A new model for diffusion bonding and its application to duplex alloys

Materials Science and Engineering A271 (1999) 458 – 468 www.elsevier.com/locate/msea

A new model for diffusion bonding and its application to duplex alloys N. Orhan a,*, M. Aksoy b, M. Eroglu b b

a Department of Metallurgy, Faculty of Technical Education, Uni6ersity of Firat, Elazig, Turkey Department of Metallurgical and Materials Engineering, Faculty of Engineering, Uni6ersity of Firat, Elazig, Turkey

Received 17 June 1998; received in revised form 8 June 1999

Abstract Diffusion bonding is an advanced bonding process in which two materials, similar or dissimilar, can be bonded in solid state. This provides to bond materials in a wide range from low carbon steels to ceramics and composites which cannot be bonded with conventional welding methods. One of the major advantages of this method is to produce new bimetal or dissimilar material couples. The process is diffusion-based and occurs in solid state and because of its increasing use as a commercial process. The estimation of final bonding time is very important but difficult without experiments for many materials. In this study, therefore, a new mathematical model is presented to predict the final bonding time for a sound bonding interface prior to bonding practice. Being different from the previous models, this model assumes a new surface morphology as a sine wave and a new creep mechanism for duplex alloys. The mechanisms operating during diffusion bonding are based on those in pressure sintering studies but here mass transfer by evaporation has been ignored. The driving forces and rate terms for those mechanisms have been altered to reflect the difference of the geometries of the two processes. Also the effect of grain size has been included in the model in case of joining fine-grained materials. In determining diffusion coefficients for duplex alloys, Darken’s equation for binary alloys has been used. Depending on this new approach, it was shown that a more realistic final bonding time could be predicted for duplex alloys by comparing the results from this new model with those from the previous ones. As a result, it was determined that the new model could be used in order to estimate the final bonding time of the duplex alloys for a sound bond interface and the relationships between its parameters safely. The predictions from this new model show a very good agreement between practice and theory. © 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Diffusion bonding; Bonding time; Duplex alloys; Creep

1. Introduction Modern manufacturing technology requires a bonding process at low temperatures and low pressures in order to avoid undesired phase transformations and large deformations for the modern materials. In many circumstances, a metal and non-metal have to be bonded together. Diffusion bonding is such a process in which two matched surfaces are held together at a temperature below the absolute melting temperature of the materials under a low pressure which does not cause a macroscopic plastic deformation in the materials for a time required to form a metallurgical bond between materials. Temperatures range between 0.5 and * Corresponding author. Fax: +90-424-2184674.

0.8 of the absolute melting points of the materials. Pressures are typically applied as some small fraction of the room temperature yield stress to avoid macroscopic deformation. Bonding times vary from a few minutes to several hours and the ratio of the roughness, asperity or height to wavelength, which are the parameters defining surface condition, is typically 1:50 [1–10]. The rate-controlling step in diffusion bonding is removal of the interfacial voids due to the surface roughness. The experimental studies on diffusion bonding are focused commonly on the bondability and interfaces of various materials and material couples, the mechanical properties of the joints and bonding times [2–18]. In the studies, pure metals, alloys, composites, ceramics and various material couples were chosen. It is seen from these studies that the researchers had to do many

0921-5093/99/$ - see front matter © 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 3 1 5 - 9

