MODELING THE EFFECT OF VAN DER WAALS

ATTRACTION ON THE INSTABILITY

OF ELECTROSTATIC CANTILEVER

AND DOUBLY-SUPPORTED NANO-BEAMS

USING MODIFIED ADOMIAN METHOD

RAHMAN SOROUSH

Engineering Group, Lahijan branch

Islamic Azad University, Lahijan, Iran

ALI KOOCHI

Engineering Group, Naein branchIslamic Azad University, Naein, Iran

ASIEH SADAT KAZEMI

Engineering Group, Bojnourd branchIslamic Azad University, Bojnourd, Iran

MOHAMADREZA ABADYAN*

Engineering Group, Chaloos branch

Islamic Azad University, Chaloos, IranAbadyan@yahoo.com

Received 18 January 2011Accepted 15 June 2011

Published 17 December 2012

A nano-scale continuum model is applied to investigating the e®ect of van der Waals (vdW)

attraction on pull-in instability of nano-beams in the presence of electrostatic forces. Two casesincluding the cantilever and doubly-supported beams are considered. The modi¯ed Adomian

decomposition (MAD) method is employed to solve the nonlinear constitutive equation of nano-

beams in the presence of vdW and electrostatic forces for the ¯rst time. The results show thatthe e®ect of vdW attraction on the instability of the doubly-supported nano-beam is weak when

compared to that of the cantilever due to the higher elastic sti®ness of the former. Basic design

parameters such as the critical de°ection and pull-in voltage of the nano-beam are computed.

The minimum initial gap and the detachment length of an actuator that does not stick to thesubstrate due to intermolecular attractions are determined. As a special case, the instability of

freestanding nano-electromechanical systems (NEMS) due to vdW attraction is investigated.

*Corresponding author.

International Journal of Structural Stability and DynamicsVol. 12, No. 5 (2012) 1250036 (18 pages)

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0219455412500368

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The MAD solutions are compared with the numerical ones and a proposed lumped model, as

well as models available from the literature.

Keywords: Van der Waals (vdW) force; modi¯ed Adomian decomposition (MAD); instability;

doubly-supported nano-beam; Cantilever nano-beam.

1. Introduction

In the past decade, beam-type nano-electromechanical systems (NEMS) have

become a common component used in developing the nano-actuators, nano-tweezers,

nano-switches, etc.1 A beam type NEMS consists of a conductive nano-electrode

suspended over a conductive substrate. Applying a voltage di®erence between the

electrode and the ground causes the electrode to de°ect and attract towards the

ground due to the presence of electrostatic forces. At the critical voltage, which is

known as the pull-in voltage, the electrode becomes unstable and pulls-in onto the

substrate.

When the initial gap between the components is very small, nano-scale phe-

nomena such as the dispersion forces and size dependency can signi¯cantly in°uence

the performance of nano-structures.1�4 Such e®ects have become the major interest

for many researchers.1�4 At sub-micrometer distances, the dispersion force between

two surfaces can be modeled via the intermolecular van der Waals (vdW) attraction

which is a function of the material properties and is proportional to the inverse cubic

power of the separation.5 Spengen et al.6 has studied the stiction in micro-switches

due to vdW forces and developed a model to predict the sensitivity to stiction. Batra

et al.7 has contemplated the pull-in behavior of micro-plates considering the vdW

attraction. Dequesnes et al.8 has calculated the e®ect of vdW intermolecular forces

on the instability of NEMS switches. Rotkin9 has obtained analytical expressions for

the e®ect of vdW forces on the pull-in gap and voltage of a nano-actuator. The

dynamic behavior of a nano-scale actuator has been investigated by Lin and Zhao10

considering the e®ect of the vdW force. Ke et al.11 have evaluated the vdW force

e®ect on the pull-in voltage of nano-tube-based NEMS. Wang et al.12 have investi-

gated the pull-in instability of nano-tweezers under the in°uence of vdW forces using

a two-parameter mass-spring model. The in°uence of vdW forces on the stability of

the electrostatic torsional NEMS actuators was analyzed by Guo and Zhao.13

Further information concerning the e®ect of vdW forces on the pull-in instability

of electromechanical systems and its modeling was available elsewhere.14�16 Rame-

zani et al.17 has applied a distributed parameter nonlinear model to studying the

instability of a cantilever NEMS due to the vdW attraction, using Green's function.

However, their method did not satisfy all the natural boundary conditions (B.C.) and

could not easily be applied to other geometries such as doubly-supported. These

shortcomings are overcome by using the modi¯ed Adomian decomposition (MAD).

