Modeling the electromechanical behavior and instability threshold of NEMS bridge in electrolyte considering the size dependency and dispersion forces

Author’s Accepted Manuscript Modeling the electromechanical behavior and instability threshold of NEMS bridge in electrolyte considering the size dependency and dispersion forces I. Karimipour, A. Kanani, A. Koochi, M. Keivani, M. Abadyan www.elsevier.com/locate/physe

PII: DOI: Reference:

S1386-9477(15)30040-0 http://dx.doi.org/10.1016/j.physe.2015.05.005 PHYSE11954

To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 11 January 2015 Revised date: 1 May 2015 Accepted date: 6 May 2015 Cite this article as: I. Karimipour, A. Kanani, A. Koochi, M. Keivani and M. Abadyan, Modeling the electromechanical behavior and instability threshold of NEMS bridge in electrolyte considering the size dependency and dispersion f o r c e s , Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2015.05.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modeling the electromechanical behavior and instability threshold of NEMS bridge in electrolyte considering the size dependency and dispersion forces I. Karimipour1, A. Kanani2, A. Koochi3, M. Keivani4 and M. Abadyan3* 1

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Faculty of engineering, Shahrekord University, Shahrekord, Iran Ionizing and Non-Ionizing Radiation Protection Research Center, Paramedical Sciences School, Shiraz University of Medical Sciences, Shiraz, Iran 3 Shahrekord Branch, Islamic Azad University, Shahrekord, Iran 4 Shahrekord University of Medical Sciences, Shahrekord, Iaran

Abstract While few studies have been conducted on modeling the pull-in instability of cantilever nanoelectromechanical systems (NEMS) in electrolyte media, no researchers has investigated this phenomenon in double-clamped NEMS. Herein, the pull-in instability of the NEMS bridge immersed in ionic liquid electrolyte media is explored for the first time considering the size effect and dispersion forces. The strain gradient elasticity in conjunction with vonKarman strain is employed to incorporate the effect of size-dependency and beam stretching in the structural model. The presence of electrochemical force field and nano-scale attractions i.e. Casimir and van der Waals forces is incorporated in the model considering the presence of the liquid media. To solve the nonlinear constitutive equation of the system two different methods including the differential transformation method (DTM) and numerical solution are employed. The proposed model is validated by comparing with the results presented in literature. Impacts of various parameters i.e. the size dependency, dispersion forces, electrolyte ion concentration and potential ratio on the instability characteristics of the NEMS bridge are discussed. The results of present theory are compared with those predicted by the classic continuum theory as well as the modified couple stress theory. Keywords NEMS bridge, Instability threshold, Electrolyte media, Strain gradient theory, Casimir force, vdW force.

1. Introduction Due to their excellent performance and low power consumption, micro/nanoelectromechanical systems (MEMS/NEMS) are increasingly used in different engineering and science branches including optics, biology, mechanics, electronics, chemistry, etc. One of *

Corresponding author: Tel/fax: +98-383-4424401; Email: [email protected]

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the most common MEMS/NEMS elements is double-clamped bridge which is frequently utilized to develop miniature devices such as actuator, accelerometer, resonator, switch, etc. [1]. A typical MEMS/NEMS bridge is constructed from a conductive double-clamped conductive beam-plate component (movable electrode) suspended over another fixed conductive component (ground electrode). The movable electrode can be excited via applying electrical force field. The electrical stimulation can result in deflection, actuation or vibration of the movable beam component, depending on the loading system and designed configuration. Generally, when the applied voltage exceeds its critical value, the pull-in instability occurs and the NEMS bridge suddenly fails. Prediction and simulation of the pullin instability of MEMS/NEMS bridge are very crucial for reliable design and fabrication of nano-devices, hence, previous researchers have studied the electromechanical response and pull-in behavior of ultra-small NEMS/MESM bridges [2-5]. With recent developments in biological, chemical and electronic sciences, NEMS/MEMS have found many applications in liquid media. Some of the promising applications of MEMS/NEMS in bio-fluid are developing sensors and manipulators for cellular handling, bio-component characterization, device motion, DNA manipulation, bio-mimetic cilia, drug delivery, etc. [6-9]. Besides biology, nano-devices such as actuators, probes, tweezers, valves, etc. are employed as precise instruments operated in ionic liquid media [10]. Moreover, usage of in liquid-immersed MEMS/NEMS have great potentials in developing supercapacitors, fuel cells, batteries, filters, micro-densitometer, micro/nano-pump, active microfluidic devices and atomic force microscopy [10-12]. In this regard some researchers have investigated the mechanical behavior of MEMS/NEMS in liquid environment. Oh et al. [8] have fabricated and characterized the oscillating bio-mimetic microfluidic device that mimics biological cilia for manipulation of microfluidics. Maali et al. [14] attempt to measure the influence of the fluid motion on the oscillating behavior of ultra-small cantilever beams immersed in viscous fluids. The pull-in performance of electrostatic parallel-plate actuators in liquid solutions was studied by Rollier et al. [6]. They have claimed that the pullin deflection of the actuators can be suppressed in liquid. Only few works have focused on the pull-in behavior of liquid-immersed NEMS in ionic environment. In ionic electrolyte media, the electrochemical field is characterized by double-layers interaction [10, 13, 15]. The double-layer appears on the surface of an electrode if the electrode is exposed to a ionic electrolyte fluid. Indeed, the double-layer concept implies presence of two parallel layers of electrical charges surrounding the electrode surface. The first layer is composed of ions adsorbed onto the electrode surface due to chemical interactions. The second layer comprises ions attracted to the first layer via the Coulomb attraction. A simple lumped model for calculating the pull-in voltage of electrostatic actuators in ionic liquid electrolytes has been presented by Boyd and Kim [10]. They incorporated the effect of double-layers electrochemical force in the pull-in model by solving the linearized Poisson–Boltzmann equation. In another work Boyd and Lee [15], have modified their model with a distributed parameter model in order to achieve more accurate results. The electromechanical behavior and frequency response of an inter-digitated silicon comb-drive actuator in various ionic liquids was investigated by Sounart et al. [13]. They present a theoretical model that predicts the characteristic actuation frequency of the system. Noghrehabadi et al. [16] theoretically investigated the static stability of nano-beams in a liquid electrolyte using a distributed force model. It should be mentioned that to the best knowledge of the authors, all the mentioned works have studied the cantilever NEMS, while no researcher has yet investigated the

