The influence of the surface energy on the instability behavior of NEMS structures in presence of intermolecular attractions

Author’s Accepted Manuscript The Influence of the surface energy on the instability behavior of NEMS structures in presence of intermolecular attractions Abed Moheb Shahedin, Amin Farokhabadi www.elsevier.com

PII: DOI: Reference:

S0020-7403(15)00306-9 http://dx.doi.org/10.1016/j.ijmecsci.2015.08.017 MS3082

To appear in: International Journal of Mechanical Sciences Received date: 14 April 2015 Revised date: 18 August 2015 Accepted date: 22 August 2015 Cite this article as: Abed Moheb Shahedin and Amin Farokhabadi, The Influence of the surface energy on the instability behavior of NEMS structures in presence of intermolecular attractions, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2015.08.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The Influence of the surface energy on the instability behavior of NEMS structures in presence of intermolecular attractions Abed Moheb Shahedina, Amin Farokhabadib,* a

Department of Aerospace Engineering, Semnan University, Semnan, Iran Department of Mechanical Engineering, Tarbiat Modares University, Tehran, Iran * Corresponding author. Email addresses: [email protected] (A. Farrokhabadi) [email protected] (A. Moheb Shahedin) b

Abstract Surface effects play a significant role in physical performance of nanostructures. Few researchers who addressed the surface effects in nano-electromechanical systems (NEMS), have exclusively focused on investigating the beam-plate configurations with planar cross-section, while no attention has been devoted to investigate these effects in NEMS fabricated from nano-wires with circular cross-section. Herein, the influence of surface layer on the instability of NEMS tweezers and cantilevers fabricated from conductive cylindrical nano-wires is demonstrated. A continuum mechanics theory is applied, in conjunction with the Euler-beam model, to obtain constitutive equations of the systems. The Gurtin-Murdoch model is employed for considering the effects of surface layer. Using the Green-Lagrange strain, the higher order surface stress components are incorporated in the governing equation. The electrostatic attraction and dispersion forces (Casimir and van der Waals) are also incorporated in the model considering the cylindrical geometry of the nano-wire. Two different approaches, i.e. numerical Runge Kutta Method (RKM) and an approximated Reduced Order Method (ROM) are employed to solve the governing equations. It is shown that the surface energy can increase the pullin voltage of the structures. Moreover, surface energy can induce a stiffening effect i.e. reduces the nano-wire deflection. Furthermore, the impact of surface layer on the stability of freestanding devices and the critical value of dispersion forces (adhesion threshold) is discussed. Keywords: Surface effects; Cylindrical nano-wire; Nano-tweezers; Nano-cantilever; Pullin instability; 1. Introduction

With recent advances increased dramatically fabricated tweezers and potential for a wide

in nanotechnology, the application of nano-wires has been in developing nano-electromechanical systems. Nano-wire cantilevers are of the most promising miniature systems that are range of applications including bio-engineering, medicine, 1

electronics, nano-scale fabrications, sensing, mass-detecting, etc. A typical NEMS cantilever is constructed from a clamped nano-wire suspended over a fixed conductive plane via a dielectric spacer. By applying voltage difference between the components, the nano-wire deflects toward the ground electrode until at a critical (pull-in) voltage, it adheres the ground. Typical nano-tweezers are composed of two parallel clamped nanowire electrodes (arms) with a distance in between. By imposing electrostatic potential between the nano-wires, the wire-tips move closer together similar to conventional tweezers, hence can be used to manipulate nano-scale objects. It is well recognized that predicting the pull-in instability voltage [1] and detachment length[2-4] are important issues for designing reliable nano-cantilevers and nano-tweezers. The pull-in instability limits the operating voltage of the cantilevers as well as the size of objects manipulated by the tweezers. As the structure scale reaches the micro/nano level, the forces such as Casimir [5] and van der Waals (vdW) [2, 6] have a lot of influence on the structures. In the previous researches, there are significant studies on the effects of van der Waals force and Casimir attraction on the instability of NEMS and nano-twezeers. Lee and Kim applied a continuum model to investigate the deflection of nano-tweezers [7]. Wang et al. [8] studied the stability of nano-tweezers using a continuum model by employing the Galerkin method. They have studied the static pull-in conditions of freestanding arms taking the van der Waals force into account. Ramezani [9] investigated the pull-in behavior of beam-plate type nano-tweezers with square cross-section arms. He applied a continuum model to obtain the constitutive equation of the nano-tweezers. Farrokhabadi et al. [10] studied the static behavior of CNT-based nano-tweezers using the Euler– Bernoulli beam theory. They could consider the effects of Coulomb and van der Waals forces on the pull-in behavior of nano-tweezers. It is also interesting for scientists to evaluate the magnitude of the Casimir force and consider its effects on the instability of nanosystems, especially electromechanical systems [11-16]. The effect of the Casimir force on the mechanical instability of plates and membranes has been simulated by Batra et al. [17-19] using the finite element method. Moghimi et al. [20] have applied the finite element method to simulate the influence of the Casimir attraction on the dynamic pull-in behavior of nanobeams. A one degree of freedom lumped parameter model has been proposed by Lin and Zhao [3, 4] to survey stiction of nanoactuators in the presence of electrostatic and Casimir attractions. Ramezani et al. [21] have used the Green's function to investigate the pull-in parameters of cantilever beam-type actuators under Casimir forces. Farrokhabadi et al. [22] studied the effects of the Casimir force on the instability and adhesion of freestanding Cylinder– Plate and Cylinder–Cylinder geometries. However, a major difference between micro- and nano-scale materials and macroscopic materials is the increase of the surface-to-volume ratio. Therefore, another important force can be affected the response of nano-scale materials and become more dominant in 2

