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Migration velocity of an elliptical inclusion in piezoelectric film
Thin Solid Films 531 (2013) 137–143
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Migration velocity of an elliptical inclusion in piezoelectric film Y. Qin, X. Wang ⁎ School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, PR China
a r t i c l e
i n f o
Article history: Received 16 March 2012 Received in revised form 22 December 2012 Accepted 3 January 2013 Available online 16 January 2013
a b s t r a c t This paper presents an analytical method to investigate on the migration velocity of an elliptical inclusion in piezoelectric film under coupled gradient stress and electric field. The effects of gradient stress, electric field, the material property and the shape parameter of inclusion on the migration velocity of the elliptical inclusion in piezoelectric material are described and discussed. © 2013 Elsevier B.V. All rights reserved.
Keywords: Piezoelectric film Elliptical inclusion Migration velocity Thermodynamics potential
1. Introduction In the past, many investigations on the use of piezoelectric materials have already been presented for active control of smart structures with light weight. The inherent electromechanical effect of piezoelectric materials has many important engineering applications [1–6]. Current examples motivating the present study are strained semiconductor laser devices [7,8], in which residual stresses induced by lattice mismatch between buried active components and surrounding materials crucially affect electronic performance. In processing of thin films, various structural defects such as inclusions and voids in the thin film are often generated, which have detrimental effects on the function of film structures [9,10]. Investigations on mechanical deformations induced by misfitting inclusions in an infinite or semi-infinite piezoelectric medium are fundamental physical and engineering problems, which can be seen in earlier researches on this subject by Nowacki [11] and Mura [12]. Because of the rapid development of piezoelectric smart structures in the last years, many related works on coupled electromechanical characteristics of piezoelectric material with inclusions are further investigated by many researchers [13–17]. Della and Shu [13] utilized the Euler–Bernoulli beam theory and Rayleigh–Ritz approximation technique to present a mathematical model for the vibration of beams with piezoelectric inclusions. Based on the linear elasticity theory and Green's function method, Kuvshinov [14] presented explicit, closedform expressions to describe electroelastic deformations due to polyhedral inclusions in uniform half-space and bi-materials. Fakri and Azrar [15] predicted the electroelastic and thermal responses of piezoelectric composites with and without voids. Lin and Sodano [16] extended the ⁎ Corresponding author. E-mail address: [email protected] (X. Wang). 0040-6090/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tsf.2013.01.013
double inclusion model to multiphase composites with piezoelectric layers, and predicted the electroelastic properties of the multifunctional composites. Based on the viscoelastic principle, Aldraihem [17] developed a comprehensive micromechanics model to estimate the effective viscoelastic properties of hybrid composites containing polymer matrix. Tang et al. [18] reported that the energy density of the nanocomposite could be significantly increased through the use of piezoelectric nanowires and a polymer with greater breakdown strength. It is seen from Ref. [19] that strain-release characteristics of composites are reinforced by nanowires on stretchable substrates. Utilizing dielectrophoretic assembly, Tomer and Randall [20] make anisotropic composites reinforced by BaTiO3 particles in a silicone elastomer thermoset polymer, and investigate the effect of electrical properties on the property of these composites. It is described in Ref. [21] that the energy-storage capability of the nanocomposite can be enhanced by the alignment of piezoelectric nanowires in the direction of the applied electric field compared to randomly oriented samples. In the above investigation into the influence of inclusions in piezoelectric materials on the mechanical and electrical characteristics of piezoelectric smart structures, the inclusion is fixed at a determinate point in piezoelectric material. In fact, the various solid films with inclusion are often subjected to severe gradient stresses induced by thermal mismatching and electric field, so that the inclusion in films may migrate [22–25]. Because one of the major damage mechanisms is from the inclusion (void) coalescence in stress concentration region induced by the motion of inclusion (void), the dynamics of stress-driven inclusion in metallic solids has been generating great research interest for the failure of metallic interconnects in integrated circuits [26–28]. Utilizing self-consistent numerical method, Gungor and Maroudas [30] describe the effects of complex external stresses on the electromigration-driven motion of morphologically stable voids in elastically deforming metallic thin films, and reveal the complex evolution of voids. It is seen in
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Ref. [31] that when the morphological stability limit is approached, the void migration speed is substantially dependent on the void size. Therefore, the inclusion migration-induced failure should also pose a great concern in piezoelectric smart structures. However, because the coupled electroelastic characteristic of piezoelectric material with inclusion is complex, the investigation on the migration of inclusion in piezoelectric film is not found in literatures so far. In this paper, we give an analytical solution for the migration of an elliptical inclusion in piezoelectric film under coupled gradient stress and electric field, based on the atomic migration principle in a diffused interface layer between inclusion and piezoelectric matrix from regions of high chemical potential to those of low chemical potential along the interface layer. The numerical examples describe the effects of gradient stress, electric field, and the material property and the shape of inclusion on the migration velocity of the elliptical inclusion in piezoelectric film. The present work may stimulate further interest on this topic because the reliability of piezoelectric smart structures is extremely sensitive to inclusion defects. 2. Basic equations and solving method Fig. 1 shows the two-dimensional (2D) model of an elliptic inclusion in piezoelectric film under coupled gradient stress and electric field, in which this 2D implementation conveys the essence of a three-dimensional (3D) problem of inclusion in piezoelectric material, and represents an inhomogeneity extending throughout the thickness along z, consistently with experimental observations in thinpiezoelectric film. In order to obtain the strict stress and electric field solutions at the interface required in the present derivation, the calculating model in Fig. 1 is based on the migration of inclusion in an infinite piezoelectric material under the combined electric and stress gradient loads, in which its realistic application in a finitewidth piezoelectric film will exist in scale effects. However, Li and Chudnovsky [32] have proved that the scale effect is negligible when the ratio of inclusion size to a characteristic size of the analytical system is less than 0.15 by using the method of numerical analyses. Because the characteristic size of the analytical system for an inclusion in a finite piezoelectric film is the width of the film, the present model can be used to calculate the inclusion motion in a realistic piezoelectric film when the inclusion size is much less than the width of film. At infinity, the model is subjected to gradient stress field qy =pX in the Y direction by thermal mismatch, where p ¼ dqy dX is a constant, a positive value of p presents a rising stress along the X-axis, while in the X-direction subjected to electric field E0. In Fig. 1, the global coordinate system OXY is fixed at the symmetric center of piezoelectric film, the local (moving) coordinate system oxy is fixed at the center of an elliptic inclusion, and xL represents a moving distance from the center
of the elliptic inclusion to the symmetric center of piezoelectric film. The relation between the two coordinate systems is given by xL = X −x and y= Y. The constitutive relations of orthotropic piezoelectric film are written as [29] 8 9 2 S11 < εx = εy ¼ 4 S13 :γ ; 0 xy
Dx Dy
0 ¼ d31
9 2 38 0 < σx = 0 0 5 σy þ 4 0 :τ ; S44 d15 xy
S13 S33 0 0 d33
8 9 < σ x = e d15 σ þ 11 0 0 : y; τxy
3 d31 E d33 5 x Ey 0 0 e33
Ex Ey
ð1aÞ
ð1bÞ
where σx, σy, τxy and εx, εy, γxy are, respectively, stresses and strains in the piezoelectric film along the main direction of coordinate system oxy, Ex, Ey and Dx, Dy are, respectively, the electric field strengths and the electric displacements along the x and y directions, Sij is the flexible coefficients of piezoelectric film, and dij and eij are the piezoelectric and dielectric constants, respectively. The inclusion in piezoelectric film may be moved by the atomic migration of an inclusion/matrix interface layer from bulk regions of high chemical potential to those of low chemical potential along the interface layer. Because the interface diffusion is generally much faster than the bulk one, only the interface diffusion is considered as the major mechanism for the inclusion motion in piezoelectric film under coupled gradient stress and electric field. In general, instable inclusions including very narrow morphologies may be split into small inclusions with stable shape to minimize the total Gibbs free energy of the system under external loads. In order to give an analytical solution for the migration velocity of an inclusion in piezoelectric film under coupled field, it is assumed in this analytical model that the inclusion is considered as an elliptic inclusion with shape parameter m, and the morphology of the elliptic inclusion is taken to be stable throughout the range of the shape parameters when the inclusion simply migrates along the length direction of film. Because the shape and size of an inclusion in piezoelectric film do not change during motion, the piezoelectric film is the source and sink for the atomic flux along the interface layer between inclusion and piezoelectric matrix. Under coupled gradient stress and electric field, atoms diffuse along the interface layer from one side of high chemical potential to the other of lower, so that the inclusion migrates in the piezoelectric matrix against the migration of the atoms. Thus, the chemical potential in the interface layer driving this atomic flux is composed of the free energy, the local elastic strain energy and the local electric energy in the interface layer, as follows μ ¼ μ 0 −Ωγi κ−Ωσ n þ ΩW p þ ΩW E ;
ð2Þ
where μ0 is the reference value of the potential, γi is the free energy of interface layer, κ is the surface curvature of the inclusion, positive for convex, Ω is the atomic volume, σn is the normal stress on the inclusion surface, Wp is the elastic strain energy density stored in the interface layer associated with an atom, as follows Wp ¼
1 ðσ ε þ σ θ εθ þ τnθ γ nθ Þ; 2 n n
ð3aÞ
and WE is the electric energy density stored in the interface layer associated with an atom, as follows WE ¼
Fig. 1. The motion of an elliptical inclusion in infinite piezoelectric material under combined gradient stress field and electric field.
