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A phenomenological description for chemically induced grain boundary migration
Acta metall. Vol. 37, No. 8, pp. 2253-2259, 1989 Printed in Great Britain. All rights reserved
A
OOOI-6160/89 $3.00 + 0.00 Copyright 0 1989 Maxwell Pergamon Macmillan plc
PHENOMENOLOGICAL DESCRIPTION FOR CHEMICALLY INDUCED GRAIN BOUNDARY MIGRATION Y. J. M. BRECHET and G. R. PURDY
Department
of Materials Science and Engineering, McMaster University, Hamilton, Ontario. Canada L8S 4L8 (Received 29 September 1988)
Abstract-The phenomenology of chemically induced grain boundary migration (CIGM) is described in a model which allows the computation of the speed of the moving grain boundary, and of the diffusion field, in a self-consistent way. Special emphasis is given to the problem of a threshold for migration, and to the question of the shape of the moving grain boundary.
R&&-On dCcrit la phtnomknologie de la migration de joints de grains induite chimiquement, par un modkle permettant de simuler la vitesse du joint en mouvement et le champ de diffusion de man&e autocohkente. On s’inttresse plus particulkkement au problZme du seuil de migration, et d la forme du joint de grains en mouvement.am
Zusammenfassung-Die Phlnomenologie der chemisch induzierten Komgrenzwanderung wird mit einem Model1 beschrieben, welches die Berechung der Geschwindigkeit der sich bewegenden Komgrenze und des
Diffusionsfeldes in selbstkonsistenter Weises erlaubt. Die Frage einer Schwelle fiir die Wanderung und die Form der bewegten Komgrenze werden insbesondere behandelt.
1. INTRODUCTION
In recent years, it has been found [l-6] that shortcircuit diffusion paths, particularly grain boundaries, can migrate, and so provide an efficient means of alloying or dealloying a polycrystalline sample. This process, called chemically (or diffusion) induced grain boundary migration, is capable of bringing the system to its homogeneous equilibrium state more quickly than the obvious mechanism of diffusion along stationary grain boundaries, necessarily followed by a volume diffusion step. (For a recent review, see [7 or 81.) The process can be seen as symmetry-breaking: a symmetric boundary has no reason to move in one direction rather than the other. Indeed, experimentally it sometimes moves in one direction, then the other, apparently at random [9, lo]. More often, because of some infinitesimal asymmetry, it begins to move in one direction, and continues, perhaps because the diffusion profile so developed creates a driving force which tends to make it proceed in the same direction. There remains much to be learned about the process. In the present work, we undertake the modelling of some particularly simple geometric cases, in the expectation that some new understanding or guidance of experiment may result.
The process of modelling CIGM can be divided into three stages: Determination of the driving force F which acts on the grain boundary [ 11, 121; Relation of the mobility A4 of the grain boundary to the speed v through v = M F. This mobility cannot be evaluated without knowing the atomistic mechanism (e.g. climb of dislocations) which makes the boundary move in response to the driving force, [13, 141. The self-consistent computation of the speed v of the grain boundary and the diffusion profile, assuming that we know the local driving force f(c), which, averaged over a typical thickness 1 on both sides of the boundary will give the driving force F as a function of the diffusion profile and the local driving force. This last step, which is a phenomenological description of CIGM, is the main one developed here, with emphasis on two special problems: if we assume no solid friction preventing the motion of the grain boundary, can we nevertheless have a threshold for its motion; and if the motion occurs, what will be the shape of the moving grain boundary? The paper is organized as follows: In Section 2 we consider the case of a foil of zero thickness (or 2253
BRECHET and PURDY:
2254
Symmetric
ConcenfrotL
Final state
profile
Asymmetric
: thermodynamic
CHEMICALLY
COnCentrotiOn
INDUCED
profile
equilibrium
Fig. 1. Two ways of attaining thermodynamic equilibrium; with and without grain boundary migration. equivalently a grain boundary of infinite diffusivity.) In 2.1 we state the assumptions underlying our approach. In 2.2 we review briefly the concept of a
driving force for grain boundary motion and develop a form appropriate to the phenomenological description. 2.3 contains a solution to the steady state one dimensional diffusion problem, and an implicit equation which gives the speed o of the grain boundary as a function of the strength and the range of the interaction which determines the driving force. In 2.4 this implicit equation is solved graphically and the existence of a threshold and the asymptotic behaviour are discussed. In Section 3 we consider the case of a finite but large diffusivity D, along the grain boundary. In 3.1 the diffusion problem is solved, and the curvature of the moving boundary is computed as a function of D,. In 3.2 we indicate how the results of Section 2 are to be modified by this curvature. In Section 4, the more general problem of the existence of a steady state shape is addressed. Finally we define the dimensionless variables which govern the existence of the phenomenon and the shape of the moving boundary, and which allow the elucidation of the different types of bchaviour expected. 2. THE ONE-DIMENSIONAL
PROBLEM
2.1. Parameters and hypotheses Figure 2 is a schematic representation of a typical alloying experiment which may lead to CIGM. Solute tThis assumption prohibits inclusion of solute trapping or equivalently a concentration discontinuity at the interface in the solution of the kinetic problem. Nevertheless it is possible to include such a discontinuity in the computation of the driving force, as indicated in a subsequent section.
