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A thermodynamic field theory for anodic bonding of micro electro-mechanical systems (MEMS)
International Journal of Engineering Science 38 (2000) 135±158
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A thermodynamic ®eld theory for anodic bonding of micro electro-mechanical systems (MEMS) Eniko T. Enikov *, James G. Boyd Department of Mechanical Engineering (M/C 251), The University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607, USA Received 27 August 1998; received in revised form 1 February 1999; accepted 3 February 1999 (Communicated by J.T. ODEN)
Abstract An anodic bond is modeled as a moving nonmaterial line forming the intersection of three material surfaces representing the unbonded conductor, the unbonded insulator, and the bonded interface. Global integral equations are written for the conservation of mass, momentum, and energy, Maxwell's equations, and the second law of thermodynamics. The global equations are then localized in the volume, the material surfaces, and the nonmaterial bond line. The second law is used to determine the thermodynamic conjugates in the thermodynamic potential and the dissipation inequality. It is demonstrated that the jump in the Poynting vector across a surface is equal to the surface Joule heating due to surface electric conduction currents. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Anodic bonding is a common method of bonding glass to conductors such as silicon or metal. The glass and conductor are brought into contact at elevated temperature and an electric potential is applied with the cathode on the glass and the anode on the conductor. The mobile positive ions, typically 2% by mass Na in Pyrex glass, diuse towards the cathode, leaving a negatively charged glass layer at the glass/conductor interface (Fig. 1). A positive charge develops on the conductor surface, and the electrostatic attraction eventually pulls the glass and conductor together with a pressure high enough to initiate a surface reaction to form a chemical bond. Unfortunately, anodic bonding is performed under harsh conditions of temperature, pressure and electric ®eld. For example, Micro Electromechanical Systems (MEMS) are often made by *
Corresponding author. Tel.: 001-312-996-6593; fax: 001-312-413-0447. E-mail address: [email protected] (E.T. Enikov)
0020-7225/00/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 9 ) 0 0 0 2 7 - 0
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Fig. 1. Anodic bonding.
bonding 300 lm thick Pyrex and silicon wafers together at 400 C with 400 V applied to the Pyrex wafer. This leads to complicated microstructural evolution including creep, ionic diusion, residual stresses, warping, and spatial instabilities such as electrical breakdown and symmetry breaking of the bond front. In particular, the extreme conditions limit the choice of polymers that can be used in MEMS [12,9]. Although anodic bonding is widely used to fabricate MEMS such as accelerometers [15], pressure sensors [6], and micropumps [12,9], there has never been a thermodynamic ®eld formulation of the process. Studies have typically consisted of simple electrical circuit analogs [2], chemistry and physics of the surface reactions [11], or one-dimensional diffusion [16] normal to plane surfaces. In the present paper we present a thermodynamic ®eld theory applicable to anodic bonding of MEMS. The work of Ancona and Tiersten [1] and Daher and Maugin [7], in which they developed the fundamental equations for semiconductors with interfaces, is directly applicable to the current formulation. However, we account for a double interface and a moving bond line. The physical domain, assumptions, and notation are introduced in Section 2. The fundamental laws are discussed in subsequent sections: the conservation of mass (Section 3), Maxwell's equations (Section 4), the conservation of momentum (Section 5), the conservation of energy (Section 6), and the second law of thermodynamics and constitutive equations (Section 7). The theory presented herein is also applicable to debonding along an insulator/conductor interface. In addition, the formulation can be modi®ed to model a wide variety of electro-me-
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
137
chanical phenomena at interfaces, including space charge accumulation and sliding at grain boundaries in ionic crystals, and a cohesive zone theory of electro-elastic fracture. 2. Assumptions, de®nitions, and notation Subscripts denote tensor components in three-dimensional space, and superscripts denote qualitative descriptions. Repeated subscripts are summed. The physical domain (Fig. 2) consists of the volumes V c , V g , and V u representing the conductor, glass, and unbonded interface. The glass and conductor are separated by surface asperities. Electrostatic pressure causes creep deformation which ¯attens these asperities, and bonding occurs when the surface roughness is suciently small. Rather than follow the motion of each asperity, the surfaces will be modeled as material surfaces rc and rg (dashed lines on Fig. 2) passing through some averaged position of the asperities. The creep deformation of the asperities is modeled using an interface creep equation. The unbonded interface ru is the union of the midpoints between rc and rg . The bond front is modeled as a nonmaterial line, c, which is the boundary of the bonded interface, rb . The bonded interface is de®ned as the region of the interface that can support nonzero tensile and shearing Cauchy stress vectors by means other than friction. It is assumed that there is no friction force between rc and rg . Although the actual conductor and glass surfaces have dierent unit normal vectors, it is assumed that rc , rg and ru have a common unit normal vector ®eld, mi , which is also the outer unit normal vector ®eld to the bond line, c (Fig. 3). This simplifying assumption can be relaxed to include cases in which ru contains corners. Materials that undergo ionic or electronic conduction across the asperities relax
Fig. 2. Geometry of bonding.
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E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
Fig. 3. Tangent, normal, and binormal vectors along the bond line.
the surface charge and electric pressure and therefore do not form an anodic bond. Therefore, it is assumed that there is no mass diusion or electronic conduction between rc and rg , although mass diusion is allowed across rb . Quasi-static conditions are assumed and the inertia terms are neglected in the momentum and energy equations. Surface properties play an important role in the fabrication and operation of MEMS. Volumes are often separated and/or bounded by thin ®lms with properties than dier greatly from the volume. Also, surfaces are often modi®ed prior to bonding, and in the case of ionic conductors such as glass, surface charge is due to surface ions. Therefore, in order to model the surface charges which accumulate on rg , it is necessary to introduce surface chemical components. A caret `^' above a symbol is used to distinguish the surface from the volume quantities. Surface vectors F^i are assumed to lie in the tangent plane so that the following transversality condition applies: 0 F^i mi :
1
Variables and their spatial derivatives are allowed to undergo jump discontinuities across all surfaces, as indicated in Fig. 4. Angled brackets are used to denote the algebraic mean of the limiting values of variables at a surface: 1 h/irc /u /c ; 2
1 h/irg /u /g ; 2
1 h/irb /c /g : 2
2
The volume and surface (tangential) gradient operators are given, respectively, by ri
o ; oxi
^ i dij ÿ mi mj o ; r oxj
3
where dij is the Kronecker delta. The material time derivative and velocity have the usual forms
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
139
Fig. 4. Jump notation.
_ dH oH vi ri H; H dt ot
vi
dui ; dt
4
where ui is the displacement. The time rate of change following the normal motion of a surface is ^oH oH v^i mi mj rj H: ^ ot ot
5
The surface material time derivative is given by ^ ^ ^_ dH oH v^i r ^ i H: H ^ dt ot
6
Because the surfaces rc ; rg and rb are material, they have the limiting velocities of the volume v^i vi :
7
This assumption is in agreement with the experiments [18] showing no growth of a transition layer between the two phases. As a result of this assumption, there will be no net mass ¯ow to or from the surfaces rc , rg , or rb , and the total mass on these surfaces will be conserved. This assumption allows for the diusional exchange of species between the surface and volume as well as diusional redistribution of mass. This is an accurate assumption for the case of Pyrex glass, because the surface barycentric velocity v^i is approximately equal to the velocity of the boro-silicate matrix, which has essentially zero diusion ¯ux.
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E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
The following integral domains are valid unless otherwise indicated: Z Z c g u integration over V V [ V [ V ; integration over r rg [ rc [ rb : V
8
r
The displacement gradient is assumed to obey the in®nitesimal strain theory, and the Lagrangian linear strain tensor is given by 1 Lij ri uj rj ui : 2
9
Finally, eijk is the permutation symbol. 3. Conservation of mass Local (dierential) forms of the mass balance are derived by starting with integral forms de®ned for an arbitrary portion of the material and then applying the rules of dierentiation of integrals with moving boundaries (transport theorems) to localize these integrals, assuming that the integrands are suciently smooth functions and the integration domain can be chosen arbitrarily. The global form of the mass balance law for component n, where 1 6 n 6 N 1, is given by Z I Z Z Z Z d d n n n n n ^ da J^i si ds p dv p^n da; q dv Ji ni da 10 q dt dt V
r
oV
or
V
r
where pn and p^n are the mass rate of production of component n per unit volume and area due to ^n the mass densities of component chemical reactions, ni the outward unit normal vector, qn and q n n n per unit volume and area, and Ji and J^i the volume and surface diusion ¯uxes of component n. Jin qn vni ÿ vi ;
n ^n ^ J^i q vni ÿ v^i :
11
Any conductor or glass mass that would actually lie between the surfaces rc and rg is assumed to lie on the surfaces rc and rg , respectively. Thus, in the model there is no conductor or glass mass within V u . Furthermore, we assume that the bonding is conducted in vacuum, thus the mass integrals over V u vanish. The bond initiation line c is not material and moves with velocity vci . The time derivative d=dt is observed following the motion of the surface r (Appendix II in Ref. [13]). By means of transport and divergence theorems ([7,13]) the integrals in Eq. (10) are rewritten as follows: Z n Z d oq n n ri q vi dv; q dv 12 ot dt V
Z oV
Jin ni da
V
Z V
ri Jin dv
Z r
mi Jin da;
13
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
141
# Z "^ n Z o^ q d ^ i ^ ^n da ^n da r vi q q ^ dt ot r
r
Z
h i ^gn ^ ^bn ^ ^cn ^ si q vci ÿ vci q vgi ÿ vci ÿ q vbi ÿ vci ds;
14
c
I
n J^i si ds
Z
^ i J^n da r i
r
or
Z
cn gn bn si J^i J^i ÿ J^i ds:
15
c
R n Eq. (15) ordinarily includes a term r J^i mi rj mj da, but this term vanishes here due to the transversality condition, Eq. (1). Eqs. (12)±(15) can be substituted into Eq. (10) to obtain the local form of the mass balance on V c ; V g ; rc ; rg ; rb and c: oqn ri qn vi ri Jin pn ot
in V c [ V g ;
^o^ qn ^ ^ i J^n mi J n p^n ^n r vi q ri ^ i i ^ ot
16
on rc [ rg [ rb ;
cn gn bn ^gn ÿ q ^bn 0 vi ÿ vci ^ qcn q si J^i J^i ÿ J^i si ^
along c;
17 18
where we used the fact that the limiting values of the surface velocities along the bond line are equal: v^ci v^gi v^bi
along c:
19
The mass conservation laws for the mixture can be obtained by summing Eqs. (16)±(18) over N 1 species and using Eq. (11) and the de®nitions of the barycentric velocities for the volume and the surface, vi
N1 X 1
q n1
qn vni ;
v^i
N1 X 1
^ q n1
^n v^ni ; q
20
resulting in oq ri qvi 0 ot
in V c [ V g ;
21
^o^ q ^ ^ 0 ri ^ vi q ^ ot
on rc [ rg [ rb ;
22
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E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
^g ÿ q ^b 0 si ^ vi ÿ vci ^ qc q
along c;
23
^ are the total volume and surface mass densities, and we have used the mass where q and q conservation laws for chemical reactions and the diusion ¯ux de®nitions N1 X
N1 X
pn 0;
n1
N1 X Jin 0;
p^n 0;
n1
n1
N1 X n J^ 0:
24
n1
Mass fractions are often more useful than densities as compositional variables. Therefore, the component balance Eqs. (16)±(18) will be rewritten in terms of mass fractions cn and c^n , which are de®ned in the volume and on the surface as ^n =^ c^n q q:
cn qn =q;
Since by de®nition cN1 1 ÿ
PN1 n1
N X cs ;
qn q and
PN1
qN1 q 1 ÿ
s1
c^N1
25 n1
^, it follows that ^n q q !