1 ðE D þ Eθ Dθ Þ 2 n n
ð3bÞ
where Dn, Dθ and En, Eθ represent, respectively, electric displacements and electric strength along the normal n and circumference θ of the interface layer.
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
In Fig. 1, J represents the atomic flux along the interface layer between inclusion and piezoelectric matrix, the number of atoms per time crossing unit length along the interface layer, which satisfies Nernst-Einstein equation, as follows D δ ∂μ J¼− i i ΩKT ∂s
ð4Þ
where Di is the diffusion coefficient of interface layer, δi is the thickness of the interface layer, K is the Boltzmann's constant, T is the absolute temperature, and s is the arc length of the interface. In practice, the interfacial diffusion coefficient should be anisotropic, which leads to faceted stable morphologies as opposed to round/elliptic ones in this model. Because the objective of this study is to give an analytical solution for the horizontal motion of an elliptical inclusion in piezoelectric film, the interfacial diffusion coefficient in the present model is taken to be isotropic, as the assumption from some investigations [33,34]. Although the approximation ignoring the anisotropic diffusion characteristic of interface is involved in the solving model, the closed-form solution with a general manner can be obtained in this paper, which explicitly gives the essential behavior for the motion of an inclusion in piezoelectric film driven by combined electric field and gradient stress. It is seen from Eq. (4) that the atomic flux in the interface layer is dependent on the variation of chemical potential μ along the arc length of interface. The mass conservation condition is given by dJ V n þ ¼0 ds Ω
ð5aÞ
the interface between inclusion and piezoelectric matrix. From Eq. (9), we have rffiffiffiffiffiffiffiffiffiffiffiffiffi ! V 1−m J¼ ρ −y þ C Ω 1þm
ð5bÞ
Vn in Eq. (5a) is the normal diffusion velocity of the interface, which is written as V n ¼ V sinφ
ð6Þ
where V is the translational velocity of the inclusion with stable morphology under the coupled mechanical and electric fields as prescribed. In the present model, the inclusion motion is horizontal due to the orientation of the external fields driving the motion of the inclusion. Substituting Eqs. (6) and (5b) into Eq. (5a) gives dJ V ¼ : dy Ω
ð7Þ
The shape for an elliptic inclusion in piezoelectric film as in Fig. 1 has the same area as a circle of radius ρ, which can be described by x¼ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm cosθ; 1−m
y¼ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m sinθ 1þm
ð8Þ
where θ is the angle between the vector radius ρ of ellipse and x-axes, and m is the shape parameter of the inclusion. The shape parameter m=0 corresponds to a circular inclusion, m→ +1 to x-direction narrow inclusion and m→ −1 to y-direction narrow inclusion, respectively. To consider the effect of the local interface changes of a moving elliptical inclusion on the atomic flux along the interface, Eq. (7) should be integrated along thepinterface between inclusion and piezoelectric ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi matrix from y to y ¼ ρ ð1−mÞ=ð1 þ mÞ, as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ ð1−mÞ=ð1þmÞ V ∫dJ ¼ ∫y dy Ω
ð9Þ
When the elliptic shape of inclusion does not vary during motion, the horizontal velocity V of the inclusion interface is a constant along
ð10Þ
where C is the undetermined constant which can be solved by the continuous condition of the atomic flux at the apex A of elliptic inclusion. Utilizing Eqs. (2), (4) and (10), the continuous condition of the atomic flux at the apex A of elliptic inclusion is expressed as " # " # rffiffiffiffiffiffiffiffiffiffiffiffiffi ! Di δi ∂κ ∂σ n ∂W p ∂W E V 1−m ¼ γi þ ρ −y þ C − − KT Ω 1þm ∂s ∂s ∂s ∂s A
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y¼ρ
:
ð1−mÞ=ð1þmÞ
ð11Þ Because ∂ κ/∂ s at the apex A of elliptic inclusion equals zero, solving Eq. (11) gives C¼
" Di δi ∂σ n KT ∂s
j −∂W∂s j −∂W∂s j p
A
#
E
ð12Þ
A
A
.
.
.
∂W p
∂σ n ∂W E where
,
and
are the gradients of normal ∂s A ∂s A ∂s A stress, strain energy density and electric energy density at the top apex A of the interface between inclusion and piezoelectric matrix. Due to the symmetrical diffusion requirement of the interface between inclusion and piezoelectric matrix as in Fig. 1, the atom flux at y = 0 should be zero at all times, as follows:
where ds ¼ −dy=ð sinφÞ
139
½ J y¼0 ¼
Vρ Ω
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m þ C ¼ 0: 1þm
ð13Þ
Substituting Eq. (12) into (13), yields ΩDi δi V¼ ρKT
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm ∂σ − n 1−m ∂s
j
A
þ
∂W p ∂s
j
∂W E þ ∂s A
j
!