GRAIN BOUNDARY
MIGRATION
atoms from an external reservoir at concentration C, diffuse into the foil with a volume diffusion coefficient D, and a grain boundary diffusion coefficient D,. For the sake of simplicity, the initial solute concentration in the solid is taken as zero, and the thermodynamic equilibrium concentration that we wish to reach is C,,. The thickness of the foil is 2L and the mobility of the grain boundary is M. The spatial coordinate z originates at the grain boundary (z = 0) and the local concentration of solute is C(z). The thickness of the grain boundary is assumed to be zero?. In addition, the grain boundary is assumed isotropic (and thus essentially structureless) within the plane of the boundary. There is another physical length in the problem: this is the thickness (1) of the layer of material on both sides of the grain boundary which will influence the motion of the boundary. If this thickness were zero, then, unless there were a jump in free energy density across the interface, there would be no driving force. However, this length cannot be less than the correlation length for the computation of the free energy, and thus, even though the concentration may be continuous through the boundary one can have a driving force due to the difference in free energy densities between the layers of thickness 1 on each side of the boundary. In an elementary kinetic calculation, as noted above, we will assign zero thickness to the boundary, and thus require that the concentration profile be continuous through the boundary. Then the asymmetry of the profile will lead to a discontinuity in the “effective concentration” and thus to a driving force. To aid in visualizing this situation, Fig. 3 shows the graphical construction for the driving force for the case of a step function (3a), an asymmetric but continuous profile, (3b), and a concentration discontinuity coupled with a diffusion field in front of the interface (3~). Anticipating the resolution of the diffusion problem, we take C(z) as a constant equal to C,, for z < 0 and a decreasing exponential for z > 0. For this kind of profile, one expects the driving force to be an average over the concentrations
wu)l
Expanding g(c) to the second order about C,, (the concentration in the boundary) gives
F=;
s
'[C(z)-C&
II
_ dz
c-co
s
=fh ‘[C(z)-Co]2dz. 0
In writing the driving force in this form we have implicitely assumed the same dependence of the free energy on both sides of the boundary. We have described only the “chemical driving force”; however,
BRECHET and PURDY:
CHEMICALLY
INDUCED
GRAIN BOUNDARY
MIGRATION
2255
_-__--( +
--_
I
8
i I
-_ Fig. 2. Schematics of an alloying experiment: D, is the diffusion coefficient along the grain boundary; D, is the bulk diffusion coefficient; C, is the external and final concentration of solute; 21 is the thickness of the layer around the boundary which will influence its motion; 2L is the thickness of the sample.
the elastic effect may be included by adding a term Y~~[C(Z)-O]~ to the free energy function for z > 0. For z < 0 there is no gradient and we will assume that the growing phase is elastically relaxed.
\
G Fig. 4. Construction
of the driving force; contribution the elastic term.
of
(0)
Therefore a more general form for F is
F=fs ;{j&(z)-Co]‘+ Yrj;C(z)‘}dz.
(1)
The first term describes the chemical driving force, and the second the additional force due to the self stress of the solute field (Fig. 4). 2.2. The nature of the driving force
Fig. 3. Computation of the driving force: the average is to be performed over the range of concentration present in the layers of thickness I around the boundary (shaded regions): (a) concentration step; (b) concentration gradient; (c) concentration gradient and discontinuity.
Thus far, in our expansion for the driving force, we have assumed a knowledge of the free energy g(c), which allows the computation of fO. At low concentrations C,,, the leading term will derive from the entropy of mixing. Because of the concavity of g(c), the geometrical construction of Fig. 3 will ensure a force of the proper sign for both alloying and dealloying. There has been a measure of controversy about the contention that purely chemical terms can lead to a driving force for grain boundary migration. However, chemical forces for interface migration are widely accepted in other instances: for example, in the discussion of the motion of planar allotrophic transformation front in an undercooled singleelement solid, or of martensitic transformation interface in an alloy, there is no possibility of a solute field stress, and local equilibrium is out of the question. Then, in the absence of a detailed atomistic model for interface migration, it is quite reasonable to write a proportionality between the velocity of the transformation interface and the free energy density difference across it. What remains controversial is the possibility that a purely entropic chemical force might act to move an interface. It is here that the thought experiment advanced by Cahn and Baluffi [13+in which CIGM is induced (or not induced) by the diffiusive flow of an isotope---needs to be pursued in the laboratory.
2256
BRECHET and PURDY:
CHEMICALLY INDUCED GRAIN BOUNDARY MIGRATION
2.3. The diffusion problem In the simplified view presented in 2.1, the grain boundary appears as a source term in the diffusion equation. In the moving frame associated with the grain boundary, the diffusion equation becomes, for steady motion.
Under the conditions which combine concentration discontinuity and a diffusion profile this result must be replaced by D Lx Mlc:
2
g!
=fo K
l
1-e-2” +----
) x
_2(3-;fx,
The general solution of this equation is
+
2 0
C(z)=Aexp
$z (
” )
+B
2.4. The existence of a threshold
+E [exp(z
r)-I]H(r)
(3)
where H is the Heaviside function. C must be finite when z + - co and therefore A = 0. When z -03, C-0, and therefore B = a/v. Assuming
,@fm C(z) = c,
C(z) = Co{ 1+ H(z)[exp(ez)-l]}.
is there-
(4)
We therefore face the following problem: Under a driving force F, a grain boundary moves with speed v = MF. Consequently a diffusion profile C(z, v) is developed. This solute profile allows us to compute, via equation (l), the force F. The system is therefore closed with respect to the self-consistent computation of the speed v. The natural dimensionless variable of the problem is
(l), (4), and (5) gives an
g&.x=[ fo+(h+d
1
-ew2”
Y)-yy
0
-2.
