Casimir force-induced instability in freestanding nanotweezers and nanoactuators made of cylindrical...

May 20, 2014 9:39 WSPC/Guidelines-IJMPB S021797921450129X

International Journal of Modern Physics BVol. 28 (2014) 1450129 (25 pages)c© World Scientific Publishing Company

DOI: 10.1142/S021797921450129X

Casimir force-induced instability in freestanding nanotweezers

and nanoactuators made of cylindrical nanowires

Amin Farrokhabadi

Department of Aerospace Engineering, Semnan University, Semnan, Iran

[email protected]

Naeimeh Abadian

Physics Department, Qom Branch, Islamic Azad University, Qom, Iran

[email protected]

Faramarz Kanjouri

Physics Department, Faculty of Science, Kharazmi University, Karaj, Iran

[email protected]

Mohamadreza Abadyan∗

Shahrekord Branch, Islamic Azad University, Shahrekord, Iran

[email protected]

Received 25 December 2013Revised 3 April 2014Accepted 8 April 2014Published 20 May 2014

The quantum vacuum fluctuation i.e., Casimir attraction can induce mechanical in-stability in ultra-small devices. Previous researchers have focused on investigating theinstability in structures with planar or rectangular cross-section. However, to the bestknowledge of the authors, no attention has been paid for modeling this phenomenon inthe structures made of nanowires with cylindrical geometry. In this regard, present workis dedicated to simulate the Casimir force-induced instability of freestanding nanoac-tuator and nanotweezers made of conductive nanowires with circular cross-section. Tocompute the quantum vacuum fluctuations, two approaches i.e., the proximity forceapproximation (for small separations) and scattering theory approximation (for largeseparations), are considered. The Euler-beam model is employed, in conjunction withthe size-dependent modified couple stress continuum theory, to derive governing equa-tions of the nanostructures. The governing nonlinear equations are solved via threedifferent approaches, i.e., using lumped parameter model, modified variation iterationmethod (MVIM) and numerical solution. The deflection of the nanowire from zero tothe final stable position is simulated as the Casimir force is increased from zero to its

∗Corresponding author.

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http://dx.doi.org/10.1142/S021797921450129X

mailto:[email protected]





A. Farrokhabadi et al

critical value. The detachment length and minimum gap, which prevent the instability,are computed for both nanosystems.

Keywords: Nanowire; Casimir force; instability; scattering theory; proximity forceapproximation.

PACS numbers: 83.60.Np, 83.35.Kt, 85.85.+j, 87.10.Ed

1. Introduction

In recent decades nanowires and nanotubes have become the most com-

mon constructive elements in fabricating nano-/micro-electromechanical systems

(NEMS/MEMS) such as nanoactuators and nanotweezers. Generally, the behav-

ior of such miniature systems is influenced by small-scale quantum interactions.

Among the small-scale interactions, Casimir attraction becomes comparable to or

even dominant over, other small-scale forces where separation between interacting

bodies is larger than few nanometers while smaller than few micrometers.1 The

Casimir force, i.e., vacuum fluctuation force has attracted increasing theoretical2–4

and experimental5–9 interest. As the dimensions of electronic and mechanical sys-

tems are reduced to the nanometer scale, the vacuum fluctuations can strongly

interfere with the electromechanical response of nanosystems.10–17 Furthermore,

the Casimir force can induce undesired instability/adhesion in freestanding nanos-

tructures during the fabrication stages. In this regard, modeling of the Casimir

force-induced instability is crucial for investigation of the performance of nanosys-

tems such as nanoactuators and nanotweezers that are made of carbon nanotubes

(CNT), metallic nanowires, etc.

Unfortunately, all the previous works in this area have focused on modeling

the Casimir-induced instability in nanodevices with planar or rectangular geome-

try. However, to the best knowledge of the authors, no attention has been paid to

investigate the instability phenomenon in nanosystems with circular cross-section

such as nanowires and nanotubes. It should be noted that a careful analysis of the

corrections due to the geometry of interacting bodies is essential to achieve highly

precise results.18–20 It has been well established that the geometry of the interact-

ing surfaces plays an important role on the strength of the Casimir attraction be-

tween bodies. The Casimir interaction has been investigated for several geometries

including parallel plates,21 plate-sphere interaction,18 parallel cylinders22 plate-

cylinder23 etc. A simple but uncontrolled method for approximating the Casimir

interaction between nonplanar geometries is the proximity force approximation

(PFA). According to the PFA, a nanosystem is treated as a sum of infinitesi-

mal parallel plates.18 Unfortunately, the total interaction of a system of particles

cannot be obtained by simply adding the forces between all pairs. Instead, one

must also consider higher-order interactions that become increasingly important in

nanoseparations. There are other approaches24–29 that can be used to more pre-

cisely approximate the vacuum fluctuation forces. Among them, the multi-scattering

approach28,29 was successfully applied by previous researchers to approximate the

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Casimir force-induced instability in freestanding nanotweezers and nanoactuators

Casimir force between different structures e.g., spheres, sphere-plane and cylinder-

plane.