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electromechanical performance of double-clamped NEMS bridges. In this regard, present work is aimed to investigate the electromechanical response and pull-in instability of the NEMS bridge immersed in ionic liquid electrolyte media. It is well-established that the electromagnetic vacuum fluctuations, i.e. dispersion forces, can significantly affect the electromechanical performance of nano-structures. Hence, incorporating the effect of dispersion forces is crucial to accurate simulation of nanostructures in liquid media. The dispersion forces between interacting bodies are generally explained as Casimir or van der Waals (vdW) attractions depending on the distance between the bodies. If the separation between the interacting bodies is sufficiently large (typically of the order of hundreds of nanometers), the retardation effect becomes significant and the fluctuation force is modeled as Casimir force. On the other hand, if the separation between the interacting bodies is typically less than few nanometers, the retardation effect is negligible and the interaction is modeled as the vdW attraction regime. The effect of dispersion forces on the pull-in instability of NEMS operating in non-liquid environments (i.e. gas or vacuum) has been studied by previous researchers [17-26]. Some pioneering papers on the pull-in instability of NEMS in non-liquid media under the presence of vdW and Casimir forces can be found in refs. [27-29]. Herein, we examine the impact of dispersion forces on the pull-in instability of liquid-immersed NEMS bridge considering the corrections due to the presence of liquid media in between the electrodes. It is well established that while molecular dynamics/mechanics is powerful for study of nanoscale systems, it is very time-consuming in modeling MEMS/NEMS structures with large number of interacting atoms. To overcome this shortcoming, nano-scale continuum mechanic models are applied to investigate the electromechanical performance of MEMS/NEMS, [15]. Since the elastic characteristics of materials in nano-scale are often size-dependent [30, 31], the applied models should be able to consider this size-dependency in constitutive equations. The size-dependency cannot be modeled by using classical continuum mechanics. However, by employing nano-scale continuum theories, the size dependent behavior of nano-structures is attributed to measurable size parameters. In this regards, the non-classical theories such as strain gradient theory [32] have been developed to consider the size phenomenon in theoretical continuum models. The strain gradient theory introduces additional elastic constants, i.e., three material length scale parameters, to interpret the size-dependent behavior of elastic solids. While some researchers have utilized strain gradient theory for analyzing MEMS/NEMS in non-liquid environments [33-41], none of them has investigated this phenomenon in liquid electrolyte media. In this work, the size-dependent pull-in instability of NEMS bridge immersed in liquid electrolytes is investigated in the presence of vdW and Casimir forces. The strain gradient theory in conjunction with Euler–Bernoulli beam model and von-Karman strain is used to derive the non-linear equilibrium equation of system considering the beam stretching. The differential transformation method (DTM) has been applied to solve the governing equation of the system as well as numerical solution. 2. Theoretical model Figure (1) shows the schematic representation of the clamped NEMS bridge in ionic liquid electrolyte which is constructed from a conductive clamped electrode suspended over another fix one (grounded electrode). The moveable electrode with a length of L, wide of b and

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thickness of h is considered. The total energy of the system is the summation of the stored strain energy and work of external forces.

