Stress induced diffusion along adhesional contact interfaces

Acta Materialia 51 (2003) 2219–2234 www.actamat-journals.com

Stress induced diffusion along adhesional contact interfaces Y. Takahashi ∗, K. Uesugi JWRI, Osaka University 11-1, Mihogaoka, Ibaraki, Osaka, Japan, 567-0047 Received 30 August 2002; received in revised form 19 December 2002; accepted 19 December 2002

Abstract The stress induced vacancy diffusion process along the bond (contact) interface between fine gold wires is numerically analyzed. The elastic stress due to the adhesional contact induces vacancy diffusion even at room temperaturs. Because of low temperatures, it is assumed in the numerical simulation that the vacancy diffusion along the adhesional contact interface (boundary self-diffusion) is predominantly produced. The residual stress is produced around the contact interface due to the adhesional elastic bonding. The stress relaxation process where the residual stress is gradually reduced by the vacancy diffusion along the interface is numerically simulated. The rate of stress relaxation increases with decreasing the radius of wire but the growth of contact width due to the vacancy diffusion during (and after) bonding is scarcely produced. It is, therefore, suggested from the numerical results that the increase of the bond strength, with time, can be governed by the relaxation process of the residual stress. Further, the activation energy for the mechanism which increases the bond-strength is experimentally obtained.  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Room temperature bonding; Adhesional contact; Surface energy; Stress relaxation; Interface self-diffusion; Vacancy diffusion; Transient behavior; Bond-strength; Gold wire

1. Introduction It is observed that the bond strength increases with the time after the adhesional bonding at the room temperature (~298 K) [1]. The increase in the bond-strength becomes striking as the bonding pressure decreases. When the adhesional contact is carried out between convex surfaces of metals such as gold wire/gold plate, the contact is produced by elastic or elastoplastic deformation, where the tensile stress occurs at the edge of the contact area,

∗

Corresponding author. E-mail address: [email protected] (Y. Takahashi).

i.e., there remains a residual stress around the contact-interface after bonding [1]. If the residual stress gradually disappears with time, or the bonded area increases after bonding, the bondstrength can increase. It is widely accepted that the vacancy diffusion (self-diffusion) in crystals (metals) is induced by the stress field under high temperatures [2]. The stress induced vacancy diffusion can occur even at room temperature to reduce the residual stress and increase the bonded area. Therefore, the vacancy diffusion can be one of the mechanisms increasing the bond-strength of the adhesional bonding. In the present study, the stress induced vacancy diffusion during (and after) the adhesional bonding

1359-6454/03/$30.00  2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00015-6

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is modeled. Because of low temperatures, the vacancy diffusion along the bond-interface layer (interface self-diffusion) is taken into account and the vacancy diffusion in bulk crystal (volume selfdiffusion) is ignored. It is also necessary to couple the vacancy diffusion with the elastic deformation. This is clear transient behavior that the local stress due to the adhesional bonding is gradually reduced by vacancy diffusion. The purpose of the present study is to understand the stress relaxation process and to comprehend the temperature, pressure and wire-radius dependences of the stress relief process after bonding. There are many studies of vacancy diffusion in the high temperature condition (T / Tm ⬎ 0.5), where T is the absolute temperature and Tm is the melting temperature in K, for example, the void shrinkage during the diffusion bonding [2,3], powder sintering [4,5] and the void growth during the creep deformation [6]. Under the high temperature condition, the transient behavior of vacancy diffusion hardly becomes a problem and the quasisteady state is usually assumed, i.e., div J = 0 or constant, where J is the diffusion flux [3–7], except for the special cases as stated below. However, as T decreases (T / Tm « 0.5), the unsteady state diffusion lasts a long time and is not neglected, i.e., the transient behavior of stress-induced diffusion needs to be considered. Raj [8] modeled the transient behavior of cavity growth on the grain boundary to understand the diffusion-induced creep rupture and analyzed the vacancy diffusion, taking into account the elastic stress field, followed by Martinez and Nix [9]. Although they ignored the surface energy gs, which is very important for the adhesion [1], they pointed out the importance of coupling the vacancy diffusion flux with the change in local elastic stress and clarified that the transient behavior cannot be neglected even in the high temperature condition when the distance between cavities (voids) is long enough. On the other hand, Takahashi et al. [10,11] analyzed the high temperature void shrinkage process under the fluctuating stress and found that the void shrinkage is facilitated when the bonding pressure was very frequently changed. The modelling methods proposed by them and the simulation results are very useful for modeling and simulating

the unsteady state stress relaxation process which is produced in the room temperature adhesional bonding.

2. Modeling 2.1. Presupposition The adhesional bonding between gold wire and gold plate is mainly treated, because the increase of the bond strength was observed in the previous study [1]. As stated above, the volume self-diffusion is ignored, i.e., D n « D b, where D n is the volume self-diffusion coefficient and D b the boundary self-diffusion coefficient. The surface self-diffusion is assumed to be produced much more rapidly, compared with the interface (boundary) self-diffusion, i.e., D s Ⰷ D b so that the boundary self-diffusion controls the unsteady state stress relaxation process, where D s is the surface self-diffusion coefficient. In other words, the surface profile with a curvature in the vicinity of the bonded interface is instantly accommodated by the surface diffusion. The atoms flowing out from the edge (tip point) of bond-interface is immediately transported by the surface diffusion to form a curvature radius r at the tip point as stated elsewhere [2,3]. Fig. 1 illustrates the elastic contact between a

Fig. 1. Schematic illustration of contact interface and stress distribution in the y direction.