In the present study, the authors demonstrate the Casimir force-induced in-

stability of freestanding nanoactuators and nanotweezers made of cylindrical

nanowires. While the effect of the van der Waals attraction, on the mechanical

instability of cylindrical nanostructures has been investigated previously,30,31 no

work has been dedicated to modeling the Casimir force-induced instability in these

structures. One of the reliable approaches to simulate the instability of nanostruc-

tures is to apply nanoscale continuum theories,32,33 which consider the size effects

in small scales. Herein, a size dependent continuum theory i.e., the modified couple

stress theory, in conjunction with Euler beam model are applied to obtain constitu-

tive governing equations of the nanosystems under the presence of Casimir force. To

solve these nonlinear constitutive equations, three different approaches, i.e., using

a lumped parameter model (LPM), modified variation iteration method (MVIM)

and a numerical solution are employed.

2. Theoretical Model

Figure 1 shows the schematic representation of two common nanostructures.

Figure 1(a) shows the typical nanostructure constructed from a cantilever free-

standing nanowire suspended over a ground plane. This configuration is commonly

observed in actuators, resonators, switches, probes, sensors, etc. The movable con-

ductive nanowire can deflect towards the fixed conductive substrate due to the

presence of the Casimir attraction.

Figure 1(b) presents another typical structure fabricated from cantilever free-

standing parallel conductive nanowires. This configuration is commonly observed

in nanotweezers, nanogrippers, nanomanipulators etc. The nanowires can deflect

toward each other due to the presence of the Casimir attraction.

In the aforementioned structures, the length and the radius of the nanowire are

L and R, respectively. The initial gap between the two nanowires or between the

ground plane and nanowire is D. In these nanostructures, the boundary conditions

are defined as no displacement and rotation as well as traction free at the free end,

i.e., no shear force and moment.

2.1. The Casimir energy

The Casimir energy per unit area for conducting parallel infinite flat plates (Epp)

separated by a distance D is (see Ref. 18):

Epp(D) = −π2hc

720D3, (1)

where h is reduced Planck’s constant (Planck’s constant divided by 2π) and c =

2.998 × 108 m/s is the speed of light. Relation (1) is the fundamental starting

point for the proximity PFA. According to the PFA, any complex configuration of

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(a)

(b)

Fig. 1. Schematic representation of two typical nanowire-based structures: (a) SEM34 and theschematic representation of cylinder–plate configuration (nanoactuator), (b) SEM35 and theschematic representation of cylinder–cylinder configuration (nanotweezers).

interacting surfaces can be treated as a sum of infinitesimal parallel plates.18 For

very short separations, the PFA gives the correct zeroth-order approximation to

the Casimir energy as

EPFA =

∫

s

EPP (D)dS = −π2hc

720

∫∫

dS

D3, (2)

where S is one of the two surfaces defining the gap. Clearly, for larger separations

and for surfaces that are not smooth, the PFA will fail. For example, significant

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discrepancies were found in measurements of the Casimir force between a sphere

and a trench array.36

Therefore, another approach is used to approximate the Casimir force in large

separations. The starting point for precisely calculating the Casimir energy in large

separations is a path integral representation for the effective action, which yields a

trace formula for the density of states.37 The precise electrodynamic Casimir energy

of two disconnected metallic surfaces is determined by the change in the photon

density of states caused by moving the surfaces from infinity to a finite distance

in vacuum.38 The total electromagnetic Casimir energy is the sum of the Dirichlet

and Neumann modes energies.39 However, in the case of large separations, only the

Dirichlet mode is dominant and thus the Neumann mode may be neglected.39 The

Casimir energy for the Dirichlet (ED) mode is evaluated as (see Refs. 37–39 for

additional details)

ED =hc

2π

∫ ∞

0

Tr ln(MM−1∞ )dq0 , (3)

where

M12(u, u′′, q0) = G0(s1(u)− s2(u

′′); q0) , G0(x, x′; q0) =

e−q0|x−x′|

4π|x− x′|.

In the above equation, the indices 1 and 2 represent the two surfaces and M−1∞ is

the functional inverse of M at a surface of separation. The information comprising

the geometry is contained in the matrixes M and M−1∞ . Furthermore, G0 is the free

space Green’s function and si(u) is a vector pointing to the ith surface parameter-

ized by the coordinate vector u, which describes a surface in three dimensions.

In the present study, to develop practical mathematical models, two different

approaches, i.e., the PFA (small separation approximation) and Dirichlet mode

(large separation approximation), are considered.

2.1.1. Cylinder–plate structure

In this case, for a small distance gap, the PFA can be used; hence, the Casimir

energy can be evaluated as28

EPFA = −π3hcL

960

√

R

2D5. (4)

By differentiating the energy, the Casimir force for the small separation approxi-

mation (SSA) can be obtained as

fcas =1

768π3hcL

√

2R

D7, (5)

where L is the length of the nanowire, R is the nanowire radius and D is the gap

distance.

For the cylinder–plate configuration, when D ≫ R, the Dirichlet boundary con-

dition is quite significant in comparison with the Neumann boundary condition.