A
:
ð14Þ
It is seen from Eq. (14) that the horizontal velocity of an inclusion in piezoelectric material is mainly dependent on the gradients of normal stress, strain energy and electric energy at the top apex A of the interface between inclusion and piezoelectric matrix. So far, the analytic expression for the stress field at the interface between inclusion and piezoelectric matrix under a gradient stress field as in Fig. 1 is not presented in literatures. Here, utilizing a loading superposition method as in Fig. 2, a gradient stress field, qy = pX, applied on a piezoelectric film with inclusion can be described from a superposition of an antisymmetric stress field of stress gradient with respect to the y-axis of the moving elliptic inclusion and a uniform stress field, as follows: qy ¼ qy1 þ qym :
ð15Þ
Based on the linear elastic theory, the stress field σijA at the apex A of the interface between inclusion and piezoelectric under coupled loadings qy = qym + qy1 is given by A A A σ ij qy ¼ pX ¼ σ ij qy1 ¼ px þ σ ij qym ¼ pxL :
ð16Þ
Thus, the gradients of normal stress at the apex A is written as ∂σ n qy ¼ pX ∂s
j
¼ A
∂σ n qym ¼ pxL ∂s
j
þ A
∂σ n qy1 ¼ px ∂s
j
:
ð17Þ
A
Because an antisymmetric gradient stress field, qy1 = px, with respect to the y-axis of the moving elliptic inclusion makes the stresses σijA(qy1 = px) at the apex A of the interface between inclusion and
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a
b
c
Fig. 2. The stress at apex A of inclusion under stress pX shown in (a) can be obtained from the superposition of an anti-symmetrically applied stress px about the y-axis shown in (b) and a uniform applied stress field with stress pxL shown in (c).
piezoelectric matrix equals zero, the stresses σijA(qy = pX) at the apex A of the interface under remote gradient stress field can be replaced by the stresses σijA(qym = pxL) at the apex A of the interface. 3. The stress and electric fields in piezoelectric materials with inclusion
Substituting (22) into Eqs. (1a) and (1b), yields
Dx ¼ −2real
3 X
d15 ηk þ e11 λk F k ″ðzk Þ;
k¼1
ð23Þ
3 X 2 d31 ηk þ d33 −e33 λk ηk F k ″ðzk Þ Dy ¼ 2real k¼1
The stress function and electric potential function of transversely isotropic piezoelectric material with inclusion, under uniform mechanical and electric loads, can be, respectively, taken as [29]
3 X εx ¼ 2real pk F k ″ðzk Þ;
εy ¼ 2real
k¼1 3 X F ðx; yÞ ¼ 2real F k ðzk Þ;
3 X ϕðx; yÞ ¼ 2real λk F k ′ðzk Þ
k¼1
ð18Þ
ð24Þ
ηk pk þ qk F k ″ðzk Þ
where
zk ¼ x þ ηk y;
λk ¼ ½
d31 ηk 3 þðd33 −d15 Þηk
ðe11 þe33 ηk Þ 2
ð19Þ
h i2 h i 4 2 2 2 2 S11 η þ ð2S13 þ S44 Þη þ S33 e11 þ e33 η −η d31 η þ ðd33 −d15 Þ ¼ 0:
ð20Þ Because of the symmetry of piezoelectric film with inclusion under uniform mechanical and electric loads, the solution for Eq. (20) is yield by η2 ¼ α 2; þ β2 i;
η3 ¼ −α 2 þ β2 i;
pk ¼ S11 ηk þ S13 −d31 ηk λk ; qk ¼ ðS13 ηk þ S33 −d33 ηk λk Þ=ηk 2
:
In the above formula, F(x,y) is the stress function, Fk(zk) is the complex function and ηk is the three complex roots (with positive imaginary parts), which satisfy the characteristic equation, as follows
η5 ¼ η2 ;
k¼1
k¼1
k¼1
where
η1 ¼ β1 i;
2εxy ¼ 2real
3 X
3 X qk ηk F k ″ðzk Þ;
η4 ¼ η1 ;
Utilizing transformation formula, the stress and electric field strength in the interface along the vector radius ρ of ellipse inclusion and the angle θ between the vector radius ρ and the x-axes are, respectively, expressed as 2
2
σ n ¼ σ x cos θ þ σ y sin θ−τxy sin2θ; τ nθ ¼ 0:5 σ x −σ y sin2θ þ τxy cos2θ
2
2
σ θ ¼ σ x sin θ þ σ y cos θ þ τxy sin2θ;
ð26Þ 2
2
ε n ¼ ε x cos θ þ ε y sin θ−0:5γ xy sin2θ; γ nθ ¼ εx −εy sin2θ þ γ xy cos2θ
2
2
ε θ ¼ ε x cos θ þ εy sin θ þ 0:5γ xy sin2θ;
ð27Þ
ð21Þ
η6 ¼ η3
ð25Þ
En ¼ Ex cosθ þ Ey sinθ; Eθ ¼ −Ex sinθ þ Ey cosθ Dn ¼ Dx cosθ þ Dy sinθ; Dθ ¼ −Dx sinθ þ Dy cosθ:
ð28Þ
where β1, β2 and α2 depend on the piezoelectric material constants, and ηk represents the conjugation of the ηk, (k=1, 2, 3). Once the roots ηk, (k=1, 2, 3) are obtained from Eq. (18) [29], the stress and electric field strength in piezoelectric film with inclusion are written as
4. Numerical examples and discussions
3 3 X X 2 σ x ¼ F ðx; yÞyy ¼ 2real ηk F k ″ðzk Þ; σ y ¼ F ðx; yÞxx ¼ 2real F k ″ðzk Þ
In example calculations, the properties of piezoelectric film are taken as:
k¼1
k¼1
3 3 X X σ xy ¼ −F ðx; yÞxy ¼ −2real ηk F k ″ðzk Þ; Ex ¼ −ϕðx; yÞx ¼ −2real λk F k ″ðzk Þ; ð22Þ k¼1
3 X Ey ¼ −ϕðx; yÞy ¼ −2real ηk λk F k ″ðzk Þ: k¼1
k¼1
Sp11 ¼ 11:6 10−12 m2 =N; p −12 2 m =N; S44 ¼ 45:0 10 dp33 ¼ 152:0 10−12 C=N;
Sp33 ¼ 14:8 10−12 m2 =N; Sp13 ¼ −4:97 10−12 m2 =N; p −12 p −12 d15 ¼ 440:0 10 C=N; d31 ¼ −60:2 10 C=N; ð29Þ ep11 ¼ 8:76 109 F=m; ep33 ¼ 3:98 109 F=m:
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
141
Here, the material properties of inclusion in the piezoelectric film are taken as: i
p
Sij ¼ αSij ;
i
p
dij ¼ αdij ;
i
p
eij ¼ αeij
ð30Þ
where α is a proportion coefficient presenting the inclusion property, when α b 1, the stiffness of inclusion is larger than the stiffness of piezoelectric matrix, and the superscripts i and p denote the material properties of inclusion and piezoelectric matrix. The influence of the inclusion property on the moving velocity is described by using the value of α = 0.001, α = 0.5, α = 2, and α = 10 in example calculations.Eq. (14) can be rewritten as
V¼
ΩDi δi ρkT
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm ∂σ f − n 1−m ∂s
j
A
∂σ e − n ∂s
j
f
þ A
∂W p ∂s
j
A
e
þ
∂W p ∂s
j
A
þ
∂W fE ∂s
j
þ A
∂W eE ∂s
j
! : A
ð31Þ In the above formula, the superscripts f and e represent the normal stress, the strain energy density and the electric energy density induced by mechanical loading and electrical loading. Utilizing the stress field and electric field of piezoelectric film with inclusion obtained from Eqs. (18) to (28), the gradient distributions of normal stress, strain energy and electric energy along the interface between inclusion and piezoelectric matrix are described in Figs. 3 and 4. Figs. 3 and 4 show that the gradient of the normal stress at the apex A of inclusion is much larger than the gradient of the strain energy and electric energy at the apex, so that it is concluded that the motion of inclusion in piezoelectric material is mainly induced by the gradient of the normal stress at the apex A of the interface layer between inclusion and piezoelectric matrix. Substituting the gradient values of normal stress, strain energy and electric energy at the apex A of the interface layer obtained from Eqs. (18) to (28) into Eq. (31), the normalized velocity of the elliptic inclusion in piezoelectric material is simplified as
V ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm p ð−B1 ⋅px L ρ þ B2 ⋅E0 Þ⋅ρS11 : 1−m
ð32Þ
where p and E0 are, respectively, the stress gradient applied on the y direction of piezoelectric material and the electric strength applied on the X direction, and the parameters B1 and B2 are obtained by solving
Fig. 4. The gradient distributions of normal stress along the interface, where the shape parameter of inclusion, m= 0.4.
