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Coherent equilibrium in alloys containing spherical precipitates
Acta metall, mater. Vol. 43, No. 5, pp. 1825 1835, 1995
~
Pergamon
0956-7151(94)00398-X
Copyright © 1995 ElsevierScienceLtd Printed in Great Britain All rights reserved 0956-7151/95 $9.50 + 0.00
COHERENT EQUILIBRIUM IN ALLOYS CONTAINING SPHERICAL PRECIPITATES
A. J. ARDELLand A. MAHESHWARI Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024-1595, U.S.A. (Received 20 May 1994; in revised form 15 September 1994)
Abstrac~Coherent equilibrium is investigated theoretically in alloys containing monodispersed spherical precipitates. The alloy system is subjected to a Dirichlet tessellation by constructing a Voronoi polyhedron around each particle, and for the purpose of calculation each polyhedron is approximated as a sphere. The basic unit of the microstructure is thus a cell consisting of concentric spheres, with the outer sphere occupied by the majority (matrix) phase. The stresses in each cell are derived analytically using as one of the boundary conditions the requirement that the displacements at the outer boundary vanish. Isotropic elasticity is assumed throughout. The equilibrium volume fraction and the coherent solubilities of the two phases are calculated as a function of the overall concentration of solute in the alloy. The relative importance of the lattice mismatch between the two phases, their elastic constants, and the curvatures of their free energies of mixing, is investigated theoretically. It is predicted that the equilibrium concentrations of both phases always increase with increasing volume fraction. The consequences of the theory on the Ni-Al and Cu-Co alloy phase diagrams are explored. Much larger effects are predicted for Ni-AI alloys than for CuzCo alloys, despite the fact that the lattice mismatch is over three times as large in the latter system. The reason is that the difference between the solute concentrations of the two phases is much bigger in CuzCo alloys. It is concluded that the influence of coherency effects will always be much greater for solid solutions in equilibrium with intermediate phases than for terminal solid solutions. Data on the solubility of Ni3AI in Ni-A1 alloys, which support the theory, are presented and discussed.
1. INTRODUCTION The idea that the solubility limits of coherent phases in thermodynamic equilibrium can differ from their counterparts in incoherent, unstressed, equilibrium dates back to the pioneering theory of Cahn on spinodal decomposition in cubic crystals [1]. Larch6 and Cahn [2] and Robin [3] later showed that thermodynamic equilibrium in stressed solids was determined not by uniformity of the familiar chemical potential, but by a quantity called the diffusion potential, which is generally a function of composition, temperature and stress [2, 4]. It was later realized by Williams [5, 6] that the coherent solubility limits within the two-phase region of a binary alloy should also depend on the volume fractions of the two phases, or equivalently, on the concentration of the alloy, C 0 These concepts have stimulated considerable theoretical work on the ramifications of coherent phase equilibria in two-phase solids [7-14]. Cahn and Larch6 [7] pointed out that the elastic contribution to the total free energy of a solid containing two phases, and/?, in coherent equilibrium depends not only on their compositions, C, and C~, and the mole fraction, z, of the/3 phase, but also on the morphology of the individual phases and their spatial distribution. They explored coherent equilibrium in a system in which the elastic free energy contribution depends only on
z and not on morphology, and in which the familiar chemical free energy of each phase is a parabolic function of composition [7]. Lee and Tao [14] have recently extended this calculation by allowing the elastic strain to vary with concentration. While these theoretical investigations provide considerable insight into the general nature of coherent equilibrium, they cannot be used in a transparent manner to predict the effects that might occur in specific alloy systems. Also, their predictions are not readily amenable to quantitative experimental verification. The theoretical efforts most closely related to the subject of this work, namely an alloy containing spherical coherent precipitates, are those of Johnson and Voorhees [9] and Pfeifer and Voorhees [12, 13]. Johnson and Voorhees solved the problem of coherent phase equilibrium in a finite closed system in which a sphere of t h e / / p h a s e of radius r is embedded entirely within a sphere of the e phase of radius r'. F o r this geometry, the mole fraction of the/~ phase, which is taken as equal to the volume fraction, is simply
Because the system of Johnson and Voorhees [9] is subjected to an external pressure in the most general case, one of the boundary conditions they imposed is
1825
1826
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS
continuity of traction on the external surface. For all their calculations, however, this pressure was taken as zero. Pfeifer and Voorhees used this approach to investigate coherent equilibrium in binary alloys in which the incoherent equilibria are characterized by congruent points [12], and to establish the parallel tangent construction as a device for constructing coherent phase diagrams [13] under certain restrictive conditions. The application by Pfeifer and Voorhees [12, 13] of the original calculations of Johnson and Voorhees [9] is equivalent to describing the morphology of the system as a dispersion of spherical particles of the fl phase embedded in spheres of the ~ phase. This is the same as subjecting the microstructure to a Dirichlet tessellation, whereby each particle of the fl phase is embedded in a Voronoi polyhedron. For ease of calculation, each polyhedral cell is assumed to be spherical. The difficulty with utilizing the solution of Johnson and Voorhees to describe coherent equilibrium in such a system is that there is no reason why the stress should fall to zero at the boundary of each cell. We believe that a better approximation is to allow the displacementin the c~phase to become zero at the boundary. The reason for this can be appreciated by considering two neighboring misfitting spherical precipitates of identical size. Even without a detailed calculation, it can be readily appreciated that the displacement in the matrix separating the two particles must be zero at the midpoint of the line joining their centers. However, the normal (radial) stress at this point is twice the stress that would obtain, at the same distance from its center, if only one particle were present. In a system containing many particles, it is reasonable to assume, as a first approximation, that the displacement will become approximately zero at the midpoints of the lines joining the particle to its nearest neighbors, i.e. along the surface of the Voronoi polyhedron surrounding it. We realize that neither the assumption of zero pressure nor zero displacement at the cell boundary can perfectly reproduce the state of stress in the matrix within it. As we will show, however, our assumption of zero displacement predicts quite different behavior for coherent equilibrium than the assumption of zero pressure, and that the admittedly limited experimental data that exist are indeed consistent with the predictions of a model of coherent equilibrium based on the state of stress that results. In this paper we develop the equations for coherent equilibrium for the tessellated microstructure described above, using the concentric sphere model of Johnson and Voorhees [9] but assuming that the ]'In the equations that follow, parameters or variables that refer to the /3 phase are indicated using a subscript or superscript ft. For the most part, no subscript is used to denote reference to the c~ phase, although there are several obvious exceptions.
displacement vanishes along the surface of the outer sphere. We then apply the results of the theory to two important types of phase diagrams, specifically those in which a terminal solid solution is in equilibrium with an intermediate ordered phase (exemplified by Ni-AI), and those in which terminal solid solutions are in equilibrium (exemplified by Cu-Co). The differences between the principal predictions of our model are contrasted with those of the model of Johnson and Voorhees, some data in the literature supporting the predictions of our model are noted, and the general conclusions that emerge are discussed. 2. DEVELOPMENT OF THE MODEL
2.1. Stresses and strains We adopt the geometry of Johnson and Voorhees [9] of concentric spheres of the ~ and fi phases, assuming spherical symmetry throughout, with radial and angular coordinates denoted by p, 0, ~b, respectively. Additionally, we assume elastically isotropic phases, that the elastic constants, molar volumes and lattice constants of the two phases are independent of concentration, and that the effects of curvature on equilibrium can be ignored. A state of equilibrium obtains if the dispersion is monodisperse, because then, even allowing for capillarity, the c~ phase is uniform in composition. The misfit responsible for the stresses and strains in the precipitate and matrix phases, E, is defined by
E-
a¢ -- a
(2)
a
where a~ and a are the lattice constants of the precipitate and matrix phasest. In this case the stresses in the precipitate are given by the solutions [15] of the differential equations
(dw
w)
2G dW-3Kt~
(3)
and (dw w) w ~p+2p +2G~p-3KeE
T~oo=V~=21,
(4)
while those in the matrix are found from the solutions to the equations [15]
Tpe=2(dw \dp
+
2 w) dw p +2G--dp
(5)
and dw
2w)
w
Too= Too= ;t ~p + P + 2G--.p
(6)
In equations (3)-(6) w is the displacement, 2 is the Lam6 constant and G is the shear modulus. Substitution of equations (3)-(6) into the equation of mechanical equilibrium under conditions of spherical
1827
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS symmetry [15], i.e. dTpp 2Tpp - 2Too + - = 0 dp p
(7)
which is valid for both phases, produces the differential equation for the displacement d Jd(p2w) } dp ( p2 do = 0.
(8)
It is shown in the Appendix that this system of equations yields the solutions for the stresses T°P- l + t / z
t/z+
T°°-- l + q z
P
(9)
(10)
t/z--~
where -- P is the hydrostatic pressure imposed by the matrix on the fl precipitate, and is given by the expression P _ 3K~E (1 + t/z)
(11)
6t/z )
r~( l -
In equations (9)-(11) t/, FI~ and 6 are defined by the equations 3K 4(;
t/ -
(12)
3K~ Ft~ = 1 ~ 4G
(13)
and 6 K~-K KF~
(14)
where K and Kt~ are the hulk moduli of the ~ and fl phases. The parameters F~ and 6 are identical to those used by Johnson and Voorhees [9], while ~/+ 1 is equal to their parameter F. The displacements in the ~ and fl phases are given by the equations w - 4G (1 + t/z)
p
oP (1 - z) w~ = 4G (1 + t/z)"
(16)
We can see immediately from equation (9) that the normal radial stress acting along the surface of the outer sphere is not zero, but is instead - P z (1 + t/) -
(1 + t/z)
2.Z Conditions for thermodynamic equilibrium
The influence of the coherency stresses on the concentrations of both phases at thermodynamic equilibrium are obtained by invoking two conditions. The first condition is that the diffusion potentials [16] be equal, which for equal partial molar volumes of the two components is represented by the equation
g~(C~) - ~,(1 - C~) = ~ ( C ~ ) - ~,,~(1 - G )
(18)
where the #i are chemical potentials of the components at the designated compositions. The other condition involves the so-called thermodynamic potentials, w, of the two phases [17], specified by the equation co -
o ~ = (e0p - e~pp,~T~op
(19)
where v
co = n0/~, (1 - Ca) - ~
(T,,)2 + ~
1
T,/T o . (20)
no is the number of atomic sites per unit volume, v is the Poisson ratio, E is the Youngs modulus and summation over repeated indices is understood. In equations (19) and (20), the stresses and strains are evaluated at the ~/fl interface (p = r). The second and third terms in equation (20) can be expressed in the form 1
u? = _ _ [T~, + 2T2o - 2v (2Te,, Too +Tool]. 2E
(21)
On substituting equations (9) and (10), with p = r, into (21), the expression for W for the ~ phase becomes p2 I +t/z) 2 + 2 ( t / z _ ~ ) 2 2E(1 + q z ) 2 ( 1
z
and
ToA.=~. = - n
H --~ P as z ~ 1. This limiting behavior makes sense to us, and demonstrates that the solution of Johnson and Voorhees is approached in the limit of zero volume fraction.
(17)
It is evident from equation (17) that the outer sphere is subjected to a hydrostatic pressure - r l that depends on the volume fraction of the fl phase; this pressure is imposed by the combined presence of all the particles in the system outside the cell containing the concentric spheres. Clearly, 13 ---, 0 as z ---, 0 and
After algebraic simplification and the use of the equalities G = E/2(1 + v) and K = E/3(1 -- 2v), reduces to the simple expression = tlP2(tlz 2 + 1) 2K(qz + 1)2.