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Fig. 2. The variation of the dimensionless Casimir force versus the parameter D/2R for thecylinder–plate geometry (nanoactuator).

Therefore, the asymptotic expression of the attractive interaction energy for struc-

tures with a large separation gap will be obtained28,39 as

ED = −hcL

D2

1

16π ln

(

D

R

) . (6)

By differentiating the energy, the Casimir force for the large separation approxima-

tion (LSA) can be obtained as

fcas =hcL

D3

1

16π ln2(

D

R

)

(

1 + 2 ln

(

D

R

))

. (7)

For the cylinder–plate configuration, the variation of the Casimir force calculated

from Eqs. (7) and that of (5) is illustrated in Fig. 2 as a function of D/2R values.

The obtained results reveal the difference between the Casimir force values obtained

from Eqs. (5) and (7).

2.1.2. Cylinder–cylinder structure

Using the PFA for this configuration, the Casimir energy can be evaluated as39

EPFA = −π3

1920hcL

√

R/D5 . (8)

By differentiating the energy, the Casimir force for the SSA can be obtained as

fcas =π3

768hcL

√

R/D7 . (9)

Considering the first term approximation for two cylinders with a large distance,

similar to case 1, the asymptotic expression of the attractive interaction energy at

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Fig. 3. The variation of the dimensionless Casimir force versus the parameter D/2R for thecylinder–cylinder geometry (nanotweezers).

a large distance can be obtained as39

ED = −

hcL/8π

D2 ln2(

D

R

)

1−2

ln

(

D

R

)

. (10)

By neglecting the secondary term (2/ ln(D/R)) and differentiating the energy, the

Casimir force for the LSA can be obtained as

fcas =hcL

4πD3 ln3(

D

R

)

(

1 + ln

(

D

R

))

. (11)

The ratio of the Casimir forces obtained from Eqs. (9) and (11) is plotted in Fig. 3 as

a function of D/2R. This figure implies a substantial difference between the Casimir

force values predicted by scattering theory and that of the PFA for a wide range

of D/2R values. The obtained results reveal the difference between the Casimir

force values obtained from Eqs. (9) and (11). By comparing Figs. 2 and 3, it can be

concluded that the PFA provides slightly lower value than scattering approach for

same D/2R value. From the mathematical point of view, PFA reveals a power-low

dependency to the separation [Eq. (9)] while a logarithmic dependence is observed

in the case of scattering approach [Eq. (11)]. Hence, Eq. (9) decays faster than

Eq. (11) by increase in D/2R parameter.

For cylinder–plate geometry, comparison between numerical solution (based on

scattering approach) and PFA shows that SSA provides acceptable results only for

k < 10 while it underestimates the Casimir energy for higher values of k.39 In the

case of cylinder–cylinder geometry, Dirichlet mode asymptotic formula (LSA) can

provide acceptable results for k > 100 where SSA is not valid.39

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2.2. Strain energy

The elastic resistance and mechanical energy of freestanding nanostructures can

be modeled according to modified couple stress continuum mechanics theory. Ac-

cording to the theory,33 the strain energy density of a continuum U can be written

as

U =1

2(σijεij +mijχij) (i, j = 1, 2, 3) . (12)

the variables of σij , εij , mij and χij are the stress tensor, strain tensor, deviatoric

part of the couple stress tensor and symmetric curvature tensor, respectively, and

defined by the following relations

σij = λεmmδij + 2µεij , (13a)

εij =1

2((∇ν)ij + (∇ν)Tij + (∇ν)Tim · (∇ν)mj) , (13b)

mij = 2l2µχij , (13c)

χij =1

2

(

(∇θ)ij + (∇θ)Tij)

, (13d)

θi =1

2(∇× ν)i , (13e)

where, λ, µ and l are the Lame constant, shear modulus and the material length

scale parameter, respectively. Also, ν and θ are displacement and rotation vectors,

respectively.

In the above relations, the material length scale parameter physically corre-

sponds to the size dependent behavior of the materials and structures at sub-micron

distances i.e., size effect. The size effect simply can be interpreted as hardening be-

havior in the elastic resistance of material (effective bending rigidity) when the di-

mensions of the structure reduce to nano/microscales.40,41 Note that experimental

works demonstrate that the size dependency is an inherent property of conductive

metals when the characteristic size of the structures is comparable to the internal

material length scale.40,41

Now, using the Euler–Bernoulli beam model for a nanowire, the displacement

field can be expressed as42

u1 = u(X)− z∂w(X)

∂X,

u2 = 0 ,

u3 = w(X) .

(14)

Considering a small deformation, and substituting relation (14) into Eq. (13a), we

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can obtain

ε11 =∂u

∂X+

1

2

(

∂w (X)

∂X

)2

− z∂2w(X)

∂X2,

ε22 = ε33 = ε12 = ε23 = ε13 = 0 , (15a)

σ11 = E

(

∂u

∂X+

1

2

(

∂w(X)

∂X

)2

− z∂2w(X)

∂X2

)

,

σ22 = σ33 = σ12 = σ23 = σ13 = 0 , (15b)

χ12 = −1

2

∂2w(X)

∂X2, χ11 = χ22 = χ33 = χ23 = χ13 = 0 , (15c)

m12 = −µl2∂2w(X)

∂X2, m11 = m22 = m33 = m23 = m13 = 0 , (15d)

where Eeff is Young’s modulus.

