Author's Accepted Manuscript

Accurate electrostatic and van der Waals pull-in prediction for fully clamped nano/micro-beams using linear universal graphs of pull-ininstability

Masoud Tahani, Amir R. Askari

PII: S1386-9477(14)00207-0DOI: http://dx.doi.org/10.1016/j.physe.2014.05.023Reference: PHYSE11619

To appear in: Physica E

Received date: 29 March 2014Revised date: 8 May 2014Accepted date: 23 May 2014

Cite this article as: Masoud Tahani, Amir R. Askari, Accurate electrostatic andvan der Waals pull-in prediction for fully clamped nano/micro-beams usinglinear universal graphs of pull-in instability, Physica E, http://dx.doi.org/10.1016/j.physe.2014.05.023

This is a PDF file of an unedited manuscript that has been accepted forpublication. As a service to our customers we are providing this early version ofthe manuscript. The manuscript will undergo copyediting, typesetting, andreview 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 whichcould affect the content, and all legal disclaimers that apply to the journalpertain.

www.elsevier.com/locate/physe

1

Accurate electrostatic and van der Waals pull-in prediction for fully

clamped nano/micro-beams using linear universal graphs of pull-in

instability

Masoud Tahani, Amir R. Askari*

Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

*Corresponding author. Tel.: +98 511 8806055; fax: +98 511 8763304.

E-mail addresses: a.r.askari@stu-mail.um.ac.ir, amaskari@gmail.com (Amir R. Askari)

Abstract

In spite of the fact that pull-in instability of electrically actuated nano/micro-beams has been

investigated by many researchers to date, no explicit formula has been presented yet which

can predict pull-in voltage based on a geometrically non-linear and distributed parameter

model. The objective of present paper is to introduce a simple and accurate formula to predict

this value for a fully clamped electrostatically actuated nano/micro-beam. To this end, a non-

linear Euler-Bernoulli beam model is employed, which accounts for the axial residual stress,

geometric non-linearity of mid-plane stretching, distributed electrostatic force and the van der

Waals (vdW) attraction. The non-linear boundary value governing equation of equilibrium is

non-dimensionalized and solved iteratively through single-term Galerkin based reduced order

model (ROM). The solutions are validated thorough direct comparison with experimental and

other existing results reported in previous studies. Pull-in instability under electrical and vdW

loads are also investigated using universal graphs. Based on the results of these graphs, non-

dimensional pull-in and vdW parameters, which are defined in the text, vary linearly versus

the other dimensionless parameters of the problem. Using this fact, some linear equations are

presented to predict pull-in voltage, the maximum allowable length, the so-called detachment

2

length, and the minimum allowable gap for a nano/micro-system. These linear equations are

also reduced to a couple of universal pull-in formulas for systems with small initial gap. The

accuracy of the universal pull-in formulas are also validated by comparing its results with

available experimental and some previous geometric linear and closed-form findings

published in the literature.

Highlights

A new supper-convergent iterative solution for nano/micro-beam pull-in analysis is introduced.

The present approach doesnt suffer from long run time. Pull-in universal graphs which accounts for the effect of van der Waals attraction are

presented.

Some linear relationships between dimensionless parameters of the problem are found. Pull-in characteristics for electrically actuated nano/micro-beams are also extracted explicitly.

Keywords

N/MEMS, vdW attraction, Pull-in instability, Detachment length, Universal pull-in graphs,

Universal pull-in formulas

1. Introduction

Stability analysis of nano/micro-systems is a very desirable research topic nowadays.

These systems have applications in many engineering fields such as communications,

automotive and robotics [1]. Nano/micro-electro-mechanical systems (N/MEMS) can be

considered as a largest collection of these systems, because of their fast response, low power

consumption, reliability and their batch fabrications [2]. Electrically actuated nano/micro-

beams represent a major structural component and plays a crucial role in many N/MEMS

3

devices [2]. One of the most important phenomena associated with electrically actuated

nano/micro-beams is pull-in instability. This instability is occurred when the input voltage

exceeds a critical value called pull-in voltage. In this manner the elastic nano/micro-beam

suddenly collapses toward the substrate. To date, lots of researchers have dealt with the

mechanical behavior of electrically actuated nano/micro-beams. Here, some of these works

are reviewed.

Nathanson et al. [3] and Taylor [4] investigated pull-in instability experimentally.

Osterberg [5] studied this instability in electrically actuated micro-beams and circular micro-

diaphragms using linear spring-mass model and presented some closed-form solutions.

Although his closed-form formulas could represent pull-in voltage in terms of system

properties explicitly, his analytical results suffered from maximum relative error of 20% in

comparison to those obtained experimentally. Tilmans and Legtenberg [6] studied free

vibration and static behaviors of a wide double clamped micro-beam using linear beam

theory. They approximated pull-in voltage and first fundamental resonance frequency of a

micro-system using the principle of minimum total potential energy and Rayleighs quotient,

respectively. They also validated their findings with experimental results. Abdel-Rahman et

al. [7] investigated the oscillatory behavior as well as pull-in instability of micro-beams

utilizing the non-linear Euler-Bernoulli beam theory in which the effect of mid-plane

stretching had been taken into account. They investigated pull-in instability and the frequency

of vibrating micro-beams about their static deflection numerically using shooting method.

