MODELING THE INFLUENCE OF SURFACE EFFECT ON INSTABILITY OF NANO-CANTILEVER IN PRESENCE OF VAN DER...

MODELING THE INFLUENCE OF SURFACE

EFFECT ON INSTABILITY OF NANO-CANTILEVER

IN PRESENCE OF VAN DER WAALS FORCE

ALI KOOCHI*, HOSSEIN HOSSEINI-TOUDESHKY

and HAMID REZA OVESY

Department of Aerospace EngineeringAmirkabir University of Technology, Tehran, Iran

*[email protected]

MOHAMADREZA ABADYAN

Mechanical Engineering Group, Shahrekord Branch

Islamic Azad University, Shahrekord, Iran

Received 28 February 2012

Accepted 12 June 2012

Published 2 April 2013

Surface e®ect often plays a signi¯cant role in the pull-in performance of nano-electromechanical

systems (NEMS) but limited works have been conducted for taking this e®ect into account.

Herein, the in°uence of surface e®ect has been investigated on instability behavior of cantilever

nano-actuator in the presence of van der Waals force (vdW). Three di®erent methods, i.e. ananalytical modi¯ed Adomian decomposition (MAD), Lumped parameter model (LPM) and

numerical solution have been applied to solve the governing equation of the system. The results

demonstrate that surface e®ect reduces the pull-in voltage of the system. Moreover, surface

energy causes the cantilever nano-actuator with the assigned parameter to de°ect as a softerstructure. It is found that while surface e®ect becomes important for low values of the cantilever

nano-actuator thickness, vdW attraction is signi¯cant for low initial gap values. Surprisingly,

the increase in the initial gap, enhances the contribution of surface e®ect in pull-in instability of

the system while reduces the contribution of vdW attraction. Furthermore, the minimum initialgap and the detachment length of the cantilever nano-actuator that does not stick to the

substrate due to vdW force and surface e®ect has been approximated. A good agreement has

been observed between the values of instability parameters predicted via these three methods.Whilst compared to the instability voltage predicted by numerical solution, the pull-in voltage

obtained by MAD series and LPM method is overestimated and underestimated, respectively.

Keywords: Surface e®ect; van der Waals (vdW) force; instability; modi¯ed Adomian decom-

position (MAD); lumped parameter model (LPM).

*Corresponding author.

International Journal of Structural Stability and DynamicsVol. 13, No. 4 (2013) 1250072 (19 pages)

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0219455412500721

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http://dx.doi.org/10.1142/S0219455412500721

1. Introduction

Recently, beam type electrostatic nano-actuators have become one of the common

components in developing nano-electromechanical systems (NEMS) such as accel-

erometer, nano-tweezers, and nano switches.1�3 Figure 1 shows a typical beam type

nano-actuator, which is constructed from a movable conducting electrode suspend-

ing over a ¯xed conductive substrate. The application of a voltage di®erence between

the electrode and the ground causes de°ection of the movable one toward the ground.

If the voltage exceeds its critical value (pull-in voltage) the moveable electrode

becomes unstable and pulls-in onto the ground.

Since the geometrical characteristics of NEMS are often of the order of several

nano-meters, the presence of intermolecular van der Waals (vdW) attraction can

highly a®ect the pull-in performance of the system. The source of vdW attraction is

intermolecular force acting between adjacent bodies. This force can cause both

(a)

(b)

Fig. 1. (a) Schematic representation of a cantilever nano-actuator considering the surface layer (b) cross

section of cantilever nano-actuator.

