Modelling solid-state diffusion bonding

Modelling solid-state diffusion bonding

Acta metall. Vol. 37, No. 9, pp. 2425-2431, 1989 Printed in Great Britain MODELLING 0001.6160/89 $3.00 + 0.00 Pergamon Press plc SOLID-STATE DIFFUS...

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Acta metall. Vol. 37, No. 9, pp. 2425-2431, 1989 Printed in Great Britain


0001.6160/89 $3.00 + 0.00 Pergamon Press plc


Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 342, England (Received 23 November 1987; in revised.form

1 February 1989)

Abstract-The increasing use of diffusion bonding as a commercial process has, over the last 15 years, been paralleled by a series of progressively more complex attempts to model the mechanisms and processes occurring during bonding. This paper describes a theoretical model for diffusion bonding which proposes a new and simplified void geometry. The mechanisms operating during diffusion bonding are based on those derived from pressure sintering studies, although the driving forces and rate terms for these mechanisms have been altered to allow for the quite different geometries of the two processes. Also included in this work is an analysis of the effect of grain size which can lead to an enhan~ment in the contribution to bonding from additional grain boundary diffusion and so may be particularly relevant when joining materials of fine grain size. The extent to which this new model offers significant developments over existing models is discussed; an initial comparison between experimental results and predictions from this new model shows that there is good agreement between practice and theory. R&m&Paralleiement a l’utilisation croissante de la soudure par diffusion en tant pro&de commercial, depuis les quinze dernibres anntes, plusieurs tentatives de plus en plus complexes on vu le jour pour modeliser les mecanismes et les processus mis en jeu dans la soudure. Cet article d&it un modele thborique de la soudure par diffusion qui propose une geometric des cavites nouvelle et simplifiee. Les mbanismes mis en jeu pendant la soudure par diffusion sont bases sur ceux que l’on tire des etudes de frittage sous pression, bien qu’on ait modifrt les forces matrices et les vitesses de ces mecanismes pour tenir compte des geometries tout a fait differentes de ces deux processus. On analyse Bgalement dans cet article l’effet de la taille des grains qui peut provoquer une contribution plus &levee a la soudure grace ii une diffusion intergranulaire supplementaire, et Ctre ainsi dun in&t particulier quand on veut souder des materiaux a petits grains. On discute tout les avantages que presente ce nouveau modele par rapport aux anciens; une premiere comparaison entre les resultats experimentaux et les previsions de ce nouveau modele montre qu’il y a un bond accord entre la theorie et la pratique. Z~~~~Die zunehmende Ve~end~g des Diffusions~ndens als technischer ProzeD war in den letzten fiinfzehn Jahren begleitet von einer Reihe von immer komplexeren Versuchen, die wghrend des Bondens ablaufenden Mechanismen und Prozesse im Model1 zu beschreiben. Die vorliegende Arbeit beschreibt ein theoretisches Model1 fiir das Diffusionsbonden, welches eine neue und vereinfachte Hohlraumgeometrie vorschlagt. Die wiihrend des Diffusionsbondens ablaufenden Mechanismen beruhen auf denjenigen, die aus Untersuchungen des Drucksintems abgeleitet wurden; ailerdjngs wurden die Terme der treibenden Krafte und der Geschwindigkeiten dieser Mechanismen gelndert, urn die ganz unterschiedlichen Geometrien der beiden Prozesse zu beriicksichtigen. Aubrdem wird der EinfluB der KomgrdBe betrachtet; diese kann zu einer Verstlrkung in der Bondbildung durch zusiitzliche Komgrenzdiffusion ffihren und so besonders wichtig werden, wenn Materialien mit kleinen Kiirnem zusammengefiigt werden. Die Verbesserungen dieses Modelles gegeniiber bestehenden Modeilen werden diskutiert; ein Vergleich von experimentellen Ergebnissen und Aussagen dieses neuen Modelles weist gute tibereinstimmung zwischen Theorie und Praxis auf.

INTRODUCTION Diffusion bonding is a solid-state welding process by which two prepared surfaces are joined at elevated t~m~rature and under applied pressure. Temperatures required for the process are usually in the range OS-O.8 of the absolute melting point of the material, pressures are typically some small fraction of the room temperature yield stress (to avoid macro~opi~ deformation), times can vary from a few minutes to several hours, and the two parameters defining the

tPresent address: Scientific Generics Limited, Kings Court, Kirkwood Road, Cambridge CB4 2PF, England.

surface condition (roughness asperity height and roughness wavelength) are typically of the ratio 1: 50. The precise values for each of these variables for a bond in a particular material are chosen such that, ideally, parent metal microstructures and properties are attained after bonding, and that there is no gross macroscopic deformation (although in practice this may occur up to a few percent). The initial contact of the asperities on the prepared surfaces creates a series of voids which subsequently may act as a weakness unless they and any resulting planar boundary are removed entirely. Therefore the aim of this work has been to re-examine the modelling of diffusion bonding and so produce a




better understanding of the mechanisms and processes that are needed to join two materials in the solid state. The assumptions of geometry in earlier work [I] have been reconsidered and con~quently a new geometry, based upon a more realistic experimentally observed void shape (an ellipse), has been adopted. 30th driving forces and rates of the mechanism resulting in bonding have been modified in the light of this new geometry. In addition, a simple statistical analysis has been included to allow bonding predictions to be made for fine-grained materials in which the contributions to bonding from grainboundary sources can be enhanced. This work was initially reported in 1984 (see Acknowledgements). PREVIOUS