But their solution did not converge for micro-systems with large initial gaps. Younis et al. [8]

developed a ROM and investigated static and free vibration behaviors as well as pull-in

instability of a double clamped micro-beam. Their alternative approach could remove the

limitations of shooting method presented by Abdel-Rahman et al.[7]. Krylov [9] studied static

and dynamic pull-in instabilities for double clamped micro-beams under distributed

4

electrostatic actuation and non-linear squeeze film damping using multi-term ROM. He used

ninth and third-orders of ROM for static and dynamic problems, respectively. He validated

the static pull-in results with 3-D coupled simulation findings obtained using IntelliSuiteTM

package. He also compared his predictions for dynamic pull-in voltage with finite difference

results. Batra et al. [10] analyzed free vibrations of micro-beams predeformed by an electric

field incorporating the effect of fringing field and finite deflections thorough simple and

computationally efficient single term ROM. They converted the boundary value governing

differential equilibrium equation to a non-linear algebraic equation using their single degree-

of-freedom (SDOF) model and solved the resulting equation numerically. Chao et al. [11]

investigated static and dynamic pull-in instabilities for double clamped micro-beams actuated

by polarized DC voltage using bifurcation analysis. Although, their model accounted for axial

residual stress, distributed electrostatic forcing term and the effect of fringing field, the

geometric non-linearity of mid-plane stretching had been neglected in their analysis.

Therefore, it cannot predict pull-in voltage for nano/micro-beams with large initial gap. They

transferred the partial differential equation of motion to an ordinary equation in time using

single-term Galerkin based ROM and used Hopf bifurcation analysis to determine static and

dynamic pull-in voltages. They simplified the problem using fifth-order Taylor's series

expansion of the electrostatic forcing term and presented some closed-form formulas for

static and dynamic pull-in voltages. Although they could provide some closed-form solutions

for static case, due to the high non-linearity involved in dynamic cases, they could present

closed-form solution only for dynamically excited systems without the effect of fringing

field. Mojahedi et al. [12] also investigated static pull-in instability using single-term ROM.

They converted the boundary value governing differential equation to an algebraic equation

using first linear and un-damped mode-shape of an un-deformed micro-beam. They solved

the resulting non-linear algebraic equation through homotopy perturbation method (HPM).

5

Their model accounted for fringing field effect, non-linearity of mid-plane stretching and

axial residual stress. It should be noted that although most of previous solutions could

represent very accurate results, to date no explicit formula for pull-in voltage has been

presented in the literature which can describe it based on a geometrically non-linear and

distributed parameter model.

By decreasing in the dimensions of electrically actuated systems from micro-scales to

nano-scales the intermolecular surface forces significantly influence on the behaviors of

nano/micro-beams. The most important forces at the scale of N/MEMS are the Casimir and

vdW attractions. The vdW force arises from the correlated oscillation of the instantaneously

induced dipole moments of the atoms placed at the close parallel conductive plates [13]. The

vdW force is a short range force in nature, but it can lead to long range effects more than 0.1

m [14]. The Casimir force can be simply understood as the long range analog of the vdW

force, resulting from the propagation of retarded electromagnetic waves [15]. The effect of

vdW and Casimir forces on the behavior of nano and micro-systems has been investigated by

many researchers. Lin and Zhao [16] studied the behavior of nano-scale actuators using a

spring-mass model considering the vdW force. They also presented some closed-form

formulas for minimum allowable gap and the detachment length in nano/micro-systems.

Ramezani et al. [17] proposed a distributed parameter model to study pull-in instability of

electrically actuated nano-cantilevers subjected to the vdW and Casimir forces. They

transferred non-linear differential equation of the model into the integral form by using the

Greens function of the clamped-free beam and the integral equation was solved analytically

using the appropriate shape function of the beam deflection. Jia et al. [18] studied the free

vibrations of nano/micro-beams with different boundary conditions under the combined

effect of the distributed electrostatic and Casimir forces for both homogenous and non-

homogenous functionally graded material with two material phases through the differential

6

quadrature method (DQM). Their model accounted for axial residual stress and the non-

linearity of mid-plane stretching. Static and dynamic pull-in instabilities of electrically

actuated nano/micro-beams in presence of the vdW and Casimir forces were investigated by

Moghimi Zand and Ahmadian [19]. They considered the geometric non-linearity of mid-

plane stretching, applied axial loading, fringing field effect, the Casimir and vdW attractions

and solved the governing equation using the non-linear finite element method (NFEM).

This paper focuses on pull-in instability of clamped-clamped nano/micro-beams actuated

by polarized DC voltage under the effect of vdW force. A non-linear Euler-Bernoulli beam

model which accounts for the effect of axial residual stress and the von Karman-type of

geometric non-linearity is utilized. The boundary value governing differential equation of

equilibrium is non-dimensionalized and reduced to an algebraic equation using a simple and

computationally efficient SDOF model. The present model obtained by approximating the

deflection field with un-damped and linear mode shape of the nano/micro-beam. The

resulting non-linear algebraic equation is also solved iteratively. To predict static pull-in

conditions, the universal graphs are presented which depict the variation of non-dimensional

pull-in parameter versus the other dimensionless parameters of the problem. These graphs

show that the non-dimensional pull-in parameter varies linearly versus the other

dimensionless parameters of the system. Based on this important finding, some simple,

efficient and accurate formulas are introduced which can predict pull-in voltage for clamped-

clamped nano/micro-beams under the combined effect of electrostatic excitation and vdW

attraction. These linear equations are also reduced to a single formula for systems with small

initial gap. It is shown that pull-in voltages predicted by this formula agree very well with

those obtained using the present iterative method and Chao's closed-form equation [11] as

well as available experimental data in the literature. The present results are also compared

with those provided by Chao's equation [11] for systems with large initial gap. It is found that

7

Chao's equation represents poor results for such systems due to the fact that, the non-linearity

of mid-plain stretching has been neglected in this formula. It is shown that using present

linear pull-in formulas can remove this inability.