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release-related and in-use stictions in nano-devices. The release-related stiction is

more crucial in manufacturing nano-structures that are made via a top-down wet

etching process. Moreover, the presence of vdW force can in°uence the performance

of nano-structures during its operation (in-use stiction). This might occur in several

operating conditions, i.e. when an AFM probe scans solid surfaces, a sensor is

placed in the vicinity of liquid surfaces, and the inertia force causes the ¯ngers of

comb-type nano-accelerometers to approach each other. Useful information about

the in°uence of vdW force on occurrence of in-use stiction problems in nano-struc-

tures can be found in Ref. 3. The pull-in instability of NEMS in the presence of

vdW attraction was analytically studied by researchers.4�9 Rotkin4 obtained ana-

lytical relation to express the e®ect of vdW forces on the pull-in voltage and pull-in

gap of a nano-actuator. Spengen et al.5 studied the stiction in MEMS due to the vdW

forces. They developed a model to predict the sensitivity to stiction. The dynamic

behavior of a nanoscale electrostatic actuator was investigated by Lin and Zhao6

considering the e®ect of the vdW force. They used a two-parameter mass-spring

model. In°uence of vdW force on pull in instability of nano-actuators was demon-

strated in Refs. 7 and 8. Dequesnes et al.9 calculated the e®ect of the vdW inter-

molecular force on the instability voltage of carbon-nanotube-based NEMS switches.

Soroush et al.10 studied the e®ect of vdW force on the instability of cantilever and

doubly cantilever nano-actuators by using modi¯ed Adomian decomposition (MAD)

method.

Along with the molecular bonding force, the in°uence of surface e®ects, i.e. the

surface residual stress and surface sti®ness, are important in modeling a nano-sized

structure due to its large value of surface area to volume ratio. It may play a crucial

role in pull-in performance of NEMS actuators however all the above-mentioned

researchers have ignored this e®ect. Gurtin and Murdoch11,12 developed a continuum

theory to model both residual surface stress and surface elasticity. This theory has

been widely applied to investigate the surface e®ects on elastic behavior of beam-type

nanostructures.13�16 Wang and Feng13 investigated the buckling of nano-beams

considering the e®ects of surface elasticity and surface residual stresses. He and

Lilley14 studied the static bending of nano-beams incorporating the surface e®ects.

Fu and Zhang15 investigated the pull-in behavior of an electrically actuated double-

clamped nano-bridge incorporating surface e®ects. The instability of electrostatic

nano switch in the presence of Casimir force and surface energy was studied by

Ma et al.16

To the best knowledge of the authors, the pull-in behavior of cantilever nano-

actuator incorporating both in°uences of surface e®ects and vdW force has not been

investigated yet. Hence in this work, the Euler�Bernoulli beam model is applied to

investigate the in°uence of these nano-scale phenomena on electromechanical

behavior of beam-type nano-actuator. Modi¯ed Adomian decomposition (MAD) is

employed to solve the nonlinear governing equation of the system. Moreover, a

lumped parameter model (LPM) is developed to simply explain the physical

Modeling the In°uence of Surface E®ect on Instability of Nano-Cantilever

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in°uence of nano-scale e®ects on pull-in performance of the system. The obtained

results are veri¯ed by comparing with those from numerical solution.

2. Theory

Figure 2 shows a free-body diagram of an incremental beam element of length dx.

The variables M , N and Q are bending moment, shear force and axial force,

respectively. According to the interaction between the surface layer and bulk

material, the contact tractions exist on the interface between the bulk material and

surface layer. The contact tractions on the element surface can generally be de¯ned

as follow:

Ti ¼ �ijnj: ð1Þ

Because of plane stress conditions in the element, the values of Tx and Tz are only

nonzero. Considering the force and moment equilibriums, one can obtain:

dQ

dxþZS

Tzdsþ qðxÞ ¼ 0; ð2aÞ

dM

dx�ZS

Txzds�Ndw

dxþQ ¼ 0; ð2bÞ

where S is the boundary of the cross-section and qðxÞ is the transverse distributed

load along the element. According to continuum theory proposed by Gurtin and

Murdoch,11 the relations between surface layer stresses and its contact tensions are:

d� ijdj

¼ Ti ði ¼ x;m;n & j ¼ x;mÞ; ð3Þ

where x is the beam length direction, n is the direction normal to the surface andm is

the direction tangent to the surface. By substituting the values of Q from Eq. (2b) in

Fig. 2. Free body diagram of an incremental element of the cantilever beam.