The first phenomenological model for diffusion bonding was by King and Owczarski [2] who proposed three bonding stages. The first was plastic deformation of the asperities leading to large-scale contact of the two surfaces and so forming the equivalent of a grain boundary. The second stage was removal of the majority of the voids at the interface as well as a simultaneous migration of the interface out of a planar orientation and away from the voids. The final stage was the elimination of the remaining isolated voids by volume diffusion. Hamilton [3], also working on the titanium alloy TidAUV, attempted to quantify the initial plastic deformation stage by representing surface roughness as a series of long ridges which collapsed and so eliminated the voids between them. A prediction of bonding rate was possible by relating the bulk strain in a representative collapsing ridge to experimental creep data; favourable agreement was found between experiment and theory. Garmong er ai. [4] extended Hamilton’s ridge analysis by modelling a representative ridge as a series of horizontal slices and summing the response of each slice to the applied stress, Their model further included diffusion considerations which led to predictions for the complete removal of their proposed long- and short-wavelength surface roughness. Again good agreement was claimed between experiment and theory although, as previously, the only alloy examined was Ti-GAlAV. The first fully comprehensive quantitative model was by Derby and Wallach [I, 51 who proposed that some or all of the same seven mechanisms that operated in pressure sintering also may occur in diffusion bonding. The original surface condition was modelled as a series of long parallel straight-sided ridges using just a single wavelength. Bonding was assumed to occur in three stages, each with its own geometry; initial instantaneous plastic flow, a second diffusion-dominant (hence time-dependent) stage based on voids with a small aspect ratio (long length and small height) and a third stage in which elimination of voids of cylindrical geometry took place.



Difficulties arose in this approach in finding exact solutions, first for the plastic deformation (use of slip-line field theory for a punch pushed into a flat surface} and the creep (extrapolation from considering the two extreme limits of elastic and of plastic behaviour) mechanisms and, secondly, for material redistribution particularly in the first two stages of bonding when the straight-sided void geometry was used. Moreover, the change in geometry from the straight-sided void to the circular void required for the final stage of bonding resulted in a discontinuity in the predicted bonding rate. Despite these drawbacks, good agreement between experimental data and theoretical predictions was obtained, confirming that there was considerable merit in using an approach based on the inclusion of all seven pressure sintering mechanisms and summing their relative contributions as bonding proceeded. Qualitative modifications to this model, including brief consideration of the effects of surface oxides and contaminants and the rapid closure of small voids by diffusional sintering, were subsequently proposed by Allen and White [6]. The most recent diffusion bonding model is that by Pilling ef al. [?I in which it is suggested that an alternative to the sintering approach is to use a diffusive creep model based on original work by Chen and Argon [8]. From the start of bonding a circular void geometry is used, based upon the assumption that contributions to bonding by surface diffusion mechanisms are always sufficient to maintain a circular void. However, this is not necessarily the case, as micrographs of void cross-sections often show (including micrographs in the paper by Pilling et al. [7]). Thus the Filling model appears to have been developed especially to model the bonding of fine-grained su~rplastic alloys (for which creep mechanisms might be expected to dominate) and good agreement for bonding superplastic Ti-4Al-4V is claimed. An analysis of the effect of grain size is also included by dividing the material surrounding a void into a series of parallel slices each of thickness equal to the grain size. In this way, the contribution from each boundary intersecting the void could be included to give the overall grain boundary diffusion rate. The present work was undertaken in order to overcome some of the approximations and assumptions inherent in the previous models. If experimental evidence of void shape is considered [7, g-111, it can be seen that an improved approximation to initial void shape might be an ellipse. Moreover, if an ellipse is used in modelling diffusion bonding, the necessity for separate stages (with their own separate geometries and the possibility of a discontinuity in bonding rate, as has been observed [5]) is removed. Also, more exact analyses of both plastic deformation and creep are possible. Thus the more rigorous approach to creep, as adopted by Pilling et al. 171,can be combined with the indusion of all seven possible bonding mechanisms, as originally proposed by







Derby and Wallach [l]. Moreover, the simplified geometry reduces the complexity of the equations used and so might provide a better basis to tackle more complex systems than the current limitation of modelling bonding only between similar single-phase materials. Geometry considerations

A surface preparation for diffusion bonding (e.g. grinding) produces a series of long parallel ridges (but not necessarily straight-sided as in the Derby and Wallach model [l]), which are typically 0.2-2 pm high for a groove width (surface roughness wavelength) of 3C70 Bum(Fig. I). The model assumes that when the surfaces are brought together, the two sets of ridges touch tip-to-tip, creating between them a series of infinitely long parallel cylinders of elliptical cross-section (Fig. 2). The closure of a section perpendicular to the longitudinal axis is modelled and, by invoking symmetry, just one quarter of an elliptical section needs to be considered. The assumption of peak-to-peak contact will resuh in calculations of maximum bonding time; in practice, shorter times might be observed. The parallel alignment geometry also creates conditions of plane strain so allowing the length of the void channels to be ignored. The alternative perpendicular alignment of the two surfaces is not considered here, partly because it does not lead to significantly greater bond strength in practice and partly because of the difficulty in modelling the resulting geometry. In this new model, it is assumed that bonding of a single wavelength of surface roughness occurs in two stages (Fig. 2). Stage 0 (retaining the convention used in pressure sintering) is plastic deformation, which is said to be instantaneous so not allowing the timedependent diffusion or creep mechanisms to contribute to bonding. Plastic deformation ceases when the contact area at the interface is sufficient to support the applied load, i.e. the localised stress on a ridge drops below the yield point of the material. Stage I is time-de~ndent and one or more of the six remaining diffusion and creep mechanisms contribute to bonding. Unlike the earlier model of Derby and Wallach [l], there is no need for a separate Stage II geometry (circular-section voids) since elliptical voids can attain circular geometry naturally without a discontinuity in bonding rate. When circular geometry is achieved, there is no driving force for the surface source mechanism and so cont~butions