The minimum allowable gap and maximum allowable length of system, the detachment

length, are also specified explicitly, through determining critical value of non-dimensional

vdW parameter. It should be noted that if vdW parameter in a nano/micro-system reaches

values greater than its critical value, this system will be pulled-in under only vdW attraction.

Therefore, such a device cannot be worked appropriately. The critical value of vdW

parameter is also predicted in terms of dimensionless properties of the system using another

universal graph. This graph also shows another linear relationship in electrically actuated

nano/micro-systems. Based on this fact, some linear formulas are also presented in this case

to predict the critical value of vdW parameter. These equations are also reduced to a single

one for systems with small initial gap. The results of this equation are also compared with

analytical findings reported by Lin and Zhao [16] using spring-mass model. It is shown that

using present pull-in formulas instead of previous closed-form solutions may be very simple

and applicable for specifying the critical dimensions and pull-in voltage in electrically

actuated nano/micro-beams.

Due to the numerous usages of clamped-clamped nano/micro-beams in N/MEMS devices,

this type of nano/micro-beams is analyzed in this paper. It is noted that, although the fully

clamped nano/micro-beam is investigated in present study, the current approach can be

applied to any boundary conditions by utilizing appropriate mode shape in the Galerkin

procedure and determining the new dimensionless parameters of system. To the best of the

authors knowledge, no previous work has been conducted in open literature.

8

2. Problem formulation

Fig. 1 shows a typical structure of N/MEMS devices, where the main components are

fixed and movable electrodes. The fixed electrode modeled as a ground plane and the

movable one modeled as a fully clamped nano/micro-beam under the combined action of the

electrostatic excitation and the vdW force. The length, width and density of nano/micro-beam

are L, b and , respectively. The initial gap between the non-actuated beam and the stationary electrode is d. Also, x, y, and z are the coordinates along the length, width and

thickness, respectively. W is deflection of the beam, I is the second moment of cross-sectional

area about the y axis, is Poissons ratio and E is the effective Youngs modulus of the nano/micro-beam which is replaced by 2/ (1 )E when 5b h> based on the plane strain theory [8].

The electrostatic excitation by polarized DC voltage V without the effect of fringing field

per unit length of the beam can be expressed as

2

22( )esbVF

d W= (1)

It is noted that the fringing field does not have a sizable effect especially for the case of

wide nano/micro-beams [11]. The vdW force per unit length of the beam takes the following

form [20]:

vdW 36 ( )Abf

d W= (2)

where is the dielectric constant of medium and A is the Hamaker constant. The Hamaker constant for two identical metal media (Ag, Au, Cu) interacting across vacuum (air) has

values in range ( ) 2030 50 10 J [20].

9

Due to the elongation of fixed-fixed nano/micro-beam which is called the mid-plane

stretching effect and the mismatch of both thermal expansion coefficient and crystal lattice

period between substrate and nano/micro-beam film which is un-avoidable in surface micro-

machining techniques, a resultant axial force is applied to the nano/micro-beam [21]

axial r aF F F= + (3)The axial force due to the mid-plane stretching effect (von Krmn-type of geometric non-

linearity) takes the following form [10]

1 202a

EbhF W dxL

= (4)and the one due to the residual stress can be defined as [21]

r rF bh= (5)where r represents the axial residual stress. So the differential equation which governs the equilibrium of the clamped-clamped nano/micro-beam subjected to the combined effect of

electrostatic excitation and the vdW force is represented as follows [8]

1 2es vdW02r

EbhEIW F W dx W F FL

= + + + (6)where prime sign denotes derivative with respect to x. The nano/micro-beam deflection is

subjected to the following kinematic boundary conditions.

(0, ) ( , )(0, ) 0, 0, ( , ) 0, 0W t W L tW t W L tx x

= = = = (7)

For convenience, the following dimensionless variables are introduced

;W xW xd L

= = (8)

Upon substitution of the dimensionless quantities given in Eq. (8) into Eq. (6) and dropping

the hats, the following result will be obtained

10

1 2 32 30 (1 ) (1 )

W W dx N WW W

= + + + (9)where

2 2 2 4

33 33 3 4

412 6 26 , , ,rF Ld v ALNh Ebh E

Lh d Eh d

= = = = (10)

In this paper , N , 3 and are non-dimensional parameters of the system and called gap, axial force, vdW and electrostatic parameters.

3. Solution procedure

Due to the high non-linearity involved in Eq. (9), a closed-form solution for this equation

cannot be found. Hence, an approximate solution will be developed through the Galerkin

weighted residual method. Based on this procedure, the nano/micro-beam deflection can be

expressed as a linear combination of a complete set of linearly independent basis functions

[22]. It is noted that these functions must satisfy all kinematic boundary conditions [22].

Therefore, linear and un-damped mode shapes of the un-deformed nano/micro-beam can be

used as these basis functions. It is proved that using only the first mode for pull-in analysis of

electrically actuated nano/micro-beams maybe very accurate [10, 12]. Hence, the deflection

of nano/micro-beam based on one-mode solution can be expressed as

( ) ( )W x w x= (11)where w is an un-known parameter determined through the Galerkin procedure and ( )x is the first linear and un-damped mode shape of the un-deformed nano/micro-beam determined

as [23]

( ) ( ) ( ) ( ) ( ){ }cosh cos sinh sinx x x x x = (12)where the parameters and for double clamped nano/micro-beam are given in Eq. (13) [23].