Figure 1. Schematic representation of clamped NEMS. 2.1. Stored strain energy The strain gradient theory introduces three length scale parameters that can model the sizedependent behavior of the ultra-small systems. By employing the strain gradient theory modified by Lam et al [32], U , the stored strain energy density in elastic materials is written as the following: 1 (1) U     p    (1) (1)  ms  s 2



ij ij

i i

ijk ijk

ij ij







in which 1 ui, j  u j ,i 2  i   mm,i

 ij 

(1) ijk 

1 3 

(2) (3)

 jk ,i   ki, j   ij,k   151 ij  mm,k  2 mk ,m 

 jk   mm,i  2 mi ,m    ki   mm, j  2 mj ,m   15  1

1 2

ijs  e jkl ul ,ki

(4)

(5)

(1) In above equations, ui ,  i , ijk , ijs ,  ij and eijk indicate components of displacement vector, dilatation gradient vector, deviatoric stretch gradient tensor, symmetric rotation gradient (1) tensor, Kronecker delta and permutation symbol, respectively. Also  ij , pi ,  ijk , mijs , are

components of Cauchy’s stress and high order stress tensors, respectively that are identified as [32]:  

 ij  2   ij 

 1  2

 

 mmij 

(6)

pi  2l02 i

(7)

(1) (1)  ijk  2l12ijk

(8)

4

mijs  2l22 ijs

(9)

In the above equations,  and  are Poisson’s ratio and shear modulus, respectively. Also l0 , l1 and l2 are additional material length scale parameters which are related to dilatation gradient vector, deviatoric stretch gradient tensor and symmetric rotation gradient tensor. The material length scale parameters can be measured using experimental techniques, molecular dynamics, etc. Some experimental measurements evaluate the material length scale parameter of single crystal and polycrystalline copper to be 12 and 5.84 μm, respectively [42, 43]. Also, the size-dependent behavior has been detected in some kinds of polymers [44]. For hardness measurement of bulk gold, it is found that the plastic length scale parameter (for indentation test and hardness behavior) of Au increases from 470 nm to 1.05 μm with increasing the Au film thickness from 500 nm to 2 μm [45]. Based on test results gathered via microhardness test, the plastic length scale parameter for metals such as Cu, Ag and Brass were determined in the range about 0.2-20 μm based on the crystallity [46]. Using microbend testing method, the plastic intrinsic material length scale of 4μm for copper and 5μm for nickel were determined [47]. All these experiments imply that when the characteristic size (thickness, diameter, etc.) of a micro/nano element is in the order of its intrinsic the material length scales (typically sub-micron), the material elastic constants highly depend on the element dimensions. Molecular dynamic simulations also could be used to compute the material length scale parameters of materials [48]. Now, based on Euler-Bernoulli beam theory, the displacement field can be written as the following: u1  u  Z

W , X

u2  0,

u3  W  X 

(10)

where u1, u2, u3 and W are the displacement field of the beam in the X, Y, Z directions and centerline beam displacement, respectively. Considering von-Karman strain and substituting the displacement field (10) in (2)-(5), the nonzero component of strain is obtained as: 2 2 (11) u 1  W  u  2W 1  W  11 

1

X

W  2W  2W , 3   X 2 X 3 X X 2 X 2 2   2u 3W W  2W  4  2W (1) (1) (1) (1)  , 113 111   Z   311  131  5  X 2 15 X 2 X 3 X X 2 

1 

 2u

Z       , 2  X  X X 2 2  X 

Z

3W



1

(1) (1) (1) (1) (1) (1) 122  133  212  221  313  331   5

 2u

2  X

1  2W

(1) (1) (1) 223  232  322  , 15 X 2 s s 12   21 

(1) 333 

Z

3W X 3



W  2W   X X 2 

1  2W 5 X 2

1  2W 2 X 2

Substituting (11) in (6)-(9), the nonzero component of stress tensor is determined as:

5

 u  2W 1  W Z    X X 2 2  X 

11  E 

  

2

(12)

  

  2u 3W W  2W  , p1  2 l02  Z   X 2 X 3 X X 2     2u

4 5

(1) 111  l12  

 X

2

Z

3W X

3



p3  2l02



2

 2u

2  X

 2W

(1) (1) (1)  223   232   322  l12 , 15 X 2 s s m12  m21   l22

X 2

W  2W  8  2W (1) (1) (1)  , 113   311  131   l12 2  X X  15 X 2

(1) (1) (1) (1) (1) 2 122  133   2(1) 12   221   313   331  5 l1  

2

 2W

2 5

(1)  333  l12

Z

3W X 3



W  2W   X X 2 

 2W X 2

2

 W X 2

Now, by substituting (11) and (12) in (1) the energy density is obtained. By integrating the energy density over the beam volume the bending strain energy, Ubend, is obtained as: (13) Ubend  UdV 

V

L  2 1  2 8 2 2   W EI  2  Al   Al   Al 0 1 2   2  15   X 2

 0

 u 1  W EA    X 2  X 

  

2

2

3  4    W    I  2l02  l12      5    X 3  

2   2u W  2W   4    A  2l02  l12    dX   5   X 2 X X 2     

2 2

In above equation, I is second cross section moment around Y axis and A is the cross section area.