Eqs. (18) to (28), as shown in Table 1. Dimensionless parameters related to each calculating terms are, respectively, taken as
V ¼
kTρ2 Sp11 V; ΩDi δi
x L ¼ xL =ρ:
ð33Þ
Fig. 5 describes the normalized motion characteristics of the rigid inclusion (α = 0.001 → 0) in piezoelectric matrix. Because the atoms in the interface layer with tensile or compressive stresses have high chemical potentials, the atoms with high stress migrate to the region with lower stress, which leads to the inclusion moving to higher stress region in piezoelectric film. As shown in Fig. 5, the shape parameter m and the position of inclusion in piezoelectric film affect the motion velocity of the inclusion; the motion velocity of inclusion is linear proportional to the distance between the center of inclusion and the zero point of the gradient stress applied on piezoelectric material; the round inclusion (m = 0) moves much slower than other inclusions with shape parameters m = − 0.4, 0.4, and 0.8; the inclusion for m = 0.4 moves with accreting velocity towards the region of higher compressive region; the inclusion for m = − 0.4 moves with accreting velocity towards the region of higher tensional stress region when the position of inclusion in piezoelectric film is x L b−20; and the inclusion for m = 0.8 moves with accreting velocity towards the region of higher tensional stress region when the position of inclusion in piezoelectric is x L > 15. Fig. 6 shows the motion characteristics of the harder inclusion (α = 0.5) in piezoelectric film. The migration velocity of the inclusion with shape parameter m =0.8 is much larger than that of inclusion with other shape parameters. When the position of inclusions in piezoelectric material is x L b5, the inclusions for m= 0.4 and m= 0.8 move with accreting velocity towards the region of higher compressive stress region. When the position of inclusions in piezoelectric material is
Table 1 The computing values of coefficients in Eq. (32) for various parameters. α 0.001 0.5 2 Fig. 3. The gradient distributions of strain energy and electric energy along the interface, where the shape parameter of inclusion, m = 0.4.
10
B1 B2 B1 B2 B1 B2 B1 B2
m= −0.8
m= −0.4
m= 0
m = 0.4
m = 0.8
0.64 −16 0.112 −1.68 0.102 0.24 0.066 0.44
2.902 210 0.936 36.78 0.368 11.56 0.097 2.58
0.06 −16 0.038 0.36 0.026 4.624 0.011 2.934
−1.824 201.8 −0.548 23.96 −0.296 11.51 −0.118 5.832
−1.04 40 −1.064 16.16 −0.906 16.12 −0.536 12.54
142
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
Fig. 5. The variation of normalized velocity of inclusion for α = 0.001 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p = 6.7 × 108 N/m3.
x L > 5, the inclusions for m= 0.4 and m = 0.8 move with accreting velocity towards the region of higher tensional stress region. The motion characteristic of the softer inclusion (α = 2) in piezoelectric matrix is shown in Fig. 7. When the position of inclusions in piezoelectric material is x L b10, the inclusions for m= 0.4 and m= 0.8 move with accreting velocity towards the region of higher compressive stress region. When the position of inclusions in piezoelectric material is x L > 10, the inclusions for m = 0.4 and m =0.8 move with accreting velocity towards the region of higher tensional stress region. The migration velocity of inclusions is linear proportional to the inclusion position, and the migration velocity of inclusions gradually increases as the inclusion position increases. Fig. 8 shows the influence of inclusion's stiffness on the motion characteristic of a circular inclusion (m = 0). The migration motion of the harder inclusions (α = 0.001, 0.5) is from the higher compressive region to the lower compressive region when the inclusions are at the compressive region of piezoelectric film, and is from the higher tensional region to the lower tensional region when the inclusions are at the tensional region of piezoelectric film. The migration motion of the softer inclusions (α = 2, 10) is from the higher tensional region of piezoelectric film to the higher compressive region, the migration velocity of the softer inclusions (α = 2, 10) in the compressive region of piezoelectric film is much lower than that in the tensional region.
Fig. 6. The variation of normalized velocity of inclusion α = 0.5 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p = 6.7 × 108 N/m3.
Fig. 7. The variation of normalized velocity of inclusion for α = 2 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p= 6.7 × 108 N/m3.
Because the gradients of normal stress, strain energy and electric energy in Eq. (14) are from the solution of an infinite piezoelectric material with inclusion, the gradients of normal stress, strain energy and electric energy at the top apex A of the interface are independent of the vector radius ρ of ellipse inclusion. Therefore, it is seen from Eq. (14) that the migration velocity of the inclusion is inversely proportional to the vector radius ρ of ellipse inclusion when the inclusion size is much less than the width of film. Generally, large-size inclusions have not stable shape, which automatically splits into small inclusions with stable shape to make the total Gibbs free energy of the system minimum. 5. Conclusion Because of the effect of electric field applied on piezoelectric material, the change of migration velocity of inclusions in piezoelectric film does not occur at the position where the gradient stress equals zero. The influence of stress gradient on the moving velocity of a slit-like inclusion (the shape parameter m → 1) along the direction of the stress gradient is larger than that of the perpendicular one (the shape parameter m → − 1), and the moving velocity of inclusion in piezoelectric film increases as the amplitude of stress gradient increases. The round inclusion (m = 0) moves much slower than elliptical inclusions with shape parameters m ≠ 0, and the effects of the
Fig. 8. The variation of normalized velocity of inclusion for m = 0 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p= 6.7 × 108 N/m3.
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
elliptical shape (aspect ratio m ≠ 0) on the motion of the inclusion are dependent on the stiffness of the inclusion. It is seen from Eq. (14) that the motion velocity of the inclusion is inversely proportional to the vector radius ρ of an elliptical inclusion. The influence of stress gradient on the moving velocity of inclusion in piezoelectric film is dependent on the hardness of the inclusion, and the influence of stress gradient on the moving velocity of inclusion decreases as the hardness of the inclusion decreases. The motion of inclusion in piezoelectric film is mainly induced by the gradient of the normal stress at the apex of the interface layer between inclusion and piezoelectric matrix. The main purpose of this paper clearly visualizes the effects of the shape and the stiffness of inclusions on the inclusion motion in piezoelectric film under coupled gradient stress and electric field, and provides a benchmark and reference for future experimental and computational investigations on the migration of inclusions in piezoelectric materials. Acknowledgements The authors wish to thank the National Science Foundation of China under numbers 11172165 and 10932007. References [1] Y. Benveniste, Proc. R. Soc. Lond. A441 (1993) 59. [2] M.L. Dunn, M. Taya, Proc. R. Soc. Lond. A443 (1993) 265. [3] K.Y. Lam, X.Q. Peng, G.R. Liu, J.N. Reddy, Smart Mater. Struct. 6 (1997) 583.