Substituting Eqs. (15a)–(15d) into Eq. (12), the strain energy density can be

defined as

Uelas =1

2

∫ L

0

∫

A

(σijεij +mijχij)dAdX

=1

2

∫ L

0

∫

A

(EeffZ2 + µl2)

(

∂2w

∂x2

)

+ Eeff

(

∂u

∂X+

1

2

(

∂w

∂X

)2)2

dAdX

=1

2(EeffI + µAl2)

∫ L

0

(

∂2w

∂X2

)2

dX + EeffA

∫ L

0

(

∂u

∂X+

1

2

(

∂w

∂X

)2)2

dX .

(16)

In the above relation, EeffI and X are the effective flexural rigidity of the nanowire

and the distance from the clamped end, respectively; µ is the shear modulus, l is a

material length scale parameter and A is the cross-sectional area of the nanostruc-

ture, respectively. In order to develop the governing equation of the nanostructure,

the constitutive material of the nanomaterial is assumed to be linear elastic and

only the static deflection of the nanostructure is considered.

2.3. Governing equation

In order to derive the governing equation of the system, the minimum energy prin-

ciple, which implies that equilibrium is achieved when the free energy reaches a

minimum value, is applied (δ(Uelas −Wcas) = 0). It is worth noting that, Wcas is

the work done by the Casimir attractive force. Therefore, the differential equation

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of longitudinal and lateral deflection of the system can be written as the following

(EeffI + µAl2)d4w

dX4−

d

dX

[{

EeffA

(

du

dX+

1

2

(

dw

dX

)2)}

dw

dX

]

= fcas , (17a)

d

dX

{

EeffA

(

du

dX+

1

2

(

dw

dX

)2)}

= 0 , (17b)

and the boundary conditions are obtained as

u(0) =

(

EeffA

(

du

dX+

1

2

(

dw

dX

)2))

∣

∣

∣

∣

X=L

= 0 , (18a)

w(0) =dw(0)

dX= 0 , (18b)

(EeffI + µAl2)d2w(L)

∂X2= (EeffI + µAl2)

d3w(L)

∂X3= 0 . (18c)

Static equilibrium of cantilever beam implies that the normal resultant along the

beam should be zero. Therefore,∫

A

σ11dA =

∫

A

Eeff

[

du

dX− Z

d2w

dX2+

1

2

(

dw

dX

)2]

dA = 0 . (19)

This leads to,

du

dX+

1

2

(

dw

dX

)2

= 0 . (20)

Substituting Eq. (20) in Eqs. (17a) and (17b), the nonlinear governing equation of

the nanostructures can be defined as

(EeffI + µAl2)d4w

∂X4= fcas , (21a)

w(0) =dw

dX(0) =

d2w

dX2(L) =

d3w

dX3(L) = 0 . (21b)

By substituting relations (5) and (7) into Eq. (21a), assuming D ± R ≈ D for

a large separation gap and algebraically simplifying the result, the dimensionless

nonlinear equation for the nanoactuator (cylinder–plate structure) is derived as

d4w

dx4=

π4γ

96

√

2

k(1− w)7SSA , (22a)

d4w

dx4=

γ

(1− w)31

ln(k(1 − w))+

γ

2(1− w)3 ln2(k(1− w))LSA , (22b)

w(0) =dw

dx(0) =

d2w

dx2(1) =

d3w

dx3(1) = 0 . (22c)

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Substituting relations (9) and (11) into (21a), and using a similar procedure, the

following dimensionless equations and boundary conditions can be obtained for the

nanotweezers (cylinder–cylinder structure) as

d4w

dx4=

π4γ

96

√

1

k(1− w)7SSA , (23a)

d4w

dx4=

2γ

(1− w)3 ln2(k(1 − w))+

2γ

(1 − w)3 ln3(k(1− w))LSA , (23b)

w(0) =dw

dx(0) =

d2w

dx2(1) =

d3w

dx3(1) = 0 . (23c)

It is worth noting that the following dimensionless parameters were used in Eqs. (22)

and (23)

x =X

L, (24a)

w =

w

DCylinder–plate

2w

DCylinder–cylinder

. (24b)

k =D

R, (24c)

γ =

hcL4

8πD4(Eeff1)Cylinder–plate

hcL4

4πD4(Eeff1)Cylinder–cylinder

. (24d)

Finally, it should be noted that under some considerations, the present model might

be used for approximating the Casimir force in nanostructures made of conductive

nanotubes. For example note that local buckling of nanotubes is not considered

in this model. Moreover, when the nanotube wall is sufficiently thin so that the

radiation from outside can look inside the nanotube, the present model requires

more corrections.

3. Solution Methods

To solve the governing equations of the nanostructures, we apply three different

approaches, i.e., the modified variational iteration method (MVIM) and numerical

solution as well as using LPM.