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

experiments to determine the time necessary for a sound bonding interface even for the same material. In other words, one of the major problems in this bonding process is to predict the final bonding time before the practice. The process depends on several mechanisms occurring in solid state which are similar to those in sintering process. Since these mechanisms are still under discussion and a sufficient model has not been found for the process yet, there is also a strong need for developing new physical and mathematical models. The studies on modeling diffusion bonding are concentrated essentially on the surface conditions, especially the nature and the shape of surface roughness and waviness which are the major factors in determining driving force and rate equations for surface diffusion, interface diffusion and creep. Determining the final bonding time and developing a realistic theoretical approach to practice by applying all possible mechanisms have also been considered in the studies [19–21]. Because the final bonding time is not certain prior to the bonding process, it is extremely important to determine this time earlier for the quality of the bonding interface and economic aspects of the method. The previous studies show that the surface profiles proposed are not so realistic to reflect the results of the dominant mechanisms exactly and the researchers state that their models are suitable only for pure metals and similar materials [18 – 20,24]. Therefore, this study aims to present a new model having a new and more realistic surface profile and giving more proper relations between the mechanisms and final bonding time for both pure metals and duplex alloys bonded by diffusion bonding. The correct choice of the surface profile has the highest priority in modeling diffusion bonding since it effects all mechanisms in the process. A new surface profile is the main difference of the model from the previous ones. Secondly, a new approach was presented for creep and interface diffusion considering the effect of the microstructure on these mechanisms. In this study, first, the previous models and diffusion bonding mechanisms are dealt with and then geometric considerations are given by focusing on the shape of the surface roughness due to the reasons mentioned above. The reasons why a sine wave has been chosen as the surface profile are explained. The following chapters give the relations between each of the dominant mechanisms and void shape forming after the initial contact of the matched surfaces. The equations derived in the study have been interpreted on a computer program written in Q-basic. The results are presented in graphics and interpreted in the last chapters. Geometric derivations are given in appendixes separately.

459

2. Diffusion bonding mechanisms Various authors [18,19] have proposed many mechanisms, which drive diffusion bonding. All these mechanisms essentially are similar. A recent model was proposed by Hill and Wallach [24]. In this study, a sintering-based approach is assumed depending on the nature of the process because of the similarity to sintering. Like Takahashi et al. [20] they have also considered the effect of grain size on diffusion bonding and the predictions have shown a good agreement with the experimental results for copper and iron. Ridley and Guo [18] proposed another model based on a lentricular void shape. All the models propose a diffusion-based approach. Thus the other mechanisms are required to define on this basis. Depending on this basis, the mechanisms of diffusion bonding in this study are considered as follows: 1. Plastic deformation: in this stage, ridges of the surface asperities deform plastically in such a way that there is no macroscopic deformation in the parts to be bonded. 2. Surface and volume diffusion: in this stage, because of the difference of the curvature of the voids at the interface of the material couple, forming after plastic deformation, there occurs a matter transfer from the parts having larger curvature to the void necks having a smaller curvature. The matter is also transferred from the bulk into the void surface by vacancy diffusion. 3. Bond interface diffusion and grain boundary diffusion for fine-grained materials: matter transfers from the bond interface to the neck of the voids due to the stress gradient at the interface and from the grain boundaries to the void surface for similar effects. 4. Coupled creep and diffusion: in this stage, diffusional creep contributes the closure of the voids. For simplicity, the following assumptions are also proposed: 1. The two surfaces are brought together to give ridgeto-ridge contact as shown in Fig. 1. In contact regions, immediate bonds are obtained. 2. During bonding, surface profile is a sine wave and they form an eye shape in contact [21] as shown in Fig. 1.

Fig. 1. The Shape of interfacial voids.

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

460

3. The ridges between the cavities are subjected to plane strain. The strain along the cavity length is zero. 4. Cavities are distributed uniformly along the interface with a spacing b, which is considered the period of the sine wave, and have a unit length in z direction. 5. The period of the sine wave, b, is considered to stay constant during bonding [19]. 6. There is no pressure in the cavities during bonding. 7. End effects are ignored.

3. Geometry consideration A surface preparation for diffusion bonding produces a series of long, nearly parallel ridges. They are not necessarily straight-sided as in the Derby and Wallach [19] and Takahashi model [20] or elliptical as in the Hill and Wallach model [24]. Ridley and Guo’s model [18] assumes a semilenticular surface for long ridges. However, the surface or interface micrographs from the experiments show a reverse curvature in the neck region of the matching surfaces. This necessitates a different surface morphology between a lenticular and a straightsided shape. Klaphaak [22] states that the surface preparation methods produce a surface similar a sine function. Takahashi [20] reported a reverse diffusion from the neck of the voids to the ridges at the beginning of diffusion bonding. These necessitate a surface morphology being convex at the contact regions of the surfaces. When sintering process is considered, it can be seen that the contact points of the spherical particles of powders have the same shape [26]. Therefore the surface profile was chosen as a sine wave the formulae of which is given in Eq. (1): y=



 

2px h 1− cos b 2

Fig. 2. The behavior of an indenture in fully plastic regime and its application to diffusion bonding.

