surface stress on the elastic wave band structure of two-dimensional phononic crystals

Physics Letters A 376 (2012) 605–609

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Physics Letters A www.elsevier.com/locate/pla

Effect of interface/surface stress on the elastic wave band structure of two-dimensional phononic crystals Wei Liu, Jiwei Chen, Yongquan Liu, Xianyue Su ∗ LTCS and Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China

a r t i c l e

i n f o

Article history: Received 4 September 2011 Received in revised form 30 October 2011 Accepted 1 November 2011 Available online 26 November 2011 Communicated by R. Wu Keywords: Phononic crystals Elastic wave band structure Multiple scattering theory Interface/surface elasticity

a b s t r a c t In the present Letter, the multiple scattering theory (MST) for calculating the elastic wave band structure of two-dimensional phononic crystals (PCs) is extended to include the interface/surface stress effect at the nanoscale. The interface/surface elasticity theory is employed to describe the nonclassical boundary conditions at the interface/surface and the elastic Mie scattering matrix embodying the interface/surface stress effect is derived. Using this extended MST, the authors investigate the interface/surface stress effect on the elastic wave band structure of two-dimensional PCs, which is demonstrated to be significant when the characteristic size reduces to nanometers. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Propagation of elastic waves in artificial periodic composites, termed as phononic crystals (PCs), has received increasing research attention in recent years [1–6]. The existence of band gaps, i.e. frequency ranges within which propagation of elastic waves is completely forbidden inside the crystals, has been predicted theoretically [1–4] and demonstrated experimentally [5,6]. PCs have also been found to posses some peculiar properties under certain conditions, e.g. negative refraction and sound focusing [7,8], subwavelength imaging [9–12] and collimation [13–17]. Therefore, various applications of PCs have been expected, for example, sound insulators, filters and waveguides. The identification of band gaps and frequency regimes where the aforementioned unique properties may occur relies on the band structure calculation results. Up to now, several methods have been proposed for calculating the elastic wave band structure of PCs, including the plane wave expansion (PWE) method [1,3,4], the finite difference time domain (FDTD) method [18,19], and the multiple scattering theory (MST) method [20–22]. Recently, the possibility of fabricating and measuring micro and nano PCs [23,24] have been demonstrated and the integration of devices based on PCs into communication and sensing systems has been anticipated in pace with the development of nanotechnologies. It has long been known that interface/surface stress may have significant effect on the mechanical and other physi-

*

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cal behaviors of smallsized materials and structures due to the high interface/surface-to-volume ratio [25–27]. Gurtin and Murdoch [28] and Gurtin et al. [29] developed the interface/surface elasticity theory to describe the interface/surface stress effect in the continuum framework and the interface/surface elasticity theory has been demonstrated to be capable of well reproducing the results of direct atomic simulations [27]. The interface/surface elasticity theory has been employed to study the size-dependent effective elastic constants of solids containing nano-inhomogeneities [30]. Some authors have investigated and demonstrated the interface/surface stress effect on the wave propagation in solids. Gurtin and Murdoch [31] investigated the effect of surface stress on plane wave propagation in homogeneous, isotropic half spaces. Wang [32] and Wang et al. [33,34] studied the diffraction of plane elastic waves by nanosized inhomogeneities and demonstrated the considerable importance of the interface/surface stress effect. To date, little attention has been paid to the interface/surface stress effect on the wave propagation in PCs, which is of importance for designing and characterizing miniaturized devices based on PCs. In this Letter, we extend the MST by incorporating the interface/surface elasticity theory and investigate the interface/surface stress effect on the elastic wave band structure of two-dimensional PCs. 2. Basic equations of interface/surface elasticity According to the interface/surface elasticity theory, the interface/surface is viewed as a negligibly thin elastic continuum which adheres to the bulk materials without slipping. The elastic constants of the interface/surface are different from those of the bulk

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materials. Assume a coherent interface Γ between two different solids Ω1 and Ω2 . The equilibrium equation of the interface Γ takes the following form [28,30]





σ 1 − σ 2 · n = −∇s · τ 1

(1)

2

where σ and σ are the stress tensor in solids Ω1 and Ω2 , respectively, and n denotes the unit normal vector to the interface Γ , with positive n being from Ω2 to Ω1 . ∇s · τ denotes the interface/surface divergence of the interface/surface stress τ . For a linear elastically isotropic interface/surface, the constitutive equation reads [27,30,32]





τ = λs tr ε s I + 2μs ε s

(2)

where λs and μs are the interface/surface elastic moduli, and I is the second-rank unit tensor in two-dimensional space. ε s is the second-rank tensor of interface/surface strains, which for a coherent interface/surface equals to the tangential strain of the bulk materials at the interface/surface. The equilibrium equations, constitutive equations and the strain-displacement equations in the bulk materials are the same as those of classical elasticity.