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fine wire and a flat plane. If Dupre´ energy of adhesion ⌬g ( = g s1 + g s2 − g i) is zero, the contact width of a h is produced (Hertz solution), where g s1 is the surface energy of wire, g s2 is the surface energy of the plane, and g i is the interface energy at the contact area. On the other hand, if ⌬g⬎ 0, the contact width of a j becomes greater than a h [12] and the tensile stress is produced at the edge of the contact area even if a compressive force F is applied to the wire [1]. If the load F is kept at fj1 for the time t = tB, (defined as the bonding time), the vacancy diffusion can occur to change the stress distribution around the bond-interface. The initial contact width aj 1 occurs at the time t = +0, immediately after F = fj 1 is applied between wire and plane and the initial stress distribution sy1 is produced on the contact interface owing to the adhesional contact. The values of aj1 and sy1 have been calculated [1]. If any vacancy diffusion does not occur and the adhesional contact is governed only by elastic deformation and ⌬g, then the two bodies of wire and plane can be separated as stated elsewhere [12]. The contact and separation are reversible. The stress distribution and the contact width aj2 at t = tB +0 (immediately after F is removed) are easily calculated [1,12]. On the other hand, when the vacancy diffusion occurs during bonding (t = 0~tB), the process between contact and separation is nonreversible. The initial stress situation sy1 changes into the stress distribution sy1⬘ at t = tB − 0 (immediately before F is reduced to zero) and also the initial contact width aj 1 changes to aj 1⬘. Although if tB is not so long, then aj1 ⬇ aj 1⬘ and sy 1 ⬇ sy 1⬘ owing to the room temperature, it is generally rather difficult to calculate the contact width aj2 at t = tB+0. It is, therefore, assumed that the bonded width aj 1⬘ at t = tB - 0 is kept even at t = tB +0, i.e., the adhesional diffusion bonding is established and also the stress distribution sy2 at t = tB + 0 can be obtained by sy 2 = sy1⬘ + sm, where sm is the Boussinesq’s stress distribution under F = fj 1 [12]. After bonding (t ⬎ tB) under F = 0, the stress relaxation process is assumed to be produced only by the interface self-diffusion. When two wires with the same radius R are in contact, the stress situation is the same between the upper and lower side but when the radius R is different as shown in Fig. 2, the stress situation will

Fig. 2. layer.

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Model of two wires in contact and contact interface

not be equal between wire 1 and wire 2. It is, however, assumed that the stress situation is approximately the same between wire 1 and wire 2 in the vicinity of the bond-interface (in ⫺X ⬍ x ⬍ X and ⫺X ⬍ y ⬍ X), if Young’s modulus E and the Poisson’s ratio n is the same between them. This assumption can hold true because half the contact width X is much smaller than R 1 or R 2.

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2.2. Diffusion equations

Fig. 2), the relationship between b and u is expressed by

The excess chemical potential ⌬m for one vacancy is given by

∂u ⫹ d⍀b ⫽ 0. ∂t

⌬m ⫽ ⫺Tn·⍀,

(1)

where Tn is the normal traction for forming one vacancy at the interface [3,7,13] and is equal to the stress sy in the y direction perpendicular to the bond interface and also ⍀ is the vacancy volume which is assumed to be equal to the atom volume. The relationship between ⌬m and the vacancy concentration C is expressed by C ⫽ Coexp(⫺⌬m / kT),

(2)

where Co is the vacancy concentration under no stress (the standard) state, k is Boltzmann’s constant. The vacancy diffusion must be produced along the interface if Tn (x) changes along the interface. The vacancies diffuse from the edge A of the bonded area to the center area O bonded and the atoms diffuse to the opposite direction of the vacancies. From Appendix A, the atom flux Jb along the bond interface is given by Db ·grad ⌬m. Jb ⫽ ⫺ kT⍀

(3)

Fig. 2 illustrates the contact between wire 1 with the radius R 1 and wire 2 with the radius R 2 and the interface diffusion layer d = db / 2, where db is the thickness of interface layer. The bonding pressure P is defined as the mean pressure applied to the wire 1 in the region of x ⬎ 0 and y ⬎ 0 (P = ⫺sB = ⫺F / 2R 1). Also, two coordinates of x and h are defined as seen in Fig. 2, where X is half contact (bonded) width. The initial width of X is expressed by Xo. From the mass conservation, div Jb = b holds true, where b is the number of atoms produced per unit time and unit volume of the interface with the thickness of db and b depends on the position x under the unsteady state [3,7–10]. 2.3. Coupling diffusion and elastic deformation The displacement u of the bulk in the y direction compensates for b. Because ⌬u·⌬x + ⍀bd·⌬x·⌬t = 0 holds good for the time increment ⌬t (see in

(4)

