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Influence of crystallography upon critical nucleus shapes and kinetics of homogeneous f.c.c.-f.c.c. nucleation—III. The influence of elastic strain energy
Acta metall. Vol. 32, No. 10, pp. 1845-1853, 1984 Printed in Great Britain. All rights reserved
0001-6160/84 $3.00+0.00 Copyright © 1984 Pergamon Press Ltd
INFLUENCE OF CRYSTALLOGRAPHY UPON CRITICAL NUCLEUS SHAPES AND KINETICS OF HOMOGENEOUS F.C.C.-F.C.C. NUCLEATION--III. THE INFLUENCE OF ELASTIC STRAIN ENERGY F. K. LEGOUESt, H. I. AARONSON and Y. W. LEE:~ Department of Metallurgical Engineering and Materials Science, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.
(Received 14 September 1983) Akstraet--The influence of crystallography upon critical nucleus shapes and kinetics of homogeneous f.c.c.-f.c.c, nucleation was studied by combining the discrete lattice point non-classical model utilized in paper II of this series with the microscopic theory of strain energy as applied by Cook and de Fontaine to compositional fluctuations in cubic lattices. This permits simultaneous consideration of the influence of anisotropic interfacial energy and anisotropic strain energy upon nucleation. This is extremely difficult to do using other models of strain energy because of mathematical problems. The model is applied to f.c.c, nuclei in AI-Cu and Cu-Co alloys. Nuclei in A1-Cu are plates because coherency strain energy predominates in this system. In Cu-Co alloys, the misft is smaller and the distortion tensor is cubic; thus the nuclei are essentially spherical. Despite this lesser effect, the influence of strain energy upon the nucleation kinetics of Co-rich precipitates in a Cu-rich Cu-Co alloy is found to be very important, reducing these kinetics by as much as five orders of magnitude. R6sum6--Nous avons &udi6 l'influence de la cristallographie sur la forme des germes critiques et sur la cin6tique de la germination homog~ne c.f.c.-c.f.c, en combinant le module non-classique de noeuds discrets utilis~ darts le deuxi+me article de cette s6rie avec la th6orie microscopique de l'6nergie de d6formation appliqu6e par Cook et de Fontaine aux fluctuations de composition dans les r6seaux cubiques. Ceci permet de consid~rer simultan6ment les influence de r6nergie interfaciale anisotrope et de l'6nergie de d&ormation anisotrope sur la germination. Ceci est extr~mement difficile fi faire en utilisant d'autres mod61es pour l'+nergie de d&ormation du fait de difficult+s math6matiques. Nous appliquons ce mod/fle ~ des germes c.f.c, dans des alliages AI-Cu et Cu-Co. Dans A1-Cu, les germes sont des plaquettes car l'6nergie de d~formation de coh6rence domine dans ce syst+me. Dans les alliages Cu-Co, le d6saccord est plus faible et le tenseur de la distorsion est cubique; les germes sont ainsi essentiellement sph6riques. Malgr~ ce moindre effet, nous avons trouv6 que l'influence de l'~nergie de d+formation sur la cin6tique de germination de pr6cipit+s riches en cobalt dans un alliage Cu-Co fiche en cuivre &ait tr+s importante et diminuait cette cin6tique de quelques cinq ordres de grandeur.
Zusammenfassung--Der EinfluB der Kristallografie auf die Form kritischer Keime und die Kinetik homogener kfz-kfz. Keimbildung wurde untersucht, indem das im Teil II dieser Serie benutzte nichtklassische Modell diskreter Gitterpunkte kombiniert wird mit der mikroskopischen Theorie der Verzerrungsenergie, wie sie von Cook und de Fontaine auf Fluktuationen in der Zusammensetzung in kubischen Gittern angewendet worden ist. Mit dieser Kombination kann der EinfluB der anisotropen Grenzflfichenund der Verzerrungsenergie auf die Keimbildung gleichzeitig behandelt werden. Mit anderen Modellen w/ire diese Aufgabe mathematisch/iuBerst schwierig. Das Modell wird auf kfz. Keime in AI-Cuund Cu-Co-Legierungen angewendet. Die Keime in Al~=u sind Platten, da die Energie in diesem System iiberwiegend durch Koh/irenzspannungen bestimmt ist. In den Cu-Co-Legierungen ist die Fehlpassung geringer, und der Distorsionstensor ist kubisch, daher sind die Keime im wesentlichen kugelf6rmig. Trotz dieses kleineren Effektes findet sich, dab der EinfluB der Verzerrungsenergie auf die Keimbildung Co-reicher Ausscheidungen in einer Cu-reichen Cu-Co-Legierung sehr wichtig ist; die Kinetik wird um fiinf Gr6Benordnungen verkleinert.
1. INTRODUCTION In the previous papers [1, 2] in this series, the shapes o f critical nuclei a n d kinetics o f nucleation were studied for h o m o g e n e o u s nucleation o f f.c.c, crystals tPresent address: IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. :[:Present address: Department of Metallurgy and Mining Engineering, University of Illinois at UrbanaChampaign, Urbana, IL 61801, U.S.A. A.M. 32/10~R
in f.c.c, matrices in b i n a r y alloy systems u n d e r the a s s u m p t i o n t h a t n o elastic strain energy is involved, i.e. t h a t the lattice p a r a m e t e r o f the two phases is the same a n d is i n d e p e n d e n t of c o m p o s i t i o n in b o t h phases. This a s s u m p t i o n n o t only ignores the effect o f strain energy o n the kinetics o f nucleation but, perh a p s m o r e importantly, neglects the effect o f anisotropy o f elastic properties o n the shape of critical nuclei. Occurrence o f plate s h a p e d G P zones is associated with large differences in the lattice p a r a m -
1845
1846
LEGOUES et al.:
INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--Ill
eter of the zones and their matrix phase; this is often explained by anisotropic elasticity [3]. The problem of calculating the elastic strain energy, W, of completely coherent precipitates has often been treated. Eshelby [4] wrote an expression for this energy as a function of Young's modulus, E and Poisson's ratio, v, when these parameters are the same in both the matrix and the precipitate. Crum showed that in this case, the expression for coherent strain energy does not depend on the shape of the precipitate [5]. Laszlo [6] derived expressions for W when E and v are different in the precipitate and the matrix (the so-called "inhomogeneous" case) for a plate, a cylinder and a sphere. Barnett et al. [7] gave a very complicated expression for W of coherent ellipsoids of revolution as a continuous function of their aspect ratio, which encompassed the particular shapes analyzed by Laszlo. Finally, Johnson et al. [8] treated the problem of cuboidal inhomogeneous precipitates. Their treatment could be applied to other shapes, such as plates, with little modification. All of these treatments consider W independently of the interfacial energy term. In other words, they apply mostly to large precipitates, when the interfacial energy term becomes negligible relative to the strain energy term. Lee et al. [9] examined one aspect of this problem by considering the interplay of the two contributing factors to the free energy of formation of an embryo, in the mathematically tractable case of ellipsoids of revolution whose interfacial and elastic strain energies were isotropic and the transformation strain was purely dilatational. Johnson and Cahn [10] have recently treated this problem using bifurcation theory. They studied the way in which morphology changes with the size of a cluster, i.e. the transition from a spherical precipitate (interfacial energy dominated) to an ellipsoid precipitate (strain energy dominated), as a function of size. Because of mathematical difficulties, only ellipsoidal shapes were considered under the isotropic elasticity assumption and with a cubic distortion tensor. It should also be noted that Cahn [11, 12] included elastic anisotropy in his continuum model and studied spinodal decomposition and certain aspects of nucleation under these conditions. The continuum model is relatively difficult to apply exactly to the nucleation problem because of the need to "cut" arbitrarily the Fourier decomposition, used in determining the nucleus concentration profile, in order to perform the integration of the free energy over the whole system [13]. In the present paper, a similar treatment to that of Cahn [13] is used but in the context of a discrete lattice point model of a compositionally inhomogeneous system, applied to f.c.c. ---, f.c.c, nucleation. The general equations for the elastic strain energy of a compositionally inhomogeneous system are based on the microscopic theory of elasticity [14], as applied by Cook and de Fontaine [t5] to the strain energy associated with ordering, de Fontaine [16, 17]
also used this thebry to calculate the strain energy of precipitates of different shapes as well as of arrays of precipitates and the influence of strain energy upon the arrangement of precipitates, de Fontaine [16, 17] noted that his calculation did not consider the interfacial energy and is thus not applicable to the study of nucleation. The Cook and de Fontaine [15] model for the strain energy of compositionally inhomogeneous systems will be added to the discrete lattice point model of Cook et al. [18] in order to treat simultaneously the interfacial energy, the chemical volume free energy and the strain energy of an embryo. Minimization of the total free energy of the system then yields a more accurate shape and free energy of formation of the critical nucleus. Although the equations for the strain energy, as developed by Cook and de Fontaine, are fairly complicated mathematically, this model is relatively easy to treat numerically and thus permits a more general treatment than has been previously utilized. In particular, it directly includes elastic as well as crystalline anisotropy, is not restricted to a specific type of nucleus shape and can treat cubic as well as tetragonal distortion tensors. 2. FREE ENERGY OF A COMPOSITIONALLY
NON-UNIFORM SYSTEM 2.1. C h e m i c a l f r e e e n e r g y
In the previous paper in this series [2], the chemical free energy of a compositionally inhomogeneous system was written as [18]
= .Iv Z AF(c,) ~vo p /.
