- Email: [email protected]
Effects of surface and boundary diffusion on void growth
Acra merall. mater. Vol. 43, No. 6, pp. 2493-2500, 1995 Copyright 0 1995 Elsevier Science Ltd 0956-7151(94)00411-O Printed in Great Britain. All rights reserved
Pergamon
0956-7151/95-$9.50+0.00
EFFECTS
OF SURFACE AND BOUNDARY ON VOID GROWTH M. D. THOULESSt
IBM Research
Division,
T. J. Watson
Research
DIFFUSION
and W. LINGER Center,
Yorktown
Heights,
NY 10598, U.S.A.
(Received 20 May 1994) Abstract-The coupled equations for boundary and surface diffusion that control the growth of a void along a grain boundary have been solved numerically for the limiting case where the slope of the cavity surface is always small. The resulting solutions are compared with asymptotic results for the same problem that have been developed earlier by other authors. It is shown that existing expressions for “crack-like” and “quasi-equilibrium” growth provide an excellent description of void growth over the entire range of possible conditions. The numerical results illustrate the transition between the two modes and the evolution of the cavity profile during growth. In particular, a wedge-shaped cavity is predicted to develop as a void grows across a grain boundary, even in the absence of crystallographic effects.
1. INTRODUCTION The thermal expansion mismatch between a silicon substrate and a metal line in an electronic component can lead to tensile stresses which may cause the nucleation and growth of voids [l-8]. Although voids
are sometimes found within grains, especially in lines subjected to electric currents [8], there are many observations of cavities developing along grain boundaries [l-7]. These voids are clearly related to those responsible for creep rupture in bulk metals and ceramics which have received extensive modelling over the last 35 years. One of the mechanisms for the growth of cavities at elevated temperatures is diffusion-matter flows from the void surface and into the grain boundary in response to the tensile component of the stress normal to the boundary. Early analyses of the phenomenon [9-111 assumed that the cavity maintained a constant surface curvature (“equilibrium shaped”), as would be consistent with a situation in which the cavity grows very slowly compared with the rate at which matter can be redistributed over the cavity surface. Chuang and Rice [12] examined the opposite limit of very fast growth and showed that the cavity would adopt a “crack-like” profile with a width proportional to (D,S,/v)“‘, where D,6,is the surface diffusivity and v is the velocity of the crack tip. Pharr and Nix [13] examined the growth of a cavity with an initial equilibrium shape for the asymptotic limit in which the grain-boundary diffusivity was much greater than the surface diffusivity, so that surface diffusion controlled the growth rate.
Their numerical analysis demonstrated that, in this limit, the cavity would evolve into the crack-like shape predicted by Chuang and Rice [12]. The more general problem in which both surface and boundary diffusivities are of comparable importance was examined by Chuang et al. [14]. Although the procedure for obtaining a general solution for the cavity growth rate and profile was outlined in this reference, the full analysis was not attempted. Instead, analytical solutions were obtained for the limiting cases of “quasiequilibrium” and “crack-like” cavity growth. The intermediate regime between these two limits was explored by examining a particular solution to the governing equation for surface diffusion in which the cavity was assumed to grow in a self-similar fashion as in the original boundary-grooving analysis of Mullins [15]. To complement the papers by Pharr and Nix [13] and Chuang et al. [14], this paper presents the results of a full numerical solution to the coupled equations governing diffusion along the surface and boundary, subject to an assumption that the dihedral angle is very small. The values for the cavity growth rate are compared with the quasi-equilibrium and crack-like solutions of Ref. [14], and the transition between the two limiting mechanisms is illustrated. Furthermore, the results reveal how the cavity profile is expected to evolve as the void grows along a grain boundary. 2. ANALYSIS The two-dimensional geometry of the void that is analysed is illustrated in Fig. 1. A through-thickness
cavity is assumed to have been nucleated on a grain boundary in a line of width I. The tip of the void lies on the grain boundary, and at a time t it is a distance a (t ) from the edge of the line. The dihedral angle is
TPresent address: Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, U.S.A. 2493
2494
THOULESS and LINIGER: EFFECTS OF SURFACE DIFFUSION ON VOID GROWTH
Y
db
/
X
Fig. 1. The geometry analysed in this paper. It is assumed that there is no variation in stress or geometry through the thickness of the line, and that the planes x = 0 and x = ! are planes along which no diffusion occurs.
following analysis, it will be assumed that the average stress on the cavitated boundary remains a ~ - - a n assumption which would be rigorously valid only for a long length of line that was pinned to the substrate at each end, but otherwise free to deform. Wth this assumption, equilibrium across the grain boundary requires that [20] a~l =
a (x) dx + 7~sin 0
where 7~ is the surface tension of the metal. The flux of atoms along the boundary, Jb(x) is given by [21] Db6b do (x) dx
Jb(x) = k T assumed to remain at a constant value qJ. The open end of the void is at the edge of the line, with the cavity surfaces making a right angle to the edge, and it is assumed that there is no diffusion along the boundaries formed by the lines y = 0 and /. These assumptions are appropriate for examining the growth of a single void nucleated at the interface between a metal line and a layer of passivation along which no diffusion occurs. They are equally appropriate for examining the growth of an array of cylindrical voids separated by a distance 2l along a grain boundary [13, 14]. Atoms flow between the surface of the void and the grain boundary in response to a tensile stress, and, in keeping with other elementary models of cavity growth, the atoms are assumed to be deposited in a uniform manner along the length of the grain boundariest. Two coupled diffusion equations have to be solved: the equations for boundary and surface diffusion. Atoms flow from the tip of the cavity and along the grain boundary in response to gradients in the component of the residual stress normal to the grain boundary, a (x). It is assumed that the total stress acting on an uncavitated boundary is ¢r~ which, for example, would be equal to E A a A T for a residual stress induced by a thermal expansion mismatch, where E is the modulus of the line, A~ is the difference in thermal expansions between the substrate and metal and AT is the drop in temperature from the zero stress condition. The effective stress acting on a boundary after cavitation is much more difficult to evaluate. It depends on factors such as the geometry of the void, the density of voids along the line and the bonding conditions between the line and the surrounding material. If a large fraction of grain boundaries contain voids then the growth of the voids will cause relaxation of the stresses [3, 18, 19]; an isolated void will have a negligible effect on the average stress in the line. For the purposes of the
(2)
where Db3 b denotes the boundary diffusivity, k is Boltzmann's constant and T is the absolute temperature. If these equations are coupled with the assumption that atoms are plated uniformly along the grain boundary, so there is no elastic distortion of the grains [i.e. the boundary flux is linear in (x)], and the condition for continuity of chemical potential at the cavity tip a [a (t)1 = ?~x [a (t)]
(3)
where x [a (t)] is the surface curvature at the cavity tip, then the equation for the flux of atoms from the cavity into the boundary is given by Jb[a ( t )] =
3Dbrb kT[1 -- a (t)]2
× { a s l - Ys[x (a (t))(l - a (t)) + sin ~O]}.
(4)
Along the surface of the cavity, atoms diffuse in response to gradients in the surface curvature. Adopting the assumption that the slope of the surface is small everywhere (which in this ease means that the dihedral angle ~ ---, 0 [14]), the differential equation for the shape of the cavity is given by [15] ay(x,t) 3t -
Dsrsysfla'y(x,t) kT dx 4
(5)
where t is the time, D, 6, is the surface diffusivity and ft is the atomic volume of the diffusing species. It is convenient to normalize this equation so that it becomes 1~ = - Y ....
(6)
where ( " ) and ( )' denote differentiation with respect to z and X, respectively, Y = y]ao tan
tThe effect of elastic distortions of the grains that may be induced by non-uniform deposition of matter along the grain boundary have been analysed by Chuang [16] and Vitek [17].