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Eq. (2a) and using Eq. (3) one can obtain:

d2M

dx2� d

dx

ZS

d�xxdx

zds� d

dxN

dw

dx

� ��ZS

d�nxdx

nzds� qðxÞ ¼ 0; ð4Þ

where nz is the projection of normal vector (to the surface) in the z direction. The

constitutive equations of the bonding surface is expressed as11,12:

�xx ¼ �0 þ E0

dux

dx; ð5aÞ

�nx ¼ �0dun

dx; ð5bÞ

where �0 is the residual surface stress and E0 is the surface elastic modulus.

The beam displacement ¯eld (including the surface layer) can be de¯ned as:

ux ¼ uðxÞ � zdwðxÞdx

; ð6aÞuz ¼wðxÞ: ð6bÞ

Considering no axial force and substituting Eq. (6) and Eq. (5) in Eq. (4), the fol-

lowing equation is obtained:

EI þ E0

ZS

z2ds

� �d4w

dx4¼ �0

ZS

n2zds

� �d2w

dx2þ qðxÞ: ð7Þ

Now, the cantilever nano-actuator shown in Fig. 1 is modeled by a cantilever beam of

length L with a uniform cross-section of thickness t and width b. Considering the

electrical and van der Waals forces, Eq. (7) can be rewritten as:

ðEIÞEffd4w

dx4¼ 2b�0

d2w

dx2þ felec þ fvdW : ð8Þ

With the following boundary conditions:

wð0Þ ¼ dw

dxð0Þ ¼ 0; at x ¼ 0;

d2w

dx2ðLÞ ¼ d3w

dx3ðLÞ ¼ 0; at x ¼ L:

ð9Þ

In the above equations the e®ective bending rigidity of the beam, (EI )eff , is de¯ned as:

ðEIÞeff ¼ EI þ 1

2E0bt

2 þ 1

6E0t

3: ð10Þ

On the right side of Eq. (8), felec and fvdW are the electrostatic and vdW forces per unit

length of the beam, respectively. Considering the ¯rst order fringing ¯eld correction,

the electrostatic force per unit length of the beam is de¯ned as17,18:

felec ¼"0bV 2

2ðh� wÞ2 1þ 0:65ðh� wÞ

b

� �; ð11Þ


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where "0 ¼ 8:854� 10�12 c2/Nm2, is the permittivity of vacuum, V is the applied

external voltage and h is the initial gap between the movable and the ground

electrode.

The e®ect of vdW molecular force is considerable at submicron separations.19 The

vdW force per unit length of the beam can be simpli¯ed as20:

fvan ¼ Ab

6�ðh� wÞ3 ; ð12Þ

where A is the Hamaker constant.

Equations (8) and (9) can be made dimensionless using the following

substitutions,

w ¼w=h; ð13aÞx ¼ x=L; ð13bÞ

� ¼ AbL4

6�h4ðEIÞeff; ð13cÞ

� ¼ "0bV 2L4

2h3ðEIÞeff; ð13dÞ

� ¼ 0:65h

b; ð13eÞ

� ¼ 2� 0bL2

ðEIÞeff: ð13fÞ

In the foregoing relations, �, � and � interpret the dimensionless values of applied

voltage, surface e®ect and vdW attraction. The substitution of these transformations

in Eq. (8) yields:

d4w

dx 4¼ �

d2w

dx 2þ �

ð1� wðxÞÞ3 þ�

ð1� wðxÞÞ2 þ��

ð1� wðxÞÞ ; ð14aÞ

wð0Þ ¼ w 0ð0Þ ¼ 0; at x ¼ 0; ð14bÞ

w 00ð1Þ ¼ w 000ð1Þ ¼ 0; at x ¼ 1: ð14cÞIn the above relations, prime denotes di®erentiation with respect to x. For con-

venience, superscript^ is eliminated in the following relations.