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Thin Solid Films journal homepage: www.elsevier.com/locate/tsf
Migration velocity of an elliptical inclusion in piezoelectric film Y. Qin, X. Wang ⁎ School of Naval Architecture, Ocean and Civil Engineering (State Key Laboratory of Ocean Engineering), Shanghai Jiaotong University, Shanghai 200240, PR China
a r t i c l e
i n f o
Article history: Received 16 March 2012 Received in revised form 22 December 2012 Accepted 3 January 2013 Available online 16 January 2013
a b s t r a c t This paper presents an analytical method to investigate on the migration velocity of an elliptical inclusion in piezoelectric film under coupled gradient stress and electric field. The effects of gradient stress, electric field, the material property and the shape parameter of inclusion on the migration velocity of the elliptical inclusion in piezoelectric material are described and discussed. © 2013 Elsevier B.V. All rights reserved.
Keywords: Piezoelectric film Elliptical inclusion Migration velocity Thermodynamics potential
1. Introduction In the past, many investigations on the use of piezoelectric materials have already been presented for active control of smart structures with light weight. The inherent electromechanical effect of piezoelectric materials has many important engineering applications [1–6]. Current examples motivating the present study are strained semiconductor laser devices [7,8], in which residual stresses induced by lattice mismatch between buried active components and surrounding materials crucially affect electronic performance. In processing of thin films, various structural defects such as inclusions and voids in the thin film are often generated, which have detrimental effects on the function of film structures [9,10]. Investigations on mechanical deformations induced by misfitting inclusions in an infinite or semi-infinite piezoelectric medium are fundamental physical and engineering problems, which can be seen in earlier researches on this subject by Nowacki [11] and Mura [12]. Because of the rapid development of piezoelectric smart structures in the last years, many related works on coupled electromechanical characteristics of piezoelectric material with inclusions are further investigated by many researchers [13–17]. Della and Shu [13] utilized the Euler–Bernoulli beam theory and Rayleigh–Ritz approximation technique to present a mathematical model for the vibration of beams with piezoelectric inclusions. Based on the linear elasticity theory and Green's function method, Kuvshinov [14] presented explicit, closedform expressions to describe electroelastic deformations due to polyhedral inclusions in uniform half-space and bi-materials. Fakri and Azrar [15] predicted the electroelastic and thermal responses of piezoelectric composites with and without voids. Lin and Sodano [16] extended the ⁎ Corresponding author. E-mail address: [email protected] (X. Wang). 0040-6090/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tsf.2013.01.013
double inclusion model to multiphase composites with piezoelectric layers, and predicted the electroelastic properties of the multifunctional composites. Based on the viscoelastic principle, Aldraihem [17] developed a comprehensive micromechanics model to estimate the effective viscoelastic properties of hybrid composites containing polymer matrix. Tang et al. [18] reported that the energy density of the nanocomposite could be significantly increased through the use of piezoelectric nanowires and a polymer with greater breakdown strength. It is seen from Ref. [19] that strain-release characteristics of composites are reinforced by nanowires on stretchable substrates. Utilizing dielectrophoretic assembly, Tomer and Randall [20] make anisotropic composites reinforced by BaTiO3 particles in a silicone elastomer thermoset polymer, and investigate the effect of electrical properties on the property of these composites. It is described in Ref. [21] that the energy-storage capability of the nanocomposite can be enhanced by the alignment of piezoelectric nanowires in the direction of the applied electric field compared to randomly oriented samples. In the above investigation into the influence of inclusions in piezoelectric materials on the mechanical and electrical characteristics of piezoelectric smart structures, the inclusion is fixed at a determinate point in piezoelectric material. In fact, the various solid films with inclusion are often subjected to severe gradient stresses induced by thermal mismatching and electric field, so that the inclusion in films may migrate [22–25]. Because one of the major damage mechanisms is from the inclusion (void) coalescence in stress concentration region induced by the motion of inclusion (void), the dynamics of stress-driven inclusion in metallic solids has been generating great research interest for the failure of metallic interconnects in integrated circuits [26–28]. Utilizing self-consistent numerical method, Gungor and Maroudas [30] describe the effects of complex external stresses on the electromigration-driven motion of morphologically stable voids in elastically deforming metallic thin films, and reveal the complex evolution of voids. It is seen in
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Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
Ref. [31] that when the morphological stability limit is approached, the void migration speed is substantially dependent on the void size. Therefore, the inclusion migration-induced failure should also pose a great concern in piezoelectric smart structures. However, because the coupled electroelastic characteristic of piezoelectric material with inclusion is complex, the investigation on the migration of inclusion in piezoelectric film is not found in literatures so far. In this paper, we give an analytical solution for the migration of an elliptical inclusion in piezoelectric film under coupled gradient stress and electric field, based on the atomic migration principle in a diffused interface layer between inclusion and piezoelectric matrix from regions of high chemical potential to those of low chemical potential along the interface layer. The numerical examples describe the effects of gradient stress, electric field, and the material property and the shape of inclusion on the migration velocity of the elliptical inclusion in piezoelectric film. The present work may stimulate further interest on this topic because the reliability of piezoelectric smart structures is extremely sensitive to inclusion defects. 2. Basic equations and solving method Fig. 1 shows the two-dimensional (2D) model of an elliptic inclusion in piezoelectric film under coupled gradient stress and electric field, in which this 2D implementation conveys the essence of a three-dimensional (3D) problem of inclusion in piezoelectric material, and represents an inhomogeneity extending throughout the thickness along z, consistently with experimental observations in thinpiezoelectric film. In order to obtain the strict stress and electric field solutions at the interface required in the present derivation, the calculating model in Fig. 1 is based on the migration of inclusion in an infinite piezoelectric material under the combined electric and stress gradient loads, in which its realistic application in a finitewidth piezoelectric film will exist in scale effects. However, Li and Chudnovsky [32] have proved that the scale effect is negligible when the ratio of inclusion size to a characteristic size of the analytical system is less than 0.15 by using the method of numerical analyses. Because the characteristic size of the analytical system for an inclusion in a finite piezoelectric film is the width of the film, the present model can be used to calculate the inclusion motion in a realistic piezoelectric film when the inclusion size is much less than the width of film. At infinity, the model is subjected to gradient stress field qy =pX in the Y direction by thermal mismatch, where p ¼ dqy dX is a constant, a positive value of p presents a rising stress along the X-axis, while in the X-direction subjected to electric field E0. In Fig. 1, the global coordinate system OXY is fixed at the symmetric center of piezoelectric film, the local (moving) coordinate system oxy is fixed at the center of an elliptic inclusion, and xL represents a moving distance from the center
of the elliptic inclusion to the symmetric center of piezoelectric film. The relation between the two coordinate systems is given by xL = X −x and y= Y. The constitutive relations of orthotropic piezoelectric film are written as [29] 8 9 2 S11 < εx = εy ¼ 4 S13 :γ ; 0 xy
Dx Dy
0 ¼ d31
9 2 38 0 < σx = 0 0 5 σy þ 4 0 :τ ; S44 d15 xy
S13 S33 0 0 d33
8 9 < σ x = e d15 σ þ 11 0 0 : y; τxy
3 d31 E d33 5 x Ey 0 0 e33
Ex Ey
ð1aÞ
ð1bÞ
where σx, σy, τxy and εx, εy, γxy are, respectively, stresses and strains in the piezoelectric film along the main direction of coordinate system oxy, Ex, Ey and Dx, Dy are, respectively, the electric field strengths and the electric displacements along the x and y directions, Sij is the flexible coefficients of piezoelectric film, and dij and eij are the piezoelectric and dielectric constants, respectively. The inclusion in piezoelectric film may be moved by the atomic migration of an inclusion/matrix interface layer from bulk regions of high chemical potential to those of low chemical potential along the interface layer. Because the interface diffusion is generally much faster than the bulk one, only the interface diffusion is considered as the major mechanism for the inclusion motion in piezoelectric film under coupled gradient stress and electric field. In general, instable inclusions including very narrow morphologies may be split into small inclusions with stable shape to minimize the total Gibbs free energy of the system under external loads. In order to give an analytical solution for the migration velocity of an inclusion in piezoelectric film under coupled field, it is assumed in this analytical model that the inclusion is considered as an elliptic inclusion with shape parameter m, and the morphology of the elliptic inclusion is taken to be stable throughout the range of the shape parameters when the inclusion simply migrates along the length direction of film. Because the shape and size of an inclusion in piezoelectric film do not change during motion, the piezoelectric film is the source and sink for the atomic flux along the interface layer between inclusion and piezoelectric matrix. Under coupled gradient stress and electric field, atoms diffuse along the interface layer from one side of high chemical potential to the other of lower, so that the inclusion migrates in the piezoelectric matrix against the migration of the atoms. Thus, the chemical potential in the interface layer driving this atomic flux is composed of the free energy, the local elastic strain energy and the local electric energy in the interface layer, as follows μ ¼ μ 0 −Ωγi κ−Ωσ n þ ΩW p þ ΩW E ;
ð2Þ
where μ0 is the reference value of the potential, γi is the free energy of interface layer, κ is the surface curvature of the inclusion, positive for convex, Ω is the atomic volume, σn is the normal stress on the inclusion surface, Wp is the elastic strain energy density stored in the interface layer associated with an atom, as follows Wp ¼
1 ðσ ε þ σ θ εθ þ τnθ γ nθ Þ; 2 n n
ð3aÞ
and WE is the electric energy density stored in the interface layer associated with an atom, as follows WE ¼
Fig. 1. The motion of an elliptical inclusion in infinite piezoelectric material under combined gradient stress field and electric field.