N X cs ;
26
s1
N X c^s ; 1ÿ
^N1 q
! N X ^ 1ÿ c^s ; q
s1
27
s1
where 1 6 s 6 N. With the mass fractions the barycentric velocity and mass diusion ¯uxes are expressed as vi
N1 X cn vni ;
Jin qcn vni ÿ vi ;
28
N1 X c^n v^ni ;
n ^c^n ^ J^i q vni ÿ v^i :
29
n1
v^i
n1
The velocities and densities of all N 1 components in the volume [8] and on the surface can be s ^, redetermined from the 4N 3 1 and 4N 1 independent variables cs ; Jis ; vi ; q and c^s ; J^i ; q spectively using the formulas qs cs q;
qN1 q 1 ÿ
N X
! cs ;
30
s1
vsi and
qcs ÿ1 Jis
vi ;
vN1 vi ÿ qÿ1 i
N X 1ÿ cs s1
!ÿ1
N X Jis s
31
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
143
!
^s c^s q ^; q
^ 1ÿ ^N1 q q
N X c^s ;
32
s1
v^si
s ^ qc^s ÿ1 J^i
v^N1 i
v^i ;
ÿ1
^ v^i ÿ q
X c^s 1ÿ s1
!ÿ1
X s1
s J^i :
33
Finally the component balance laws, Eqs. (16)±(18), are rewritten in terms of the mass fractions as qc_s ri Jis ps ;
in V c [ V g
^ i J^s mi J s p^s ; ^c^_s r q i i cs
gs
34
on rc [ rg [ rb
bs
^c c^gs q ^g ÿ c^bs q ^b 0 si J^i J^i ÿ J^i si ^ vi ÿ vci ^ ccs q
35 along c:
36
4. Maxwell's equations Although we allow for surface free charge, we do not include surface electric ®eld E^i , surface ^ i , or surface magnetic ®eld strength H^ i . Faraday's law in integral form is electric displacement D Z Z d Bi ni da Ei ki ds 0; 37 dt S
oS
where Ei is the convected electric ®eld, Bi the convected magnetic induction, S an arbitrary surface, and ki the unit vector tangent to the line. The Gauss' law can be written in terms of the electric displacement vector Di and the volume and surface free charge densities qf and q^f as Z Z Z f f q^ da q dv Di ni da: 38 r
V
oV
Ampere's law can be written in terms of the eective magnetic ®eld strength (magnetomotive ^ i as intensity) Hi and the volume and surface conduction currents Ii and I Z Z Z Z ^ i si ds Ii ni da d Di ni da Hi ki ds: I 39 dt r\S
S
S
oS
If S is a closed surface, the integral of Ei in Eq. (37) is identically zero [10] and one can derive the Gauss±Faraday law Z Bi ni da 0 oV
40
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E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
for any volume V . The transport and divergence theorems can be used to localize Eqs. (37)±(40) [7]:
Bi eijk Ek;j 0; in V c [ V g
eijk mj Ek 0; on rg [ rc [ rb ;
mj Dj q^f ; on rg [ rc [ rb ;
Di;i qf ; in V c [ V g
eijk Hk;j ÿ Di Ii ; in V c [ V g Bi;i 0; in V c [ V g where
^ i ; on rg [ rc [ rb ; eijk mj Hk I
mi Bi 0; on rg [ rc [ rb ;
41 42 43 44
denotes the convective derivative:
Ui
oUi rj Ui vj Ui rj vj ÿ Uj rj vi : ot
45
The eect of the material on the electromagnetic ®elds can be introduced in various ways. We use the statistical formulation for the nonconvected variables electric ®eld Ei , polarization Pi , electric displacement Di and magnetic induction Bi which can be formally obtained from the above equations through the substitutions: E i Ei ;
Bi B i ;
Di 0 Ei Pi ;
Hi Hi ÿ eijk vj Dk ;
46
where 0 is the permittivity of free space, Hi 1=l0 Bi is the magnetic ®eld strength (current potential) observed in the stationary (laboratory) frame, and eijk vj Bk is assumed negligible. Substitution of Eq. (46) and the assumption oBi =ot 0 into Eqs. (41) and (42) results in eijk Ek;j 0; in V c [ V g
eijk mj Ek 0 on rg [ rc [ rb ;
0 Ei;i qf ÿ Pi;i ; in V c [ V g
0 mj Dj q^f on rg [ rc [ rb :
47 48
The Maxwell equations in V u are simpli®ed by the assumption that all ®elds there are uniform in mi direction except displacement, velocity and electric potential. Consider the portion of the double surface shown in Fig. 5. The jump conditions for the electric displacement vector Eq. (48) on rg and rc yield 0 mi Eig ÿ Eiu q^g q^p ÿ mi Pig ;
0 mi Eiu ÿ Eic q^c ÿ q^p ;
49
where q^p mi Piu
50
is surface polarization charge due to the presence of polarizable matter between rc and rg . This is somewhat contradictory to the statement that the bonding is performed in vacuum. If
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
145
Fig. 5. Interfacial separation and polarization charge.
that was the case q^p will be zero. But it has been suggested [2] that the surface roughness greatly reduces the eective electrostatic pressure by dropping the potential across the gap. This eect is modeled herein as a surface dipole sheet between the two adjacent bulk materials [14,17]. We require Eiu to be perpendicular to rc and use the notation E? for the magnitude of the electric ®eld in V u Eiu E? mi :
51
The conservation of charge can be derived from the full set of Maxwell's equations, but not after using the present simplifying assumptions. Therefore, the charge conservation which is enforced by Maxwell's equations, will instead be established through a separate integral postulate Z I Z Z d d f f ^ i si ds 0; q^ da q dv Ii ni da I 52 dt dt r
V
oV
or
followed by localization using the divergence and transport theorems in a manner similar to Eqs. (16)±(18). oqf ri qf vi ri Ii 0 in V c [ V g ; ot ^ o^ qf ^ ^ i mi Ii 0 on rc [ rg [ rb ; ^ iI vi q^f r ri ^ ^ ot
^c I ^g ÿ I ^ b ^ si I vi ÿ vci ^ qc q^g ÿ q^b 0 along c: i i i
53 54 55
146
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
5. Conservation of momentum 5.1. Linear momentum For small currents and negligible magnetization in the quasi-electrostatic approximation the electrostatic body force fie is given by fie qf Ei Pj rj Ei ;
56
fie can also be written as the divergence of the electrostatic stress tensor Tije fje ri Tije ;
57
where 1 1 Tije 0 Ei Ej Pi Ej ÿ 0 Ek Ek dij Di Ej ÿ 0 Ek Ek dij : 2 2
58
Only zeroth order singularities (discontinuities) are assumed for the stress ®eld, so surface tension is not included on r. The quasi-static global balance of linear momentum for an arbitrary volume element is given by Z 0
Tij Tije ni da;
59
oV
where Tij is the Cauchy stress tensor. The above equation is a generalized version of the momentum balance, where the electrostatic body force has been substituted with the surface integral of the electrostatic stress tensor. To obtain the local form of momentum balance we apply the divergence theorem to Eq. (59) for the case when the volume V contains a surface r where the stress tensors have discontinuity, Z 0 oV
Tij Tije ni da
Z
ri Tij ri Tije dv
V
Z
mi Tij Tije da:
60
r
In deriving the mass balance we assumed that there was no mass within V u . Nevertheless, in deriving the equilibrium equations we will assume that V u can support a compressive Cauchy stress. This assumption is used in the cohesive zone theories of fracture mechanics. Localization of Eq. (60) yields 0 ri Tij ri Tije in V c [ V g [ V u ;
61
0 mi Tij Tije on rc [ rg [ rb :
62
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
Because V u is very thin, oV u rc [ rg . Applying Eq. (60) to V u results in Z Z u ue 0 mi Tij Tij da mi Tiju Tijue da rc [rg
147
63
ru
or in local form, 0 mi Tiju Tijue ru
on ru :
64
From Eqs. (62) and (64) it can be seen that the total stress vector (traction) is continuous across any of the surfaces mi Tiju Tijue mi Tijg Tijge mi Tijc Tijce :
65
However, mi Tij and mi Tije are not necessarily continuous across a surface because a surface charge creates a surface force distribution which gives rise to a jump in the electric stress tensor. From Eqs. (62), (64) and (65) it follows that 0 mi Tij Tije gc
across V u :
66
5.2. Angular momentum The quasi-static global balance of angular momentum for an arbitrary volume element containing discontinuity surfaces r is given by Z 0 eijk xj nl Tlk Tlke da: 67 oV
Applying the divergence theorem to Eq. (67) yields 0 eijk Tjk Tjke
in V c [ V g [ V u
68
which stipulates that Tij Tije is symmetric. The tensor Pi Ej is symmetric in Vg because the glass is assumed to be an isotropic material. Furthermore, Tjke is zero in V c , the conductor. Thus, in both materials Tije is symmetric and from Eq. (68) it follows that the Cauchy stress tensor is also symmetric. Thus, the conservation of angular momentum simpli®es to the trivial result Tjk Tkj ;
Tjke Tkje
in V c [ V g [ V u :
69
Even if the glass was not isotropic and the electric stress was therefore not symmetric, it may be assumed that the electromagnetic body couples are small. This assumption is usually used in the theory of piezoelectric fracture, which includes very high electric ®elds near crack tips in anisotropic materials. The conservation of angular momentum at the interfaces rc ; rg and rb is trivially satis®ed as well.