+ ~5a2~ [c(p + r ) - c(p)] 2
(1)
where Nv is the number of atoms per unit volume, ce is the probability of having an atom of type B at site p, AF(cp) is the free energy, relative to a system of composition Co, that one atom at site p would have in compositionally homogeneous surroundings, K is the gradient energy coefficient and a is the lattice parameter. The summation over p is done over all relevant sites in the system and the summation over r is done over all the nearest neighbors of site p. The case of a regular solution, nearest-neighbors model was ' treated in the previous paper and will be used here. In this case K -
kT, a 2
(2)
12
AF(cp) = k T Fop in cp + (1 - cp) In (1 - Cp) (1 -- Co) L co -- 2kT~(c, -- Co)-'[
(3)
I
where k is Boltzmann's constant, T,, is the critical temperature of the miscibility gap and co is the initial and average composition of the matrix.
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--Ill
2.2. Elastic strain energy
and
Cook and de Fontaine [15] used the microscopic theory [14] to obtain the elastic strain energy of a compositionally inhomogeneous system. With this theory and the harmonic approximation, they wrote the elastic free energy of a Bravais lattice as
F
=
W + ~'f(p)u,(p) P I
/
i
+ ~ Y~ ¢o(p,p )u,(p).:(p ) (4) p,p'
where W is the initial free energy of the undistorted standard state, f ( p ) , the initial force acting in the direction i is equal to
ui(p) = ~ U,(h)exp[ik(h)x(p)]
Ou,(p)
where
02F
c~o(p,p')
=
Ou~(p)Oui(p,)
(6)
where the parameters u~(p) are the Cartesian coordinates of the displacements of the atoms from their sites on the reference lattice and p and p ' define lattice sites. The reference lattice is chosen by Cook and de Fontaine [15] to be the average lattice of the solid solution. Thus it follows that the variable is the variation of solute concentration from its mean value. In the present case, where one nucleus in an infinite matrix will be treated, the average composition is also the initial and average matrix composition and thus the solute variable is q ( p ) = c - e0, and the elastic energy will be calculated relative to a standard state corresponding to a homogeneous system of composition c0 and having the average lattice parameter. In order to relate W a n d f ( p ) to q(p) they expanded the elastic energy in powers of q(p) and u~(p) [15].
AF = ½~ [dp(p,p')q(p )q(p') p,p'
+ 2d~,(p,p")u,(p)q (p') + d~o(p,p')ui(p)uj(p')].
(7)
The parameters tk (p,p'), dpo(p,p') are called the solute, solute-lattice and lattice coupling parameters respectively. Cook and de Fontaine [15] were then able to restrict consideration to solute distributions obeying cyclic boundary conditions in a domain of volume V = vL 3, where v is the volume of one unit cell of the Bravais lattice and L is the size the cyclic domain. The number of atoms in the domain is equal to N = zL 3 where z is the number of atoms in one unit cell. The cyclic boundary condition permits the transformation of the double summation in equation (7) to a single summation over the allowed sites in the first Brillouin Zone of reciprocal space by using the Fourier polynomials
q(p) = ~, Q(h)exp[ik(h)x(p)] h~R
(8)
k(h),
the
wave
vector,
is
given
by
k(h) = 2nh,b,, where b, are the translation vectors of the reciprocal lattice. The symbol h is an element of the triplets h~ defined by h, = mJL and the summation in equations (8) and (9) is over the domain R corresponding to the allowed sites in the first Brillouin zone. The Fourier coefficients are given by
Q(h)=l
~ q(p)exp[-ik(h)x(p)]
(10)
U i ( h ) = l ~ u,(p)exp[~ik(h)x(p)].
(11)
and
(5)
and the coefficient dpo(p,p') is given by
(9)
heR
OF :,(p) -
1847
The summation in equations (10) and (11) is done over all p in the cyclic domain. By substitution of q (p) and of ui(p) into equation (7) and by writing the equilibrium condition 0F
0u*
= 0,
(12)
Cook and de Fontaine eventually obtained N
AFe~= 5 ~ V(h)Q*(h)Q(h)
(13)
hER
where
V(h) = O(h) - ~,(h)Oo(h)~*.
(14)
• (h), ~i(h) and ~o(h) are respectively the Fourier transforms of the coupling parameters q~(p,p'), q~,(p,p') and ~o(P,P')" The coupling parameters are obtained as described by Seitz et al. [16] from the Born-Huang parameters [14]. It is important to note that AFeI depends on both the direction and magnitude of the wave vector and hence takes into account elastic anisotropy and the discrete nature of the system. The most important drawback of this model, as compared with the previously described continuum models is that, since it treats solute atoms as "defects" and precipitates as compositional wave fluctuations, it is not able to take into account the difference in elastic properties between the matrix and the precipitate; only the elastic constants of the matrix are employed. However, it enables the treatment of a tetragonal distortion tensor, which proves very difficult to do with any other model because of mathematical difficulties.
2.3. Total free energy The total free energy of a compositionally inhomogeneous system, relative to the free energy of an homogeneous system of composition Co may be written as AFrot = AFc~ + AFel.
(15)
This of course assumes that the two energies are
1848
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPE~-III
separate and additive, with no interaction between them. Equation (15) must be an approximation, especially on an atomic scale where strain energy changes the distance between atoms and thus presumably modifies the value of the chemical bonds. Nonetheless, since there seems to be no simple means of describing such "cross-terms", this approximation will be used. In order to make the chemical and elastic notations consistent, AF,~ is rewritten as a summation over p instead of over h substituting equation (10) into equation (13) AFt, = 1 ~ ( c e - - C o ) ( C ¢ - C o ) V(h)exp[ik(h)(xp,-- xe) 1.
(16)
h¢:O
Let us now write I.,,e, = ~ V(h)exp[ik(h)(xp, - xp)]. he0
(17)
Ipe, is independent of Cp and depends only on the vectorial distance (x¢ - xe) between the two points p and p ' . Thus, lee, can be independently calculated for each pair of points p, p ' and can indeed be considered as an "elastic interaction parameter" between them. The equation for the total free energy now becomes
eL
2a
I
p'
(18)
J
It should be noted at this point that the T c used in calculating the chemical interfacial energy is that of the incoherent miscibility gap. In the systems to be treated in this paper, namely Cu-Co and A1-Cu, this circumstance requires the following actions. For Cu-Co, the incoherent solvus is experimentally known; hence the Tc to be used in equation (18) is readily obtained [19]. In the A1-Cu system, the experimental GP zone solvus represents the coherent solvus. The incoherent T~ is estimated from this information through the following equation, written by Cahn [11] for finding the miscibility gap = F" + q 2 y ( c -- Co)2
(19)
where ¢ represent the free energy function when strain energy is included, F ' is the strain-free free energy function, t/is the misfit and y = (Cll +
2c,2)(c,i - c,2)
A ¢ = - 2k T~c°h ( c - Co)2 + q 2 y ( c - Co)2 + kT
,
(l-c))
c In Cca+ (1 - c) In ~ (
-2kT~¢°hl+kT
= (c - Co)[q2Y
, (1 -- c ) ] c l n C" + ( 1 -cc l m ~ 0f
(21)
Thus Tincoh = T ocoh + - r/: ~. Y
(22)
For the A1-Cu system, when the GP solvus is fitted to follow the regular solution model, T~°h is found to be equal to 990K. For this system, r/=0.12, c , = 12.4 x 10" dynes/cm 2 and c12 = 9.34 x 1011 dynes/cm2; thus r / 2 y = l . 1 0 x 10~°dynes/cm2/atom and therefore T~ c°h= 1925 K. 2.4. Calculation o f the critical nucleus concentration profile and its free energy o f formation
As was done in the previous paper of this series [2], the equilibrium concentration profile of the critical nucleus is obtained by minimization of the total free energy of the system ~AFtot = 0 ~cp
for all p.
(23)
The following system of non-linear equations is thus obtained under the assumptions of the regular solution model and nearest-neighbor interactions
ICe--Ce+rl
+ z (c,- co)(c
regular solution model
(20)
Cll
where cH and c~2 are elastic constants. It is important to note, with Cahn, that ¢ is only useful for determining the coherent solvus and cannot be used otherwise (such as for determining the shape or concentration profile of the critical nucleus), because of problems arising from elastic anisotropy. With the
~F(p)
2K
~c(p) ~- Ice+'- 2Cp +
c e r]
+ ~ (ce, - co)lee, = 0. p'
(24)
This system of equations is solved simultaneously to obtain the concentration profile of the critical nucleus. The free energy of formation of the critical nucleus is then found by substituting the Cp'S into the expression for AFt°, equation (18).