nano-scale structures, which greatly alter the mechanical properties of nano-materials such as strength, fracture toughness, elastic moduli, etc. [23]. Mechanically speaking, there are two distinct but critical effects due to nano-scale free surfaces. The first effect is that of surface residual stress due to the fact that atoms lying at the material surfaces have a different bonding configuration as compared to bulk atoms. Because these atoms are therefore not at an energy minimum, surface stresses exist which serve to cause these atoms to deform in order to find their minimum energy configuration [24, 25]. Surface elasticity is another effect that occurs due to the lack of bonding neighbors for surface atoms. Again because surface atoms have a different bonding environment than atoms that lie within the bulk material, the elastic properties (stiffness in particular) of surfaces differ from those of an idealized bulk material, and the effects of the difference between surface and bulk elastic properties become magnified as the surface area to volume ratio increases with decreasing structural dimension [26, 27]. As a result, the crucial influences of surface layer i.e. residual surface stress and surface elasticity [28, 29] on the behavior of the nano-structures might not be negligible. While molecular dynamic simulation might be used to modeling the surface layer in nanoelements, this method is very time consuming and cannot easily be applied for modeling the response of systems with large number of atoms. A continuum theory has been developed by Gurtin and Murdoch [30] to model both the residual surface stress and the surface elasticity in miniature structures. In this theory, surface effect on the flexural deformation is considered by using the Young–Laplace equation. This theory has been widely applied to investigate the surface effect on elastic behavior of beam-type nanostructures [31-34]. He and Lilley [31] have studied the static bending of nano-beams incorporating the surface effect. Wang and Feng [32] have investigated the buckling and vibration of nano-beams by considering the effect of surface elasticity and surface residual stress. Recently, some researchers have investigated the instability of NEMS structures by considering the effect of surface energy. Fu and Zhang [33] have applied a modified continuum model to investigate the pull-in behavior of electrically actuated double-clamped nano-bridges incorporating the surface effect. Ma et al. [35] have studied the effect of surface energies and Casimir force on the instability parameters of cantilever NEMS switches. Koochi et al. [36] analyzed the effect of residual stress and surface elasticity on pull-in behavior of cantilever nano-actuators. However, both the later works have used simplified boundary conditions for solving the governing equation of the structures [35, 36]. Wang and Wang [37] investigated the pull in instability of nanoswitches under the electrostatic and intermolecular Casimir attractions. Their analysis was based on the geometrically nonlinear Euler-Bernoulli beam theory with considering the surface energy. Ansari et al. [38] have done a comprehensive investigation on the size-dependent pull-in instability of geometrically non-linear rectangular nano-plates including surface stress effect undergoing hydrostatic and electrostatic actuations. Shaat 3

and Mohamed [39] developed a size-dependent electrostatic model for micro-actuated beams by considering the microstructure and surface energy effect. Recently, Ansari et al. [40] have investigated the pull-in instability characteristics of hydrostatically and electrostatically actuated circular nano-plates including surface stress effect. By reviewing the mentioned researches, it seems that more investigations are required for precise modeling of the surface layer in NEMS. To the best knowledge of the authors no researchers have investigated the effect of surface layer on the electromechanical instability of nano-tweezers. Furthermore, while previous efforts have exclusively focused on the modeling of nano-cantilevers with planar rectangular cross-section geometry, no attention has been paid to investigate surface effect in nano-cantilevers fabricated from cylindrical conductive wires with circular cross-section. In this regard, the main aim of the present study is modeling the surface layer in electromechanical nano-tweezers and nano-cantilevers. For this purpose, the Gurtin-Murdoch theory in conjunction with the Green-Lagrange strains employed for considering the effect of surface layer. Song et al. [41] proposed this modified theory which considers the residual stress effect on the bulk stiffness and bulk stress consequently. This modified theory is in more agreement with the experiments conducted on cantilever nano-wires. It is wellestablished that the distribution and strength of the physical forces (Coulomb, Casimir and van der Waals attractions) in NEMS depends on the geometry of interacting components. Herein, the electrostatic and dispersion forces corresponding to cylindercylinder (for nano-tweezers) and cylinder-plate (for nano-cantilever) geometries are incorporated in the governing equation. Neglecting the shear deformation effects on the pull-in behavior of nano-wire made structures, it is assumed that the aspect ratio of length to diameter of nano-wire is larger than 60 [42] in this paper. Therefore, the governing elasticity equation of structure is derived using continuum Euler-Bernoulli beam theory as well as the nonlinear Green-Lagrangian strain assumption in conjunction with Hamilton principle of minimum energy. To solve the constitutive equations, two different approaches, i.e. an approximated Reduced Order Method (ROM) as well as numerical Runge Kutta Method (RKM) is employed. 2. Theoretical Model