In recent years, some promising approximation analytical methods such as the

homotopy perturbation,18,19 modi¯ed Adomian decomposition,20�23 etc., were pro-

posed to solve the nonlinear engineering problems. The potential of the homotopy

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perturbation method in analyzing the instability of cantilever beam-type NEMS

were investigated in previous works.24,25

In this study, the MAD is applied to studying the e®ect of the vdW attraction on

the instability of both cantilever and doubly-supported nano-beams using a dis-

tributed parameter model. The analytical solution obtained is compared with the

lumped model and numerical data as well as results reported in the literature.

2. Governing Equation

Figures 1(a) and 1(b) show a schematic diagram of the cantilever and doubly-sup-

ported actuators, respectively. In order to study the nano-structures, the molecular

dynamics (MD) approach has been employed. However, this method is very time-

consuming and cannot be easily used in complex structures with extremely large

number of atoms. An alternative reliable approach to simulate the instability of

(a)

(b)

Fig. 1. Schematic representation of the nano-beam: (a) Cantilever (b) Doubly-supported.

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nano-structures is to apply the nano-scale continuum models. In this study, the

nano-structure will be modeled using a nano-scale continuum model. The actuators

are modeled by a beam of length L with a uniform rectangular cross-section of width

w and thickness h suspended over a conductive substrate with an initial gap of g

between the movable and the ground electrode.

The electrostatic force per unit length of the beam, felec, enhanced with the ¯rst

order fringing correction, can be expressed as26:

felec ¼"0wV 2

2ðg� UÞ2 1þ 0:65ðg� UÞ

w

� �; ð1Þ

where U is the de°ection of the actuator,X is the distance from the clamped end, "0 ¼8:854� 10�12 C2N�1m�2 is the permittivity of vacuum and V is the applied voltage.

The e®ect of the vdW intermolecular force is considered for separations below

20 nm.26 The vdW force per unit length of the beam is5

fvdW ¼ Aw

6�ðg� UÞ3 ; ð2Þ

where A is the Hamaker constant. In reality, the intermolecular interaction between

the two surfaces highly depends on the dielectric properties of the surfaces and also

on the geometric parameters.27,28

To ¯nd the governing equation of the nano-beam, Hamilton's principle is applied,

which implies equilibrium when the free energy reaches its minimum value. There-

fore, one can obtain:

�W ¼ �ðWelas �Welec �WvdWÞ

¼ �1

2

Z L

0

EeffId2U

dX 2

!2

dX �Z L

0

felecUðXÞdX �Z L

0

fvdWUðXÞdX !

¼ 0; ð3Þ

where Welas is the elastic energy and Welec and WvdW are the works done by the

electrical and vdW forces, respectively. In the above equation, I and Eeff are the

moment of inertia of the beam cross-section and the e®ective beam material modulus,

respectively. Note that Eeff equals to Young's modulus E for narrow beams (w < 5 h)

and plate modulus E/(1�� 2) for wide beams (w > 5 h), where � is the Poisson ratio.29

As there are no de°ection and rotation at the ¯xed end and the bending moment

and shear force are absent at the free end of the beam, the boundary value problem

for the nano-beam can be de¯ned as:

EeffId4U

dX 4¼ felec þ fvdW; ð4aÞ

Uð0Þ ¼ dUð0ÞdX

¼ d2UðLÞdX 2

¼ d3UðLÞdX 3

¼ 0; for cantilever beam ð4bÞ

Uð0Þ ¼ dU

dXð0Þ ¼ UðLÞ ¼ dU

dXðLÞ ¼ 0: for doubly-supported beam ð4cÞ

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With the substitutions of u ¼ U/g and x ¼ X/L, Eq. (4) becomes:

d4u

dx4¼ �

ð1� uðxÞÞ3 þ�

ð1� uðxÞÞ2 þ��

ð1� uðxÞÞ ; ð5aÞ

uð0Þ ¼ u 0ð0Þ ¼ u 00ð1Þ ¼ u 000ð1Þ ¼ 0; for cantilever beam ð5bÞuð0Þ ¼ u 0ð0Þ ¼ uð1Þ ¼ u 0ð1Þ ¼ 0: for doubly-supported beam ð5cÞ

In the above equations, the dimensionless parameters �, � and � are de¯ned as

� ¼ AwL4

6�g4EeffI; ð6aÞ

� ¼ "0wV 2L4

2g3EeffI; ð6bÞ

� ¼ 0:65g

w: ð6cÞ

2.1. Limitations of the model

In order to achieve precise results, the limitations of the presented model should be

considered. It is noticeable that the constitutive material of the nano-cantilever is

assumed linear elastic and only the static de°ection of the nano-beam is taken into

account.