determination of maximum bonding time, which can be longer than real bonding time, but causes a sound bonding interface. As pressure is applied, ridges are deformed instantaneously and when the contact area becomes large enough to support the applied load, plastic deformation ceases. This is a time-independent manner. The deformation amount has been determined using Sharp’s study [23] on the behavior of an indenture in fully plastic regime (Fig. 2). Sharp’s equation is: Pk =

4. Plastic deformation It has been assumed that the surfaces are brought into contact in a ridge-to-ridge and valley-to-valley manner as shown in Fig. 1. This assumption gives the

2apsy 2a+a ln

  aE 3syR

(2)

where Pk is the indentation pressure or bonding pressure in our case, sy is the yield strength of the material at the room temperature, E is Young’s Modulus, R is the radius of curvature of the indenter or the radius of the curvature of the sine wave in our case. E should be considered as the softer material’s Young’s modulus in case of dissimilar material couple. Here, the curvature of the void ridge can be determined from the second derivative of the surface profile equation as: y; =

(1)

where h is the average height of the ridges, x is the distance in x direction and b is the width of the void or the period of the sine wave. In our model, the bonding is assumed to occur in two stages. In the first stage, the first contact and bonds form because of the applied pressure and resulting plastic deformation. This kind of plastic deformation is instantaneous and thus not allowing the time-dependent diffusion or creep. In the second stage, diffusionbased mechanisms run.

' 

1 3

2hp 2 b2

(3)

The inverse of this equation gives the radius of curvature of the void ridge (R): R=

b2 2hp 2

(4)

Thus, the contact width of the void ridge after plastic deformation can be found as: a=

   2psy

2Pk (Ea/3syb 2) 2+ ln 2hp 2



1/2

(5)

This equation can be solved by interpolation. Refering Fig. 3 and assuming the sides of the voids slide relative to the deformation in the width of the voids provided that the distances of the void centers remain constant during plastic deformation, the height of the voids (h0) can be obtained as:

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

h0 =

hb0 b

(6)

The void volume after plastic deformation (V0) is determined as: (7)

V0 = h0b0

 

2DsVds dms kTr dr

(8)

where dms/dr is the chemical potential gradient, r is the radius of the void neck which is given in the next chapter, k is the Boltzman constant, ds is the thickness of the free surface layer, Ds is surface diffusion coefficient and V is atomic volume. From the Gibbs–Thomson relationship: ms =Vg(K)

(9)

where K is the curvature of the surface profile and g is the free surface energy. Differentiating this equation gives: dms =Vg dK= −Vg



1 1 − r R



  dV dt

=

s

(10)

Fig. 3. The shape of the interfacial void after plastic deformation.



2dsDsV 1 1 − kTr r R



(11)

The matter transferred by surface source–volume diffusion is given as: dV dt

The activation for the diffusion around the free surfaces of the voids is caused by the difference of the curvature between tip and void surface. This difference leads to a chemical potential gradient given by Gibbs– Thomson relationship. In addition, a surface diffusion also occurs through the volume adjacent to the surface beside the diffusion along a thin surface layer along the free surfaces. Matter is transferred from the point of least curvature to the point of greatest or to the void neck. The matter transferred around surfaces of voids (Js) has been determined using Nernst – Einstein [11] equation and found as: Js =

Substituting this into the Eq. (8) we get the matter transferred around the surfaces of the voids. Therefore:

 

5. Diffusion around surface areas

461

=

sv



2dsD6 Vr 1 1 − kTr r R



(12)

Total matter transferred by the surface source mechanisms, hence, can be estimated by:

      dV dt

=

st

dV dt

+

s

dV dT

(13)

sv

where the term of (1/r −1/R) is the aspect ratio and surface diffusion ceases when this ratio goes to unity. The material transferred by surface source diffusion is assumed to be distributed around the void surface, thus the changes both the height and width of the voids can be deduced, respectively, as below, considering the length of surface profile from Appendix B:

 

  dh0 dt

dV dt = 4l s

s

(14)

and

     db0 dt

=

s

dV dt

s

b0 h0

(15)

where l is the length of a quarter of the surface profile and given from the Appendix B as: l=

'    2b0 1+

h0p 2 b0

2

(16)