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E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
6. Conservation of energy The ®rst law of thermodynamics states that the rate of change of the sum of kinetic, internal and free electromagnetic energies is equal to the sum of the power of forces acting on the system, heat supplied to the system and the electromagnetic energy ¯ux, which is the Poynting vector (see Ref. [10] for example). The quasi-static global form of the ®rst law for an arbitrary volume element containing discontinuity surfaces r is given by Z h Z Z Z i d d 1 0 ^ da Tji Tjie vi ÿ qj ÿ ejik Ei Hk nj da ÿ q^i si ds; Ek Ek q dv dt dt 2 r
V
oV
or
70 where and ^ are the internal energies per unit mass and per unit area, and qi and q^i are the heat ¯ux vectors per unit volume and area. The surface transport Theorem (14) and Eq. (19) can be used to rewrite the ®rst term in Eq. (70) as Z Z Z Z Z c g u b d ^ da ^_ da ^_ da ^_ da ^_ da dt r rc rg ru rb Z si ^ vi ÿ vci ^c ^g ^u ÿ ^b ds; 71 c
where the surface energy ^u contains the polarization and elastic energy of the unbonded region V u . Eq. (71) can be used to rewrite Eq. (70) as: Z Z Z Z Z c g u b ^_ da ^_ da ^_ da ^_ da si ^ vi ÿ vci ^c ^g ^u ÿ ^b ds rc
Z V
rg
ru
c
rb
oEk 0 Ek Ek Ek;j vj Ek Ek rj vj q_ dv ot 2 0
0
Z h i Tji Tjie rj vi ÿ rj qj ÿ ejik rj Ei Hk Ei rj Hk dv V
Z
Z ^ i q^i da ÿ si ^ ÿ qj ÿ ejik Ei Hk mj ÿ r qci q^gi ÿ q^bi ds;
r
72
c
where we have used the momentum balance equations and the fact that r are material surfaces to eliminate the divergence of the stress tensors. Eqs. (41)±(43) can be used to localize Eq. (72):
q_ Tji rj vi Ei Pj rj vi Ei Ii Ei Pi ÿ ri qi oPi _ rj Pi vj Ei Ii ÿ ri qi Tij Lij Ei ot
in V c [ V g
73
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
149
If one de®nes polarization per unit mass, pi Pi =q, Eq. (73) can be rewritten as q_ Tij L_ ij ÿ ri qi Ii Ei qEi p_ i ;
74
since, due to Eq. (21), q
d o oPi rj vj Pi : Pi =q Pi rj Pi vj Pi rj vj ot dt ot
75
The remaining terms in the global energy balance, Eq. (72), include integrals along c, rc , rg , rb and the thin region V u . Localization along c yields h i si ^ 76 vi ÿ vci ^c ^g ^u ÿ ^b q^ci q^gi ÿ q^bi 0 along c: The ®nal terms in Eq. (72) refer to the surfaces: Z rc
c ^_ da
Z
g ^_ da
rg
Z
u ^_ da
ru
Z
b ^_ da
rb
Z 0 0 oEk 0 Ek Ek rj Ek vj Ek Ek rj vj dv ot 2 Vu
Z h
i Tji Tjie rj vi ÿ rj qj ÿ ejik rj Ei Hk Ei rj Hk dv
Vu
Z ^ i q^i da: ÿ qj ÿ ejik Ei Hk mj ÿ r
77
r
Recall that across the thickness of V u we assumed that all ®elds are uniform except displacement, velocity and electric potential. Thus it is convenient to rewrite some of the volume integrals in Eq. (77) as boundary integrals Z rc
c ^_ da
Z rg
g ^_ da
Z ru
u ^_ da
Z
b ^_ da
rb
Z 0 0 oEk 0 Ek Ek rj Ek vj Ek Ek rj vj dv ot 2 Vu
Z h
i mj Tji Tjie vi ÿ mj qj ÿ ejik Ei Hk mj da
oV u
Z ^ i q^i da: ÿ qj ÿ ejik Ei Hk mj ÿ r r
78
150
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
Using Eq. (46) the convected Poynting vector can be rewritten as 1 0 eijk Ej Hk eijk Ej Hk Tije vj ÿ Ek Ek Ek Pk vi : 2
79
The Poynting theorem can be simpli®ed by dropping the time variation of the magnetic ®eld as well as the Joule heating term because there are no conduction or convection currents in V u , i.e. ÿ ejik Ei Hk ;j Hi
oBi oDi oDi Ei Ii Ei Ei in V u ; ot ot ot
80
where Ii Ii qf vi . Eqs. (79) and (80) can be used to cancel out the energy of the free space in Eq. (78) to obtain Z Z Z Z c g u b ^_ da ^_ da ^_ da ^_ da rc
rg
Z
mj Tjiu n_ i
ru
ÿ
quj mj
rb
Piu Eiu n_ j mj
ru [rb
Z
Eiu
oPiu n mk da ot k
^ i q^i da; ÿ qj ÿ ejik Ei Hk mj ÿ r
81
n_ i vi gc ;
82
r
where ni ui gc n0i ;
where n0i is a constant. The surface energy balance, Eq. (81), can be rewritten in terms of Eqs. (50), (82) and (51) as Z Z Z Z c g u b ^_ da ^_ da ^_ da ^_ da rc
rg
Z ru [rb
Z
ru
rb
! p o^ q mj Tjiu n_ i ÿ quj mj q^p E? n_ j mj E? nk mk da ot
^ i q^i da: ÿ qj ÿ ejik Ei Hk mj ÿ r
83
r
The last integral in Eq. (83) contains the jump in the normal component of the Poynting vector, which can be simpli®ed as follows: mi eijk Ej Hk mi eijk hEj iHk mi eijk Ej hHk i:
84
The jump conditions (41) and (43) can be used to further simplify Eq. (84), resulting in ^ j: mi eijk Ej Hk ÿhEj iI
85
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
151
Thus, it can be seen that the electromagnetic energy ¯ux crossing a surface is reduced by the magnitude of the Joule heating due to surface conduction currents. This result can be used in the localization of Eq. (83) c ^ i q^i I ^ i hEi i c ^_ ÿqj rc mj ÿ r r
on rc
86
g ^ i q^i I ^ i hEi i g ^_ ÿqj rg mj ÿ r r
on rg ;
87
p
u o^ q ^_ mj Tjiu n_ i q^p E? n_ j mj E? n mk ot k
on ru ;
88 p
b q ^ i q^i I ^ i hEi i b q^p E? n_ j mj E? o^ ^_ mj Tjib n_ i ÿ qj rb mj ÿ r n mk ; r ot k
on rb :
89
Introducing ^ut as the total surface energy along the unbonded interface, ^ut ^c ^g ^u ;
90
Eqs. (86)±(89) and (76) can be rewritten as p
ut o^ q ^_ ÿ mi qi gc mi Tiju n_ j q^p E? n_ j mj E? n mk ot k c
g
^ hEi i c I ^ hEi i g ÿ r ^ i q^c ÿ r ^ i q^g I r r i i i i
on ru ;
91 p
b q ^ b hEi i b ÿ r ^ i q^b q^p E? n_ j mj E? o^ ^_ ÿmi qi rb mi Tiju n_ j I n mk r i i ot k
h i si ^ vi ÿ vci ^ut ÿ ^b q^ci q^gi ÿ q^bi 0
along c:
on rb ;
92 93
7. Second law of thermodynamics and constitutive equations For simplicity we assume that rc , rg , and ru are in thermal equilibrium and therefore have the same temperature distribution hc hg hu :
94
The entropies on rc , rg , and ru are therefore added in a manner similar to the energies, Eq. (90). We extend the global [3] Clausius±Duhem inequality to include the surface terms, neglecting radiative heating, as
152
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
d dt
! Z Z N X d 1 ^ ^ ls Jis da gut da gb da = ÿ ni qi ÿ dt h s1 V c [V g ru oV rb ! I N X s 1 ^s Jbi si ds; q^i ÿ ÿ l ^ h s1 Z
d qg dv dt
Z
95
or
where g and ^ g are the entropies per unit mass and area, h and ^h are the volume and surface temperatures, and ls are the reduced chemical potentials [8] ls ls ÿ lN1 :
96
The global inequality (95) can be localized in a manner similar to the mass balance, resulting in: ! N X ls Jis ri qi 1 ÿ qi ri in V c [ V g 97 ri qg_ = ÿ h h h s1 "" ut
^_ = ÿ mi g
N X 1 ls Jis qi ÿ h s1
!##g c
! # N c g X ^ ^ q 1 q cs gs cs gs i i ^i ^i ^ J^i l ^ J^i ÿr r u l ^hu ^ h s1 "
on ru 98
"" b
^_ = ÿ mi g
N X 1 ls Jis qi ÿ h s1
!## ^i r rb
"
si ^ vi ÿ
vci ^ gut
1 ^ = si u ÿg ^ h b
ÿ
1 ^b h
ÿ
q^ci
ÿ
ÿ q^bi
q^gi
N X s1
N X s1
N X 1 bs ^bs ^ Ji l ^u s1 h
cs ^cs Jbi l
!#
bs
^bs J^i l
! ^i ÿr
N X gs ^gs Jbi l
q^bi ^hb
! on rb
99
!
s1
along c:
100
Inequalities (97)±(100) will now be used to constrain the form of the constitutive equations. The internal energies are assumed to have the following arguments: s g; Lrev ij ; pi ; c
in V c [ V g ;
101
^ut ^ut ^ ^p ; c^cs ; c^gs on rc [ rg [ ru ; gut ; ni ; nirr i ;q
102
^b ^b ^ gb ; c^bs
103
on rb ;
where the Lagrangian strain tensor is decomposed into reversible and irreversible parts irr Lij Lrev ij Lij :
104
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
153
The following transformation is used to replace the entropy in the list of arguments with the temperature and the polarization with the electric ®eld: s w h; Lrev ij ; Ei ; c ÿ hg ÿ Ei pi
in V c [ V g ;
105
u ut ^ ut ^ ^cs ; c^gs ^ut ÿ ^ w h; ni ; nirr h ^ g ÿ E? q^p nk mk i ; E? ; c b b b ^ b ^ w h ; c^bs ^b ÿ ^ h ^ g
on rc [ rg [ ru
on rb ;
106 107
^ are the Helmholtz free energy per unit mass and per unit area. The derivation of where w and w the thermodynamic constraints requires the rate form of Eqs. (105)±(107): _ ÿ hg_ _ ÿ w_ hg
d Ei pi dt
in V c [ V g ;
ut d u ut ut _ u ut _ ^_ ^ ^ _ h ^ g ^ ÿ w h^ g ÿ E? q^p nk mk dt
108
on rc [ rg [ ru ;
b b b b _b b ^_ ^ ^ h ^ on rb : h^ g_ ^_ ÿ w g
109
110
Also, it is useful to rewrite Eq. (97) as s X N N X l qi ri h s qhg_ = ÿ ri qi ls ri Jis : h Ji ri h h s1 s1
111
Using Eqs. (105) and (74) in Eq. (111) we obtain s X N N X l ÿ1 s _ _ _ _ ÿq w gh Tij Lij ÿ Pi Ei ÿ h qi ri h ÿ h Ji ri ÿ ls ri Jis = 0: h s1 s1
112
Inequality (112) can simpli®ed further by applying the chain rule of dierentiation to the Helmholtz free energy, N X ow _ ow _ ow _ irr ow _ ow s _ qw q h q rev Lij ÿ q rev Lij q c_ ; Ei q oh oLij oLij oEi cs s1
113
and substituting this result and the volume mass balance, Eq. (34), into inequality (113) to obtain
154
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
! ! ow ow ow _ irr ow _ _ _ Ei Ii Ei ÿq g h Tij ÿ q rev Lij q rev Lij ÿ Pi q oh oLij oLij oEi s N N N X X X l ow ri h s s s s s ÿ l c_ ÿ ÿq l p ÿ qi P 0: ÿ h Ji ri s h oc h s1 s1 s1
114
The last PN two terms in inequality (114) can be rewritten in terms of the conductive heat ¯ux (qi ÿ s1 ls Jis ): ! s N N N X X l ri h ri h X s s s ÿ qi ÿ l Ji Jis ri ls : ÿ h J i ri ÿ ÿ qi h h h s1 s1 s1
115
Inequality (114) must be satis®ed for all independent thermodynamic processes. This constraint [4] results in the following thermodynamic conjugates: Tij q
ow ; oLrev ij
gÿ
ow ; oh
Pi ÿq
ow ; oEi
ls
ow : ocs
116
In the case of ionic conduction (for the glass region V g ) the electric current can be expressed with the diusion mass currents using the component valence zn , molar mass Mn and the Faraday's constant F: Igi
N s N X X z zN1 s ÿ J F ~zs Jis ; s i N1 M M s1 s1
117
where s z zN1 ÿ : ~z F Ms MN1 s
Eqs. (115)±(118) can be used to reduce inequality (114) to ! N N N X X ri h X irr s s Tij L_ ij ÿ qi ÿ l Ji Jis ri ls /~zs ÿ ls p s = 0 ÿ h s1 s1 s1
118
in V g :
119
In the conductor, V c , the conduction current is independent of mass ¯uxes, thus the entropy inequality is ! N N N X X ri h X irr Tij L_ ij ÿ qi ÿ ls Jis Jis ri ls Ici Ei ÿ ls ps = 0 in V c : 120 ÿ h s1 s1 s1 The thermodynamic conjugates and reduced dissipation inequalities will now be derived for the ut b surfaces in a bmanner similar to the volumes. We substitute ^g_ and ^g_ from Eqs. (109) and (110) ut and ^_ and ^_ from Eqs. (91) and (92) into inequalities (98) and (99) to obtain:
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
155
ut _ u ut ^_ ÿ ^ ^ c hEi i c I ^ g hEi i g h ^ ÿw g mi Tiju n_j ÿ E_ ? q^p ni mi ÿ mi qi gc I r r i i "" !##g N N c g ^ ^u X X u u 1 ^ q q^i ri h 1 cs ^cs gs ^ ^i ^gs J^i = 0 h ÿ i mi ls Jis ÿ ^h r l Ji l qi ÿ u u ^ ^h ^ h s1 s1 h c
g
on r [ r [ r
c
u
121
and b _b b ^_ ÿ ^ ^ b hEi i b h ^ ÿw g ÿ mi qi rb I r i "" !## ! bs ^bs N N b ^ ^b X X ^ J l b b ^ h r q 1 i i ^ ^i =0 h ÿ i b ÿ ^h r mi lbs Jibs qi ÿ b h ^ ^ h h s1 s1 b r
on rb
122
The surface mass balance, Eq. (35), can be used to rewrite the last terms in inequalities (121) and (122) as ! ! s! s ^s J^i ^s ^ l l l s s s s ^ ^i ^i ^ i J^ ^hr ^i ^s r ^s ÿ^ hr ^ hr qc^_ ÿ mi Jis p^s ; 123 J^i l J^i l i ^ ^ ^ h h h which can then be used in Eqs. (121) and (122), combining the jumps of the volume diusions in a single jump, to obtain X X ut X cs X gs _ u ut ^_ ÿ ^ ^c c^_ l ^g c^_ l ^cs q ^gs ÿ ^cs p^cs ÿ ^gs p^gs h ^ l l ÿw g mi Tiju n_j ÿ E_ ? q^p ni mi q N
s1
"" ^ c hEi i c I ^ g hEi i g ÿ mi I r r i ÿ q^gi ÿ c
N X s1
! gs
^gs J^i l
qi ÿ
N X s1
!