3. Solution Methods

3.1. Modi¯ed adomian decomposition (MAD)

The basic idea of MAD is explained in Ref. 21. In order to apply MAD, the boundary

value problem is solved using an in¯nite converged series. The details of the method

and mathematical computations are explained in Appendix A. Brie°y, the analytical

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MAD solution of Eq. (14) can be obtained as follows:

wðxÞ ¼ � 1

2!C1x

2 � 1

3!C2x

3 þ 1

4!ðð�þ � þ ��Þ � C1�Þx4

� 1

5!C2�x

5 � 1

6!ðð3�þ 2� þ ��ÞC1 þ �ð�þ � þ ��ÞC1�

2Þx6

� 1

7!C2ðð3�þ 2� þ ��Þ þ �2Þx7 þ � � � ; ð15Þ

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

equation from the boundary conditions at x ¼ L i.e. using Eq. (14c). For any given

�, �, � and �, Eq. (15) can be used to obtain the pull-in parameters of the cantilever

nano-actuator. The instability in Eq. (15) occurs when d�ðx ¼ 1Þ=dw ! 0. The pull-

in voltage of the system can be determined via plotting the w versus �.

3.2. Lumped parameter model (LPM )

A lumped parameter model simpli¯es the behavior of a distributed system into

discrete elements under acceptable assumptions. Although simple LPM may not

provide accurate values, it is very useful to understand physical aspects of the

phenomena. In this regards, a lumped parameter model is developed in Appendix B.

According to the proposed LPM, the relation between the dimensionless beam tip

de°ection, wtip, and the applied voltage, �, can be rewritten as

� ¼ �2wtipð1� wtipÞ2 � �ð1� 0:5� þ ffiffiffi�

psinh

ffiffiffi�

p � coshffiffiffi�

p Þð1� wtipÞ�1

ð1� 0:5� þ ffiffiffi�

psinh

ffiffiffi�


p Þð1þ �ð1� wtipÞÞ: ð16Þ

The pull-in parameters of the cantilever nano-actuator actuator can be obtained

from Eq. (16) by setting d�=dwtip ¼ 0.

3.3. Numerical solution

In order to verify the analytical results, the cantilever nano-actuator is numerically

simulated and the results are compared with those obtained via MAD and LPM. The

nonlinear governing di®erential equation (Eq. (14)) is solved with the boundary value

problem solver of MAPLE commercial software. The dsolve command is applied to

solve the boundary value problems. A ¯nite di®erence technique using Richardson

extrapolation algorithm is used to ¯nd the numerical solution. The step size of the

parameter variation is chosen based on the sensitivity of the parameter to the tip

de°ection. The pull-in parameters can be determined via the slope of the w-� graphs.

4. Results and Discussion

In order to validate the proposed model and to compare with results in the literature,

the pull-in voltage of typical cantilever micro-actuators (� ¼ � ¼ 0) is calculated in

this section. A comparison between the pull-in voltage values obtained by present

model and those reported in the literature22�24 is presented in Table 1. This table


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reveals that the di®erence between obtained values is within the range of those of

other methods presented in the literature.

As an example for case study, typical silver-made cantilever nano-actuators are

investigated in the following sub-sections. For (001) silver, the Young's modulus E,

surface stress � 0 and surface modulus E0 are 76GPa, 0.89N/m and 1.22N/m,

respectively.14 The value of Hamaker constant is A ¼ 3:5� 10�19J.

4.1. Softening e®ect of surface layer

Figure 3 shows the variation of de°ection of a typical cantilever nano-actuator when

the applied voltage increases from zero to pull-in value. The geometrical character-

istics of the cantilever nano-actuator i.e. width, thickness, length and initial gap are

250, 35, 1,000 and 50 nm, respectively. As seen from this ¯gure, surface e®ect

increases the de°ection of the nano-actuator for a given applied voltage value. It

should be noted that while e®ect of surface modulus usually leads to a hardening

behavior of the material, surface stress causes the system to deform softer. However,

since the thickness of the surface layer is negligible, the surface modulus cannot

considerably increase the bending rigidity and the softening behavior is prominent.

Table 1. Pull-in voltage comparison for cantilever beam with � ¼ � ¼ 0, E ¼ 77GPaand � ¼ 0:33. The length, thickness and initial gap are 300, 1 and 2.5 m, respectively.