11

4.7300, 0.9825 = = (13)Here, ( )x is normalized such that the parameter w describes the mid-point deflection of nano/micro-beam. Hence, the parameter can be determined as [24]

0.6297 = (14)Next, we multiply Eq. (9) by ( )x , substitute Eq. (11) into the resulting equation, integrate the outcome from 0x = to 1 and obtain

1 13 2 31 3 30 0

/ (1 ) / (1 )w w w dx w dx + = + (15)where

1 1 1 21 30 0 0

,dx N dx dx = = (16)The deflection of nano/micro-beam under electrical and vdW loads can be determined by

solving the algebraic Eq. (15). In this paper, this equation is solved iteratively. To this end,

the non-linear Eq. (15) is rewritten in the following form

( )1 0w w + =` (17)where

( ) 1 13 2 33 30 0/ (1 ) / (1 )w w w dx w dx = ` (18)At the first step, the non-linear terms in Eq. (15) (i.e. ( )w` ) are calculated when 0 0w = as

( )1 1 13 2 30 3 0 0 3 0 30 0 0/ (1 ) / (1 )w w dx w dx dx = = + ` (19)Substituting 0` from Eq. (19) into Eq. (17), yields the first estimation of mid-point deflection

(i.e. 1w ) as

01

1w =

` (20)

12

The next estimation of ( )w` (i.e. 1` ) will be obtained by substituting 1w from Eq. (20) into Eq. (18) as

1 13 2 31 3 1 1 3 10 0

/ (1 ) / (1 )w w dx w dx = ` (21)Substituting 1` from Eq. (21) into Eq. (17), leads to the second estimation of mid-point

deflection (i.e. 2w ). This iterative procedure is continued till the convergence is achieved or

pull-in is happened. The convergence criteria is defined as

( ) 61 / 10i i iw w w (22)and the pull-in will be happened if

1iw (23)

It is noted that the integrals 1 20

/ (1 )iw dx and 1 30 / (1 )iw dx should be calculated numerically and repeated at the each step.

4. Results and discussion

4.1. Comparison and validation

To validate the present iterative approach, a poly-silicon micro-beam with the geometric

and material properties listed in Table 1 with four different lengths is considered. Pull-in

voltages for these four cases are compared with experimental observations provided by

Tilmans and Legtenberg [6] and DQM findings reported by Kuang and Chen [25] in Table 2.

It should be noted that in this comparison, the effect of vdW force has been neglected

(i.e. 3 0 = ). Based on the results of Table 2, the accuracy of present itrative approach can be observed.

Another validation is preformed for micro-beams with 3 0N = = and some different gap parameters in Fig. 2. The present results are compared with those obtained by Abdel-Rahman

13

et al [7] in this figure. It should be noted that they solved the similar problem using shooting

method, but their solution could not capture pull-in instbility especially for cases with large

gap parameter. The effect of vdW force is also neglegted in this comparison.

The accuracy and simplicity of present approach as well as its ability in capturing pull-in

instability for cases with large gap parameter can be observed from Fig. 2. Consider another

poly-silicon micro-beam with properties listed in Table 3. Pull-in voltage for this micro-beam

without the effect of vdW force is calculated as PI 45.80 VV = which agrees very well with

PI 45.62 VV = extracted from Fig. 4 of Krylovs paper [9]. It is noted that Krylov considered

both distributed electrostatic force and distributed force originating from the air squeeze film

pressure. However, he set all time derivatives to zero for obtaining the results of this figure

which means that the air pressure effect has been neglected in it. Fig. 3 shows the variation of

mid-point deflection versus the input voltage for this micro-beam. It should be noted that

although we use 1w for pull-in condition in our solution, this instability can be observed when the slope of deflection-voltage graph reaches infinity [12]. Therefore, in such graphs

we reduce the upper limits of vertical axis from 1 to the first value which can depict this

infinity slope for a better presentation.

It should be noted that Krylov [9] multiplied both sides of the governing equation by the

denominator of electrostatic forcing term and solved the resulting boundary value equation

through a ninth-order ROM, but we solved the problem without this multiplication. It is

worth noting that this multiplication adds significant effects of higher-order modes in the

reduced order equation(s). Therefore, it is essential to account for higher-order modes in such

condition.

The present single-mode technique has been also used in some previous studies [10, 12].

Batra et al. [10] studied oscillatory behavior of a micro-beam predeformed by an electric field

numerically and Mojahedi et al. [12] investigated pull-in instability through HPM. It is noted

14

that the non-linear algebraic equilibrium equation has been solved through a new iterative

procedure in this study. One of the benefits of present approach is its short run time which

provides us to determine the pull-in voltages for a large number of nano/micro-systems.

Based on this ability, we can plot some universal graphs which depict the variation of pull-in

parameter (i.e. PI ) versus the other non-dimensional parameters of the system. These universal graphs show the linear relationship between pull-in parameter and the other

dimensionless parameters of the problem. Using this important finding, pull-in voltage for a

nano/micro-system can be determined without the need for solving the non-linear governing

equation. This procedure is explained more in the next two sections.

Another comparison is also performed here to show the accuracy of present iterative

approach when the effect of vdW force has been taken into account. To this end, another

nano/micro-beam with non-dimensional properties 0N = and 6 = is considered. Fig. 4 shows the variation of pull-in parameter ( PI ) versus the vdW parameter. The present findings are also compared with NFEM results obtained by Moghimi Zand and Ahmadian

[19] in Fig. 4. Based on this figure and the previous comparisons, it can be observed that the

combination of single-mode ROM and present iterative solution can be treated as a promising

tool for pull-in problems in electrically actuated nano/micro-beams.