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3.1. Modified variational iteration method

Based on MVIM, the solution is approximated with possible unknowns. Afterwards,

the corrected function is constructed using general Lagrange multiplier that can be

identified via the variational theory.43 In order to solve the constitutive equations

using MVIM, the governing differential equations (22) and (23) are transformed by

Taylor expansion to

d4w

dx4=

∞∑

j=0

aj [⌢

w(x)]j . (25)

By applying MVIM, solution of Eqs. (22) and (23) is reduced to (see Appendix for

details)

w(x) =c12x2 +

c26x3 +

a024

x4 +a1c1720

x6 +a1c25040

x7 +a2c

21

6720x8 + · · · . (26)

The unknown parameters of c1 and c2 can be obtained using the natural boundary

conditions.

3.2. Numerical solution

Besides the analytical results, the governing equations of each nanostructure are

solved numerically and the obtained results are compared with those obtained via

MVIM. The nonlinear governing differential equations (22) and (23) are solved with

the boundary value problem solver of MAPLE 15 commercial software. The step

size of the parameter variation is chosen based on the sensitivity of the parameter

to the tip deflection. The instability parameters can be determined via the slope of

the w–γ graphs.

3.3. Lumped parameter model

In order to develop an appropriate LPM, the elastic response of each nanowire is

modeled by a linear spring with a stiffness of 8EeffI/L3 (Refs. 14 and 44) as shown

in Fig. 4. This model assumes the uniform Casimir forces along the length of the

nanowires.

Considering the Eqs. (19) and (20), the governing equations of the LPM for

each nanostructure tip deflection are obtained as

8wtip =

π4γ

96

√

2

k(1 − wtip)7SSA

γ

(1 − wtip)3 ln(k(1 − wtip))+

γ

2(1 − wtip)3 ln2(k(1 − wtip))LSA

Cylinder–plate ,

π4γ

96

√

1

k(1 − wtip)7SSA

2γ

(1 − wtip)3 ln2(k(1 − wtip))+

2γ

(1 − wtip)3 ln3(k(1 − wtip)); LSA

Cylinder–cylinder .

(27)

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(a)

(b)

Fig. 4. Schematic configuration of LPMs for (a) nanoactuator and (b) nanotweezers.

3.4. Instability analysis

By sufficient increasing in Casimir force, the wire tip deflection exceeds its maximum

stable deflection and the instability occurs. From the mathematical point of view,

if the value of parameter γ exceeds its critical value, γcr, the nanowire adheres to

the ground/another wire.

For any given γ, where γ ≤ γcr, one can find analytical and numerical solutions

for w. However, when γ > γcr, no solution exists for w. This means that the

instability occurs and the nanowire collapses onto the ground. For both numerical

and MVIM solutions, the values of γcr and the corresponding nanowire critical tip

deflection, wcr, can be determined by plotting the dimensionless Casimir attraction

γ versus the dimensionless nanowire tip deflection wtip(1). Note that at the onset

of instability, the slope of this curve becomes zero. This reveals the possibility of

further increase in the nanowire deflection even without any change in the Casimir

attraction.

In the case of the LPM, the instability parameters can be obtained from Eq. (27)

by setting dγ/dwtip = 0.

3.5. Pros and cons of solving methods

It should be noted that numerical solution is a powerful and time-consuming ap-

proach to solve the nonlinear governing equation of complex systems especially for

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highly nonlinear singular differential equations such as this case. However, numerical

solution cannot provide closed-form solutions. On the other hand MVIM provides

finite series solutions instead of just a long list of numbers. This gives us some in-

sight about physical aspects and nature of the equation. Furthermore, MVIM series

solutions facilitate the parametric study and can be used to better understanding

the effect of various parameters on the response of the system. It should be men-

tioned that the simple LPM is not reliable for determining the physical behavior

of the system. Although the simple LPMs do not provide precise values, they are

very practical in understanding physical aspects of the phenomena without undue

mathematical complexities.

4. Results and Discussion

In this section, the instability of the abovementioned nanostructures (Fig. 1) due

to the presence of the Casimir force is demonstrated. Two applied cases, including

the SSA and LSA, are considered and discussed.

4.1. Nanoactuator

Figure 5 shows the variation of the centerline deflection of a cantilever actuator

[Fig. 1(a)] in the presence of vacuum fluctuations. Figure 5(a) corresponds to the

SSA assumption while Fig. 5(b) presents the result of the LSA assumption. As

shown, increasing the dimensionless Casimir force (γ) results in increasing the

nanowire’s dimensionless deflection (x). At the instability point, the Casimir attrac-

tion overcomes the elastic resistance of the nanowire; hence the nanowire collapses

and adheres to the ground plane.

Figures 6 and 7 show the relation between the parameter γ and the tip dis-

placement of the nanowire; wtip = w(x = 1). Results obtain via three different

Fig. 5. The centerline deflection of the cantilever nanowire for different values of γ, (a) the SSAfor k = 5, (b) the LSA for k = 100.