3. COMPUTER CONSIDERATIONS
Although it is not customary to include such material, in the present case the problem of solving numerically the system of non-linear equations is difficult enough to require some discussion. 3. I. Reasons and drawbacks f o r the choice o f a cyclic domain
The cyclic domain condition was introduced in order to be able to limit the number of atoms considered, i.e. to limit the number of wavelengths to be taken into account in the first Brillouin zone. For the cases treated by Cook and de Fontaine [15, 20] and by Seitz and de Fontaine [16, 17], this is a straightforward choice since they studied ordering and arrays of precipitates where this condition di-
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES~III rectly applies. In the present paper, a single nucleus in an infinite matrix is studied. Thus the cyclic boundary condition does not, in principle, apply. This problem can be solved two ways. First, the cyclic domain can be taken as infinite. In this case, an integration over the first Brillouin zone rather than a summation has to be performed in equation (13) and p is summed to infinity [20]. This is not a problem because all the terms in the summation are proportional to (Cp - Co) and thus tend toward zero as p goes to infinity. The difficulty with this method arises from the need to calculate the Ipp, matrix as an integral over the first Brillouin zone, namely
Ipp, = 2z
V(h )exp[ik(h )(xp, - xp)ldvh.
(25)
This integral does not converge rapidly and extremely small values of the increment dvh have to be chosen in order to calculate it numerically. The point is then reached where the "machine errors" arising from the summation of a very large number of terms become important. Further, the number of Ipp, to be calculated is also very large: Ipp, eventually goes to zero as p ' is moved farther away from p, but this convergence is also very slow. Thus, the numerical calculation of Ipp, is very lengthy and the computer time required becomes prohibitive. The second way to consider the problem of the cyclic domain is to place nominally arbitrary limits on the matrix domain surrounding the nucleus. This is a reasonable because the nucleus never becomes so large as to change noticeably the matrix composition in the domain. Since as soon as the radius of the nucleus becomes larger than about two lattice parameters the number of non-linear equations becomes too large to be solved, these conditions are readily followed. Ipp, can then be calculated exactly with a reasonable expenditure of computer time. This alternative was chosen over the first one for just this reason.
3.2. Comparisons with the "'no-elasticity" case It seems at first that the system of equations to be solved is not much different from the one previously described [2]. Indeed, the only term added is linear, i.e. (% - Co)Ipp,,which does not make the system much more complicated. Nonetheless, the following additional problems are encountered. In paper II, most of the non-diagonal terms in the system of equations are equal to zero because only nearest-neighbor interactions are considered. In the present case, all of the non-diagonal terms are non-zero because lpp, converges to zero very slowly. Although this is not a theoretical or even a programming problem, it does consume much computer time and thus restricts the number of calculations feasible. Furthermore, in paper II all the symmetries of the f.c.c, lattice were considered, thus reducing greatly the number of equations to be solved. In the present case, it is not possible to utilize all the symmetries since it is
1849
expected that the shape of the nucleus wilt evolve to that of a plate or a needle, neither of which follows the symmetries of the cube. In order to be able to model plates and needles, it was decided to "break" the symmetry by allowing the z axis not to be equivalent to the x and y axes. This doubles the number of equations and renders the programming more complex.
3.3. Problems associated with the solving of large numbers of non-linear equations The system of equations is solved by a Newton-Raphson method. This method requires a "first guess". This was found to become increasingly difficult as the number of equations increases. The larger the system, the closer the "first guess" must be to the actual solution. Hence reasonable first guesses were obtained from the "no-elasticity" case (already solved in paper II); misfit between the two lattices considered was then progressively increased; for each iteration, the solution of the previous one was used as the "first guess". This method, though time consuming because the increment of misfit has to be small in order to ensure convergence, was the only feasible way found to solve the problem. Because of the iterative method, it is impossible to find any bifurcation point (a la Johnson and Cahn [10]). Indeed, if the shape of the nucleus does not change continuously but bifurcates, or "jumps" to another shape when the misfit or the size of the nucleus reaches a certain value, this method will not give the absolute minimum in the activation free energy but a metastable type of nucleus.
3.4. Calculation and utilization of lpp, Ipp, must be calculated for every pair of points in the system. Consider a system composed of 12 x 12 x 12 f.c.c, unit cells (as will be the case in all later calculations); even when all operating symmetries are employed, there are close to 100,000 Ipp, to evaluate. Although once the Ipp, matrix has been calculated, it can be stored and used later without any further computation, this procedure still requires about 2-4 h of c.p.u, time on a VAX11/780 computer. Thus, even though it would probably be interesting, it was impractical to treat elastic parameters as variables and to study their influence on the shape of the nucleus. Instead only A1-Cu GP zones and Co-rich precipitates in Cu-Co alloys were analyzed. 4. RESULTS The only free parameter in the method, once the Born-Huang coupling parameters have been determined for the host metal, is the distortion tensor r/ [16, 17]. The coupling parameters have been determined experimentally for a certain number of metals including A1 [21] and Cu [22]. Thus the only influence of the type of defect on the elastic strain energy of the
1850
LEGOUES et al.:
INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--Ill
system will be through the value of the ~/tensor. The distortion can be determined experimentally by measuring the lattice parameter change upon introduction of the defects [17]. However, t, the tetragonality ratio itself seems somewhat more difficult to secure experimentally. It was demonstrated by Seitz and de Fontaine [ 17] that the tetragonality ratio is the determining factor in the shape of the duster. If t > l, the shape will be a disk, if t < l, the shape will be a needle, while for t = 1 the shape could be a cube or a sphere. Since only average values of r/are known through experimental evidence [23], it will be necessary to assume "reasonable" values of t; in the case of Cu-Co, the micrographs presented in paper IV [19] in this series show that the precipitates have roughly a spherical symmetry. Thus the tetragonality ratio will be set at 1. In the A1-Cu case, GP zones are experimentally known to be extremely thin plates [3] (possibly not more than one or two atomic layers thick). In this case, it seem reasonable to assume that the critical nucleus will also be a thin plate (although possibly with a different aspect ratio); thus t will be taken as oo.
4. I. Influence of elasticity on critical nucleus shape and concentration profile 4.1.1. The Cu-Co case: cubic defect. The numerical value of the misfit r/was varied parametrically at constant tetragonality ratio t = 1, from zero to about
3 ~ (the misfit between the copper and the cobalt lattices is about 1.7~). This permits a parametric study of the influence of strain energy on nucleus shape, concentration profile and free energy of formation. Figure 1 shows the evolution of the critical nucleus concentration profile as the misfit increases. It can be seen that, even at relatively high misfits, there is very little anisotropy. Hence the shape of the critical nucleus remains nearly spherical (though it is actually a very slightly flattened sphere). Thus the main influence of strain energy upon the nucleus shape in this case is due to the concomitant change in the undercooling. The introduction of misfit changes the phase diagram, as described by Cahn [13,24], so as to decrease To. Thus, at constant composition, an increase in the value of the misfit results in a decrease in undercooling and, consequently, an increase of the size of the critical nucleus, as seen in Fig. 1. Although convergence problems prevented calculations for misfits exceeding 3~, Fig. 1 does show that the shape tends toward that of a plate with increasing misfit, with the solute concentration decreasing faster in the (001) than in the (100) direction. Because of the strong anisotropy in the elastic properties of copper, it is indeed to be expected that at larger misfits a plate-like shape should be obtained [16, 17]. 4.1.2. Al-Cu Case: tetragonal defect. As in the preceeding subsection, the value of the misfit was
= 1 . 0 5 °/e
strain energy)
r/ = O ( no 1.0
1.0'
0.8
O.B
06
0.6
0.4
0.4.
0.2
0.2
I 1
• •e=# 2
l
I I
I 3
°°et' 2
I 3
= 1.7 o/. ( strain energy
1.0
corresponding
to
'1.0 q
II o+
~/= 2.5 °/°
CuCo) 0.6
08
0.6
0.6
0.4
0.4
0.2
0.2
t I
~.
I
!
|~0-
I
1
2
3
1
2
3
R/a
Fig. 1. Evolution o f the critical nucleus concentration profile as a function of the value o f the misfit, using the B o r n - H u a n g parameters o f Cu and a tetragonal distortion of t = l, at T = 560°C. The crosses
represents points in the (100) direction, the open circles points in the (001) direction and the solid circles represent other directions.
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--III
1.0 ~ 0.8
~7=0 {n0 strain energy)
1.0 0.8
0.6
0.6
0.4
0.4
0.2
1
2
+~
3
0117"
4
r/:9.6"/,
\ \~\
o+ / V..\. 0.4
2
\
1.0 ~ - I , /
\<1oo>
o+
~
\)o,,
•
2
3
~ =12% (correspondingto the strain of AI-Cu CP
+oo+,,
0.4
/ /,o., \ \ :oo1,\3 1
~7= 6,0 o/"
0.2
\.