(1)
0)
X = X/Ct 0
= Dsfsys~t/kTag
THOULESS and LINIGER: EFFECTS OF SURFACE DIFFUSION ON VOID GROWTH and a0 = a (0), the initial length of the cavity. To determine the evolution of the cavity, equation (6) must be solved subject to four boundary conditions imposed by the geometry and the assumed condition of no flux a t x = 0 Y'(0, ~) = 0
Y"'(O, z) = 0
~) = - 1
Jb[a (t)], and the surface diffusional flux at the tip of the cavity, Js[a (t)] Js[a (t ), t] = Jb[a (t ), t ]/2.
(9a)
Using equation (4) and the relationship between the surface flux and curvature [15], this becomes, in non-dimensional form
Y"'IA (~), z]
Y ( A (z), z) = 0
Y'(A ( r ) ,
2495
(7)
3A I tr - 1 - 2 ,.[L--A('r)] ~c°s~k
Y"[A (z), z] L--~A--~ J
(9b)
where A ( z ) = a ( t ) / a o. An initial condition for the shape of the cavity at time t = 0 is also required. For the purposes of this analysis, the cavity is assumed to have an initially uniform curvature so, under the small-slope approximation, the initial condition becomes
where L = l/ao, A = Dbbb/Ds6 ~ and a = ao~l/?, sin ~k. This equation introduces a non-linearity into the problem even though the original expressions had been simplified to a linear form.
Y (X, O) = (1 -- X2)/2.
A summary of the numerical technique devised to solve this non-linear moving boundary problem is provided in the Appendix. Only the results and their physical significance are provided here. The solutions given in this section are dependent on three parameters; the normalized width of the line, L, the ratio of the boundary to surface diffusivities, A, and the applied stress. This last quantity can be conveniently normalized by the equilibrium stress, ae, that must be applied to prevent the initial cavity from sintering
(8)
The governing equation, the boundary conditions and the initial condition [equations (6)-(8)] are essentially the same as those studied in Refs [13] and [14]. However, the essential difference between the present analysis and the earlier ones is the final condition required to solve what is a moving-boundary problem. It is this aspect of the problem that introduces analytical complexities. If the length of the cavity A (z) were a known function then, for any fixed z, the three equations would represent a linear-boundary value problem which could be readily solved. However, A (z) is unknown and must be determined as part of the solution; this requires the imposition of another condition. Chuang et aL [14] imposed the requirement that the cavity grew in a self-similar shape with the length of the cavity being proportional to t 1/4. This form was suggested by the original Mullins' solution of equation (6) for a grain boundary groove [15] and, although it does lead to a solution that interpolates between the two limiting asymptotes, it has been shown elsewhere that a grain boundary groove exhibits a different growth law when growing under a constant stress [22, 23]. Pharr and Nix [13] assumed that grain-boundary diffusion was infinitely rapid so that a finite cavity growth rate (governed by surface diffusion) is obtained even in the absence of a chemical potential gradient along the boundary. The surface curvature of the cavity tip then adjusts itself to match the applied stress and provides the fifth condition required for solving the problemt. In the analysis presented in this paper, the fifth condition is introduced rigorously by an interface condition imposed by the requirement of conservation of matter, linking the boundary diffusional flux,
3. RESULTS
a~/ae=a{l+L-1
} -' - z .
Numerical solutions for the cavity growth rate will be compared to the analytical solutions of Chuang et al, [14] for the two limiting cases of growth: the "quasi-equilibrium" mode and the "crack-like" mode. In the notation of the present paper, the quasi-equilibrium growth rate is given by equation (61) of Ref. [14] as
V-
dA (r) 9 L A ( Z A (z) -- 1) dz = 7 ~ A ( z ) 2 [ L - A ( z ) ] 2"
(lla)
For an isolated void on a boundary, in which the uncavitated ligament changes only negligibly as the cavity grows, this equation reduces to V-
9 AY, 4 A (T)L
(llb)
when Z >>1. The crack-like growth rate is given by equation (64) of Ref. [14] as 27 A3 V = 64 [L _ A (z )]3
[(
)
8ZL i r a I 1+-~--
(12a)
which reduces to V --- 1.84(--~L-~-) 3/a
"['In the text of Ref. [13], this condition is treated as one of the four boundary conditions; the third condition of equation (7) is invoked later in the numerical analysis.
(I0)
(12b)
for an isolated void on a grain boundary when Z L / A >>1.
THOULESS and LINIGER:
2496
2 0.2
--
EFFECTS OF SURFACE DIFFUSION
ON VOID GROWTH
large values of CA/L, the initial equilibrium con@uration is immediately perturbed by the formation of a “nose” along the grain boundary. This nose establishes regions along the surface where the curvature changes sign, and atoms flow not only to the grain boundary but also to the central portion of the cavity which decreases in width. This third mode of growth is exactly analogous to the results obtained in the analysis of Pharr and Nix [ 131which are valid for the limit of A/L + 03.
quasi-equilibrium
I
I
111,l
2
4
6
6 10
20
Position of cavity tip, a(f)/a,
Fig. 2. Normalized velocity plotted against the location of the cavity tip for different values of the parameter ZA/L. Only the values of A vary between these plots; L = 1600 and Z = 10 for all the curves. Superimposed on this figure are the analytical expressions for quasi-equilibrium and crack-like growth [equations 12(b) and 13(b)] valid for an isolated cavity.
3.1. Isolated void
In Fig. 2, the results for the cavity growth rate are plotted in a normalized form, FL/AZ against A (r) for a case in which A (7)~ L. Equations (11 b) and (12b) are also plotted on this figure. It can be seen that for relatively low values of the quantity ZA/L, less than about 0.2, the initial growth rate is described very well by the analytical equation for quasi-equilibrium growth [equation (1 lb)]. The velocity of the cavity tip along the boundary decreases as the cavity grows, owing to the extra matter that must be removed from the cavity surface for an increment of growth, until it is approximately equal to the growth rate for the crack-like mode under the same conditions [equation (13b)]. This latter mode then dominates, and the velocity remains independent of the cavity size because the width remains constant and the amount of matter that must be transported along the boundary per increment of advance does not vary. At large values of CA/L, the initial growth rate does not follow any existing analytical expression, but the cavity again decelerates to a velocity described by equation (12b). The transition between the two modes of growth can also be illustrated by examining cavity profiles as they evolve from the initial equilibrium configuration. Successive profiles for three different values of the parameter ZA/L are shown in Fig. 3. When CA/L is very small, the cavity maintains a shape very closely approximating the equilibrium configuration. At an intermediate value of this parameter, the initial growth occurs with the cavity maintaining an equilibrium shape, but at a critical size the cavity starts to elongate along the boundary and grow in a crack-like fashion. It should be noted that during this transition there are no inflections along the surface of the cavity and the flux of atoms along the surface is always directed towards the grain boundary. At very
3.2. Interacting cavity While the behaviour of an isolated cavity yields interesting insight into the transition between the growth modes, it is also instructive to examine the effects caused by the cavity interacting with the boundaries (or other cavities). Figure 4 shows specific results for the velocity of the cavity tip when the width of the line is only twenty times the initial cavity size. Equations (I la) and (12a) are also plotted as a comparison to the numerical results, and it can be seen that, together, they provide an excellent description of the behaviour. As is the case for the isolated cavity, the velocity initially decays but then, with the influence of the far boundary, both the velocity and acceleration of the cavity tip increase dramatically. The evolution of the profile of the void as it grows across the line is shown in Fig. 5. Whether the cavity initially evolves smoothly into a crack-like profile or develops a “nose” depends on the level of the stress and the ratio of the boundary to surface diffusivities (Fig. 3). The particularly intriguing aspect of Fig. 5 is the development of the profile once the cavity starts to accelerate under the influence of the remote boundary. The surfaces of the void are then flat and inclined at a constant angle to the grain boundary so as to form a uniform wedge as also observed in calculations by Martinez and Nix [24]. As the cavity approaches the far boundary, the shape of the tip looks very reminiscent of a shock wave.