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(a) (b)

(c)

Fig. 6. The variation of γ versus the tip displacement of the nanowire for SSA using k = 2.5,k = 5 and k = 10, (a) MVIM, (b) numerical solution, (c) LPM.

solution methods, i.e., the MVIM, numerical and LPM approach are presented in

these figures. Figures 6(a)–6(c) correspond to the PFA for the SSA case using and

Figs. 7(a)–7(c) correspond to the scattering method for LSA case, respectively. As

shown, when the parameter γ reaches its critical value, γcr, the tip deflection of the

nanowire reaches its maximum stable value, wcr. As seen from these figures, increase

in k value leads to significant increase in γcr of the nanoactuator. The dependency

of γcr on k physically reveals that increasing the nanowire diameter results in the

higher rigidity of the nanowires hence allows the designers to decrease the initial

gap more or increase the wire length more without worrying about the occurrence

of the instability. On the other hand, increasing the initial separation reduces the

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(a) (b)

(c)

Fig. 7. The variation of γ versus the tip displacement of the nanowire for LSA using k = 50,k = 100 and k = 200, (a) MVIM, (b) Numerical Solution, (c) LPM.

impact of quantum vacuum fluctuations which permit the fabricators to increase

the length or reduce the diameter of the nanowire(s).

It should be noted that increase in k value does not produce substantial change

in wcr of the system. From the mathematical point of view, value of wcr determined

by PFA [Eqs. (22a) and (23a)] is independent of k parameter while there is a slight

dependency to k parameter in the case of scattering approach [Eqs. (22b) and

(23b)].

4.2. Nanotweezers

Figure 8 shows the variation of the centerline deflection of the arms of freestand-

ing nanotweezers (parallel nanowires) in the presence of the Casimir attraction.

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(a) (b)

Fig. 8. Tip displacement variation of the nanowires for different values of γ: (a) SSA for k = 5,(b) LSA for k = 100.

Figure 8(a) corresponds to the SSA case and Fig. 8(b) illustrates the nanowire

deflection determined by using the LSA assumption.

As seen, increasing the vacuum fluctuations (γ) leads to increase in the arms

deflection (x). When the Casimir attraction overcomes the elastic resistance of the

arms, instability occurs and the arms adhere to each other.

Furthermore, the variation of the tip displacement of the nanowires as a function

of Casimir force (γ) is shown in Figs. 9 and 10 for two different cases, i.e., the

PFA and scattering theory aproximation, respectively. Similar to what is mentioned

about the nanoactuator, when the parameter γ exceeds its critical value, γcr, the

nanowires tip deflection exceeds its maximum stable value, wcr and the instability

occurs. Furthermore, while increase in k value results in increase in γcr, no significant

change is observed in wcr of the nanowires.

Interestingly, comparison between Figs. 5–10 show higher values of γcr for the

tweezers (parallel cylinders geometry) in comparison with the actuator (cylinder–

plate geometry) with the same k value. This means that the structural stability

of the nanotweezers is less affected by the Casimir force for systems with equal

separations.

4.3. The detachment length and minimum gap

The maximum permissible length of the nanowire Lmax, which is required to prevent

stiction, is called the detachment length.14,15 On the other hand, if the length of the

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(a) (b)

(c)

Fig. 9. The variation of γ versus the tip displacement of the nanowires for SSA using k = 2.5,k = 5 and k = 10, (a) MVIM, (b) numerical solution, (c) LPM.

nanowire is known, then there is a minimum gap Dmin, which prevents stiction due

to the Casimir force. If the length of the nanowire exceeds Lmax or the gap reduced

less than Dmin, then the nanowire sticks to the ground or the other nanowire.

Thus, the maximum length and minimum gap are very important in design and

fabrication of nanostructures. By considering two different values for parameter

k e.g., 200 and 10 for LSA and SSA, respectively, and substituting γcr into the

definition of γ, Eq. (24d), the values of Lmax and Dmin can be determined. Table 1

shows the values of Lmax and Dmin for the nanostructures computed via numerical

solution method.

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(a) (b)

(c)

Fig. 10. The variation of γ versus the tip displacement of the nanowires for LSA using k = 50,k = 100 and k = 200, (a) MVIM, (b) numerical solution, (c) LPM.

Table 1. Detachment length, Lmax and minimum gap Dmin of freestanding nanostructures, wherek = 200 for the LSA and k = 10 for the SSA.

Lmax Dmin

ActuatorSSA 6.56× 106D(EeffI + µAl2)0.25 1.53 × 10−7L(EeffI + µAl2)−0.25

(Cylinder–plate)LSA 8.13× 106D(EeffI + µAl2)0.25 1.23 × 10−7L(EeffI + µAl2)−0.25

TweezersSSA 6.01× 106D(EeffI + µAl2)0.25 1.66 × 10−7L(EeffI + µAl2)−0.25

(cylinder–cylinder)LSA 8.43× 106D(EeffI + µAl2)0.25 1.19 × 10−7L(EeffI + µAl2)−0.25

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(a) (b)

Fig. 11. Variation of the nanowire detachment length as a function of the wire diameter andinitial gap (a) actuator (cylinder–plate), (b) tweezers (cylinder–cylinder).