(23)
On using equation (21) for the precipitate phase, the equation for u?~ is simply p2 W, = ~ . (24) It remains to evaluate the right-hand side of equation (19). The radial components of the strain are readily calculated from the relationship epp = dw/dp. Evaluating epp at p = r from equation (15), using equation (16), and recalling that
1828
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS
T ~v = -- P, we find
2.3. Equation o f state Jbr coherent equilibrium
(epp - - t, ts " ) T t~ - -
-PP'-PP
(25)
tlP 2
K(qz + l)
Using equations (23)-(25), (19) can now be written as
no[#, (1 -- C=) --/.z~(l -- C~)] = (% - e~)r~
Johnson and Voorhees refer to the equation that describes the dependence of z on C°, and the other relevant parameters, as the equation of state for the coherent system. In establishing the equation of state for our model, we use their reduced concentrations
+ % - v
_
Y~=-1+2
_p2
- 2Ke(tlz + 1)2 {~/(1 + 6Vl,)(rlZ 2 - 2rlz - 1) -
(r/z +
1) 2}
_
C~-Co~ Cots - Co~
_
-1+
AC~ ~ 2AC'o
Ya = 1 + 2 Cts -- Cots _ 1 + 2 ACf*
Co~ - Co~
(26)
aCo
(34)
(35)
and
which on substituting (11) becomes n0[#l (1 - C~) - #~(1 - Ce)] = 'Jr'x
(27)
WT_ 9cZKts {--6 (r/z)2 + 2*'/z + 1} 2Fts (1 -- 6qz) 2
(28)
W = 1+ 2
where
C o _ Cots
C o _ Co~ - 1+ 2 - Cots - Co~ AC0
(36)
where
To obtain the final expression for the equilibrium concentration it is necessary to select an appropriate reference or comparison state for the system. It is conventional to choose the state of incoherent equilibrium for this purpose [9, 15], so that the equilibrium concentrations in the reference state are the incoherent solubility limits for both phases. We denote the reference chemical potentials by/~o, and the equilibrium incoherent solubilities by the subscript O, so that equations (18) and (27) for the reference state (with E = O) become
Co~ - Co~ = &Co.
(37)
The substitution of equations (28), (31) and (32) into (34) and (35) produces the equations for Y= and Y/~ given by
A { - 6 (~z)2 + 2~z + 1} L = - 1+
(1 - 6rtz) 2
(38)
and Yt~= 1 + ( 1 + ( )
A{ - 3 (qz) 2 + 2qz + 1} (1 - -
&lz ) 2
(39)
where #02 ( C 0 ~ ) -- ,/~01 (1 -- C0c~)
= #gz(Cots)-/lg~(l - C0~)
A-
(29)
9E2K~
(40)
z~FBACo
and and iz0~(1 -- Co=) - #o~1(1 -- Coil) = 0.
(30)
On subtracting equations (29) and (30), respectively, from (18) and (27), and assuming that the differences between the concentrations in the coherent and reference states are small, we obtain the results [9] AC~ = C= - C0~-
kI) T
z~(c0B- c0o)
Cots-
/.~AC~
(41)
The equation of state for the system is obtained by invoking the condition of conservation of mass, namely zCts + (1 - z)C~ = C o
(42)
On substituting equations (34)-(39) into (42), we obtain after some algebra the equation (32)
Zt~
Z~
(31)
and AC~ = G -
C = z_~_ 1.
&/2(26 - (A)z 3 + r/{A(2( - &l) -
~/6 2(1 +
W)
-
40 }z 2
where Z~ and Xts are given by [9] nokT
I
c~1n72]
no 02F
(33)
Z-CoO-_-Co) l + ~-~nC - N o8C 2 where k is Boltzmann's constant, T is the temperature, 72 is the activity coefficient of component 2 in the stress-free state, No is Avogadro's n u m b e r and F i s the chemical free energy. In equation (33), all the terms are evaluated at the incoherent solubility limits of both phases.
+ {A(~ + 2 , ) + - (1 + W ) = 0.
2&l(l + W) + 2}z + A (43)
The roots of equation (43) determine the volume fraction of the fl phase in coherent equilibrium with the ~ phase. These roots determine the values of Y~ and Yts by substitution into equations (38) and (39), which in turn establish the coherent solubilities of the and fl phases via equations (34) and (35).
ARDELL and MAHESHWARI:
COHERENT EQUILIBRIUM IN ALLOYS
1.0 r-
3. RESULTS
0.20.1
t/ -
l-v 2(1 - 2v)
(44)
/ / i
S°o °
." ~oSo ,*" # °.¢
0.6
/ ,... •"
Z
#°o °
/ s',"
.... ~.-
0.4
..: , ;
:o #,%*
."4""
i:"t/.~"
0.2 0
-,.i~" i
-1.5 -1,0 -0,5
I
0
0.5
..
1.0
I
1.5
,
I
2.0
W Fig. 1. The dependence ofz on W predicted by equation (43) for 6 = 0, ~ = 0 and q = 1, with A varying from 0 to 0.25.
and
3 + v -- 4vl~
.,.:.'"
A--O/ /
3.1. General features o f the solutions The principal objective of this work is to translate the predictions of the preceding calculations to the kinds of effects that might be observable experimentally. We therefore make no attempt to explore the broad spectrum of effects that the theory is capable of predicting. It is nevertheless useful to generalize to a limited extent, so that some sense of the magnitudes of the effects expected can be appreciated. The parameters in the theory can assume a large range of values, the complete exploration of which becomes a formidable task. Since, from equations (12) and (13), t/ and F~ can be expressed as
1829
(45)
FB - 2(1 -- 2v/~) it is clear that they exist over the ranges 1/2 < t / < oo and 3/'2 < Ft~ < oc, the upper limits obtaining when v = 1/2. F o r most materials, however, t/is unlikely to exceed 7 and Fa is unlikely to exceed 8. Since for many practical cases of interest Kt~ and K will probably differ by no more than a factor of two, a realistic range of values for 6 is, from equation (14), - 1/3 < c5 < 2/3. F o r a terminal solid solution in equilibrium with an intermediate ordered intermetallic compound, the range of the parameter ( will be, from equation (41), - 1 < ~ < 0, the lower limit applying to a line c o m p o u n d (Zr~ = oc), and the upper limit governed by the fact that the curvature of the function F(C~) will be smaller than that for the solid solution. For terminal solid solutions in equilibrium the range o f possibilities is greater, but ~ is unlikely to exceed unity by an appreciable amount. The parameter A is always positive, and can in principle be quite large. However, the insertion of representative values of the parameters into equation (40) indicates that A = 0.25 is a realistic upper limit. Over the ranges of the parameters discussed herein, the types of complications described by Johnson and Voorhees regarding whether there are multiple solutions to the equation of state (43), or whether the solutions represent absolute minima or maxima of the free energy of the system, do not arise; all the solutions are single-valued and represent energy minima. Figure 1 illustrates the variation of z with W when 6 = 0 (both phases have identical compressibilities), = 0 (identical curvatures of the free energy functions), q = 1 (or v = 0.2), and A varies from 0 to 0.25. The solutions are, of course, physically meaningful
tW. C. Johnson (private communication) has pointed out that under certain circumstances, solutions for which W > 1 can be physically meaningful.
only for z lying in the range 0 ~ < z ~ < l , and for practical purposes this must occur also for values of W satisfying - 1 ~< W ~< 1 because C ° cannot exceed C0~ initially if the system is to lie within the two-phase field of the conventional phase diagram. We see, however, that W exceeds unity when z exceeds ~ 0.9 for A = 0.1, and when z exceeds ~ 0.7 for A = 0.25. Such solutions expose the limitations of the model, which cannot be expected to predict realistic alloy behavior at large volume fractions owing to the restrictions of the concentric sphere geometry. The issue of W exceeding unity for 0 ~< z ~< 1 did not arise in the model of Johnson and Voorhees because they explored coherent phase equilibria only in systems characterized by zero external pressure. It is easy to show that the imposition of finite pressure on the system shifts the curves o f z (W) in such a way that z can exist mathematically over the range 0 ~ < z ~ < l for W < - I as well as W > l . Clearly, the range of parameters over which either model yields physically meaningful solutions must be chosen, and evaluated, with some caret. The dependencies of Y~ and Y~ on W for 6 = 0, = 0 and r / = i, with A varying from 0 to 0.25, are shown in Fig. 2. The curves in Fig. 2 illustrate another general feature of our model, namely that Y~ and Y~ both always increase as W increases, indicating through equations (34) and (35) that AC~ and AC~ always increase as C O increases. This behavior is exactly opposite to that predicted by the model of Johnson and Voorhees [9]. Since z exists mathematically for W > 1, Y~ and Y~ exist as well, but have no physical significance in this regime. The dependence of the coherent solubility limits on W i s shown in Fig. 3 for 6 = 0, A = 0.25, ~/= 1 and several values of ~ between - 1 and 0.5. The data are presented in the form AC~/ACo and AC~/ACo vs W, and are limited to values of W ~ 0 the dependence of AC e on W [Fig. 3(b)] is larger than that of AC~ [Fig. 3(a)], while the opposite is true when
ARDELL and MAHESHWARI:
1830 2.0-
1.0 0,5
~,2
To calculate X;, from equation (33) we used the thermodynamic model of Calderon et al. [22], in which ~ 2F/~C 2 for the matrix phase is represented by the equation
^=o
2F
RT
OC 2
2M-
C0(1-C0)
6N(1 - 2C0)
(46)
0.25
-o.5
-I
[211.
0 *.,. . . ......... ..-:~'2 . .1. . . .°. . ,
v. o
-l.O/
temperature dependence of 6 is evidently quite small
5 /
y
_
COHERENT EQUILIBRIUM IN ALLOYS
.....
......... 0.2
•,.--:-:'.':LC'::;':--- . . . . . .
o 1
^=o
.,5
-2.0 ' ' ' ' ' ' -2.0-I,5-1.0-0,5
'
'
'
'
0 0.5 W
'
'
'
'
'
'
l.O 1.5 2.0
Fig. 2. The dependencies of Y~ and Y# on W for ~ = 0, ~ = 0 and ~ / = l, with A varying from 0 to 0.25.
< 0; for ~ = 0 the variations of z~C~ and ACs~ with W are identical. We reiterate that both AC= and AC B increase as C 0 increases. This behavior is characteristic of all the solutions of the equation of state of our model, and is perhaps the major difference between the predictions of the model of Johnson and Voorhees and ours; it is the prediction that can be most readily tested experimentally. The behavior predicted when 6 varies is shown in Fig. 4 for ~/ = 1, A = 0.25 and ~ = 0 (AC~ and AC# are equal in this case). We see that when 6 = 2/3 the variation of AC=,~ is much stronger than when 6 = - 1 / 3 . The variation of AC,. n with W when ~1 varies from 1 to 7 is shown in Fig. 5 for 6 = 0, ~ = 0 and A = 0.25. It is evident that the predicted changes in coherent solubility can be quite large when the parameters fi and q have values close to their expected upper limits. We reiterate that the variations of AC=,sj with W seen in Figs 4 and 5 are about as large as we are ever likely to encounter in real systems because of the large value of A ( = 0.25) used in the calculations. This will become more apparent in the next sections, where we apply the predictions of the model to two real alloy systems, C u - C o and the Ni-rich portion of the Ni-A1 phase diagram. 3.2. The Ni-rich region of the N i- A I phase diagram The constants used in predicting coherent phase equilibria in this alloy system are GNi~AI)= G~, = 75.2 G P a [18, 19], K;, = 176.3 GPa [19], K~i3AI=K;,,= 173.3GPa [20], E =0.0047 [21] and a~. = 0.355 nm [21]; we refer here to the N i A 1 solid solution as the 7 phase. These yield the values ~/ = 1.76, F# = 2.73 and 6 = -0.00607. The temperature dependencies of the physical constants has been ignored in the following calculations; except for those of a and 6 they have not been measured, and the
where M = -- 104361 + 3.29574T and N = 41291.4; both quantities are expressed in J/mol. The remaining quantities needed to complete the calculations are 2F/~C2 for the 7' phase, and the incoherent solubility limits of both phases. In the absence of a reliable thermodynamic model for the free energy of the 7' phase, we have made the assumption that X;,,= 2L,, so that ~ = - 0 . 5 at all temperatures. As can be expected from the results in Fig. 3, this assumption primarily affects the composition of the y' phase, and the effect on the calculated values of z and AC;, is minor. A varies from 0.0596 at 400°C to 0.2392 at 1300°C.