4. DISCUSSION
Perhaps the most important results of the previous section are the demonstration that the analytical solutions of Chuang et al. [14] for the two limiting modes of cavity growth provide an excellent description for the entire process, even in the transition region where both models are equally appropriate. This last point is particularly important as there appears to be a general perception that crack-like growth is fast whereas quasi-equilibrium growth is slow. In fact, this analysis illustrates that the cavity tip decelerates as the growth makes the transition into a crack-like mode. Based on calculations of relaxation times for surface perturbations, Chuang et al. [14] recognised that there would be a transition in growth modes of a cavity. In the normalized notation of this paper, they
THOULESS and LINIGER:
EFFECTS OF SURFACE DIFFUSION
ON VOID GROWTH
0
5 10 15 Distance along grain boundary, x/a,
20
0
5 10 15 Distance along grain boundary, da,
20
ai
E
g 0.2 z ’
0.1 0 r 0
15 5 10 Distance along grain boundary, x/a,
20
Fig. 3. Void profiles for three different values of ZA/L. In all cases L = 1000; the other parameters are: (a) C = 2, A = 5, (b) Z = 10, A = 20 and (c) I: = 10, A = 500.
AM 43/6X
2497
2498
THOULESS and LINIGER:
100
/
-.
numerical
-.
- - -
0.1
1
I
crack-like
1’1’1
solution
quasi-equilibrium
EFFECTS OF SURFACE DIFFUSION 1
limit
limit
_I IIIIII
I-
2
4 6 0 IO Position of cavity tip, a(rJh,
20
Fig. 4. Normalized velocity plotted against the location of the cavity tip for different values of the parameter ZA/L. Only the values of A vary between these plots; L = 20 and Z = 10 for all the curves. Superimposed on this figure are the analytical expressions for quasi-equilibrium and crack-like growth [equations 12(a) and 13(a)] valid for interacting cavities.
with the conditions of equations (13). In a similar fashion, an expression for the transition under the influence of the remote boundary can be obtained from equations (1 la) and (12a). Furthermore, since A (z) 2 1, equation (14) shows that quasi-equilibrium growth can only occur if CA/L is significantly less than one. The other rationalisation for the transition condition can be obtained from expressions for the width of the cavity. Whereas, an equilibrium shaped cavity has a maximum width, w (t ) = 0.5 a(t ) tan $, at x = 0, so that W=w/a,tan$=A(r)/2
(15)
as $ -+ 0, Chuang and Rice [12] showed that the width of a crack-like cavity was independent of its length and depends on the velocity I,$/=
I/-l’3
(16)
as I++-
showed that if A (r)‘V<<6
(13a)
the cavity will grow in a quasi-equilibrium mode with the velocity given by equation (ll), whereas if A (~)~?‘>>6
(l3b)
the cavity will grow in a crack-like mode with the velocity given by equation (12). Essentially identical conditions for the transition can be obtained by two other approaches. These make use of the analytical results for the cavity velocity and profile [12, 141, and are confirmed by the numerical results of the previous section. It is clear from Fig. 2 that an approximate condition on which limiting mode best describes the cavity growth can be given by equating the two velocities of equations 1l(b) and 12(b), so that quasiequilibrium growth occurs if A (z)<< 1.2m.
(14)
This can be shown, by substitution back into one of the expressions for the cavity growth rate, to be consistent ,
,
”
,
u!L
0
ON VOID GROWTH
IO 15 5 Distance along grain boundary, x/a,
”
=
0.15
20
Fig. 5. Evolution of the cavity profile as a void extends across a grain boundary, L = 20.
0. The numerical results (Fig. 3) show excellent agreement with this equation, and it appears that a cavity can be described by the quasi-equilibrium equations until its width is approximately that of a crack-like cavity growing at the same rate. At this point the crack-like mode takes over. Equations (15) and (16) then predict a transition when A3(7) I/ u 8, which is consistent with the previous two equations. Furthermore, equation (11 b) shows that if ZA/L > 2.6,
(17)
the width appropriate for crack-like growth is less than the initial width of the void, so, as discussed in Pharr and Nix [13], a “nose” of the appropriate width develops from the tip of the cavity. Equation (16) also provides a rationalization for the intriguing shape that is predicted by the analysis for a cavity that grows across a grain boundary. Figure 5 shows that the cavity is predicted to form a wedge with the apex at the opposite boundary from where it was nucleated. Equation (12a) states that the velocity is inversely proportional to the cube of the uncavitated portion of the grain boundary. If this result is combined with equation (16) which shows that the steady-state width of a crack-like cavity is inversely proportional to the cube root of the velocity, it can be seen that the width of the cavity is predicted to be directly proportional to the distance from the far boundary. Quantitatively, the slope of the resulting wedge can be obtained as ,,/m. Rather remarkably, considering that in such a geometry, the cavity does not develop a steady-state width, this result appears to be in excellent agreement with the numerical analysis. Furthermore, there are experimental observations of wedge-like cavities extending along a grain boundary [25,26]. Although this form of cavity has been ascribed to crystallographic effects [27], it is intriguing that such a shape is predicted to evolve in an isotropic system.
THOULESS and LINIGER:
EFFECTS OF SURFACE DIFFUSION ON VOID GROWTH
5. CONCLUSIONS The major conclusions of this paper are that the asymptotic results of Chuang et al. [14] provide an excellent description for the growth of cavities over the entire range o f growth. The numerical results illustrate the transition between the two limiting growth modes, and show that it can be rationalized by matching the asymptotic expressions. Computational difficulties limited the comparison between the asymptotic results and the full solution of the coupled boundary- and surface-diffusion equations to the linearised form of these equations. However, Chuang et al. [14] provide asymptotic results for the case when ~ >>0. The form of these expressions are identical to the linearised versions, except for a function of ~k only. It therefore seems a reasonable conjecture that similar agreement would be obtained were a full numerical solution developed without the restriction of ~b---, 0. In connection with this, it is noted that there was no restriction on ~, in the numerical analysis of Pharr and Nix [13], which showed excellent agreement with the asymptotic resuits in the limit of L/A---~ ~ . Finally, attention should be drawn once again to the predicted profile of a void that grows across a boundary. It appears that a straight-sided wedge shape may evolve even in the absence of any crystallographic effects. Furthermore, despite the lack of steady-state width, this profile can be rationalized on the basis o f the analytical solutions of Refs [12] and [141. REFERENCES
1. C. Y. Li, R. D. Black and W. R. LaFontaine, Appl. Phys. Lett. 53, 31-33 (1988). 2. F. G. Yost, D. E. Amos and A. D. Romig Jr, in Proe. IEEE 27th Annual Reliability Phys. Syrup. IEEE, pp. 193-201, New York (1989). 3. F. G. Yost, Scripta metall, mater. 23, 1323--1328 (1989). 4. T. D. Sullivan, Appl. Phys. Lett. 55, 2399-2401 (1989). 5. M. A. Korhonen, C. A. Paszkiet and C. Y. Li, J. appl. Phys. 69, 8083-8091 (1991). 6. S. M. Hu, Appl. Phys. Lett. 59, 2685-2687 (1991). 7. P. Borgesen, J. K. Lee, R. Gleixner and C. Y. Li, Appl. Phys. Lett. 60, 1706-1708 (1992). 8. O. Kraft, S. Bader, J. E. Sanchez Jr and E. Arzt, Mat. Res. Soc. Syrup. Proc. 308, 199-204 (1993). 9. D. Hull and D. E. Rimmer, Phil. Mag. 4, 673-687 (1959). I0. M. V. Speight and J. E. Harris, Metal Sci. J. 1, 83-85 (1967). 11. R. Raj and M. F. Ashby, Acta metall. 23, 653-666 (1975). 12. T. J. Chuang and J. R. Rice, Acta metall. 21, 1625-1628 (1973). 13. G. M. Pharr and W. D. Nix, Actametall. 27, 1615-1631 (1979). 14. T. J. Chuang, K. I. Kagawa, J. R. Rice and L. B. Sills, Acta metall. 27, 265-284 (1979). 15. W. Mullins, J. appl. Phys. 28, 333-339 (1957). 16. T. J. Chuang, J. Am. Ceram. Soc. 65, 93-103 (1982). 17. V. Vitek, Acta metall. 26, 1345-1356 (1978). 18. B. F. Dyson, Metals Sci. 10, 349-353 (1976). 19. J. R. Rice, Acta metall. 29, 675-681 (1981).