Fig. 1. Modelled surface-long

parallel ridges




Fig. 2. Stages of void closure: (a) initial geometry, Stage 0, showing maximum and minimum radii of curvature; (b) void geometry during Stage I showing dimensions of the unit cell.

from these will cease. However, the use of elliptical geometry allows these surface source mechanisms to reactivate naturally if the aspect ratio of the voids alters from unity as a result of contributions to bonding from interface sources and creep mechanisms which can operate continuously throu~out Stage I regardless of void geometry. This again offers an improvement over previous models. Using the manner of material redistribution indicated in Fig. 3, it is possible to derive a series of expressions for the rates of each mechanism and the rates of geometric (void shape) changes for a given set of process variables. The geometric changes are expressed in terms of bonded area growth and void height change. The neck radius of the ellipse is derived in terms of the major and minor semi-axes of the ellipse and its rate of change does not have to be independently as in the previous model [l]. The rate of void shrinkage is expressed in terms of the rate of change of bonded area, ci, and the rate of change in void height, h. The former could also be expressed in terms of the rate of change of the major semi-axis, (;, of the ellipse.

Fig. 3. Routes of material transfer: (a) surface source mechanisms; (b) interface source mechanisms; (c) bulk deformation mechanisms.




As stated above, the symmetry of the bondline is such that only one quarter of a void needs to be modelled in order to follow how bonding proceeds. This forms the basis of the approach which follows. Also, the plane strain geometry adopted assumes unit depth perpendicular to the original grinding marks.

MECHANISMS OPERATING As in the previous model by Derby and Wallach [I, 51, this new model considers that the m~hanisms occurring during bonding are analogous to those in pressure sintering 112, 131, namely: 0. plastic yielding deforming

1. 2. 3. 4. 5. 6.

an original contacting asperity. surface diffusion from a surface source to a neck; volume diffusion from a surface source to a neck; evaporation from a surface source to condensation at a neck; grain boundary diffusion from an interfacial source to a neck; volume diffusion from an interfacial source to a neck; power-law creep.

The routes of material transfer of these mechanisms have been summarised in Fig. 3. The driving force for mechanisms 1 to 3 is a difference in surface curvature, now assumed to be the difference in maximum and minimum radii of the ellipse. For the interfacial source mechanisms, the rate is determined by the chemical potential gradient along the bond interface, which is in turn dependent upon the void neck radius (minimum radius of curvature of the ellipse) and the applied pressure. Power-law creep and plastic deformation are similarly driven primarily by the applied pressure, although the latter ceases above a critical contact area as described below.

c Q

Fig. 4. Plastic deformation slip line field for collapse of asperities (after Johnson er al. [14]).

2. no discontinuity in the slip-line field across the bond interface; 3. the required notch radius of curvature is equal to the neck radius of the void. Using the analysis of Johnson et al. [141, the stress is longitudinally distributed across the neck and this stress distribution is given by qyj=2K[l




Thus the maximum stress is predicted to be at the centre of the bonded area, i.e. at y = a. From this, it can be shown (see Appendix 1) that the bonded area after yield, +id, is given by

The void height after plastic yield, h$&, is calculated using a volume conservation argument (see Appendix 1) to give

(3) Plastic deformation (Stage 0) The contact area of asperities, though initially small, will rapidly grow until the applied load can be supported, i.e. the local stress falls below the material’s yield stress. An analysis of the defo~ation of a s~met~cally deep-notched specimen has been proposed by Johnson et al. 1141such that the plastic region is localised around the core-see Fig. 4. This is a close approximation to asperity contact and can be used to calculate the bonded area for a given applied load after yielding. There are three assumptions necessary in this approach: 1. no work hardening-a

reasonable approximation because of the high temperature of bonding and the microscopically small amount of plastic deformation that occurs;


~l~usion from surface source to a neck (Stage I) ~~hanisms 1,2 and 3 are all driven by differences in chemical potential arising from changes in curvature of a free surfaces. Matter is transferred from the point of least curvature to the point of greatest, this being the sharp neck of the void at the bond interface. The rates of these mechanisms will hence tend to zero as the aspect ratio of the void tends to unity, i.e. as the void changes from elliptical to circular crosssection (assuming plane strain, hence unit depth, as stated previously). The driving forces for mechanisms 1 and 2 were derived previously [I] from the Kuczynski analysis

HILL and WALLACH: MODELLING SOLID-STATE DIFFUSION BONDING [ 151of the sintering of parallel wires. The volume flux was thus given by 3=--.