0.5 -
(a)
0.4 ~=~ AC(~ 0.3
S
AC° 0.2
,
I
~
I
I
0 W
0.5
I
-"1.5 -1.0 -0.5
0.5
0.5
.l~..: ~"
0.1 n
5
o°°* .0
,
I
,
I
1.0
].5
(b) 0.5
.°#°
0.4 .#
~ . 0.3
¢." j° •° • "
~C° 0.2
-"
°°°°
0.1 0
•"
.....
...."'"
° ~, •
.... ,
I
,
r
-1.5 -1.0 -0.5
•
°°°°". o
°°..° °°.°° .°o°
J r,
0 W
.° -0.5 •
~-I ,
,
0.5
I
1,0
,
I
1.5
Fig. 3. The dependencies of the normalized coherent solubility limits AC~/ACo (a) and ACyACo (b) on W for 6 = 0, r/ = 1 and A = 0.25,/with ~"varying from - 1 to 0.5.
ARDELL and MAHESHWARI:
COHERENT EQUILIBRIUM IN ALLOYS
0.6
~
~ = 2/3
0.5 L-
/
AC
0.4
Z". //
AC°
0.3
I/3
t
.....
#° f°
.,,=" ,°o.*"
~'.... j.y;.--" I 0.1 ' I - .5 -1.0 -0.5
I
0 W
,
L
,
0.5
I
,
1.0
l
1.5
Fig. 4. The dependencies of the normalized coherent solubility limits of both phases on W for ~ = 0, r/= 1 and A = 0.25, with c~varying from - 1/3 to 2/3 (AC~ = AC/~= ~C~.~ in this case)
We believe that the equilibrium incoherent solubilities of the 5' and 7' phases are not well established. The reason is that 7' precipitates remain fully coherent with the 7 matrix to very large sizes, approaching several micrometers [23, 2@ Consequently, true incoherent phase equilibrium is unlikely to be attained in experiments involving prolonged aging within the two-phase field of initially supersaturated solid solutions, or very slow cooling from the solutiontreatment temperature. These are the kinds of experiments used by early investigators, such as Alexander and Vaughan [25] and Taylor and Floyd [26] to establish the solubility limits of the 7 solid solution. We do not concur with the conclusion of Nash et aI. [27] that the incoherent solubility of A1 in Ni is known from the work of Alexander and Vaughan [25].
0.8
-
11=7 0.6
4
Ac=,l~ AC o
0,4 I
0.21
0
i
,
I
I
-1.5 -1.0 -0.5
I
J
0 W
0,5
i
J
1.0
i
i
1,5
Fig. 5. The dependencies of the normalized coherent solubility limits of both phases on W for 5 = 0, ~ = 0 and A = 0.25, with ~ varying from I to 7 (AC~ = ACt~ = AC~.I~in this case).
1831
On the other hand, the coherent solubility of the 7' phase has been measured [28], and the literature on this topic recently reviewed [29]. No account has been taken of the possible effect of z in these experiments, so the solubilities measured can be thought of as averages over a relatively small range of z, the values of which have generally been less than 0.15. For purposes of this work, we have taken the coherent solubilities summarized by Ardell [29] as the solubility limits at z = 0 [we call this quantity C:(0)] and used them to calculate Co;. from 400 to 1300'C; equation (28) with z -= 0 was substituted into (31) for this calculation, assuming that Z;. is identical for the coherent and incoherent phases over the small range of compositions involved, The data of Taylor and Floyd [26] on the composition of the 5" phase were taken as Co:,,, although they too are very likely to be coherent solubilities over the temperature range of the calculations. We note here that C0~.is the counterpart of the incoherent solubility limit calculated by Rastogi and Ardell [28], but is roughly a factor of 4 smaller than their values because they used a molar volume of Ni3A1 approx. 4 times larger than the atomic volumes used here [the atomic volume is equivalent to 1in o , which enters the calculation through equation (33)]. The goal of our calculations is to demonstrate the consequences of coherent phase equilibrium over a practical range of alloy concentrations. It is completely impractical to perform aging experiments on alloys in which the overall concentration exceeds the maximum solubility limit of A1 in Ni, which is approx. 20.5% at 1300'~C [27]. Hence, the maximum value of C°j in all the calculations, C ~ , was taken as 0.205. The maximum value of W, Wm, varies with temperature, of course, and at each temperature C~. was computed from the maximum value of z corresponding to this value of Win. [We call this quantity C~.(zm), where z m is the maximum possible value of z at a given temperature.] In summary, then, equation (43) was used to calculate the dependence of z on W. Realistic values of z exist over a range 0 < z < Zm (where z m is determined using C 0m _- 0.205), which establishes Wm from equation (36). The values of C:.(0) and C:, (0) (the value o f C v at z = 0), and C:,(zm) and C;.,(Zm) (the value of C v at z = z~,) were then calculated from equations (34) and (35), via (38) and (39). The results of the calculations are shown in Fig. 6. It is apparent that the effect of coherency strains on the phase equilibrium are quite significant for the matrix phase, the difference between C.(Zm) and C:(0) exceeding 1% at the lower temperatures. This difference diminishes as the temperature increases, but is still large at 1100°C. The effect on the solubility of the 5" phase is smaller, but this is a consequence of the fact that we have chosen L., = 2 L, in the calculations. Two other features of interest are briefly examined. The first is the relative change in volume fraction that can occur on loss of coherency, which produces a new
1832
ARDELL and MAHESHWARI:
COHERENT EQUILIBRIUM IN ALLOYS
----ic,(
c,,0,../-g
t
,
0,05
O,10
"ti i
114,, ,
400
O.15
0.20
0.25
Fig. 6. The Ni-rich portion of the binary N i ~ l phase diagram. Co,,and Ccv,are the incoherent solubility limits of the 7 and 7' phases. C~(0) and C~,(zm) define the coherent solubility limits of the 7 phase at z = 0 and z = z,,, and C:,,(0) and C:,,(Zm) do likewise for the 7' phase. The open and filled symbols represent the coherent solubility limits of alloys containing l 1.1 and 13.8% AI, respectively, estimated from the data of Gentry and Fine [30].
volume fraction z~ of incoherent 7'- This change is illustrated by plotting z/z~ vs z in Fig. 7, and is always positive in Ni-A1 alloys. Hence, if loss of coherency were to occur in Ni-A1 alloys, the volume fraction of 7' would increase because z~>z. Since z/z~ approaches zero as z ~ 0, the fractional increase in volume fraction on loss of coherency can be quite large for small supersaturations. F o r more concentrated alloys, in which z is fairly large to begin with, the increase in volume fraction on loss of coherency
is much smaller ( ~ 1 0 %
at 400°C to ~ 2 5 %
at
1100°C). It is also of interest to evaluate the pressures acting on the precipitate and matrix phases within the basic cell. It is expected from equations (11) and (17) that both P and FI increase as z increases, and these pressures can be calculated using values of the appropriate parameters. The results of such calculations are shown in Fig. 8, where it is seen that FI can become quite large in Ni-A1 alloys for values of z approach-
1.0[
2.0
0.9 900
0.8
1100 °C
o
0.7
1
,
5
a.
~
0.6 -0.5
1.0
0.4 0.3 0.5
0.2 0.1 0
,
I
I
0 O.l 0.2 0.3 0,4 0.5 0.6 0.7 Z
Fig. 7. Illustrating the predicted change in volume fraction of 7' precipitates in Ni AI alloys on loss of coherency. The ratio of the incoherent and coherent volume fractions, z i / z , is plotted against z for the three temperatures indicated.
0
'
i i
i
l
0 0,I 0.2 0.3 0,4 0.5 0.6 0.7 z
Fig. 8. Illustrating the variations of the hydrostatic pressure with volume fractions of 7" precipitates in N i ~ l alloys. P is the pressure on the precipitates and FI is the pressure on the cell.
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM 1N ALLOYS ing z m. However, for volume fractions up to about 0.3, 13 is generally < 700 MPa. The effects of hydrostatic pressures of this magnitude on physical properties (lattice constants, for example) are small. At significantly larger volume fractions the quantitative predictions of the theory must be viewed with caution.
1833
700
(a )
O~
,,'"'
co=
C=(O)~ (Cu) ,~j~"~" C"(zm)
A
3.3. Cu-Co alloys The values of the physical constants used in predicting coherent phase equilibrium in this alloy system are Gc, = 4 2 . 1 G P a [31], Kcu= 137.1GPa [31], and K c o = 2 0 4 . 1 G P a [32], e = - 0 . 0 1 4 9 [33], and a c u = 0 . 3 6 1 5 n m [34]; the temperature dependencies of these parameters have been ignored because not all of them are known. For Cu-rich C u - C o alloys q = 2.442, Ft~ = 4.636 and 6 = 0.104. The thermodynamic model of Hasebe and Nishizawa [35] was used to compute ~ 2F/~C2 from equation (46), with M = 33,846 J/mol and N = 2467 J/mol. Equation (33) was then used to calculate 7,~ and Zl;. The parameters ~ and A are thus relatively strongly temperature-dependent, primarily through the temperature dependencies of Z~ and Z#- The maximum solubility of Co in Cu is ~ 8 . 4 % , hence C0m = 0.084. The equilibrium incoherent solubilities of the c¢ and fl phases were estimated from the data compiled by Nishizawa and Ishida [36]. A ranges from 1.263 x 10 4 at 400°C to 8.475 x 10 3 at II00°C, and ~ varies from 3.4026 to 0.4071 over the same range of temperature. The results of the calculations are shown in Fig. 9. Since the differences between the solubility limits for coherent and incoherent equilibrium are quite small, data on both terminal solid solutions are presented only over the range 500 700°C. This range of temperatures is representative of the vast majority of investigations on the decomposition of supersaturated C u - C o alloys. It is quite evident that the elastic energy contribution to coherent phase equilibria in this alloy system is much smaller than we might expect, considering that e is much larger than in the Ni-A1 system (E =0.0149, cf. 0.0047). The reason for this is that the values of A are almost 500 times smaller than in the Ni-A1 alloys (this is largely because the difference C0,-C0~ is nearly unity in C u - C o alloys, and roughly an order of magnitude smaller in Ni AI alloys). If the Curich portion of the phase diagram were displayed over the entire range of C°o, i.e. to C ° = 0.08, the difference between C~(0) and C~(Zm) [Fig. 9(a)] would not be distinguishable. The effect of coherent equilibrium on Ct~ is substantially larger, but is nevertheless small [the abscissa in Fig. 9(a) is expanded considerably over that in Fig. 9(b)]. We conclude that the variation of both C~ and C~ with C ~°:ois probably too small to be detected experimentally in experiments on the aging of supersaturated Cu Co alloys.
v I--=
=/" /
500
=
¢c.)+ cco)
J
, , , I ....
I ....
1 ....
I ....
I , , , , I
0.002 0,003 0.004 0,005 0,006 0,007 0,008
e4o (b) 700
=============================
c /
":::::::::"':%(z.,) :::::::,//
6so..