√

inc(0) uinc (r N ) + N (r N ) = u

2

(λ + 2μ)∇∇ · u − μ∇ × ∇ × u + ρω u = 0



anNζ Jnζ (r N )

nζ

=





anζ Jnζ (r) + bnζ Hnζ (r)

(4)

nζ





Jn2 (r) = ∇ × z J n (β r )e inφ

β

(5)

and

Hn1 (r) = ∇ H n (α r )e

inφ



Hn3 (r) =

β



bnM ζ  Hn ζ  (r M )

(8)

M = N n ζ 

B = TA

(9)

where the elastic Mie scattering matrix T = {tnζ n ζ  } can be obtained from the boundary conditions at the interface/surface. For PCs consisting of identical scatterers, T is independent of the lattice position. The scattered field from scatterer M can be translated to the incident field at scatterer N by using the following relation [22]

Hnζ (r M ) =



∇ × ∇ × zH n (β r )e



G nζ n ζ  (R M − R N )Jn ζ  (r N )

(10)

n ζ 

where G nζ n ζ  (R M − R N ) is given by

 G nζ n ζ  (R) =



H n −n (α R )e −i (n −n)φ R , H n −n (β R )e

−i (n −n)φ

R

ζ = ζ = 1

, ζ = ζ  = 2, 3

(11)

In addition, by using Bloch’s theorem one can relate the expansion coefficients anζ at different lattice positions

anMζ = e ik·(RM −RN ) anNζ

(12)

n ζ 

tn ζ  n ζ 

n ζ 

× anN ζ 



e ik·(RM −RN ) G n ζ  nζ (R M − R N ) − δnn δζ ζ 



M = N

=0

(13)

where δ is the Kronecker delta, and thus the elastic wave band structure of 2D PCs can be obtained by solving the following secular equation



Hn2 (r) = ∇ × zH n (β r )e inφ 1

 

tered field expansion coefficients B = {bnNζ }:

 



  ∇ × ∇ × z J n (β r )e inφ 

N ( 0)

an ζ  Jn ζ  (r N ) +

where k is the Bloch wave vector. Since the elastic wave band structure is concerned, the external incident wave in Eq. (8) vanishes. Substituting Eqs. (9)–(12) into Eq. (8) yields

where Jnζ (r) and Hnζ (r) are defined as [22]

1



(3)

where u is the displacement, λ and μ the Lamé constants, and ρ the mass density of the medium. In the cylindrical coordinate system (r , φ, z), the general solution of Eq. (3) is of the following form [22]

Jn3 (r) =

(7)

The scattered wave by each scatterer can be related to the incident wave at the same scatterer by the following relation between the incident field expansion coefficients A = {anNζ } and the scat-

The MST has already been successfully applied to calculating the band structure of various PCs [20–22] and it features excellent convergence and capability of handling PCs with mixing solid and fluid components. Compared with the other methods, the MST explicitly utilizes the boundary conditions at the interface/surface, and thus offers the potential to conveniently embody the interface/surface stress effect. Based on the interface/surface elasticity, here we extend the MST and investigate the interface/surface stress effect on the elastic wave band structure of 2D PCs. To be complete, the fundamentals of the MST are outlined in this section. Consider two-dimensional PCs consisting of aligned cylindrical inclusions or pores periodically embedded in a matrix. The timeharmonic elastic wave equation in linear, isotropic, homogeneous media reads



usc M (r M )

where r N and r M denote the position of the same spatial point measured from scatterers N and M, respectively. Using the general solution in Eq. (4), Eq. (7) can be recast into the following form

n ζ 

3.1. Fundamentals of MST

Jn1 (r) = ∇ J n (α r )e inφ

 M = N

3. MST with interface/surface stress effect taken into account

u(r) =

√

where α = ω ρ /(λ + 2μ), β = ω ρ /μ, J n (x) is the Bessel function, and H n (x) is the Hankel function of the first kind. The index ζ in Eq. (4), running from 1 to 3, represents three wave modes, i.e. the longitudinal mode (ζ = 1) and the two shear modes (ζ = 2, 3). The first and second term of the right-hand side of Eq. (4) stand for the incoming and outgoing waves, respectively. According to the MST, in a multiple scattering system, the incident wave at each scatterer N located at the lattice position R N , uinc N , is equal to the sum of the scattered waves from all the other scatterers plus the possible external incident wave uinc(0) . This physical picture can be mathematically expressed as

inφ





(6)

det

n ζ 



tn ζ  n ζ  gn ζ  nζ (k) − δnn δζ ζ 

= 0

(14)