The width for narrow and wide cases are 0.5 and 50 m, respectively.

Case Pull-in voltage (V )

Ref. 22 Ref. 23 Ref. 24 Numerical MAD

Narrow beam 1.20 1.21 1.29 1.24 1.27

Wide beam 2.25 2.27 2.37 2.27 2.31

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Neglecting surface effectConsidering surface effect

w

x

V=

Pull-in

2.737

2

0

(a)

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


w

x

Pull-in

V= 0

2

3.211

(b)

Fig. 3. In°uence of surface e®ect on cantilever nano-actuator tip de°ection for di®erent values of applied

voltage (t ¼ 35 nm, h ¼ 50 nm) (a) Numerical (b) MAD, and (c) LPM.

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This softening e®ect becomes more signi¯cant by further increasing the de°ection of

the nano-actuator and concomitant curvature enhancement.

Note that the softening e®ect is valid in this case: (001) silver-made structure with

a positive surface modulus. The surface layer/modulus might behave di®erent under

di®erent conditions. In the cases of materials with negative surface modulus (such as

(111) silver), one may obtain opposite result, i.e. surface layer may provide a stif-

fening e®ect. Generally, considering the size dependency of material characteristics

and nonclassic continuum theories such as modi¯ed coupled stress, and strain gra-

dient might cause totally di®erent results.

4.2. In°uence of geometrical parameters

Figure 4 shows the in°uence of the nano-scale phenomena on pull-in voltages (VPI)

as a function of initial gap. As seen, the consideration of surface energy and/or vdW

attraction decreases the pull-in voltages of the nano-actuator. Increasing the initial

gap reduces the in°uence of vdW attraction due to the inverse cubic dependency of

vdW force to the distance. On the other hand, enhancing h increases the surface

e®ect as the result of enhancing the curvature at higher h values. Figure 4 reveals

that while vdW force is the dominant factor for low values of initial gap, the surface

energy is prominent in high gap values.

In order to better illustrate the importance of incorporating vdW force and sur-

face e®ect in pull-in models, the computational error due to neglecting nano-scale

phenomena are presented in Fig. 5. The vertical axis of this ¯gure reveals the dif-

ference between pull-in voltages (�VPI) computed incorporating nano-scale phe-

nomena and those calculated neglecting the phenomena. As seen, the interaction

between the surface energy and vdW force causes a decreasing�increasing trend in

computational error if both e®ects are ignored.

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


w

x

Pull-in

V= 0

2

2.399

(c)

Fig. 3. (Continued)


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10 15 20 25 30 35 40 45 500

1

2

3

4Neglecting surface effect and vdW forceConsidering vdW force and neglecting surface effectConsidering surface effect and neglecting vdW forceConsidering surface effect and vdW force

VPI (Volt)

h (nm)

(a)

10 15 20 25 30 35 40 45 500

1

2

3


VPI (Volt)

h (nm)

(b)

10 15 20 25 30 35 40 45 500

1

2

3


VPI (Volt)

h (nm)

(c)

Fig. 4. In°uence of surface e®ects and vdW force on pull-in voltage for varying initial gap (t ¼ 35 nm) (a)

Numerical, (b) MAD, and (c) LPM.

20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Neglecting surface effect and vdW foreNeglecting surface effectNeglecting vdW force

∆VPI

(Volt)

h (nm)

(a)

20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3Neglecting surface effect and vdW foreNeglecting surface effectNeglecting vdW force

h (nm)

∆VPI

(Volt)

(b)

Fig. 5. Variation of computational error due to neglecting nano-scale phenomena. (a) Numerical, (b)

MAD, and (c) LPM.

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The variation of the pull-in voltage of the cantilever nano-actuator is demon-

strated in Fig. 6 as a function of the beam thickness. As seen from this ¯gure,

decreasing the actuator thickness results in a decrease in the pull-in voltage of the

system. Interestingly, this ¯gure reveals that the e®ect of surface energies on pull-in

performance of thin nano-actuator is more profound in comparison with thick

nano-actuator. Note that in this case, the e®ect of vdW force on instability voltage is

not important due to the large value of initial gap (h ¼ 50 nm).