1.0 ~ " &
tkl~O0
1851
3
4
I
\+\ ...... 2
3
4
5
R/o Fig. 2. Evolution of the critical nucleus concentration profile as a function of the value of the misfit, using the Born-Huang parameters of A1 and a tetragonal distortion tensor of t = ~ , at T = -25°C. The crosses represent points in the (100) direction, the open circles are points in the (001) direction, the open triangles are points in the (001) direction, the closed triangles refer to the (011) direction and the solid circles represent other directions. varied parametrically, but now up to 12~, the actual misfit between the pure A1 and Cu lattices [23]. Figure 2 shows the evolution of the concentration profile in this case. The continuous lines are fictitious but help to show the evolution of the shape since they represent the concentration profiles in different directions. Figure 2 clearly shows that, as the misfit increases, the shape of the critical nucleus evolves from a sphere to a disk. This can be seen by comparing the concentration profiles in the (100) and the (001) directions. The "thickness" (radius in the (001) direction) is seen to become much smaller than the "width" (radius in the (100) direction) as the misfit increases. Figure 3 represents an estimate of the shape of the critical nucleus constructed from the c = 0.5 isoconcentrate. When the misfit is 12~, the plate morphology is clearly displayed. Another conclusion to be drawn from Fig. 2 is that the plate morphology does not develop until the misfit is about 9.5~o. Figure 4 represents the variation of the critical nucleus concentration profile at constant misfit (corresponding to the misfit between A1 and Cu), as a function of the temperature, or undercooling. It can be noted that, for such a large misfit (12%), the plate-like shape appears even at very small sizes. The aspect ratio, though, increases with the size: at T = 0.07 T~, the aspect ratio is about 2/0.5 = 4 . At T = 0 . 1 5 T~, it is about 3/0.5 = 6. Thus, at smaller sizes, the influence of interfacial energy becomes progressively more
important, but strain energy still plays a major role in determining the critical nucleus shape.
4.2. Influence o f strain energy on kinetics o f nucleation in Cu-Co This study was done only for C u - C o because it is the system which was studied experimentally [19] and the general behavior displayed should not vary qualitatively from one system to another. 4.2.1. Influence o f the strain energy' on the radius and the free energy of formation o f the critical nucleus. Figure 5 shows a comparison of the variations of R* and AF* with T/T+ at Co = 0.01 when elasticity is
Fig. 3. Approximate shape of one octant of the critical nucleus ofa GP zone in A1-Cu at co = 0.02 and T = -75°C.
1852
LEGOUES et al.:
INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--III
lO
10 08
0.8
T:O.13T¢
T=0.17T¢
0.6
0.6 0.4
OA I ~ ' ~ & L ~ + 1
2
3
4
5
2
3
4
5
1.0
1.0 0.8
1
/+~
T =0.23 Tc
+ ~
0.8
T=O.28Tc
0.6 04
0.4
0.2
0.2 t) . 1
2
3
4
5
t
2
3
4
5
R/O Fig. 4. Variation of the critical nucleus concentration profile in AI-I%Cu at constant misfit of 12%, as a function of supersaturation. The crosses represent points in the (100) direction, the open circles are points in the (001) direction, the open triangles are points in the (110) direction, the closed triangles refer to the (011) direction and the closed circles represent other directions.
neglected and when it is not. As expected, when the nucleus is very small, the influence of strain energy becomes negligible as compared with that of the interfacial energy and the two cases converge. 4.2.2. Influence of strain energy on kinetics of nucleation in Cu-Co. Calculation of the rate of homogeneous nucleation, particularly in Cu-rich Cu-Co alloys, is detailed in paper IV of this series [19]. In this section, equations (1)-(4) of paper IV will be used to calculate nucleation rates in Cu-Co when elastic strain energy is taken into account and when it is not. Figures 6-8 show this comparison. As expected from the behavior observed in Fig. 5, the main difference between the curves "with" and "without" strain energy occurs at small undercoolings, when the crit-
ical nucleus becomes large enough so that strain energy becomes a major barrier to nucleation. It can be seen that at low undercoolings (where experimental studies are feasible--see paper IV) strain energy decreases the rate of nucleation by up to 5 orders of magnitude. 5. SUMMARY
Cook and de Fontaine have applied the microscopic theory of elasticity to compositional fluc20
I~-~\\
18
16 to 14 --without coherency strata energy ----wilh coherency strain energy
16]
\"/i /
t 12
14 24
12
20
//~/ 10
~0
o~
e
/
/~~/
12 08
a
--
4
04
2 --
2 I
01
I
02
co = W=th --------Without
o
I
Q3
01
Q2
03
TIT c
I oo
ol
0.01 strain
energy
strain
I 02
energy
I ..... o3
I o4
T/T c
Fig. 5. Variation of the critical radius and of the free energy
Fig. 6. Comparison of the rate of nucleation as a function
of formation of the critical nucleus with temperature, with and without strain energy, in Cu-l%Co.
of temperature when strain energy is taken into account and when it is not in Cu-l%Co.
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--III 20
20--
18
18
16
16 ..~
14--
1853
14
ro
~ 12--
12
~. 10-%
4
¢o = 0 0 0 8
-
--With 2
0
-
Without
I
0.0
0.1
co = 0005
4 --
strain
-----
8
--With
energy
I 02
strain
2
strain energy
I 03
I 04
T~ Tc
0
Without
O0
energy
strain
I
I
01
02
energy
I
I
0.3
04
T/r
Fig. 7. Comparison of the rate of nucleation as a function of temperature when strain energy is taken into account and when it is not in Cu-0.8%Co.
Fig. 8. Comparison of the rate of nucleation as a function of temperature when strain energy is taken into account and when it is not in Cu-0.5%Co.
t u a t i o n s in cubic crystals [15, 20]. T h e i r model is used in the present p a p e r in order to study the influence of coherency strain energy u p o n critical nucleus shapes a n d c o n c e n t r a t i o n profiles a n d o n the kinetics o f nucleation. It is f o u n d that, when the distortion tensor is tetragonal, t h e shape of the critical nucleus evolves from a sphere to a plate-like shape as the value o f the misfit increases, consistently with w h a t can be deduced from experimental observations on G P zones in A1-Cu alloys. W h e n the distortion tensor is cubic, the shape remains spherical, at least at the low value of the misfit c o r r e s p o n d i n g to the actual value in C u - C o alloys. The influence of coherency strain energy u p o n the kinetics of nucleation was studied in C u - C o alloys. As expected, this influence is p r e d o m i n a n t at low undercoolings, when the critical nucleus is large, i.e. when the influence of interfacial energy is diminished. In these alloys, coherency strain energy reduces nucleation rates by a b o u t five orders o f magnitude; this is quite substantial, considering the small misfit ( a b o u t 1.7%) between the nucleus a n d m a t r i x phases.
2. F. K. LeGoues, Y. W. Lee and H. I. Aaronson, Acta metall. 32, 1837 (1984). 3. A. Kelly and R. B. Nicholson, Progress in Materials Science p. 149. Macmillan, New York (1963). 4. J. D. Eshelby, Proc. R. Soc., A241, 376 (1957). 5. F. R. N. Nabarro, Proc. R. Soc. A175, 519 (1940). 6. F. Laszlo, J. Iron Steel Inst. 164, 5 (1950). 7. D. M. Barnett, J. K. Lee, H. I. Aaronson and K. C. Russell, Scripta metall. 8, 1447 (1974). 8. W. C. Johnson, Y. Y. Earmme and J. K. Lee, J. appl. Mech. 47, 781 (1980). 9. J. K. Lee, D. M. Barnett and H. I. Aaronson, Metall. Trans. 8A, 963 (1977). 10. W. C. Johnson and J. W. Cahn, unpublished research. 11. J. W. Cahn. Acta metall. 10, 907 (1962). 12. J. W. Cahn, Acta metall. 9, 795 (1962). 13. J. W. Cahn, The Mechanism of Phase Transformations in Crystalline Solids, p. 1. Institute of Metals, London (1969). 14. M. Born and K. Huang, Dynamical Theory of Crvstal Lattices. Oxford Univ. Press (1954). 15. H. E. Cook and D. de Fontaine, Acta metall. 17, 915 (1969). 16. E. Seitz, D. de Fontaine and F. Plesset, Lattice Statics Calculations of Elastic Strain Effects Due to Small Clusters of Defects in Cubic Metals. I. Theoretical and Computational Considerations. UCLA Report ENG-7689 (1976). 17. E. Seitz and D. de Fontaine, Lattice Statics Calculations of Elastic Strain Effects Due to Small Clusters of Defects in Cubic Metals. II. Application to Self Energies and Interaction Energies. UCLA Report ENG-7690 (1976). 18. H. E. Cook, D. de Fontaine and J. E. Hilliard, Acta metall. 17, 765 (11969). 19. F. K. LeGoues and H. I. Aaronson, Acta metall. 32, 1855 (1984). 20. H. E. Cook and D. de Fontaine, Acta metall. 19, 607 (1971). 21. G. Gilat and R. M. Nicklow, Phys. Rev. 143, 487 (1966). 22. E. C. Svenson, B. N. Brockhouse and J. M. Rowe, Phys. Rev. 155, 619 (1962). 23. W. B. Pearson, Handbook of Lattice Spacings and Structures of Metals and Alloys. Pergamon Press, Oxford (1958). 24. J. W. Cahn, Acta metall. 10, 179 (1962).
Acknowledgements--Appreciation is expressed to the Division of Materials Research of the National Science Foundation for support of this research through Grant DMR-8007567 and to Dr J. W. Cahn (NBS), Dr H. E. Cook (Ford Motor Co.) and to Professor D. de Fontaine (Univ. of California at Berkeley) for valuable discussions.
REFERENCES
1. F. K. LeGoues, H. I. Aaronson, Y. W. Lee and G. J. Fix, Proc. Int. Conf. on Solid-Solid Phase Transformations (edited by H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman), p. 427. Metall. Soc. A.I.M.E., Warrendale, PA (1983).