2499
20. R. M. Cannon and W. C. Carter, J. Am. Ceram. Soc. 72, 1550-1555 (1989). 21. C. Herring, J. appl. Phys. 21, 437-445 (1950). 22. M. D. Thouless, Acta metall, mater. 41, 1057-1064 (1993). 23. F. Y. G6nin, W. W. Mullins and P. Wynblatt, Acta metall, mater. 41, 3541-3547 (1993). 24. L. Martinez and W. D. Nix, Metall. Trans. 13A, 427-437 (1982). 25. T. D. Sullivan and L. A. Miller, Mat. Res. Soc. Symp. Proc. 308, 237-248 (1993). 26. H. Kaneko, M. Hasunuma, A. Sawabe, T. Kawanoue, Y. Kohanawa, S. Komatsu and M. Miyauchi, Proc. IEEE 28th Annual Reliability Phys. Syrup. IEEE, pp. 194-199, New York (1990). 27. J. Douglas Jr and T. M. Gallie Jr, Duke Math. J. 22, 557-000 (1955). NOMENCLATURE a (t) length of cavity a0 initial length of cavity (t = 0) A (z) normalized cavity length [a (t)/a0] Ob~ b boundary diffusivity D~bs surface diffusivity Jb flUX of atoms along the grain boundary Js flux of atoms along the void surface k Boltzmann's constant l width of line L normalized width of line (l/ao) t time T absolute temperature v velocity of cavity tip Ida (t)/dt ] V normalized velocity of cavity tip IdA (Q/dz] w width of cavity W normalized width of cavity (w/a o tan ~,) x co-ordinate along the grain boundary X normalized co-ordinate along the grain boundary
(x/ao) y Y ?~
co-ordinate along the length of the line normalized co-ordinate along the length of the line (y/ao tan ~b) surface energy
A
DbSb/Ds5 s
x tr~ tre
surface curvature average stress acting across the width of the line stress required to prevent the initial cavity from sintering { ?s sin ~, [1 + (L - l)/cos ~]/1 } a~ I/y~ sin ¢ a~/G = a [ 1 + (L - 1)/cos ~ ]- I normalized time, Ds6sf~t/kTa 4 dihedral angle at cavity tip atomic volume of diffusing species
a Z z ~b [2
APPENDIX
The partial differential equation (6), the boundary conditions [equation (7)], and the interface condition [equation (9b)], together represent a moving boundary value problem for the unknowns Y(X, z) and A (z). This Appendix summarises the numerical technique for solving this problem starting from the initial condition [equation (8)]. Following an idea first proposed by Douglas and Gallic [27], the spatial variable X was uniformly discretized so that X~=iAX,
i = 0 , 1,2 . . . . .
(AI)
where A X = d (O)/m, and m is an integer. By definition, the scaled initial cavity length is unity, so that X,, = A (0)= 1. The time variable was discretized non-uniformly, with 30 = 0 and J
~j= ~ Axu, j = [ , 2 . . . . . #=1
(A2)
2500
THOULESS and LINIGER:
x0 x,
.”
.
.
.
.
.
.
I
j&
&+,
.
...”
. . . . .
EFFECTS OF SURFACE DIFFUSION
A
Fig. Al. Grid points showing how the time intervals, ~~- rj_ , , are defined by the time taken for the cavity tip to reach successive values of X,, separated by equal intervals.
ON VOID GROWTH
ing paragraph) was solved to obtain a preliminary void profile Y $‘)(X,,rj). The result Y $) was then substituted into the discrete verston of the interface condition [equation (9b)] which, for v iv,,, was not satisfied but produced a non-vanishing residue Rj!‘). A new guess A77 + ‘) was then made in an effort to reduce the size of this residue, IRf’)I. The process was then repeated until “convergence” was obtained, i.e. until IRy)I had become smaller than a prescribed tolerance. The new guesses were first found by a search procedure which changed Ar; in such a direction that IRy)I decreased. Then, after a sign change in Ry) was observed, the final value AT, was obtained by the secant method. The differential equation (6) was discretized implicitly (backward) with respect to 7. If Y, and Y; denote, respectively, numerical approximations of Y (X, , TV) and Y (Xi, TV_ ,), then equation (6) was replaced by Y,- Y; = --p(Y,+, -4Y;+,+6Y,-4Y,_,+ i = 2,3,
The time steps Azj = rj - rj_ , were chosen in such a way that the moving boundary X = A (2) passed diagonally through the grid _ points [(X,,,+j =.$ir,], j = 0, 1, . .-., where Aj = A (tj) (Fig. Al). Determmmg the position of the moving boundary then amounted to finding the time-step sequence {AT.]. The time points, (t,} themselves were-then computed kern equation (A2). The advantage of this technique was that the last two boundary conditions of equation (7) and the interface. condition [equation (9b)] were always formulated at a grid point of the spatial variable X where a numerical value of Y is defined, rather than in between grid points where Y would have to have been approximated by interpolation. Starting from the initial condition [equation (8)], the moving boundary problem was solved by stepping forward from one time level, 4, to the next. Altogether, an integer number r time steps were carried out, after which the moving boundary was located at X))1+,= (m + r)AX. In order to avoid computational difficulties, r was chosen in such a way that the cavity length X,,,+-1was shorter than the full length L. At each time step, a sequence {Ary)), v = 1,2, . ., v,,, of approximations to Ar, was produced. For each Arf’), the boundary-value problem (discretized as shown in the follow-
Y;_,),
., rn + j - 2,
(A3)
where p = p, = As,/AX4, a discretization which is accurate to first order in 7 but to second order in X. The four boundary conditions [equation (7)] were replaced by the second-order approximations
-SY,+
-3y,+4y,-
Y*=O,
BY,-24Y,+
14Y,-3Y,=O,
Ym+l = 0, 3Y,+j-4Y,+j_,
+ Y,,,+,_,= -2AX,
(A4)
respectively. Equations (A3) and (A4) represent m + j + 1 equations for the m + j + 1 unknowns Y,, Y,, . ., Y,,,,,. Finally, the second-order discretizations Yi,, “(2Y,+j-5Y,+,-,+4Y,+,-*-3Y,+,-,)lAXZ, YC+, =(5Y,,,+,-
1gY,+,_,+24Y,,,+,_z - 14Ym+j_3+ 3Y,,,+j_4)/AX’
were used for Y”[A (T), 71 and Y”‘[A (r), the interface condition of equation (9b).