2.2. Work of external forces Considering the distribution of external forces per unit length of the beam (fext), the work by the external forces, Vext, can be obtained as: Vext 

L W

0 0

fext (X )dWdX

(14)

The external forces is the summation of electrochemical and dispersion forces. These forces are determined in the following subsections. 2.2.1. Electrochemical force The electrochemical force is the sum of the electrical force Fe and chemical (or osmotic) force Fc which can be written as (15) and (16), respectively [15]: 1 (15) 2 F     e

2

0

  z e Fc  2n K BTb cosh( 0 )  1 K T   B

6

(16)

where ε0 is the permittivity of vacuum, ε is the relative permittivity of the dielectric medium, KB is the Boltzmann constant, n∞ is the bulk concentration, T is the absolute temperature, e is the electronic charge, z0 is the absolute value of the valence, and b is the width of electrode. In the above equation ψ is the electric potential of the liquid-immersed electrodes (which is the summation of applied potential and zeta potential for each electrode) and can be obtained from the Poisson–Boltzmann equation as the following [15]: 2 

2 z0en

 0

z e sinh( 0 ) K BT

(17)

For small potentials (17) can be linearized to the linear Poisson–Boltzmann equation (Debye– Hückel approximation): (18) d 2 2   dZ 2  (Z=0 )   1

 (Z=g)   2

where g is the initial gap between two electrodes,  2  2e2 z02n /  0K BT and 1 /  is the Debye length. By solving (18) and substituting the obtained solution in (15)-(17), the electrochemical force, fEC, is obtained as: f EC  Fe  Fc 

 2  2 2  2 cosh  ( g  W ( X ))  1  ( )   1   2sinh  ( g  W ( X ))  1  2 2

b 0 1

2

(19)

The electrochemical force can be attractive or repulsive, depending on the dominant parameters. 2.2.2. Dispersion force The dispersion forces per unit length of the beam are defined considering the vdW and Casimir force regimes. Both Casimir and vdW attractions are connected with the existence of zero-point vacuum oscillations of the electromagnetic field [26] and can be expressed in terms of quantum confinement effects when a pair of parallel plates is brought close together. For close parallel conductive plates, the virtual photon emitted by an atom of one surface reaches an atom of the other surface during its lifetime. The vdW force arises from the correlated oscillations of the instantaneously induced dipole moments of these atoms. However, when the separation between the two surfaces is large enough, the virtual photons emitted by an atom of the first surface cannot reach the second one during its lifetime. In this case, the interaction between two surfaces can be described by Casimir force (retarded vdW). This retardation effect is a result of the finite propagation speed of the electromagnetic field. If the separation is well below the plasma (for metals)/absorption (for dielectrics) wavelength of the surface material, the retardation is not significant. Thus the Casimir attractions become more effective at longer distances than the vdW forces [26]. Based on what mentioned, two interaction regimes is defined: First, the large separation regime in which the Casimir force is dominant (typically above several tens of nanometers [49-52]). Casimir force between two perfectly conducting planar geometries separated by medium with constant refractive index is calculated as a limit case of a three-layer model using the piston approach [50]. Using piston approach, the Casimir

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attraction per unit length, fCas , of the movable conductive electrode of NEMS bridge operating in a liquid media is obtained as [52]: 2 (20)  b fCas 

240n  g  W  X  

4

where ħ=1.055×10-34Js is Planck’s constant divided by 2π and n is the refractive index of fluid, respectively. The second regime is the small separation regime (typically below several tens of nanometers [49-51]), in which the vdW force is the dominant attraction. In this case, the attraction between two ideal surfaces is proportional to the inverse cube of the separation. Considering three layer interaction (two metals separated by liquid medium), the vdW force per unit length, fvdW, of the movable conductive electrode of the NEMS bridge is [53]: (21) A (n)b fvdW 

6 ( g  W ( X ))

3

where A(n) is the Hamaker constant for three-layer interaction which is a function of refractive index (n) of the liquid media. For conductive metals (such as Ag, Cu, Au, etc.) interacting across water-based media, the Hamaker constant is about 10-40×10-20J [53]. It should be noted that the van der Waals interaction potential is largely insensitive to variations in electrolyte concentration and pH, and so may be considered as fixed in a first approximation [53]. 2.3. Dimensionless governing Equation To derive the governing equation of longitudinal and lateral deflection of the nano-bridge, the Hamilton principle is employed which minimizes the total energy of the system. It mathematically implies: (22)  Ubend  Vext   0 where δ indicates variations symbol. Using (13), (14) and (22), the governing equation of the system can be derived as the following  X



  u 1 W   EA      X 2  X  

 X

  