Fig. 1 shows the schematic representation of two different types of nano-structures fabricated from nano-wire. A typical cantilever NEMS switch fabricated form nano-wire is presented in Figs 1(a). Figs 1(b) represent typical nanotweezers made of cantilever parallel conductive nano-wires. The surface energy due to surface layers has a dominant effect on the electromechanical response of these nano-structures. The schematic of surface layer on the nano-wire is illustrated in Fig. 1(c). The length and the radius of the nano-wires are L and R, respectively. The initial gap between the two nano-wires (for nanotweezers) or between the ground and nano-wire (for switch) is denoted by D. 4

(a)

(b)

(c) Fig.1. Schematic representation of typical nano-wire based structures, (a) Cantilever NEMS, (b) Nanotweezers, (c) Surface layer

5

2.1 Governing Equations 2.1.1 Bulk strain energy The strain energy of nano-structure can be written as L

Ub 

1  σijεij  dAdx, 2  0A

where, the

and

(1)

are stress and strain tensor respectively.

In a number of recent investigations, researchers have used the Timoshenko beam model [42-47] to study the shear deformation effects on the response of carbon nanotube made structures. In a performed research by Chang and Lee [42], they compared the obtained frequency based on Timoshenko beam model with the studies based on Euler beam model and deduced that if the ratio of length to diameter increased to 60, the influence of the shear deformation and rotary inertia on the mode shape and the resonant frequencies can be neglected and there is not much discrepancy between Timoshenko and EulerBernoulli beam model. As a result the authors assume that the aspect ratio of length to diameter of nano-wire have to be larger than 60. Therefore the shear deformation effects can be neglected and the Euller-Bernoulli beam theory will be highly acceptable. For an Euler–Bernoulli beam, the displacement field can be expressed as [48] u x  z

w  x  x

, u y  0, u z  w  x  .

(2)

Considering the nonlinear Green-Lagrangian strain, the strain and stress field, of nanostructure can be written as ε xx  z

2 w  x  x

σ xx  E(z

2



1  2 w(x) 1 w(x) 2 (z ) ( ) ,  ε yy  ε zz  ε xy  ε yz  ε zx  0, 2 x 2 2 x

2w  x  x

2



1  2 w(x) 1 w(x) 2 z  ( ) ),  σ yy  σ zz  σ xy  σ yz  σ zx  0, 2 x 2 2 x

(3)

(4)

where, E is the Young's modulus. Substituting Eqs. (3) and (4) in Eq. (1), the strain energy of bulk can be defined as

6

UB 

1 L  2w 1  2 w 2 1 w 2 2  E[  z  (  z )  ( ) ]  dAdx   2 0 A  x 2 2 x 2 2 x  1 L  2 w 2 1 * 2w 4 1 w 4 1  2 w w 2  EI( )  EI ( )  E A( )  EI( )  dx,  2 0  x 2 4 x 2 4 x 2 x 2 x 

(5)

where, I is the nano-wire's moment of inertia. 2.1.2 Strain energy in the surface layer According to the surface elasticity theory, the strain energy in the surface layer can be written as L

US 

1 (τijεij )dsdx. 2 0 A

(6)

According to the continuum theory proposed by Gurtin and Murdoch [30, 49], the inplane components of the surface stress tensor can be defined as τij  μ 0  u i, j  u j,i    λ0  τ0  u k,k δij  τ0  δij  u j,i  ,  

(7)

where, and are the surface elastic constants, and is the residual surface stress. The out-of-plane components of the surface stress tensor are defined by [49] τ ni  τ0  u n,i . 

(8)

By substituting relation (2) in Equations (7) and (8), one obtains  2w 1 2w 1 w  xx  0  E0  z 2  ( z 2 )2  ( )2  , 2 2 x  x x 

(9)

where, is the surface elastic modulus (P-wave modulus), which can be determined from atomistic calculations [30]. By substituting Eqs. (9) and (3) in Eq. (6) the surface energy can be written as

7

US 

 2w 1 1 L  2 w 2 1 w 2  w {   z  (  z )  ( )   0 n 2Z ( ) 2  0 2 2   2 0 A  x 2 x 2 x  x 2

 2w 1 2w 1 w   E 0  z 2  (z 2 ) 2  ( ) 2  }dsdx 2 x 2 x   x 1 L 1 w 2 1 2w 2 w 2 2w 2   { 0 C0 ( )  0 I0 ( 2 )  0S0 ( )  E 0 I0 ( 2 ) 2 0 2 x 2 x x x 2 2 1  w 1 w 1  w w 2  E 0 I*0 ( 2 ) 4  E 0S0 ( ) 4  E 0 I0 ( 2 ) }dx, 4 x 4 x 2 x x

(10)

where, I*0   z4ds ,

I0   z 2ds ,

S

S

S0   n 2z ds , S

C0   ds. S

2.1.3 Work of external force By considering the distribution of external forces per unit length of the beam, the performed work by these external forces can be obtained as L

Wext  f ext w  x  dx.

