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, [email protected]
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.
Modeling the E®ect of Van Der Waals Attraction
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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.
Modeling the E®ect of Van Der Waals Attraction
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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.
Modeling the E®ect of Van Der Waals Attraction
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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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