The displacement u consists of the rigid displacement ur and the elastic displacement ue, i.e., u = ur + ue. The rigid displacement ur is independent of the position x and also the value of ⫺X·⌬ur gives the volume outflow, d·⍀·Jb |x = 1·⌬t, from the bond interface to the surface through the tip point, i.e., d⍀ ∂ur ⫽ ⫺ [Jb]at ∂t X

x=1

⫽

(5)

d⍀Db ·grad(sy|x=1,h=0), ⫺ X kT where sy is the stress in the y direction. On the other hand, the elastic displacement ue and the partial differential of ue with respect to time t are obtained by Eqs. (A-8) and (A-9), respectively as shown in Appendix B. From Eq. (A-5) in Appendix B, the normal traction Tn (x) is given by Tn(x) ⫽ sy |h=0 ⫽

⌰ X2

冘 ⬁

{cn(⫺a2n)cosanx}

(6)

n

gs ⫹ sin a, r where an = (4n - 3)p / 2. The parameters of cn and ⌰ are referred to in Appendix B. The change in Tn (x) with time is obtained by the change in the parameter cn with time, as stated in Appendix B. The increment ⌬cm of each cm is expressed by

再 冘 ⬁

⌬cm ⫽

冎

1 2 (c a3)⫺cma3m ⌬t, t a2mn ⫽ 1 n n

(7)

where am = (4 m − 3)p / 2, m is the natural number and t can be be defined as the time constant given by Eq. (A-14) in Appendix B [8]. Because the initial values of cm (for t = 0 or t = tB + 0) is calculated as explained in Appendix C, the values of cm for 0 ⬍ t ⱕ tB − 0 or t ⬎ tB can be obtained from Eq. (7), i.e., cm at t = t + ⌬t = cm at t = t + ⌬cm (step 1). The parameters cm at t + ⌬t (m = 1, 2, …⬁)

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give the stress distribution at t + ⌬t by Eq. (6) (step 2). The atom flux Jb at t + ⌬t is also calculated by Eq. (3) (step 3). By repeating the steps of 1 ~ 3, the stress relaxation process due to the interface self-diffusion can be calculated. However, the bond width X and the surface curvature radius r are also changed by the atom outflow d·Jb |x = 1 from the tip point. It is, therefore, necessary to take this into account. 2.4. Geometry for diffusion process Fig. 3 shows the models for the contact between different wires. Fig. 3 (a) is for radius R 1 ⫽ (or =) radius R 2 and Fig. 3 (b) is for R 2 = ⬁. These models are used for calculating the initial neck radius r with half the dihedral angle a = p / 2, when the initial contact width Xo is given. The value of Xo is obtained as the adhesional contact width aj [1]. The neck of width Xo is defined as the contact area although the neck area shifts from the x axis. The values of xo and r and the position (coordinate) C(xc, yc) were converged by repeating the following manner so that the area GHF (= So) can be equal to the area FED (= So∗) in Fig. 3(a), i.e., the triangle area ABC (= S ABC) = S 1 + S 2 + S 3 (= S t) can be established, where S 1, S 2 and S 3 are shown in Fig. 3 (see Appendix D). If SABC ⬍ ( ⬎ ) St for a given xo, then So∗ ⬍ ( ⬎ ) So, i.e., the radius r should be increased (decreased). After SABC = St, if Xo ⬎ ( ⬍ ) xc − r for a given xo, the value of xo has to be increased (decreased) and r and (xc, yc) have to be calculated again for the changed xo. For R 2 = ⬁ of Fig. 3(b), the radius r should be calculated so that the area AOEC (= SAOEC) can be equal to St (= S 1 +S 3) for a given xo (if SAOEC ⬎ St (So∗ ⬎ So), it is required to decrease the value of r. Fig. 4 shows the change in the geometrical situation for the time increment ⌬t due to the interface self-diffusion. From Appendix E, 2X(t)⌬ur = the area JKE⬘D⬘ is always established. Because the distance x(t) of the point F from the y axis is given by x(t) = (R 21 - y 21)1 / 2, r(t) of circle C(xc, yc) is converged by the repetition method shown in Fig. 3 (Appendix D). After ⌬ur at t = t is calculated, the distance x(t + ⌬t) is obtained by x(t + ⌬t) = {R 21 − (y 1 − ⌬ur)2}1 / 2 as shown in Fig. 4. The radius

Fig. 3. Model of wire contact to take the mass conservation into account. (a) for R 1 ⫽ R 2 and (b) for R 2 = ⬁.

r(t + ⌬t) of circle C⬘ is calculated for this x(t + ⌬t) in the same manner. The half the bonded width at t = t + ⌬t is thus obtained by X(t + ⌬t) = xc − r(t + ⌬t). In addition, for R 2 = ⬁, the coordinate y 1 at t = t + ⌬t is given by y 1(t + ⌬t) = y 1(t) − 2⌬ur.

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Fig. 4. Model when increasing half the contact width X due to the ridge displacement ur. The area GHF = (1) + (2) + (3) + (4). The area GHF + (1) + (5) + (6) = (4) + the area JKE⬘D⬘. Thus, 2X(t) ⌬ur = {(5) + (1) + (3)} + {(6) + (1) + (2)} = the area JKE⬘D⬘. This model becomes more important, as T and tB increase.

Table 1 shows the material constants of pure gold. The half dihedral angle a was assumed to be p / 2 according to the model of Fig. 3, although the equilibrium angle is defined as ao = cos−1(g i / 2gs) and also, a can tend to be smaller than ao as P increases [3].