(Co)
600~_..
a (cu) + 13(co)
I
,
I
,
I
,
~:::::.:
I
~
I
,
500
0.982 0.984 0.986 0.988 0.990 0.992 0.994
Fig. 9. The Cu-rich (a) and Co-rich (b) portions of the Cu-Co phase diagram. C0, in (a) and C0~ in (b) are the incoherent solubility limits taken from the compilation of Nishizawa and Ishida [36]. C~(0) and C~(zm) in (a) indicate the range of expected coherent solubility limits of Co in Cu for alloys containing up to 8.4% Co. C¢(0) and Ct~(zm) in (b) indicate the range of concentrations of Co in the Co-rich precipitates for the same alloys as in (a). The abscissa in (a) is considerably expanded over that in (b). 4. DISCUSSION There are two principal consequences of the calculations presented in this paper. The first is that both C, and C~ always increase as C o increases. This is perhaps the most significant difference between our results and those of Johnson and Voorhees [9] and Pfeifer and Voorhees [12], and follows directly from our choice of boundary conditions in the calculations of the stresses within each cell. The second consequence is that the difference between coherent and incoherent equilibrium is much greater for a solid
1834
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS
0.11
0.10
CAI 0.09 / / -
•
o.111
42s
• [] o
0.111 0.138 0.138
450 425 450
0.08 0
0.5
1.0
1.5 2.0 t- I/,I (s-I13)
2.5
3.0
3.5
Fig. 10. The variation of A1 concentration, CAI, as a function of the reciprocal cube root of the aging time in two binary N i ~ l alloys aged at the temperatures indicated. The changes observed occurred during the later stages (t/> 8 h) of ~;' precipitation in the two alloys. Data of Gentry and Fine [30].
the theory and uncertainties in the precise location of the incoherent solvus. The important point is that C~ for C°1=0.138 exceeds that for C ~ = 0 . 1 1 1 , as would be predicted by the theory, although by a far greater amount than expected. The data of Gentry and Fine clearly suggest that careful experiments on the coarsening of 7' precipitates at low aging temperatures have the potential of providing quantitative data on the variation of C,/with C 0, not only in Ni-A1 alloys, but in other binary Ni-base alloys in which E is even larger. Experiments of this kind are in progress. The results of the calculations on Cu-Co alloys suggest that the practical consequences of coherent equilibrium on terminal solid solutions are negligible. Nucleation experiments conducted at very small undercoolings from the incoherent solvus might be affected because nucleation rates are extremely sensitive to undercooling, but the predicted variations of C~ and Ct~ with C 0 would not be detectable. 5. SUMMARY AND CONCLUSIONS
solution in equilibrium with an intermediate phase than with another terminal solid solution. The parameter AC0 = C0~- C0~ plays the decisive role here. Having demonstrated that C. can vary by amounts exceeding 1 at.% in Ni Al alloys, it is of considerable interest to explore the literature to see if there is any evidence that C increases as C ~,l increases. The data of Gentry and Fine [30] on the variation of AI concentration with aging time provide convincing evidence that this is indeed the case. Gentry and Fine used magnetic measurements to follow the variation of AI concentration, CA~, with aging time in two Ni-A1 alloys containing 11.1 and 13.8% A1. They aged both alloys at low temperatures, 425 and 450°C, and their data for aging times exceeding 8 h are reproduced in Fig. 10, where CA~ is plotted vs t--~/3, in accord with the type of time dependence expected during coarsening of the 7' particles at very long aging times [37, 38]. Irrespective of the mechanism of precipitate growth over the range of aging times investigated, it is quite clear that CA~ at both aging temperatures is substantially larger for the alloy containing 13.8% A1 than it is for the alloy containing 1l.1% AI. We believe that this is a manifestation of coherent equilibrium, because the behavior is otherwise difficult to rationalize. If we assume that the data of Gentry and Fine are consistent with y' coarsening at the longer aging times, and assume that the dependence of CA~on t - 1/3 is linear, as depicted in Fig. 10, we obtain values of C~,for the four sets of data by extrapolating the linear portions to t J/3= 0, as shown. This analysis produced the four data points shown in the Ni-A1 phase diagram (Fig. 6). It is evident that the values of C~ for C°l = 0.111 are small with respect to the calculated phase boundaries. However, quantitative agreement is not really expected, given the approximations of
A theory of coherent equilibrium has been developed for alloys containing monodisperse spherical coherent precipitates. The microstructure is first subjected to a Dirichlet tessellation, producing a Voronoi polyhedron surrounding each particle. Capillarity is ignored. The basic assumption of the theory is that the state of stress in each polyhedron can be calculated by approximating each polyhedron as a sphere, and assuming that the displacements at the surface of each sphere vanish. It is then possible to calculate the dependence of C~ on z using the principles and procedures established by Johnson and Voorhees [9]. The theory predicts that C~ will always increase with C °, or equivalently, with z. Calculations have been performed to estimate the magnitudes of the changes of the solubility in N i A 1 alloys containing 7' precipitates and in Cu Co alloys containing Co precipitates. The effect of coherent equilibrium is much larger in the Ni AI system than in the Cu-Co system, despite the much larger value ore for f.c.c. Co precipitates. This is attributed to the strong influence of C0~~- C0~ on the important parameter A. The calculations indicate that the variation of C~ with C o in the Ni-A1 system is large enough, exceeding 1 at.% at low temperatures, to be measurable. Examination of the data of Gentry and Fine [30] indicates that C~ increases with increasing AI concentration in Ni-A1 alloys aged at low temperatures, providing experimental support for the theory.
Acknowledgements--The authors
express their gratitude to the National Science Foundation for financial support for this research under Grant #DMR 9212536. These calculations were inspired by stimulating communications with Professor J. Frade of the University of Aveiro, Portugal. We also thank Professor W. C. Johnson of the University of Virginia for helpful comments on the implications of the theory.
A R D E L L and MAHESHWARI:
C O H E R E N T EQUILIBRIUM IN ALLOYS
REFERENCES 1. 2. 3. 4.
J. W. Cahn, Acta metall. 10, 907 (1962). F. Larch6 and J. W. Cahn, Acta metall. 21, 1051 (1973). P. Y. Robin, Am. Miner. 59, 1299 (1974). W. W. Mullins and R. F. Sekerka, J. chem. Phys. 82, 5192 (19853. 5. R. O. Williams, Metall. Trans. IIA, 247 (1980). 6. R. O. Williams, Calphad 8, 1 (1984). 7. J. W. Cahn and F. Lareh+, Acta metall. 32, 1915 (1984). 8. A. L. Roitburd, Soy. Phys. Solid State 27, 598 (1985). 9. W. C. Johnson and P. W. Voorhees, Metall. Trans. 18A, 1213 (19873. 10. Z.-K. Liu and J. /~gren, Acta metall, mater. 38, 561 (1990). 11. W. C. Johnson and W. H. Mfiller, Acta metall, mater. 39, 89 (1991). 12. M. J. Pfeifer and P. W. Voorhees, Metall. Trans. 22A, 1921 (1991). 13. M. J. Pfeifer and P. W. Voorhees, Acta metall, mater. 39, 2001 (199l). 14. J. K. Lee and W. Tao, Aeta metall, mater. 42, 569 (1994). 15. P. W. Voorhees and W. C. Johnson, J. chem. Phys. 84, 5108 (1986). 16. F. Larch~ and J. W. Cahn, Aeta metall. 26, 1579 (1978). 17. W. C. Johnson and P. W. Voorhees, J. appl. Phys. 59, 2735 (1986). 18. E. Z. Vintaikin, Soy. Phys. Dokl. 11, 9l (19663. 19. H. Pottebohm, G. Neite and E. Nembach, Mater. Sei. Engng 60, 189 (1983). 20. F. X. Kayser and C. Stassis, Physica status solidi 64, 335 (1981). 21. E. Nembach and G. Neite, Prog. Mater. Sci. 29, 177 (1985). 22. H. A. Calderon, P. W. Voorhees, J. k Murray and G. Kostorz, Aeta metall, mater. 42, 991 (1994). 23. A. J. Ardell and R. B. Nicholson, Acta metall. 14, 1295 (1966). 24. T. Miyazaki, H. Imamura and T. Kozakai, Mater. Sci. Engng 54, 9 (1982). 25. W. O. Alexander and N. B. Vaughan, J. Inst. Metals 61, 247 (1937). 26. A. Taylor and R. W. Floyd, J. Inst. Metals 81, 25 (1952 53). 27. P. Nash, M. F. Singleton and J. L. Murray, Phase Diagrams of Binary Nickel Alloys (edited by P. Nash), p. 3. ASM International, Materials Park, Ohio (1991). 28. P. K. Rastogi and A. J. Ardell, Acta metall. 17, 595 (1969). 29. A. J. Ardell, in Experimental Methods of Phase Diagram Determination (edited by J. E. Morral, R. S. Schiffman and S. M. Merchant), p. 578. TMS, Warrendale, Pa (1994). 30. W. Gentry and M. E. Fine, Acta metall. 20, 181 (1972). 31. A. Kelly and G. W. Groves, Crystallography and Crystal Defects, p. 163. Addison-Wesley, Reading, Mass. (1970). 32. R. D. Dragsdorf, J. appl. Phys. 31, 434 (1960). 33. V. Gerold and H.-M. Pham, Z. Metallk. 71,286 (1980). 34. B. D. Cullity, Elements o f X - R a y DifJ?action, 2nd edn, p. 506. Addison-Wesley, Reading, Pa (1978). 35. M. Hasebe and T. Nishizawa, Calphad 4, 83 (1980). 36. T. Nishizawa and K. Ishida, Bull. Alloy Phase Diag. 5, 161 (1984). 37. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1962). 38. A. J. Ardell, Acta metall. 16, 511 (19683.
1835
APPENDIX The general solution to equation (8) has the form p2w - A'p 3 + B '
(A1)
where A ' and B ' are integration constants. Since w must approach 0 as p---,0, B ' - 0 . The solution inside the precipitate phase is therefore (A2)
wr~ = Ap
where A ' is denoted here by A. In the matrix phase w takes the form D
w = Bp + p 2
(A3)
where the integration constants A ' and B ' are now denoted by B and D, respectively. The constants A, B and D can be evaluated by imposing
the conditions of continuity of displacement and traction across a coherent interface [I5], and the requirement that w vanish at p - r'. Substitution of equation (A2) into (3) and (4), and equation (A3) into (5) and (6) yields the following equations for the stresses in the precipitate and matrix phases, namely T ~,, ~ -- T ool~= T ,I,4,t~ = (3ZI*+ 2Gt~)A _ 3EKt~= 3KI~(A - E) (A4) T,,, = (3Z + 2G )B -
4GD 4GD - ~ = 3KB - p3 p-
(A5)
and 2GD Too = T ~ = 3KB +- p3
(A6)
where we have made use of the equality 32 + 2G = 3K for both phases. Invoking the condition of continuity of traction at the interface (p = r), using equations (A4) and (A5), leads to the relationship 3KB -
4GD r~ = 3KI~(A - E )
(A7)
while continuity of displacement, using equations (A2) and (A3), produces the result D = (A - B ) r 3.
(AS)
On utilizing the condition w = 0 at p = r' in equation (A3), we obtain the result r'3\ B A=B l--r3)=z(Z-1) (A9) where we have made use of equation (l). Using equations (A7) (A9) and the parameters defined in equations (12) (14), the constants A and B can be expressed as A
3Kl~ (1 -- z) 4GFI~(1 6~lz)
(A10)
and B-
- 3Kj~z
4cr~(1 -
(A11)
a~i
Since the particles of the fl phase are subjected to a hydrostatic pressure P, i.e. T~,p(p = r) = - P
(A12)
we obtain, using equations (A5), (AS), (A10) and (AI 1) the expression for P presented in equation (I 1).