71
(A5)
which occur in
Pergamon
0956-7151/95-$9.50+0.00
EFFECTS
OF SURFACE AND BOUNDARY ON VOID GROWTH M. D. THOULESSt
IBM Research
Division,
T. J. Watson
Research
DIFFUSION
and W. LINGER Center,
Yorktown
Heights,
NY 10598, U.S.A.
(Received 20 May 1994) Abstract-The coupled equations for boundary and surface diffusion that control the growth of a void along a grain boundary have been solved numerically for the limiting case where the slope of the cavity surface is always small. The resulting solutions are compared with asymptotic results for the same problem that have been developed earlier by other authors. It is shown that existing expressions for “crack-like” and “quasi-equilibrium” growth provide an excellent description of void growth over the entire range of possible conditions. The numerical results illustrate the transition between the two modes and the evolution of the cavity profile during growth. In particular, a wedge-shaped cavity is predicted to develop as a void grows across a grain boundary, even in the absence of crystallographic effects.
1. INTRODUCTION The thermal expansion mismatch between a silicon substrate and a metal line in an electronic component can lead to tensile stresses which may cause the nucleation and growth of voids [l-8]. Although voids
are sometimes found within grains, especially in lines subjected to electric currents [8], there are many observations of cavities developing along grain boundaries [l-7]. These voids are clearly related to those responsible for creep rupture in bulk metals and ceramics which have received extensive modelling over the last 35 years. One of the mechanisms for the growth of cavities at elevated temperatures is diffusion-matter flows from the void surface and into the grain boundary in response to the tensile component of the stress normal to the boundary. Early analyses of the phenomenon [9-111 assumed that the cavity maintained a constant surface curvature (“equilibrium shaped”), as would be consistent with a situation in which the cavity grows very slowly compared with the rate at which matter can be redistributed over the cavity surface. Chuang and Rice [12] examined the opposite limit of very fast growth and showed that the cavity would adopt a “crack-like” profile with a width proportional to (D,S,/v)“‘, where D,6,is the surface diffusivity and v is the velocity of the crack tip. Pharr and Nix [13] examined the growth of a cavity with an initial equilibrium shape for the asymptotic limit in which the grain-boundary diffusivity was much greater than the surface diffusivity, so that surface diffusion controlled the growth rate.
Their numerical analysis demonstrated that, in this limit, the cavity would evolve into the crack-like shape predicted by Chuang and Rice [12]. The more general problem in which both surface and boundary diffusivities are of comparable importance was examined by Chuang et al. [14]. Although the procedure for obtaining a general solution for the cavity growth rate and profile was outlined in this reference, the full analysis was not attempted. Instead, analytical solutions were obtained for the limiting cases of “quasiequilibrium” and “crack-like” cavity growth. The intermediate regime between these two limits was explored by examining a particular solution to the governing equation for surface diffusion in which the cavity was assumed to grow in a self-similar fashion as in the original boundary-grooving analysis of Mullins [15]. To complement the papers by Pharr and Nix [13] and Chuang et al. [14], this paper presents the results of a full numerical solution to the coupled equations governing diffusion along the surface and boundary, subject to an assumption that the dihedral angle is very small. The values for the cavity growth rate are compared with the quasi-equilibrium and crack-like solutions of Ref. [14], and the transition between the two limiting mechanisms is illustrated. Furthermore, the results reveal how the cavity profile is expected to evolve as the void grows along a grain boundary. 2. ANALYSIS The two-dimensional geometry of the void that is analysed is illustrated in Fig. 1. A through-thickness
cavity is assumed to have been nucleated on a grain boundary in a line of width I. The tip of the void lies on the grain boundary, and at a time t it is a distance a (t ) from the edge of the line. The dihedral angle is
TPresent address: Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI 48109, U.S.A. 2493
2494
THOULESS and LINIGER: EFFECTS OF SURFACE DIFFUSION ON VOID GROWTH
Y
db
/
X
Fig. 1. The geometry analysed in this paper. It is assumed that there is no variation in stress or geometry through the thickness of the line, and that the planes x = 0 and x = ! are planes along which no diffusion occurs.
following analysis, it will be assumed that the average stress on the cavitated boundary remains a ~ - - a n assumption which would be rigorously valid only for a long length of line that was pinned to the substrate at each end, but otherwise free to deform. Wth this assumption, equilibrium across the grain boundary requires that [20] a~l =
a (x) dx + 7~sin 0
where 7~ is the surface tension of the metal. The flux of atoms along the boundary, Jb(x) is given by [21] Db6b do (x) dx
Jb(x) = k T assumed to remain at a constant value qJ. The open end of the void is at the edge of the line, with the cavity surfaces making a right angle to the edge, and it is assumed that there is no diffusion along the boundaries formed by the lines y = 0 and /. These assumptions are appropriate for examining the growth of a single void nucleated at the interface between a metal line and a layer of passivation along which no diffusion occurs. They are equally appropriate for examining the growth of an array of cylindrical voids separated by a distance 2l along a grain boundary [13, 14]. Atoms flow between the surface of the void and the grain boundary in response to a tensile stress, and, in keeping with other elementary models of cavity growth, the atoms are assumed to be deposited in a uniform manner along the length of the grain boundariest. Two coupled diffusion equations have to be solved: the equations for boundary and surface diffusion. Atoms flow from the tip of the cavity and along the grain boundary in response to gradients in the component of the residual stress normal to the grain boundary, a (x). It is assumed that the total stress acting on an uncavitated boundary is ¢r~ which, for example, would be equal to E A a A T for a residual stress induced by a thermal expansion mismatch, where E is the modulus of the line, A~ is the difference in thermal expansions between the substrate and metal and AT is the drop in temperature from the zero stress condition. The effective stress acting on a boundary after cavitation is much more difficult to evaluate. It depends on factors such as the geometry of the void, the density of voids along the line and the bonding conditions between the line and the surrounding material. If a large fraction of grain boundaries contain voids then the growth of the voids will cause relaxation of the stresses [3, 18, 19]; an isolated void will have a negligible effect on the average stress in the line. For the purposes of the
(2)
where Db3 b denotes the boundary diffusivity, k is Boltzmann's constant and T is the absolute temperature. If these equations are coupled with the assumption that atoms are plated uniformly along the grain boundary, so there is no elastic distortion of the grains [i.e. the boundary flux is linear in (x)], and the condition for continuity of chemical potential at the cavity tip a [a (t)1 = ?~x [a (t)]
(3)
where x [a (t)] is the surface curvature at the cavity tip, then the equation for the flux of atoms from the cavity into the boundary is given by Jb[a ( t )] =
3Dbrb kT[1 -- a (t)]2
× { a s l - Ys[x (a (t))(l - a (t)) + sin ~O]}.
(4)
Along the surface of the cavity, atoms diffuse in response to gradients in the surface curvature. Adopting the assumption that the slope of the surface is small everywhere (which in this ease means that the dihedral angle ~ ---, 0 [14]), the differential equation for the shape of the cavity is given by [15] ay(x,t) 3t -
Dsrsysfla'y(x,t) kT dx 4
(5)
where t is the time, D, 6, is the surface diffusivity and ft is the atomic volume of the diffusing species. It is convenient to normalize this equation so that it becomes 1~ = - Y ....
(6)
where ( " ) and ( )' denote differentiation with respect to z and X, respectively, Y = y]ao tan
tThe effect of elastic distortions of the grains that may be induced by non-uniform deposition of matter along the grain boundary have been analysed by Chuang [16] and Vitek [17].