3. Experimental The adhesional bonding between gold wire with R = 50 mm and gold plate with the thickness of 130 mm was carried out to investigate the effect of temperature on the stress relaxation process. The

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Table 1 Material constants of pure gold used in the present study Name of properties (unit)

Symbols

Poisson’s ratio Melting point (K) Thickness of bond-interface (m) Atomic volume (m3) Surface tension (Jm⫺2) Boundary (Interface) tension (Jm⫺2) Shear modulus (Nm⫺2)

n Tm db ⍀ gs gb G(T)

Young’s modulus (Nm⫺2) Boundary self-diffusion coefficient (m2/s) Frequency factor of Db (m2/s) Activation energy of Db (J/mol) Yield stress (Nm⫺2)

E(T) Db Db0 Qb sY(T)

Correction coefficient for sy(T)

⌫(T)

a

Value 0.42 1336 5.76×10⫺10 1.70×10⫺29 1.485 0.36 G(T) = aG (T/(Tm)2+bG (T/Tm) +cG, where aG = ⫺9.88215 × 109 bG = ⫺6.20764 × 109 cG = 3.10090 × 1010 E(T) = 2(1+v) G(T) Db = Db0exp(⫺Qb/RT) 1.0 × 10−5 0.870 × 105 sY(T) = 1.0 × 108 G(T)/G(T0)×⌫(T) sy(To) = 9.93 × 107, T0 = 300 (K) ⌫(T) = a⌫(T/Tm)3+b⌫ (T/Tm)2 + c⌫(T/Tm)+d⌫ where a⌫ = 0.0, b⌫ = ⫺0.854965, c⌫ = 0.342016, d⌫ = 0.95905

Ref. No. [1] [14] a

[15] [16] [16] [17,18]

[15,18] [15] [1,19] [1,20] [18,19,21]

It was assumed that db ⬇ 2b, where b is Burger’s vector15.

purity of gold was 99.99 mass %. At first, these surfaces were irradiated by Ar ion beam under the acceleration voltage of 2 kV and the atmosphere pressure of 4 × 10⫺6 Pa to remove surface contamination and activate the bonding surfaces. The bonding was carried out at the room temperature (298±3 K) after moving them from the irradiation chamber to the bonding chamber, using a transfer rod. The atmosphere pressure was less than 1 × 10⫺8 Pa during bonding. The length of wire was about 20 mm and the center of the wire was pressed (f ⬇ 500 N / m) by the bonding tool. The length of the bonded area was 1 ~ 1.2 mm which was defined as the bonded length. After bonding, the specimens were heated and kept at the designated temperatures (298, 348 or 373 K) within ± 3 K for the period tr. After the relaxation (annealing) process, the peel tests were carried out at the room temperature. The bonded area was instantly fractured without peeling and the bondstrength Fp appeared to be proportional to the bonded length. Fp was, therefore, assumed to be the average bond strength of the bonded area [1]. The fractured surfaces of Au plates were observed

by a scanning electron microscope (SEM) after peel tests to measure the bonded area.

4. Results and discussion The increase in the bond-strength Fp (pull or peel strength) with the holding time tr was observed after the room temperature adhesional bonding [1]. Two reasons are considered; the increase of bonded area with tr and the residual stress relaxation around the bonded area. As stated below, in the region of T = 298~373 K, the bonded area hardly increases after bonding. Therefore, in the present study, the stress relaxation process is mainly discussed. Fig. 5 shows the calculated results of the stress distribution (the stress normal to the bonded interface), i.e., the change in Tn (x) of Eq. (6) with tr. The bonding condition is T = 300 K, P = 5 MPa, and tB = 60 s. The materials are gold wires with the radius R = 50 µm. The initial bonded width Xo is 0.93 µm for the adhesional elastic contact (= aj for wire / wire contact) [1,12]. The points

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Fig. 5. Stress distribution on bond-interface changing with time (calculated results). (a) immediately before force is reduced (t=tB0), (b) immediately after force is reduced to zero (t= tB + 0), (c) tr = 1.0 week, (d) tr = 1.0 week, (d) tr = 1.0 month and (f) tr = 1.0 year.

A and A⬘ are the edge (tip point) of the bonded area. The stress distribution hardly changes for t = 0 ~ 60 s (3.0 x 10⫺9 t s) and the width AA⬘ hardly changes from the initial width 2Xo according to the calculated results. The vacancy diffusion mainly results in the elastic displacement followed

by the stress relaxation as shown in Fig. 5. The stress distributions (a) for t = tB ⫺0 and (b) for t = tB + 0 have a high frequency oscillation because they are calculated by Fourier series of Eq. (6) from n = 1–100. In other words, the finite Fourier series inevitably exhibits high frequency oscil-