2

4  2  u 1  W    A  2l02  l12      5  X 2  X 2  X   

     u 1  W    EA   X 2  X   

  

2

  

2 

4  2  u 1  W    A  2l02  l12      5  X 2  X 2  X    4

  0  

  

 W      X   

(23a)

2 

(23b)

6

  2 8 2 2   W  2 4 2 W  EI   A  2l0  l1  l2   4   I  2l0  l1  6  fext 15 5  X    X  

with the following boundary conditions: u (0)  u ( L) 

  u 1  W    X  X 2  X 

W  0  W  L  

W  0  X



  

2

W  L  X

 0   X 0, L



3W  0  X

3



(24a) 3W  L  X 3

0

8

(24b)

It should be noted that, by solving equation (23a) with boundary conditions of (24a) we can proved that:  u 1  W EA    X 2  X 

  

2

4  2  u 1  W    A  2l02  l12      5  X 2  X 2  X   

  

2

L

2

EA  W   dx  2L  x   0



(25)

Now, by substituting (19) and (20 or 21) in (23b) and then nondimensionalization of the relations, the governing equation and boundary condition can be derived as (26) and (27): 1  w 2  d 2w d 4w d 6w  D2      dx  2  0  x   dx dx 4 dx6   2 n  1 2     cosh0 1 w   2  1  n 2 sinh  1  w  0   1 w

D1

 

w  0   w  0   w  0   0 w 1  w 1  w 1  0

(26)

(27)

where the dimensionless parameters are identified as: x

X L

(28-a)

w

W g

(28-b)

 AbL4   6 g 4 EI n     2 bL4  5  240  g EI

vdW (n = 3)

(28-c) Casimir (n = 4)

1/2

 b 0    gEI 

  1  L2  

s 

g b

(28-e)

12 E  h l2 

2

  2 1 0   g g  

  6  h

(28-d)

(28-f) (28-g) (28-h)

2

(28-i)

2 2   l    l  D1  1  s  30  0   8  1   15    15   l2    l2    

9

(28-j)

 2 2   l   s   l0  D2   5   2 1     30 L h 2   l2   l2        

(28-k)

In above relations,  n ,  , s and 0 interpret the dimensionless values of dispersion forces, beam electrode voltage, size-effect, and bulk ion concentration, respectively. The dimensionless parameter  indicates the ratio of potential on the ground electrode over that on the beam electrode. Note that the integral term in Eq. (26) corresponds to the axial extension effect. The bending of double-clamped beams involves generally a stretching. When the maximum deflection is less than the thickness, small deflection can be considered valid, and the stretching can be neglected [54]. But for NEMS bridges, the gap can be larger than the beam thickness, so that the maximum deflection at middle point is larger than the thickness. Considering the invalidity of small deflection, it is required to take axial stress of the double-clamped beam into account. It should be noted that relation (26) turns to that of the classical theory, by setting the l0, l1 and l2 equal to zero. Furthermore, the size-dependent modified couple stress beam theory can be obtained by considering l0=l1=0 and l2=l. 3. Solution Methods 3.1. Differential transformation method (DTM) The basic idea and the fundamental theorems of the DTM and its applicability are given in [55, 56]. The concept of DTM is based on series expansion. For the convenience of readers, the details of the DTM in solving the governing equation (26) are explained in Appendix A. Using DTM, the deflection of the NEMS bridge is determined as the following series solution: (29) w  x   C x 2  C x 4  C x5  1

[e

20

e

20

2

3

 2][360bD1   a 2 (40a  96b  100c)  15 n ]  30  2[ 2  e0  e 0  1] 20

20

x6  ...

10800(e  e  2) D2 In relation (29), constants C1, C2 and C3 are determined using boundary conditions at x=1 (see Appendix A). In above relation, n=3 denotes the vdW attraction and n=4 corresponds to the Casimir force. The instability occurs if dw( x  0.5) / d    . The instability parameters of the system can be determined from the slope of the w -  graphs by plotting w vs.  .

3.2. Numerical solution method In addition with the analytical method, the deflection of the NEMS bridge is numerically simulated using MAPLE software. By numerically solving the differential equations, the deflection of the nano-bridge is determined. When the instability occurs, no solution exists for (26) and the pull-in parameters of the system can be determined by plotting the mid deflection vs. the applied force. By using the parameters u  uL / g 2 , equations (23a) and (23b) can be dimensionless as the followings system of equations:

10

  u w 2   2  u w 2    ( )   D2  ( )   0  2 2 x  x x  x   x 2  x

(30)

 u  w 2  4w 6w  2 u  w 2   2w n  D2   2    D2 [2    ] 2  4 6 2  x  x  x  x        x x x x 1 wn  2 1 2     cosh 0 1 w  2  1  2 sinh 0 1 w  