0001-6160/84 $3.00+0.00 Copyright © 1984 Pergamon Press Ltd
INFLUENCE OF CRYSTALLOGRAPHY UPON CRITICAL NUCLEUS SHAPES AND KINETICS OF HOMOGENEOUS F.C.C.-F.C.C. NUCLEATION--III. THE INFLUENCE OF ELASTIC STRAIN ENERGY F. K. LEGOUESt, H. I. AARONSON and Y. W. LEE:~ Department of Metallurgical Engineering and Materials Science, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A.
(Received 14 September 1983) Akstraet--The influence of crystallography upon critical nucleus shapes and kinetics of homogeneous f.c.c.-f.c.c, nucleation was studied by combining the discrete lattice point non-classical model utilized in paper II of this series with the microscopic theory of strain energy as applied by Cook and de Fontaine to compositional fluctuations in cubic lattices. This permits simultaneous consideration of the influence of anisotropic interfacial energy and anisotropic strain energy upon nucleation. This is extremely difficult to do using other models of strain energy because of mathematical problems. The model is applied to f.c.c, nuclei in AI-Cu and Cu-Co alloys. Nuclei in A1-Cu are plates because coherency strain energy predominates in this system. In Cu-Co alloys, the misft is smaller and the distortion tensor is cubic; thus the nuclei are essentially spherical. Despite this lesser effect, the influence of strain energy upon the nucleation kinetics of Co-rich precipitates in a Cu-rich Cu-Co alloy is found to be very important, reducing these kinetics by as much as five orders of magnitude. R6sum6--Nous avons &udi6 l'influence de la cristallographie sur la forme des germes critiques et sur la cin6tique de la germination homog~ne c.f.c.-c.f.c, en combinant le module non-classique de noeuds discrets utilis~ darts le deuxi+me article de cette s6rie avec la th6orie microscopique de l'6nergie de d6formation appliqu6e par Cook et de Fontaine aux fluctuations de composition dans les r6seaux cubiques. Ceci permet de consid~rer simultan6ment les influence de r6nergie interfaciale anisotrope et de l'6nergie de d&ormation anisotrope sur la germination. Ceci est extr~mement difficile fi faire en utilisant d'autres mod61es pour l'+nergie de d&ormation du fait de difficult+s math6matiques. Nous appliquons ce mod/fle ~ des germes c.f.c, dans des alliages AI-Cu et Cu-Co. Dans A1-Cu, les germes sont des plaquettes car l'6nergie de d~formation de coh6rence domine dans ce syst+me. Dans les alliages Cu-Co, le d6saccord est plus faible et le tenseur de la distorsion est cubique; les germes sont ainsi essentiellement sph6riques. Malgr~ ce moindre effet, nous avons trouv6 que l'influence de l'~nergie de d+formation sur la cin6tique de germination de pr6cipit+s riches en cobalt dans un alliage Cu-Co fiche en cuivre &ait tr+s importante et diminuait cette cin6tique de quelques cinq ordres de grandeur.
Zusammenfassung--Der EinfluB der Kristallografie auf die Form kritischer Keime und die Kinetik homogener kfz-kfz. Keimbildung wurde untersucht, indem das im Teil II dieser Serie benutzte nichtklassische Modell diskreter Gitterpunkte kombiniert wird mit der mikroskopischen Theorie der Verzerrungsenergie, wie sie von Cook und de Fontaine auf Fluktuationen in der Zusammensetzung in kubischen Gittern angewendet worden ist. Mit dieser Kombination kann der EinfluB der anisotropen Grenzflfichenund der Verzerrungsenergie auf die Keimbildung gleichzeitig behandelt werden. Mit anderen Modellen w/ire diese Aufgabe mathematisch/iuBerst schwierig. Das Modell wird auf kfz. Keime in AI-Cuund Cu-Co-Legierungen angewendet. Die Keime in Al~=u sind Platten, da die Energie in diesem System iiberwiegend durch Koh/irenzspannungen bestimmt ist. In den Cu-Co-Legierungen ist die Fehlpassung geringer, und der Distorsionstensor ist kubisch, daher sind die Keime im wesentlichen kugelf6rmig. Trotz dieses kleineren Effektes findet sich, dab der EinfluB der Verzerrungsenergie auf die Keimbildung Co-reicher Ausscheidungen in einer Cu-reichen Cu-Co-Legierung sehr wichtig ist; die Kinetik wird um fiinf Gr6Benordnungen verkleinert.
1. INTRODUCTION In the previous papers [1, 2] in this series, the shapes o f critical nuclei a n d kinetics o f nucleation were studied for h o m o g e n e o u s nucleation o f f.c.c, crystals tPresent address: IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, U.S.A. :[:Present address: Department of Metallurgy and Mining Engineering, University of Illinois at UrbanaChampaign, Urbana, IL 61801, U.S.A. A.M. 32/10~R
in f.c.c, matrices in b i n a r y alloy systems u n d e r the a s s u m p t i o n t h a t n o elastic strain energy is involved, i.e. t h a t the lattice p a r a m e t e r o f the two phases is the same a n d is i n d e p e n d e n t of c o m p o s i t i o n in b o t h phases. This a s s u m p t i o n n o t only ignores the effect o f strain energy o n the kinetics o f nucleation but, perh a p s m o r e importantly, neglects the effect o f anisotropy o f elastic properties o n the shape of critical nuclei. Occurrence o f plate s h a p e d G P zones is associated with large differences in the lattice p a r a m -
1845
1846
LEGOUES et al.:
INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--Ill
eter of the zones and their matrix phase; this is often explained by anisotropic elasticity [3]. The problem of calculating the elastic strain energy, W, of completely coherent precipitates has often been treated. Eshelby [4] wrote an expression for this energy as a function of Young's modulus, E and Poisson's ratio, v, when these parameters are the same in both the matrix and the precipitate. Crum showed that in this case, the expression for coherent strain energy does not depend on the shape of the precipitate [5]. Laszlo [6] derived expressions for W when E and v are different in the precipitate and the matrix (the so-called "inhomogeneous" case) for a plate, a cylinder and a sphere. Barnett et al. [7] gave a very complicated expression for W of coherent ellipsoids of revolution as a continuous function of their aspect ratio, which encompassed the particular shapes analyzed by Laszlo. Finally, Johnson et al. [8] treated the problem of cuboidal inhomogeneous precipitates. Their treatment could be applied to other shapes, such as plates, with little modification. All of these treatments consider W independently of the interfacial energy term. In other words, they apply mostly to large precipitates, when the interfacial energy term becomes negligible relative to the strain energy term. Lee et al. [9] examined one aspect of this problem by considering the interplay of the two contributing factors to the free energy of formation of an embryo, in the mathematically tractable case of ellipsoids of revolution whose interfacial and elastic strain energies were isotropic and the transformation strain was purely dilatational. Johnson and Cahn [10] have recently treated this problem using bifurcation theory. They studied the way in which morphology changes with the size of a cluster, i.e. the transition from a spherical precipitate (interfacial energy dominated) to an ellipsoid precipitate (strain energy dominated), as a function of size. Because of mathematical difficulties, only ellipsoidal shapes were considered under the isotropic elasticity assumption and with a cubic distortion tensor. It should also be noted that Cahn [11, 12] included elastic anisotropy in his continuum model and studied spinodal decomposition and certain aspects of nucleation under these conditions. The continuum model is relatively difficult to apply exactly to the nucleation problem because of the need to "cut" arbitrarily the Fourier decomposition, used in determining the nucleus concentration profile, in order to perform the integration of the free energy over the whole system [13]. In the present paper, a similar treatment to that of Cahn [13] is used but in the context of a discrete lattice point model of a compositionally inhomogeneous system, applied to f.c.c. ---, f.c.c, nucleation. The general equations for the elastic strain energy of a compositionally inhomogeneous system are based on the microscopic theory of elasticity [14], as applied by Cook and de Fontaine [t5] to the strain energy associated with ordering, de Fontaine [16, 17]
also used this thebry to calculate the strain energy of precipitates of different shapes as well as of arrays of precipitates and the influence of strain energy upon the arrangement of precipitates, de Fontaine [16, 17] noted that his calculation did not consider the interfacial energy and is thus not applicable to the study of nucleation. The Cook and de Fontaine [15] model for the strain energy of compositionally inhomogeneous systems will be added to the discrete lattice point model of Cook et al. [18] in order to treat simultaneously the interfacial energy, the chemical volume free energy and the strain energy of an embryo. Minimization of the total free energy of the system then yields a more accurate shape and free energy of formation of the critical nucleus. Although the equations for the strain energy, as developed by Cook and de Fontaine, are fairly complicated mathematically, this model is relatively easy to treat numerically and thus permits a more general treatment than has been previously utilized. In particular, it directly includes elastic as well as crystalline anisotropy, is not restricted to a specific type of nucleus shape and can treat cubic as well as tetragonal distortion tensors. 2. FREE ENERGY OF A COMPOSITIONALLY
NON-UNIFORM SYSTEM 2.1. C h e m i c a l f r e e e n e r g y
In the previous paper in this series [2], the chemical free energy of a compositionally inhomogeneous system was written as [18]
= .Iv Z AF(c,) ~vo p /.