4.3. E®ect of surface layer on detachment length and minimum

gap of the cantilever nano-actuator

The variation of dimensionless pull-in voltage of the nano-actuator as a function of �

has been demonstrated in Fig. 7 for two di®erent h=b values. This ¯gure reveals that

in the presence of vdW forces, pull-in voltage decreases with increasing the surface

e®ect. The points of intersection between the curves and the horizontal axis (�)

correspond to the critical value of vdW force i.e. �C . Once the value of vdW

attraction exceeds its critical value, the cantilever nano-beam adheres to the ground

even without applying any voltage di®erence. Note that �C determines the minimum

beam/ground separation (minimum gap) and maximum permissible length

(detachment length) of freestanding cantilever nano-actuator that guarantees no

occurrence of stiction due to the presence of vdW attraction.

Figure 8 shows the in°uence of surface e®ects on dimensionless instability voltage

of the system in the presence of vdW attraction for h=b ¼ 1. As seen in this ¯gure, the

pull-in voltage of the cantilever nano-actuator decreases with increasing the surface

e®ect. Similarly, Fig. 8 reveals that the presence of vdW forces reduces the instability

voltage of the nano-actuator. The boundary line in Fig. 8 presents the critical values

of the vdW force (�C). Figure 8 also shows that increasing the � values leads to

decreasing the critical value of vdW force (�CÞ). This interestingly implies that

20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Neglecting surface effect and vdW foreNeglecting surface effectNeglecting vdW force

h (nm)

∆VPI(Volt)

(c)

Fig. 5. (Continued)


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the surface e®ect decrease the detachment length while increase the minimum gap

of freestanding actuator. Substituting the values of �C into the de¯nition of �

(Eq. (13c)); the detachment length, Lmax, and minimum gap, hmin, are obtained as:

Lmax ¼h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6�ðEIÞeff�C

Ab

4

r; ð17aÞ

hmin ¼L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAb

6�ðEIÞeff�C

4

s; ð17bÞ

where �C can be approximated from the boundary line in Fig. 8 as the following

relations:

�c ¼ 1:203� 0:077�; Numerical ð18aÞ�c ¼ 1:231� 0:057�; MAD ð18bÞ�c ¼ 0:0842� 0:043�: LPM ð18cÞ

35 40 45 50 55 60 65 70 75 80 85 90 95 1000

2

4

6

8

10

12

14

16

18


VPI (Volt)

t (nm)

(a)

35 40 45 50 55 60 65 70 75 80 85 90 95 1000

2

4

6

8

10

12

14

16

18


VPI (Volt)

t (nm)

(b)

35 40 45 50 55 60 65 70 75 80 85 90 95 1000

2

4

6

8

10

12

14

16

18

20

Neglecting surface effect and vdW forceConsidering vdW force and neglecting surface effectConsidering surface effect and neglecting vdW forceConsidering surface effect and vdW force

VPI (Volt)

t (nm)

(c)

Fig. 6. In°uence of surface e®ects on pull-in voltage of cantilever nano-actuator for varying thickness

values (h ¼ 50 nm). (a) Numerical, (b) MAD, and (c) LPM.

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0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Numerical, η=0Numerical, η=2MAD, η=0MAD, η=2LPM, η=0LPM, η=2

βPI

α

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Numerical, η=0Numerical, η=2MAD, η=0MAD, η=2LPM, η=0LPM, η=2

βPI

α

(b)

Fig. 7. E®ect of vdW force (�) on pull-in voltage with and without considering surface residual stress(a) h=b ¼ 0:1 (b) h=b ¼ 1. In this ¯gure, � ¼ 0 corresponds to neglecting the surface e®ect.


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It should be noted that this linear approximation is valid only for low � values and

for higher � values, a nonlinear trend is obtained.