~
Pergamon
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Copyright © 1995 ElsevierScienceLtd Printed in Great Britain All rights reserved 0956-7151/95 $9.50 + 0.00
COHERENT EQUILIBRIUM IN ALLOYS CONTAINING SPHERICAL PRECIPITATES
A. J. ARDELLand A. MAHESHWARI Department of Materials Science and Engineering, University of California, Los Angeles, CA 90024-1595, U.S.A. (Received 20 May 1994; in revised form 15 September 1994)
Abstrac~Coherent equilibrium is investigated theoretically in alloys containing monodispersed spherical precipitates. The alloy system is subjected to a Dirichlet tessellation by constructing a Voronoi polyhedron around each particle, and for the purpose of calculation each polyhedron is approximated as a sphere. The basic unit of the microstructure is thus a cell consisting of concentric spheres, with the outer sphere occupied by the majority (matrix) phase. The stresses in each cell are derived analytically using as one of the boundary conditions the requirement that the displacements at the outer boundary vanish. Isotropic elasticity is assumed throughout. The equilibrium volume fraction and the coherent solubilities of the two phases are calculated as a function of the overall concentration of solute in the alloy. The relative importance of the lattice mismatch between the two phases, their elastic constants, and the curvatures of their free energies of mixing, is investigated theoretically. It is predicted that the equilibrium concentrations of both phases always increase with increasing volume fraction. The consequences of the theory on the Ni-Al and Cu-Co alloy phase diagrams are explored. Much larger effects are predicted for Ni-AI alloys than for CuzCo alloys, despite the fact that the lattice mismatch is over three times as large in the latter system. The reason is that the difference between the solute concentrations of the two phases is much bigger in CuzCo alloys. It is concluded that the influence of coherency effects will always be much greater for solid solutions in equilibrium with intermediate phases than for terminal solid solutions. Data on the solubility of Ni3AI in Ni-A1 alloys, which support the theory, are presented and discussed.
1. INTRODUCTION The idea that the solubility limits of coherent phases in thermodynamic equilibrium can differ from their counterparts in incoherent, unstressed, equilibrium dates back to the pioneering theory of Cahn on spinodal decomposition in cubic crystals [1]. Larch6 and Cahn [2] and Robin [3] later showed that thermodynamic equilibrium in stressed solids was determined not by uniformity of the familiar chemical potential, but by a quantity called the diffusion potential, which is generally a function of composition, temperature and stress [2, 4]. It was later realized by Williams [5, 6] that the coherent solubility limits within the two-phase region of a binary alloy should also depend on the volume fractions of the two phases, or equivalently, on the concentration of the alloy, C 0 These concepts have stimulated considerable theoretical work on the ramifications of coherent phase equilibria in two-phase solids [7-14]. Cahn and Larch6 [7] pointed out that the elastic contribution to the total free energy of a solid containing two phases, and/?, in coherent equilibrium depends not only on their compositions, C, and C~, and the mole fraction, z, of the/3 phase, but also on the morphology of the individual phases and their spatial distribution. They explored coherent equilibrium in a system in which the elastic free energy contribution depends only on
z and not on morphology, and in which the familiar chemical free energy of each phase is a parabolic function of composition [7]. Lee and Tao [14] have recently extended this calculation by allowing the elastic strain to vary with concentration. While these theoretical investigations provide considerable insight into the general nature of coherent equilibrium, they cannot be used in a transparent manner to predict the effects that might occur in specific alloy systems. Also, their predictions are not readily amenable to quantitative experimental verification. The theoretical efforts most closely related to the subject of this work, namely an alloy containing spherical coherent precipitates, are those of Johnson and Voorhees [9] and Pfeifer and Voorhees [12, 13]. Johnson and Voorhees solved the problem of coherent phase equilibrium in a finite closed system in which a sphere of t h e / / p h a s e of radius r is embedded entirely within a sphere of the e phase of radius r'. F o r this geometry, the mole fraction of the/~ phase, which is taken as equal to the volume fraction, is simply
Because the system of Johnson and Voorhees [9] is subjected to an external pressure in the most general case, one of the boundary conditions they imposed is
1825
1826
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS
continuity of traction on the external surface. For all their calculations, however, this pressure was taken as zero. Pfeifer and Voorhees used this approach to investigate coherent equilibrium in binary alloys in which the incoherent equilibria are characterized by congruent points [12], and to establish the parallel tangent construction as a device for constructing coherent phase diagrams [13] under certain restrictive conditions. The application by Pfeifer and Voorhees [12, 13] of the original calculations of Johnson and Voorhees [9] is equivalent to describing the morphology of the system as a dispersion of spherical particles of the fl phase embedded in spheres of the ~ phase. This is the same as subjecting the microstructure to a Dirichlet tessellation, whereby each particle of the fl phase is embedded in a Voronoi polyhedron. For ease of calculation, each polyhedral cell is assumed to be spherical. The difficulty with utilizing the solution of Johnson and Voorhees to describe coherent equilibrium in such a system is that there is no reason why the stress should fall to zero at the boundary of each cell. We believe that a better approximation is to allow the displacementin the c~phase to become zero at the boundary. The reason for this can be appreciated by considering two neighboring misfitting spherical precipitates of identical size. Even without a detailed calculation, it can be readily appreciated that the displacement in the matrix separating the two particles must be zero at the midpoint of the line joining their centers. However, the normal (radial) stress at this point is twice the stress that would obtain, at the same distance from its center, if only one particle were present. In a system containing many particles, it is reasonable to assume, as a first approximation, that the displacement will become approximately zero at the midpoints of the lines joining the particle to its nearest neighbors, i.e. along the surface of the Voronoi polyhedron surrounding it. We realize that neither the assumption of zero pressure nor zero displacement at the cell boundary can perfectly reproduce the state of stress in the matrix within it. As we will show, however, our assumption of zero displacement predicts quite different behavior for coherent equilibrium than the assumption of zero pressure, and that the admittedly limited experimental data that exist are indeed consistent with the predictions of a model of coherent equilibrium based on the state of stress that results. In this paper we develop the equations for coherent equilibrium for the tessellated microstructure described above, using the concentric sphere model of Johnson and Voorhees [9] but assuming that the ]'In the equations that follow, parameters or variables that refer to the /3 phase are indicated using a subscript or superscript ft. For the most part, no subscript is used to denote reference to the c~ phase, although there are several obvious exceptions.
displacement vanishes along the surface of the outer sphere. We then apply the results of the theory to two important types of phase diagrams, specifically those in which a terminal solid solution is in equilibrium with an intermediate ordered phase (exemplified by Ni-AI), and those in which terminal solid solutions are in equilibrium (exemplified by Cu-Co). The differences between the principal predictions of our model are contrasted with those of the model of Johnson and Voorhees, some data in the literature supporting the predictions of our model are noted, and the general conclusions that emerge are discussed. 2. DEVELOPMENT OF THE MODEL
2.1. Stresses and strains We adopt the geometry of Johnson and Voorhees [9] of concentric spheres of the ~ and fi phases, assuming spherical symmetry throughout, with radial and angular coordinates denoted by p, 0, ~b, respectively. Additionally, we assume elastically isotropic phases, that the elastic constants, molar volumes and lattice constants of the two phases are independent of concentration, and that the effects of curvature on equilibrium can be ignored. A state of equilibrium obtains if the dispersion is monodisperse, because then, even allowing for capillarity, the c~ phase is uniform in composition. The misfit responsible for the stresses and strains in the precipitate and matrix phases, E, is defined by
E-
a¢ -- a
(2)
a
where a~ and a are the lattice constants of the precipitate and matrix phasest. In this case the stresses in the precipitate are given by the solutions [15] of the differential equations
(dw
w)
2G dW-3Kt~
(3)
and (dw w) w ~p+2p +2G~p-3KeE
T~oo=V~=21,
(4)
while those in the matrix are found from the solutions to the equations [15]
Tpe=2(dw \dp
+
2 w) dw p +2G--dp
(5)
and dw
2w)
w
Too= Too= ;t ~p + P + 2G--.p
(6)
In equations (3)-(6) w is the displacement, 2 is the Lam6 constant and G is the shear modulus. Substitution of equations (3)-(6) into the equation of mechanical equilibrium under conditions of spherical
1827
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS symmetry [15], i.e. dTpp 2Tpp - 2Too + - = 0 dp p
(7)
which is valid for both phases, produces the differential equation for the displacement d Jd(p2w) } dp ( p2 do = 0.
(8)
It is shown in the Appendix that this system of equations yields the solutions for the stresses T°P- l + t / z
t/z+
T°°-- l + q z
P
(9)
(10)
t/z--~
where -- P is the hydrostatic pressure imposed by the matrix on the fl precipitate, and is given by the expression P _ 3K~E (1 + t/z)
(11)
6t/z )
r~( l -
In equations (9)-(11) t/, FI~ and 6 are defined by the equations 3K 4(;
t/ -
(12)
3K~ Ft~ = 1 ~ 4G
(13)
and 6 K~-K KF~
(14)
where K and Kt~ are the hulk moduli of the ~ and fl phases. The parameters F~ and 6 are identical to those used by Johnson and Voorhees [9], while ~/+ 1 is equal to their parameter F. The displacements in the ~ and fl phases are given by the equations w - 4G (1 + t/z)
p
oP (1 - z) w~ = 4G (1 + t/z)"
(16)
We can see immediately from equation (9) that the normal radial stress acting along the surface of the outer sphere is not zero, but is instead - P z (1 + t/) -
(1 + t/z)
2.Z Conditions for thermodynamic equilibrium
The influence of the coherency stresses on the concentrations of both phases at thermodynamic equilibrium are obtained by invoking two conditions. The first condition is that the diffusion potentials [16] be equal, which for equal partial molar volumes of the two components is represented by the equation
g~(C~) - ~,(1 - C~) = ~ ( C ~ ) - ~,,~(1 - G )
(18)
where the #i are chemical potentials of the components at the designated compositions. The other condition involves the so-called thermodynamic potentials, w, of the two phases [17], specified by the equation co -
o ~ = (e0p - e~pp,~T~op
(19)
where v
co = n0/~, (1 - Ca) - ~
(T,,)2 + ~
1
T,/T o . (20)
no is the number of atomic sites per unit volume, v is the Poisson ratio, E is the Youngs modulus and summation over repeated indices is understood. In equations (19) and (20), the stresses and strains are evaluated at the ~/fl interface (p = r). The second and third terms in equation (20) can be expressed in the form 1
u? = _ _ [T~, + 2T2o - 2v (2Te,, Too +Tool]. 2E
(21)
On substituting equations (9) and (10), with p = r, into (21), the expression for W for the ~ phase becomes p2 I +t/z) 2 + 2 ( t / z _ ~ ) 2 2E(1 + q z ) 2 ( 1
z
and
ToA.=~. = - n
H --~ P as z ~ 1. This limiting behavior makes sense to us, and demonstrates that the solution of Johnson and Voorhees is approached in the limit of zero volume fraction.
(17)
It is evident from equation (17) that the outer sphere is subjected to a hydrostatic pressure - r l that depends on the volume fraction of the fl phase; this pressure is imposed by the combined presence of all the particles in the system outside the cell containing the concentric spheres. Clearly, 13 ---, 0 as z ---, 0 and
After algebraic simplification and the use of the equalities G = E/2(1 + v) and K = E/3(1 -- 2v), reduces to the simple expression = tlP2(tlz 2 + 1) 2K(qz + 1)2.