+ ~5a2~ [c(p + r ) - c(p)] 2
(1)
where Nv is the number of atoms per unit volume, ce is the probability of having an atom of type B at site p, AF(cp) is the free energy, relative to a system of composition Co, that one atom at site p would have in compositionally homogeneous surroundings, K is the gradient energy coefficient and a is the lattice parameter. The summation over p is done over all relevant sites in the system and the summation over r is done over all the nearest neighbors of site p. The case of a regular solution, nearest-neighbors model was ' treated in the previous paper and will be used here. In this case K -
kT, a 2
(2)
12
AF(cp) = k T Fop in cp + (1 - cp) In (1 - Cp) (1 -- Co) L co -- 2kT~(c, -- Co)-'[
(3)
I
where k is Boltzmann's constant, T,, is the critical temperature of the miscibility gap and co is the initial and average composition of the matrix.
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--Ill
2.2. Elastic strain energy
and
Cook and de Fontaine [15] used the microscopic theory [14] to obtain the elastic strain energy of a compositionally inhomogeneous system. With this theory and the harmonic approximation, they wrote the elastic free energy of a Bravais lattice as
F
=
W + ~'f(p)u,(p) P I
/
i
+ ~ Y~ ¢o(p,p )u,(p).:(p ) (4) p,p'
where W is the initial free energy of the undistorted standard state, f ( p ) , the initial force acting in the direction i is equal to
ui(p) = ~ U,(h)exp[ik(h)x(p)]
Ou,(p)
where
02F
c~o(p,p')
=
Ou~(p)Oui(p,)
(6)
where the parameters u~(p) are the Cartesian coordinates of the displacements of the atoms from their sites on the reference lattice and p and p ' define lattice sites. The reference lattice is chosen by Cook and de Fontaine [15] to be the average lattice of the solid solution. Thus it follows that the variable is the variation of solute concentration from its mean value. In the present case, where one nucleus in an infinite matrix will be treated, the average composition is also the initial and average matrix composition and thus the solute variable is q ( p ) = c - e0, and the elastic energy will be calculated relative to a standard state corresponding to a homogeneous system of composition c0 and having the average lattice parameter. In order to relate W a n d f ( p ) to q(p) they expanded the elastic energy in powers of q(p) and u~(p) [15].
AF = ½~ [dp(p,p')q(p )q(p') p,p'
+ 2d~,(p,p")u,(p)q (p') + d~o(p,p')ui(p)uj(p')].
(7)
The parameters tk (p,p'), dpo(p,p') are called the solute, solute-lattice and lattice coupling parameters respectively. Cook and de Fontaine [15] were then able to restrict consideration to solute distributions obeying cyclic boundary conditions in a domain of volume V = vL 3, where v is the volume of one unit cell of the Bravais lattice and L is the size the cyclic domain. The number of atoms in the domain is equal to N = zL 3 where z is the number of atoms in one unit cell. The cyclic boundary condition permits the transformation of the double summation in equation (7) to a single summation over the allowed sites in the first Brillouin Zone of reciprocal space by using the Fourier polynomials
q(p) = ~, Q(h)exp[ik(h)x(p)] h~R
(8)
k(h),
the
wave
vector,
is
given
by
k(h) = 2nh,b,, where b, are the translation vectors of the reciprocal lattice. The symbol h is an element of the triplets h~ defined by h, = mJL and the summation in equations (8) and (9) is over the domain R corresponding to the allowed sites in the first Brillouin zone. The Fourier coefficients are given by
Q(h)=l
~ q(p)exp[-ik(h)x(p)]
(10)
U i ( h ) = l ~ u,(p)exp[~ik(h)x(p)].
(11)
and
(5)
and the coefficient dpo(p,p') is given by
(9)
heR
OF :,(p) -
1847
The summation in equations (10) and (11) is done over all p in the cyclic domain. By substitution of q (p) and of ui(p) into equation (7) and by writing the equilibrium condition 0F
0u*
= 0,
(12)
Cook and de Fontaine eventually obtained N
AFe~= 5 ~ V(h)Q*(h)Q(h)
(13)
hER
where
V(h) = O(h) - ~,(h)Oo(h)~*.
(14)
• (h), ~i(h) and ~o(h) are respectively the Fourier transforms of the coupling parameters q~(p,p'), q~,(p,p') and ~o(P,P')" The coupling parameters are obtained as described by Seitz et al. [16] from the Born-Huang parameters [14]. It is important to note that AFeI depends on both the direction and magnitude of the wave vector and hence takes into account elastic anisotropy and the discrete nature of the system. The most important drawback of this model, as compared with the previously described continuum models is that, since it treats solute atoms as "defects" and precipitates as compositional wave fluctuations, it is not able to take into account the difference in elastic properties between the matrix and the precipitate; only the elastic constants of the matrix are employed. However, it enables the treatment of a tetragonal distortion tensor, which proves very difficult to do with any other model because of mathematical difficulties.
2.3. Total free energy The total free energy of a compositionally inhomogeneous system, relative to the free energy of an homogeneous system of composition Co may be written as AFrot = AFc~ + AFel.
(15)
This of course assumes that the two energies are
1848
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPE~-III
separate and additive, with no interaction between them. Equation (15) must be an approximation, especially on an atomic scale where strain energy changes the distance between atoms and thus presumably modifies the value of the chemical bonds. Nonetheless, since there seems to be no simple means of describing such "cross-terms", this approximation will be used. In order to make the chemical and elastic notations consistent, AF,~ is rewritten as a summation over p instead of over h substituting equation (10) into equation (13) AFt, = 1 ~ ( c e - - C o ) ( C ¢ - C o ) V(h)exp[ik(h)(xp,-- xe) 1.
(16)
h¢:O
Let us now write I.,,e, = ~ V(h)exp[ik(h)(xp, - xp)]. he0
(17)
Ipe, is independent of Cp and depends only on the vectorial distance (x¢ - xe) between the two points p and p ' . Thus, lee, can be independently calculated for each pair of points p, p ' and can indeed be considered as an "elastic interaction parameter" between them. The equation for the total free energy now becomes
eL
2a
I
p'
(18)
J
It should be noted at this point that the T c used in calculating the chemical interfacial energy is that of the incoherent miscibility gap. In the systems to be treated in this paper, namely Cu-Co and A1-Cu, this circumstance requires the following actions. For Cu-Co, the incoherent solvus is experimentally known; hence the Tc to be used in equation (18) is readily obtained [19]. In the A1-Cu system, the experimental GP zone solvus represents the coherent solvus. The incoherent T~ is estimated from this information through the following equation, written by Cahn [11] for finding the miscibility gap = F" + q 2 y ( c -- Co)2
(19)
where ¢ represent the free energy function when strain energy is included, F ' is the strain-free free energy function, t/is the misfit and y = (Cll +
2c,2)(c,i - c,2)
A ¢ = - 2k T~c°h ( c - Co)2 + q 2 y ( c - Co)2 + kT
,
(l-c))
c In Cca+ (1 - c) In ~ (
-2kT~¢°hl+kT
= (c - Co)[q2Y
, (1 -- c ) ] c l n C" + ( 1 -cc l m ~ 0f
(21)
Thus Tincoh = T ocoh + - r/: ~. Y
(22)
For the A1-Cu system, when the GP solvus is fitted to follow the regular solution model, T~°h is found to be equal to 990K. For this system, r/=0.12, c , = 12.4 x 10" dynes/cm 2 and c12 = 9.34 x 1011 dynes/cm2; thus r / 2 y = l . 1 0 x 10~°dynes/cm2/atom and therefore T~ c°h= 1925 K. 2.4. Calculation o f the critical nucleus concentration profile and its free energy o f formation
As was done in the previous paper of this series [2], the equilibrium concentration profile of the critical nucleus is obtained by minimization of the total free energy of the system ~AFtot = 0 ~cp
for all p.
(23)
The following system of non-linear equations is thus obtained under the assumptions of the regular solution model and nearest-neighbor interactions
ICe--Ce+rl
+ z (c,- co)(c
regular solution model
(20)
Cll
where cH and c~2 are elastic constants. It is important to note, with Cahn, that ¢ is only useful for determining the coherent solvus and cannot be used otherwise (such as for determining the shape or concentration profile of the critical nucleus), because of problems arising from elastic anisotropy. With the
~F(p)
2K
~c(p) ~- Ice+'- 2Cp +
c e r]
+ ~ (ce, - co)lee, = 0. p'
(24)
This system of equations is solved simultaneously to obtain the concentration profile of the critical nucleus. The free energy of formation of the critical nucleus is then found by substituting the Cp'S into the expression for AFt°, equation (18).