W. Liu et al. / Physics Letters A 376 (2012) 605–609

where

gn ζ  nζ (k) =



e ik·RN G n ζ  nζ (R N )

(15)

The sum in the right-hand side of Eq. (15) is conditionally convergent and difficult to be directly evaluated with high accuracy. Alternatively, it can be calculated using the method by Chin et al. [35].

 anζ Jnζ (r) + bnζ Hnζ (r)

(16)

nζ



cnζ Jnζ (r)

(17)

nζ

When the interface stress effect is taken into account, the boundary conditions at the interface of the matrix and the scatterer consist of the interface equilibrium conditions described in Eq. (1) and displacement continuity conditions, i.e.









u1 (r) − u2 (r) r =a = 0

(18)

σ 1 (r) − σ 2 (r) · er |r =a = −∇s · τ

(19)

where er is the unit vector along the radial direction. If the scatterers of 2D PCs are cylindrical voids, only Eq. (19) is used and the stress tensor in the scatterer σ 2 (r) vanishes. If one of the two components of the PC is a fluid, i.e. if μ1 = 0 or μ2 = 0, then the displacement continuity conditions in Eq. (18) should be replaced by the continuity of only the radial displacement. In the cylindrical coordinate system (r , φ, z), the surface divergence in Eq. (19) can be written in component form as following

∇s · τ =

1 ∂ τφφ r ∂φ

eφ +

1 ∂ τφ z r ∂φ

ez −

τφφ r

er

(20)

s εzz =

ε

s zφ

1 ∂ uφ r ∂φ ∂ uz

+

where

A 11 = n J n (α1 a) − α1 a J n+1 (α1 a), B 11 = nH n (α1 a) − α1 aH n+1 (α1 a), C 11 = n J n (α2 a) − α2 a J n+1 (α2 a),

A 21 = in J n (β1 a) B 21 = inH n (β1 a) C 21 = in J n (β2 a)

A 12 = in J n (α1 a),

A 22 = β1 a J n+1 (β1 a) − n J n (β1 a)

B 12 = inH n (α1 a),

B 22 = β1 aH n+1 (β1 a) − nH n (β1 a)

C 12 = in J n (α2 a),

C 22 = β2 a J n+1 (β2 a) − n J n (β2 a)

A 1 = (2n + 2) J n+1 (β1 a) − β1 a J n+2 (β1 a) B 1 = (2n + 2) H n+1 (β1 a) − β1 aH n+2 (β1 a) C 1 = (2n + 2) J n+1 (β2 a) − β2 a J n+2 (β2 a)





A 13 = (2 + k1 )μ1 n2 − n J n (α1 a)

  − λ1 (2n + 2) + 2μ1 (2n + 1) − μ1k1 α1 a J n+1 (α1 a)

A 23 = (2 + k1 )μ1 in(n − 1) J n (β1 a) − (2 + k1 )μ1 inβ1 a J n+1 (β1 a)





B 13 = (2 + k1 )μ1 n2 − n H n (α1 a)

  − λ1 (2n + 2) + 2μ1 (2n + 1) − μ1k1 α1 aH n+1 (α1 a) + (λ1 + 2μ1 )α12 a2 H n+2 (α1 a)

B 23 = (2 + k1 )μ1 in(n − 1) H n (β1 a)

− (2 + k1 )μ1 inβ1 aH n+1 (β1 a)   C 13 = 2μ2 n2 − n J n (α2 a)   − λ2 (2n + 2) + 2μ2 (2n + 1) α2 a J n+1 (α2 a) + (λ2 + 2μ2 )α22 a2 J n+2 (α2 a) C 23 = 2μ2 in(n − 1) J n (β2 a) − 2μ2 β2 a J n+1 (β2 a) A 14 = (2 − nk1 )μ1 in(n − 1) J n (α1 a)