ls Jis
h ÿ ^h h
N
N
s1
s1
u
##g ÿ c
q^ci
ÿ
N
s1
N X s1
! cs ^cs J^i l
^ i ^hu r u h^
u
N N N X X X ^ i^ h r cs gs ^ il ^ il ^cs ÿ ^gs P 0 ^s gc ÿ J^i r J^i r Jis ls ÿ l u ÿ mi ^ h s1 s1 s1
g
on r [ r [ ru
124
for the unbonded interface and N N X X b bs bs _b b ^_ ÿ ^ ^ b hEi i b ^ c^_ l ^ ÿ ^bs p^bs I h ^ ÿw l g q r i
"" ÿ mi ÿ mi
N X s1
s1
!
s1
b N X hÿ^ h s s qi ÿ l Jj h s1
^bs rb ÿ Jis ls ÿ l
##
N X s1
ÿ
q^bi
ÿ
rb bs ^ lbs P 0 J^i r^
N X s1
! bs ^bs J^i l
on rb
^ i ^hb r ^hb 125
156
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158 ut
^ in Eq. (124) according to for the bonded interface. We next expand the free energy potential w the list of arguments assumed in Eq. (103) and subsequently in Eq. (107): ! ! ^ ut ^ ut ow ow u ^_u u g h mi Tij ÿ n_ j ÿ u ^ onj o^ h
! ^ ut ow p q^ ni mi E_ ? ÿ oE? ! ! N N ^ ut ^ ut ^ ut X X ow o w ow cs c cs _ g gs _ gs ^ c hEi i c I ^ g hEi i g ^l ^l ^ c^ ÿ ^ c^ ÿ irr n_ j irr I ÿ cs ÿ q gs ÿ q r r i i o^ c o^ c onj s1 s1 "" ! u ##g N N N ^ X X X ^ i ^hu r h ÿ h cs gs c g cs gs s s ^ ^ ^ Ji ÿ ^ Ji ÿ mi qi ÿ l Ji ÿ q^i q^i ÿ l l ^hu h s1 s1 s1 c ÿ
N X
^cs p^cs ÿ l
s1
N N N N X X X X cs gs ^ il ^ il ^gs p^gs ÿ mi Jis ls ÿ l ^cs ÿ ^gs P 0: ^s gc ÿ J^i r J^i r l s1
s1
s1
126
s1
Inequality (126) must apply for all processes, which leads to the thermodynamic conjugates ut
^ ow ^ ÿ u ; g o^ h ut
ut
mi Tiju
^ ow ; onj
ut
ut
^ ow ni mi q^ ÿ ; oE?
^ ow ^l ^ cs ; q o^ c c cs
p
ut
^ ow ^l ^ gs ; q o^ c g gs
127
and the residual dissipation inequality ut
^ ow ^ c hEi i c I ^ g hEi i g ÿ irr n_ j irr I r r i i onj "" ! u ##g N X hÿ^ h s s ÿ mi qi ÿ l Ji ÿ h s1 c
ÿ
N X
^cs p^cs ÿ l
s1
q^ci
q^gi
ÿ
N X s1
cs ^cs J^i l
! N X ^ i ^hu r gs ^gs ^ Ji ÿ l ^hu s1
N N N N X X X X cs gs ^ il ^ il ^gs p^gs ÿ mi Jis ls ÿ l ^cs ÿ ^gs = 0 ^s gc ÿ J^i r J^i r l s1
s1
on rc [ rg [ ru :
s1
s1
128
Repeating the above procedure for the bonded interface yields the thermodynamic conjugates b
^ ow ^ g ÿ b; o^ h b
b
^ ow ^ l ^ bs ; q o^ c b bs
and the residual dissipation inequality
129
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
"" ^ b hEi i b I r i
ÿ mi
b N X hÿ^ h s s qi ÿ l Ji h s1
!## ÿ rb
q^bi
! N X ^ i ^hb r bs ^bs ^ Ji ÿ l ^hb s1
N N N X X X bs bs bs bs s s ^ il ^ p^ ÿ mi Ji l ÿ l ^bs = 0 ^ rb ÿ J^i r ÿ l s1
s1
157
on rb :
130
s1
The surface current on the glass is related to the surface mass diusion by N X gs ~zs J^i ;
^g I i
131
s1
and the average electric ®eld on rg is given by hEi irg ÿri h/irg :
132
With these results and the normal and surface decomposition of the derivative operator, Eq. (3), we see that, due to the transversality condition, Eq. (1), only the surface gradient of the potential creates Joule heat, since by de®nition the surface currents are along the interface ^g ÿ hEi irg I i
N N X X gs gs ^ i h/i g : ~zs J^i ri h/irg ÿ ~zs J^i r r s1
133
s1
Eq. (133) can be used in Eqs. (128) and (130) to rewrite the residual dissipation inequality for the unbonded interface as ÿ
N N ^ ut X X ow gs cs gs s _ j irr I ^ i ^ ^ il ^ c hEi i c ÿ ^ ^cs J J^i r n r l ~ z h/i ÿ r rg i i irr onj s1 s1 "" ! ! ## g u N N N ^ X X X ^ i ^h0u r h ÿ h cs gs c g cs ^ gs ^ s s ^ Ji ÿ ^ Ji q^i q^i ÿ ÿ mi qi ÿ l Ji l l ^hu h s1 s1 s1 c
ÿ
N N N X X X ^cs p^cs ÿ ^gs p^gs ÿ mi Jis ls ÿ l ^s gc 0 l l s1
s1
s1
on rc [ rg [ ru
134
and for the bonded interface as ÿ
N X s1
ÿ
"" bs ^ i ^ J^i r lbs
q^bi
ÿ
N X s1
~zs h/irb ÿ mi !
bs ^bs J^i l
! ## N ^hb X h ÿ qi ÿ ls Jis h s1
rb
b N N X X ^ i^ h r bs bs ^ p^ ÿ mi Jis ls ÿ l ^s rb = 0 ÿ l b ^ h s1 s1
on rb
135
158
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
One can identify the generalized ¯uxes and their conjugate forces for the dissipative processes in inequalities (119), (120), (134), (135), and (100). Inequality (100) requires that the entropy production due to bonding or debonding and the heat conduction into c must be positive semide®nite. The velocity of c due to bonding or debonding is given by the dierence 8 < < 0 debonding; si vci ÿ v^i 0 neitherbondingnordebonding; 136 : > 0 bonding: gb , to the bonding or debonding velocity is equal to the The thermodynamic conjugate, ^ gut ÿ ^ dierence between the limiting values of the entropy of the unbonded and bonded interfaces at c. The bonding or debonding velocities can be written as a function of ^gut ÿ ^gb . If the interfacial entropy is a continuous function at c, then 0 ^gut ÿ ^gb . In this case, the bonding or debonding velocities could be written as a function of the separation ni . This approach is used in the thermodynamic cohesive zone theory of fracture mechanics [5] in which the assumption of self-similar crack growth is used to relate the crack propagation rate to the interfacial separation. References [1] M.G. Ancona, H.F. Tiersten, Fully macroscopic description of bounded semiconductors with an application to the si-sio2 interface, Phys. Rev. B 22 (12) (1980) 6104±6119. [2] T. Anthony, Anodic bonding of imperfect surfaces, J. Appl. Phys. 54 (5) (1983) 2419±2428. [3] R. Bowen, Theory of Mixtures, vol. 3, Continuum Physics, Academic Press, New York, 1976. [4] B.D. Coleman, M. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47 (2) (1967) 597±613. [5] F. Costanzo, D.H. Allen, A continuum thermodynamic analysis of cohesive zone models, Internat. J. Engrg. Sci. 33 (15) (1995) 2197±2219. [6] A. Cozma, B. Puers, Characterization of the electrosatic bonding of silicon and pyrex glass, J. Micromech. Microengrg. 5 (1995) 98±102. [7] N. Daher, G.A. Maugin, Deformable semiconductors with interfaces: basic continuum equations, Internat. J. Engrg. Sci. 25 (9) (1987) 1093±1129. [8] D.G.B. Edelen, Mass balance laws and the decomposition, evolution and stability of chemical systems, Internat. J. Engrg. Sci. 13 (1975) 763±784. [9] P.J. Hesketh, Y. Lin, J.G. Boyd, S. Zsivanovic, J. Cunnen, Y. Ming, J. Stetter, S.M. Lunte, G.S. Wilson, Biosensors and micro¯uidic systems, in: International Symposium on Aerospace, Nagoya, Japan, 1996. [10] K. Hutter, A.A.F. Van De Ven, Field Matter Interactions in Thermoelastic Solids, Lecture Notes in Physics, Springer, Berlin, 1978. [11] P. Jorgensen, Eect of an electric ®eld on silicon oxidation, J. Chem. Phys. 37 (1962) 874±877. [12] Y.C. Lin, P.J. Hesketh, J.G. Boyd, S.M. Lunte, G.S. Wilson, Characteristics of a polyimide microvalve, in: SolidState Sensor and Actuator Workshop, Hilton Head, South Carolina, June 2-6, 1996, pp. 113±116. [13] G. Maugin, Continuum Mechanics of Electromagnetic Solids, North-Holand, Amsterdam, The Netherlands, 1988. [14] W.K.H. Panofsky, Classical Electricity and Magnetism, Addison-Wesley, Reading, MA, 1962. [15] E. Peeters, S. Vergote, B. Puers, W. Sansen, A combined silicon fusion and glass/silicon anodic bonding process for uniaxial capacitive accelerometer, J. Micromech. Microengrg. 2 (1992) 176±179. [16] E.H. Snow, M.E. Dumesnil, Space-charge polarization in glass ®lms, J. Appl. Phys. 37 (5) (1965) 2123±2131. [17] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [18] G. Wallis, D. Pomerantz, Field assisted glass-metal sealing, J. Appl. Phys. 10 (10) (1969) 3946±3949.