1 ðE D þ Eθ Dθ Þ 2 n n
ð3bÞ
where Dn, Dθ and En, Eθ represent, respectively, electric displacements and electric strength along the normal n and circumference θ of the interface layer.
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
In Fig. 1, J represents the atomic flux along the interface layer between inclusion and piezoelectric matrix, the number of atoms per time crossing unit length along the interface layer, which satisfies Nernst-Einstein equation, as follows D δ ∂μ J¼− i i ΩKT ∂s
ð4Þ
where Di is the diffusion coefficient of interface layer, δi is the thickness of the interface layer, K is the Boltzmann's constant, T is the absolute temperature, and s is the arc length of the interface. In practice, the interfacial diffusion coefficient should be anisotropic, which leads to faceted stable morphologies as opposed to round/elliptic ones in this model. Because the objective of this study is to give an analytical solution for the horizontal motion of an elliptical inclusion in piezoelectric film, the interfacial diffusion coefficient in the present model is taken to be isotropic, as the assumption from some investigations [33,34]. Although the approximation ignoring the anisotropic diffusion characteristic of interface is involved in the solving model, the closed-form solution with a general manner can be obtained in this paper, which explicitly gives the essential behavior for the motion of an inclusion in piezoelectric film driven by combined electric field and gradient stress. It is seen from Eq. (4) that the atomic flux in the interface layer is dependent on the variation of chemical potential μ along the arc length of interface. The mass conservation condition is given by dJ V n þ ¼0 ds Ω
ð5aÞ
the interface between inclusion and piezoelectric matrix. From Eq. (9), we have rffiffiffiffiffiffiffiffiffiffiffiffiffi ! V 1−m J¼ ρ −y þ C Ω 1þm
ð5bÞ
Vn in Eq. (5a) is the normal diffusion velocity of the interface, which is written as V n ¼ V sinφ
ð6Þ
where V is the translational velocity of the inclusion with stable morphology under the coupled mechanical and electric fields as prescribed. In the present model, the inclusion motion is horizontal due to the orientation of the external fields driving the motion of the inclusion. Substituting Eqs. (6) and (5b) into Eq. (5a) gives dJ V ¼ : dy Ω
ð7Þ
The shape for an elliptic inclusion in piezoelectric film as in Fig. 1 has the same area as a circle of radius ρ, which can be described by x¼ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm cosθ; 1−m
y¼ρ
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m sinθ 1þm
ð8Þ
where θ is the angle between the vector radius ρ of ellipse and x-axes, and m is the shape parameter of the inclusion. The shape parameter m=0 corresponds to a circular inclusion, m→ +1 to x-direction narrow inclusion and m→ −1 to y-direction narrow inclusion, respectively. To consider the effect of the local interface changes of a moving elliptical inclusion on the atomic flux along the interface, Eq. (7) should be integrated along thepinterface between inclusion and piezoelectric ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi matrix from y to y ¼ ρ ð1−mÞ=ð1 þ mÞ, as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ ð1−mÞ=ð1þmÞ V ∫dJ ¼ ∫y dy Ω
ð9Þ
When the elliptic shape of inclusion does not vary during motion, the horizontal velocity V of the inclusion interface is a constant along
ð10Þ
where C is the undetermined constant which can be solved by the continuous condition of the atomic flux at the apex A of elliptic inclusion. Utilizing Eqs. (2), (4) and (10), the continuous condition of the atomic flux at the apex A of elliptic inclusion is expressed as " # " # rffiffiffiffiffiffiffiffiffiffiffiffiffi ! Di δi ∂κ ∂σ n ∂W p ∂W E V 1−m ¼ γi þ ρ −y þ C − − KT Ω 1þm ∂s ∂s ∂s ∂s A
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
y¼ρ
:
ð1−mÞ=ð1þmÞ
ð11Þ Because ∂ κ/∂ s at the apex A of elliptic inclusion equals zero, solving Eq. (11) gives C¼
" Di δi ∂σ n KT ∂s
j −∂W∂s j −∂W∂s j p
A
#
E
ð12Þ
A
A
.
.
.
∂W p
∂σ n ∂W E where
,
and
are the gradients of normal ∂s A ∂s A ∂s A stress, strain energy density and electric energy density at the top apex A of the interface between inclusion and piezoelectric matrix. Due to the symmetrical diffusion requirement of the interface between inclusion and piezoelectric matrix as in Fig. 1, the atom flux at y = 0 should be zero at all times, as follows:
where ds ¼ −dy=ð sinφÞ
139
½ J y¼0 ¼
Vρ Ω
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1−m þ C ¼ 0: 1þm
ð13Þ
Substituting Eq. (12) into (13), yields ΩDi δi V¼ ρKT
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm ∂σ − n 1−m ∂s
j
A
þ
∂W p ∂s
j
∂W E þ ∂s A
j
!