0
Equation (6) can be discussed graphically for two extreme cases of a purely elastic (fO = 0) and a purely chemical (q. = 0) driving force. These extremes show qualitatively differing behaviour: 2.4.1. The purely chemical case. Here, equation (6) reduces to D”
gives b = C,,, and a = UC,. The solution fore
Combining equations implicit equation in x
2q;y.!$. 1
1-e-” x 1
MlCSl
.x= (
1+1-e-2”_2.5 2x
(7)
>
This result is graphically illustrated in Fig. 5(a). If D,/MfCif, is larger than a critical value (which corresponds to too small a mobility or driving force, or too large a volume diffusion coefficient), v = 0 is the only solution; the grain boundary does not move. If D,/MlCzf, is smaller than this critical value, there are non-zero solutions to equation (6), and the grain boundary can assume steady state motion. Note that v jumps from zero to a finite value when the threshold is reached. Immediately above the threshold, two non-zero solutions exist, and we take this to imply the possibility of a branching kinetic instability, leading to jerky motion of the boundary [Fig. S(b)]. 2.4.2. Purely elastic case. For any diffusion profile, the chemical term will be present. This is not the case for the elastic term. The strain energy per unit volume is given by qf YC; D,/2v. If this energy is larger than the critical value 6 (the interaction energy of the dislocations within a coherency-breaking wall), and if sufficient matrix dislocations are available to completely relax the coherency strains, the elastic term will vanish. This mechanism also introduces a threshold, but one of completely different nature. In the case of a pure elastic driving force, equation (6) reduces to 1 --emZX DV MI&/,z Y ‘x=2x.
tIf a 6’ singularity were allowed on the right hand side of (2), it would be possible to have a concentration discontinuity at the interface as well as a diffusion gradient in front of it, as shown in Fig. 3(c).
. X
(8)
This equation is graphically illustrated in Fig. 5(c). Near v = 0, where the coherency loss condition is most likely to be fulfilled, we have v = MYqzCi. Therefore the condition for coherency loss is D,/2M > .s. If DJ2M > E there is no threshold, and the speeds starts from zero and increases parabolically with n,C,.
BRECHET and PURDY:
CHEMICALLY
INDUCED
GRAIN BOUNDARY
MIGRATION
2257
3. THE CASE OF A FINITE BOUNDARY DIFFUSION COEFFICIENT
In the preceeding section we assumed that the concentration along the boundary was equilibrated instantaneously, and this allowed the treatment of the planar case as a one-dimensional problem. In sections 2 and 3, we allow D, < ox, which introduces the problem of the shape of the grain boundary. The shape is a history of the diffusion profile along the grain boundary; the parts of the boundary closer to the surface will start moving before those which are farther away, they will be the first regions saturated with solute (Fig. 6). Therefore one expects the boundary to move “against its curvature” [12], as is observed experimentally. The problem is now more complex: We solve the diffusion problem and, among the possible shapes, select the one which is compatible with the history of diffusion along the grain boundary. Then the velocity v is found self-consistently as for the planar case.
i b)
3. I. The diffusion problem
IC)
Y
1
The shape sketched in Fig. 6 is strongly reminiscent of the “Ivantsov problem”, in which a paraboloid generates a diffusion field, [15]. We will show that an approximate solution is indeed a parabola. Let R be the radius of curvature at the tip. The diffusion equation is written in parabolic coordinated [16]
/
‘c-
X
Fig. 5. Graphical solutions of the equations giving selfconsistently the velocity V. (a) Purely chemical case: 0 corresponds to a driving force under the threshold: no migration occurs; @ is the threshold situation; @ is the above threshold case: two non-xero speeds are possible. (b) Purely chemical case: the velocity versus the driving force showing the possibility of a branching kinetic instability. The dashed line shows qualitatively the effect of a concentration jump at the boundary: the branching instability exists only for a finite range of driving forces. (c) Graphical solution for the purely elastic case.
If D,/2M > e, there is a range of forbidden speeds, and the grain boundary moves only if qi YC; D,/2v is large enough. To summarize: in the purely chemical case, whatever the value of D,, one expects a threshold for migration; in the purely elastic case, depending on the magnitude of D,/A4, a threshold may or may not exist.
Outside the parabola, C = C,, and inside at infinite distance (e = 0), C = 0. The Peclet number is defined by VR p=20,
Along the parabola (S = l), at steady state we must have C = C,. (Because diffusion is allowed in the boundary, if the boundary, if the boundary concentration were inhomogeneous a flux would result and the diffusion field would not be steady.) The solution for the diffusion field in parabolic coordinates is
r; CO ep” dt
C(Yrl)=
I
p epp dt
for
c
s0
11 (al
(b)
(c)
IdI
Fig. 6. The motion of a grain boundary for the case of finite surface diffusion coefficient D,. (a) Initial situation. (b) The grain boundary starts moving at the surface. (c) The grain boundary moves against its own curvature. (d) Effect of a finite thickness on the shape of a moving grain boundary.
(10)
2258
BRECHET and PURDY:
CHEMICALLY
INDUCED
3.2. The velocity
GRAIN BOUNDARY
Integrating
As in the planar case, the driving force will be computed through an average over a layer of thickness I, which means an average over 5 between 1 = 5 and 5 = l/R,/Hq ‘. in consequence, V depends on the abscissa along the parabola: strictly speaking the parabola is not a steady shape. However, if R & L, q $ 1, the translational speed is approximately constant. Qualitatively the effect of finite thickness will be to bend the corners of the parabola [Fig. 6(d)]. We have then with equation (1)
MIGRATION
with respect to time gives
z(x, t) = kQfYC;t
-$Yq;c;
. (14)
X
The first term on the right hand of (14) is the drift of the shape at speed v, and the second term gives the approximate shape at time t. The curvature at the tip is estimated from
I-l/R j-z;
s1
M(W)-C'O)=
c(t)=ld5. (11)
+ Yd
Knowing that u = MF, equations (10) and (11) again yield a self-consistent system to determine the velocity. The radius of curvature R is much larger than the thickness of the sample 2L, and we can expand C in (11) for large Peclet numbers to show that the relative correction to the speed at the surface of the foil relative to that at the centre is of order (L/R)=. Similarly the correction to the speed due to curvature compared to the speed of the planar interface is of order (f/R). Both corrections are negligible for the limits considered here and the speed of the curved boundary is essentially the same as the speed of the planar boundary under the same driving force. 3.3. The curvature of the grain boundary As noted in the introduction to Section 3, the curvature of the boundary depends on the kinetics necessary to ensure the concentration C, along the entire boundary. We can estimate this time dependence as follows: for simplicity, we assume a purely elastic driving force, so that t’ - MY t/,’ C(x, T)
(15) One finds that the radius of curvature will be infinite for D, = co, and that the higher the velocity the more curved the boundary. Note that to perform this estimate of the radius of curvature we have assumed R + L. If the foil is too thick, or if the grain boundary moves too fast, it will becomes more and more difficult to provide solute to the centre of the boundary. At the limit the grain boundary will migrate at the surface only [3]. Obviously, there is then no steady state. The purpose of the next section is to find a criterion for the existence of a steady shape for the moving boundary.