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A thermodynamic ®eld theory for anodic bonding of micro electro-mechanical systems (MEMS) Eniko T. Enikov *, James G. Boyd Department of Mechanical Engineering (M/C 251), The University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607, USA Received 27 August 1998; received in revised form 1 February 1999; accepted 3 February 1999 (Communicated by J.T. ODEN)
Abstract An anodic bond is modeled as a moving nonmaterial line forming the intersection of three material surfaces representing the unbonded conductor, the unbonded insulator, and the bonded interface. Global integral equations are written for the conservation of mass, momentum, and energy, Maxwell's equations, and the second law of thermodynamics. The global equations are then localized in the volume, the material surfaces, and the nonmaterial bond line. The second law is used to determine the thermodynamic conjugates in the thermodynamic potential and the dissipation inequality. It is demonstrated that the jump in the Poynting vector across a surface is equal to the surface Joule heating due to surface electric conduction currents. Ó 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction Anodic bonding is a common method of bonding glass to conductors such as silicon or metal. The glass and conductor are brought into contact at elevated temperature and an electric potential is applied with the cathode on the glass and the anode on the conductor. The mobile positive ions, typically 2% by mass Na in Pyrex glass, diuse towards the cathode, leaving a negatively charged glass layer at the glass/conductor interface (Fig. 1). A positive charge develops on the conductor surface, and the electrostatic attraction eventually pulls the glass and conductor together with a pressure high enough to initiate a surface reaction to form a chemical bond. Unfortunately, anodic bonding is performed under harsh conditions of temperature, pressure and electric ®eld. For example, Micro Electromechanical Systems (MEMS) are often made by *
Corresponding author. Tel.: 001-312-996-6593; fax: 001-312-413-0447. E-mail address: [email protected] (E.T. Enikov)
0020-7225/00/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 2 2 5 ( 9 9 ) 0 0 0 2 7 - 0
136
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
Fig. 1. Anodic bonding.
bonding 300 lm thick Pyrex and silicon wafers together at 400 C with 400 V applied to the Pyrex wafer. This leads to complicated microstructural evolution including creep, ionic diusion, residual stresses, warping, and spatial instabilities such as electrical breakdown and symmetry breaking of the bond front. In particular, the extreme conditions limit the choice of polymers that can be used in MEMS [12,9]. Although anodic bonding is widely used to fabricate MEMS such as accelerometers [15], pressure sensors [6], and micropumps [12,9], there has never been a thermodynamic ®eld formulation of the process. Studies have typically consisted of simple electrical circuit analogs [2], chemistry and physics of the surface reactions [11], or one-dimensional diffusion [16] normal to plane surfaces. In the present paper we present a thermodynamic ®eld theory applicable to anodic bonding of MEMS. The work of Ancona and Tiersten [1] and Daher and Maugin [7], in which they developed the fundamental equations for semiconductors with interfaces, is directly applicable to the current formulation. However, we account for a double interface and a moving bond line. The physical domain, assumptions, and notation are introduced in Section 2. The fundamental laws are discussed in subsequent sections: the conservation of mass (Section 3), Maxwell's equations (Section 4), the conservation of momentum (Section 5), the conservation of energy (Section 6), and the second law of thermodynamics and constitutive equations (Section 7). The theory presented herein is also applicable to debonding along an insulator/conductor interface. In addition, the formulation can be modi®ed to model a wide variety of electro-me-
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
137
chanical phenomena at interfaces, including space charge accumulation and sliding at grain boundaries in ionic crystals, and a cohesive zone theory of electro-elastic fracture. 2. Assumptions, de®nitions, and notation Subscripts denote tensor components in three-dimensional space, and superscripts denote qualitative descriptions. Repeated subscripts are summed. The physical domain (Fig. 2) consists of the volumes V c , V g , and V u representing the conductor, glass, and unbonded interface. The glass and conductor are separated by surface asperities. Electrostatic pressure causes creep deformation which ¯attens these asperities, and bonding occurs when the surface roughness is suciently small. Rather than follow the motion of each asperity, the surfaces will be modeled as material surfaces rc and rg (dashed lines on Fig. 2) passing through some averaged position of the asperities. The creep deformation of the asperities is modeled using an interface creep equation. The unbonded interface ru is the union of the midpoints between rc and rg . The bond front is modeled as a nonmaterial line, c, which is the boundary of the bonded interface, rb . The bonded interface is de®ned as the region of the interface that can support nonzero tensile and shearing Cauchy stress vectors by means other than friction. It is assumed that there is no friction force between rc and rg . Although the actual conductor and glass surfaces have dierent unit normal vectors, it is assumed that rc , rg and ru have a common unit normal vector ®eld, mi , which is also the outer unit normal vector ®eld to the bond line, c (Fig. 3). This simplifying assumption can be relaxed to include cases in which ru contains corners. Materials that undergo ionic or electronic conduction across the asperities relax
Fig. 2. Geometry of bonding.
138
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
Fig. 3. Tangent, normal, and binormal vectors along the bond line.
the surface charge and electric pressure and therefore do not form an anodic bond. Therefore, it is assumed that there is no mass diusion or electronic conduction between rc and rg , although mass diusion is allowed across rb . Quasi-static conditions are assumed and the inertia terms are neglected in the momentum and energy equations. Surface properties play an important role in the fabrication and operation of MEMS. Volumes are often separated and/or bounded by thin ®lms with properties than dier greatly from the volume. Also, surfaces are often modi®ed prior to bonding, and in the case of ionic conductors such as glass, surface charge is due to surface ions. Therefore, in order to model the surface charges which accumulate on rg , it is necessary to introduce surface chemical components. A caret `^' above a symbol is used to distinguish the surface from the volume quantities. Surface vectors F^i are assumed to lie in the tangent plane so that the following transversality condition applies: 0 F^i mi :
1
Variables and their spatial derivatives are allowed to undergo jump discontinuities across all surfaces, as indicated in Fig. 4. Angled brackets are used to denote the algebraic mean of the limiting values of variables at a surface: 1 h/irc /u /c ; 2
1 h/irg /u /g ; 2
1 h/irb /c /g : 2
2
The volume and surface (tangential) gradient operators are given, respectively, by ri
o ; oxi
^ i dij ÿ mi mj o ; r oxj
3
where dij is the Kronecker delta. The material time derivative and velocity have the usual forms
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
139
Fig. 4. Jump notation.
_ dH oH vi ri H; H dt ot
vi
dui ; dt
4
where ui is the displacement. The time rate of change following the normal motion of a surface is ^oH oH v^i mi mj rj H: ^ ot ot
5
The surface material time derivative is given by ^ ^ ^_ dH oH v^i r ^ i H: H ^ dt ot
6
Because the surfaces rc ; rg and rb are material, they have the limiting velocities of the volume v^i vi :
7
This assumption is in agreement with the experiments [18] showing no growth of a transition layer between the two phases. As a result of this assumption, there will be no net mass ¯ow to or from the surfaces rc , rg , or rb , and the total mass on these surfaces will be conserved. This assumption allows for the diusional exchange of species between the surface and volume as well as diusional redistribution of mass. This is an accurate assumption for the case of Pyrex glass, because the surface barycentric velocity v^i is approximately equal to the velocity of the boro-silicate matrix, which has essentially zero diusion ¯ux.
140
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
The following integral domains are valid unless otherwise indicated: Z Z c g u integration over V V [ V [ V ; integration over r rg [ rc [ rb : V
8
r
The displacement gradient is assumed to obey the in®nitesimal strain theory, and the Lagrangian linear strain tensor is given by 1 Lij ri uj rj ui : 2
9
Finally, eijk is the permutation symbol. 3. Conservation of mass Local (dierential) forms of the mass balance are derived by starting with integral forms de®ned for an arbitrary portion of the material and then applying the rules of dierentiation of integrals with moving boundaries (transport theorems) to localize these integrals, assuming that the integrands are suciently smooth functions and the integration domain can be chosen arbitrarily. The global form of the mass balance law for component n, where 1 6 n 6 N 1, is given by Z I Z Z Z Z d d n n n n n ^ da J^i si ds p dv p^n da; q dv Ji ni da 10 q dt dt V
r
oV
or
V
r
where pn and p^n are the mass rate of production of component n per unit volume and area due to ^n the mass densities of component chemical reactions, ni the outward unit normal vector, qn and q n n n per unit volume and area, and Ji and J^i the volume and surface diusion ¯uxes of component n. Jin qn vni ÿ vi ;
n ^n ^ J^i q vni ÿ v^i :
11
Any conductor or glass mass that would actually lie between the surfaces rc and rg is assumed to lie on the surfaces rc and rg , respectively. Thus, in the model there is no conductor or glass mass within V u . Furthermore, we assume that the bonding is conducted in vacuum, thus the mass integrals over V u vanish. The bond initiation line c is not material and moves with velocity vci . The time derivative d=dt is observed following the motion of the surface r (Appendix II in Ref. [13]). By means of transport and divergence theorems ([7,13]) the integrals in Eq. (10) are rewritten as follows: Z n Z d oq n n ri q vi dv; q dv 12 ot dt V
Z oV
Jin ni da
V
Z V
ri Jin dv
Z r
mi Jin da;
13
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
141
# Z "^ n Z o^ q d ^ i ^ ^n da ^n da r vi q q ^ dt ot r
r
Z
h i ^gn ^ ^bn ^ ^cn ^ si q vci ÿ vci q vgi ÿ vci ÿ q vbi ÿ vci ds;
14
c
I
n J^i si ds
Z
^ i J^n da r i
r
or
Z
cn gn bn si J^i J^i ÿ J^i ds:
15
c
R n Eq. (15) ordinarily includes a term r J^i mi rj mj da, but this term vanishes here due to the transversality condition, Eq. (1). Eqs. (12)±(15) can be substituted into Eq. (10) to obtain the local form of the mass balance on V c ; V g ; rc ; rg ; rb and c: oqn ri qn vi ri Jin pn ot
in V c [ V g ;
^o^ qn ^ ^ i J^n mi J n p^n ^n r vi q ri ^ i i ^ ot
16
on rc [ rg [ rb ;
cn gn bn ^gn ÿ q ^bn 0 vi ÿ vci ^ qcn q si J^i J^i ÿ J^i si ^
along c;
17 18
where we used the fact that the limiting values of the surface velocities along the bond line are equal: v^ci v^gi v^bi
along c:
19
The mass conservation laws for the mixture can be obtained by summing Eqs. (16)±(18) over N 1 species and using Eq. (11) and the de®nitions of the barycentric velocities for the volume and the surface, vi
N1 X 1
q n1
qn vni ;
v^i
N1 X 1
^ q n1
^n v^ni ; q
20
resulting in oq ri qvi 0 ot
in V c [ V g ;
21
^o^ q ^ ^ 0 ri ^ vi q ^ ot
on rc [ rg [ rb ;
22
142
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
^g ÿ q ^b 0 si ^ vi ÿ vci ^ qc q
along c;
23
^ are the total volume and surface mass densities, and we have used the mass where q and q conservation laws for chemical reactions and the diusion ¯ux de®nitions N1 X
N1 X
pn 0;
n1
N1 X Jin 0;
p^n 0;
n1
n1
N1 X n J^ 0:
24
n1
Mass fractions are often more useful than densities as compositional variables. Therefore, the component balance Eqs. (16)±(18) will be rewritten in terms of mass fractions cn and c^n , which are de®ned in the volume and on the surface as ^n =^ c^n q q:
cn qn =q;
Since by de®nition cN1 1 ÿ
PN1 n1
N X cs ;
qn q and
PN1
qN1 q 1 ÿ
s1
c^N1
25 n1
^, it follows that ^n q q !