(31)

D1

 

The system of nonlinear differential equations (Eqs. ( 30) and ( 31)) is solved numerically. The step size of the parameter variation is chosen based on the sensitivity of the parameter to the maximum deflection (mid-length deflection for clamped beams). The pull-in parameters can be determined via the slope of the w -β graphs. 4. Result and discussion In the following, typical nano-bridge with the geometrical characteristics of L  25h and   0.1 are considered. The Young’s modulus E, and shear modulus μ are selected as 169 GPa and 65.8 GPa, respectively. 4.1. Comparison with literature To the best knowledge of the authors the electromechanical behavior of clamped nanoactuators immersed in the ionic liquid electrolyte have not been modeled yet. Indeed, only cantilever configuration is considered by few researchers [15, 16]. So in this section a cantilever nano-actuators is simulated using presented model and the obtained results are compared with literature [15, 16]. The governing equation of the cantilever nano-actuator was obtained based on classical theory (l0=l1=l2=0) and neglecting dispersion forces. Figure (2) shows the influences of the applied voltage,  , on the normalized tip deflection, w(x=1), of the cantilever beam for n  0,   0.1, 0  1 . It can be observed that the normalized tip deflection would increase with an increase in the input voltage. This figure reveals that the DTM results are in very good agreement with the finite element solution [13] and modified Adomian results [16].

11

Figure 2 .Variation of the normalized tip deflection of cantilever nano-actuator as a function of applied voltage.

4.2. Deflection and stability Figure 3 shows the variation of typical clamped nano-actuator deflection when the applied voltage increases from zero to pull-in value. The vertical axis reveals the deflection of the nano-beam while the horizontal axis reveals the dimensionless length of the beam. As seen, increasing the applied voltage increases the deflection of the nano-actuator. When the applied voltage exceeds its critical value, βPI, then no solution exists and the pull-in instability occurs. Note that the operation distance of the system is limited by this instability. This figure shows that the nano-beam has an initial deflection even when no voltage applied which is due to the presence of dispersion forces. It is observed that the results of DTM are in good agreement with those of numerical method. The relative error of presented methods with respect to the numerical solution is less that 1%.

(a) (b) Figure 3. Deflection of the clamped NEMS for different values of applied voltage from 0 to pull in voltage; (a) Casimir force (b) vdW force. ( l0 / l2  l1 / l2  h / l2  1 , n  500 and 0  1.5,   0.1 ). 4.3. Influence of dispersion forces If the gap between the beam and the ground is of the order of several nanometers, the effect of dispersion forces must be taken into account. The effect of dispersion forces on pull-in voltage of the nano-bridge is presented in Figure 4. As seen, increasing the dispersion forces leads to a decrease in the pull-in voltage of the system. Interestingly, the intersection point of the curves and the horizontal axis corresponds to the critical value of dispersion forces in liquid media; if the nano-beam is close enough to the ground, dispersion forces can induce stiction even without any electrostatic force.

12

(a) (b) Figure 4. Effects dispersion force on pull-in voltage a) Casimir, b) vdW for l0 / l2  l1 / l2  h / l2  1 , 0  1.5,   0.1 . Influence of dispersion forces on the pull-in deflection (mid-point deflection at pull-in) of the system is presented in Figure 5. This figure shows that while increase in the Casimir force slightly reduces the pull-in deflection of the actuator, increasing the vdW attraction augments the pull-in deflection of the system.

(a) (b) Figure 5. Effects dispersion forces on pull-in deflection a) Casimir b) vdW for l0 / l2  l1 / l2  h / l2  1 , 0  1.5,   0.1 . 4.4. Influence of size effect Variation of the pull-in voltage (  PI ) of the NEMS bridge is demonstrated in Figure 6 as a function of size effect parameter, h/l2, for different theories. This figure shows that reducing h/l2 results in enhance the instability voltage of the system. This means size effect provides a hardening behavior that enhances the elastic resistance and consequently pull-in voltage of the nano-device. On the other hand, with increase in the beam thickness, results of strain gradient theory approaches to those of classic theory (horizontal lines). Figure 7 represents the influence of size effect on the instability deflection ( wPI ) of the nano-bridge. As seen,

13

with increasing h/l2 the pull-in deflection of the nano-bridge increases, although in classical theory the pull-in deflection is independent of the size effect.

(a) (b) Figure 6. Pull-in voltage for different models as a function of size effect parameter (h/l2) and neglecting dispersion forces, (a) Numerical, (b) DTM. l0 / l2  l1 / l2  1 , 0  1.5,   0.1,n  0 .