− (2 + k1 )μ1 inα1 a J n+1 (α1 a)   A 24 = (2 − nk1 )μ1 n − n2 J n (β1 a) + (2 − nk1 )μ1nβ1 a J n+1 (β1 a) − μ1 β12 a2 J n+2 (β1 a) − (2 + k1 )μ1 inα1 aH n+1 (α1 a)   B 24 = (2 − nk1 )μ1 n − n2 H n (β1 a)

ur

+ (2 − nk1 )μ1nβ1 aH n+1 (β1 a)

r

− μ1 β12 a2 H n+2 (β1 a) C 14 = 2μ2 in(n − 1) J n (α2 a) − 2μ2 inα2 a J n+1 (α2 a)

∂z



∂ uφ = εφ z = + 2 r ∂φ ∂z s

(22)

B 14 = (2 − nk1 )μ1 in(n − 1) H n (α1 a)

where eφ and ez are unit vectors along the azimuthal and z direction, respectively. And the components of the surface strain ε s can be obtained as s εφφ =

A 14 an1 + A 24 an2 + B 14 bn1 + B 24 bn2 = C 14 cn1 + C 24 cn2

+ (λ1 + 2μ1 )α12 a2 J n+2 (α1 a)

and that in the scatterer is



A 13 an1 + A 23 an2 + B 13 bn1 + B 23 bn2 = C 13 cn1 + C 23 cn2

A 2 an3 + B 2 bn3 = C 2 cn3

As seen in the preceding section, the determination of the elastic Mie scattering matrix is critical to the calculation of the band structure of PCs. For macroscopic PCs where the interface/surface stress effect is neglected, T is calculated utilizing the classical boundary conditions, i.e. the displacement and traction continuity conditions at the interface [22], or the traction free conditions at the surface. However, when the interface/surface stress effect is taken into account as the characteristic size of PCs reduces to nanometers, the nonclassical boundary conditions described by the interface/surface elasticity theory should be employed to obtain the elastic Mie scattering matrix T, which is derived here. Consider a scatterer with the radius a, and let ρi , λi and μi denote the mass density and Lamé constants of the matrix (i = 1) and the scatterer (i = 2), respectively. The displacement in the matrix can be written as



A 12 an1 + A 22 an2 + B 12 bn1 + B 22 bn2 = C 12 cn1 + C 22 cn2

A 1 an3 + B 1 bn3 = C 1 cn3

3.2. Elastic Mie scattering matrix embodying the interface/surface stress effect

u2 (r) =

Eqs. (18) and (19) combined with Eqs. (2), (20) and (21) give rise to the following equations:

A 11 an1 + A 21 an2 + B 11 bn1 + B 21 bn2 = C 11 cn1 + C 21 cn2

R N =0

u1 (r) =

607

1

1 ∂ uz







C 24 = 2μ2 n − n2 J n (β2 a) + 2μ2nβ2 a J n+1 (β2 a) (21)

− μ2 β22 a2 J n+2 (β2 a)

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W. Liu et al. / Physics Letters A 376 (2012) 605–609

Fig. 1. Sketch of the 2D square lattice PC composed of aligned infinite holes embedded in an aluminum matrix.

A 2 = 2(1 − nk2 )μ1 n(n + 1) J n+1 (β1 a)

  − μ1 (3n + 4) − n2k2 β1 a J n+2 (β1 a)

+ μ1 β12 a2 J n+3 (β1 a) B 2 = 2(1 − nk2 )μ1 n(n + 1) H n+1 (β1 a)

  − μ1 (3n + 4) − n2k2 β1 aH n+2 (β1 a) + μ1 β12 a2 H n+3 (β1 a)

C 2 = 2μ2 n(n + 1) J n+1 (β2 a) − μ2 (3n + 4)β2 a J n+2 (β2 a)

Fig. 2. Band structure of the 2D square lattice PC for in-plane modes with surface A. (For interpretation of the references to color in this figure, the reader is referred to the web version of this Letter.)