4. CONDITIONS FOR THE EXISTENCE OF A STEADY SHAPE
If there exists a steady shape z(x), then the iterative scheme defined by
(12)
is the speed a time T of a point of abscissa x (Fig. 7). For small curvature, C(x, T) can be estimated from the first harmonic of the transient solution of the diffusion equation for a sheet of length 2L [17]. C(x,T)-C,,[l-icoszexp(gT)].
(13)
In writing (13), we have set the length 2L of the boundary to the thickness 2L of the sheet, and the curvilinear coordinate rl to the Cartesian coordiante x.
MIGi:TION
I
3= 0
MIGRATION
z(x)
L
x
Fig. 7. Schematic of a highly curved grain boundary Section 4.
Fig. 8. Behaviour of a system as a function of the two dimensionless variables of the problem:
D
or 2 for elastic case 2ME > D, MY@;L
for elastic case . >
BRECHET
and PURDY:
CHEMICALLY
INDUCED
should converge. This is a generalization of equation (14) in which the length of the grain boundary becomes lim (2L,). n-m Conversely, if this suite does not converge, then there will be no steady state shape. From equation (16) 32~y~:C;
Z,+,(O) =
L2 = TL
n’D,
’
’
But LZ+, >LZ+Z;:+,(o)=L*+(rL;)2. Therefore if we define the suite A,, by A ,,+,=L;+Y*A; A,=
L*
with
GRAIN
BOUNDARY
MIGRATION
2259
Each parameter governs a different aspect of the system behaviour; if x is larger than a critical value, the grain boundary will not move; if p is smaller than a critical value, no steady shape will be attained and the migration, if it occurs, will be a surface phenomenon; if B is larger than this critical value, the grain boundary will migrate against its curvature and may be able to reach a steady state shape with radius R/L proportional to fl. For the same material then, samples with differing thickness should show different behaviour. For sufficiently thin specimens, we expect to find threshold behaviour for migration, depending on the nature of the driving force. If the force is chemical in nature, the present treatment suggests that a branching instability may develop, leading to irregular migration. It is hoped that the results of this modelling exercise will stimulate further experimental study, in regimes
where
limiting
or transition
behaviour
might
be expected.
The divergence of A, will imply the divergence of L, and therefore the nonexistence of a steady state. It can be. shown that the suite A, coverges if and only if
Acknowledgements-This research was supported by the Natural Sciences and Engineering Research Council of Canada. We are grateful to J. D. Embury for stimulating discussions.
REFERENCES L’<$
We have then an upper bound on the value L,, the thickness above which there will be no steady state; in other words if L>L,=L
K’D 64v
it is certain that there will be no steady state, but surface migration only. As might be expected, this upper bound (and likely the real threshold as well) increases with D, and decreases with v (and therefore with the driving force and the mobility).
5.
6. I.
5. CONCLUSIONS
We can summarize the preceeding description of CIGM as follows. The problem is governed by two dimensionless parameters
8. 9. 10.
D” 11. 12.
a=Mf,l D” or 2Mc for driving forces of elastic origin (
>
13.
and 14. 15.
DS Or MY?@;L
for the elastic driving force . >
16. 17.
M. Hillert and G. R. Purdy, Acta metall. 26, 333 (1978). L. Chongmo and M. Hillert. Actu metall. 29, 1949 (1981). L,. Chongmo and M. Hillert, Acta mefall. 30, 1113 (1982). J. R. Michael and D. B. Williams. Proc. Conf on Interface Migration and Control of Microstructure (edited by Pande, King, Smith and Walter), p. 73. Am. Sot. Metals, Metals Metals Park, Ohio (1986). G. R. Purdy, K. Tashiro, Proc. Conf on Interfuce Migration and Control of Microsnucture (edited by Pande, King, Smith and Walter), p. 33. Am. Sot. Metals, Metals Park. Ohio (1986). K. Tashiro, G. R. Purdy. Scripfa mefall. 21, 361 (1987). P. G. Shewmon, G. Meyrick, Proc. Conf on Interface Migration and Control of Microstructure (edited by Pande, King, Smith and Walter), p. 7. Am. Sot. Metals, Metals, Metals Park, Ohio (1986). C. Handwerker. in Dijiision Phenomena in Thin Films (edited bv D. Gunta). Noves (1988). G. Meyrick, V. Siyer and-P. G. Shewmon, Acta metall. 33, 273, (1985). J. D. Pan and R. W. Baluffi, Acfn me/all. 30, 861 (1982). M. Hillert, Scripta mefall. 17, 237 (1983). G. R. Purdy, in Proc. Symp. on Solute-defecr Inteructions (edited by Saimoto, Purdy and Kidson), p, 382, Pergamon Press: Oxford (1986): J. W. Cahn and R. W. Baluffi, Scripra metah. 13, 503 (I 979). R. W. Baluffi and J. W. Cahn, Acta metall. 29, 493 (1981). W. P. Ivantsov, Dohl. Abad. Nauks SSSR 58, 567 (1947). G. Horvay and J. W. Cahn, Acta metall. 9, 695 (1961). J. Crank, The Mathematics of D$iision. Oxford Univ. Press (1970).