3. COMPUTER CONSIDERATIONS
Although it is not customary to include such material, in the present case the problem of solving numerically the system of non-linear equations is difficult enough to require some discussion. 3. I. Reasons and drawbacks f o r the choice o f a cyclic domain
The cyclic domain condition was introduced in order to be able to limit the number of atoms considered, i.e. to limit the number of wavelengths to be taken into account in the first Brillouin zone. For the cases treated by Cook and de Fontaine [15, 20] and by Seitz and de Fontaine [16, 17], this is a straightforward choice since they studied ordering and arrays of precipitates where this condition di-
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES~III rectly applies. In the present paper, a single nucleus in an infinite matrix is studied. Thus the cyclic boundary condition does not, in principle, apply. This problem can be solved two ways. First, the cyclic domain can be taken as infinite. In this case, an integration over the first Brillouin zone rather than a summation has to be performed in equation (13) and p is summed to infinity [20]. This is not a problem because all the terms in the summation are proportional to (Cp - Co) and thus tend toward zero as p goes to infinity. The difficulty with this method arises from the need to calculate the Ipp, matrix as an integral over the first Brillouin zone, namely
Ipp, = 2z
V(h )exp[ik(h )(xp, - xp)ldvh.
(25)
This integral does not converge rapidly and extremely small values of the increment dvh have to be chosen in order to calculate it numerically. The point is then reached where the "machine errors" arising from the summation of a very large number of terms become important. Further, the number of Ipp, to be calculated is also very large: Ipp, eventually goes to zero as p ' is moved farther away from p, but this convergence is also very slow. Thus, the numerical calculation of Ipp, is very lengthy and the computer time required becomes prohibitive. The second way to consider the problem of the cyclic domain is to place nominally arbitrary limits on the matrix domain surrounding the nucleus. This is a reasonable because the nucleus never becomes so large as to change noticeably the matrix composition in the domain. Since as soon as the radius of the nucleus becomes larger than about two lattice parameters the number of non-linear equations becomes too large to be solved, these conditions are readily followed. Ipp, can then be calculated exactly with a reasonable expenditure of computer time. This alternative was chosen over the first one for just this reason.
3.2. Comparisons with the "'no-elasticity" case It seems at first that the system of equations to be solved is not much different from the one previously described [2]. Indeed, the only term added is linear, i.e. (% - Co)Ipp,,which does not make the system much more complicated. Nonetheless, the following additional problems are encountered. In paper II, most of the non-diagonal terms in the system of equations are equal to zero because only nearest-neighbor interactions are considered. In the present case, all of the non-diagonal terms are non-zero because lpp, converges to zero very slowly. Although this is not a theoretical or even a programming problem, it does consume much computer time and thus restricts the number of calculations feasible. Furthermore, in paper II all the symmetries of the f.c.c, lattice were considered, thus reducing greatly the number of equations to be solved. In the present case, it is not possible to utilize all the symmetries since it is
1849
expected that the shape of the nucleus wilt evolve to that of a plate or a needle, neither of which follows the symmetries of the cube. In order to be able to model plates and needles, it was decided to "break" the symmetry by allowing the z axis not to be equivalent to the x and y axes. This doubles the number of equations and renders the programming more complex.
3.3. Problems associated with the solving of large numbers of non-linear equations The system of equations is solved by a Newton-Raphson method. This method requires a "first guess". This was found to become increasingly difficult as the number of equations increases. The larger the system, the closer the "first guess" must be to the actual solution. Hence reasonable first guesses were obtained from the "no-elasticity" case (already solved in paper II); misfit between the two lattices considered was then progressively increased; for each iteration, the solution of the previous one was used as the "first guess". This method, though time consuming because the increment of misfit has to be small in order to ensure convergence, was the only feasible way found to solve the problem. Because of the iterative method, it is impossible to find any bifurcation point (a la Johnson and Cahn [10]). Indeed, if the shape of the nucleus does not change continuously but bifurcates, or "jumps" to another shape when the misfit or the size of the nucleus reaches a certain value, this method will not give the absolute minimum in the activation free energy but a metastable type of nucleus.
3.4. Calculation and utilization of lpp, Ipp, must be calculated for every pair of points in the system. Consider a system composed of 12 x 12 x 12 f.c.c, unit cells (as will be the case in all later calculations); even when all operating symmetries are employed, there are close to 100,000 Ipp, to evaluate. Although once the Ipp, matrix has been calculated, it can be stored and used later without any further computation, this procedure still requires about 2-4 h of c.p.u, time on a VAX11/780 computer. Thus, even though it would probably be interesting, it was impractical to treat elastic parameters as variables and to study their influence on the shape of the nucleus. Instead only A1-Cu GP zones and Co-rich precipitates in Cu-Co alloys were analyzed. 4. RESULTS The only free parameter in the method, once the Born-Huang coupling parameters have been determined for the host metal, is the distortion tensor r/ [16, 17]. The coupling parameters have been determined experimentally for a certain number of metals including A1 [21] and Cu [22]. Thus the only influence of the type of defect on the elastic strain energy of the
1850
LEGOUES et al.:
INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--Ill
system will be through the value of the ~/tensor. The distortion can be determined experimentally by measuring the lattice parameter change upon introduction of the defects [17]. However, t, the tetragonality ratio itself seems somewhat more difficult to secure experimentally. It was demonstrated by Seitz and de Fontaine [ 17] that the tetragonality ratio is the determining factor in the shape of the duster. If t > l, the shape will be a disk, if t < l, the shape will be a needle, while for t = 1 the shape could be a cube or a sphere. Since only average values of r/are known through experimental evidence [23], it will be necessary to assume "reasonable" values of t; in the case of Cu-Co, the micrographs presented in paper IV [19] in this series show that the precipitates have roughly a spherical symmetry. Thus the tetragonality ratio will be set at 1. In the A1-Cu case, GP zones are experimentally known to be extremely thin plates [3] (possibly not more than one or two atomic layers thick). In this case, it seem reasonable to assume that the critical nucleus will also be a thin plate (although possibly with a different aspect ratio); thus t will be taken as oo.
4. I. Influence of elasticity on critical nucleus shape and concentration profile 4.1.1. The Cu-Co case: cubic defect. The numerical value of the misfit r/was varied parametrically at constant tetragonality ratio t = 1, from zero to about
3 ~ (the misfit between the copper and the cobalt lattices is about 1.7~). This permits a parametric study of the influence of strain energy on nucleus shape, concentration profile and free energy of formation. Figure 1 shows the evolution of the critical nucleus concentration profile as the misfit increases. It can be seen that, even at relatively high misfits, there is very little anisotropy. Hence the shape of the critical nucleus remains nearly spherical (though it is actually a very slightly flattened sphere). Thus the main influence of strain energy upon the nucleus shape in this case is due to the concomitant change in the undercooling. The introduction of misfit changes the phase diagram, as described by Cahn [13,24], so as to decrease To. Thus, at constant composition, an increase in the value of the misfit results in a decrease in undercooling and, consequently, an increase of the size of the critical nucleus, as seen in Fig. 1. Although convergence problems prevented calculations for misfits exceeding 3~, Fig. 1 does show that the shape tends toward that of a plate with increasing misfit, with the solute concentration decreasing faster in the (001) than in the (100) direction. Because of the strong anisotropy in the elastic properties of copper, it is indeed to be expected that at larger misfits a plate-like shape should be obtained [16, 17]. 4.1.2. Al-Cu Case: tetragonal defect. As in the preceeding subsection, the value of the misfit was
= 1 . 0 5 °/e
strain energy)
r/ = O ( no 1.0
1.0'
0.8
O.B
06
0.6
0.4
0.4.
0.2
0.2
I 1
• •e=# 2
l
I I
I 3
°°et' 2
I 3
= 1.7 o/. ( strain energy
1.0
corresponding
to
'1.0 q
II o+
~/= 2.5 °/°
CuCo) 0.6
08
0.6
0.6
0.4
0.4
0.2
0.2
t I
~.
I
!
|~0-
I
1
2
3
1
2
3
R/a
Fig. 1. Evolution o f the critical nucleus concentration profile as a function of the value o f the misfit, using the B o r n - H u a n g parameters o f Cu and a tetragonal distortion of t = l, at T = 560°C. The crosses
represents points in the (100) direction, the open circles points in the (001) direction and the solid circles represent other directions.
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--III
1.0 ~ 0.8
~7=0 {n0 strain energy)
1.0 0.8
0.6
0.6
0.4
0.4
0.2
1
2
+~
3
0117"
4
r/:9.6"/,
\ \~\
o+ / V..\. 0.4
2
\
1.0 ~ - I , /
\<1oo>
o+
~
\)o,,
•
2
3
~ =12% (correspondingto the strain of AI-Cu CP
+oo+,,
0.4
/ /,o., \ \ :oo1,\3 1
~7= 6,0 o/"
0.2
\.