6. Diffusion around interface The pressure uniformly applied exerts a stress gradient around the interface due to the voids. This stress gradient results in a chemical potential which causes diffusion. The interface can be thought as a grain boundary for similar materials. In the case of a dissimilar materials couple, diffusion is interface diffusion and in this situation a new diffusion coefficient must be determined. When pressure is applied, a compressive stress gradient occurs at the regions adjacent to the void neck and it becomes zero midway between cavities as can be seen from Fig. 4. Therefore, diffusion takes place only in the stress gradient region. In the region where stress is constant, only power low creep occurs. Chen and Argon [25] have proposed an effective diffusion length and Ridley and Guo [18] used this

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

462

The average normal stress in the stress gradient region is then determined by: spbg s1 =





 

3 3g 1 1 − (b−b0 − B) − rB 2 2 B r 3g 3 B+ − (b−b0 − B) rB 2 2





(22)

The atom fluxes through a grain boundary and the volume around the interface is, respectively: Fig. 4. Stress distribution along interface.

Jb = −

characteristic length in their model. In this new model this length is considered as the border of the stress gradient region. The effective diffusion length (Ld ) is: Ld =

Dbdb Vsp kTo%

(17)

where db is the thickness of the grain boundary, o% is the power law creep deformation rate and given as: o%= A

 sp G

n

(18)

where A is the creep constant. The atom flux along the interface can be estimated by using Nernst –Einstein equation. This flux (Jb ) is: Jb = −

   d bD b VkT

dmb(r) dr

(19)

where dmb(r) ds =V dr dr ds is the stress gradient and can be estimated by dr following the way given by Exener and co-workers [26]. The necessary boundary conditions to solve Eq. (18) are as follows (Fig. 4): ds =0 dr

b0 +B 2

 à à 1 1 b at r = 0 s= g − à r R 2 à b0 b Ì s dr = s1B− r at 5r5 0+B 2 2 à s=s11 at r \a +B à à spb = s1B+s11(b − b0 −B) à B=b0 when Ld \b0, and B = Ld when Ld Bb0 Å (20)

&

at

 

r=

Solving the equation for these boundary conditions the stress gradient along the interface can be found as: g g 3B s1 − − ds B r = (21) dr B2

 



J6 =



3Dbdb g g s1 − − B r VkTB



3D6R g g s1 − − VkTB B r





(23) (24)

For volume diffusion the limiting area is assumed as the radius of the curvature of the sine wave. Therefore:

    dV dt dV dt

= − 2Jb − 4J6

b

=

b



6 g g (D d + 2D6R) s1 − − BkT b b B r



(25) (26)

This equation can be solved readily if the radius of the tip is known. There are many factors influencing the void tip radius such as surface tension, surface diffusivity, void size and geometry. In this study, the approach of Bross and Exner [26] in the sintering process was used to calculate minimum tip radius. It is found approximately from the tangent of the angle between the x-axis and void surface in Fig. 5 as: r= 2a

h0 b0

(27)

The changes in the height and width of the voids caused by the material transferred by interface source diffusion can be written as below, respectively, considering the geometric relations given in Appendix C:

    dh0 dt db0 dt

=−

b

b

=−

   

1 dV b0 dt

b

b0 dh0 h0 dt

b



Fig. 5. Void neck and ridge radius.

(28) (29)

 

 

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

dV dh0 1 c= b0 c dt 2 dt

463

(36)

If a constant volume is considered, the mass transferred into the void due to the creep of one slice is: dVi = dhi Xi =

 ))

h0 si n SA dtXi m G

(37)

Thus: m

dVc = 4 lim % dVi

Fig. 6. The interfacial geometry for power law creep.

(38)

i=1 n“

7. Power law creep mechanism

Consequently, it can be found as:

Because creep occurs at elevated temperatures during diffusion bonding, micro asperities also contribute to void closure. Macrocreep is undesirable because of specimen distortion. As the width of the void changes, the stress and consequently deformation rate also change. In finding the contribution of the creep in the model, the surface profile contour was taken as the base and a slice approach was used. As can be seen in Fig. 6, the void ridge is divided into m slices. m can be any integer satisfying the accuracy required. The stress acting on the ith slice having the thickness of h/m (i =1, 2, 3,…, m) can be written as: si =



b0 s Xi p

(30)