2ADR y rkT

This expression must be adapted for each of the mechanisms 1 to 3. For surface diffusion (mechanism 1), the area A in the assumed two-dimensional geometry becomes a thin surface layer of thickness 6, while the diffusion coefficient becomes that for surface diffusion, D,. To allow for the previously discussed decrease in rate as the aspect ratio of the void tends to unity, a reduction factor 1

=~W',Y I





[ h2





Note that r in equation (4) has been replaced by r, for the assumed geometry. For volume diffusion (mechanism 2), the area through which the flux occurs is the area of the neck, while the appropriate diffusion coefficient is that for volume diffusion. Hence equation (4) now becomes ci=-..-2RD,y kT



For mechanism 4, the area through which flux occurs is a boundary layer of thickness 6, and, if the appropriate diffusion coefficient is included, the expression becomes

and similarly for mechanism 5


‘, ‘h (---> must be included. This factor equals zero when the maximum and minimum radii of curvature are the same, i.e. for a cylindrical void. It can be rewritten in terms of the major and minor semi-axes of the ellipse and included into equation (4) to give 3

and the volume transfer rates were determined in the previous model [1] by adapting original work by Johnson [17] to give




c __h

[ h*



p., has been halved to include only its contribution to matter transfer within the cell being modelled since the grain boundary being considered is coincident with the edge of the cell, i.e. is also contributing to the cell quadrant below that being modelled. The rates of change of bonded area and void height are the same for both mechanisms 4 and 5 and are given by

-_‘I a


ci,= -a[b(z-l)+a]





The expression for the rate of mass transfer by evaporation and condensation (mechanism 3) was derived by Derby [5] from work by Kingery and Berg [16] which, when combined with the reduction factor, gives

The rates of void closure are identical for all three mechanisms and are given by


where the subscript i denotes the particular mechanism (1, 2 or 3). Equations (8) and (9) are derived in Appendix 2. Grain boundary dzjiision from an interface source to a neck (Stage I)

The grain boundary mechanisms, 4 and 5, both have the same source and sink although their routes of material transfer are different. The interface is considered to be a normal high-angle grain boundary

where the subscript i denotes the particular bonding mechanism (4 or 5). Detailed derivations are shown in Appendix 3. Power law creep mechanism (Stage I)

Because of the high temperatures typically used to effect diffusion bonding in reasonably short times, microcreep of asperities also contributes to void closure. Macrocreep is undesirable because of specimen distortion and so the process conditions are chosen to avoid this for any given material. Approximate plasticity solutions for elliptical hole growth were developed by McClintock [18] from work by Berg [19] on the deformation of an elliptical hole in a viscous material. Hancock [20] has further extended the McClintock work to describe the growth of an elliptical hole in plane strain under power-law creep. It is this last analysis that has been modified for use in the current model although, because it was developed for the growth of a single hole in an infinite material, it is not an exact solution for a diffusion bond geometry where there will be an interaction between neighbouring voids. Such interaction decreases as bonding proceeds and the voids become smaller in size. The radius of an ellipse was defined by Berg [ 191as the average of the major and minor semi-axes. Using this and the terms derived by Hancock [20] for, firstly,




the relationship between original and new ellipse radii for a given strain and, secondly, the relationship for the change in eccentricity for the same strain, an equation can be determined (see Appendix 4) for the rate of change of height of an ellipse.

The two equations hence become

(11) and (12) for pd and p5





and (15)

If a similar derivation is carried out for the rate of change of the major semi-axis, it implies that this is increasing, i.e. that the bonded area is decreasing. This would be expected in the case of a single void in an infinite material; a decrease in aspect ratio, and a decrease in void volume. However, where there are a series of voids interacting, such as at a bond interface, this behaviour will result in two possibilities. Firstly, the material between the voids may be forced away from the bond interface in order to accommodate the decreasing bonded area; this is in direct opposition to the applied stress and so is energetically unlikely. Secondly, as the voids get longer, the bonded area will remain constant but the void centres will move apart. Not only is this also unlikely but it is not found ex~rimentally; examinations of bonded specimens reveal the void spacings to be very similar to the original surface wavelengths. In light of the above, power-law creep has been modelled as a two-stage process. The first part is a change in void height as detailed above. The second stage is a filling-in of the void by assuming that the ridge height decreases by the same amount as the void height, and the displaced material is moved into the void. This increases the bonded area, giving the rate of change as dc=

-~[b(~- I)+a].


where (90-4) is the angle between the applied pressure and each grain boundary under consideration (see Fig. 5). vd is divided by 2 if cfiequals 0 or n/2 radians to allow for the fact that only one quadrant of a respresentative void is modelled and so at these two values of 4, at which two quadrants coincide, just half of the flux should be included. All additional grain boundaries are assumed to meet the free surface perpendicularly; qualitative statistical examinations of micrographs through diffusion bonds have shown that the majority of intersections are close to right angles. The number of grain boundaries intercepting the ellipse in the modelled cell is assumed to be equivalent to a quarter of the circu~erence of the ellipse divided by the grain size (plus one). Using the geometrical construction shown in Fig. 5, the equivalent circle angle, 8, is given by nb GS





where nb = the number of the grain boundary. angle 4 is then given by



fitant? (E )


See Appendix 4 for the complete derivation.

!a ;

Equivalent Circle


Eflect of grain size

In the initial stages of bonding, it is probable that a void will be intersected by grain boundaries other than those formed by the contacting surfaces. Accordingly, more than one grain boundary wili contribute to void elimination, as was allowed for by Pilling ef a!. [7]. However, since the chemical potential driving force for the grain boundary diffusion mechanisms is partly dependent upon the angle between the applied pressure and a particular grain boundary, its actual contribution will be dependent on its precise orientation with respect to the applied pressure. The driving force will be a maximum when the two are perpendicular and a minimum when parallel. A statistical approach has, therefore, been developed leading to an increase in the grain boundary diffusion contribution and so allowing for a change in bonding rate with grain size.