A
OOOI-6160/89 $3.00 + 0.00 Copyright 0 1989 Maxwell Pergamon Macmillan plc
PHENOMENOLOGICAL DESCRIPTION FOR CHEMICALLY INDUCED GRAIN BOUNDARY MIGRATION Y. J. M. BRECHET and G. R. PURDY
Department
of Materials Science and Engineering, McMaster University, Hamilton, Ontario. Canada L8S 4L8 (Received 29 September 1988)
Abstract-The phenomenology of chemically induced grain boundary migration (CIGM) is described in a model which allows the computation of the speed of the moving grain boundary, and of the diffusion field, in a self-consistent way. Special emphasis is given to the problem of a threshold for migration, and to the question of the shape of the moving grain boundary.
R&&-On dCcrit la phtnomknologie de la migration de joints de grains induite chimiquement, par un modkle permettant de simuler la vitesse du joint en mouvement et le champ de diffusion de man&e autocohkente. On s’inttresse plus particulkkement au problZme du seuil de migration, et d la forme du joint de grains en mouvement.am
Zusammenfassung-Die Phlnomenologie der chemisch induzierten Komgrenzwanderung wird mit einem Model1 beschrieben, welches die Berechung der Geschwindigkeit der sich bewegenden Komgrenze und des
Diffusionsfeldes in selbstkonsistenter Weises erlaubt. Die Frage einer Schwelle fiir die Wanderung und die Form der bewegten Komgrenze werden insbesondere behandelt.
1. INTRODUCTION
In recent years, it has been found [l-6] that shortcircuit diffusion paths, particularly grain boundaries, can migrate, and so provide an efficient means of alloying or dealloying a polycrystalline sample. This process, called chemically (or diffusion) induced grain boundary migration, is capable of bringing the system to its homogeneous equilibrium state more quickly than the obvious mechanism of diffusion along stationary grain boundaries, necessarily followed by a volume diffusion step. (For a recent review, see [7 or 81.) The process can be seen as symmetry-breaking: a symmetric boundary has no reason to move in one direction rather than the other. Indeed, experimentally it sometimes moves in one direction, then the other, apparently at random [9, lo]. More often, because of some infinitesimal asymmetry, it begins to move in one direction, and continues, perhaps because the diffusion profile so developed creates a driving force which tends to make it proceed in the same direction. There remains much to be learned about the process. In the present work, we undertake the modelling of some particularly simple geometric cases, in the expectation that some new understanding or guidance of experiment may result.
The process of modelling CIGM can be divided into three stages: Determination of the driving force F which acts on the grain boundary [ 11, 121; Relation of the mobility A4 of the grain boundary to the speed v through v = M F. This mobility cannot be evaluated without knowing the atomistic mechanism (e.g. climb of dislocations) which makes the boundary move in response to the driving force, [13, 141. The self-consistent computation of the speed v of the grain boundary and the diffusion profile, assuming that we know the local driving force f(c), which, averaged over a typical thickness 1 on both sides of the boundary will give the driving force F as a function of the diffusion profile and the local driving force. This last step, which is a phenomenological description of CIGM, is the main one developed here, with emphasis on two special problems: if we assume no solid friction preventing the motion of the grain boundary, can we nevertheless have a threshold for its motion; and if the motion occurs, what will be the shape of the moving grain boundary? The paper is organized as follows: In Section 2 we consider the case of a foil of zero thickness (or 2253
BRECHET and PURDY:
2254
Symmetric
ConcenfrotL
Final state
profile
Asymmetric
: thermodynamic
CHEMICALLY
COnCentrotiOn
INDUCED
profile
equilibrium
Fig. 1. Two ways of attaining thermodynamic equilibrium; with and without grain boundary migration. equivalently a grain boundary of infinite diffusivity.) In 2.1 we state the assumptions underlying our approach. In 2.2 we review briefly the concept of a
driving force for grain boundary motion and develop a form appropriate to the phenomenological description. 2.3 contains a solution to the steady state one dimensional diffusion problem, and an implicit equation which gives the speed o of the grain boundary as a function of the strength and the range of the interaction which determines the driving force. In 2.4 this implicit equation is solved graphically and the existence of a threshold and the asymptotic behaviour are discussed. In Section 3 we consider the case of a finite but large diffusivity D, along the grain boundary. In 3.1 the diffusion problem is solved, and the curvature of the moving boundary is computed as a function of D,. In 3.2 we indicate how the results of Section 2 are to be modified by this curvature. In Section 4, the more general problem of the existence of a steady state shape is addressed. Finally we define the dimensionless variables which govern the existence of the phenomenon and the shape of the moving boundary, and which allow the elucidation of the different types of bchaviour expected. 2. THE ONE-DIMENSIONAL
PROBLEM
2.1. Parameters and hypotheses Figure 2 is a schematic representation of a typical alloying experiment which may lead to CIGM. Solute tThis assumption prohibits inclusion of solute trapping or equivalently a concentration discontinuity at the interface in the solution of the kinetic problem. Nevertheless it is possible to include such a discontinuity in the computation of the driving force, as indicated in a subsequent section.