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lations, because the maximum tensile stress smax is theoretically infinite in elastic adhesional contact. The high frequency oscillation of t = tB + 0 becomes larger than that of (a) because of the force reduction. However, as seen in Fig. 5(c), the high frequency oscillation disappears at t = 1 day (4.3 × 10−5 t s). The maximum tensile stress smax is observed as a horn on the inside of the tip points in the beginning but gradually decreases with time and is finally equal to the value of gs / r in Fig. 5(f) which is approximately under the steady state (1.6 × 10−2 τ s). The time required to reach the steady state appears to depend on the initial stress distribution and the value of gs / r and cannot be decided by the parameter t, although t becomes a good index for the transient behavior [8]. Fig. 6 shows the change in smax with time after bonding as the ratio against smax at tr = 0 which is the maximum tensile stress immediately after removing P. The stress relaxation for wire/plane bonding (b) is slower than that of wire/wire bonding (a), because the initial contact width aj (⬇ 1.42 µm) for wire/plane contact is larger than that (⬇ 0.93 µm) of wire/wire contact. As the wire radius R decreases, the stress smax rapidly decreases and the decreasing rate of smax is striking for R less than submicrons. This means that the stress relaxation for nano scale bonding (R ⬍1 µm) is instantly produced even at the room temperature [12]. From the calculated result of R = 50 µm for wire/plane in Fig. 6(b), it is found that smax / smax (tr = 0) becomes to be less than 40% when tr ⬎ 0.6 × 106 s (⬇ one week). This implies that a clear increase in the bond strength is observed one week after bonding if it is due to the stress relaxation by the vacancy diffusion along the bond interface. Fig. 7 shows the influence of temperature on the stress relaxation process after the elastic adhesional bonding. The stress relaxation rate naturally increases with T. The decrease of the maximum tensile stress ⌬smax can be arranged with the square root of tr. Because the stress distribution is approximated to the Fourier series with high frequency oscillations, the change in smax cannot be perfectly proportional to the square root of tr but the approximation of ⌬smax ⬇ a tr 1 / 2 is possible in the beginning of the stress relaxation process,

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Fig. 6. Maximum tensile stress smax decreasing with time after elastic adhesional contact. (a) for wire/wire contact and (b) for wire/plane contact.

where a is a proportional constant. If this is so, the increase in the bond strength can be proportional to the square root of tr. As tr increases, the relationship of ⌬smax ⬇ a tr 1 / 2 does not hold good, as suggested in Fig. 7, because the decrease in smax is quite the transient behavior which is saturated into the steady state (smax = gs·sina / r). In taking into account only the elastic deformation, the residual stress increases with the bonding pressure P, i.e., the stress relaxation process becomes longer with increasing P because the contact width 2Xo increases with P. Actually, as P

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Fig. 7. Effect of temperature on the stress relaxation process after elastic adhesional contact between wire and plane. Plastic deformation is ignored.

increases, the plastic deformation contributes more to the adhesional bonding [1] and the relaxation process must be shorter. Therefore, the elasto-plastic adhesional contact [1] needs to be considered for the effect of P on the stress relaxation process. As seen in Fig. 5, the high tensile stresses are locally produced inside the edge of the bonded area at t = 0 ~ tB +0. It is necessary to limit these initial tensile stress values for high bonding pressures, i.e., the peak of the tensile stress has to be reduced by the plastic deformation. The limitation is given by von Mises yield criterion syield given by Eq. (8) for the plane strain problem. syield ⫽

冊 冑冉 2

3

1⫹

p s, 2 Y

(8)

where sY is the yield stress under the uniaxial stress condition (see Table 1). The initial stress distribution sinitial for the elasto-plastic adhesional contact (t = 0) and half the elasto-plastic contact width aep was calculated as stated elsewhere [1]. Fig. 8 shows the change (calculated results) in the stress distribution Tn (ξ) with time for the elasto-plastic adhesional contact. The stress distribution of Fig. 8(a) scarcely changes from the initial stress distribution sinitial during bonding (t = 0 ~ tB ⫺0,), owing to a short bonding time (tB = 60 s). The stress distribution sb of Fig. 8(b) is changed by removing the force F ( = 2RP). The stress distri-

bution sb was calculated as follows; at first, Boussinesq’s stress distribution was modified in the same manner for calculating the stress distribution sinitial [1]. The stress distribution sb at t = tB +0 was given by sb = sa + sm, where sm is Boussinesq’s stress distribution limited by Eq. (8) and sa is the stress distribution at t = tB - 0 which is shown in Fig. 8(a). The approximation of sb to the Fourier series was carried out as stated in Appendix C (兰 Tn (ξ) dx = 0 holds at t = tB + 0). The maximum tensile stress smax at t = tB + 0 becomes twice as large as syield, owing to adding sm to sa. This implies that the plastic deformation is induced and the two bodies in contact can locally be separated at the edge so that smax can be reduced to syield. It is not so easy to calculate the amount of separation and the stress distribution after separation. It was, therefore, assumed in the present study that the separation does not occur and Fig. 8(b) is kept immediately after bonding and the stress relaxation after tB is produced only by the vacancy diffusion. Although the plastic deformation must instantly reduce the tensile stress to the level of the yield criterion, the relative change in smax with tr must be not so different. The infinite elastic tensile stress can be limited even by the plasticity correction as mentioned above. As shown in Fig. 8(c) and (d), the residual stress can be reduced with time. Because the initial contact Xo (= aep ⬇ 1.36 µm) is greater than aj = 0.93 µm for the elastic adhesional contact, the relaxation rate becomes slower, compared with the elastic adhesional contact. However, if the plastic deformation is taken into account, the stress relaxation time becomes shorter with increasing P as seen in Fig. 9. Fig. 10 shows the effect of temperature on the stress relaxation process (calculated results) after elasto-plastic adhesional bonding. Because the stress distribution at t = tB + 0 has not such large components of the high frequency oscillations (see Fig. 8(b)), the decrease in smax is clearly proportional to the square root of time, compared with Fig. 7. It is found from Eq. (7) that the time tr required for the interface self-diffusion to reduce smax by the same quantity can be proportional to the time parameter t. If the change in the bond width 2X