Fig. 5. Representative void for a material with grain size smaller than the void. In this example, diffusion from six additional grain boundaries enhance void closure. The geometric construction for obtaining the angle between a grain boundary and the applied stress is also shown.




and the radius of curvature at the point of intercept, r,, is

Temp. 500 1.0




II ,



l°C) BOO








I, =

(21) >Hence the angle, (90-4) between the grain boundary and the applied pressure can be determined and so values of p4 and p5 calculated for each grain boundary intersecting the surface of the void. These can be summed to give an overall rate for the contribution from grain boundary diffusion to void closure. (


THE COMPLETE MODEL It is assumed that the seven possible bonding mechanisms listed above operate in such a way that their contribution to bonding can simply be summed together to arrive at the overall amount of bonding. By inputting given process conditions and data for any particular material, it is possible to predict the final bonded area that is likely to result, as well as observe the contribution to void closure from each individual mechanism. This can be useful because once a dominant mechanism has been identified, the bonding conditions or material properties can be optimised to enhance bonding. The new model described in this paper represents an advance on previous models in several areas. When compared to the Derby and Wallach model [l, 51, the new model has a number of improvements, namely:


better slip-line field analysis of plastic deformation; -a more precise modelling of the geometry, which is more representative of the voids observed in practice; -a more realistic analysis of the creep void closure contribution; -the elimination of the discontinuity that previously existed towards final void closure [5]; -a greater flexibility for mechanisms to reactivate should the geometry permit. The Pilling model [7], a development of the Chen and Argon [8] work on the coalescence of voids, omitted the surface diffusion contributions by assuming a somewhat unrealistic void shape and, as stated previously, for some materials this can be a major factor in bonding. Furthermore, their model should over-predict the contribution from the grain boundary mechanism since the additional boundaries are assumed to be contributing equally (no grain boundary orientation effect is included). Notwithstanding these drawbacks, however, it is recognised that the Pilling [7] model was developed primarily to allow predictions when diffusion bonding the superplastic alloy Ti-6A14V; it was not intended to be generally applicable to all materials as the case of the model of Derby and Wallach [I, 51.







Fig. 6. Predictions of the mainframe computer model for diffusion bonding copper under the process conditions shown (one process condition, in this case temperature, can be varied). For all mainframe maps, the number in each area identifies the dominant bonding mechanism: 0 plastic flow; 1 surface diffusion (surface source); 2 volume diffusion (surface source); 3 evaporation/condensation; 4 grain boundary diffusion (interface source); 5 volume diffusion (interface source); 6 power-law creep.

Implementation of the model

Modelling is undertaken using an iterative computer program which produces two alternative forms of graphical output. In the first (see Fig. 6), the axes are fractional-bonded area against one of the five process parameters (temperature, pressure, surface roughness height, surface roughness aspect ratio or grain size) with time plotted as a variable contour. When one process parameter is set as a variable displayed on the x-axis, the remaining four are set to specific values which are input with the other data. The resulting graphs are diffusion-bonding maps with each discrete area indicating a dominant mechanism. On these maps, the number Q-6) in each area identifies the dominant bonding mechanism; the seven bonding mechanisms were listed earlier. The second form of output (see Fig. 7) is less complex, the axes being fractional area bonded against time. The advantage of this format is that it is possible to see clearly the relative rates of the various contributing mechanisms. With the first type of graph, this information is not shown and, in an extreme case, the dominant mechanism might be contributing only marginally more than one seventh of the total bonding rate. For this second form, all five process parameters must be defined in the input data. Using the model, it is possible to predict the time necessary to achieve a certain amount of bonding under given bonding conditions. The model is applicable to any single phase, similar-to-similar metal bond provided the appropriate materials’ data are known and providing the effects of surface oxides and contaminant films are ignored.



Temp. PC) 600

700 I


600 I

900 I

1000 1



50 Bonding

100 time




Fig. 7. Predictions of the microcomputer model for diffusion bonding copper under the process conditions shown. For all the microcomputer maps, the contributions to bonding from the three mechanism families are indicated by: T-total amount of bonding; G-interface source family- S-surface source family; C+reep


and plastic deformation.


I ,


0, I 0.7

I 0.6

I 0.6


I 1.0


Fig. 8. Predictions of the model for diffusion bonding copper compared to the experimental results of Ohashi and Hashimoto [24] for a bonding time of 4min.