N X cs ;
26
s1
N X c^s ; 1ÿ
^N1 q
! N X ^ 1ÿ c^s ; q
s1
27
s1
where 1 6 s 6 N. With the mass fractions the barycentric velocity and mass diusion ¯uxes are expressed as vi
N1 X cn vni ;
Jin qcn vni ÿ vi ;
28
N1 X c^n v^ni ;
n ^c^n ^ J^i q vni ÿ v^i :
29
n1
v^i
n1
The velocities and densities of all N 1 components in the volume [8] and on the surface can be s ^, redetermined from the 4N 3 1 and 4N 1 independent variables cs ; Jis ; vi ; q and c^s ; J^i ; q spectively using the formulas qs cs q;
qN1 q 1 ÿ
N X
! cs ;
30
s1
vsi and
qcs ÿ1 Jis
vi ;
vN1 vi ÿ qÿ1 i
N X 1ÿ cs s1
!ÿ1
N X Jis s
31
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
143
!
^s c^s q ^; q
^ 1ÿ ^N1 q q
N X c^s ;
32
s1
v^si
s ^ qc^s ÿ1 J^i
v^N1 i
v^i ;
ÿ1
^ v^i ÿ q
X c^s 1ÿ s1
!ÿ1
X s1
s J^i :
33
Finally the component balance laws, Eqs. (16)±(18), are rewritten in terms of the mass fractions as qc_s ri Jis ps ;
in V c [ V g
^ i J^s mi J s p^s ; ^c^_s r q i i cs
gs
34
on rc [ rg [ rb
bs
^c c^gs q ^g ÿ c^bs q ^b 0 si J^i J^i ÿ J^i si ^ vi ÿ vci ^ ccs q
35 along c:
36
4. Maxwell's equations Although we allow for surface free charge, we do not include surface electric ®eld E^i , surface ^ i , or surface magnetic ®eld strength H^ i . Faraday's law in integral form is electric displacement D Z Z d Bi ni da Ei ki ds 0; 37 dt S
oS
where Ei is the convected electric ®eld, Bi the convected magnetic induction, S an arbitrary surface, and ki the unit vector tangent to the line. The Gauss' law can be written in terms of the electric displacement vector Di and the volume and surface free charge densities qf and q^f as Z Z Z f f q^ da q dv Di ni da: 38 r
V
oV
Ampere's law can be written in terms of the eective magnetic ®eld strength (magnetomotive ^ i as intensity) Hi and the volume and surface conduction currents Ii and I Z Z Z Z ^ i si ds Ii ni da d Di ni da Hi ki ds: I 39 dt r\S
S
S
oS
If S is a closed surface, the integral of Ei in Eq. (37) is identically zero [10] and one can derive the Gauss±Faraday law Z Bi ni da 0 oV
40
144
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
for any volume V . The transport and divergence theorems can be used to localize Eqs. (37)±(40) [7]:
Bi eijk Ek;j 0; in V c [ V g
eijk mj Ek 0; on rg [ rc [ rb ;
mj Dj q^f ; on rg [ rc [ rb ;
Di;i qf ; in V c [ V g
eijk Hk;j ÿ Di Ii ; in V c [ V g Bi;i 0; in V c [ V g where
^ i ; on rg [ rc [ rb ; eijk mj Hk I
mi Bi 0; on rg [ rc [ rb ;
41 42 43 44
denotes the convective derivative:
Ui
oUi rj Ui vj Ui rj vj ÿ Uj rj vi : ot
45
The eect of the material on the electromagnetic ®elds can be introduced in various ways. We use the statistical formulation for the nonconvected variables electric ®eld Ei , polarization Pi , electric displacement Di and magnetic induction Bi which can be formally obtained from the above equations through the substitutions: E i Ei ;
Bi B i ;
Di 0 Ei Pi ;
Hi Hi ÿ eijk vj Dk ;
46
where 0 is the permittivity of free space, Hi 1=l0 Bi is the magnetic ®eld strength (current potential) observed in the stationary (laboratory) frame, and eijk vj Bk is assumed negligible. Substitution of Eq. (46) and the assumption oBi =ot 0 into Eqs. (41) and (42) results in eijk Ek;j 0; in V c [ V g
eijk mj Ek 0 on rg [ rc [ rb ;
0 Ei;i qf ÿ Pi;i ; in V c [ V g
0 mj Dj q^f on rg [ rc [ rb :
47 48
The Maxwell equations in V u are simpli®ed by the assumption that all ®elds there are uniform in mi direction except displacement, velocity and electric potential. Consider the portion of the double surface shown in Fig. 5. The jump conditions for the electric displacement vector Eq. (48) on rg and rc yield 0 mi Eig ÿ Eiu q^g q^p ÿ mi Pig ;
0 mi Eiu ÿ Eic q^c ÿ q^p ;
49
where q^p mi Piu
50
is surface polarization charge due to the presence of polarizable matter between rc and rg . This is somewhat contradictory to the statement that the bonding is performed in vacuum. If
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
145
Fig. 5. Interfacial separation and polarization charge.
that was the case q^p will be zero. But it has been suggested [2] that the surface roughness greatly reduces the eective electrostatic pressure by dropping the potential across the gap. This eect is modeled herein as a surface dipole sheet between the two adjacent bulk materials [14,17]. We require Eiu to be perpendicular to rc and use the notation E? for the magnitude of the electric ®eld in V u Eiu E? mi :
51
The conservation of charge can be derived from the full set of Maxwell's equations, but not after using the present simplifying assumptions. Therefore, the charge conservation which is enforced by Maxwell's equations, will instead be established through a separate integral postulate Z I Z Z d d f f ^ i si ds 0; q^ da q dv Ii ni da I 52 dt dt r
V
oV
or
followed by localization using the divergence and transport theorems in a manner similar to Eqs. (16)±(18). oqf ri qf vi ri Ii 0 in V c [ V g ; ot ^ o^ qf ^ ^ i mi Ii 0 on rc [ rg [ rb ; ^ iI vi q^f r ri ^ ^ ot
^c I ^g ÿ I ^ b ^ si I vi ÿ vci ^ qc q^g ÿ q^b 0 along c: i i i
53 54 55
146
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
5. Conservation of momentum 5.1. Linear momentum For small currents and negligible magnetization in the quasi-electrostatic approximation the electrostatic body force fie is given by fie qf Ei Pj rj Ei ;
56
fie can also be written as the divergence of the electrostatic stress tensor Tije fje ri Tije ;
57
where 1 1 Tije 0 Ei Ej Pi Ej ÿ 0 Ek Ek dij Di Ej ÿ 0 Ek Ek dij : 2 2
58
Only zeroth order singularities (discontinuities) are assumed for the stress ®eld, so surface tension is not included on r. The quasi-static global balance of linear momentum for an arbitrary volume element is given by Z 0
Tij Tije ni da;
59
oV
where Tij is the Cauchy stress tensor. The above equation is a generalized version of the momentum balance, where the electrostatic body force has been substituted with the surface integral of the electrostatic stress tensor. To obtain the local form of momentum balance we apply the divergence theorem to Eq. (59) for the case when the volume V contains a surface r where the stress tensors have discontinuity, Z 0 oV
Tij Tije ni da
Z
ri Tij ri Tije dv
V
Z
mi Tij Tije da:
60
r
In deriving the mass balance we assumed that there was no mass within V u . Nevertheless, in deriving the equilibrium equations we will assume that V u can support a compressive Cauchy stress. This assumption is used in the cohesive zone theories of fracture mechanics. Localization of Eq. (60) yields 0 ri Tij ri Tije in V c [ V g [ V u ;
61
0 mi Tij Tije on rc [ rg [ rb :
62
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
Because V u is very thin, oV u rc [ rg . Applying Eq. (60) to V u results in Z Z u ue 0 mi Tij Tij da mi Tiju Tijue da rc [rg
147
63
ru
or in local form, 0 mi Tiju Tijue ru
on ru :
64
From Eqs. (62) and (64) it can be seen that the total stress vector (traction) is continuous across any of the surfaces mi Tiju Tijue mi Tijg Tijge mi Tijc Tijce :
65
However, mi Tij and mi Tije are not necessarily continuous across a surface because a surface charge creates a surface force distribution which gives rise to a jump in the electric stress tensor. From Eqs. (62), (64) and (65) it follows that 0 mi Tij Tije gc
across V u :
66
5.2. Angular momentum The quasi-static global balance of angular momentum for an arbitrary volume element containing discontinuity surfaces r is given by Z 0 eijk xj nl Tlk Tlke da: 67 oV
Applying the divergence theorem to Eq. (67) yields 0 eijk Tjk Tjke
in V c [ V g [ V u
68
which stipulates that Tij Tije is symmetric. The tensor Pi Ej is symmetric in Vg because the glass is assumed to be an isotropic material. Furthermore, Tjke is zero in V c , the conductor. Thus, in both materials Tije is symmetric and from Eq. (68) it follows that the Cauchy stress tensor is also symmetric. Thus, the conservation of angular momentum simpli®es to the trivial result Tjk Tkj ;
Tjke Tkje
in V c [ V g [ V u :
69
Even if the glass was not isotropic and the electric stress was therefore not symmetric, it may be assumed that the electromagnetic body couples are small. This assumption is usually used in the theory of piezoelectric fracture, which includes very high electric ®elds near crack tips in anisotropic materials. The conservation of angular momentum at the interfaces rc ; rg and rb is trivially satis®ed as well.