Results reveal that pull-in voltage obtained by MAD series is much closer to the

numerical value in comparison with the results provided by LPM. The presented

¯gures show that while MAD series overestimates the instability voltage, the

LPM underestimates the pull-in voltage of the system. Comparing analytical MAD

with LPM reveals that MAD overcomes the low precision of LPM in determining

instability voltage. On the other hand, the LPM has the advantage of providing

simple closed-form approximation formula that is useful for engineers and

designers.

(a) (b)

(c)

Fig. 8. E®ect of vdW force (�) and surface e®ect (�) on pull-in voltage of cantilever nano-actuator forh=b ¼ 1, (a) Numerical, (b) MAD, and (c) LPM.

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5. Conclusions

In this study, the in°uence of surface layer on pull-in performance of cantilever nano-

actuator was investigated incorporating the e®ect of vdW force. The nonlinear

governing equation was solved using three di®erent approaches, i.e. using analytical

MAD, lumped parameter model and numerical solution. For a (001) silver-made

structure with a positive surface modulus, it was found that:

. Surface e®ect provides a softening behavior on electromechanical response of the

cantilever nano-actuator. For a given applied voltage, incorporating of the surface

e®ect in calculating the beam de°ection leads to a lower value of computed

de°ection. Furthermore, the surface e®ect reduces the pull-in voltage of cantilever

nano-actuator due to the softening e®ect. The reduction is more prominent for thin

nano-actuators in comparison with thick actuators. Note that this deteriorating

e®ect becomes more signi¯cant when the initial gap is increased.

. Increasing the actuator's initial gap diminishes the vdW attraction but enhances

the surface e®ect. The interaction between the surface e®ect and molecular force

causes a decreasing–increasing trend in computational error when the pull-in

voltage is calculated with neglecting these nano-scale e®ects.

. Enhancing the surface e®ect leads to decreasing the critical value of vdW force.

This implies that the surface e®ect decreases the detachment length while

increases the minimum gap of a freestanding nano-actuator.

. While analytical MAD solution overestimates the instability voltage, the proposed

LPM underestimates the pull-in voltage of the system. Comparing analytical

MAD with LPM reveals that MAD overcomes the low precision of LPM. On the

other hand, the LPM has the advantage of providing simple closed-form

approximation for engineers and NEMS designers.

Acknowledgment

The authors are greatly thankful to Professor Randolph Rach for his kind attention

and valuable comments on this study.

Appendix A: Modi¯ed Adomian decomposition (MAD) method

In order to solve Eq. (16) by MAD analytical method, consider the following fourth-

order boundary-value problem:

yð4ÞðxÞ ¼ fðx; yÞ þ �yð2ÞðxÞ; 0 � x � Lb; ðA:1aÞyð0Þ ¼�0; y 0ð0Þ ¼ �1: ðA:1bÞ

Employing MAD method,25,26 the dependent variable in Eq. (A.1) is written as:

yðxÞ ¼X1n¼0

ynðxÞ: ðA:2Þ


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According to Ref. 27 and using the relations (A.1) and (A.2), the following

recursive equations can be provided:

y0ðxÞ ¼ �0;

y1ðxÞ ¼ �1xþ 1

2C1x

2 þ 1

3!C2x

3

þZ x

0

Z x

0

Z x

0

Z x

0

A0 þ �d2

dx2½y0�

� �dx dx dx dx;

ynþ1ðxÞ ¼Z x

0

Z x

0

Z x

0

Z x

0

An þd2

dx2½yn�

� �dx dx dx dx:

ðA:3Þ

In the above relations the coe±cient Ak is determined from the nonlinear part

of the function fðx; yÞ. It can further be presented as the following convenient

equations26,27:

An ¼Xnv¼1

Cðv;nÞhvðf0Þ; ðA:4Þ

where

Cðv;nÞ ¼Xpi

Yvi¼1

1

k!f kipi ; ðA:5Þ

and ki is the number of repetition in the fpi, the values of pi are selected from the

following range by combination without repetition,

Xvi¼1

kipi ¼ n; n > 0; 0 � i � n; 1 � pi � n� vþ 1: ðA:6Þ

In Eq. (A.4), hvðf0Þ is calculated by di®erentiating the nonlinear terms of f , v

times with respect to g at ¼ 0, and can be represented as follows:

hvðf0Þ ¼dv

dgv½fðÞ�¼0: ðA:7Þ

Now by using relations (A.7), the series terms yn are obtained from recursive

relations (A.3) as following:

y0 ¼ 1

y1 ¼1

2!C1x

2 þ 1

3!C2x

3 � 1

4!ð�þ � þ ��Þx4

y2 ¼1

4!C1�x

4 þ 1

5!C2�x

5 þ 1

6!ðð3�þ 2� þ ��ÞC1 þ �ð�þ � þ ��ÞÞx6

þ ð3�þ 2� þ ��Þ 1

7!C2x

7 � 1

8!ð�þ � þ ��Þx8

� �

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y3 ¼1

6!C1�

2x6 þ 1

7!C2�

2x7 � 1

8!ð�2ð�þ � þ ��Þ

þ 2�C1ð3�þ 2� þ ��Þ þ C 21ð36�þ 18� þ 6��ÞÞx8

� 1

9!ðC1C2ð120�þ 60� þ 20��Þ þ 2�C2ð3�þ 2� þ ��ÞÞx9

þ 1

10!½2�ð3�þ 2� þ ��Þð�þ � þ ��Þ þ ð3�þ 2� þ ��Þ2C1

þ ð�þ � þ ��Þð30C1ð6�þ 3� þ ��Þ � 20C 22Þ�x10

þ 1

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

� 1

12!½ð3�þ 2� þ ��Þ2ð�þ � þ ��Þ þ 1680ð�þ � þ ��Þ2ð6�þ 3� þ ��Þ�x12

� � � ðA:8ÞAnd ¯nally the solution of Eq. (16) by substituting Eq. (A.8) in (A.2) and sub-

stituting y ¼ 1� w can be summarized to (15).

Appendix B: Developing a lumped parameter model (LPM)

In order to develop a simple lumped parameter model, the nano-cantilever actuator

shown in Fig. 1 is replaced by a one-dimensional simple structure (Fig. B.1). The

structure is constructed from a linear spring with sti®ness of K . The model assumes

uniform force distribution (q) along the beam. In order to determine the elastic

sti®ness of the cantilever nano-actuator in the presence of surface e®ect, consider

a uniform load distribution (q ¼Constant) being applied to the beam. Hence, the

governing equation of the beam is obtained as

ðEIÞeffd4w

dx4� 2�0b

d2w

dx2¼ q:

wðoÞ ¼ w 0ð0Þ ¼ 0:

w 00ðLÞ ¼ w 000ðLÞ ¼ 0:

ðB:1Þ

Fig. B.1. Schematic representation of the lumped parameter model for a cantilever nano-actuator.


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The general solution of Eq. (B.1) can be determined as:

w ¼ C1e

ffiffiffiffiffiffiffiffiffi2�0b

ðEIÞeff

qx þ C2e

�ffiffiffiffiffiffiffiffiffi2�0b

ðEIÞeff

qx þ C3xþ C4 �

q

4�0bx2; ðB:2Þ

where C1, C2, C3 and C4 can be de¯ned using boundary conditions.

The elastic sti®ness of the structure, K , is determined from Eq. (B.1) as the

following:

K ¼ qL

wðx ¼ LÞ ¼ðEIÞeff�2

L3ð1� 0:5� þ � sinhffiffiffi�


p Þ : ðB:3Þ

Considering that the LPM simulates only the tip de°ection of the beam (wtip) and

by using relations (13), (14) and (B.3), the following dimensionless relation is

obtained for explaining the electromechanical behavior of the proposed LPM:

Kwtip ¼�

ð1� wtipÞ3þ �

ð1� wtipÞ2þ ��

ð1� wtipÞ; ðB:4Þ

where

K ¼ �2

1� 0:5� þ ffiffiffi�

psinh

ffiffiffi�


p : ðB:5Þ

Equation (B.4) can be easily rewritten to the new form of relation (16).

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