VeriJication of the model

Both the previous models of Derby and Wallach [l, 51 and Pilling [7j, claim good agreement with experimental data. However, such agreement must he dependent upon three factors. Firstly whether the model itself is indeed precise, secondly the accuracy of the input data used by the model and, thirdly, the accuracy or the reliability of the experimental results to which it is being compared. The Derby and Wallach model, in particular, has heen rigorously compared to two materials (copper and iron) and good agreement was reported [9,21,22]. A detailed analysis has heen made [23] of precisely which experimentally measured surface roughness parameter most accurately correspond to the surface roughness parameters used in the model for representing surface wavelength and peak-to-valley height. It was evident that it is not possible to use the same single parameter in all circumstances; a more detailed discussion of the most appropriate parameter is given in Appendix 5. For the purpose of the verification in this work, averaged values (in each case based upon several similar preparations for a given material) have heen used in the examples that follow. As shown in Appendix 5, these averaged values do not necessarily give the best fit but, in the absence of the full original data from previous work, their agreement is consistently good over a range of bonding conditions and materials. In the case of the Pilling (71 comparison with experimental data, the initial void size was assumed to he comparable to the grit size used in surface preparation. This is likely to overestimate the actual void size by approximately an order of magnitude and so the Pilling model will overestimate the extent of bonding. A direct comparison between previous models is, therefore, not straight-forward. Nonetheless, results from the new model presented in this paper have

been compared to those from the model of Derby and Wallach [l, 51 using the modified description of surface roughness and good agreement, within experimental error, has been found. When prediction from the model are compared to experimental data from three sources [ 11,21,24] for the extent of bonding in three materials (f.c.c. copper, b.c.c. a-iron and f.c.c. y-iron), agreement to within 10% is observed (Figs 8-l 1). The materials data used for these three materials is given in Table 1. Given the inherent errors in measuring the actual bonded area and surface roughness parameters, combined with small errors in published materials’ properties data, this agreement is extremely encouraging. Similar correlation has been found with a range of other pure and alloyed materials (e.g. Type 316 stainless steel in Fig. 12). This demonstrates that the model would seem to be capable of realistically Temp. (“C) 600



0.5 Homologous




Fig. 9. Predictions of the model for diffusion bonding cc-iron compared to the experimental results of Derby [9] for bonding times of 10min (A) and 60 min (a).





Temp. 1°C) 1000



Temp. (‘Cl 1200

* to







0.66 Homologous



Fig. 10. Predictions of the model for diffusion bonding y-iron compared to the experimental results of Derby [9] for









bonding times of 10 min (A) and 60 min (a).

Fig. 11. Predictions of the model for diffusion bonding En8 steel compared to the experimental results of Thornton [ 1l] for a bonding time of 20 min.

predicting bonding behaviour for any materials which are not unduly complicated by the presence of stable oxides or contamination and which do not form intermetallic phases. A detailed investigation of the effect of grain size variations on the rate of bonding has revealed that there was a significant increase in the contribution from grain boundary diffusion once the grain size is less than approximately 10 pm. This agrees with the suggestion by Allen and White f6] that a fine-grained material will bond only moderately faster than a coarse-grained material, even in an optimum case.

The overall magnitude of the increase in bonding rate with decreasing grain size is dependent upon the particular materials and bonding conditions. An example of a relatively large increase due to a finegrain size is shown in Fig. 13 for copper, but for y-iron, Fig. 14 (under equivalent bonding conditions, i.e. the same fractions of absolute melting point and of yield stress), the effect is smaller. Such fine-grain sizes (less than IOpm) are found in superplastic materials which, increasingly, are being used in combined diffusion bonding and su~~~astic forming operations.

Table 1. Data for model (see Ref. 1251) Value Parameter Atomic volume Burgers vector Density Melting point Shear modulus at 300 K





nm3 nm kg m-l K GPa

1.18 x low2 0.248 7.87 x lo3 1810 64 4.48 X lo-

1.21 X 10-2 0.258 7.69 X IO’ 1810 81 5.05 X 10-b

1.18 x 1O-2 0.256 8.96 x 10’ 1356 42.1 3.97 X 10-4

3 X IO-’

2 X 10--x

2.17 0.756



Temperature coefficient of shear modulus Yield stress, normalised in terms of shear modulus Surface energy Interface energy





Jm-’ Jme2


m*s-’ kJ mol-’ m*s-’ kJ mol-’ m2 s-I kJ mol-’ nm nm

2 X 10-q 251 I.1 X to-* 174 10 241 0.1 0.1

I .75


Volume pre-exponential Volume activation energy Boundary pre-exponential Boundary activation energy Surface pre-exponential Surface activation energy Boundary layer thickness Surface layer thickness

1.8 x 10-S

270 7.5 X 10-4 159 0.4 222 0.1 0.1

2 X to- 5 197 5 X 10-I 104 7 X 10-6 79.1 0.1 0.1

Pmw law creep

Activation energy Constant Exponent

as volume diffusion activation energy m


7 x 10” 6.9

4.3 X 10’ 4.5

7.4 X 10’ 4.8







Temp. I’C) 600



A new model for diffusion bonding has been proposed with several features which are improvements over earlier models. The use of an elliptical void shape:

(a) is more















Fig. 12. Predictions of the model for diffusion bonding Type 316 stainless steel compared to the results of Hill (231for bonding times of 5 min (A), 20 min (m) and 60 min (a).

f El

0.6 -




z x L 0.2



100 Grain size


Fig. 13. Predictions of the model for diffusion bonding copper showing the effect of grain size. The time contours are for 1, 10 and 60min.