4.4. Comparison between the PFA results and scattering method

approximation

For comparison between the results obtained by the PFA and scattering method,

two common freestanding nanostructures are investigated as a case study. Consider

the typical nanoactuator and nanotweezers made from cantilever gold nanowires.

The gold nanowire has Young’s modulus of 80 GPa.45 Figure 11 depicts the vari-

ation of the nanowire detachment length as a function of the wire diameter and

initial gap numerically obtained by the scattering (LSA) and PFA theories (SSA).

Figures 11(a) and 11(b) present the results for the nanoresonator and nanotweezers,

respectively (Size effect is neglected).

As shown in Fig. 11, the scattering theory is more conservative, i.e., it predicts

lower Lmax and higher Dmin values for these nanostructures. This means that the

PFA overestimates the critical values of these nanostructures. Note that the sepa-

ration values in Fig. 11 (D > 50 nm) is substantially greater than the acceptable

range for small separation assumption (SSA). Indeed, the higher-order interactions

are so significant that the total interaction of the system of particles cannot be

obtained by simply adding the forces between all pairs. Hence PFA cannot produce

reliable values of Casimir force in this range (D > 50 nm) and one should use

scattering approach instead for achieving acceptable results. This should be consid-

ered in theoretical modeling of nanostructures in order to achieve precise, reliable

simulation results.

4.5. Effect of length scale parameter (size effect)

In this section, size effect i.e., the effect of the length scale parameter l on the

maximum length of nanostructures is examined. The size effect can be interpreted

as the hardening behavior in the elastic resistance of material (effective bending

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(a) (b)

Fig. 12. Variation of Lmax as a function of the length scale parameter l and initial gap D usingthe scattering method approximation: (a) cylinder–plate, (b) cylinder–cylinder.

rigidity) when the dimensions of the structure reduce to ultra-small scales. In order

to demonstrate the effect of length scale parameter, two aforementioned nanostruc-

tures are considered as a case study. Figure 12 illustrates the numerically computed

variation of the Lmax of typical 2 nm diameter nanowires as a function of length

scale parameter (l) at three different gap distances (D). The obtained results depict

that Lmax and γcr increase significantly by considering the size effect. As seen, the

size effect plays an important role on the instability of nanostructures. This physi-

cally arises from the hardening behavior of the elastic resistance of the material in

ultra-small scales.

The prediction of the Casimir force-induced instability is a critical subject in

design AFM probes and NEMS devices. With decrease in distance between the

AFM probe and sample surfaces, the probe jumps into contact with the surfaces

and renders its imaging performance. Similarly, a NEMS switch/actuator/resonator

might adhere to its substrate even without an applied voltage, if the minimum gap

between the nanowire/nanotube and substrate is not considered. In this regard,

present model helps the designer and manufacturers to predict and avoid undesir-

able adhesion and failure of the nanodevices during manufacturing or operation.

Furthermore this makes the designers able to precisely determining the reliable

range of operation and limitation of nanosystems.

Furthermore, while molecular dynamics (MD) and molecular mechanics (MM)

could be applied to study the effect of van der Waals force on the nanostructures,

none of the commercial MD/MM softwares are yet able to incorporate Casimir at-

traction. Hence the proposed time-consuming continuum approach might be worthy

for analyzing and designing of nanostructures with large number of atoms.

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5. Conclusions

Herein, continuum mechanics was applied for modeling the effect of Casimir at-

traction acting upon the instability of freestanding nanostructures fabricated from

cylindrical nanowires. The PFA and the asymptotic scattering approximation were

used to model the Casimir force for small and large separations, respectively. Ob-

tained results show that the Casimir attraction can induce a significant insta-

bility in freestanding nanostructures at submicron separations. We have found

that:

(1) In comparison with the asymptotic scattering approximation, using the PFA

for modeling the case of large separation leads to lower critical values for the

Casimir force and deflection.

(2) In comparison with the cylinder–plate configuration, the structural stability of

the cylinder–cylinder geometry is less affected by the Casimir force for systems

with equal separations.

(3) The detachment length (Lmax) and minimum gap (gmin) parameters of the

nanostructures were determined, which are critical parameters in the design

and fabrication of nanostructures.

(4) It is found that the length scale parameter increases the Lmax values while

decreasing the Dmin values of the respective systems.

(5) Good agreement between the numerical and MVIM analytical solutions was

observed. Moreover, proposed LPM is able to simulate the instability behavior

of the nanosystems without mathematical complexity.

The prediction of the Casimir force-induced instability is a critical subject

in design nanosystems. Present model helps the designers and manufacturers

to precisely predict the reliable range of operation and limitation of nano-

systems and to avoid undesirable adhesion/failure during manufacturing pro-

cesses. To the best knowledge of the authors, none of the commercial MD/MM

softwares are yet able to incorporate Casimir attraction, hence the pro-

posed approach can be useful for quantifying the impact of Casimir force on

nanodevices.