GRAIN BOUNDARY
MIGRATION
atoms from an external reservoir at concentration C, diffuse into the foil with a volume diffusion coefficient D, and a grain boundary diffusion coefficient D,. For the sake of simplicity, the initial solute concentration in the solid is taken as zero, and the thermodynamic equilibrium concentration that we wish to reach is C,,. The thickness of the foil is 2L and the mobility of the grain boundary is M. The spatial coordinate z originates at the grain boundary (z = 0) and the local concentration of solute is C(z). The thickness of the grain boundary is assumed to be zero?. In addition, the grain boundary is assumed isotropic (and thus essentially structureless) within the plane of the boundary. There is another physical length in the problem: this is the thickness (1) of the layer of material on both sides of the grain boundary which will influence the motion of the boundary. If this thickness were zero, then, unless there were a jump in free energy density across the interface, there would be no driving force. However, this length cannot be less than the correlation length for the computation of the free energy, and thus, even though the concentration may be continuous through the boundary one can have a driving force due to the difference in free energy densities between the layers of thickness 1 on each side of the boundary. In an elementary kinetic calculation, as noted above, we will assign zero thickness to the boundary, and thus require that the concentration profile be continuous through the boundary. Then the asymmetry of the profile will lead to a discontinuity in the “effective concentration” and thus to a driving force. To aid in visualizing this situation, Fig. 3 shows the graphical construction for the driving force for the case of a step function (3a), an asymmetric but continuous profile, (3b), and a concentration discontinuity coupled with a diffusion field in front of the interface (3~). Anticipating the resolution of the diffusion problem, we take C(z) as a constant equal to C,, for z < 0 and a decreasing exponential for z > 0. For this kind of profile, one expects the driving force to be an average over the concentrations
wu)l
Expanding g(c) to the second order about C,, (the concentration in the boundary) gives
F=;
s
'[C(z)-C&
II
_ dz
c-co
s
=fh ‘[C(z)-Co]2dz. 0
In writing the driving force in this form we have implicitely assumed the same dependence of the free energy on both sides of the boundary. We have described only the “chemical driving force”; however,
BRECHET and PURDY:
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MIGRATION
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_-__--( +
--_
I
8
i I
-_ Fig. 2. Schematics of an alloying experiment: D, is the diffusion coefficient along the grain boundary; D, is the bulk diffusion coefficient; C, is the external and final concentration of solute; 21 is the thickness of the layer around the boundary which will influence its motion; 2L is the thickness of the sample.
the elastic effect may be included by adding a term Y~~[C(Z)-O]~ to the free energy function for z > 0. For z < 0 there is no gradient and we will assume that the growing phase is elastically relaxed.
\
G Fig. 4. Construction
of the driving force; contribution the elastic term.
of
(0)
Therefore a more general form for F is
F=fs ;{j&(z)-Co]‘+ Yrj;C(z)‘}dz.
(1)
The first term describes the chemical driving force, and the second the additional force due to the self stress of the solute field (Fig. 4). 2.2. The nature of the driving force
Fig. 3. Computation of the driving force: the average is to be performed over the range of concentration present in the layers of thickness I around the boundary (shaded regions): (a) concentration step; (b) concentration gradient; (c) concentration gradient and discontinuity.
Thus far, in our expansion for the driving force, we have assumed a knowledge of the free energy g(c), which allows the computation of fO. At low concentrations C,,, the leading term will derive from the entropy of mixing. Because of the concavity of g(c), the geometrical construction of Fig. 3 will ensure a force of the proper sign for both alloying and dealloying. There has been a measure of controversy about the contention that purely chemical terms can lead to a driving force for grain boundary migration. However, chemical forces for interface migration are widely accepted in other instances: for example, in the discussion of the motion of planar allotrophic transformation front in an undercooled singleelement solid, or of martensitic transformation interface in an alloy, there is no possibility of a solute field stress, and local equilibrium is out of the question. Then, in the absence of a detailed atomistic model for interface migration, it is quite reasonable to write a proportionality between the velocity of the transformation interface and the free energy density difference across it. What remains controversial is the possibility that a purely entropic chemical force might act to move an interface. It is here that the thought experiment advanced by Cahn and Baluffi [13+in which CIGM is induced (or not induced) by the diffiusive flow of an isotope---needs to be pursued in the laboratory.
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CHEMICALLY INDUCED GRAIN BOUNDARY MIGRATION
2.3. The diffusion problem In the simplified view presented in 2.1, the grain boundary appears as a source term in the diffusion equation. In the moving frame associated with the grain boundary, the diffusion equation becomes, for steady motion.
Under the conditions which combine concentration discontinuity and a diffusion profile this result must be replaced by D Lx Mlc:
2
g!
=fo K
l
1-e-2” +----
) x
_2(3-;fx,
The general solution of this equation is
+
2 0
C(z)=Aexp
$z (
” )
+B
2.4. The existence of a threshold
+E [exp(z
r)-I]H(r)
(3)
where H is the Heaviside function. C must be finite when z + - co and therefore A = 0. When z -03, C-0, and therefore B = a/v. Assuming
,@fm C(z) = c,
C(z) = Co{ 1+ H(z)[exp(ez)-l]}.
is there-
(4)
We therefore face the following problem: Under a driving force F, a grain boundary moves with speed v = MF. Consequently a diffusion profile C(z, v) is developed. This solute profile allows us to compute, via equation (l), the force F. The system is therefore closed with respect to the self-consistent computation of the speed v. The natural dimensionless variable of the problem is
(l), (4), and (5) gives an
g&.x=[ fo+(h+d
1
-ew2”
Y)-yy
0
-2.