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Fig. 8. Change in stress distribution on bond-interface after elasto-plastic adhesional bonding. Plasticity correction is taken into account.

with time is negligible small and other geometrical parameters of R and r are approximately constant during the stress relaxation process, then tr required to produce the same value of ⌬smax can be proportional to the time constant t. In other words, Db·E·tr / T can be constant, i.e., Eq. (9) is established [13].

冉 冊

Qb T , ⫽ ADb ⫽ A·Dboexp ⫺ trE RT

(9)

where A is a constant, Dbo the frequency factor of the interface self-diffusion coefficient Db, Qb the

activation energy for the interface self-diffusion, and R is the gas constant (see Table 1). If the change in Young’s modulus E with T can be ignored, compared with the term of exp (⫺Qb / RT),

冉 冊

Qb T / tr ⬀ exp ⫺ RT

(10)

is obtained. In the region where ⌬smax ⬇ a·tr 1 / 2 is established in Fig. 10, for T = 300 K, the time taken to reduce smax down to 400 MPa and Young’s modu-

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Fig. 9. Effect of bonding pressure on stress relaxation process. Plasticity correction is taken into account.

Fig. 11 shows the experimental results of the change in the bond-strength Fp with the square root of tr. The adhesional bonding was carried out at T = 298 ± 3 K and only the annealing (holding) temperature was changed in the range of T = 298 ~ 373 K. Although the experimental results are scattered, Fp increases with time and the increase in Fp is proportional to the square root of tr. After the time tr required to obtain Fp = 45 N / m marked by the dotted line in Fig. 11 is substituted into Eq. (10), an activation energy Q is obtained from (T / tr) ⫺(1 / T) plots. This is the activation energy of the mechanism which increases the bond-strength with time. The value is Q = 76 kJ / mol. The value of Q is smaller than that of Qb = 87 kJ / mol indicated in Table 1. If the interface self-diffusion is controlled by the vacancy diffusion,then the activation energy Qb is given by Qb = Qf + Qm, where Qf is the formation energy of vacancy and Qm is the migration energy of vacancy [22,23]. In experimental tests, Ar ion beam irradiation for removing the surface contamination was carried out before bonding. The Ar ion irradiation can increase the vacancy concentration Co (see Eq. (2)). If this is so, the value of Qf can decrease resulting in reducing the value of Qb to 76 kJ/mol. The value of 76 kJ/mol is also much less than the volume self-diffusion (170~174 kJ/mol) [15,22,23]. This implies that the interface self-diffusion contributes to the increase in Fp and the volume self-diffusion can be ignored.

Fig. 10. Effect of temperature on stress relaxation process when plastic deformation is taken into account.

lus E are, respectively, tr = 1.21 × 106 s and E = 8.27 × 104 MPa at T = 300 K. On the other hand, at T = 350 K, tr = 1.0 × 104 s and E = 8.15 × 104 MPa are obtained from Fig. 10 and Table 1. After substituting these values into Eq. (9) and eliminating A·Dbo, we obtain Qb ⬇ 87 kJ/mol, which is equal to the amount of the activation energy of Db (see Table 1). If E is ignored, then we obtain Qb = 86.7 kJ / mol. Also, even if the results in Fig. 7 is used, the same result is obtained.

Fig. 11. Change in bond strength with time (experimental results). The bond-strength Fp is equal to the fracturing force for the bonded length of 1 mm.

Y. Takahashi, K. Uesugi / Acta Materialia 51 (2003) 2219–2234

The contour area bonded Ac of the experimental points (a–d) were measured by SEM. The amount of Ac (⬇6 × 10⫺9 m2) appeared to be constant, independent of time. The contour area was larger than the real contact area owing to the surface roughness due to Ar ion irradiation [24]. The initial contact width 2Xo under P = 5 MPa T = 298 K was also scattered between 4.3 and 5.3 µm because of the surface roughness. The bond-width 2aep for elasto-plastic contact of Au wire/Au plate contact is calculated as 3.4 mm [1]. Although the experimental results are slightly larger than the calculated result, the experimental results are roughly in agreement with the calculated value. This implies that the plastic deformation contributes to the initial contact. There remains the possibility that the visco plastic (power law creep) deformation partially contributes to reduce the maximum tensile stress smax but the activation energy measured in the present study Q = 76 kJ / mol is lower than that of the power law creep at the middle (or low) temperatures [25–27]. Therefore, the increase in Fp is not derived from the contribution of the power law creep, i.e., it is suggested that the stress relaxation process due to the vacancy diffusion along the bond-interface increases the bond strength.

5. Conclusions The stress relief process due to the stress induced vacancy diffusion along the bond (contact) interface has been discussed. When the elastic stress and strain exist around the contact interface between fine wires (or small particles and powders), the stress induced vacancy diffusion can be produced even under room temperature to increase the bond (adhesional contact) strength.