(11)

0

It is worth noting that the external force, in Eq. (11) can be determined by the summation of dispersion forces (Casimir attraction or van der Waals force) with the electrostatic fluctuation as   f x f ext  f elec  x    Cas .  f vdW  x 

(12)

In the following subsection, the derivation of the mentioned external force will be presented for each structure. 2.1.3.1

Electrostatic attraction

The nano-wires are considered as perfect cylindrical conductors. To calculate the electrical forces acting on the nano-wire, a capacitance model may be used which is described in two following sections.

8

2.1.3.1.1 Cantilever NEMS For this type of structure, the nano-wire and plate are considered as perfect conductors. For a conductive cylinder with infinite length, the electrostatic energy per unit length is given by [50] E elec 

0 V 2 1 C(D)V 2  , D 2 arccosh(1  ) R

(13)

where, V and ε0 are the applied voltage and the permittivity of vacuum. Hence, the electrostatic force per unit length, felec, will be obtained by [51] f elec 

dE elec  dD

0 V 2 D(D  2R) arccosh 2 (1 

D ) R

,

(14)

where, V is the applied voltage. Note that by applying DC voltage, the nano-wire deflects and the distance between the nano-wire and the ground reduces to D-w. Regarding to this point and by considering D±R≈D, the electrostatic force per unit length of the deflected nano-wire can be further simplified as f elec 

0 V 2

Dw (D  w)arccosh ( ) R



2

0 V 2

Dw (D  w)ln (2 ) R

.

2

(15)

2.1.3.1.2 Nano-tweezers For this type of structures, the nano-wires are considered as perfect cylindrical conductors. Therefore, the electrostatic energy per unit length is obtained by [10] 1 E elec  C(D)V 2  2

0 V 2 2  D  D  ln 1     1  1   R  R   

.

(16)

Hence, the electrostatic force per unit length can be defined by f elec 

dE elec dD

0 V 2   D D 2R   D2  2DR  ln 1   1  D     R R

2

.

(17)

where, V is the applied voltage. 9

In a similar deduction to NEMS, by applying the external voltage on the nano-structure, the moveable arms will deflect to each other in magnitude to reduce their between gap from to . If both nano-wires have the same geometry and material properties, their deflections will be equal ( ). Thus, by replacing with in relation (17) and considering that (R) is much less than the distance between nano-wires, it deduced f elec 

0 V 2  2  (D  2w) ln   D  2w      R

2.1.3.2

2

.

(18)

Casimir Force

The total electromagnetic Casimir energy is the sum of energies of the Dirichlet and Neumann modes. However, in the case of large separations, the Dirichlet mode is dominant. As a result, the Neumann mode can be neglected [52]. 2.1.3.2.1 NEMS Cantilever For the NEMS cantilever configuration, when D>>R, the Dirichlet boundary condition is quite significant in comparison with the Neumann boundary condition. Therefore, the asymptotic expression of the attractive interaction energy for structures with a large separation gap will be obtained [53, 54] as D E Cas 

hcL D2

1 D 16 ln   R

,

(19)

where, ̅ is the reduced Planck’s constant (Planck’s constant divided by 2π) equal to

h  1.05457×10-34 J.s and c = 2.998×108 m/s is the speed of light. By differentiating the energy, the Casimir force can be obtained as f Cas 

D dE Cas hcL  3 dD D

1 hcL 1  3 . D D D 8 ln   16 ln 2   R R

(20)

We remark that by applying the external voltage on the nano-wire, it will deflect toward the ground to reduce the gap in between them from D to D - w.

10

f Cas 

  D  w  1  ln  R   . D  w      ln 2    R 

hc 16  D  w 

3

(21)

2.1.3.2.2 Nano-tweezers Considering the first term approximation for two nano-wires with a large distance, similar to cantilever NEMS, the asymptotic expression of the attractive interaction energy at a large distance can be obtained as [53] D E Cas 

hcL D 8D 2 ln 2   R

.

(22)

By differentiating the energy and considering the deflection of the nano-wires, w1 and w2, the Casimir force per unit length for the large separation approximation can be obtained as f Cas 

D dE Cas  dD

  D  1  ln    .  D    R  4D ln   R   hc

3

3

(23)

To derive Eq. (22), it should be noted that by applying the external voltage on the nanowires, the arms deflect towards each other to reduce the gap in between them from D to D – w1 – w2. If both arms of the nano-tweezers have the same geometry and material properties, then their deflections will be equal (w1 = w2 = w). Therefore, the Casimir force can be obtained as f Cas 