0
Equation (6) can be discussed graphically for two extreme cases of a purely elastic (fO = 0) and a purely chemical (q. = 0) driving force. These extremes show qualitatively differing behaviour: 2.4.1. The purely chemical case. Here, equation (6) reduces to D”
gives b = C,,, and a = UC,. The solution fore
Combining equations implicit equation in x
2q;y.!$. 1
1-e-” x 1
MlCSl
.x= (
1+1-e-2”_2.5 2x
(7)
>
This result is graphically illustrated in Fig. 5(a). If D,/MfCif, is larger than a critical value (which corresponds to too small a mobility or driving force, or too large a volume diffusion coefficient), v = 0 is the only solution; the grain boundary does not move. If D,/MlCzf, is smaller than this critical value, there are non-zero solutions to equation (6), and the grain boundary can assume steady state motion. Note that v jumps from zero to a finite value when the threshold is reached. Immediately above the threshold, two non-zero solutions exist, and we take this to imply the possibility of a branching kinetic instability, leading to jerky motion of the boundary [Fig. S(b)]. 2.4.2. Purely elastic case. For any diffusion profile, the chemical term will be present. This is not the case for the elastic term. The strain energy per unit volume is given by qf YC; D,/2v. If this energy is larger than the critical value 6 (the interaction energy of the dislocations within a coherency-breaking wall), and if sufficient matrix dislocations are available to completely relax the coherency strains, the elastic term will vanish. This mechanism also introduces a threshold, but one of completely different nature. In the case of a pure elastic driving force, equation (6) reduces to 1 --emZX DV MI&/,z Y ‘x=2x.
tIf a 6’ singularity were allowed on the right hand side of (2), it would be possible to have a concentration discontinuity at the interface as well as a diffusion gradient in front of it, as shown in Fig. 3(c).
. X
(8)
This equation is graphically illustrated in Fig. 5(c). Near v = 0, where the coherency loss condition is most likely to be fulfilled, we have v = MYqzCi. Therefore the condition for coherency loss is D,/2M > .s. If DJ2M > E there is no threshold, and the speeds starts from zero and increases parabolically with n,C,.
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MIGRATION
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3. THE CASE OF A FINITE BOUNDARY DIFFUSION COEFFICIENT
In the preceeding section we assumed that the concentration along the boundary was equilibrated instantaneously, and this allowed the treatment of the planar case as a one-dimensional problem. In sections 2 and 3, we allow D, < ox, which introduces the problem of the shape of the grain boundary. The shape is a history of the diffusion profile along the grain boundary; the parts of the boundary closer to the surface will start moving before those which are farther away, they will be the first regions saturated with solute (Fig. 6). Therefore one expects the boundary to move “against its curvature” [12], as is observed experimentally. The problem is now more complex: We solve the diffusion problem and, among the possible shapes, select the one which is compatible with the history of diffusion along the grain boundary. Then the velocity v is found self-consistently as for the planar case.
i b)
3. I. The diffusion problem
IC)
Y
1
The shape sketched in Fig. 6 is strongly reminiscent of the “Ivantsov problem”, in which a paraboloid generates a diffusion field, [15]. We will show that an approximate solution is indeed a parabola. Let R be the radius of curvature at the tip. The diffusion equation is written in parabolic coordinated [16]
/
‘c-
X
Fig. 5. Graphical solutions of the equations giving selfconsistently the velocity V. (a) Purely chemical case: 0 corresponds to a driving force under the threshold: no migration occurs; @ is the threshold situation; @ is the above threshold case: two non-xero speeds are possible. (b) Purely chemical case: the velocity versus the driving force showing the possibility of a branching kinetic instability. The dashed line shows qualitatively the effect of a concentration jump at the boundary: the branching instability exists only for a finite range of driving forces. (c) Graphical solution for the purely elastic case.
If D,/2M > e, there is a range of forbidden speeds, and the grain boundary moves only if qi YC; D,/2v is large enough. To summarize: in the purely chemical case, whatever the value of D,, one expects a threshold for migration; in the purely elastic case, depending on the magnitude of D,/A4, a threshold may or may not exist.
Outside the parabola, C = C,, and inside at infinite distance (e = 0), C = 0. The Peclet number is defined by VR p=20,
Along the parabola (S = l), at steady state we must have C = C,. (Because diffusion is allowed in the boundary, if the boundary, if the boundary concentration were inhomogeneous a flux would result and the diffusion field would not be steady.) The solution for the diffusion field in parabolic coordinates is
r; CO ep” dt
C(Yrl)=
I
p epp dt
for
c
s0
11 (al
(b)
(c)
IdI
Fig. 6. The motion of a grain boundary for the case of finite surface diffusion coefficient D,. (a) Initial situation. (b) The grain boundary starts moving at the surface. (c) The grain boundary moves against its own curvature. (d) Effect of a finite thickness on the shape of a moving grain boundary.
(10)
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3.2. The velocity
GRAIN BOUNDARY
Integrating
As in the planar case, the driving force will be computed through an average over a layer of thickness I, which means an average over 5 between 1 = 5 and 5 = l/R,/Hq ‘. in consequence, V depends on the abscissa along the parabola: strictly speaking the parabola is not a steady shape. However, if R & L, q $ 1, the translational speed is approximately constant. Qualitatively the effect of finite thickness will be to bend the corners of the parabola [Fig. 6(d)]. We have then with equation (1)
MIGRATION
with respect to time gives
z(x, t) = kQfYC;t
-$Yq;c;
. (14)
X
The first term on the right hand of (14) is the drift of the shape at speed v, and the second term gives the approximate shape at time t. The curvature at the tip is estimated from
I-l/R j-z;
s1
M(W)-C'O)=
c(t)=ld5. (11)
+ Yd
Knowing that u = MF, equations (10) and (11) again yield a self-consistent system to determine the velocity. The radius of curvature R is much larger than the thickness of the sample 2L, and we can expand C in (11) for large Peclet numbers to show that the relative correction to the speed at the surface of the foil relative to that at the centre is of order (L/R)=. Similarly the correction to the speed due to curvature compared to the speed of the planar interface is of order (f/R). Both corrections are negligible for the limits considered here and the speed of the curved boundary is essentially the same as the speed of the planar boundary under the same driving force. 3.3. The curvature of the grain boundary As noted in the introduction to Section 3, the curvature of the boundary depends on the kinetics necessary to ensure the concentration C, along the entire boundary. We can estimate this time dependence as follows: for simplicity, we assume a purely elastic driving force, so that t’ - MY t/,’ C(x, T)
(15) One finds that the radius of curvature will be infinite for D, = co, and that the higher the velocity the more curved the boundary. Note that to perform this estimate of the radius of curvature we have assumed R + L. If the foil is too thick, or if the grain boundary moves too fast, it will becomes more and more difficult to provide solute to the centre of the boundary. At the limit the grain boundary will migrate at the surface only [3]. Obviously, there is then no steady state. The purpose of the next section is to find a criterion for the existence of a steady shape for the moving boundary.