4.2. Universal graphs for pull-in instability

Fig. 5 shows the variation of pull-in parameter versus the other non-dimensional

parameters of the problem. Based on this figure, pull-in parameter varies linearly versus axial

force and vdW parameters and almost linearly versus the gap parameter.

Due to the linear variation of pull-in parameter, one can fit some lines to the results of Fig.

5. These lines and their corresponding equations are presented in Fig. 6. Due to the fact that

PI varies linearly versus axial force and vdW parameters and almost linearly versus the gap

15

parameter, Fig. 6 depicts the variation of PI versus N and 3 at some constant gap parameters. Although the slopes of these lines are a little different for small gap parameters,

the lines become parallel by increasing in the value of this parameter. It is worth noting that

this fact can also show almost linear variation of PI versus . It is noted that gap parameter in most of electrically actuated nano/micro-beams is quite

small (e.g. 0 6 ) [5, 6, 26, 27]. Therefore, a universal pull-in formula for such systems can be presented. Due to the non-linearity involved in the variation of pull-in parameter

versus the gap parameter for systems with large gap parameter, it is better to employ the

equations presented in Fig. 6 and a linear interpolation between their results for predicting

pull-in instability more accurately. The universal pull-in formula is

3PI 1.66 1.5 1.41 69.4N = + + (24)It is to be noted that if bN N= , where bN is the non-dimensional buckling load, the

nano/microsystem will be buckled and the pull-in parameter reaches zero. It may be noted

that bN for double clamped nano/micro-beams is around -40.

Consider a poly-silicon micro-beam with properties presented in Table 1. The results of

universal pull-in formula (i.e. Eq. (24)) have been compared with experimental observations

provided by Tilmans and Legtenberg [6] in Table 4 as well as those obtained through present

iterative approach. These results are also compared with those provided by Chao's closed-

form equation [11]. It should be noted that the effect of mid-plane stretching has been

neglected in Chao's formula. The Chao's closed-form equation is [11]

ChaoPI

0.927783 82.2287 2.02165

NV += (25)

where

16

24

3 ,

2rF LbL N

EId EI = = (26)

It should be noted that by increasing in the gap parameter, it is better to employ the

equations presented in Fig. 6 and linear interpolation between their results. To explain this

procedure more, consider the previous micro-beam with length 510mL = , initial gap

4md = and the other properties presented in Table 1. The gap parameter for this case is

42.66 = which is placed between the categories of 24 = and 54 = in Fig. 6(b). Pull-in parameter for this case can be calculated as PI 231.38 = through linear interpolation between PI 192.24 = and PI 255.17 = which are related to 24 = and 54 = , respectively. Pull-in voltage which corresponds to PI 231.38 = , can also be calculated as PI 48.04VV = for this poly-silicon micro-beam. The calculated pull-in voltage is also compared with that

obtained by present iterative approach and Chao's closed-form equation (i.e. Eq. (25)) in

Table 5.

One can observe the accuracy of pull-in equations presented in Fig. 6 for large gap

parameter systems. The poor prediction provided by Chao's equation (i.e. Eq. (25)) is because

of neglecting the mid-plane stretching effect in this formula.

To validate the universal pull-in formula when the effect of vdW force is taken into

account, consider a gold nano/micro-beam with properties listed in Table 6. Hamaker

constant for this system has been presented as 2044 10 JA = by Buks and Roukes [28]. Pull-in voltages obtained by universal pull-in formula (i.e. Eq. (24)) and the present

iterative approach for a system with properties presented in Table 6 are compared with each

other in Table 7. Based on the results of this table, one can observe the accuracy of Eq. (24)

for nano/micro-beams actuated by both electrical and vdW loadings. It should be noted that

the accuracy of this equation will be increased, by increasing in the applied axial force.

17

Based on the results of Tables 4 and 7, one can observe the accuracy of Eq. (24) for

systems with small gap parameter. Therefore, this equation has been introduced for predicting

pull-in voltage for nano/micro-systems with small initial gap. It is noted that one can utilize

universal graphs presented in Fig. 6 for systems with large gap parameter.

4.3. Universal graphs for pull-in instability under only vdW attraction

Pull-in instability can be observed in a nano/micro-beam excited by only vdW attraction.

To describe such conditions, the electrostatic parameter is set to zero (i.e. 0 = ) and the variation of mid-point deflection versus the vdW parameter is plotted in Fig. 7. Based on this

figure, by increasing in the values of vdW parameter, the deflection of nano/micro-beam may

be increased and when this parameter reaches its critical value, the nano/micro-beam

suddenly collapses toward its substrate. It is noted that the infinity slop can be also observed

in this case.

The maximum allowable length (the detachment length) and the minimum allowable gap

for an electrically actuated nano/micro-system can be determined using the critical value of

vdW parameter. Due to the linear relationships which can be observed in pull-in instability

under electrical load, a linear variation of critical vdW parameter (i.e. Cr3 ) versus N and

can be guessed. Fig. 8 illustrates the variation of Cr3 versus N in some constant gap parameters.