(a) (b) Figure 7. Pull-in deflection for different models as a function of size effect parameter (h/l2),and neglecting dispersion forces (a) Numerical, (b) DTM. l0 / l2  l1 / l2  1 , 0  1.5,   0.1,n  0 . 4.5. Effects of ion concentration The effect of ion concentration parameter ( 0 ) on the instability behavior of the nano-bridge is shown in Figure 8 considering the presence of dispersion forces. This figure reveals that the pull-in voltage enhances by increasing 0 . The results of Fig 8 demonstrate that the augmentation of ions in the vicinity of electrodes surface increases the instability voltage. Figure 9 shows the effect of ion concentration on the pull-in deflection of nano-bridge. As seen, increase in ion concentration can decrease the pull-in deflection. Indeed, Figures 8 and

14

9 reveal that increase in the Debye length of the electrolyte enhances the instability voltage while reduces the maximum stable deflection of the system.

(a) (b) Figure 8. Effects of ion concentration on the pull-in voltage for l0 / l2  l1 / l2  h / l2  1,n  500 ,   0.1 (a) Casimir (b) vdW.

(a) (b) Figure 9. Effects of ion concentration on the pull-in deflection for l0 / l2  l1 / l2  h / l2  1,n  500 ,   0.1 (a) Casimir (b) vdW. 4.6. Effect of potential ratio Figure 10 shows the influence of the potential ratio () on mid-point deflection of the nanobridge for different size parameter values. As seen, deflection of the double-clamped electrode is always positive for any amount of  value. Note that this trend is different from what observed for cantilever beam where the beam free-end deflection can be positive or negative depending on  value [16]. This figure implies that for both repulsive and attractive electrochemical forces, the double-clamped beam deflects downward. This difference is the result of nonlinear stretching term that induces stiffening effect in the NEMS bridge.

15

(a) (b) Figure 10. Influence potential ratio and size effect on normalized mid deflection (a) Casimir (b) vdW for l0 / l2  l1 / l2  h / l2  1 , 0  1.5,n  50 ,   10 . 5. Conclusion Herein, the strain gradient theory has been employed to investigate the size-dependent pull-in instability of double-clamped NEMS bridge immersed in liquid electrolyte, incorporating the effect of dispersion forces. It is found that:  The dispersion forces reduce the pull-in voltage of the nano-bridge. While Casimir force reduces the pull-in deflection of the actuator, vdW attraction increases the instability deflection of the system.  For ultra-thin nano-bridge that the thickness is comparable with the material length scale parameters, size effect increases the pull-in voltage due to the stiffening effect.  Augmentation of bulk ion concentration increases the pull-in voltage of the NEMS bridge while decreases the pull-in deflection of the system.  Nano-bridge can only bend down for any potential ratio value. This trend is different from that reported for nano-cantilevers.  The nonlinear stretching term induces stiffening effect in the NEMS bridge. Hence, for both repulsive and attractive electrochemical forces, the double-clamped beam deflects downward.  The results obtained using Differential transformation method is in good agreement with those of numerical method.

16

Appendix A. Differential transformation method To apply DTM for solving the governing equation of the actuator, the differential transform of the kth derivative of arbitrary function f(x) is defined as: (A.1) 1  d k f ( x)  F (k ) 

 k !  dx k

  x  x 0

where f(x) is the original function and F(k) is the transformed function. The differential inverse transform of F(k) is defined as  )A.2) k f ( x) 

 F (k )( x  x0 )

k 0

The differential transformation relations for functional operations can be found in Ref. [52]. To solve the governing equation of the system using DTM, we multiply both sides of the equation (26) by (1  w)4 sinh 2 (0 (1  w)) . After some elaborations and using y  1  w , one can rewrite equation (26) as: (A.3) 4 6  1 dy 2  d 2 y  d y d y 4 4   (e20 y  e20 y  2)  D1 y 4  D2 y  y     dx  4 6  0  dx   dx 2  x  x   (e20 y  e20 y  2)  2  2 y 4  (e0 y  e0 y )  ( 2 1)   

  

where  4 Casimir    3 y vdW

)A.4(

Now, the deflection of the nano-structures can be considered as: w( x) 

)A.5(



 W (k ) x k  W (0)  W (1) x  W (2) x 2  W (3) x3  W (4) x 4  ....

k 0

where W is the transformation function. By apply DTM on equation (A.3), the following equation is obtained:  k  R   j m p  k    ( F ( R)  F ( R )  2 ( R ))  Y ( q ) Y p  q Y m  p Y j  m               1 2             R 0    j 0   m 0 p 0 q 0



 

 k  R  j  4  k  R  j  3 k  R  j  2  k  R  j  1 Y  k  R  j  4 

    D1  

 k R   j m p       Y (q)Y  p  q   Y  m  p   Y  j  m    k  R  j  6  k  R  j  5  k  R  j  4     j 0   m 0 p 0 q 0   