A
:
ð14Þ
It is seen from Eq. (14) that the horizontal velocity of an inclusion in piezoelectric material is mainly dependent on the gradients of normal stress, strain energy and electric energy at the top apex A of the interface between inclusion and piezoelectric matrix. So far, the analytic expression for the stress field at the interface between inclusion and piezoelectric matrix under a gradient stress field as in Fig. 1 is not presented in literatures. Here, utilizing a loading superposition method as in Fig. 2, a gradient stress field, qy = pX, applied on a piezoelectric film with inclusion can be described from a superposition of an antisymmetric stress field of stress gradient with respect to the y-axis of the moving elliptic inclusion and a uniform stress field, as follows: qy ¼ qy1 þ qym :
ð15Þ
Based on the linear elastic theory, the stress field σijA at the apex A of the interface between inclusion and piezoelectric under coupled loadings qy = qym + qy1 is given by A A A σ ij qy ¼ pX ¼ σ ij qy1 ¼ px þ σ ij qym ¼ pxL :
ð16Þ
Thus, the gradients of normal stress at the apex A is written as ∂σ n qy ¼ pX ∂s
j
¼ A
∂σ n qym ¼ pxL ∂s
j
þ A
∂σ n qy1 ¼ px ∂s
j
:
ð17Þ
A
Because an antisymmetric gradient stress field, qy1 = px, with respect to the y-axis of the moving elliptic inclusion makes the stresses σijA(qy1 = px) at the apex A of the interface between inclusion and
140
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
a
b
c
Fig. 2. The stress at apex A of inclusion under stress pX shown in (a) can be obtained from the superposition of an anti-symmetrically applied stress px about the y-axis shown in (b) and a uniform applied stress field with stress pxL shown in (c).
piezoelectric matrix equals zero, the stresses σijA(qy = pX) at the apex A of the interface under remote gradient stress field can be replaced by the stresses σijA(qym = pxL) at the apex A of the interface. 3. The stress and electric fields in piezoelectric materials with inclusion
Substituting (22) into Eqs. (1a) and (1b), yields
Dx ¼ −2real
3 X
d15 ηk þ e11 λk F k ″ðzk Þ;
k¼1
ð23Þ
3 X 2 d31 ηk þ d33 −e33 λk ηk F k ″ðzk Þ Dy ¼ 2real k¼1
The stress function and electric potential function of transversely isotropic piezoelectric material with inclusion, under uniform mechanical and electric loads, can be, respectively, taken as [29]
3 X εx ¼ 2real pk F k ″ðzk Þ;
εy ¼ 2real
k¼1 3 X F ðx; yÞ ¼ 2real F k ðzk Þ;
3 X ϕðx; yÞ ¼ 2real λk F k ′ðzk Þ
k¼1
ð18Þ
ð24Þ
ηk pk þ qk F k ″ðzk Þ
where
zk ¼ x þ ηk y;
λk ¼ ½
d31 ηk 3 þðd33 −d15 Þηk
ðe11 þe33 ηk Þ 2
ð19Þ
h i2 h i 4 2 2 2 2 S11 η þ ð2S13 þ S44 Þη þ S33 e11 þ e33 η −η d31 η þ ðd33 −d15 Þ ¼ 0:
ð20Þ Because of the symmetry of piezoelectric film with inclusion under uniform mechanical and electric loads, the solution for Eq. (20) is yield by η2 ¼ α 2; þ β2 i;
η3 ¼ −α 2 þ β2 i;
pk ¼ S11 ηk þ S13 −d31 ηk λk ; qk ¼ ðS13 ηk þ S33 −d33 ηk λk Þ=ηk 2
:
In the above formula, F(x,y) is the stress function, Fk(zk) is the complex function and ηk is the three complex roots (with positive imaginary parts), which satisfy the characteristic equation, as follows
η5 ¼ η2 ;
k¼1
k¼1
k¼1
where
η1 ¼ β1 i;
2εxy ¼ 2real
3 X
3 X qk ηk F k ″ðzk Þ;
η4 ¼ η1 ;
Utilizing transformation formula, the stress and electric field strength in the interface along the vector radius ρ of ellipse inclusion and the angle θ between the vector radius ρ and the x-axes are, respectively, expressed as 2
2
σ n ¼ σ x cos θ þ σ y sin θ−τxy sin2θ; τ nθ ¼ 0:5 σ x −σ y sin2θ þ τxy cos2θ
2
2
σ θ ¼ σ x sin θ þ σ y cos θ þ τxy sin2θ;
ð26Þ 2
2
ε n ¼ ε x cos θ þ ε y sin θ−0:5γ xy sin2θ; γ nθ ¼ εx −εy sin2θ þ γ xy cos2θ
2
2
ε θ ¼ ε x cos θ þ εy sin θ þ 0:5γ xy sin2θ;
ð27Þ
ð21Þ
η6 ¼ η3
ð25Þ
En ¼ Ex cosθ þ Ey sinθ; Eθ ¼ −Ex sinθ þ Ey cosθ Dn ¼ Dx cosθ þ Dy sinθ; Dθ ¼ −Dx sinθ þ Dy cosθ:
ð28Þ
where β1, β2 and α2 depend on the piezoelectric material constants, and ηk represents the conjugation of the ηk, (k=1, 2, 3). Once the roots ηk, (k=1, 2, 3) are obtained from Eq. (18) [29], the stress and electric field strength in piezoelectric film with inclusion are written as
4. Numerical examples and discussions
3 3 X X 2 σ x ¼ F ðx; yÞyy ¼ 2real ηk F k ″ðzk Þ; σ y ¼ F ðx; yÞxx ¼ 2real F k ″ðzk Þ
In example calculations, the properties of piezoelectric film are taken as:
k¼1
k¼1
3 3 X X σ xy ¼ −F ðx; yÞxy ¼ −2real ηk F k ″ðzk Þ; Ex ¼ −ϕðx; yÞx ¼ −2real λk F k ″ðzk Þ; ð22Þ k¼1
3 X Ey ¼ −ϕðx; yÞy ¼ −2real ηk λk F k ″ðzk Þ: k¼1
k¼1
Sp11 ¼ 11:6 10−12 m2 =N; p −12 2 m =N; S44 ¼ 45:0 10 dp33 ¼ 152:0 10−12 C=N;
Sp33 ¼ 14:8 10−12 m2 =N; Sp13 ¼ −4:97 10−12 m2 =N; p −12 p −12 d15 ¼ 440:0 10 C=N; d31 ¼ −60:2 10 C=N; ð29Þ ep11 ¼ 8:76 109 F=m; ep33 ¼ 3:98 109 F=m:
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
141
Here, the material properties of inclusion in the piezoelectric film are taken as: i
p
Sij ¼ αSij ;
i
p
dij ¼ αdij ;
i
p
eij ¼ αeij
ð30Þ
where α is a proportion coefficient presenting the inclusion property, when α b 1, the stiffness of inclusion is larger than the stiffness of piezoelectric matrix, and the superscripts i and p denote the material properties of inclusion and piezoelectric matrix. The influence of the inclusion property on the moving velocity is described by using the value of α = 0.001, α = 0.5, α = 2, and α = 10 in example calculations.Eq. (14) can be rewritten as
V¼
ΩDi δi ρkT
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm ∂σ f − n 1−m ∂s
j
A
∂σ e − n ∂s
j
f
þ A
∂W p ∂s
j
A
e
þ
∂W p ∂s
j
A
þ
∂W fE ∂s
j
þ A
∂W eE ∂s
j
! : A
ð31Þ In the above formula, the superscripts f and e represent the normal stress, the strain energy density and the electric energy density induced by mechanical loading and electrical loading. Utilizing the stress field and electric field of piezoelectric film with inclusion obtained from Eqs. (18) to (28), the gradient distributions of normal stress, strain energy and electric energy along the interface between inclusion and piezoelectric matrix are described in Figs. 3 and 4. Figs. 3 and 4 show that the gradient of the normal stress at the apex A of inclusion is much larger than the gradient of the strain energy and electric energy at the apex, so that it is concluded that the motion of inclusion in piezoelectric material is mainly induced by the gradient of the normal stress at the apex A of the interface layer between inclusion and piezoelectric matrix. Substituting the gradient values of normal stress, strain energy and electric energy at the apex A of the interface layer obtained from Eqs. (18) to (28) into Eq. (31), the normalized velocity of the elliptic inclusion in piezoelectric material is simplified as
V ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffi 1þm p ð−B1 ⋅px L ρ þ B2 ⋅E0 Þ⋅ρS11 : 1−m
ð32Þ
where p and E0 are, respectively, the stress gradient applied on the y direction of piezoelectric material and the electric strength applied on the X direction, and the parameters B1 and B2 are obtained by solving
Fig. 4. The gradient distributions of normal stress along the interface, where the shape parameter of inclusion, m= 0.4.