(1)
0)
X = X/Ct 0
= Dsfsys~t/kTag
THOULESS and LINIGER: EFFECTS OF SURFACE DIFFUSION ON VOID GROWTH and a0 = a (0), the initial length of the cavity. To determine the evolution of the cavity, equation (6) must be solved subject to four boundary conditions imposed by the geometry and the assumed condition of no flux a t x = 0 Y'(0, ~) = 0
Y"'(O, z) = 0
~) = - 1
Jb[a (t)], and the surface diffusional flux at the tip of the cavity, Js[a (t)] Js[a (t ), t] = Jb[a (t ), t ]/2.
(9a)
Using equation (4) and the relationship between the surface flux and curvature [15], this becomes, in non-dimensional form
Y"'IA (~), z]
Y ( A (z), z) = 0
Y'(A ( r ) ,
2495
(7)
3A I tr - 1 - 2 ,.[L--A('r)] ~c°s~k
Y"[A (z), z] L--~A--~ J
(9b)
where A ( z ) = a ( t ) / a o. An initial condition for the shape of the cavity at time t = 0 is also required. For the purposes of this analysis, the cavity is assumed to have an initially uniform curvature so, under the small-slope approximation, the initial condition becomes
where L = l/ao, A = Dbbb/Ds6 ~ and a = ao~l/?, sin ~k. This equation introduces a non-linearity into the problem even though the original expressions had been simplified to a linear form.
Y (X, O) = (1 -- X2)/2.
A summary of the numerical technique devised to solve this non-linear moving boundary problem is provided in the Appendix. Only the results and their physical significance are provided here. The solutions given in this section are dependent on three parameters; the normalized width of the line, L, the ratio of the boundary to surface diffusivities, A, and the applied stress. This last quantity can be conveniently normalized by the equilibrium stress, ae, that must be applied to prevent the initial cavity from sintering
(8)
The governing equation, the boundary conditions and the initial condition [equations (6)-(8)] are essentially the same as those studied in Refs [13] and [14]. However, the essential difference between the present analysis and the earlier ones is the final condition required to solve what is a moving-boundary problem. It is this aspect of the problem that introduces analytical complexities. If the length of the cavity A (z) were a known function then, for any fixed z, the three equations would represent a linear-boundary value problem which could be readily solved. However, A (z) is unknown and must be determined as part of the solution; this requires the imposition of another condition. Chuang et aL [14] imposed the requirement that the cavity grew in a self-similar shape with the length of the cavity being proportional to t 1/4. This form was suggested by the original Mullins' solution of equation (6) for a grain boundary groove [15] and, although it does lead to a solution that interpolates between the two limiting asymptotes, it has been shown elsewhere that a grain boundary groove exhibits a different growth law when growing under a constant stress [22, 23]. Pharr and Nix [13] assumed that grain-boundary diffusion was infinitely rapid so that a finite cavity growth rate (governed by surface diffusion) is obtained even in the absence of a chemical potential gradient along the boundary. The surface curvature of the cavity tip then adjusts itself to match the applied stress and provides the fifth condition required for solving the problemt. In the analysis presented in this paper, the fifth condition is introduced rigorously by an interface condition imposed by the requirement of conservation of matter, linking the boundary diffusional flux,
3. RESULTS
a~/ae=a{l+L-1
} -' - z .
Numerical solutions for the cavity growth rate will be compared to the analytical solutions of Chuang et al, [14] for the two limiting cases of growth: the "quasi-equilibrium" mode and the "crack-like" mode. In the notation of the present paper, the quasi-equilibrium growth rate is given by equation (61) of Ref. [14] as
V-
dA (r) 9 L A ( Z A (z) -- 1) dz = 7 ~ A ( z ) 2 [ L - A ( z ) ] 2"
(lla)
For an isolated void on a boundary, in which the uncavitated ligament changes only negligibly as the cavity grows, this equation reduces to V-
9 AY, 4 A (T)L
(llb)
when Z >>1. The crack-like growth rate is given by equation (64) of Ref. [14] as 27 A3 V = 64 [L _ A (z )]3
[(
)
8ZL i r a I 1+-~--
(12a)
which reduces to V --- 1.84(--~L-~-) 3/a
"['In the text of Ref. [13], this condition is treated as one of the four boundary conditions; the third condition of equation (7) is invoked later in the numerical analysis.
(I0)
(12b)
for an isolated void on a grain boundary when Z L / A >>1.
THOULESS and LINIGER:
2496
2 0.2
--
EFFECTS OF SURFACE DIFFUSION
ON VOID GROWTH
large values of CA/L, the initial equilibrium con@uration is immediately perturbed by the formation of a “nose” along the grain boundary. This nose establishes regions along the surface where the curvature changes sign, and atoms flow not only to the grain boundary but also to the central portion of the cavity which decreases in width. This third mode of growth is exactly analogous to the results obtained in the analysis of Pharr and Nix [ 131which are valid for the limit of A/L + 03.
quasi-equilibrium
I
I
111,l
2
4
6
6 10
20
Position of cavity tip, a(f)/a,
Fig. 2. Normalized velocity plotted against the location of the cavity tip for different values of the parameter ZA/L. Only the values of A vary between these plots; L = 1600 and Z = 10 for all the curves. Superimposed on this figure are the analytical expressions for quasi-equilibrium and crack-like growth [equations 12(b) and 13(b)] valid for an isolated cavity.
3.1. Isolated void
In Fig. 2, the results for the cavity growth rate are plotted in a normalized form, FL/AZ against A (r) for a case in which A (7)~ L. Equations (11 b) and (12b) are also plotted on this figure. It can be seen that for relatively low values of the quantity ZA/L, less than about 0.2, the initial growth rate is described very well by the analytical equation for quasi-equilibrium growth [equation (1 lb)]. The velocity of the cavity tip along the boundary decreases as the cavity grows, owing to the extra matter that must be removed from the cavity surface for an increment of growth, until it is approximately equal to the growth rate for the crack-like mode under the same conditions [equation (13b)]. This latter mode then dominates, and the velocity remains independent of the cavity size because the width remains constant and the amount of matter that must be transported along the boundary per increment of advance does not vary. At large values of CA/L, the initial growth rate does not follow any existing analytical expression, but the cavity again decelerates to a velocity described by equation (12b). The transition between the two modes of growth can also be illustrated by examining cavity profiles as they evolve from the initial equilibrium configuration. Successive profiles for three different values of the parameter ZA/L are shown in Fig. 3. When CA/L is very small, the cavity maintains a shape very closely approximating the equilibrium configuration. At an intermediate value of this parameter, the initial growth occurs with the cavity maintaining an equilibrium shape, but at a critical size the cavity starts to elongate along the boundary and grow in a crack-like fashion. It should be noted that during this transition there are no inflections along the surface of the cavity and the flux of atoms along the surface is always directed towards the grain boundary. At very
3.2. Interacting cavity While the behaviour of an isolated cavity yields interesting insight into the transition between the growth modes, it is also instructive to examine the effects caused by the cavity interacting with the boundaries (or other cavities). Figure 4 shows specific results for the velocity of the cavity tip when the width of the line is only twenty times the initial cavity size. Equations (I la) and (12a) are also plotted as a comparison to the numerical results, and it can be seen that, together, they provide an excellent description of the behaviour. As is the case for the isolated cavity, the velocity initially decays but then, with the influence of the far boundary, both the velocity and acceleration of the cavity tip increase dramatically. The evolution of the profile of the void as it grows across the line is shown in Fig. 5. Whether the cavity initially evolves smoothly into a crack-like profile or develops a “nose” depends on the level of the stress and the ratio of the boundary to surface diffusivities (Fig. 3). The particularly intriguing aspect of Fig. 5 is the development of the profile once the cavity starts to accelerate under the influence of the remote boundary. The surfaces of the void are then flat and inclined at a constant angle to the grain boundary so as to form a uniform wedge as also observed in calculations by Martinez and Nix [24]. As the cavity approaches the far boundary, the shape of the tip looks very reminiscent of a shock wave.