(23)
On using equation (21) for the precipitate phase, the equation for u?~ is simply p2 W, = ~ . (24) It remains to evaluate the right-hand side of equation (19). The radial components of the strain are readily calculated from the relationship epp = dw/dp. Evaluating epp at p = r from equation (15), using equation (16), and recalling that
1828
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS
T ~v = -- P, we find
2.3. Equation o f state Jbr coherent equilibrium
(epp - - t, ts " ) T t~ - -
-PP'-PP
(25)
tlP 2
K(qz + l)
Using equations (23)-(25), (19) can now be written as
no[#, (1 -- C=) --/.z~(l -- C~)] = (% - e~)r~
Johnson and Voorhees refer to the equation that describes the dependence of z on C°, and the other relevant parameters, as the equation of state for the coherent system. In establishing the equation of state for our model, we use their reduced concentrations
+ % - v
_
Y~=-1+2
_p2
- 2Ke(tlz + 1)2 {~/(1 + 6Vl,)(rlZ 2 - 2rlz - 1) -
(r/z +
1) 2}
_
C~-Co~ Cots - Co~
_
-1+
AC~ ~ 2AC'o
Ya = 1 + 2 Cts -- Cots _ 1 + 2 ACf*
Co~ - Co~
(26)
aCo
(34)
(35)
and
which on substituting (11) becomes n0[#l (1 - C~) - #~(1 - Ce)] = 'Jr'x
(27)
WT_ 9cZKts {--6 (r/z)2 + 2*'/z + 1} 2Fts (1 -- 6qz) 2
(28)
W = 1+ 2
where
C o _ Cots
C o _ Co~ - 1+ 2 - Cots - Co~ AC0
(36)
where
To obtain the final expression for the equilibrium concentration it is necessary to select an appropriate reference or comparison state for the system. It is conventional to choose the state of incoherent equilibrium for this purpose [9, 15], so that the equilibrium concentrations in the reference state are the incoherent solubility limits for both phases. We denote the reference chemical potentials by/~o, and the equilibrium incoherent solubilities by the subscript O, so that equations (18) and (27) for the reference state (with E = O) become
Co~ - Co~ = &Co.
(37)
The substitution of equations (28), (31) and (32) into (34) and (35) produces the equations for Y= and Y/~ given by
A { - 6 (~z)2 + 2~z + 1} L = - 1+
(1 - 6rtz) 2
(38)
and Yt~= 1 + ( 1 + ( )
A{ - 3 (qz) 2 + 2qz + 1} (1 - -
&lz ) 2
(39)
where #02 ( C 0 ~ ) -- ,/~01 (1 -- C0c~)
= #gz(Cots)-/lg~(l - C0~)
A-
(29)
9E2K~
(40)
z~FBACo
and and iz0~(1 -- Co=) - #o~1(1 -- Coil) = 0.
(30)
On subtracting equations (29) and (30), respectively, from (18) and (27), and assuming that the differences between the concentrations in the coherent and reference states are small, we obtain the results [9] AC~ = C= - C0~-
kI) T
z~(c0B- c0o)
Cots-
/.~AC~
(41)
The equation of state for the system is obtained by invoking the condition of conservation of mass, namely zCts + (1 - z)C~ = C o
(42)
On substituting equations (34)-(39) into (42), we obtain after some algebra the equation (32)
Zt~
Z~
(31)
and AC~ = G -
C = z_~_ 1.
&/2(26 - (A)z 3 + r/{A(2( - &l) -
~/6 2(1 +
W)
-
40 }z 2
where Z~ and Xts are given by [9] nokT
I
c~1n72]
no 02F
(33)
Z-CoO-_-Co) l + ~-~nC - N o8C 2 where k is Boltzmann's constant, T is the temperature, 72 is the activity coefficient of component 2 in the stress-free state, No is Avogadro's n u m b e r and F i s the chemical free energy. In equation (33), all the terms are evaluated at the incoherent solubility limits of both phases.
+ {A(~ + 2 , ) + - (1 + W ) = 0.
2&l(l + W) + 2}z + A (43)
The roots of equation (43) determine the volume fraction of the fl phase in coherent equilibrium with the ~ phase. These roots determine the values of Y~ and Yts by substitution into equations (38) and (39), which in turn establish the coherent solubilities of the and fl phases via equations (34) and (35).
ARDELL and MAHESHWARI:
COHERENT EQUILIBRIUM IN ALLOYS
1.0 r-
3. RESULTS
0.20.1
t/ -
l-v 2(1 - 2v)
(44)
/ / i
S°o °
." ~oSo ,*" # °.¢
0.6
/ ,... •"
Z
#°o °
/ s',"
.... ~.-
0.4
..: , ;
:o #,%*
."4""
i:"t/.~"
0.2 0
-,.i~" i
-1.5 -1,0 -0,5
I
0
0.5
..
1.0
I
1.5
,
I
2.0
W Fig. 1. The dependence ofz on W predicted by equation (43) for 6 = 0, ~ = 0 and q = 1, with A varying from 0 to 0.25.
and
3 + v -- 4vl~
.,.:.'"
A--O/ /
3.1. General features o f the solutions The principal objective of this work is to translate the predictions of the preceding calculations to the kinds of effects that might be observable experimentally. We therefore make no attempt to explore the broad spectrum of effects that the theory is capable of predicting. It is nevertheless useful to generalize to a limited extent, so that some sense of the magnitudes of the effects expected can be appreciated. The parameters in the theory can assume a large range of values, the complete exploration of which becomes a formidable task. Since, from equations (12) and (13), t/ and F~ can be expressed as
1829
(45)
FB - 2(1 -- 2v/~) it is clear that they exist over the ranges 1/2 < t / < oo and 3/'2 < Ft~ < oc, the upper limits obtaining when v = 1/2. F o r most materials, however, t/is unlikely to exceed 7 and Fa is unlikely to exceed 8. Since for many practical cases of interest Kt~ and K will probably differ by no more than a factor of two, a realistic range of values for 6 is, from equation (14), - 1/3 < c5 < 2/3. F o r a terminal solid solution in equilibrium with an intermediate ordered intermetallic compound, the range of the parameter ( will be, from equation (41), - 1 < ~ < 0, the lower limit applying to a line c o m p o u n d (Zr~ = oc), and the upper limit governed by the fact that the curvature of the function F(C~) will be smaller than that for the solid solution. For terminal solid solutions in equilibrium the range o f possibilities is greater, but ~ is unlikely to exceed unity by an appreciable amount. The parameter A is always positive, and can in principle be quite large. However, the insertion of representative values of the parameters into equation (40) indicates that A = 0.25 is a realistic upper limit. Over the ranges of the parameters discussed herein, the types of complications described by Johnson and Voorhees regarding whether there are multiple solutions to the equation of state (43), or whether the solutions represent absolute minima or maxima of the free energy of the system, do not arise; all the solutions are single-valued and represent energy minima. Figure 1 illustrates the variation of z with W when 6 = 0 (both phases have identical compressibilities), = 0 (identical curvatures of the free energy functions), q = 1 (or v = 0.2), and A varies from 0 to 0.25. The solutions are, of course, physically meaningful
tW. C. Johnson (private communication) has pointed out that under certain circumstances, solutions for which W > 1 can be physically meaningful.
only for z lying in the range 0 ~ < z ~ < l , and for practical purposes this must occur also for values of W satisfying - 1 ~< W ~< 1 because C ° cannot exceed C0~ initially if the system is to lie within the two-phase field of the conventional phase diagram. We see, however, that W exceeds unity when z exceeds ~ 0.9 for A = 0.1, and when z exceeds ~ 0.7 for A = 0.25. Such solutions expose the limitations of the model, which cannot be expected to predict realistic alloy behavior at large volume fractions owing to the restrictions of the concentric sphere geometry. The issue of W exceeding unity for 0 ~< z ~< 1 did not arise in the model of Johnson and Voorhees because they explored coherent phase equilibria only in systems characterized by zero external pressure. It is easy to show that the imposition of finite pressure on the system shifts the curves o f z (W) in such a way that z can exist mathematically over the range 0 ~ < z ~ < l for W < - I as well as W > l . Clearly, the range of parameters over which either model yields physically meaningful solutions must be chosen, and evaluated, with some caret. The dependencies of Y~ and Y~ on W for 6 = 0, = 0 and r / = i, with A varying from 0 to 0.25, are shown in Fig. 2. The curves in Fig. 2 illustrate another general feature of our model, namely that Y~ and Y~ both always increase as W increases, indicating through equations (34) and (35) that AC~ and AC~ always increase as C O increases. This behavior is exactly opposite to that predicted by the model of Johnson and Voorhees [9]. Since z exists mathematically for W > 1, Y~ and Y~ exist as well, but have no physical significance in this regime. The dependence of the coherent solubility limits on W i s shown in Fig. 3 for 6 = 0, A = 0.25, ~/= 1 and several values of ~ between - 1 and 0.5. The data are presented in the form AC~/ACo and AC~/ACo vs W, and are limited to values of W ~
ARDELL and MAHESHWARI:
1830 2.0-
1.0 0,5
~,2
To calculate X;, from equation (33) we used the thermodynamic model of Calderon et al. [22], in which ~ 2F/~C 2 for the matrix phase is represented by the equation
^=o
2F
RT
OC 2
2M-
C0(1-C0)
6N(1 - 2C0)
(46)
0.25
-o.5
-I
[211.
0 *.,. . . ......... ..-:~'2 . .1. . . .°. . ,
v. o
-l.O/
temperature dependence of 6 is evidently quite small
5 /
y
_
COHERENT EQUILIBRIUM IN ALLOYS
.....
......... 0.2
•,.--:-:'.':LC'::;':--- . . . . . .
o 1
^=o
.,5
-2.0 ' ' ' ' ' ' -2.0-I,5-1.0-0,5
'
'
'
'
0 0.5 W
'
'
'
'
'
'
l.O 1.5 2.0
Fig. 2. The dependencies of Y~ and Y# on W for ~ = 0, ~ = 0 and ~ / = l, with A varying from 0 to 0.25.
< 0; for ~ = 0 the variations of z~C~ and ACs~ with W are identical. We reiterate that both AC= and AC B increase as C 0 increases. This behavior is characteristic of all the solutions of the equation of state of our model, and is perhaps the major difference between the predictions of the model of Johnson and Voorhees and ours; it is the prediction that can be most readily tested experimentally. The behavior predicted when 6 varies is shown in Fig. 4 for ~/ = 1, A = 0.25 and ~ = 0 (AC~ and AC# are equal in this case). We see that when 6 = 2/3 the variation of AC=,~ is much stronger than when 6 = - 1 / 3 . The variation of AC,. n with W when ~1 varies from 1 to 7 is shown in Fig. 5 for 6 = 0, ~ = 0 and A = 0.25. It is evident that the predicted changes in coherent solubility can be quite large when the parameters fi and q have values close to their expected upper limits. We reiterate that the variations of AC=,sj with W seen in Figs 4 and 5 are about as large as we are ever likely to encounter in real systems because of the large value of A ( = 0.25) used in the calculations. This will become more apparent in the next sections, where we apply the predictions of the model to two real alloy systems, C u - C o and the Ni-rich portion of the Ni-A1 phase diagram. 3.2. The Ni-rich region of the N i- A I phase diagram The constants used in predicting coherent phase equilibria in this alloy system are GNi~AI)= G~, = 75.2 G P a [18, 19], K;, = 176.3 GPa [19], K~i3AI=K;,,= 173.3GPa [20], E =0.0047 [21] and a~. = 0.355 nm [21]; we refer here to the N i A 1 solid solution as the 7 phase. These yield the values ~/ = 1.76, F# = 2.73 and 6 = -0.00607. The temperature dependencies of the physical constants has been ignored in the following calculations; except for those of a and 6 they have not been measured, and the
where M = -- 104361 + 3.29574T and N = 41291.4; both quantities are expressed in J/mol. The remaining quantities needed to complete the calculations are 2F/~C2 for the 7' phase, and the incoherent solubility limits of both phases. In the absence of a reliable thermodynamic model for the free energy of the 7' phase, we have made the assumption that X;,,= 2L,, so that ~ = - 0 . 5 at all temperatures. As can be expected from the results in Fig. 3, this assumption primarily affects the composition of the y' phase, and the effect on the calculated values of z and AC;, is minor. A varies from 0.0596 at 400°C to 0.2392 at 1300°C.