In the present model, the e®ect of curvature on de°ections, i.e. ¯nite kinematics, is

not considered and it should be corrected for short beams with L > 10g.30 However,

previous researches reveal that Eq. (5) has enough accuracy to model small to large

beam de°ections when the initial gap is less than the beam length (i.e. g/L < 1,

which is common in most practical cases).31

When the beam width w is su±ciently larger than g, Eq. (2) o®ers acceptable

results.27,32 In this study, only the cantilevers that are wider than the separation

(g/w � 1) are considered. Furthermore, the charges distribution and molecular force

are assumed uniform along the beam length. Therefore, further improvement of the

model requires inclusion of screening and quantum e®ects especially for modeling

semi-conducting MEMS33 or nano-tube-based NEMS.34

3. Lumped Model

The lumped model has been adopted by previous researchers for investigating the

instability of cantilever electromechanical systems.35 This model assumes the elec-

trostatic and intermolecular forces to be uniform along the beam. In this model, u is

identical to the tip de°ection, utip, for cantilevers and the mid-length de°ection, umid,

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for doubly-supported beams. Hence, � may be rewritten as17

� ¼ 8utipð1� utipÞ2 � �ð1� utipÞ�1

1þ �ð1� utipÞ; ð7Þ

Similarly, we can obtain Eq. (8) for the doubly-supported nano-beam

� ¼ 384umidð1� umidÞ2 � �ð1� umidÞ�1

1þ �ð1� umidÞ: ð8Þ

The pull-in parameters of NEMS can be obtained from Eqs. (7) and (8) by setting

d�/dutip ¼ 0 and d�/dumid ¼ 0, respectively, for given values of � and g/w. Simi-

larly, the instability point of the free-standing actuator can be determined by setting

� ¼ 0 and d�/dutip ¼ 0 (cantilever case) or d�/dumid ¼ 0 (doubly-supported case).

4. Modi¯ed Adomian Decomposition

In order to apply the MAD to study the pull-in behavior of NEMS, the transfor-

mation y ¼ 1� u is adopted to rewrite Eq. (5) into:

d4y

dx¼ � �

yðxÞ3 ��

yðxÞ2 ���

yðxÞ ; ð9aÞ

yð0Þ ¼ 1; y 0ð0Þ ¼ 0; y 00ð1Þ ¼ 0; y 000ð1Þ ¼ 0; for cantilever ð9bÞyð0Þ ¼ 1; y 0ð0Þ ¼ 0; yð1Þ ¼ 1; y 0ð1Þ ¼ 0: for doubly-supported ð9cÞ

Based on the MAD, the de°ection of the NEMS in Eq. (9) can be represented as23:

yðxÞ ¼X1n¼0

yn ¼ 1þ C1x2

2!þ C2x3

3!

� L�4 �X1n¼0

fn;3 þ �X1n¼0

fn;2 þ ��X1n¼0

fn;1

" #: ð10Þ

The di®erential operator L4 and corresponding inverse L�4 are de¯ned as

L4 ¼ d4

dx4ð Þ; L�4 ¼

Z x

0

Z x

0

Z x

0

Z x

0

ð Þdx dxdxdx: ð11Þ

In Eq. (10), the functions fn;k, which approximate the nonlinear term y�kn , are

determined through the MAD's polynomials23:

fn;k ¼Xnv¼1

Cðv;nÞhv;kðy0Þ; ðn > 0Þ; ð12aÞ

Cðv;nÞ ¼Xpi

Yvi¼1

1

q!yqipi ;

Xvi¼1

qipi ¼ n; 0 � i � n; 1 � pi � n� vþ 1

!; ð12bÞ

hv;kðy0Þ ¼dv

dy �0

½y�k0 Þ�; ð12cÞ

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where qi is the number of repetition of the ypi and the values of pi are selected from

the above range by combination without repetition. Expanding Eqs. (12) and sub-

stituting into Eq. (10), the recursive relations of Eq. (10) can be written as follows23:

y0 ¼ 1;

y1 ¼ C1x2

2!þ C2x3

3!þ L�4 �f0;3 þ �f0;2 þ ��f0;1

� �;

ynþ1 ¼ L�4½�fn;3 þ �fn;2 þ ��fn;1�:

ð13Þ

From the above recursive relations, the solution of Eq. (10) is obtained as:

y0 ¼ 1;

y1 ¼ 1

2!C1x

2 þ 1

3!C2x

3 � 1

4!ð�þ � þ ��Þx4;

y2 ¼ ð3�þ 2� þ ��Þ 1

6!C1x

6 þ 1

7!C2x

7 � 1

8!ð�þ � þ ��Þx8

� �;

y3 ¼ � C 21

8!ð36�þ 18� þ 6��Þx8 � C1C2

9!ð120�þ 60� þ 20��Þx9

þ 1

10!½ð3�þ 2� þ ��Þ2C1 þ ð�þ � þ ��Þ

� ð6�þ 3� þ ��Þð30C1 � 20C 22Þ�x10

þ 1

11!½ð3�þ 2� þ ��Þ2C2 þ 70C2ð�þ � þ ��Þð6�þ 3� þ ��Þ�x11:

ð14Þ

Accordingly, the solution of Eq. (5) can be summarized as:

uðxÞ ¼ � 1

2!C1x

2 � 1

3!C2x

3 þ 1

4!ð�þ � þ ��Þx4 � C1

6!ð3�þ 2� þ ��Þx6

� C2

7!ð3�þ 2� þ ��Þx7 þ 1

8!½6C 2

1ð6�þ 3� þ ��Þ

þ ð�þ � þ ��Þð3�þ 2� þ ��Þ�x8 þ 20C1C2

9!ð6�þ 3� þ ��Þx9

� 1

10!½ð3�þ 2� þ ��Þ2C1 þ ð�þ � þ ��Þ

� ð6�þ 3� þ ��Þð30C1 � 20C 22Þ�x10

� 1

11!½ð3�þ 2� þ ��Þ2C2 þ 70C2ð�þ � þ ��Þð6�þ 3� þ ��Þ�x11:

ð15Þ

where the constants C1 and C2 can be determined by solving the resulting algebraic

equations from the boundary conditions at x ¼ Lb, i.e. using Eqs. (5b) or (5c) for the

cantilever and doubly-supported beam-type NEMS, respectively.

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For any given values of �, � and �, Eq. (15) can be used to obtain the pull-in

parameters of the cantilever as well as the doubly-supported beam-type NEMS. The

instability of the cantilever and doubly-supported actuators occurs when

d�ðx ¼ 1Þ/du ! 0 and d�ðx ¼ 0:5Þ/du ! 0, respectively. The pull-in voltage of the

nano-beam can be determined by plotting � vs. u. For freestanding cantilever and

doubly-supported nano-beams, the critical de°ection can be obtained from the

condition d�ðx ¼ 1Þ/du ! 0 and d�ðx ¼ 0:5Þ/du ! 0, respectively.

5. Numerical Solution

In order to study the pull-in behavior of nano-structures, the boundary value pro-

blem (Eq. (5)) is solved numerically using the MAPLE commercial software. The

\dsolve" command has been applied to ¯nd a numerical solution for the ordinary

di®erential equation. The step size of the parameter variation is selected based on the

sensitivity of the parameter to the tip de°ection. Furthermore, the numerical results

are compared with those of the analytical MAD solutions as well as the lumped

parameter model in the following sections.

6. Veri¯cation

In order to verify the series solution obtained, typical beams with g/w ¼ 0:5 will be

numerically simulated with the results compared with those of the MAD. Table 1

shows the comparison between the numerical solution and MAD ones. Higher

accuracy can be obtained by including more terms in the series solution uðxÞ. Byusing six terms, the global error is less than 1%, which is within the excellent range

for most engineering applications. Therefore, six terms are selected in the following

section for the convenience of calculation and acceptable error.

To compare with the literature, the pull-in parameters of typical cantilever micro-

actuators were calculated. Geometrical dimensions and material properties of the

beams are listed in Table 2. A comparison between the pull-in voltages obtained by

MAD and those of the literature24,25,36�38 is presented in Table 3. As can be seen, the

error between the MAD and numerical results is within the range of those available

in the literature. Furthermore, Table 4 presents a comparison between the proposed

MAD solution, other reported methods17,39 and the experiment preformed by pre-

vious researchers,40 for which the length, width, thickness and initial gap are equal

20, 5, 0.057 and 0.092mm, respectively. As can be seen, the proposed solutions are

reliable in predicting the pull-in voltage of the electromechanical systems.