4. CONDITIONS FOR THE EXISTENCE OF A STEADY SHAPE
If there exists a steady shape z(x), then the iterative scheme defined by
(12)
is the speed a time T of a point of abscissa x (Fig. 7). For small curvature, C(x, T) can be estimated from the first harmonic of the transient solution of the diffusion equation for a sheet of length 2L [17]. C(x,T)-C,,[l-icoszexp(gT)].
(13)
In writing (13), we have set the length 2L of the boundary to the thickness 2L of the sheet, and the curvilinear coordinate rl to the Cartesian coordiante x.
MIGi:TION
I
3= 0
MIGRATION
z(x)
L
x
Fig. 7. Schematic of a highly curved grain boundary Section 4.
Fig. 8. Behaviour of a system as a function of the two dimensionless variables of the problem:
D
or 2 for elastic case 2ME > D, MY@;L
for elastic case . >
BRECHET
and PURDY:
CHEMICALLY
INDUCED
should converge. This is a generalization of equation (14) in which the length of the grain boundary becomes lim (2L,). n-m Conversely, if this suite does not converge, then there will be no steady state shape. From equation (16) 32~y~:C;
Z,+,(O) =
L2 = TL
n’D,
’
’
But LZ+, >LZ+Z;:+,(o)=L*+(rL;)2. Therefore if we define the suite A,, by A ,,+,=L;+Y*A; A,=
L*
with
GRAIN
BOUNDARY
MIGRATION
2259
Each parameter governs a different aspect of the system behaviour; if x is larger than a critical value, the grain boundary will not move; if p is smaller than a critical value, no steady shape will be attained and the migration, if it occurs, will be a surface phenomenon; if B is larger than this critical value, the grain boundary will migrate against its curvature and may be able to reach a steady state shape with radius R/L proportional to fl. For the same material then, samples with differing thickness should show different behaviour. For sufficiently thin specimens, we expect to find threshold behaviour for migration, depending on the nature of the driving force. If the force is chemical in nature, the present treatment suggests that a branching instability may develop, leading to irregular migration. It is hoped that the results of this modelling exercise will stimulate further experimental study, in regimes
where
limiting
or transition
behaviour
might
be expected.
The divergence of A, will imply the divergence of L, and therefore the nonexistence of a steady state. It can be. shown that the suite A, coverges if and only if
Acknowledgements-This research was supported by the Natural Sciences and Engineering Research Council of Canada. We are grateful to J. D. Embury for stimulating discussions.
REFERENCES L’<$
We have then an upper bound on the value L,, the thickness above which there will be no steady state; in other words if L>L,=L
K’D 64v
it is certain that there will be no steady state, but surface migration only. As might be expected, this upper bound (and likely the real threshold as well) increases with D, and decreases with v (and therefore with the driving force and the mobility).
5.
6. I.
5. CONCLUSIONS
We can summarize the preceeding description of CIGM as follows. The problem is governed by two dimensionless parameters
8. 9. 10.
D” 11. 12.
a=Mf,l D” or 2Mc for driving forces of elastic origin (
>
13.
and 14. 15.
DS Or MY?@;L
for the elastic driving force . >
16. 17.
M. Hillert and G. R. Purdy, Acta metall. 26, 333 (1978). L. Chongmo and M. Hillert. Actu metall. 29, 1949 (1981). L,. Chongmo and M. Hillert, Acta mefall. 30, 1113 (1982). J. R. Michael and D. B. Williams. Proc. Conf on Interface Migration and Control of Microstructure (edited by Pande, King, Smith and Walter), p. 73. Am. Sot. Metals, Metals Metals Park, Ohio (1986). G. R. Purdy, K. Tashiro, Proc. Conf on Interfuce Migration and Control of Microsnucture (edited by Pande, King, Smith and Walter), p. 33. Am. Sot. Metals, Metals Park. Ohio (1986). K. Tashiro, G. R. Purdy. Scripfa mefall. 21, 361 (1987). P. G. Shewmon, G. Meyrick, Proc. Conf on Interface Migration and Control of Microstructure (edited by Pande, King, Smith and Walter), p. 7. Am. Sot. Metals, Metals, Metals Park, Ohio (1986). C. Handwerker. in Dijiision Phenomena in Thin Films (edited bv D. Gunta). Noves (1988). G. Meyrick, V. Siyer and-P. G. Shewmon, Acta metall. 33, 273, (1985). J. D. Pan and R. W. Baluffi, Acfn me/all. 30, 861 (1982). M. Hillert, Scripta mefall. 17, 237 (1983). G. R. Purdy, in Proc. Symp. on Solute-defecr Inteructions (edited by Saimoto, Purdy and Kidson), p, 382, Pergamon Press: Oxford (1986): J. W. Cahn and R. W. Baluffi, Scripra metah. 13, 503 (I 979). R. W. Baluffi and J. W. Cahn, Acta metall. 29, 493 (1981). W. P. Ivantsov, Dohl. Abad. Nauks SSSR 58, 567 (1947). G. Horvay and J. W. Cahn, Acta metall. 9, 695 (1961). J. Crank, The Mathematics of D$iision. Oxford Univ. Press (1970).