148
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
6. Conservation of energy The ®rst law of thermodynamics states that the rate of change of the sum of kinetic, internal and free electromagnetic energies is equal to the sum of the power of forces acting on the system, heat supplied to the system and the electromagnetic energy ¯ux, which is the Poynting vector (see Ref. [10] for example). The quasi-static global form of the ®rst law for an arbitrary volume element containing discontinuity surfaces r is given by Z h Z Z Z i d d 1 0 ^ da Tji Tjie vi ÿ qj ÿ ejik Ei Hk nj da ÿ q^i si ds; Ek Ek q dv dt dt 2 r
V
oV
or
70 where and ^ are the internal energies per unit mass and per unit area, and qi and q^i are the heat ¯ux vectors per unit volume and area. The surface transport Theorem (14) and Eq. (19) can be used to rewrite the ®rst term in Eq. (70) as Z Z Z Z Z c g u b d ^ da ^_ da ^_ da ^_ da ^_ da dt r rc rg ru rb Z si ^ vi ÿ vci ^c ^g ^u ÿ ^b ds; 71 c
where the surface energy ^u contains the polarization and elastic energy of the unbonded region V u . Eq. (71) can be used to rewrite Eq. (70) as: Z Z Z Z Z c g u b ^_ da ^_ da ^_ da ^_ da si ^ vi ÿ vci ^c ^g ^u ÿ ^b ds rc
Z V
rg
ru
c
rb
oEk 0 Ek Ek Ek;j vj Ek Ek rj vj q_ dv ot 2 0
0
Z h i Tji Tjie rj vi ÿ rj qj ÿ ejik rj Ei Hk Ei rj Hk dv V
Z
Z ^ i q^i da ÿ si ^ ÿ qj ÿ ejik Ei Hk mj ÿ r qci q^gi ÿ q^bi ds;
r
72
c
where we have used the momentum balance equations and the fact that r are material surfaces to eliminate the divergence of the stress tensors. Eqs. (41)±(43) can be used to localize Eq. (72):
q_ Tji rj vi Ei Pj rj vi Ei Ii Ei Pi ÿ ri qi oPi _ rj Pi vj Ei Ii ÿ ri qi Tij Lij Ei ot
in V c [ V g
73
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
149
If one de®nes polarization per unit mass, pi Pi =q, Eq. (73) can be rewritten as q_ Tij L_ ij ÿ ri qi Ii Ei qEi p_ i ;
74
since, due to Eq. (21), q
d o oPi rj vj Pi : Pi =q Pi rj Pi vj Pi rj vj ot dt ot
75
The remaining terms in the global energy balance, Eq. (72), include integrals along c, rc , rg , rb and the thin region V u . Localization along c yields h i si ^ 76 vi ÿ vci ^c ^g ^u ÿ ^b q^ci q^gi ÿ q^bi 0 along c: The ®nal terms in Eq. (72) refer to the surfaces: Z rc
c ^_ da
Z
g ^_ da
rg
Z
u ^_ da
ru
Z
b ^_ da
rb
Z 0 0 oEk 0 Ek Ek rj Ek vj Ek Ek rj vj dv ot 2 Vu
Z h
i Tji Tjie rj vi ÿ rj qj ÿ ejik rj Ei Hk Ei rj Hk dv
Vu
Z ^ i q^i da: ÿ qj ÿ ejik Ei Hk mj ÿ r
77
r
Recall that across the thickness of V u we assumed that all ®elds are uniform except displacement, velocity and electric potential. Thus it is convenient to rewrite some of the volume integrals in Eq. (77) as boundary integrals Z rc
c ^_ da
Z rg
g ^_ da
Z ru
u ^_ da
Z
b ^_ da
rb
Z 0 0 oEk 0 Ek Ek rj Ek vj Ek Ek rj vj dv ot 2 Vu
Z h
i mj Tji Tjie vi ÿ mj qj ÿ ejik Ei Hk mj da
oV u
Z ^ i q^i da: ÿ qj ÿ ejik Ei Hk mj ÿ r r
78
150
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
Using Eq. (46) the convected Poynting vector can be rewritten as 1 0 eijk Ej Hk eijk Ej Hk Tije vj ÿ Ek Ek Ek Pk vi : 2
79
The Poynting theorem can be simpli®ed by dropping the time variation of the magnetic ®eld as well as the Joule heating term because there are no conduction or convection currents in V u , i.e. ÿ ejik Ei Hk ;j Hi
oBi oDi oDi Ei Ii Ei Ei in V u ; ot ot ot
80
where Ii Ii qf vi . Eqs. (79) and (80) can be used to cancel out the energy of the free space in Eq. (78) to obtain Z Z Z Z c g u b ^_ da ^_ da ^_ da ^_ da rc
rg
Z
mj Tjiu n_ i
ru
ÿ
quj mj
rb
Piu Eiu n_ j mj
ru [rb
Z
Eiu
oPiu n mk da ot k
^ i q^i da; ÿ qj ÿ ejik Ei Hk mj ÿ r
81
n_ i vi gc ;
82
r
where ni ui gc n0i ;
where n0i is a constant. The surface energy balance, Eq. (81), can be rewritten in terms of Eqs. (50), (82) and (51) as Z Z Z Z c g u b ^_ da ^_ da ^_ da ^_ da rc
rg
Z ru [rb
Z
ru
rb
! p o^ q mj Tjiu n_ i ÿ quj mj q^p E? n_ j mj E? nk mk da ot
^ i q^i da: ÿ qj ÿ ejik Ei Hk mj ÿ r
83
r
The last integral in Eq. (83) contains the jump in the normal component of the Poynting vector, which can be simpli®ed as follows: mi eijk Ej Hk mi eijk hEj iHk mi eijk Ej hHk i:
84
The jump conditions (41) and (43) can be used to further simplify Eq. (84), resulting in ^ j: mi eijk Ej Hk ÿhEj iI
85
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
151
Thus, it can be seen that the electromagnetic energy ¯ux crossing a surface is reduced by the magnitude of the Joule heating due to surface conduction currents. This result can be used in the localization of Eq. (83) c ^ i q^i I ^ i hEi i c ^_ ÿqj rc mj ÿ r r
on rc
86
g ^ i q^i I ^ i hEi i g ^_ ÿqj rg mj ÿ r r
on rg ;
87
p
u o^ q ^_ mj Tjiu n_ i q^p E? n_ j mj E? n mk ot k
on ru ;
88 p
b q ^ i q^i I ^ i hEi i b q^p E? n_ j mj E? o^ ^_ mj Tjib n_ i ÿ qj rb mj ÿ r n mk ; r ot k
on rb :
89
Introducing ^ut as the total surface energy along the unbonded interface, ^ut ^c ^g ^u ;
90
Eqs. (86)±(89) and (76) can be rewritten as p
ut o^ q ^_ ÿ mi qi gc mi Tiju n_ j q^p E? n_ j mj E? n mk ot k c
g
^ hEi i c I ^ hEi i g ÿ r ^ i q^c ÿ r ^ i q^g I r r i i i i
on ru ;
91 p
b q ^ b hEi i b ÿ r ^ i q^b q^p E? n_ j mj E? o^ ^_ ÿmi qi rb mi Tiju n_ j I n mk r i i ot k
h i si ^ vi ÿ vci ^ut ÿ ^b q^ci q^gi ÿ q^bi 0
along c:
on rb ;
92 93
7. Second law of thermodynamics and constitutive equations For simplicity we assume that rc , rg , and ru are in thermal equilibrium and therefore have the same temperature distribution hc hg hu :
94
The entropies on rc , rg , and ru are therefore added in a manner similar to the energies, Eq. (90). We extend the global [3] Clausius±Duhem inequality to include the surface terms, neglecting radiative heating, as
152
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
d dt
! Z Z N X d 1 ^ ^ ls Jis da gut da gb da = ÿ ni qi ÿ dt h s1 V c [V g ru oV rb ! I N X s 1 ^s Jbi si ds; q^i ÿ ÿ l ^ h s1 Z
d qg dv dt
Z
95
or
where g and ^ g are the entropies per unit mass and area, h and ^h are the volume and surface temperatures, and ls are the reduced chemical potentials [8] ls ls ÿ lN1 :
96
The global inequality (95) can be localized in a manner similar to the mass balance, resulting in: ! N X ls Jis ri qi 1 ÿ qi ri in V c [ V g 97 ri qg_ = ÿ h h h s1 "" ut
^_ = ÿ mi g
N X 1 ls Jis qi ÿ h s1
!##g c
! # N c g X ^ ^ q 1 q cs gs cs gs i i ^i ^i ^ J^i l ^ J^i ÿr r u l ^hu ^ h s1 "
on ru 98
"" b
^_ = ÿ mi g
N X 1 ls Jis qi ÿ h s1
!## ^i r rb
"
si ^ vi ÿ
vci ^ gut
1 ^ = si u ÿg ^ h b
ÿ
1 ^b h
ÿ
q^ci
ÿ
ÿ q^bi
q^gi
N X s1
N X s1
N X 1 bs ^bs ^ Ji l ^u s1 h
cs ^cs Jbi l
!#
bs
^bs J^i l
! ^i ÿr
N X gs ^gs Jbi l
q^bi ^hb
! on rb
99
!
s1
along c:
100
Inequalities (97)±(100) will now be used to constrain the form of the constitutive equations. The internal energies are assumed to have the following arguments: s g; Lrev ij ; pi ; c
in V c [ V g ;
101
^ut ^ut ^ ^p ; c^cs ; c^gs on rc [ rg [ ru ; gut ; ni ; nirr i ;q
102
^b ^b ^ gb ; c^bs
103
on rb ;
where the Lagrangian strain tensor is decomposed into reversible and irreversible parts irr Lij Lrev ij Lij :
104
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
153
The following transformation is used to replace the entropy in the list of arguments with the temperature and the polarization with the electric ®eld: s w h; Lrev ij ; Ei ; c ÿ hg ÿ Ei pi
in V c [ V g ;
105
u ut ^ ut ^ ^cs ; c^gs ^ut ÿ ^ w h; ni ; nirr h ^ g ÿ E? q^p nk mk i ; E? ; c b b b ^ b ^ w h ; c^bs ^b ÿ ^ h ^ g
on rc [ rg [ ru
on rb ;
106 107
^ are the Helmholtz free energy per unit mass and per unit area. The derivation of where w and w the thermodynamic constraints requires the rate form of Eqs. (105)±(107): _ ÿ hg_ _ ÿ w_ hg
d Ei pi dt
in V c [ V g ;
ut d u ut ut _ u ut _ ^_ ^ ^ _ h ^ g ^ ÿ w h^ g ÿ E? q^p nk mk dt
108
on rc [ rg [ ru ;
b b b b _b b ^_ ^ ^ h ^ on rb : h^ g_ ^_ ÿ w g
109
110
Also, it is useful to rewrite Eq. (97) as s X N N X l qi ri h s qhg_ = ÿ ri qi ls ri Jis : h Ji ri h h s1 s1
111
Using Eqs. (105) and (74) in Eq. (111) we obtain s X N N X l ÿ1 s _ _ _ _ ÿq w gh Tij Lij ÿ Pi Ei ÿ h qi ri h ÿ h Ji ri ÿ ls ri Jis = 0: h s1 s1
112
Inequality (112) can simpli®ed further by applying the chain rule of dierentiation to the Helmholtz free energy, N X ow _ ow _ ow _ irr ow _ ow s _ qw q h q rev Lij ÿ q rev Lij q c_ ; Ei q oh oLij oLij oEi cs s1
113
and substituting this result and the volume mass balance, Eq. (34), into inequality (113) to obtain
154
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
! ! ow ow ow _ irr ow _ _ _ Ei Ii Ei ÿq g h Tij ÿ q rev Lij q rev Lij ÿ Pi q oh oLij oLij oEi s N N N X X X l ow ri h s s s s s ÿ l c_ ÿ ÿq l p ÿ qi P 0: ÿ h Ji ri s h oc h s1 s1 s1
114
The last PN two terms in inequality (114) can be rewritten in terms of the conductive heat ¯ux (qi ÿ s1 ls Jis ): ! s N N N X X l ri h ri h X s s s ÿ qi ÿ l Ji Jis ri ls : ÿ h J i ri ÿ ÿ qi h h h s1 s1 s1
115
Inequality (114) must be satis®ed for all independent thermodynamic processes. This constraint [4] results in the following thermodynamic conjugates: Tij q
ow ; oLrev ij
gÿ
ow ; oh
Pi ÿq
ow ; oEi
ls
ow : ocs
116
In the case of ionic conduction (for the glass region V g ) the electric current can be expressed with the diusion mass currents using the component valence zn , molar mass Mn and the Faraday's constant F: Igi
N s N X X z zN1 s ÿ J F ~zs Jis ; s i N1 M M s1 s1
117
where s z zN1 ÿ : ~z F Ms MN1 s
Eqs. (115)±(118) can be used to reduce inequality (114) to ! N N N X X ri h X irr s s Tij L_ ij ÿ qi ÿ l Ji Jis ri ls /~zs ÿ ls p s = 0 ÿ h s1 s1 s1
118
in V g :
119
In the conductor, V c , the conduction current is independent of mass ¯uxes, thus the entropy inequality is ! N N N X X ri h X irr Tij L_ ij ÿ qi ÿ ls Jis Jis ri ls Ici Ei ÿ ls ps = 0 in V c : 120 ÿ h s1 s1 s1 The thermodynamic conjugates and reduced dissipation inequalities will now be derived for the ut b surfaces in a bmanner similar to the volumes. We substitute ^g_ and ^g_ from Eqs. (109) and (110) ut and ^_ and ^_ from Eqs. (91) and (92) into inequalities (98) and (99) to obtain:
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
155
ut _ u ut ^_ ÿ ^ ^ c hEi i c I ^ g hEi i g h ^ ÿw g mi Tiju n_j ÿ E_ ? q^p ni mi ÿ mi qi gc I r r i i "" !##g N N c g ^ ^u X X u u 1 ^ q q^i ri h 1 cs ^cs gs ^ ^i ^gs J^i = 0 h ÿ i mi ls Jis ÿ ^h r l Ji l qi ÿ u u ^ ^h ^ h s1 s1 h c
g
on r [ r [ r
c
u
121
and b _b b ^_ ÿ ^ ^ b hEi i b h ^ ÿw g ÿ mi qi rb I r i "" !## ! bs ^bs N N b ^ ^b X X ^ J l b b ^ h r q 1 i i ^ ^i =0 h ÿ i b ÿ ^h r mi lbs Jibs qi ÿ b h ^ ^ h h s1 s1 b r
on rb
122
The surface mass balance, Eq. (35), can be used to rewrite the last terms in inequalities (121) and (122) as ! ! s! s ^s J^i ^s ^ l l l s s s s ^ ^i ^i ^ i J^ ^hr ^i ^s r ^s ÿ^ hr ^ hr qc^_ ÿ mi Jis p^s ; 123 J^i l J^i l i ^ ^ ^ h h h which can then be used in Eqs. (121) and (122), combining the jumps of the volume diusions in a single jump, to obtain X X ut X cs X gs _ u ut ^_ ÿ ^ ^c c^_ l ^g c^_ l ^cs q ^gs ÿ ^cs p^cs ÿ ^gs p^gs h ^ l l ÿw g mi Tiju n_j ÿ E_ ? q^p ni mi q N
s1
"" ^ c hEi i c I ^ g hEi i g ÿ mi I r r i ÿ q^gi ÿ c
N X s1
! gs
^gs J^i l
qi ÿ
N X s1
!