Based on the results of Fig. 8, one can see parallel lines for large gap parameters and

almost parallel lines for systems with small initial gaps. Therefore, it can be concluded that

the critical vdW parameter just like the pull-in parameter varies linearly versus axial force

and almost linearly versus the gap parameter. Since initial gap in most of the electrically

actuated nano/micro-systems is quite small (e.g. 0 6 ), one can represent another universal pull-in formula for such systems. This formula is called universal vdW pull-in

18

formula in this paper. It is noted that one can utilize the results of Fig. 8 and a linear

interpolation between them for cases with large gap parameter through the procedure

explained in previous section. The universal vdW pull-in formula is

Cr3 1.2 0.62 50.29N = + + (27)

It should be noted that if the parameters N and in Eq. (27) are set to zero, the critical vdW parameter will be obtained through a distributed linear beam model without the effect of

axial residual stress. Using Eq. (10) one can obtain the minimum allowable gap and the

detachment length for such model as

3 34 4max min

1/ , /L d Eh A d L A Eh = =^ ^ (28)

where

2.239=^ (29)Lin and Zhao [16] also determined such equation based on a linear spring-mass model. They

calculated^ analytically as

2.121=^ (30)It is noted that the small difference between these two coefficients (i.e. the present ^ and that

obtained by Lin and Zhao [16]) is due to the fact that they had been used lump model instead

of a distributed parameter model.

It is worth noting that although Eqs. (24) and (27) can be useful for predicting pull-in

voltage and the critical dimensions for most of electrically actuated nano/micro-beams, the

more accurate results can be obtained by solving Eq. (9) through present iterative approach.

In other words, the present universal formulas are useful for the first step in designing

19

N/MEMS devices and more accurate simulations as well as some experimental validations

must be employed for finalizing the design.

5. Conclusions

The main goal of present study was to provide explicit formulas to predict pull-in voltage,

detachment length and minimum allowable gap for a fully clamped electrically actuated

nano/micro-beam affected by vdW attraction. To this end, a non-linear Euler-Bernoulli beam

model was utilized which accounts for the axial residual stress, the geometric non-linearity of

mid-plane stretching as well as the inherent non-linearity of electrical and vdW loadings. The

boundary value differential equation that governs the nano/micro-beam equilibrium was

reduced to a non-linear algebraic equation through single term Galerkin based ROM and

solved iteratively. The model's predictions for pull-in voltages were compared with some

previous semi-analytical and numerical results as well as experimental observations available

in the literature. Furthermore, the variation of non-dimensional pull-in and vdW parameters

versus the other dimensionless parameters of the system was investigated using some

universal graphs. It was observed that the non-dimensional pull-in and vdW parameters vary

linearly versus the other dimensionless parameters of the problem. Based on this fact, some

linear formulas were introduced to predict pull-in voltage, detachment length and minimum

allowable gap for clamped-clamped nano/micro-beams. These linear equations were also

reduced to a couple of universal pull-in formulas for systems with small initial gap. The

results of introduced universal pull-in formulas were also compared and validated with those

obtained by some previous closed-form equations and present iterative method as well as

available experimental data in the literature.

20

Acknowledgement

The authors wish to express appreciation to Research Deputy of Ferdowsi University of

Mashhad for supporting this project by grant No.: 24331-19/9/91.

References

[1] S.D. Senturia, Microsystem Design, Kluwer Academic Publishers, Dordrecht, 2001.

[2] M.I. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York,

2011.

[3] H.C. Nathanson, W.E. Newell, R.A. Wickstrom, J.R. Davis, The Resonant Gate

Transistor, IEEE T. Electron Dev., 14 (1967) 117-133.

[4] G.I. Taylor, The Coalescence of Closely Spaced Drops When They are at Different

Electric Potentials, Proc. of Roy. Soc. A, 1968, pp. 423-434.

[5] P.M. Osterberg, Electrostatically Actuated Microelectromechanical Test Structures for

Material Property Measurement, Massachusetts Institute of Technology, Cambridge,

Massachusetts, USA, 1995.

[6] H.A.C. Tilmans, R. Legtenberg, Electrostatically Driven Vacuum-Encapsulated

Polysilicon Resonators - Part II:Theory and Performance, Sensor Actuat. A-Phys., 45

(1994) 67-84.

21

[7] E.M. Abdel-Rahman, M.I. Younis, A.H. Nayfeh, Characterization of the Mechanical

Behavior of an Electrically Actuated Microbeam, J. Micromech. Microeng., 12 (2002)

759-766.

[8] M.I. Younis, E.M. Abdel-Rahman, A.H. Nayfeh, A Reduced-Order Model for Electrically

Actuated Microbeam-Based MEMS, J. Microelectromech S., 12 (2003) 672-680.

[9] S. Krylov, Lyapunov Exponents as a Criterion for the Dynamic Pull-In Instability of

Electrostatically Actuated Microstructures, Int. J. Nonlin. Mech., 42 (2007) 626-642.

[10] R.C. Batra, M. Porfiri, D. Spinello, Vibrations of Narrow Microbeams Predeformed by

an Electric Field, J. Sound Vib., 309 (2008) 600-612.

[11] P.C.P. Chao, C.W. Chiu, T.H. Liu, DC Dynamic Pull-In Predictions for a Generalized

ClampedClamped Microbeam Based on a Continuous Model and Bifurcation

Analysis, J. Micromech. Microeng., 18 (2008) 115008.

[12] M. Mojahedi, M. Moghimi Zand, M.T. Ahmadian, Static Pull-In Analysis of

Electrostatically Actuated Microbeams Using Homotopy Perturbation Method, Appl.

Math. Model., 34 (2010) 1032-1041.

[13] E.M. Lifshitz, The Theory of Molecular Attractive Forces between Solids, Sov. Phys.

JETP, 2 (1956) 73-83.

[14] H. Liu, S. Gao, S. Niu, L. Jin, Analysis on the Adhesion of Microcomb Structure in

MEMS, Int. J. Appl. Electrom., 33 (2010) 979984.