Appendix A: Excess chemical potential and interface self-diffusion equation The excess chemical potential ⌬m is expressed by ⌬m ⫽ ⫺kT ln C / Co⬇⫺kT⌬C / Co,

(A-1)

2231

where ⌬C = C⫺Co is small enough. Vacancy flux Jva (number of vacancy per unit area per unit time: atoms/m2/second) is expressed by Jva = ⫺(Dva / ⍀) grad C = ⫺(Dva / ⍀) grad ⌬C. The self-diffusion coefficient D is equal to DvaC (⬇Dva Co). Therefore, Jva = ⫺(Dva / Co / ⍀) grad (⌬C / Co = (D / ⍀) grad (⌬m / kT). As the direction of the atom flux is opposite to that of the vacancy flux (J = ⫺Jva), J is given by J ⫽ ⫺(D / kT⍀) grad ⌬m.

(A-2)

Appendix B: Stress function and coupling diffusion and elastic deformation We introduce the stress function with the term of the surface energy gs, expressed by

冘 ⬁

⌽(x, h) ⫽

{fn(1 ⫹ anh)exp(⫺anh)cos(anx)}

n⫽1

⫹

gs 2 2 X (x sina ⫹ h2 cosa), 2r (A-3)

where ξ =x / X, h = y / X, X is half the bond width (OA in Fig. 2), a half the dihedral angle at the tip point A in Fig. 2, r the curvature radius in the vicinity of the tip point, an = (4n - 3)p / 2, n the natural number, fn = ⌰·Cn. The parameter cn is normalized by

冘 ⬁

ancn ⫽ 1,

(A-4)

n⫽1

and ⌰ is obtained by the force balance of X 1 sy|h = 0dx = F in the y direction. 2 0 Three stress components sx, sy, and txy are obtained from sx = ∂2⌽ / ∂y2, sy = ∂2⌽ / ∂x 2, and txy = ∂2 ⌽ / ∂x∂y, respectively. For example, sy is expressed by

冕

冘 ⬁

sy ⫽

⌰ ⫺a2ncn(an·h ⫹ 1)exp X2n ⫽ 1

gs (⫺anh)·cosanx ⫹ sina. r

(A-5)

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Also, the coefficient ⌰ is given by

冉

冊

1 gs ⌰ ⫽ X X sina⫺ F . r 2

(A-11) (A-6)

Equation (A-5) means that the stress due to the surface tension, sy\␰ = ␩ = 0 is always works at the tip point. Because of the plane strain condition (⑀z = 0), the y component of strain, ⑀y =1 / E {sy n(sz + sx)} is given by

冘 ⬁

ey ⫽ ⫹

⌰(1 ⫹ v) {⫺cn·a2ncosanx·exp(⫺anh)·(1⫺2v ⫹ anh)} E X2 n ⫽ 1

(1⫺v2)gs v(v ⫹ 1)gs sina⫺ cosa, E r Er

(A-7) where E is the Young’s modulus and n is the Poisson’s ratio. The elastic displacement ue 兰⑀y dy = 兰 ⑀y·X·dh is given by

冘 ⬁

ue ⫽ ⫹ ⬗

冕

⌰(1 ⫹ v) {c a cosanx·exp(⫺anh)·(2⫺2v ⫹ anh)} E X n⫽1 n n

Because ∂ue / ∂t = ∂u / ∂t − ∂ur / ∂t, Eq. (A-12) therefore holds.

冘 再冘

冉 冊

⬁

2⌰(1 ⫺ v2) ∂cn an cosanx· ⫽ E X n⫽1 ∂t d⍀Db⌰ k T X4

⬁

cn a3n⫺

n⫽1

冘 ⬁

cna4ncosanx

n⫽1

冎

(A-12)

After both sides of Eq. (A-12) is multiplied by cos amx and integrated from x = 0 to x = 1 by x and the orthogonalization leads to the rate equation which should be solved;

再 冘 ⬁

冎

∂cm 1 2 (c a3)⫺cma3m , ⫽ ∂t t a2mn ⫽ 1 n n

(A-13)

where t has a dimension of time and is given by t⫽

2(1⫺v2)kTX3 . Ed⍀Db

(A-14)

{(1⫺v2)sin a⫺v(v ⫹ 1)cosa}gsX h, E r (1⫺2v ⫹ anh)exp(⫺anh)dh ⫽ ⫺a⫺1 n (2⫺2v ⫹ anh)exp(⫺anh).