Thus the strain rate of a slice is: o; = SA

))

si n G

(31)

where S is the sign, G is the shear modulus of the material and n is the exponent of creep. From the definition of true strain and strain rate:

))

dhi s n dhi = = doi =o; dt= SA i dt hi (h/m) G dh0 SA dt

&) ) 0

Xi dy

(39)

0

8. The calculation of the deformation rate for duplex alloys The deformation rate for duplex ductile alloys can be determined using the equation proposed by Tanaka et al.[27]. Therefore, the deformation rate for duplex alloys is given as: o; = (1− f)o 1 + fo 2

(40)

where f is the rate of the second phase. The deformation rates of each phase can be given as follows, respectively: o1 = K1(s1 − fKEx)n1

(41)

o2 = K2(s2 − (1− f)KEx)n2

(42)

where n1 and n2 are creep exponents of first and second phases, respectively. K1 and K2 are creep constants and K is shape factor.

9. Effect of grain size

si dh0 G

(33)





b0 2y cos − 1 1− i 2p h0

(34)

) )&>   

Combining Eqs. (32) and (33):

 

h

n

From the surface profile equation and Fig. 6: Xi =

) )&

dV si n c= 4b0SA dt G

(32)

It can be obtained by: h

 

dh0 s n c= SA p dt G

h

0

b0

?

b0 2y cos − 1 1 − i 2p h0

n

dy

(35)

This gives the rate of collapse of bonded surface due to the creep. From Appendix D, considering the volume change we get:

In a polycrystalline material, grains around free surfaces can be considered different for whether material is superplastic. For a superplastic or fine-grained material, free surfaces of voids are intersected by several grain boundaries through which atoms diffuse into the cavity thus contributing void closure since grain size in these materials is smaller than 10 mm. This effect can be determined by simplifying the grain boundaries as straight lines directed towards the center of the equivalent circle at about right angles to the free surfaces of the voids. The grain boundaries intersect the profile of free surfaces with a spacing ‘d’ apart. d is the grain diameter at the same time as shown in Fig. 7. Therefore the number of the grain boundaries (N) around the profile is determined as:

 

464

N =INT

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

4Requ d

(43)

where Req is the radius of the equivalent circle and the angle of the one quarter of the profile. The rate of volume change of the void by the grain boundary diffusion around the grain boundaries intersecting the voids can be determined from the following equation:

 





dV 12Vdgb g g gb= s11 − − dt kTBi Bi Req

(44)

where, by referring to Fig. 7: Req =

h0 b 20 + 2 8h0



b0 u =arctan 2(Req − h0) u ui = d s1i =







0.5bsp cos fi Xi

(45) (46) (47) (48)

ui =p −u− ui

(49)

B cos f

(50)

Bi =

Table 1 Materials parameters for a-Ti and b-Ti and Ti–6Al–4V (after Guo and Ridley [18]) Parameter

a-Ti

b-Ti

Atomic volume (m3) Burgers vector (m) Shear modulus at 300 K (mN m−2) Temperature coefficient of shear modulus (K−1) Surface energy (J m−2) Volume diffusion coefficient (pre-exponential) (m2 s−1) Volume diffusion activation energy (kJ mol−1) Grain boundary diffusion coefficient (pre-exponential) (m2 s−1) Grain boundary width (m) Grain boundary diffusion activation energy (kJ mol−1)

1.76×10−29 2.95×10−10 4.36×104

1.81×10−29 2.86×10−10 2.05×103

6.2×10−4

2.6×10−4

1.0 8.6×10−10

1.0 1.9×10−7

150

153

6×10−7

9×10−8

5.9×10−10 97

5.72×10−10 153

Ti–6Al–4V (stress dependence of strain rate) Strain rate o =Ac(s/G)n Strain rate power law creep Ac =AdbDb0 exp(−Qgb/ constant RT)(G/KT) Exponent n = 1.43 Creep constant A% =1.2×10−9 sy= 9.4×108 Yield stress ×(1–4×10−4×(T−300))

From the surface profile equation for a slice: Xi =

     b0 2yi cos 1− 2p h0

(51)

where F1 and F2 are fractures of the first and second phase, respectively, and D1 and D2 are diffusion coefficient of the first and the second phases.