ls Jis
h ÿ ^h h
N
N
s1
s1
u
##g ÿ c
q^ci
ÿ
N
s1
N X s1
! cs ^cs J^i l
^ i ^hu r u h^
u
N N N X X X ^ i^ h r cs gs ^ il ^ il ^cs ÿ ^gs P 0 ^s gc ÿ J^i r J^i r Jis ls ÿ l u ÿ mi ^ h s1 s1 s1
g
on r [ r [ ru
124
for the unbonded interface and N N X X b bs bs _b b ^_ ÿ ^ ^ b hEi i b ^ c^_ l ^ ÿ ^bs p^bs I h ^ ÿw l g q r i
"" ÿ mi ÿ mi
N X s1
s1
!
s1
b N X hÿ^ h s s qi ÿ l Jj h s1
^bs rb ÿ Jis ls ÿ l
##
N X s1
ÿ
q^bi
ÿ
rb bs ^ lbs P 0 J^i r^
N X s1
! bs ^bs J^i l
on rb
^ i ^hb r ^hb 125
156
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158 ut
^ in Eq. (124) according to for the bonded interface. We next expand the free energy potential w the list of arguments assumed in Eq. (103) and subsequently in Eq. (107): ! ! ^ ut ^ ut ow ow u ^_u u g h mi Tij ÿ n_ j ÿ u ^ onj o^ h
! ^ ut ow p q^ ni mi E_ ? ÿ oE? ! ! N N ^ ut ^ ut ^ ut X X ow o w ow cs c cs _ g gs _ gs ^ c hEi i c I ^ g hEi i g ^l ^l ^ c^ ÿ ^ c^ ÿ irr n_ j irr I ÿ cs ÿ q gs ÿ q r r i i o^ c o^ c onj s1 s1 "" ! u ##g N N N ^ X X X ^ i ^hu r h ÿ h cs gs c g cs gs s s ^ ^ ^ Ji ÿ ^ Ji ÿ mi qi ÿ l Ji ÿ q^i q^i ÿ l l ^hu h s1 s1 s1 c ÿ
N X
^cs p^cs ÿ l
s1
N N N N X X X X cs gs ^ il ^ il ^gs p^gs ÿ mi Jis ls ÿ l ^cs ÿ ^gs P 0: ^s gc ÿ J^i r J^i r l s1
s1
s1
126
s1
Inequality (126) must apply for all processes, which leads to the thermodynamic conjugates ut
^ ow ^ ÿ u ; g o^ h ut
ut
mi Tiju
^ ow ; onj
ut
ut
^ ow ni mi q^ ÿ ; oE?
^ ow ^l ^ cs ; q o^ c c cs
p
ut
^ ow ^l ^ gs ; q o^ c g gs
127
and the residual dissipation inequality ut
^ ow ^ c hEi i c I ^ g hEi i g ÿ irr n_ j irr I r r i i onj "" ! u ##g N X hÿ^ h s s ÿ mi qi ÿ l Ji ÿ h s1 c
ÿ
N X
^cs p^cs ÿ l
s1
q^ci
q^gi
ÿ
N X s1
cs ^cs J^i l
! N X ^ i ^hu r gs ^gs ^ Ji ÿ l ^hu s1
N N N N X X X X cs gs ^ il ^ il ^gs p^gs ÿ mi Jis ls ÿ l ^cs ÿ ^gs = 0 ^s gc ÿ J^i r J^i r l s1
s1
on rc [ rg [ ru :
s1
s1
128
Repeating the above procedure for the bonded interface yields the thermodynamic conjugates b
^ ow ^ g ÿ b; o^ h b
b
^ ow ^ l ^ bs ; q o^ c b bs
and the residual dissipation inequality
129
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
"" ^ b hEi i b I r i
ÿ mi
b N X hÿ^ h s s qi ÿ l Ji h s1
!## ÿ rb
q^bi
! N X ^ i ^hb r bs ^bs ^ Ji ÿ l ^hb s1
N N N X X X bs bs bs bs s s ^ il ^ p^ ÿ mi Ji l ÿ l ^bs = 0 ^ rb ÿ J^i r ÿ l s1
s1
157
on rb :
130
s1
The surface current on the glass is related to the surface mass diusion by N X gs ~zs J^i ;
^g I i
131
s1
and the average electric ®eld on rg is given by hEi irg ÿri h/irg :
132
With these results and the normal and surface decomposition of the derivative operator, Eq. (3), we see that, due to the transversality condition, Eq. (1), only the surface gradient of the potential creates Joule heat, since by de®nition the surface currents are along the interface ^g ÿ hEi irg I i
N N X X gs gs ^ i h/i g : ~zs J^i ri h/irg ÿ ~zs J^i r r s1
133
s1
Eq. (133) can be used in Eqs. (128) and (130) to rewrite the residual dissipation inequality for the unbonded interface as ÿ
N N ^ ut X X ow gs cs gs s _ j irr I ^ i ^ ^ il ^ c hEi i c ÿ ^ ^cs J J^i r n r l ~ z h/i ÿ r rg i i irr onj s1 s1 "" ! ! ## g u N N N ^ X X X ^ i ^h0u r h ÿ h cs gs c g cs ^ gs ^ s s ^ Ji ÿ ^ Ji q^i q^i ÿ ÿ mi qi ÿ l Ji l l ^hu h s1 s1 s1 c
ÿ
N N N X X X ^cs p^cs ÿ ^gs p^gs ÿ mi Jis ls ÿ l ^s gc 0 l l s1
s1
s1
on rc [ rg [ ru
134
and for the bonded interface as ÿ
N X s1
ÿ
"" bs ^ i ^ J^i r lbs
q^bi
ÿ
N X s1
~zs h/irb ÿ mi !
bs ^bs J^i l
! ## N ^hb X h ÿ qi ÿ ls Jis h s1
rb
b N N X X ^ i^ h r bs bs ^ p^ ÿ mi Jis ls ÿ l ^s rb = 0 ÿ l b ^ h s1 s1
on rb
135
158
E.T. Enikov, J.G. Boyd / International Journal of Engineering Science 38 (2000) 135±158
One can identify the generalized ¯uxes and their conjugate forces for the dissipative processes in inequalities (119), (120), (134), (135), and (100). Inequality (100) requires that the entropy production due to bonding or debonding and the heat conduction into c must be positive semide®nite. The velocity of c due to bonding or debonding is given by the dierence 8 < < 0 debonding; si vci ÿ v^i 0 neitherbondingnordebonding; 136 : > 0 bonding: gb , to the bonding or debonding velocity is equal to the The thermodynamic conjugate, ^ gut ÿ ^ dierence between the limiting values of the entropy of the unbonded and bonded interfaces at c. The bonding or debonding velocities can be written as a function of ^gut ÿ ^gb . If the interfacial entropy is a continuous function at c, then 0 ^gut ÿ ^gb . In this case, the bonding or debonding velocities could be written as a function of the separation ni . This approach is used in the thermodynamic cohesive zone theory of fracture mechanics [5] in which the assumption of self-similar crack growth is used to relate the crack propagation rate to the interfacial separation. References [1] M.G. Ancona, H.F. Tiersten, Fully macroscopic description of bounded semiconductors with an application to the si-sio2 interface, Phys. Rev. B 22 (12) (1980) 6104±6119. [2] T. Anthony, Anodic bonding of imperfect surfaces, J. Appl. Phys. 54 (5) (1983) 2419±2428. [3] R. Bowen, Theory of Mixtures, vol. 3, Continuum Physics, Academic Press, New York, 1976. [4] B.D. Coleman, M. Gurtin, Thermodynamics with internal state variables, J. Chem. Phys. 47 (2) (1967) 597±613. [5] F. Costanzo, D.H. Allen, A continuum thermodynamic analysis of cohesive zone models, Internat. J. Engrg. Sci. 33 (15) (1995) 2197±2219. [6] A. Cozma, B. Puers, Characterization of the electrosatic bonding of silicon and pyrex glass, J. Micromech. Microengrg. 5 (1995) 98±102. [7] N. Daher, G.A. Maugin, Deformable semiconductors with interfaces: basic continuum equations, Internat. J. Engrg. Sci. 25 (9) (1987) 1093±1129. [8] D.G.B. Edelen, Mass balance laws and the decomposition, evolution and stability of chemical systems, Internat. J. Engrg. Sci. 13 (1975) 763±784. [9] P.J. Hesketh, Y. Lin, J.G. Boyd, S. Zsivanovic, J. Cunnen, Y. Ming, J. Stetter, S.M. Lunte, G.S. Wilson, Biosensors and micro¯uidic systems, in: International Symposium on Aerospace, Nagoya, Japan, 1996. [10] K. Hutter, A.A.F. Van De Ven, Field Matter Interactions in Thermoelastic Solids, Lecture Notes in Physics, Springer, Berlin, 1978. [11] P. Jorgensen, Eect of an electric ®eld on silicon oxidation, J. Chem. Phys. 37 (1962) 874±877. [12] Y.C. Lin, P.J. Hesketh, J.G. Boyd, S.M. Lunte, G.S. Wilson, Characteristics of a polyimide microvalve, in: SolidState Sensor and Actuator Workshop, Hilton Head, South Carolina, June 2-6, 1996, pp. 113±116. [13] G. Maugin, Continuum Mechanics of Electromagnetic Solids, North-Holand, Amsterdam, The Netherlands, 1988. [14] W.K.H. Panofsky, Classical Electricity and Magnetism, Addison-Wesley, Reading, MA, 1962. [15] E. Peeters, S. Vergote, B. Puers, W. Sansen, A combined silicon fusion and glass/silicon anodic bonding process for uniaxial capacitive accelerometer, J. Micromech. Microengrg. 2 (1992) 176±179. [16] E.H. Snow, M.E. Dumesnil, Space-charge polarization in glass ®lms, J. Appl. Phys. 37 (5) (1965) 2123±2131. [17] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [18] G. Wallis, D. Pomerantz, Field assisted glass-metal sealing, J. Appl. Phys. 10 (10) (1969) 3946±3949.