4. DISCUSSION
Perhaps the most important results of the previous section are the demonstration that the analytical solutions of Chuang et al. [14] for the two limiting modes of cavity growth provide an excellent description for the entire process, even in the transition region where both models are equally appropriate. This last point is particularly important as there appears to be a general perception that crack-like growth is fast whereas quasi-equilibrium growth is slow. In fact, this analysis illustrates that the cavity tip decelerates as the growth makes the transition into a crack-like mode. Based on calculations of relaxation times for surface perturbations, Chuang et al. [14] recognised that there would be a transition in growth modes of a cavity. In the normalized notation of this paper, they
THOULESS and LINIGER:
EFFECTS OF SURFACE DIFFUSION
ON VOID GROWTH
0
5 10 15 Distance along grain boundary, x/a,
20
0
5 10 15 Distance along grain boundary, da,
20
ai
E
g 0.2 z ’
0.1 0 r 0
15 5 10 Distance along grain boundary, x/a,
20
Fig. 3. Void profiles for three different values of ZA/L. In all cases L = 1000; the other parameters are: (a) C = 2, A = 5, (b) Z = 10, A = 20 and (c) I: = 10, A = 500.
AM 43/6X
2497
2498
THOULESS and LINIGER:
100
/
-.
numerical
-.
- - -
0.1
1
I
crack-like
1’1’1
solution
quasi-equilibrium
EFFECTS OF SURFACE DIFFUSION 1
limit
limit
_I IIIIII
I-
2
4 6 0 IO Position of cavity tip, a(rJh,
20
Fig. 4. Normalized velocity plotted against the location of the cavity tip for different values of the parameter ZA/L. Only the values of A vary between these plots; L = 20 and Z = 10 for all the curves. Superimposed on this figure are the analytical expressions for quasi-equilibrium and crack-like growth [equations 12(a) and 13(a)] valid for interacting cavities.
with the conditions of equations (13). In a similar fashion, an expression for the transition under the influence of the remote boundary can be obtained from equations (1 la) and (12a). Furthermore, since A (z) 2 1, equation (14) shows that quasi-equilibrium growth can only occur if CA/L is significantly less than one. The other rationalisation for the transition condition can be obtained from expressions for the width of the cavity. Whereas, an equilibrium shaped cavity has a maximum width, w (t ) = 0.5 a(t ) tan $, at x = 0, so that W=w/a,tan$=A(r)/2
(15)
as $ -+ 0, Chuang and Rice [12] showed that the width of a crack-like cavity was independent of its length and depends on the velocity I,$/=
I/-l’3
(16)
as I++-
showed that if A (r)‘V<<6
(13a)
the cavity will grow in a quasi-equilibrium mode with the velocity given by equation (ll), whereas if A (~)~?‘>>6
(l3b)
the cavity will grow in a crack-like mode with the velocity given by equation (12). Essentially identical conditions for the transition can be obtained by two other approaches. These make use of the analytical results for the cavity velocity and profile [12, 141, and are confirmed by the numerical results of the previous section. It is clear from Fig. 2 that an approximate condition on which limiting mode best describes the cavity growth can be given by equating the two velocities of equations 1l(b) and 12(b), so that quasiequilibrium growth occurs if A (z)<< 1.2m.
(14)
This can be shown, by substitution back into one of the expressions for the cavity growth rate, to be consistent ,
,
”
,
u!L
0
ON VOID GROWTH
IO 15 5 Distance along grain boundary, x/a,
”
=
0.15
20
Fig. 5. Evolution of the cavity profile as a void extends across a grain boundary, L = 20.
0. The numerical results (Fig. 3) show excellent agreement with this equation, and it appears that a cavity can be described by the quasi-equilibrium equations until its width is approximately that of a crack-like cavity growing at the same rate. At this point the crack-like mode takes over. Equations (15) and (16) then predict a transition when A3(7) I/ u 8, which is consistent with the previous two equations. Furthermore, equation (11 b) shows that if ZA/L > 2.6,
(17)
the width appropriate for crack-like growth is less than the initial width of the void, so, as discussed in Pharr and Nix [13], a “nose” of the appropriate width develops from the tip of the cavity. Equation (16) also provides a rationalization for the intriguing shape that is predicted by the analysis for a cavity that grows across a grain boundary. Figure 5 shows that the cavity is predicted to form a wedge with the apex at the opposite boundary from where it was nucleated. Equation (12a) states that the velocity is inversely proportional to the cube of the uncavitated portion of the grain boundary. If this result is combined with equation (16) which shows that the steady-state width of a crack-like cavity is inversely proportional to the cube root of the velocity, it can be seen that the width of the cavity is predicted to be directly proportional to the distance from the far boundary. Quantitatively, the slope of the resulting wedge can be obtained as ,,/m. Rather remarkably, considering that in such a geometry, the cavity does not develop a steady-state width, this result appears to be in excellent agreement with the numerical analysis. Furthermore, there are experimental observations of wedge-like cavities extending along a grain boundary [25,26]. Although this form of cavity has been ascribed to crystallographic effects [27], it is intriguing that such a shape is predicted to evolve in an isotropic system.
THOULESS and LINIGER:
EFFECTS OF SURFACE DIFFUSION ON VOID GROWTH
5. CONCLUSIONS The major conclusions of this paper are that the asymptotic results of Chuang et al. [14] provide an excellent description for the growth of cavities over the entire range o f growth. The numerical results illustrate the transition between the two limiting growth modes, and show that it can be rationalized by matching the asymptotic expressions. Computational difficulties limited the comparison between the asymptotic results and the full solution of the coupled boundary- and surface-diffusion equations to the linearised form of these equations. However, Chuang et al. [14] provide asymptotic results for the case when ~ >>0. The form of these expressions are identical to the linearised versions, except for a function of ~k only. It therefore seems a reasonable conjecture that similar agreement would be obtained were a full numerical solution developed without the restriction of ~b---, 0. In connection with this, it is noted that there was no restriction on ~, in the numerical analysis of Pharr and Nix [13], which showed excellent agreement with the asymptotic resuits in the limit of L/A---~ ~ . Finally, attention should be drawn once again to the predicted profile of a void that grows across a boundary. It appears that a straight-sided wedge shape may evolve even in the absence of any crystallographic effects. Furthermore, despite the lack of steady-state width, this profile can be rationalized on the basis o f the analytical solutions of Refs [12] and [141. REFERENCES
1. C. Y. Li, R. D. Black and W. R. LaFontaine, Appl. Phys. Lett. 53, 31-33 (1988). 2. F. G. Yost, D. E. Amos and A. D. Romig Jr, in Proe. IEEE 27th Annual Reliability Phys. Syrup. IEEE, pp. 193-201, New York (1989). 3. F. G. Yost, Scripta metall, mater. 23, 1323--1328 (1989). 4. T. D. Sullivan, Appl. Phys. Lett. 55, 2399-2401 (1989). 5. M. A. Korhonen, C. A. Paszkiet and C. Y. Li, J. appl. Phys. 69, 8083-8091 (1991). 6. S. M. Hu, Appl. Phys. Lett. 59, 2685-2687 (1991). 7. P. Borgesen, J. K. Lee, R. Gleixner and C. Y. Li, Appl. Phys. Lett. 60, 1706-1708 (1992). 8. O. Kraft, S. Bader, J. E. Sanchez Jr and E. Arzt, Mat. Res. Soc. Syrup. Proc. 308, 199-204 (1993). 9. D. Hull and D. E. Rimmer, Phil. Mag. 4, 673-687 (1959). I0. M. V. Speight and J. E. Harris, Metal Sci. J. 1, 83-85 (1967). 11. R. Raj and M. F. Ashby, Acta metall. 23, 653-666 (1975). 12. T. J. Chuang and J. R. Rice, Acta metall. 21, 1625-1628 (1973). 13. G. M. Pharr and W. D. Nix, Actametall. 27, 1615-1631 (1979). 14. T. J. Chuang, K. I. Kagawa, J. R. Rice and L. B. Sills, Acta metall. 27, 265-284 (1979). 15. W. Mullins, J. appl. Phys. 28, 333-339 (1957). 16. T. J. Chuang, J. Am. Ceram. Soc. 65, 93-103 (1982). 17. V. Vitek, Acta metall. 26, 1345-1356 (1978). 18. B. F. Dyson, Metals Sci. 10, 349-353 (1976). 19. J. R. Rice, Acta metall. 29, 675-681 (1981).