10. Diffusion coefficient for duplex alloys In determining the diffusion coefficient for duplex alloys, Darken’s equation [28] for binary alloys is used. Darken gave the following equation by assuming each phase behaves independently according to a fixed reference system: D = F1D1 + F2D2

Fig. 7. Schematic of the grain boundaries around a void.

(52)

11. Comparison of the predictions with the experimental data As mentioned above, various studies have recently been done on diffusion bonding, yet it is difficult to find very much data on a specific material. Therefore, for the comparison with both previous models and experimental data, the current model is applied to Ti–6Al– 4V alloy, a well-known duplex alloy on which there are many available data. A super duplex alloy is used to show the effect of both grain size and pressure, which are major factors of diffusion bonding, on the process. The material parameters used in the model are presented in Table 1 for the material mentioned [18]. As can be seen in Fig. 8 that the bonding time decreases with the increase in the bonding pressure, and the bonded area at the initial stage of the bonding is a very small part of the total area. This is partially due to the bigger difference between the radius of neck and ridge which causes a decrease in mass transfer as a result of the distribution of mass to the all free surface of void by surface diffusion. This result is different

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

465

Fig. 10. The dependence of the bonding time on grain size. Fig. 8. Predictions of the bonding time with respect to the bonding pressure for Ti – 6Al – 4V; temperature 927°C, grain size 5 mm, surface roughness height 30 mm, wavelength 60 mm.

when compared with the other models. The deviations from the curves which are indicated with the letter ‘a’, show the decrease in the effect of grain boundary diffusion. This is because of the decrease in the number of grain boundaries intersecting the free surface of void as a result of the decrease in the length of the surface of void by mass transfer. The deviation decreases depending on the pressure which is one of the major factors accelerating diffusional creep and grain boundary diffusion. The decrease in deviation with the increasing bonding pressure infers that diffusional creep becomes more effective than grain boundary diffusion with higher pressures. These mechanisms, on the other hand, excite surface diffusion by changing the shape of the void. Consequently, closure of the void slows down. In Fig. 9 the temperature – bonding time relationship at constant pressure is given. The bonding time shortens with the increasing temperature as stated in the

Fig. 9. The dependence of bonding time on temperature.

previous studies [1–27]. The proximity in total bonding times between 990 and 970°C arises from the difference in diffusion coefficients of the different microstructures. The bigger amount of phase with a lower surface and grain boundary diffusion coefficients in 990°C results in the deceleration of diffusion at this temperature thus decreasing the contribution of grain boundary in void closure. It also decreases the contribution of grain boundary diffusion in void closure. The effects of grain size in total final time was computed for the grain size below 15 mm considering the grain sizes of superplastic materials to be less then 10 mm. From Fig. 10, it can be seen that the smaller the grain size the shorter final bonding time is since more grain size means more high diffusion paths, and more mass transfer. This result is in good agreement with the literature [18–20,24]. The model also gives the relationship between the surface roughness and bonding time as convenient to literature and expectation as can be seen in Fig. 11. The final time gets longer when the surface roughness is bigger. The combinations of pressures and times to reach a sound bond at 927°C for Ti–6Al–4V of grain size 5 mm and constant surface roughness are shown in Fig. 8. The results were compared with the ones of the diffusion bonding models of Pilling and Ridley, Guo and Ridley [18], and Chen and Argon [25] and experimental data of Takahashi et al. [20] for which a sound bond is defined as one having a lap shear strength which is greater than 95% of that of the parent metal. The current model predicts essentially the correct relationship between bonding pressure and time and the times required to produce a sound bond, but tends to overestimate the area fraction bonded during the bonding process (Fig. 12). The deviations are uncertain but could reflect the reliability of the diffusion data at hand.

466

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

Fig. 11. The dependence of the bonding time on surface roughness.