  D  2w    1  ln  1    . R D   2w    3 3 D  4  D  2w  ln  1   D   R hc

2.1.3.3

(24)

van der Waals attraction

The Lennard-Jones potential is a suitable model to describe van der Waals interaction between bodies [55]. A continuum model has been established to compute the attractive van der Waals energy by double-volume integral of Lennard-Jones potential [56]. 2.1.3.3.1 Cantilever NEMS For the nano-wire suspended over the half space, by integrating over two volumes and assuming (R
E vdW  

2C1,212 R

AR 2  3 , 3D

3D3

(25)

where, Ā is the Hamaker constant which is in the range of 0.4-4×10-19 J [57]. Now the molecular force per unit length, fvdW, can be obtained from (25) as f vdW 

dE vdW AR 2  . dD D4

(26)

Note that by deflecting the cantilever nano-wire the initial gap between the moveable nano-wire and fixed ground reduces from D to D-w. As a result, the van der Waals force due to the deflection of nano-wire can be written as f vdW 

AR 2

 D  w 4

.

(27)

2.1.3.3.2 Nano-tweezers For two nano-wires suspended over the space with the distance of D, by integrating over two volumes and assuming (R
A R . 24D3/2

(28)

Now the intermolecular force per unit length, fvdW, can be obtained from (28) as f vdW 

dE vdW A R  . dD 16D5/2

(29)

Note that by deflecting the arms their in-between gap reduces from g to D-w1-w2. By assuming the identical geometry of the arms, the displacements of w1, w2 will be the same (w1=w2=w). Therefore, similar to what mentioned about the electrical force, the van der Waals force is derived from equation (29) by replacing D with D-2w f vdW 

A R 16  D  2w 

5/2

.

(30)

12

2.1.4 Equilibrium Equations In order to derive the governing equation of the system, the minimum energy principle, which implies equilibrium when the free energy reaches a minimum value, is applied and the following equation is obtained

  UT  Wext   0.

(31)

where, is the total strain energy of bulk and surface and is the total work done by external forces. Considering Eq. (31) and the neglecting the higher order terms one can be obtained as L

0

f Cas (x)  1 4w 1 2w (EI  E 0 I0  0 I0 ) 4  (0S0  0 C0 ) 2  f elec (x)    wdx 2 2 x x  f vdW (x)  L

 1 w 1 3 w     (0S0  0 C0 )  (EI  E 0 I0  0 I0 ) 3  w   2 x 2 X   0 

(32)

L

 1  2 w   w     (EI  E 0 I0  0 I0 ) 2       0.  2 x   X   0 

Hence the governing equation of the system can be derived as fCas (x) 1 4w 1 2w (EI  E0 I0  0 I0 ) 4  (0S0  0C0 ) 2  f elec (x)   2 2 x x f vdW (x) w  0 

dw d2 w  0   2  L   0, dx dx

3 1 1   dw  d w  S   C L  EI  E I   I L  0.   0 0 0 0 0 0  0 0  3   2 2   dx   dx

(33-a)

(33-b)

(33-c)

It is worth to note that the extended relations in this section have considered the surface energy influences on the bending stiffness and of the structures according to some recent investigation as it is has been discussed in subheading 2.1.2. Therefore, the relations are approachable for any specified alternative elastic modulus which is gain according to the size of the structure. 2.1.4.1

Dimensionless Governing equation

For cylindrical nano-wires with circular section we have 13

I

R 4 , 4

I0  R 3 ,

S0  4R,

C0  2R

(34)

Therefore, the governing equation and boundary condition of the nano-structures are obtained as 4 f Cas  x  1 d2w  1 d w 4 3 3  R E   R E   R   5  R   f elec  x    , 0 0 0  4 2 2 dx 4  dx  f vdW  x 

w  0 

5R0

dw d2 w  0   2  L   0, dx dx

(35-a)

(35-b)

dw 1 3  d3 w 1 4 3 L   R E   R E  R 0  3  L   0.    0 dx 2 4  dx

(35-c)

It is also possible to simplify the variables in the obtained governing equation as well as boundary conditions to derive dimensionless relations as X

x , L

W



k

w , D

(36-b)

20R0 L2 , R 4 E  4R 3E 0  2R 30

2 



(36-a)

80 V 2 L4



D2 R 4 E  4R 3E 0  2R 30



(36-c)



,

hcL5

8D4 R 4 E  4R 3E 0  2R 30

2D , R



(36-d)

,

(36-e)

(36-f)

14

 AR 2 L4 Nanocantilever  5 4 3 3  D R E  4R E 0  2R 0  . A RL4   8D7/2 R 4 E  4R 3 E  2R 3 Nanotweezers 0 0 







(36-g)



As a result, the dimensionless form of the governing equation of each nano-structures including the surface effect, Casimir or van der Waals force as well as Coulomb electrostatic parameters come as β η   2 1  W  ln 2 k 1  W     1  W 4     k 1  W  γ 1+2ln     2  β      2 1  W  ln 2  k 1  W  3  2 k 2 1  W  ln   2 1  W   4 2  d W d W    4 2 β η dX dX   5  2 1  2W  ln 2  k 1  2W  2 1  2W 2       k 1  2W   γ 1+ln     2  β      2 1  2W  ln 2  k 1  2 W  3 3 k 1  2W  ln  1  2W   2  