 

 k  R  j  3 k  R  j  2  k  R  j  1 Y  k  R  j  6 

  N r 1  1    L  1 Y  L  1 r  L  Y  r  L      D2   r   r 1 L  0  

 

 k R   j m p     Y (q )Y  p  q   Y  m  p   Y  j  m     k  R  j  2  k  R  j  1 Y  k  R  j  2      j 0   m 0 p 0 q 0   

  k 

  

k s m



p

    F3 (s)   F4 (s)  ( 2  1) (s)  2 2    Y (q)Y  p  q Y  m  p Y  k  s  m  

s 0 

(A.6)

m 0 p 0 q 0



17

    

    

where Casimir  4  F1  k   F2  k   2  k      k  3   F1     F2     2     Y  k    vdW   0 F1 (0)  exp  2 0Y (0)  , F2 (0)  exp  2 0Y (0)  ,

F3 (0)  exp   0Y (0)  , F1 (k)  F3 (k) 

2 0 k

0 k

(A.7)

F4 (0)  exp   0Y (0)  ,

k

 iY  i F1 (k  i) ,

F2 (k) 

i 1

k

 iY  i F3 (k  i) ,

F4 (k) 

i 1

2 0 k  0 k

k

 iY  i F2 (k  i) , i 1

k

 iY  i F4 (k  i) i 1

The differential transformations of boundary conditions for nano-clamped is as the following: Y (0)  1, Y (1)  0, Y (3)  0

(A.8-a)

n

n

n

k 0

k 0

k 0

 Y (k )  1,  kY (k )  0 ,  k (k  1)(k  2)Y (k )  0

(A.8-b)

By solving equations (A.6) and (A.8a), and assuming Y (2)  C1 , Y (4)  C2 and Y (5)  C3 , the deflection of the nano-bridge can be obtained as: (e20  e20 )(360bD1   C12 (40C1  96C2  100C3 )  15 n )  720bD1      C12 (80C1  192C2  200C3 )  30 n  30  2 ( 2   e0   e 0  1)  Y  6  10800(e20  e20  2) D2

Y (7) 

(A.9) (A.10)

C3 D1 42 D2

where constants C1, C2 and C3 can be determined using boundary conditions from equation(A.8-b).

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Figure Captions Figure 1. Schematic representation of clamped NEMS. Figure 2 .Variation of the normalized tip deflection of cantilever nano-actuator as a function of applied voltage.

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Figure 3. Deflection of the clamped NEMS for different values of applied voltage from 0 to pull in voltage; (a) Casimir force (b) vdW force. ( l0 / l2  l1 / l2  h / l2  1 , n  500 and 0  1.5,   0.1 ). Figure 4. Effects dispersion force on pull-in voltage a) Casimir, b) vdW for l0 / l2  l1 / l2  h / l2  1 , 0  1.5,   0.1 . Figure 5. Effects dispersion forces on pull-in deflection a) Casimir b) vdW for l0 / l2  l1 / l2  h / l2  1 , 0  1.5,   0.1 . Figure 6. Pull-in voltage for different models as a function of size effect parameter (h/l2) and neglecting dispersion forces, (a) Numerical, (b) DTM. l0 / l2  l1 / l2  1 , 0  1.5,   0.1,n  0 . Figure 7. Pull-in deflection for different models as a function of size effect parameter (h/l2),and neglecting dispersion forces (a) Numerical, (b) DTM. l0 / l2  l1 / l2  1 , 0  1.5,   0.1,n  0 . Figure 8. Effects of ion concentration on the pull-in voltage for l0 / l2  l1 / l2  h / l2  1,n  500 ,   0.1 (a) Casimir (b) vdW. Figure 9. Effects of ion concentration on the pull-in deflection for l0 / l2  l1 / l2  h / l2  1,n  500 ,   0.1 (a) Casimir (b) vdW. Figure 10. Influence potential ratio and size effect on normalized mid deflection (a) Casimir (b) vdW for l0 / l2  l1 / l2  h / l2  1 , 0  1.5,n  50 ,   10 .

Modeling the electromechanical behavior and instability of NEMS bridge in electrolyte: Influence of size dependency and dispersion forces I. Karimipour1, A. Kanani2, A. Koochi3 and M. Abadyan3 1

Faculty of engineering, Shahrekord University, Shahrekord, Iran Shahrekord Branch, Islamic Azad University, Shahrekord, Iran 3 Islamic Azad University, Isfahan, Iran

2

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Deflection and electromechanical instability of NEMS bridge due to the electrochemical double-layer force Highlights    

The pull-in instability of the NEMS bridge immersed in ionic liquid electrolyte media is explored for the first time. The dispersion forces i.e. Casimir and van der Waals attractions are included in the model considering the corrections due to the liquid media (Three layer interaction). The strain gradient elasticity is employed to incorporate the effect of sizedependency and beam stretching in the structural model. To solve the nonlinear constitutive equation of the system two different methods including the differential transformation method (DTM) and numerical solution are employed.

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