7. Results

7.1. Cantilever case

The in°uences of intermolecular attraction on the pull-in de°ection, uPI , and voltage,

�PI , of the cantilever NEMS are shown in Figs. 2 and 3, respectively. Note that uPI

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Tab

le1.

Variation

oftypical

NEMSde°ection

obtained

byMAD.TherelativeerrorbetweenMAD

andnumerical

solution

sisreportedin

(%).Theanalytical

solution

convergesto

thenumerical

solution

asthenumber

oftheselected

term

sincrease.

Case

Numerical

MAD

2Terms

Error

3Terms

Error

4Terms

Error

5Terms

Error

6Terms

Error

7Terms

Error

Can

tilever

(�¼

�¼

g/w¼

0:5)

0.20

620.14

5329

.53

0.18

5410

.09

0.20

003.01

0.20

460.78

0.20

580.19

0.2061

0.05

Dou

bly-supported

(�¼

�¼

20,g/w¼

0:5)

0.16

910.12

1128

.38

0.14

7412

.83

0.16

204.02

0.16

542.19

0.16

810.59

0.1684

0.41

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Fig. 2. E®ects of vdW force on uPI of cantilever NEMS for g/w ¼ 0 and 1.

Table 4. Pull-in voltage comparison for cantilever MEMS.

Pull-in voltage (V )

Present study

(MAD)

Green's

function [17]

Di®erential

quadrature [39]

Experimental

measurement [40]

Pull-I voltage (V) 71.8 73.7 66.4 68.5

Relative error

with experiment (%)

5 8 3 —

Table 2. Geometrical parameters and material properties of the nano-beamused in comparison.

Case Material Properties Geometrical Dimensions

E (GPa) � Lð�mÞ wð�mÞ hð�mÞ gð�mÞNarrow beam 77 0.33 300 0.5 1 2.5Wide beam 77 0.33 300 50 1 2.5

Table 3. Pull-in voltage comparison for the nano-beam with vdW force neglected.

Case Pull-in voltage (V )

Ref. 36 Ref. 37 Ref. 38 Ref. 17 Ref. 24, 25 Numerical MAD

Narrow beam 1.23 1.20 1.21 1.29 1.21 1.24 1.27

Wide beam 2.27 2.25 2.27 2.37 2.16 2.27 2.31

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corresponds to the tip de°ection of the cantilever NEMS at the onset of instability.

From the ¯gures, we observe that as the vdW force increases, the pull-in de°ection

and voltage of the actuators will be decreased. These ¯gures also show that as

the fringing ¯eld increases, the value uPI increases while the value �PI decreases.

Note that g/w ¼ 0 reveals an asymptotical limit for very wide MEMS with very

small gaps.

Neglecting intermolecular forces is a common practice in the literature for

microsystems. Figure 4 shows the variation of pull-in parameters of the cantilever

NEMS when the e®ects of vdW force is neglected, i.e., � ¼ 0. However, the inter-

molecular force cannot be neglected at sub-micrometer separations. As can be seen

from this ¯gure, the fringing ¯eld increases the cantilever tip pull-in de°ection and

reduces the pull-in voltage considerably, which therefore should not be neglected in

the analysis of MEMS instability.

The relation between the vdW force � and the NEMS tip de°ection, u, in the

absence of voltage di®erence is presented in Fig. 5. When the electrode/ground

separation is su±ciently small, the electrode may collapse onto the substrate due to

the vdW force. In other words, when � exceeds its critical value, �C , no solution

exists for u and instability occurs.

The maximum length Lmax of the electrode that stiction does not occur is called

the detachment length.24 Alternatively, for a known electrode length, there is a

minimum electrode/ground gap, gmin, to ensure that the electrode does not adhere to

the substrate as a result of intermolecular force.24 The detachment length and

minimum gap are basic design parameters and can be obtained by �C . The critical

Fig. 3. E®ects of vdW force on the �PI in cantilever NEMS for g/w ¼ 0 and 1.

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Fig. 5. Relationship between � and the tip de°ection, u, of the cantilever freestanding NEMS.

Fig. 4. E®ect of the fringing ¯eld on the pull-in parameters of cantilever NEMS neglecting vdW force

i.e. � ¼ 0.