W  0 

dW  0   0, dX

 d3 W d2 W dW  1     3   1  0, 2 dX  dX  dX

 Cantilever  vdW  ,

 Cantilever  Casimir  ,  Tweezers  vdW  ,

(37-a)

 Tweezers  Casimir  ,

(Geometrical B.C. at fixed end) (Natural B.C. at free end)

(37-b)

(37-c)

The non-dimensional parameters and account for the surface effect, electrostatic force, Casimir force, the gap between arms and the van der Waals attraction respectively. 3. Solution methods

It is obvious that no exact solution exists for Eq. (37), due to high nonlinearity of the problem. Hence, two different approaches are utilized in this section to determine the dimensionless pull-in voltage ( ) and the pull-in deflection ( ) i.e. the voltage and wire tip deflection at the pull-in point. 15

3.1 Runge-Kutta Method (RKM) The MATHEMATICA commercial software is employed to numerically solve the system of equations. It uses the Runge–Kutta numerical method [58] to reduce error and adjust the progress steps accordingly. The formula for the fourth order Runge-Kutta method is given below. Consider the problem W  f  X, W  X   W  X0   α

Defining

to be the independent variable step size and

h Wi 1  Wi  (k1  2k 2  2k 3  k 4 ) 6

where,

(38)

to

(39)

are defined as follow,

k1  f (X, W(X)) k 2  f (X  ΔX , W  X   k1 ΔX ) 2 2 k 3  f (X  ΔX , W  X   k 2 ΔX ) 2 2 k 4  f (X  ΔX, W  X   k 3ΔX).

(40)

In order to obtain an approximate solution for a higher-order differential equation by using RKM, it is converted into a system of first order ODEs. 3.2 Reduce Order Method (ROM) To apply the reduced-order method [59], Eq. (18) is discretized into a finite-degree-offreedom system. This discretization leads to a system of algebraic equation which can be solved numerically to obtain the final solution. For this purpose the displacement is expressed as a linear combination of a complete set of linearly independent basis functions ϕi(x) in the form of n

W  X    qi i  X , i 1

(41)

where, the index i refers to the number of modes included in the simulation. We use the linear mode shapes of the nano-beam as basic functions in the Galerkin procedure. Using Taylor expansion for external forces, electrostatic force and Casimir or van der Waals attraction, and then substituting Eq. (41) into Eq. (37), multiplying the result by ϕ , and integrating the outcome from X = 0 to 1, Eq. (37-a) for cantilever NEMS and van der Waals attraction is converted to the following form as a function of parameter . 16

0  1qi  2  qi   3  qi   4  qi   0 2

The constants of

3

4

(42)

can be defined as

 β 1 0    2  η i dx  2ln  k   0 1 1   1 1 2  1 iv ii 1  i i   ξ i i   β  3   4η  i dx  2   ln  k  2ln  k    0 0 0

  3  3ln  k   ln 2  k   1 3 4  2   β   10ηln k  i dx     2ln 4  k     0 

(43)

  12  18ln  k   11ln 2  k   3ln 3  k    1 4 3    20η  β    i dx  6ln 5  k     0    60  120ln  k   105ln 2  k   50ln 3  k   12ln 4  k    1 5  4   35η  β    i dx.  24ln 6  k     0 

For the other case, the obtained coefficients of

are mentioned in appendix.

4. Results and discussion

In this section, the obtained results for each structure in the presence of different attraction force will be represented. 4.1 Cantilever NEMS The variations of the non-dimensional applied voltage versus the tip displacement for different values of surface effect parameter, and as a gap parameter is represented in Fig.2. Increasing the voltage difference between electrodes, the nano-wire bends over the ground plate and finally collapses in a certain voltage so-called pull-in voltage. It is worth to note that, the presence of Casimir and van der Waals attractions have been considered in obtained results in Figs.2 (a) and (b) respectively. The obtained results reveal that existence of surface effect increases the pull-in voltage and deflection of the NEMS structure dramatically. These results depict that here is an initial tip displacement in the structure even when no voltage is applied which is due to the presence of dispersion forces.

17

(b)

(a) Fig.2. The curves of

(voltage) versus (a)

for different surface effect parameter and (vdW) and (b) (Casimir).

(gap parameter),

In another attempt the variation of pull-in voltage parameter versus the gap parameter is illustrated in Fig. 3 neglecting any intermolecular attraction. As it is prospected, increasing the gap parameter of leads to an increment in voltage parameter. However, it is obvious that the existence of surface effect increases the pull-in voltage parameter of NEMS structure and has a hardening effect on the ultra-small structures.

Fig.3. The curves of

versus

for

,

and

The variation of intermolecular attraction versus the tip displacement, are plotted in Figs.4 (a) and (b) for van der Waals parameter, and Casimir parameter, 18

respectively in various amounts of surface parameter, .

as well as the gap parameter

(a)

(b)

Fig.4. The curves of molecular attraction versus for different values of surface effect parameter and , freestanding with (a) van der Waals and (b) Casimir force.