2499
20. R. M. Cannon and W. C. Carter, J. Am. Ceram. Soc. 72, 1550-1555 (1989). 21. C. Herring, J. appl. Phys. 21, 437-445 (1950). 22. M. D. Thouless, Acta metall, mater. 41, 1057-1064 (1993). 23. F. Y. G6nin, W. W. Mullins and P. Wynblatt, Acta metall, mater. 41, 3541-3547 (1993). 24. L. Martinez and W. D. Nix, Metall. Trans. 13A, 427-437 (1982). 25. T. D. Sullivan and L. A. Miller, Mat. Res. Soc. Symp. Proc. 308, 237-248 (1993). 26. H. Kaneko, M. Hasunuma, A. Sawabe, T. Kawanoue, Y. Kohanawa, S. Komatsu and M. Miyauchi, Proc. IEEE 28th Annual Reliability Phys. Syrup. IEEE, pp. 194-199, New York (1990). 27. J. Douglas Jr and T. M. Gallie Jr, Duke Math. J. 22, 557-000 (1955). NOMENCLATURE a (t) length of cavity a0 initial length of cavity (t = 0) A (z) normalized cavity length [a (t)/a0] Ob~ b boundary diffusivity D~bs surface diffusivity Jb flUX of atoms along the grain boundary Js flux of atoms along the void surface k Boltzmann's constant l width of line L normalized width of line (l/ao) t time T absolute temperature v velocity of cavity tip Ida (t)/dt ] V normalized velocity of cavity tip IdA (Q/dz] w width of cavity W normalized width of cavity (w/a o tan ~,) x co-ordinate along the grain boundary X normalized co-ordinate along the grain boundary
(x/ao) y Y ?~
co-ordinate along the length of the line normalized co-ordinate along the length of the line (y/ao tan ~b) surface energy
A
DbSb/Ds5 s
x tr~ tre
surface curvature average stress acting across the width of the line stress required to prevent the initial cavity from sintering { ?s sin ~, [1 + (L - l)/cos ~]/1 } a~ I/y~ sin ¢ a~/G = a [ 1 + (L - 1)/cos ~ ]- I normalized time, Ds6sf~t/kTa 4 dihedral angle at cavity tip atomic volume of diffusing species
a Z z ~b [2
APPENDIX
The partial differential equation (6), the boundary conditions [equation (7)], and the interface condition [equation (9b)], together represent a moving boundary value problem for the unknowns Y(X, z) and A (z). This Appendix summarises the numerical technique for solving this problem starting from the initial condition [equation (8)]. Following an idea first proposed by Douglas and Gallic [27], the spatial variable X was uniformly discretized so that X~=iAX,
i = 0 , 1,2 . . . . .
(AI)
where A X = d (O)/m, and m is an integer. By definition, the scaled initial cavity length is unity, so that X,, = A (0)= 1. The time variable was discretized non-uniformly, with 30 = 0 and J
~j= ~ Axu, j = [ , 2 . . . . . #=1
(A2)
2500
THOULESS and LINIGER:
x0 x,
.”
.
.
.
.
.
.
I
j&
&+,
.
...”
. . . . .
EFFECTS OF SURFACE DIFFUSION
A
Fig. Al. Grid points showing how the time intervals, ~~- rj_ , , are defined by the time taken for the cavity tip to reach successive values of X,, separated by equal intervals.
ON VOID GROWTH
ing paragraph) was solved to obtain a preliminary void profile Y $‘)(X,,rj). The result Y $) was then substituted into the discrete verston of the interface condition [equation (9b)] which, for v iv,,, was not satisfied but produced a non-vanishing residue Rj!‘). A new guess A77 + ‘) was then made in an effort to reduce the size of this residue, IRf’)I. The process was then repeated until “convergence” was obtained, i.e. until IRy)I had become smaller than a prescribed tolerance. The new guesses were first found by a search procedure which changed Ar; in such a direction that IRy)I decreased. Then, after a sign change in Ry) was observed, the final value AT, was obtained by the secant method. The differential equation (6) was discretized implicitly (backward) with respect to 7. If Y, and Y; denote, respectively, numerical approximations of Y (X, , TV) and Y (Xi, TV_ ,), then equation (6) was replaced by Y,- Y; = --p(Y,+, -4Y;+,+6Y,-4Y,_,+ i = 2,3,
The time steps Azj = rj - rj_ , were chosen in such a way that the moving boundary X = A (2) passed diagonally through the grid _ points [(X,,,+j =.$ir,], j = 0, 1, . .-., where Aj = A (tj) (Fig. Al). Determmmg the position of the moving boundary then amounted to finding the time-step sequence {AT.]. The time points, (t,} themselves were-then computed kern equation (A2). The advantage of this technique was that the last two boundary conditions of equation (7) and the interface. condition [equation (9b)] were always formulated at a grid point of the spatial variable X where a numerical value of Y is defined, rather than in between grid points where Y would have to have been approximated by interpolation. Starting from the initial condition [equation (8)], the moving boundary problem was solved by stepping forward from one time level, 4, to the next. Altogether, an integer number r time steps were carried out, after which the moving boundary was located at X))1+,= (m + r)AX. In order to avoid computational difficulties, r was chosen in such a way that the cavity length X,,,+-1was shorter than the full length L. At each time step, a sequence {Ary)), v = 1,2, . ., v,,, of approximations to Ar, was produced. For each Arf’), the boundary-value problem (discretized as shown in the follow-
Y;_,),
., rn + j - 2,
(A3)
where p = p, = As,/AX4, a discretization which is accurate to first order in 7 but to second order in X. The four boundary conditions [equation (7)] were replaced by the second-order approximations
-SY,+
-3y,+4y,-
Y*=O,
BY,-24Y,+
14Y,-3Y,=O,
Ym+l = 0, 3Y,+j-4Y,+j_,
+ Y,,,+,_,= -2AX,
(A4)
respectively. Equations (A3) and (A4) represent m + j + 1 equations for the m + j + 1 unknowns Y,, Y,, . ., Y,,,,,. Finally, the second-order discretizations Yi,, “(2Y,+j-5Y,+,-,+4Y,+,-*-3Y,+,-,)lAXZ, YC+, =(5Y,,,+,-
1gY,+,_,+24Y,,,+,_z - 14Ym+j_3+ 3Y,,,+j_4)/AX’
were used for Y”[A (T), 71 and Y”‘[A (r), the interface condition of equation (9b).
71
(A5)
which occur in