Appendix C: Calculation method of initial stress distribution

(A-8) Therefore, we obtain

冉 冊 ∂ue ∂t

(A-9)

at h=0

冉 冊

冘 ⬁

⫽

2⌰(1⫺v2) ∂cn an cosanx· . E X n⫽1 ∂t

冘 ⬁

(A-10)

We also obtain Eq. (A-11) from Eq. (4). d⍀Db ∂u ⫽ ⫺d⍀·div Jb ⫽ ⫺ ·div·grad(sy|x=1,h=0) ∂t kT

冘 ⬁

d⍀Db⌰ c a4cosanx ⫽⫺ k T X4 n ⫽ 1 n n

冘 k

S⫽

After sy in Eq. (5) is substituted by Eq. (A-5), we obtain d⍀Db ⌰ ∂ur c a3. ⫽⫺ ∂t k T X4 n ⫽ 1 n n

The initial stress distribution on the bond interface was approximated to the Fourier series Tno(ξ) by using the least squares method, i.e., the values of cm were calculated to minimize the value of Eq. (A-15). {Tn(xi)⫺Tno(xi)}2,

(A-15)

i⫽1

where Tn (x) is an approximate equation of Tno (x), k = 1000 and xi = i / k. The values of cm can be calculated so that ∂S / ∂cm = 0 for m = 1 ~ N. The natural number N should be ⬁ theoretically, but if N ⱖ 50 then a sufficient accuracy was obtained in calculation. In the present study, a proper integer of 50 ~ 100 was adopted for N. The method of least squares cannot be applied for the approximation of the stress distribution immediately after bonding (F = 0), because ⌰ / X 2 = (gs / r·sina and all cn become zero if the least

Y. Takahashi, K. Uesugi / Acta Materialia 51 (2003) 2219–2234

squares method is adopted. The stress distribution immediately after bonding was, therefore, calculated as follows. The stress distribution s 3 immediately after the force F is reduced from f 1 to f 2 is given by s 3 = s 1 - s 2, where s 1 and s 2 are the stress distribution for F 1 = f 1 and Boussinesq’s stress distribution for force reduction (F 2 = f 1 f 2) [1], respectively. They are approximated by

冘 N

sj ⫽ (⌰j / X ) 2

2 n n,j

(⫺a c ·cosanx)

(A-16)

n⫽1

⫹ (gs / r)sina,

s2 ⫽ F2 / {p(X2o⫺x2)1/2}.

(A-17)

Because of s 3 = s 1 - s 2,

冉

gs f1 sina⫺ r 2X

冘

冊冘 N

{⫺a2n(cn,1⫺cn,2)cosanx}⫺

n⫽1

N

f2 2X

冘

⌰3 (⫺a2ncn,3·cosanx) ⫹ (gs / r)sina X2 n ⫽ 1 N

⫺a2ncn,2cosanx ⬅

n⫽1

(A-18) is obtained. After both sides of Eq. (A-18) are multiplied by cos amx, the orthogonalization leads to the value of each cm,3; cm,3 ⫽

再

y1 ⫽ R1⫺dc1 ⫽ 冑R21⫺x2o,

X2 ⌰1 f2 c 2 (cm,1⫺cm,2)⫺ ⌰3 X 2X m,2

y2 ⫽ ⫺R2 ⫹ dc2 ⫽

⫺冑R22⫺x2o, yc ⫽

y22⫺y21 ⫹ (R1 ⫹ r)2⫺(R2 ⫹ r)2 , and 2(y2⫺y1)

xc ⫽ ((R1 ⫹ r)2⫺(y1⫺yc)2)1/2 are obtained, where d c1 and d c2 are referred in Fig. 3. Also, S 1, S 2 and S 3 are, respectively, given by S 1 = R 21·q 1 / 2, S 2 = R 22·q 2 / 2, and S 3 = r 2·q 3 / 2, where q 1 is given by >

where ⌰j = X{X(gs / r)sina - Fj / 2}, F 3 = f 2, and j = 1, 2, or 3. Boussinesq’s stress distribution along the x axis is theoretically given [1] by

2233

>

>

>

cosq1 ⫽ AC·AO / (兩AC兩·兩AO兩) ⫽

冑x

(A-20)

y1⫺yc 2 c

⫹ (yc⫺y1)2.

Angle q 2 and angle q 3 can be obtained in the same manner. > > Because of SABC = 兩CA × CB兩 / 2, SABC is given by SABC ⫽

sinq3 2 冑xc ⫹ (yc⫺y1)2冑x2c ⫹ (yc⫺y2)2. 2

(A-21)

It is requred to determine the values of r and xo so that S ABC = St (= S 1 + S 2 + S 3) and Xo + r = xc. For Fig. 3(b) of R2 = ⬁, yc = r and SAOEC = (y1⫺yc)·xc + xc·yc. 2

(A-19)

冎

2 gs ⫹ 3 · ·sina . am r Equation (A-19) corresponds to cn,j in Eq. (A-16) for j = 3. Therefore, by using Eq. (A-19), we obtain the stress distribution s 3 immediately after reducing the force from f 1 to f 2.

Appendix D: Calculation of geometry parameters in Fig. 3 In Fig. 3(a), if proper values are, respectively, given to r and xo,

Appendix E: Mass conservation taking into account the rigid displacement ⌬ur In Fig. 4, the number of (1)~(6) expresses the area of each region. The area GHF = the area FDE = (1) + (2) + (3) + (4) is established from the assumption of the model as shown in Fig. 3(a). Also, the area G⬘F⬘H⬘ = (4) + the area JKE⬘D⬘ because of So = So∗ (see Fig. 3). The rigid displacement ⌬ur·X(t) = the area HH⬘JD = the area GG⬘KE. Therefore, (5) + (1) + (3) + (2) + (1) + (6) = the area JKE⬘D⬘, i.e., 2⌬ur·X is equal to the atom outflow (area JKE⬘D⬘) for the time increment ⌬t. If the model of Fig. 4 is used, the mass consevation always holds good.

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