0.5 -
(a)
0.4 ~=~ AC(~ 0.3
S
AC° 0.2
,
I
~
I
I
0 W
0.5
I
-"1.5 -1.0 -0.5
0.5
0.5
.l~..: ~"
0.1 n
5
o°°* .0
,
I
,
I
1.0
].5
(b) 0.5
.°#°
0.4 .#
~ . 0.3
¢." j° •° • "
~C° 0.2
-"
°°°°
0.1 0
•"
.....
...."'"
° ~, •
.... ,
I
,
r
-1.5 -1.0 -0.5
•
°°°°". o
°°..° °°.°° .°o°
J r,
0 W
.° -0.5 •
~-I ,
,
0.5
I
1,0
,
I
1.5
Fig. 3. The dependencies of the normalized coherent solubility limits AC~/ACo (a) and ACyACo (b) on W for 6 = 0, r/ = 1 and A = 0.25,/with ~"varying from - 1 to 0.5.
ARDELL and MAHESHWARI:
COHERENT EQUILIBRIUM IN ALLOYS
0.6
~
~ = 2/3
0.5 L-
/
AC
0.4
Z". //
AC°
0.3
I/3
t
.....
#° f°
.,,=" ,°o.*"
~'.... j.y;.--" I 0.1 ' I - .5 -1.0 -0.5
I
0 W
,
L
,
0.5
I
,
1.0
l
1.5
Fig. 4. The dependencies of the normalized coherent solubility limits of both phases on W for ~ = 0, r/= 1 and A = 0.25, with c~varying from - 1/3 to 2/3 (AC~ = AC/~= ~C~.~ in this case)
We believe that the equilibrium incoherent solubilities of the 5' and 7' phases are not well established. The reason is that 7' precipitates remain fully coherent with the 7 matrix to very large sizes, approaching several micrometers [23, 2@ Consequently, true incoherent phase equilibrium is unlikely to be attained in experiments involving prolonged aging within the two-phase field of initially supersaturated solid solutions, or very slow cooling from the solutiontreatment temperature. These are the kinds of experiments used by early investigators, such as Alexander and Vaughan [25] and Taylor and Floyd [26] to establish the solubility limits of the 7 solid solution. We do not concur with the conclusion of Nash et aI. [27] that the incoherent solubility of A1 in Ni is known from the work of Alexander and Vaughan [25].
0.8
-
11=7 0.6
4
Ac=,l~ AC o
0,4 I
0.21
0
i
,
I
I
-1.5 -1.0 -0.5
I
J
0 W
0,5
i
J
1.0
i
i
1,5
Fig. 5. The dependencies of the normalized coherent solubility limits of both phases on W for 5 = 0, ~ = 0 and A = 0.25, with ~ varying from I to 7 (AC~ = ACt~ = AC~.I~in this case).
1831
On the other hand, the coherent solubility of the 7' phase has been measured [28], and the literature on this topic recently reviewed [29]. No account has been taken of the possible effect of z in these experiments, so the solubilities measured can be thought of as averages over a relatively small range of z, the values of which have generally been less than 0.15. For purposes of this work, we have taken the coherent solubilities summarized by Ardell [29] as the solubility limits at z = 0 [we call this quantity C:(0)] and used them to calculate Co;. from 400 to 1300'C; equation (28) with z -= 0 was substituted into (31) for this calculation, assuming that Z;. is identical for the coherent and incoherent phases over the small range of compositions involved, The data of Taylor and Floyd [26] on the composition of the 5" phase were taken as Co:,,, although they too are very likely to be coherent solubilities over the temperature range of the calculations. We note here that C0~.is the counterpart of the incoherent solubility limit calculated by Rastogi and Ardell [28], but is roughly a factor of 4 smaller than their values because they used a molar volume of Ni3A1 approx. 4 times larger than the atomic volumes used here [the atomic volume is equivalent to 1in o , which enters the calculation through equation (33)]. The goal of our calculations is to demonstrate the consequences of coherent phase equilibrium over a practical range of alloy concentrations. It is completely impractical to perform aging experiments on alloys in which the overall concentration exceeds the maximum solubility limit of A1 in Ni, which is approx. 20.5% at 1300'~C [27]. Hence, the maximum value of C°j in all the calculations, C ~ , was taken as 0.205. The maximum value of W, Wm, varies with temperature, of course, and at each temperature C~. was computed from the maximum value of z corresponding to this value of Win. [We call this quantity C~.(zm), where z m is the maximum possible value of z at a given temperature.] In summary, then, equation (43) was used to calculate the dependence of z on W. Realistic values of z exist over a range 0 < z < Zm (where z m is determined using C 0m _- 0.205), which establishes Wm from equation (36). The values of C:.(0) and C:, (0) (the value o f C v at z = 0), and C:,(zm) and C;.,(Zm) (the value of C v at z = z~,) were then calculated from equations (34) and (35), via (38) and (39). The results of the calculations are shown in Fig. 6. It is apparent that the effect of coherency strains on the phase equilibrium are quite significant for the matrix phase, the difference between C.(Zm) and C:(0) exceeding 1% at the lower temperatures. This difference diminishes as the temperature increases, but is still large at 1100°C. The effect on the solubility of the 5" phase is smaller, but this is a consequence of the fact that we have chosen L., = 2 L, in the calculations. Two other features of interest are briefly examined. The first is the relative change in volume fraction that can occur on loss of coherency, which produces a new
1832
ARDELL and MAHESHWARI:
COHERENT EQUILIBRIUM IN ALLOYS
----ic,(
c,,0,../-g
t
,
0,05
O,10
"ti i
114,, ,
400
O.15
0.20
0.25
Fig. 6. The Ni-rich portion of the binary N i ~ l phase diagram. Co,,and Ccv,are the incoherent solubility limits of the 7 and 7' phases. C~(0) and C~,(zm) define the coherent solubility limits of the 7 phase at z = 0 and z = z,,, and C:,,(0) and C:,,(Zm) do likewise for the 7' phase. The open and filled symbols represent the coherent solubility limits of alloys containing l 1.1 and 13.8% AI, respectively, estimated from the data of Gentry and Fine [30].
volume fraction z~ of incoherent 7'- This change is illustrated by plotting z/z~ vs z in Fig. 7, and is always positive in Ni-A1 alloys. Hence, if loss of coherency were to occur in Ni-A1 alloys, the volume fraction of 7' would increase because z~>z. Since z/z~ approaches zero as z ~ 0, the fractional increase in volume fraction on loss of coherency can be quite large for small supersaturations. F o r more concentrated alloys, in which z is fairly large to begin with, the increase in volume fraction on loss of coherency
is much smaller ( ~ 1 0 %
at 400°C to ~ 2 5 %
at
1100°C). It is also of interest to evaluate the pressures acting on the precipitate and matrix phases within the basic cell. It is expected from equations (11) and (17) that both P and FI increase as z increases, and these pressures can be calculated using values of the appropriate parameters. The results of such calculations are shown in Fig. 8, where it is seen that FI can become quite large in Ni-A1 alloys for values of z approach-
1.0[
2.0
0.9 900
0.8
1100 °C
o
0.7
1
,
5
a.
~
0.6 -0.5
1.0
0.4 0.3 0.5
0.2 0.1 0
,
I
I
0 O.l 0.2 0.3 0,4 0.5 0.6 0.7 Z
Fig. 7. Illustrating the predicted change in volume fraction of 7' precipitates in Ni AI alloys on loss of coherency. The ratio of the incoherent and coherent volume fractions, z i / z , is plotted against z for the three temperatures indicated.
0
'
i i
i
l
0 0,I 0.2 0.3 0,4 0.5 0.6 0.7 z
Fig. 8. Illustrating the variations of the hydrostatic pressure with volume fractions of 7" precipitates in N i ~ l alloys. P is the pressure on the precipitates and FI is the pressure on the cell.
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM 1N ALLOYS ing z m. However, for volume fractions up to about 0.3, 13 is generally < 700 MPa. The effects of hydrostatic pressures of this magnitude on physical properties (lattice constants, for example) are small. At significantly larger volume fractions the quantitative predictions of the theory must be viewed with caution.
1833
700
(a )
O~
,,'"'
co=
C=(O)~ (Cu) ,~j~"~" C"(zm)
A
3.3. Cu-Co alloys The values of the physical constants used in predicting coherent phase equilibrium in this alloy system are Gc, = 4 2 . 1 G P a [31], Kcu= 137.1GPa [31], and K c o = 2 0 4 . 1 G P a [32], e = - 0 . 0 1 4 9 [33], and a c u = 0 . 3 6 1 5 n m [34]; the temperature dependencies of these parameters have been ignored because not all of them are known. For Cu-rich C u - C o alloys q = 2.442, Ft~ = 4.636 and 6 = 0.104. The thermodynamic model of Hasebe and Nishizawa [35] was used to compute ~ 2F/~C2 from equation (46), with M = 33,846 J/mol and N = 2467 J/mol. Equation (33) was then used to calculate 7,~ and Zl;. The parameters ~ and A are thus relatively strongly temperature-dependent, primarily through the temperature dependencies of Z~ and Z#- The maximum solubility of Co in Cu is ~ 8 . 4 % , hence C0m = 0.084. The equilibrium incoherent solubilities of the c¢ and fl phases were estimated from the data compiled by Nishizawa and Ishida [36]. A ranges from 1.263 x 10 4 at 400°C to 8.475 x 10 3 at II00°C, and ~ varies from 3.4026 to 0.4071 over the same range of temperature. The results of the calculations are shown in Fig. 9. Since the differences between the solubility limits for coherent and incoherent equilibrium are quite small, data on both terminal solid solutions are presented only over the range 500 700°C. This range of temperatures is representative of the vast majority of investigations on the decomposition of supersaturated C u - C o alloys. It is quite evident that the elastic energy contribution to coherent phase equilibria in this alloy system is much smaller than we might expect, considering that e is much larger than in the Ni-A1 system (E =0.0149, cf. 0.0047). The reason for this is that the values of A are almost 500 times smaller than in the Ni-A1 alloys (this is largely because the difference C0,-C0~ is nearly unity in C u - C o alloys, and roughly an order of magnitude smaller in Ni AI alloys). If the Curich portion of the phase diagram were displayed over the entire range of C°o, i.e. to C ° = 0.08, the difference between C~(0) and C~(Zm) [Fig. 9(a)] would not be distinguishable. The effect of coherent equilibrium on Ct~ is substantially larger, but is nevertheless small [the abscissa in Fig. 9(a) is expanded considerably over that in Fig. 9(b)]. We conclude that the variation of both C~ and C~ with C ~°:ois probably too small to be detected experimentally in experiments on the aging of supersaturated Cu Co alloys.
v I--=
=/" /
500
=
¢c.)+ cco)
J
, , , I ....
I ....
1 ....
I ....
I , , , , I
0.002 0,003 0.004 0,005 0,006 0,007 0,008
e4o (b) 700
=============================
c /
":::::::::"':%(z.,) :::::::,//
6so..