Eqs. (18) to (28), as shown in Table 1. Dimensionless parameters related to each calculating terms are, respectively, taken as
V ¼
kTρ2 Sp11 V; ΩDi δi
x L ¼ xL =ρ:
ð33Þ
Fig. 5 describes the normalized motion characteristics of the rigid inclusion (α = 0.001 → 0) in piezoelectric matrix. Because the atoms in the interface layer with tensile or compressive stresses have high chemical potentials, the atoms with high stress migrate to the region with lower stress, which leads to the inclusion moving to higher stress region in piezoelectric film. As shown in Fig. 5, the shape parameter m and the position of inclusion in piezoelectric film affect the motion velocity of the inclusion; the motion velocity of inclusion is linear proportional to the distance between the center of inclusion and the zero point of the gradient stress applied on piezoelectric material; the round inclusion (m = 0) moves much slower than other inclusions with shape parameters m = − 0.4, 0.4, and 0.8; the inclusion for m = 0.4 moves with accreting velocity towards the region of higher compressive region; the inclusion for m = − 0.4 moves with accreting velocity towards the region of higher tensional stress region when the position of inclusion in piezoelectric film is x L b−20; and the inclusion for m = 0.8 moves with accreting velocity towards the region of higher tensional stress region when the position of inclusion in piezoelectric is x L > 15. Fig. 6 shows the motion characteristics of the harder inclusion (α = 0.5) in piezoelectric film. The migration velocity of the inclusion with shape parameter m =0.8 is much larger than that of inclusion with other shape parameters. When the position of inclusions in piezoelectric material is x L b5, the inclusions for m= 0.4 and m= 0.8 move with accreting velocity towards the region of higher compressive stress region. When the position of inclusions in piezoelectric material is
Table 1 The computing values of coefficients in Eq. (32) for various parameters. α 0.001 0.5 2 Fig. 3. The gradient distributions of strain energy and electric energy along the interface, where the shape parameter of inclusion, m = 0.4.
10
B1 B2 B1 B2 B1 B2 B1 B2
m= −0.8
m= −0.4
m= 0
m = 0.4
m = 0.8
0.64 −16 0.112 −1.68 0.102 0.24 0.066 0.44
2.902 210 0.936 36.78 0.368 11.56 0.097 2.58
0.06 −16 0.038 0.36 0.026 4.624 0.011 2.934
−1.824 201.8 −0.548 23.96 −0.296 11.51 −0.118 5.832
−1.04 40 −1.064 16.16 −0.906 16.12 −0.536 12.54
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Fig. 5. The variation of normalized velocity of inclusion for α = 0.001 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p = 6.7 × 108 N/m3.
x L > 5, the inclusions for m= 0.4 and m = 0.8 move with accreting velocity towards the region of higher tensional stress region. The motion characteristic of the softer inclusion (α = 2) in piezoelectric matrix is shown in Fig. 7. When the position of inclusions in piezoelectric material is x L b10, the inclusions for m= 0.4 and m= 0.8 move with accreting velocity towards the region of higher compressive stress region. When the position of inclusions in piezoelectric material is x L > 10, the inclusions for m = 0.4 and m =0.8 move with accreting velocity towards the region of higher tensional stress region. The migration velocity of inclusions is linear proportional to the inclusion position, and the migration velocity of inclusions gradually increases as the inclusion position increases. Fig. 8 shows the influence of inclusion's stiffness on the motion characteristic of a circular inclusion (m = 0). The migration motion of the harder inclusions (α = 0.001, 0.5) is from the higher compressive region to the lower compressive region when the inclusions are at the compressive region of piezoelectric film, and is from the higher tensional region to the lower tensional region when the inclusions are at the tensional region of piezoelectric film. The migration motion of the softer inclusions (α = 2, 10) is from the higher tensional region of piezoelectric film to the higher compressive region, the migration velocity of the softer inclusions (α = 2, 10) in the compressive region of piezoelectric film is much lower than that in the tensional region.
Fig. 6. The variation of normalized velocity of inclusion α = 0.5 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p = 6.7 × 108 N/m3.
Fig. 7. The variation of normalized velocity of inclusion for α = 2 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p= 6.7 × 108 N/m3.
Because the gradients of normal stress, strain energy and electric energy in Eq. (14) are from the solution of an infinite piezoelectric material with inclusion, the gradients of normal stress, strain energy and electric energy at the top apex A of the interface are independent of the vector radius ρ of ellipse inclusion. Therefore, it is seen from Eq. (14) that the migration velocity of the inclusion is inversely proportional to the vector radius ρ of ellipse inclusion when the inclusion size is much less than the width of film. Generally, large-size inclusions have not stable shape, which automatically splits into small inclusions with stable shape to make the total Gibbs free energy of the system minimum. 5. Conclusion Because of the effect of electric field applied on piezoelectric material, the change of migration velocity of inclusions in piezoelectric film does not occur at the position where the gradient stress equals zero. The influence of stress gradient on the moving velocity of a slit-like inclusion (the shape parameter m → 1) along the direction of the stress gradient is larger than that of the perpendicular one (the shape parameter m → − 1), and the moving velocity of inclusion in piezoelectric film increases as the amplitude of stress gradient increases. The round inclusion (m = 0) moves much slower than elliptical inclusions with shape parameters m ≠ 0, and the effects of the
Fig. 8. The variation of normalized velocity of inclusion for m = 0 at different motion distance x L along the X direction, where, E0 = 3 × 102 Vol/mm and p= 6.7 × 108 N/m3.
Y. Qin, X. Wang / Thin Solid Films 531 (2013) 137–143
elliptical shape (aspect ratio m ≠ 0) on the motion of the inclusion are dependent on the stiffness of the inclusion. It is seen from Eq. (14) that the motion velocity of the inclusion is inversely proportional to the vector radius ρ of an elliptical inclusion. The influence of stress gradient on the moving velocity of inclusion in piezoelectric film is dependent on the hardness of the inclusion, and the influence of stress gradient on the moving velocity of inclusion decreases as the hardness of the inclusion decreases. The motion of inclusion in piezoelectric film is mainly induced by the gradient of the normal stress at the apex of the interface layer between inclusion and piezoelectric matrix. The main purpose of this paper clearly visualizes the effects of the shape and the stiffness of inclusions on the inclusion motion in piezoelectric film under coupled gradient stress and electric field, and provides a benchmark and reference for future experimental and computational investigations on the migration of inclusions in piezoelectric materials. Acknowledgements The authors wish to thank the National Science Foundation of China under numbers 11172165 and 10932007. References [1] Y. Benveniste, Proc. R. Soc. Lond. A441 (1993) 59. [2] M.L. Dunn, M. Taya, Proc. R. Soc. Lond. A443 (1993) 265. [3] K.Y. Lam, X.Q. Peng, G.R. Liu, J.N. Reddy, Smart Mater. Struct. 6 (1997) 583.
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