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values of � for cantilever NEMS is:

�C ¼ 1:234: ð16Þ

Substituting the value of �C into the de¯nition of � (Eq. (6a)), the detachment

length and minimum gap are obtained as:

Lmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:602�EEffh3g4

A

4

r; gmin ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:660AL4

�EEffh3

4

s; ðNumerical solutionÞ ð17aÞ

Lmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:617�EEffh3g4

A

4

r; gmin ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:621AL4

�EEffh3

4

s; ðMAD methodÞ ð17bÞ

Lmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:421�EEffh3g4

A

4

r; gmin ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2:372AL4

�EEffh3

4

s: ðLumped modelÞ ð17cÞ

7.2. Doubly-supported case

Figures 6 and 7 show the relation between the vdW � and pull-in parameters (uPI

and �PI) of the doubly-supported actuator for di®erent � values. Note that uPI

corresponds to the mid-length de°ection of the doubly-supported nano-beam at the

onset of instability.

Fig. 6. E®ects of vdW force on uPI of doubly-supported actuator for g/w ¼ 0 and 1.

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Increasing the vdW attraction reduces the pull-in parameters. However, a com-

parison of Figs. 3 and 7 reveals that the e®ect of vdW force on the instability of

doubly-supported nano-beam is weak compared with that of the cantilever one. This

is due to the higher elastic sti®ness of the doubly-supported structure.

Figure 8 shows the variation of pull-in parameters of doubly-supported actuator

neglecting the intermolecular attractions, i.e. � ¼ 0. Comparing this ¯gure with

Fig. 4 reveals higher values of �PI are obtained for the doubly-supported NEMS

compared to the cantilever one. This is attributed to the higher elastic sti®ness of the

doubly-supported structure. Moreover, Figs. 4 and 8 show that when the separation

of the movable electrode from the ¯xed plane is comparable to its width, the fringing

¯eld becomes very signi¯cant and cannot be ignored.

The relation between the vdW force � and the NEMS mid-length de°ection, u, in

the absence of voltage di®erence is presented in Fig. 9. This ¯gure reveals that the

beam has initial de°ection due to the presence of intermolecular force even without

the presence of the voltage. Collapse occurs when vdW force reaches values greater

than its critical one (i.e. �C). This ¯gure and Fig. 5 show that MAD solution is very

close to that of numerical ones.

The value of �C for the doubly-supported beam-type NEMS is:

�C ¼ 50:750 ð18ÞA comparison of �C for the doubly-supported beam-type NEMS (Eq. (18)) with

that of the cantilever (Eq. (16)) reveals that stiction of the doubly-supported

actuator occurs at a lower electrode/substrate gap (g). Similar to what was

Fig. 7. E®ects of vdW force on �PI of doubly-supported actuator for g/w ¼ 0 and 1.

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Fig. 9. Relationship between � and the tip de°ection, u, of the doubly-supported freestanding NEMS.

Fig. 8. E®ect of fringing ¯eld on the pull-in parameters of doubly-supported NEMS neglecting vdW force

i.e. � ¼ 0.

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mentioned for Eq. (17), the detachment length and minimum gap of the doubly-

supported NEMS are obtained using relations (18) and (6a), namely,

Lmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi25:047�EEffh3g4

A

4

r; gmin ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0399AL4

�EEffh3

4

sðNumerical solutionÞ; ð19aÞ

Lmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi25:375�EEffh3g4

A

4

r; gmin ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0394AL4

�EEffh3

4

sðMAD methodÞ; ð19bÞ

Lmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi20:256�EEffh3g4

A

4

r; gmin ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0494AL4

�EEffh3

4

sðLumped modelÞ: ð19cÞ

The results reveal that the MAD solutions overestimate the values of uPI and �PI

for the NEMS. In order to achieve more accurate results, one can consider more terms

in the series expansion of Eq. (15).

8. Conclusions

In this article, the modi¯ed Adomian decomposition has been applied to investi-

gating the e®ect of van der Waals forces on the nonlinear pull-in behavior of the

cantilever and doubly-supported actuators. The results reveal that as the inter-

molecular force increases, the pull-in de°ection and voltage of the actuators will be

decreased. Furthermore, the critical value of the intermolecular force at the onset of

instability for the freestanding nano-beam was calculated. It is found that the e®ect

of van der Waals force on the instability of the doubly-supported beam-type actuator

is weak compared with that of the cantilever due to higher elastic sti®ness of the

former. The analytical solution was also compared with the result of the lumped

parameter model and numerical solution as well as those published in the literature.

The proposed modi¯ed Adomian decomposition has signi¯cantly overcome the

shortcomings of the lumped model and produced results with relatively small errors

compared with those reported in the literature. For this reason, it is suitable for use

in parametric studies.

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