(Co)
600~_..
a (cu) + 13(co)
I
,
I
,
I
,
~:::::.:
I
~
I
,
500
0.982 0.984 0.986 0.988 0.990 0.992 0.994
Fig. 9. The Cu-rich (a) and Co-rich (b) portions of the Cu-Co phase diagram. C0, in (a) and C0~ in (b) are the incoherent solubility limits taken from the compilation of Nishizawa and Ishida [36]. C~(0) and C~(zm) in (a) indicate the range of expected coherent solubility limits of Co in Cu for alloys containing up to 8.4% Co. C¢(0) and Ct~(zm) in (b) indicate the range of concentrations of Co in the Co-rich precipitates for the same alloys as in (a). The abscissa in (a) is considerably expanded over that in (b). 4. DISCUSSION There are two principal consequences of the calculations presented in this paper. The first is that both C, and C~ always increase as C o increases. This is perhaps the most significant difference between our results and those of Johnson and Voorhees [9] and Pfeifer and Voorhees [12], and follows directly from our choice of boundary conditions in the calculations of the stresses within each cell. The second consequence is that the difference between coherent and incoherent equilibrium is much greater for a solid
1834
ARDELL and MAHESHWARI: COHERENT EQUILIBRIUM IN ALLOYS
0.11
0.10
CAI 0.09 / / -
•
o.111
42s
• [] o
0.111 0.138 0.138
450 425 450
0.08 0
0.5
1.0
1.5 2.0 t- I/,I (s-I13)
2.5
3.0
3.5
Fig. 10. The variation of A1 concentration, CAI, as a function of the reciprocal cube root of the aging time in two binary N i ~ l alloys aged at the temperatures indicated. The changes observed occurred during the later stages (t/> 8 h) of ~;' precipitation in the two alloys. Data of Gentry and Fine [30].
the theory and uncertainties in the precise location of the incoherent solvus. The important point is that C~ for C°1=0.138 exceeds that for C ~ = 0 . 1 1 1 , as would be predicted by the theory, although by a far greater amount than expected. The data of Gentry and Fine clearly suggest that careful experiments on the coarsening of 7' precipitates at low aging temperatures have the potential of providing quantitative data on the variation of C,/with C 0, not only in Ni-A1 alloys, but in other binary Ni-base alloys in which E is even larger. Experiments of this kind are in progress. The results of the calculations on Cu-Co alloys suggest that the practical consequences of coherent equilibrium on terminal solid solutions are negligible. Nucleation experiments conducted at very small undercoolings from the incoherent solvus might be affected because nucleation rates are extremely sensitive to undercooling, but the predicted variations of C~ and Ct~ with C 0 would not be detectable. 5. SUMMARY AND CONCLUSIONS
solution in equilibrium with an intermediate phase than with another terminal solid solution. The parameter AC0 = C0~- C0~ plays the decisive role here. Having demonstrated that C. can vary by amounts exceeding 1 at.% in Ni Al alloys, it is of considerable interest to explore the literature to see if there is any evidence that C increases as C ~,l increases. The data of Gentry and Fine [30] on the variation of AI concentration with aging time provide convincing evidence that this is indeed the case. Gentry and Fine used magnetic measurements to follow the variation of AI concentration, CA~, with aging time in two Ni-A1 alloys containing 11.1 and 13.8% A1. They aged both alloys at low temperatures, 425 and 450°C, and their data for aging times exceeding 8 h are reproduced in Fig. 10, where CA~ is plotted vs t--~/3, in accord with the type of time dependence expected during coarsening of the 7' particles at very long aging times [37, 38]. Irrespective of the mechanism of precipitate growth over the range of aging times investigated, it is quite clear that CA~ at both aging temperatures is substantially larger for the alloy containing 13.8% A1 than it is for the alloy containing 1l.1% AI. We believe that this is a manifestation of coherent equilibrium, because the behavior is otherwise difficult to rationalize. If we assume that the data of Gentry and Fine are consistent with y' coarsening at the longer aging times, and assume that the dependence of CA~on t - 1/3 is linear, as depicted in Fig. 10, we obtain values of C~,for the four sets of data by extrapolating the linear portions to t J/3= 0, as shown. This analysis produced the four data points shown in the Ni-A1 phase diagram (Fig. 6). It is evident that the values of C~ for C°l = 0.111 are small with respect to the calculated phase boundaries. However, quantitative agreement is not really expected, given the approximations of
A theory of coherent equilibrium has been developed for alloys containing monodisperse spherical coherent precipitates. The microstructure is first subjected to a Dirichlet tessellation, producing a Voronoi polyhedron surrounding each particle. Capillarity is ignored. The basic assumption of the theory is that the state of stress in each polyhedron can be calculated by approximating each polyhedron as a sphere, and assuming that the displacements at the surface of each sphere vanish. It is then possible to calculate the dependence of C~ on z using the principles and procedures established by Johnson and Voorhees [9]. The theory predicts that C~ will always increase with C °, or equivalently, with z. Calculations have been performed to estimate the magnitudes of the changes of the solubility in N i A 1 alloys containing 7' precipitates and in Cu Co alloys containing Co precipitates. The effect of coherent equilibrium is much larger in the Ni AI system than in the Cu-Co system, despite the much larger value ore for f.c.c. Co precipitates. This is attributed to the strong influence of C0~~- C0~ on the important parameter A. The calculations indicate that the variation of C~ with C o in the Ni-A1 system is large enough, exceeding 1 at.% at low temperatures, to be measurable. Examination of the data of Gentry and Fine [30] indicates that C~ increases with increasing AI concentration in Ni-A1 alloys aged at low temperatures, providing experimental support for the theory.
Acknowledgements--The authors
express their gratitude to the National Science Foundation for financial support for this research under Grant #DMR 9212536. These calculations were inspired by stimulating communications with Professor J. Frade of the University of Aveiro, Portugal. We also thank Professor W. C. Johnson of the University of Virginia for helpful comments on the implications of the theory.
A R D E L L and MAHESHWARI:
C O H E R E N T EQUILIBRIUM IN ALLOYS
REFERENCES 1. 2. 3. 4.
J. W. Cahn, Acta metall. 10, 907 (1962). F. Larch6 and J. W. Cahn, Acta metall. 21, 1051 (1973). P. Y. Robin, Am. Miner. 59, 1299 (1974). W. W. Mullins and R. F. Sekerka, J. chem. Phys. 82, 5192 (19853. 5. R. O. Williams, Metall. Trans. IIA, 247 (1980). 6. R. O. Williams, Calphad 8, 1 (1984). 7. J. W. Cahn and F. Lareh+, Acta metall. 32, 1915 (1984). 8. A. L. Roitburd, Soy. Phys. Solid State 27, 598 (1985). 9. W. C. Johnson and P. W. Voorhees, Metall. Trans. 18A, 1213 (19873. 10. Z.-K. Liu and J. /~gren, Acta metall, mater. 38, 561 (1990). 11. W. C. Johnson and W. H. Mfiller, Acta metall, mater. 39, 89 (1991). 12. M. J. Pfeifer and P. W. Voorhees, Metall. Trans. 22A, 1921 (1991). 13. M. J. Pfeifer and P. W. Voorhees, Acta metall, mater. 39, 2001 (199l). 14. J. K. Lee and W. Tao, Aeta metall, mater. 42, 569 (1994). 15. P. W. Voorhees and W. C. Johnson, J. chem. Phys. 84, 5108 (1986). 16. F. Larch~ and J. W. Cahn, Aeta metall. 26, 1579 (1978). 17. W. C. Johnson and P. W. Voorhees, J. appl. Phys. 59, 2735 (1986). 18. E. Z. Vintaikin, Soy. Phys. Dokl. 11, 9l (19663. 19. H. Pottebohm, G. Neite and E. Nembach, Mater. Sei. Engng 60, 189 (1983). 20. F. X. Kayser and C. Stassis, Physica status solidi 64, 335 (1981). 21. E. Nembach and G. Neite, Prog. Mater. Sci. 29, 177 (1985). 22. H. A. Calderon, P. W. Voorhees, J. k Murray and G. Kostorz, Aeta metall, mater. 42, 991 (1994). 23. A. J. Ardell and R. B. Nicholson, Acta metall. 14, 1295 (1966). 24. T. Miyazaki, H. Imamura and T. Kozakai, Mater. Sci. Engng 54, 9 (1982). 25. W. O. Alexander and N. B. Vaughan, J. Inst. Metals 61, 247 (1937). 26. A. Taylor and R. W. Floyd, J. Inst. Metals 81, 25 (1952 53). 27. P. Nash, M. F. Singleton and J. L. Murray, Phase Diagrams of Binary Nickel Alloys (edited by P. Nash), p. 3. ASM International, Materials Park, Ohio (1991). 28. P. K. Rastogi and A. J. Ardell, Acta metall. 17, 595 (1969). 29. A. J. Ardell, in Experimental Methods of Phase Diagram Determination (edited by J. E. Morral, R. S. Schiffman and S. M. Merchant), p. 578. TMS, Warrendale, Pa (1994). 30. W. Gentry and M. E. Fine, Acta metall. 20, 181 (1972). 31. A. Kelly and G. W. Groves, Crystallography and Crystal Defects, p. 163. Addison-Wesley, Reading, Mass. (1970). 32. R. D. Dragsdorf, J. appl. Phys. 31, 434 (1960). 33. V. Gerold and H.-M. Pham, Z. Metallk. 71,286 (1980). 34. B. D. Cullity, Elements o f X - R a y DifJ?action, 2nd edn, p. 506. Addison-Wesley, Reading, Pa (1978). 35. M. Hasebe and T. Nishizawa, Calphad 4, 83 (1980). 36. T. Nishizawa and K. Ishida, Bull. Alloy Phase Diag. 5, 161 (1984). 37. I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1962). 38. A. J. Ardell, Acta metall. 16, 511 (19683.
1835
APPENDIX The general solution to equation (8) has the form p2w - A'p 3 + B '
(A1)
where A ' and B ' are integration constants. Since w must approach 0 as p---,0, B ' - 0 . The solution inside the precipitate phase is therefore (A2)
wr~ = Ap
where A ' is denoted here by A. In the matrix phase w takes the form D
w = Bp + p 2
(A3)
where the integration constants A ' and B ' are now denoted by B and D, respectively. The constants A, B and D can be evaluated by imposing
the conditions of continuity of displacement and traction across a coherent interface [I5], and the requirement that w vanish at p - r'. Substitution of equation (A2) into (3) and (4), and equation (A3) into (5) and (6) yields the following equations for the stresses in the precipitate and matrix phases, namely T ~,, ~ -- T ool~= T ,I,4,t~ = (3ZI*+ 2Gt~)A _ 3EKt~= 3KI~(A - E) (A4) T,,, = (3Z + 2G )B -
4GD 4GD - ~ = 3KB - p3 p-
(A5)
and 2GD Too = T ~ = 3KB +- p3
(A6)
where we have made use of the equality 32 + 2G = 3K for both phases. Invoking the condition of continuity of traction at the interface (p = r), using equations (A4) and (A5), leads to the relationship 3KB -
4GD r~ = 3KI~(A - E )
(A7)
while continuity of displacement, using equations (A2) and (A3), produces the result D = (A - B ) r 3.
(AS)
On utilizing the condition w = 0 at p = r' in equation (A3), we obtain the result r'3\ B A=B l--r3)=z(Z-1) (A9) where we have made use of equation (l). Using equations (A7) (A9) and the parameters defined in equations (12) (14), the constants A and B can be expressed as A
3Kl~ (1 -- z) 4GFI~(1 6~lz)
(A10)
and B-
- 3Kj~z
4cr~(1 -
(A11)
a~i
Since the particles of the fl phase are subjected to a hydrostatic pressure P, i.e. T~,p(p = r) = - P
(A12)
we obtain, using equations (A5), (AS), (A10) and (AI 1) the expression for P presented in equation (I 1).