The critical values of dispersion force at this state are very important in NEMS design. The critical values of ,

and

are determined by setting

and

.

Detachment length [2-4] is an important engineering design parameter in nano-structures which is the maximum permissible length of the freestanding beam which guarantees that the beam does not adhere to the ground due to the intermolecular force. The detachment length can be directly determined [36] using Eq. (36-g) and Eq. (36-e) for nanocantilevers in presence of the van der Waals ( ) and Casimir ( ) attractions, respectively as bellow, L vdw 

4

LCas  5



D5 R 4 E  4R 3E 0  2R 3 0 AR

2



,

D4 82 R 4 E  322 R 3E 0  162 R 30 hc

(44)

.

(45)

The critical values of and are listed in Table. 1. This interestingly implies that the surface effect increases the critical values for intermolecular forces and the detachment length consequently.

19

Table 1 The critical intermolecular attractions for nano-cantilever Surface Parameter ( ) Critical vdW ( Critical Casimir (

) )

RKM ROM RKM ROM

0

0.75

1.5

0.9391 0.9400 3.8553 3.8613

1.2011 1.2050 4.9312 4.9501

1.4547 1.4701 5.9724 6.0389

As shown, when the parameters and reaches their critical values, the tip deflection of the nano-wire reaches its maximum stable value . Furthermore, the obtained results reveal that the surface effect increases the maximum values of dispersion force parameters. The variation of the pull-in voltage of the cantilever nano-actuator as a function of the beam radius is represented in Fig. 5. As can be seen, assuming the constant initial gap (D=50 nm) and decreasing the nano-wire's radius results in the pull-in voltage enhancement of the nano-structure. Interestingly, this figure reveals that the effect of the surface energies on the pull-in performance of thin nano-actuator is more profound in comparison with thick nano-actuator

(a)

(b)

Fig.5. Influence of surface effect on the pull-in voltage of nano-actuator for different radius values

4.2 Nano-tweezers The electrostatic behavior of nano-tweezers is shown in Fig. 6. The voltage difference versus the tip displacement for different values of surface effect has been plotted in nondimensional forms so as a gap parameter. Because of the van der Waals and Casimir attractions which have been considered in Figs. 6(a) and (b) respectively, an initial tip displacement in the nano-tweezers structure is obvious while the applied 20

voltage is zero. This initial deflection is more significant in the presence of van der Waals force.

(a) Fig.6. The curves of

(b)

versus

(voltage) for different surface effect parameter and (a) (vdW) and (b) (Casimir).

(gap parameter),

Similar to the obtained results in previous subsection for cantilever NEMS, the pull-in voltage of the nano-tweezers and its corresponding deflection increase in higher amounts of surface effect parameter. The pull-in voltage parameter versus the gap parameter is illustrated in Fig. 7 in absence of any intermolecular attraction. The figure shows that the increment of gap parameter increases the pull-in voltage. Moreover, the existence of surface effect increases the pull in voltage parameter of tweezers structure similar to cantilever NEMS.

Fig.7. The curves of

versus

for =0 (solid line), =0.75 (dash line) and =1.5 (dash-dot line).

21

The displacement variations of the tweezers in the tips are shown versus the van der Waals and the Casimir parameters in Figs.8 (a) and (b) respectively, according to the RKM and ROM for different values of surface parameter, and gap parameter, .

(b)

(a)

Fig.8. The curves of molecular attraction versus for different surface effect parameter and freestanding with (a) van der Waals and (b) Casimir force.

,

As shown, when the parameter reaches its critical value , the tip deflection of the nano-wire reaches its maximum stable value . The critical values of and listed in Table. 2 give the detachment length using Eq. (36-g) and Eq. (36-e) for nano-tweezers, respectively as below,

L vdw 

4

LCas  5



D7/2 8R 4 E  32R 3E 0  16R 30



(46)

A R



D4 82 R 4 E  322 R 3E 0  162 R 30 hc

.

(47)

The surface effect will increase the critical values and the detachment length as well. Table 2 The critical intermolecular attractions for nano-tweezers Surface Parameter ( ) Critical vdW ( Critical Casimir (

) )

RKM ROM RKM ROM

0

0.75

1.5

1.4034 1.4073 6.4375 6.4462

1.7953 1.8041 8.2338 8.2639

2.1745 2.2010 9.9722 10.0815

22

Finally, the variation of the pull-in voltage of the nano-tweezers structure as a function of the beam radius is represented in Fig.9 while the initial gap, D before applying the voltage difference is considered to be 50 nm. As can be seen from the figures, decreasing the arms radius increases the pull-in voltage of the system. Similar to cantilever NEMS, in nano-tweezers the effect of surface energies on pull-in performance of thin nanoactuator is more profound in comparison with thick nano-actuator. Moreover, in the same nano-wire radiuses and identical amount of surface effect the pull-in voltage parameter is higher remarkably in presence of van der Waals attraction compared to Casimir consideration.