22

[15] S.K. Lamoreaux, The Casimir Force: Background, Experiments, and Applications, Rep.

Prog. Phys., 68 (2005) 201-236.

[16] W.H. Lin, Y.P. Zhao, Dynamic Behavior of Nanoscale Electrostatic Actuators, Chin.

Phys. Lett., 20 (2003) 2070-2073.

[17] A. Ramezani, A. Alasty, J. Akbari, Closed-Form Solutions of the Pull-In Instability in

Nano-Cantilevers under Electrostatic and Intermolecular Surface Forces, Int. J. Solids

Struct., 44 (2007) 4925-4941.

[18] X.L. Jia, J. Yang, S. Kitipornchai, C.W. Lim, Free Vibration of Geometrically Nonlinear

Micro-Switches under Electrostatic and Casimir Forces, Smart Mater. Struct., 19

(2010) 115028.

[19] M. Moghimi Zand, M.T. Ahmadian, Dynamic Pull-In Instability of Electrostatically

Actuated Beams Incorporating Casimir and van der Waals Forces, J. Mech. Eng. Sci.,

224 (2010) 2037-2047.

[20] J.N. Israelachvili, Intermolecular and Surface Forces, 3rd ed., Elsevier, University of

California, Santa Barbara, California, 2011.

[21] J. Qian, C. Liu, D.C. Zhang, Y.P. Zhao, Residual Stresses in Microelectromechanical

System, Journal of Mechanical Strength, 23 (2001) 393-401.

[22] J.N. Reddy, Energy Principles and Variational Methods in Applied Mechanics, John

Wiley and Sons, New York, 2002.

[23] B. Balachandran, E. Magrab, Vibrations, 2nd ed., Cengage Learning, Toronto, 2009.

23

[24] A.R. Askari, M. Tahani, Analytical Approximations to Nonlinear Vibration of a

Clamped Nanobeam in Presence of The Casimir Force, Int. J. Aerosp. Lightweight

Struct., 2 (2012) 317-334.

[25] J.H. Kuang, C.J. Chen, Dynamic Characteristics of Shaped Micro-Actuators Solved

Using the Differential Quadrature Method, J. Micromech. Microeng., 14 (2004) 647-

655.

[26] A.H. Nayfeh, M.I. Younis, E.M. Abdel-Rahman, Reduced-Order Models for MEMS

Applications, Nonlinear Dynam., 41 (2005) 211-236.

[27] R.C. Batra, M. Porfiri, D. Spinello, Review of Modeling Electrostatically Actuated

Microelectromechanical Systems, Smart Mater. Struct., 16 (2007) R23-31.

[28] E. Buks, M.L. Roukes, Stiction, Adhesion Energy, and the Casimir Effect in

Micromechanical Systems, Phys. Rev. B, (2001) 63033402.

24

Figure captions

Fig. 1. Schematic of an electrically actuated nano/micro-beam under the effect of vdW force.

Fig. 2. Comparison between present results and those obtained by Abdel-Rahman et al. [7] through

shooting method. Solid lines depict present findings and markers display shooting method results for a

micro-beam with 3 0N = = .

Fig. 3. Comparison between present results and those obtained through ninth-order ROM [9] for a

poly-silicon micro-beam with properties presented in Table 3.

Fig. 4. Comparison between present results and those obtained through NFEM [19] for a nano/micro-

beam with dimensionless properties 0N = and 6 = .

Fig. 5. Universal graphs for pull-in instability.

Fig. 6. Fitted lines to the results of universal graphs and their corresponding equations. (a) 0N = and (b) 3 0 = .

Fig. 7. pull-in instability under vdW attraction.

Fig.8. Fitted lines to the results of vdW pull-in universal graph and their corresponding equations.

25

Fig. 1. Schematic of an electrically actuated nano/micro-beam under the effect of vdW force.

0 30 60 90 120 150 1800

0.16

0.32

0.48

0.64

w

43

1. =1.52. =103. =304. =50

1 2

Fig. 2. Comparison between present results and those obtained by Abdel-Rahman et al. [7] through

shooting method. Solid lines depict present findings and markers display shooting method results for a

micro-beam with 3 0N = = .

26

0 10 20 30 40 500

0.125

0.25

0.375

0.5

V (V)

w

PresentKrylov [9]

Fig. 3. Comparison between present results and those obtained through ninth-order ROM [9] for a

poly-silicon micro-beam with properties presented in Table 3.

0 10 20 30 40 500

20

40

60

80

3

PI

PresentMoghimi Zand and Ahmadian [19]

Fig. 4. Comparison between present results and those obtained through NFEM [19] for a nano/micro-

beam with dimensionless properties 0N = and 6 = .

27

0 30 60 90 120 1500

125

250

375

500

PI

0 8 16 24 32 400

105

210

315

420

3

PI

-20 -4 12 28 44 600

125

250

375

500

N

PI

0 30 60 90 120 1500

105

210

315

420

PI

(a) (b)

(d)(c)

062454

96

1503 = 0, =

0624

54

96

150

N = 0, =

3 = 0, N = 6040200

-20 403020100N = 0, 3 =

Fig. 5. Universal graphs for pull-in instability.