12. Conclusions In this study a new diffusion bonding model has been established. The model based on a more realistic surface profile and takes into account most of the

Fig. 12. Comparison of predictions of present model with predictions and experimental results of Guo et al. [18], Takahashi et al. [20] and Chen et al. [25] for Ti–6Al–4V, temperature 927°C, grain size 5 mm, surface roughness height 30 mm, and wavelength 60 mm.

possible mechanisms for diffusion bonding. In the model, effective diffusion distance, effects of grain size, phase proportions, and stress state are considered quantitatively. In this new model there is no need to change the geometry of the voids from elliptical-like to circular as in the Ridley and Guo [18] or Derby and Wallach [19] models. In such a geometry, the surface diffusion stops as the shape of the voids have become circular. However, the geometry proposed in this model allows the surface source mechanisms to reactivate naturally even if the aspect ratio of the voids reaches to unity giving a reverse matter transfer and as a result of the contributions from creep and interface source mechanisms it remains activated. The model is able to reflect the effects of the process conditions like pressure, temperature, time, surface quality and microstructure such as grain size, and phase proportions to a certain extent. The model especially reflects the effect of grain size, pressure, diffusion coefficient or microstructure and roughness much more better than the previous models [18–20]. The model also shows that the most effective mechanisms in diffusion bonding are creep and grain boundary diffusion. The bonding parameters such as pressure and temperature are the most effective parameters in the process. The model also shows that the surface profile chosen effects the determination of the final bonding time very much since it is the fundamental of the mechanisms running the process. The model also considers the effect of microstructure on diffusion bonding. In determining diffusion coefficients Darken’s Equation (Eq. (52)) is used therefore it is possible to provide the contribution of present phases to diffusion rate in correct proportions. The model reflects this contribution with a very small difference between the bonding times at 970 and 990°C. Beta phase which has a lower grain boundary diffusion coefficient than alpha phase caused in a decrease of the positive effect of the temperature on diffusion bonding, as can be seen in Fig. 9. Compared with the other models it is seen that the model shows the effect of the grain size more precisely. From Fig. 10 it can be seen that a grain size higher than 10 mm has no effect on diffusion bonding time. This is also a very good consistency with the experimental studies [1–10]. Determining the surface diffusion for the contour of the sine wave can provide new approach for this mechanism on condition that the boundary conditions are considered carefully. Using the model the time to form a sound bond can be predicted for given bonding conditions. The changes in interfacial void volume and volume shape can be obtained as a function of bonding time. The

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

467

predictions from the model have been compared with the experimental results available for Ti – 6Al –4V and show good agreement.

Appendix A. Nomenclature a

A b0 b B d Db Ds D6 E F G h h0 hi J l m n Pk Q R r S T t Vb g d db ds u uI fI m r p o; si sp sy V

half of the bonded interface, equivalent to half of the bonded area if unit thickness considered material coeffecient for creep the width of the void after plastic deformation period of the sine curve or initial wavelength effective diffusion distance for the calculations grain size grain diffusion coefficient surface diffusion coeffecient volume diffusion coefficient Young’s modulus the fraction of the phases shear modulus average roughness height or the amplitude of the sine wave the height of the void after plastic deformation the height of the ith slice of the void ridge atom flux the length of the surface profile the power law creep exponent of the first phase the power law creep exponent of the second phase bonding pressure activation energy radius of the curvature of the void ridge the distance in y direction for the neck radius sign absolute temperature time void volume surface free energy thickness boundary layer thickness surface layer thickness the arc angle of the equivalent circle the arc angle of the ith slice the angle of the ith slice to the horizontal chemical potential radius of the curvature of the void neck pi (3.14159) power law creep strain rate the stress on the ith slice applied stress yield stress atomic volume

Fig. 13. Geometry of surface and interface diffusion.

Appendix B. Geometric derivations for surface source mechanisms By definition, the shape of the void changes while the volume remains constant. The volume of the matter transferred is a thin layer of thickness h. From the Fig. 13 the matter transferred: dV=dhl

(53)

where l is the length of the surface contour. This length can be found from the integration of the surface profile equation:

  

l= 2b0 1+

h0p 2 b0

2 1/2

(54)

During surface diffusion, the volume of the tip remains constant. Consequently: V6 =

b0h0 = constant 2

(55)

Differentiating this with respect to time gives:

     db0 dh0 = dt dt

b0 h0

(56)

Appendix C. Geometric derivations for interface source mechanisms By definition, the volume of the material in the unit cell remains constant but the shape and volume of the unit cell alter. From Fig. 13 if a layer with a thickness of h is transferred into the void, the volume of the material transferred can be found as: dV= b0dh Differentiating this with respect to time:

(57)

   

N. Orhan et al. / Materials Science and Engineering A271 (1999) 458–468

468

dV dt

= b0

b

dh0 dt

(58)