+ μ2 β22 a2 J n+3 (β2 a) where k1 = (λs + 2μs )/(μ1 a) and k2 = μs /(μ1 a) are two dimensionless parameters characterizing the interface/surface stress effect. Solving Eq. (22) yields the relation between the coefficients {anζ } and {bnζ }, and thus the elastic Mie scattering matrix T embodying the interface/surface stress effect. 4. Numerical results and discussions The size-dependence of the interface/surface stress effect on the band structure of 2D PCs can be directly identified from the involvement of the scatterer radius a in the two dimensionless parameters k1 and k2 . Interface/surface elastic constants can be obtained from atomic simulations [27,36] and experiments. Existing results show that interface/surface elastic constants can be either positive or negative depending on the crystallographic structure, and the absolute values of μs /μ1 and λs /μ1 are on the order of angstroms [27,36]. For macroscopic PCs with large scatterer radius a, k1 and k2  1, the interface/surface effect can be neglected and the elastic Mie scattering matrix T reduces to that with no interface/surface stress effect. However, as the radius of the scatterer shrinks to nanometers, k1 and k2 would become significant and the interface/surface stress effect cannot be neglected. Here, using the extended MST, we demonstrate the surface stress effect on the band structure of 2D PCs. Consider a 2D PC composed of aligned infinite holes embedded in an aluminum matrix and arranged in square lattice, which is sketched in Fig. 1. The radius of the hole a and the lattice constant d are set to be 4.1 nm and 10 nm, respectively, resulting a filling ratio f of 0.53. The material parameters of aluminum are mass density ρ = 2697 kg/m3 , Lamé constants λ = 52.09 GPa and μ = 34.7 GPa. Two sets of surface elastic constants, which were obtained via molecular dynamics simulation by Miller and Shenoy [27], are employed and denoted as surface A (λs = 3.489 N/m, μs = −6.218 N/m) and surface B (λs = 6.842 N/m, μs = −0.376 N/m), respectively. The elastic wave band structure of 2D PCs can be decoupled into two kinds of modes: in-plane modes and out-of-plane modes. Fig. 2 shows the band structure of the 2D square lattice PC for in-plane modes with surface A, with the blue triangles denoting the conventional results without surface stress effect while the red dots representing the results with surface stress effect. It can be

Fig. 3. Band structure of the 2D square lattice PC for out-of-plane modes with surface A.

clearly observed in Fig. 2 that the surface stress effect significantly influences the band structure, and the influence is more prominent in the relatively high frequency regime. For the 2D square lattice PC with surface A, the frequency bands for in-plane modes shift downwards as the surface stress effect is considered. In addition, when the surface stress effect is taken into account, there exists a complete band gap for in-plane modes extending the frequency from 1.3160 to 1.6439 THz. However, if the surface stress is neglected, the band gap is from 1.4591 to 1.7489 THz, i.e. the first band gap for in-plane modes becomes lower and wider due to the surface stress effect. Shown in Fig. 3 is the band structure of the 2D square lattice PC for out-of-plane modes with surface A. Like the case of in-plane modes, the frequency bands for out-of-plane modes also shift downwards. The first complete band gap for out-of-plane modes also becomes wider as the surface stress effect is taken into account. It extends from 1.1001 to 1.4701 THz without sur-

W. Liu et al. / Physics Letters A 376 (2012) 605–609

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and its value for surface B is considerably small, and thus the surface stress effect on the band structure here is not that obvious. However, as evidenced in Figs. 2–4, surface stress effect would be significant in many cases and should be taken into account in the design and characterization of miniaturized PC devices. 5. Conclusions In summary, the interface/surface elasticity is employed to describe the nonclassical conditions at the interface/surface and the MST is accordingly extended to investigate the interface/surface stress effect on the elastic wave band structure of 2D PCs in this Letter. The elastic Mie scattering matrix embodying the interface/surface stress effect is obtained. Numerical examples for a 2D PC show that the surface stress effect on the band structure would be significant as the characteristic size reduces to nanometers. Acknowledgements This work is supported by the National Natural Science Foundation of China under grant No. 90916007. Fig. 4. Band structure of the 2D square lattice PC for in-plane modes with surface B.

References

Fig. 5. Band structure of the 2D square lattice PC for out-of-plane modes with surface B.

face stress effect, and shifts to being from 0.9959 to 1.4030 THz as the surface stress effect is present. The elastic wave band structure of the 2D square lattice PC for in-plane modes with surface B is plotted in Fig. 4 and that for out-of-plane modes is presented in Fig. 5. Again, it is manifested in Fig. 4 that the surface stress would considerably affect the band structure. However, in contrast to those in Fig. 2, the frequency bands for in-plane modes shift upwards as the surface stress effect is present. Additionally, with the surface stress being considered, the first complete band gap becomes narrower, extending from 1.5337 to 1.8055 THz, which is different from the case of surface A. It is noted in Fig. 5 that in the considered frequency range the difference between the results with and without surface stress effect is much tinier than those in Figs. 2–4. The reason is that for out-of-plane modes only the surface elastic constant μs is involved

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