Appendix A. Modified Variation Iteration Method

To illustrate the basic idea of the method,46 the following general nonlinear system

is considered as

L1[u(ζ)] +N1[u(ζ), v(ζ)] = 0 ,

L2[u(ζ)] +N2[u(ζ), v(ζ)] = 0 .(A.1)

where L1[u(ζ)] and L2[u(ζ)] are linear differential operators and N1[u(ζ), v(ζ)] and

N2[u(ζ), v(ζ)] are nonlinear analytic operators. The basic character of the method

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is to construct a correction functional for the system as follows

un+1(ζ) = un(ζ) +

∫ ζ

0

λ1(τ){L1[un(ζ)] +N1[un(ζ), vn(ζ)]}dτ ,

vn+1(ζ) = vn(ζ) +

∫ ζ

0

λ2(τ){L2[vn(ζ)] +N2[un(ζ), vn(ζ)]}dτ .

(A.2)

The λ1(τ) and λ1(τ) are general Lagrange multipliers which can be identified opti-

mally by the variational theory. Furthermore, un and vn are the nth approximate

solutions and un and vn represent restricted variations, i.e., δun = 0 and δvn = 0.

Calculating the variation of (A.2) and noting that δu(τ) = 0 and δv(τ) = 0, the

Lagrange multipliers are produced.46

Now, the described relation of (25) is reduced to a set of integral–differential

equations by assuming w′′′′(x) = f(w, x) and w′′′(x) = u(x):

w′′(x) = w′′(0) +

∫ x

0

u(τ)dτ ,

w′(x) = w′(0) + w′′(0)x+

∫ x1

0

∫ x

0

u(τ)dτdx1 .

(A.3)

Afterward, the following system of integral–differential equations can be considered

as

u′(x) = f(w, x) ,

w′(x) = w′(0) + w′′(0)x+ F (u(x)) .(A.4)

where F (u(x)) =∫ x1

0

∫ x

0u(τ)dτdx1. By inspiring of Eq. (A.2), correctional functions

are constructed as

wn+1(x) = wn(x) +

∫ x

0

λ1(s){w′n(x)− F [un(s))]ds ,

un+1(x) = un(x) +

∫ x

0

λ2(τ){un(s)− f(wn(s))}ds .

(A.5)

The F (un(s)) and wn(s) are considered as restricted variations, i.e., δF (un(s)) = 0

and δwn(s) = 0. The Lagrange multipliers can be identified as λ1 = λ2 = 1.46

Finally, the iterations can be written in the following form:

wn+1(x) = wn(x) +

∫ x

0

{w′n(s)− F (un(s))}ds ,

un+1(x) = un(x) +

∫ x

0

{un(s)− f(wn(s))}ds .

(A.6)

As initial guess, we select w0(x) = 0 and u0(x) = 0. By assuming w′′0 (0) = c1

and also considering physical boundary conditions of (22c) and (23c), it could be

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deduced;

wn+1(x) =c12x2 +

1

2

∫ x

0

(x − τ)2un(τ)dτ ,

un+1(x) = c2 + a0x+

∫ x

0

∞∑

j=1

aj [w(s)]j

ds .

(A.7)

By considering two terms expansion, one can obtain series solution of (26) from

the iterative system of (A.5). Based on the natural boundary conditions and the

applied theory, the Eq. (26) is finally obtained as:

Case 1. Cylinder–plate (SSA)

w =1

k3/2

(

1

2!C1k

3/2x2 +1

3!C2k

3/2x3 +1.435

4!γkx4 +

7.5336

6!C1γkx

6

+10.0448

7!C2γkx

7 +18.0175

8!γ2k1/2x8 + · · ·

)

.

Case 2. Cylinder–plate (LSA)

w =1

ln5(k)

[

1

2!C1 ln

5(k)x2 +1

3!C2 ln

5(k)x3 +1

4!(0.5 ln3(k) + 0.1 ln4(k))γx4

+1

6!C1(1.5 ln

2(k) + 3.75 ln3(k) + 4.5 ln4(k))γx6

+1

7!C2(2 ln

2(k) + 5 ln3(k) + 6 ln4(k))γx7

+1

8!(1.25 + 5.625l ln(k) + 10 ln2(k) + 7.5 ln3(k))γ2x8 + · · ·

]

.

Case 3. Cylinder–cylinder (SSA)

w =1

k3/2

(

1

2!C1k

3/2x2 +1

3!C2k

3/2x3 +1.0147

4!γkx4 +

5.3271

6!C1γkx

6

+7.1027

7!C2γkx

7 +9.0087

8!γ2k1/2x8 + · · ·

)

.

Case 4. Cylinder–cylinder (LSA)

w =1

ln7(k)

[

1

2!C1 ln

7(k)x2 +1

3!C2 ln

7(k)x3 +1

4!(2 ln4(k) + 2 ln5(k))γx4

+1

6!C1(9 ln

3(k) + 15 ln4(k) + 9 ln5(k))γx6

+1

7!C2(12 ln

3(k) + 20 ln4(k) + 12 ln5(k))γx7

+1

8!(30 + 80l ln(k) + 80 ln2(k) + 30 ln3(k))γ2x8 + · · ·

]

.

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Casimir force-induced instability in freestanding nanotweezers and nanoactuators made of cylindrical...

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