochemical pull-in values β significantly increase, whereas the corresponding values of tip deflection decrease. For different values of non-dimensional voltage ratio φ₂/φ₁, we observe very different values of the electrochemical force β and similar tip deflections, typically included in range 0.2 < u(1) < 0.5. Furthermore, for low values of φ₂/φ₁ a very high electrochemical force is required to cause the pull-in instability.

In Table 1, upper and lower pull-in parameters are reported for the specific set of parameters. Increasing the magnitude of the intermolecular surface forces, namely for higher values of coefficients α_W and α_C, the pull-in electrochemical values β always significantly decrease, whereas the corresponding tip displacements show different trends depending on the ionic concentration and voltage ratio. In particular, for high ionic concentration and very small φ₂/φ₁ the pull-in deflection significantly increases with intermolecular forces, contrary to corresponding β reduction. Otherwise, for ξ₀ = 0.5 and φ₂/φ₁ = 0.5 the pull-in deflection slightly decreases with intermolecular forces.

In Table 2 critical values of βu, βl, ru and rl are shown for various values of the nondimensional support stiffness k and ionic concentration ξ₀. These estimates reveal that increasing values of the stiffness parameter k, raise the pull-in electrochemical force without producing significant effect on the pull-in tip deflection.

Approximated Analytical Relations for the Pull-in Parameters

In this section, the lower and upper estimates computed by using the theoretical approach described above, are exploited to obtain approximated analytical relations for pull-in electrochemical force β and corresponding tip deflection r for nanocantilever beam immersed in liquid electrolytes. In particular, the variations of the parameters β and r, with the ionic concentration ξ₀, fixed φ₂/φ₁ = 0.1 and negligible intermolecular forces, are described by the following approximated relations, which hold for 0 < ξ₀ ≤ 2

$$\beta \text{ = }\frac{-138.056+995.632{\xi }_0+3615.06{\xi }_0^2}{1419.31-563.898{\xi }_0+0.426288{\xi }_0^2},\text{ (14)}$$

$$r\text{ = }\frac{36.3559+31.9245{\xi }_0-17.7879{\xi }_0^2}{82.0033+66.736{\xi }_0-19.4946{\xi }_0^2}.\text{ (15)}$$

Numerical coefficients in relations (14) and (15) are set by interpolating the lower and upper bounds provided by the analytical method by using the function Nonlinear Model Fit available in Mathematica®. The analytical estimates have been computed varying the ionic concentration ξ₀ from 0.2 to 1 with a step of 0.2 and from 1 to 2 with a step of 0.1, for fixed φ₂/φ₁ = 0.1 and negligible intermolecular forces for a rigidly clamped cantilever. In Figure 5a and Figure 5b, the solid curves represent the approximated relations (14) and (15), whereas lower and upper bounds, βl and βu in Figure 5a, rl and ru in Figure 5b, are indicated by small circles and point. We observe that the approximated curves are perfectly bounded by analytical predictions, thus ensuring the accuracy of the approximated relations (14) and (15).

Conclusion

In this work, the present analytical method is extended to investigate the pull-in instability for a micro- nanocantilever actuator immersed in an electrolytic solution under the influence of the ionic concentration and the non-dimensional voltage ratio. The analytical method provides lower and upper bounds for the pull-in parameters of the initial BVP, combining the effects of vdW and Casimir forces with the influence of the electrochemical force. The accuracy is proved comparing analytical predictions with the numerical results provided by the shooting method. At first, we have observed that while the intermolecular force is always attractive, the electrochemical force can be attractive or repulsive depending on the magnitude of the electric and osmotic forces. Indeed, even if the electric force is always attractive and osmotic force always repulsive, their sum, namely the electrochemical force, is not defined at priori. Then, the pull-in instability may appear only when the total force, namely the sum of intermolecular and electrochemical forces, is attractive, bending the mobile electrode toward the substrate. On the contrary, a repulsive total force pulls apart the two electrodes and no pull-in instability is possible. Furthermore, it is noted that the combined effects of the flexible boundary conditions and intermolecular surface forces significantly reduce the pull-in voltage, whereas a higher ionic concentration increase the pull-in voltage delaying the pull-in instability. Therefore, a model complete with all these effects are required to avoid overestimation of the pull-in instability which may cause dangerous failure of the device. In this sense, some approximate relations are also provided to help the design of MEMS and NEMS actuators by simplify the prediction of pull-in parameters in order to guarantee the proper operation of these devices in various applications, avoiding unexpected faults.

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