(a)

(b)

Fig.9. Influence of surface effect on pull-in voltage of nano-actuator for varying radius values

5. Conclusion

In present study, a continuum mechanics was applied to obtain the constitutive equations of nano-tweezers and cantilever NEMS constructed from conductive nano-wires. Using the Gurtin-Murdoch model as well as the Green-Lagrange strain theory, the influence of surface energy on the pull-in instability of both structures was investigated in the presence of the electrostatic, Casimir and van der Waals attractions. The Runge Kutta numerical method (RKM) and an analytical approximated Reduced Order Method (ROM) were employed to solve the governing equations. The results revealed that, -

The structures are instable at a critical value of dispersion force and the detachment length is determined directly using this critical value. The surface effect increases the critical dispersion force and the detachment length consequently. 23

-

Comparing the graphs, while the initial deflection in the presence of van der Waals force is more dominant than Casimir attraction in both cantilever NEMS and nanotweezers structures, this phenomenon is highly significant in the nano-tweezers.

-

Generally, it is obvious that in the all obtained results, the surface energy has a noticeable effect on the instability behavior of NEMS. The existence of surface effect increases the pull-in voltage and deflection parameters of the cantilever NEMS and tweezers under the electrostatic excitation as well as freestanding devices. As a result, it has a hardening effect on these structures.

-

In the freestanding state, the surface effect increases the maximum values of dispersion force parameters in both structures. Furthermore, the influence of the surface energies is more profound in the thin nano-wire structures in comparison with the thick nano-wire actuators in both structures.

-

Comparing both applied solution methods in this study reveals that the semianalytical solution of ROM has an excellent agreement with the obtained results by RKM numerical approach so the least difference is visible especially in lower surface effects.

Appendix Cantilever/Casimir  β  1  2ln  k / 2    1 0    2  γ    dx  2ln 2  k / 2     i   0  2ln  k  1 1   1   1  2ln  k / 2  1  6ln  k / 2    1 2 1 iv ii 1  i i   ξ i i   β  3     γ   i dx 2 3 2ln 2  k / 2    0   ln  k  2ln  k    ln  k / 2  0 0

  3  3ln  k   ln 2  k    3  9ln  k / 2   13ln 2  k / 2   12ln 3  k / 2    1  2   β    γ   i3dx 4    2ln 4  k  2ln k / 2        0   12  18ln  k   11ln 2  k   3ln 3  k    β    5  6ln  k    1 4  3     i dx   12  42ln  k / 2   71ln 2  k / 2   77ln 3  k / 2   60ln 4  k / 2    0   γ   6ln 5  k / 2        60  120ln 6  k / 2  ln  k   105ln 2  k   50ln 3  k   12ln 4  k    β     24ln 6  k    1 5  4     i dx   60  240ln  k / 2   465ln 2  k / 2   580ln 3  k / 2   522ln 4  k / 2   360ln 5  k / 2    0    γ   24ln 6  k / 2     

24

Tweezers/vdW 1 1 β 0    2  η i dx 2  ln  k   0 1 1 1   2 1  5  2 iv ii 1  i i   ξ i i   β  3  2   η i dx   2  0 0   ln  k  ln  k   0

  6  6ln  k   2ln 2  k   35  1  2   β    η i3dx 4   4  ln  k     0   48  72ln  k   44ln 2  k   12ln 3  k   105  1 3   β  η  4 dx   4  i 3ln 5  k     0   120  240ln  k   210ln 2  k   100ln 3  k   24ln 4  k   1155  1  4   β  η  i5 dx  6   16 3ln  k     0 

Tweezers/Casimir  β  1  ln  k / 2    1 0    2  γ 3    dx  ln  k / 2     i  2ln  k    0 1 1   2  ln  k    3+5ln  k / 2   3ln 2  k / 2    1 2 iv ii 1  i i   ξ i i   β  3   2γ    i dx   ln 4  k / 2    ln  k   0 0   0   3  3ln  k   ln 2  k    12  27ln  k / 2   26ln 2  k / 2   12ln 3  k / 2    1 3  2    2β   2γ     i dx     ln 4  k  ln 5  k / 2     0     12  18ln  k   11ln 2  k   3ln 3  k     4β      3ln 5  k    1 4  3     i dx   60  168ln  k / 2   213ln 2  k / 2   154ln 3  k / 2   60ln 4  k / 2    0   4γ   3ln 6  k / 2        60  120ln 6  k / 2  ln  k   105ln 2  k   50ln 3  k   12ln 4  k     2β    6   3ln  k    1 5  4     i dx  30  100ln  k / 2   155ln 2  k / 2   145ln 3  k / 2   87ln 4  k / 2   30ln 5  k / 2    0    +8γ   ln 7  k / 2     

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Highlights -

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Considering the influence of surface layer on the instability of NEMS tweezers and cantilevers fabricated from conductive cylindrical nano-wires using the Green-Lagrange strain. Applying a continuum mechanics theory in conjunction with the Euler-beam model for obtaining the constitutive equations of the systems. Incorporating the electrostatic attraction and dispersion forces (Casimir and van der Waals) in the model response.

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