realistic than straight-sided or cylindrical voids which have been used previously; W allows more exact analyses of the bonding mechanisms, in particular plastic deformation and power law creep; (c) permits reactivation of the surface diffusion mechanism whenever the void shape becomes non-cylindrical Cd) is generally less complex. The effect of grain size has been allowed for by including the contributions to bonding from grain boundaries intersecting the original voids, as has been done in the past [7]. However, in the proposed new model, allowance has been made for grain boundary orientation. This is recognition of the effect on the chemical potential driving force for grain boundary diffusion of the angle between the applied pressure and a grain boundary. The new model is capable of accurate predictions when compared to experimental data although, as in all modelling, it is necessary to consider carefully the accuracy of the input to the model. The model has shown that seven mechanisms can contribute to bonding; the extent to which any one contributes is a function of both the material and the process conditions. Implemented as a computer pro~amme, the model rapidly enables a precise prediction of the extent of bonding to be determined, thereby virtually eliminating the need to experimentally optimise bonding conditions. It can be applied to any similar-to-similar material bond providing that the required data are known for that material. This new simplified model also should facilitate the development of more complex models capable of considering dissimilar joining or two-phase material bonding. ~c~~~~ledge~~rs_The authors would like to thank Professor R. W. K. Honeycombe for provision of laboratory facilities, the University of Cambridge computing Service for the use of the IBM 3081 computer and related peripherals, Dr E. Yoffe, Dr J. Hancock, Dr B. Derby and Dr S. Timothy for their helpful discussions. The financial support of both the Science and Engineering Research Council and The Welding Institute is gratefully acknowledged. This work was reported orally in 1984 at The Institute of Metals meeting on Diffusion Bonding.

0 1

I 10



Grain size (microns)

Fig. 14. Predictions of the model for diffusion bonding y-iron showing the effect of grain size. The time contours are for 1 and IOmin.

1. B. Derby and E. R. Wallach, Mei& Sei. 16,49 (1982). 2. W. H. King and W. A. Owczarski, Weld. J. Iies. Suppl. 46, 289 (1967).



3. C. H. Hamilton, Titanium Science and Technology (edited by R. I. Jaffee and H. M. [email protected], Vol. I, pp. 625648, Plenum Press, New York (1973). 4. G. Garmong, N. E. Paton and A. S. Argon, MetaN. Trans. 6A, 1269 (1975). 5. B. Derby and E. R. Wallach, Metal Sci. 18,427 (1984). 6. D. J. Allen and A. A. L. White, Proc. Conf. on The Joining of Metal: Practice and Performance, Warwick. institution of Metallurgists (1981). 7. J. Pilling, D. W. Livesey, J. B. Hawkyard and N. Ridley, Metal Sci. 18, 117 (1984). 8, I.-W. Chen and A. S. Argon, Acta metall. 29, 1759





and since Von Mises assumed that


Equation (2) is solved numerically by letting the equation have the form

(1981). 9. B. Derby and E. R. Wallach, J. Mater, Sci. 19, 3140

(1984). 10. V. S. Gostomel’skii, E. S. Karakozov and A. P. Ternovskii, Soviet Phys. Dokl. 25, 125 (1980). Il. C. E. Thornton, Ph.D. thesis, Univ. of Cambridge (1983). 12. M. F. Ashby, Acta metall. 22, 275 (1974). 13. D. S. Wilkinson and M. F. Ashby, Acta metall. 23,1277 (1975). 14. W. Johnson, R. Sowerby and R. D. Venter, Plane Strain Slip Line Fieldsfor Metal Defurmafion Processes

119- 121. Pergamon Press, Oxford (1982). 15. G. C. Kuczynski. Tram Am. Inst. Min. Engrs 185, 169

Initially, the value of a, _ , is set to an arbitrarily small value (say, 1 x 10-‘sm). The solution then converges (within twelve iterations) to a constant value equal to aYield. The void height can be calculated using a volume conservation argument. Thus, to obtain the void height after plastic yield, h,, , the final and initial volumes of material are equated to give

(1949). 16. W. D. Kingery and M. Berg, J. appl. Fhys. 26, 1205

(1955). 17. D. L. Johnson, 1. appl. Fhys. 40, 192 (1969). 18. F. A. McClintock, J. appl. Mech. pp. 363-371 (1968). 19. C. A. Berg, Proc. 4th U.S. National Cong. Appl. Mech. A.S.M.E., Vol. 2, pp. 885-892 (1962). 20. J. W. Hancock, Metals Sci. 10, 319 (1976). 21. B. Derby, Ph.D. thesis, Univ. of Cambridge (1981). 22. B. Derby and E. R. Wallach, J. Mater. Sci. 19, 3149 (1984). 23. A. D. Hill, Ph.D. thesis, Univ. of Cambridge (1984). 24. 0. Ohashi and T. Hashimoto, J. Japan Weld. Sac. 45,


In practice, a similar iterative approach to that used for equation (2) needs to be adopted to allow convergence to the void height at plastic yield, namely




b - i(b -a,) J

47; J. Japan Weld. Sot. 45, (1976). 25. H. J. Frost and M. F. Ashby, Deformation mechanism Maps, Pergamon Press, Oxford (1982).



Geometry Derivations For Surface Source Mechanisms



Plastic Deformation

Stress distribution between the edge of the void and the midpoint of the neck, i.e. from y = 0 to a, is given by (see Fig. 4) *,,=?K[l





By integration between limits y = 0 and y = a, the mean stress at the interface is

(22) The driving force is the applied pressure. Thus, if the bonded area within the unit cell is width a, then the interfacial pressure can be defined as DL _I p,=“_L (23) a

By definition, the shape of the void changes while the volume remains constant. The volume of matter transferred [see Fig. 15(a)] is half of a thin layer




1 ;

6V = f [h&z + cSh + 6&h].