1.0 ~ " &
tkl~O0
1851
3
4
I
\+\ ...... 2
3
4
5
R/o Fig. 2. Evolution of the critical nucleus concentration profile as a function of the value of the misfit, using the Born-Huang parameters of A1 and a tetragonal distortion tensor of t = ~ , at T = -25°C. The crosses represent points in the (100) direction, the open circles are points in the (001) direction, the open triangles are points in the (001) direction, the closed triangles refer to the (011) direction and the solid circles represent other directions. varied parametrically, but now up to 12~, the actual misfit between the pure A1 and Cu lattices [23]. Figure 2 shows the evolution of the concentration profile in this case. The continuous lines are fictitious but help to show the evolution of the shape since they represent the concentration profiles in different directions. Figure 2 clearly shows that, as the misfit increases, the shape of the critical nucleus evolves from a sphere to a disk. This can be seen by comparing the concentration profiles in the (100) and the (001) directions. The "thickness" (radius in the (001) direction) is seen to become much smaller than the "width" (radius in the (100) direction) as the misfit increases. Figure 3 represents an estimate of the shape of the critical nucleus constructed from the c = 0.5 isoconcentrate. When the misfit is 12~, the plate morphology is clearly displayed. Another conclusion to be drawn from Fig. 2 is that the plate morphology does not develop until the misfit is about 9.5~o. Figure 4 represents the variation of the critical nucleus concentration profile at constant misfit (corresponding to the misfit between A1 and Cu), as a function of the temperature, or undercooling. It can be noted that, for such a large misfit (12%), the plate-like shape appears even at very small sizes. The aspect ratio, though, increases with the size: at T = 0.07 T~, the aspect ratio is about 2/0.5 = 4 . At T = 0 . 1 5 T~, it is about 3/0.5 = 6. Thus, at smaller sizes, the influence of interfacial energy becomes progressively more
important, but strain energy still plays a major role in determining the critical nucleus shape.
4.2. Influence o f strain energy on kinetics o f nucleation in Cu-Co This study was done only for C u - C o because it is the system which was studied experimentally [19] and the general behavior displayed should not vary qualitatively from one system to another. 4.2.1. Influence o f the strain energy' on the radius and the free energy of formation o f the critical nucleus. Figure 5 shows a comparison of the variations of R* and AF* with T/T+ at Co = 0.01 when elasticity is
Fig. 3. Approximate shape of one octant of the critical nucleus ofa GP zone in A1-Cu at co = 0.02 and T = -75°C.
1852
LEGOUES et al.:
INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--III
lO
10 08
0.8
T:O.13T¢
T=0.17T¢
0.6
0.6 0.4
OA I ~ ' ~ & L ~ + 1
2
3
4
5
2
3
4
5
1.0
1.0 0.8
1
/+~
T =0.23 Tc
+ ~
0.8
T=O.28Tc
0.6 04
0.4
0.2
0.2 t) . 1
2
3
4
5
t
2
3
4
5
R/O Fig. 4. Variation of the critical nucleus concentration profile in AI-I%Cu at constant misfit of 12%, as a function of supersaturation. The crosses represent points in the (100) direction, the open circles are points in the (001) direction, the open triangles are points in the (110) direction, the closed triangles refer to the (011) direction and the closed circles represent other directions.
neglected and when it is not. As expected, when the nucleus is very small, the influence of strain energy becomes negligible as compared with that of the interfacial energy and the two cases converge. 4.2.2. Influence of strain energy on kinetics of nucleation in Cu-Co. Calculation of the rate of homogeneous nucleation, particularly in Cu-rich Cu-Co alloys, is detailed in paper IV of this series [19]. In this section, equations (1)-(4) of paper IV will be used to calculate nucleation rates in Cu-Co when elastic strain energy is taken into account and when it is not. Figures 6-8 show this comparison. As expected from the behavior observed in Fig. 5, the main difference between the curves "with" and "without" strain energy occurs at small undercoolings, when the crit-
ical nucleus becomes large enough so that strain energy becomes a major barrier to nucleation. It can be seen that at low undercoolings (where experimental studies are feasible--see paper IV) strain energy decreases the rate of nucleation by up to 5 orders of magnitude. 5. SUMMARY
Cook and de Fontaine have applied the microscopic theory of elasticity to compositional fluc20
I~-~\\
18
16 to 14 --without coherency strata energy ----wilh coherency strain energy
16]
\"/i /
t 12
14 24
12
20
//~/ 10
~0
o~
e
/
/~~/
12 08
a
--
4
04
2 --
2 I
01
I
02
co = W=th --------Without
o
I
Q3
01
Q2
03
TIT c
I oo
ol
0.01 strain
energy
strain
I 02
energy
I ..... o3
I o4
T/T c
Fig. 5. Variation of the critical radius and of the free energy
Fig. 6. Comparison of the rate of nucleation as a function
of formation of the critical nucleus with temperature, with and without strain energy, in Cu-l%Co.
of temperature when strain energy is taken into account and when it is not in Cu-l%Co.
LEGOUES et al.: INFLUENCE OF CRYSTALLOGRAPHY ON NUCLEUS SHAPES--III 20
20--
18
18
16
16 ..~
14--
1853
14
ro
~ 12--
12
~. 10-%
4
¢o = 0 0 0 8
-
--With 2
0
-
Without
I
0.0
0.1
co = 0005
4 --
strain
-----
8
--With
energy
I 02
strain
2
strain energy
I 03
I 04
T~ Tc
0
Without
O0
energy
strain
I
I
01
02
energy
I
I
0.3
04
T/r
Fig. 7. Comparison of the rate of nucleation as a function of temperature when strain energy is taken into account and when it is not in Cu-0.8%Co.
Fig. 8. Comparison of the rate of nucleation as a function of temperature when strain energy is taken into account and when it is not in Cu-0.5%Co.
t u a t i o n s in cubic crystals [15, 20]. T h e i r model is used in the present p a p e r in order to study the influence of coherency strain energy u p o n critical nucleus shapes a n d c o n c e n t r a t i o n profiles a n d o n the kinetics o f nucleation. It is f o u n d that, when the distortion tensor is tetragonal, t h e shape of the critical nucleus evolves from a sphere to a plate-like shape as the value o f the misfit increases, consistently with w h a t can be deduced from experimental observations on G P zones in A1-Cu alloys. W h e n the distortion tensor is cubic, the shape remains spherical, at least at the low value of the misfit c o r r e s p o n d i n g to the actual value in C u - C o alloys. The influence of coherency strain energy u p o n the kinetics of nucleation was studied in C u - C o alloys. As expected, this influence is p r e d o m i n a n t at low undercoolings, when the critical nucleus is large, i.e. when the influence of interfacial energy is diminished. In these alloys, coherency strain energy reduces nucleation rates by a b o u t five orders o f magnitude; this is quite substantial, considering the small misfit ( a b o u t 1.7%) between the nucleus a n d m a t r i x phases.
2. F. K. LeGoues, Y. W. Lee and H. I. Aaronson, Acta metall. 32, 1837 (1984). 3. A. Kelly and R. B. Nicholson, Progress in Materials Science p. 149. Macmillan, New York (1963). 4. J. D. Eshelby, Proc. R. Soc., A241, 376 (1957). 5. F. R. N. Nabarro, Proc. R. Soc. A175, 519 (1940). 6. F. Laszlo, J. Iron Steel Inst. 164, 5 (1950). 7. D. M. Barnett, J. K. Lee, H. I. Aaronson and K. C. Russell, Scripta metall. 8, 1447 (1974). 8. W. C. Johnson, Y. Y. Earmme and J. K. Lee, J. appl. Mech. 47, 781 (1980). 9. J. K. Lee, D. M. Barnett and H. I. Aaronson, Metall. Trans. 8A, 963 (1977). 10. W. C. Johnson and J. W. Cahn, unpublished research. 11. J. W. Cahn. Acta metall. 10, 907 (1962). 12. J. W. Cahn, Acta metall. 9, 795 (1962). 13. J. W. Cahn, The Mechanism of Phase Transformations in Crystalline Solids, p. 1. Institute of Metals, London (1969). 14. M. Born and K. Huang, Dynamical Theory of Crvstal Lattices. Oxford Univ. Press (1954). 15. H. E. Cook and D. de Fontaine, Acta metall. 17, 915 (1969). 16. E. Seitz, D. de Fontaine and F. Plesset, Lattice Statics Calculations of Elastic Strain Effects Due to Small Clusters of Defects in Cubic Metals. I. Theoretical and Computational Considerations. UCLA Report ENG-7689 (1976). 17. E. Seitz and D. de Fontaine, Lattice Statics Calculations of Elastic Strain Effects Due to Small Clusters of Defects in Cubic Metals. II. Application to Self Energies and Interaction Energies. UCLA Report ENG-7690 (1976). 18. H. E. Cook, D. de Fontaine and J. E. Hilliard, Acta metall. 17, 765 (11969). 19. F. K. LeGoues and H. I. Aaronson, Acta metall. 32, 1855 (1984). 20. H. E. Cook and D. de Fontaine, Acta metall. 19, 607 (1971). 21. G. Gilat and R. M. Nicklow, Phys. Rev. 143, 487 (1966). 22. E. C. Svenson, B. N. Brockhouse and J. M. Rowe, Phys. Rev. 155, 619 (1962). 23. W. B. Pearson, Handbook of Lattice Spacings and Structures of Metals and Alloys. Pergamon Press, Oxford (1958). 24. J. W. Cahn, Acta metall. 10, 179 (1962).
Acknowledgements--Appreciation is expressed to the Division of Materials Research of the National Science Foundation for support of this research through Grant DMR-8007567 and to Dr J. W. Cahn (NBS), Dr H. E. Cook (Ford Motor Co.) and to Professor D. de Fontaine (Univ. of California at Berkeley) for valuable discussions.
REFERENCES
1. F. K. LeGoues, H. I. Aaronson, Y. W. Lee and G. J. Fix, Proc. Int. Conf. on Solid-Solid Phase Transformations (edited by H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman), p. 427. Metall. Soc. A.I.M.E., Warrendale, PA (1983).