Rearranging and changing the sign due to the decrease in height gives:

  dh0 dt

=−

b

 

1 dV b0 dt

(59) b

The volume of the material in the unit cell is: Vuc =

b0h0 = constant 2

(60)

Differentiating with respect to time we get:

     db0 dt

dh0 dt

=

b

h0 b0

b

(61)

Appendix D. Geometric derivations for creep mechanism During creep, while void surfaces comes nearer, the width of the voids remains constant. Considering the volume of the void, the alteration in height of the void can be obtained. The volume of the void can be derived as follows: 1 V6 = b0h0 2

(62)

Differentiating this with respect to time gives:

  dV dt

 

1 dh0 = b0 2 dt c

(63) c

Rearranging this equation for h0, gives:

    dh0 dt

=

c

2 dV b0 dt

(64) c

.

References [1] R.A. Nichting, G.R. Edwards, D.L. Olson, B.B. Rath, Advances in Welding Science and Tech., in: Proc. Int. Conf. on Trends in Welding Research, 1986, pp. 733 – 738. [2] M. Dupeux, W. Chuangeng, P. Willemin, Acta Metall. Mater. 41 (11) (1993) 3071 – 3076. [3] H. Andrejewski, K.F. Badawi, B. Roland, Weld. J. 72 (9) (1993) 435s – 439s. [4] Y. Takahashi, K. Inoue, K. Nishiguchi, Acta Metall. Mater. 41 (11) (1993) 3077 – 3084. [5] D.W. Walsh, J.P. Balaguer, Advances in Welding Science and Tech., TWR 86 (1986). [6] F.A. Calvo, J. Mater. Sci. 28 (1988) 2273 – 2280. [7] P. Lefroncois, B. Criqui, M. Hourcade, Weld. J. (1993) 313s–320s. [8] V. Movchan, L.G. Fedan, V.S. Gaidawakin, Metals Sci. Heat Treat. 29 (1987) 553 – 555. [9] S. Mukae, K. Nisho, H. Baba, Yasetsu Gakkai Rounbunshu 6 (1988) 22 – 29. [10] V.A. Bachin, E.A. Goritskaya, I.E. Tikhonova, Weld. Prod. 33 (1986) 16 – 18. [11] F.A. Calvo, A. Urena, M.A. Gomez, J. Mater. Sci. 24 (1988) 4152 – 4159. [12] A. Sunwoo, Scripta Metall. Mater. 31 (4) (1994) 407–412. [13] G.R. Kamat, Weld. J. 67 (6) (1988) 44 – 46. [14] A. Wisbey, P.G. Partridge, Mater. Sci. Tech. 3 (11) (1993) 945 – 953. [15] R. Dubrovsky, Advances in Welding Science and Tech., in: Proc. Int. Conf. on Trends in Welding Research., 1986, pp. 733–738. [16] G. Haufler and H. Mayer, Proc. Conf. High Temp. Alloys for Gas Turbines and Other Appl., 1986, pp. 801 – 810. [17] Y. Meahara, Y. Komizo, T.G. Langdon, Mater. Sci. Tech. 4 (1988) 669 – 674. [18] H. Ridley, Z.X. Guo, Mater. Sci. Tech. 3 (1985) 945–953. [19] B. Derby, E.R. Wallach, Metal Sci. 18 (1984) 427. [20] Y. Takahashi, T. Sakaki, H. Uzaka, Weld. World 27 (3) (1991) 100 – 113. [21] N. Orhan, PhD Thesis, University of Firat, 1996. [22] D.J. Klaphaak, Carbide Tool J. 19 (5) (1987) 14 – 16. [23] S.J. Sharp, M.F. Ashby, N.A. Felec, Acta Metall. Mater. 41 (3) (1993) 685 – 692. [24] E.R. Wallach, A. Hill, Acta Metall. 37 (9) (1989) 2425–2437. [25] I.W. Chen, A.S. Argon, Acta Metall. 27 (1981) 1759–1768. [26] P. Bross, H.E. Exner, Acta Metall. 27 (1979) 1013 –1020. [27] M. Tanaka, Acta Metall. Mater. 39 (7) (1991) 1548–1554. [28] D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, second ed., 1993.