(25) (26)

The term 6&h is negligibly small compared

to the other terms and so has been neglected in equation (26). Since the material is actually moved in the manner shown in Fig. IS(b), the volumes of the initial and final voids can be equated $h(c+&)=;c(h



Hence h& = cd h.


Substituting expression (28) into equation (26) gives

Equating F, and 6, the bonded area necessary to support the load can be shown to be Fb -_y


If this is differentiate (24)


with respect to time and rearranged 6 =:

4” xc .






-L ff-

Geometry derivation From Hancock [20]



a) Sh


For Creep mechanisms



i and




1_ 6h

If substitutions are made for (7, o, and a, and equations (35) and (36) are then differentiated with respect to time (37)

Fig. 15. Material redistribution for surface source mechanisms: (a) the thin layer of material, half of which is moved; (b) the actual change in geometry due to the re~st~bution.

and (38)


The volume of the void is constant and is given by


VCDnll = f eh = %(bh - ah).


which can be differentiated to give This can be differentiated rearranged to give

with respect

to time and



where R and M can be expressed in terms of c and h such that






c-h M=--. c+h

The substituting R, h4, & and h? into (40) gives APPENDIX



Geometry 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. If a layer of material of thickness 6h is moved into the void, then the volume of material transferred is given by 6V = a&h.


The value of ci can be calculated by a volume conservation argument in a similar manner to equation (14) in which, because the volume of material in the unit cell remains constant

If this is differentiated with respect to time, then P=ah’.

Rearranging and changing the sign, since h is decreasing, gives h’=


a The volume of material in the unit cell equals


Vmmt=bh +h.

which can be arranged to give -

Substituting c = b -a, differentiating with respect to time and rearranging gives d=



Since c = b -a, equation (33) can be expanded and then differentiated with respect to time 0=66--~6b+$kh+~6a

VCDnl, = bh - ; ch.




The model uses a single wavelength and a single amplitude (peak-to-valley height) to describe the surfaces being bonded. In earlier work [23], the various experimentally measured surface roughness parameters were considered in order to determine which would be most appropriate to use. For wavelength, either the average, A,, or the root mean square (r.m.s.), !9, can be used since the model is not particularly sensltlve and . ^ to .small changes . in wavelength . _ also tor many surtaces these two wavelengtns are not very





d = rate of change of a with time a+, = magnitude of a after plastic yield A = area through which diffusive flux occurs b = width of unit cell being modelled

Temp. (VI 600



c= D= Da = D. = 0: = GS =




0.5 Homologous



Fig. 16. Predictions of the model for diffusion bonding a-iron compared to the experimental results of Derby [9] for bonding times of IOmin (A) and 60min (0).

different in magnitude. However, there are significantly more choices for peak-to-valley height and the value of this parameter is more critical. It was concluded [23] firstly that extreme values (e.g. the maximum peak-to-valley height) are not representative and secondly that more than one parameter may be needed (for example, if a surface has been ground or polished rather than prepared with a cutting tool such as a lathe). In general terms, it is suggested that the optimum or most representative amplitude parameter for a ground surface is twice the r.m.s. roughness, 2R,, while that for a lathed surface is the average roughness depth, R,,. The effect of choosing the most appropriate amplitude parameter can be seen, for example, by comparing Fig. 16 with Fig. 9, the difference being that R,, was used in the former and 2R, in the latter. The result is improved agreement between the experimental data and predictions from the experimental model. Thus it is possible that the comparisons shown in Fig. 8-14 might be improved if the full original surface metrology data were available, so allowing the appropriate amplitude parameters to be used. It must be noted, however, that many of the benefits from modelling lie in the relative predictions as various parameters are altered and not in absolute values alone. Clearly, improvements in describing surfaces (especially those with complex geometries such as produced by grinding) could lead to slightly better predictions. Nonetheless, any modelling invokes assumptions and the averaged values used at present appear to give good agreement over a range of materials, their preparation and bonding conditions.

NOMENCLATURE a = length of bonded interface, equivalent . . .^ bonded area if umt thickness considered

major semi-axis of ellipse diffusion coefficient boundary diffusion coefficient surface diffusion coefficient volume diffusion coefficient grain size h = height of unit cell, equivalent to minor semiaxis of ellipse /i = rate of change of h with time h y,cld= magnitude of h after plastic yield k = Boltzmann’s constant K = factor in Von Mises yield criterion m = power-law creep exponent M = eccentricity of ellipse &f = rate of chanee of M with time M, = original eccentricity of ellipse nb = the number of the grain boundary under consideration P = applied bonding pressure P, = pressure at bond interface P, = vapour pressure r = radius of curvature re = radius of curvature on major semi-axis rh = radius of curvature on minor semi-axis r, = radius of curvature at point of emergence of grain boundary R = average ellipse radius R = rate of change of R with time R, = original average ellipse radius T = temperature of bonding (degrees Kelvin) I’,,,,, = volume which remains constant 3 = rate of change of a volume f V) with respect to time y = distance across neck between voids y = surface energy 6, = boundary layer thickness S, = surface layer thickness 6c = change in c 6h = change in h 6V = volume of material being transferred t = average strain i = rate of change of t with time 0 = equivalent circle angle n = pi = 3.142 p = density of material cY= yield stress ca = stress along interface us = stress perpendicular to interface do = orientation of grain boundary with respect to applied stress Q = atomic volume



1, 2, 3, 4, 5, 6, 7 as subscripts


identify the bonding