28

0 8 16 24 32 400

105

210

315

420

3

PI

PI = -1.4093 + 69.3, = 0PI = -1.4983 + 78.22, = 6PI = -1.7863 + 114.4, = 24PI = -2.0493 + 184.98, = 54PI = -2.1693 + 286.94, = 96PI = -2.2923 + 419.02, = 150

-20 -4 12 28 44 600

125

250

375

500

N

PI

PI = 1.677N + 69.62, = 0PI = 1.655N + 79.02, = 6PI = 1.518N + 116.1, = 24PI = 1.373N + 186.3, = 54PI = 1.289N + 287.9, = 96PI = 1.242N + 419.7, = 150

(a)

(b)

Fig. 6. Fitted lines to the results of universal graphs and their corresponding equations. (a) 0N = and (b) 3 0 = .

29

0 10.2 20.4 30.6 40.8 510

0.1

0.2

0.3

0.4

0.5

3

w

N = 0, = 0

Fig. 7. pull-in instability under vdW attraction.

-20 -4 12 28 44 600

65

130

195

260

N

3Cr

Present itrative appraoch, = 0Present itrative appraoch, = 6Present itrative appraoch, = 24Present itrative appraoch, = 54Present itrative appraoch, = 96Present itrative appraoch, = 150

3Cr = 1.202 N + 50.29, = 03Cr = 1.194 N + 54.04, = 63Cr = 1.128 N + 68.96, = 243Cr = 1.013 N + 99.32, = 543Cr = 0.923 N + 145, = 963Cr = 0.868 N + 205.1, = 150

Fig.8. Fitted lines to the results of vdW pull-in universal graph and their corresponding equations.

30

Graphical abstract

Universal graphs for pull-in instability of micro-beam based MEMS devices are presented. These graphs show some interesting linear relationships between dimensionless parameters of the system.

0 30 60 90 120 1500

125

250

375

500

Gap parameter

Dim

ensi

onle

ss p

ull-i

n vo

ltage

0 8 16 24 32 400

105

210

315

420

vdW parameter

Dim

ensi

onle

ss p

ull-i

n vo

ltage

-20 -4 12 28 44 600

125

250

375

500

Axial force parameter

Dim

ensi

onle

ss p

ull-i

n vo

ltage

0 30 60 90 120 1500

105

210

315

420

Gap parameter

Dim

ensi

onle

ss p

ull-i

n vo

ltage

0624

54

96

150

N = 0, =

(c) (d)

(b)(a)

3 = 0, =

3 = 0, N = N = 0, 3 =6040200

-20

010203040

150

96

542460

31

Table captions

Table 1. Geometric and material properties of the poly-silicon micro-beam.

Table 2. A comparison between pull-in voltages (V) calculated by different methods for micro-beams

with the geometric and material properties presented in Table 1.

Table 3. Geometric and material properties of the Krylov's poly-silicon micro-beam [9].

Table 4. Validation of universal pull-in formula for a poly-silicon micro-beam with properties

presented in Table 1.

Table 5. Comparisons between pull-in voltages calculated using the results of Fig. 6 and that obtained

by present iterative approach and Chao's closed-form equation for a poly-silicon micro-beam with

properties presented in Table 1, length 510mL = and initial gap 4md = .

Table 6. Geometric and material properties of the gold nano/micro-beam.

Table 7. Validation of universal pull-in formula for a gold nano/micro-beam with properties listed in

Table 6.

32

Table 1

Geometric and material properties of the poly-silicon micro-beam.

(m)b (m)h (m)d (GPa)E 3(kg/m ) (MPa) 100 1.5 1.18 151 0.3 2332 6

Table 2

A comparison between pull-in voltages (V) calculated by different methods for micro-beams with the

geometric and material properties presented in Table 1.

(m)L Experiment [6] DQM [25] Present iterative approach

210 27.95 0.05 28.10 28.00 310 13.78 0.03 14.00 13.97 410 9.13 0.02 8.90 8.85 510 6.57 0.02 6.40 6.35

Table 3

Geometric and material properties of the Krylov's poly-silicon micro-beam [9].

(m)L (m)b (m)h (m)d (GPa)E 3(kg/m ) (MPa) 300 20 2 2 169 0.28 2332 0

33

Table 4

Validation of universal pull-in formula for a poly-silicon micro-beam with properties presented in

Table 1.

(m)L

PI (V)V

Experiment

[6]

Present iterative

approach

Universal pull-in

formula (Eq. (24))

Chao's closed-form

formula (Eq. (25))

210 27.95 0.05 28.00 28.15 27.61 310 13.78 0.03 13.97 14.08 13.90 410 9.13 0.02 8.85 8.88 8.83 510 6.57 0.02 6.35 6.36 6.36

Table 5

Comparisons between pull-in voltages calculated using the results of Fig. 6 and that obtained by

present iterative approach and Chao's closed-form equation for a poly-silicon micro-beam with

properties presented in Table 1, length 510mL = and initial gap 4md = .

Interpolated pull-in

voltage

Present iterative

approach Chao's closed-form formula (Eq. (25))

48.04 V 47. 95 V 39.71 V

Table 6

Geometric and material properties of the gold nano/micro-beam.

(m)L (m)b (nm)h (GPa)E 3(kg/m ) 20 (10 J)A 5 1 100 80 0.42 19300 44

34

Table 7

Validation of universal pull-in formula for a gold nano/micro-beam with properties listed in Table 6.

(nm)d (MPa)r Present iterative approach Universal pull-in formula (Eq. (24))

15

0 0.43 0.44

10 0.49 0.49

20 0.54 0.54

20

0 1.11 1.12

10 1.17 1.17

20 1.22 1.22

25

0 1.69 1.70

10 1.76 1.76

20 1.83 1.83

0 5.06 5.11

50 10 5.24 5.28

20 5.42 5.45