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γ′ Precipitate morphology formed under the influence of elastic interaction energies in nickel-base alloys
Materials Science and Engineering, 78 (1986) 87-94
~,' P r e c i p i t a t e M o r p h o l o g y F o r m e d Energies in Nickel-base Alloys
87
under the Influence
o f E l a s t i c Interaction
MINORU DOI and TORU MIYAZAKI Department of Materials Science and Engineering, Metals Section, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466 (Japan) (Received July 12, 1985; in revised form August 6, 1985)
ABSTRACT
The morphological change o f ~" precipitates in some nickel-base alloys was investigated by means o f transmission electron microscopy. In Ni-Si, N i - A l and Nimonic 115 alloys, each o f the ~[' cuboids which were closely aligned along (100> directions changed into a plate during coarsening. In an N i - S i Al alloy, however, the ~,' particles were not aligned along a certain crystallographic direction, and they remained spherical during coarsening. The ~,' spheres in an N i - S i - A l alloy had dissolved uniformly into the "y matrix during reversion, and some spheres were left in a sparse and random distribution throughout the matrix. In Ni-Si, N i - A l and N i - A l - T i alloys, however, the cuboidal or plate-like "y' particles dissolved very locally into the ~[ matrix during reversion, and some particles were left in groups in the matrix. The elastic interaction energies between the "y' precipitates played an essential role in the development o f such types o f precipitate morphology as the above. 1. INTRODUCTION It is widely recognized nowadays that the 7' precipitate morphology (i. e. the shape and distribution of 7' precipitate particles) greatly influences the properties of nickel-base superalloys. Therefore, one subject which is important to the development of new types of superalloy is h o w to produce the desirable ~,' precipitate morphology. The morphology of a coherent 7' particle is determined by minimizing the sum o f the elastic strain energy of the particle, the surface energy o f the particle and the elastic interaction energy between particles. 0025-5416/86/$3.50
The elastic interaction arises from the overlap of the strain fields of the individual 7' particles. Knowledge of the effect of elastic interaction energies on the precipitate morphology has been rapidly advancing [1-9]. It used to be thought that the elastic interaction energy does n o t affect the shape of the individual precipitate particle b u t has a marked influence only on the distribution of precipitate particles in the matrix. However, our group has recently found the extraordinary phenomenon that a single ~/' cuboid in some nickel-base alloys splits into a pair of parallel plates (i.e. a doublet) or into eight small cuboids (i.e. an ogdoad) during coarsening [ 6 - 9 ] . This is a typical example which clearly indicates that the elastic interaction energy also has an essential effect on the shape of the ~/' precipitates. From our previous results [ 6 - 9 ] , we expect that the elastic interaction energy has a marked influence on the formation and stability of a particular variety of precipitate morphology in addition to the influence on the splitting. In the present study the morphological changes of ~' precipitates during aging at a temperature considerably lower than the 7' solvus line and those during reversion are investigated b y means of transmission electron microscopy (TEM). Such morphological changes are discussed b y means of energy calculations based on microelasticity theory.
2. EXPERIMENTAL PROCEDURES Ni-7at.%Si-7at.%A1, Ni-12at.%Si, Ni-14at.%Si, Ni-8at.%A1-5at.%Ti, Ni-12at.%A1 and Nimonic 115 alloys were © Elsevier Sequoia/Printed in The Netherlands
88
used in the present study. Each alloy was quenched into iced water after homogenizing and then aged at a temperature considerably lower than the ~/' solvus line to ensure that a large n u m b e r of 7' particles precipitated close together. Furthermore, some of the aged alloys were subjected to a reversion treatment at a temperature higher than the aging temperature but lower than the ~/' solvus line. Thin foil specimens for TEM were prepared by electropolishing the samples. TEM observations of the 7' precipitate morphology were performed at an accelerating voltage of 200 kV. The energetically stable morphology of 7' precipitates was numerically calculated with a computer on the basis of microelasticity theory. "
3. EXPERIMENTAL RESULTS
3.1. Morphological change o f closely distributed 7' particles Figure 1 shows the morphological change of 7' precipitate particles in Nimonic 115
Fig. 1. TEM images of closely distributed T' particles in Nimonic 115 superalloy aged at 1273 K (a) for 3.0 X 10 2 s and (b) for 2.4 × 103 s.
superaUoy during aging at 1273 K. A large number of fine 7' particles were formed by aging for only 3.0 X 102 s, as shown in Fig. l(a). During coarsening by the further aging at 1273 K, the 7' particles became cuboidal and aligned along <100 > directions. Finally, t h e y changed their shapes to plates (see Fig. l(b)); the aspect ratio p ( - c / a ) 1 Such a decreases from 1 to less than 5" morphological change was also observed in Ni-14at.%Si alloy, as shown in Fig. 2. Figure 3 illustrates the relationship between the size and the aspect ratio p of 7' particles in Ni-12at.%A1 alloy. As a result of aging at 1133 K, a sparse distribution formed (indicated by open circles in the figure) and splitting t o o k place; after aging at 1063 K, closely distributed precipitates formed (indicated by full circles in the figure). This figure clearly indicates that, when the ~/' particles are closely distributed in the T matrix, each particle can exist as a cuboid (i.e. p = 1) with a largest size o f about 0.05 pm. When the particle size exceeded about 0.1 pm, each of the closely distributed
Fig. 2. TEM images of closely distributed ~,' particles in Ni-14at.%Si alloy aged at 973 K (a) for 2.500 x 105 s and (b) for 5.000 x 105 s.
89
particles changed into plates (i.e. p < ½), as indicated by the open arrow in the figure. When the 7' particles were sparsely distributed, however, they retained a cuboidal shape up to a b o u t 0.3 #m. In this case the aspect ratio decreases from 1 to less than z
1.0
--O CO00 ~
000-
\ ¢9
&
- 00- -O-CO- -0 --0 O- O- -0 ....
0 .....
0 ......
because o f the splitting of a single cuboid into a pair of plates, as indicated by the broken arrow in the figure. In Ni-7at.%Si-7at.%A1 alloy the 7' particles are not aligned along a certain crystallographic direction even when they are closely distributed, as shown in Fig. 4. Furthermore, they can grow increasingly larger as spheres.
I
\
Ill
SplIQIn|
[]
3.2. Morphological change during reversion
Fig. 3. Relationship b e t w e e n the size and the aspect ratio p (-~c/a) o f 7' particles in Ni-12at.%A1 alloy: o, sparsely distributed 7' particles ( 1133 K) ; 0, closely distributed 3/ particles (1063 K).
Figure 5 shows the morphological change of 7' particles in Ni-12at.%Si alloy before and after reversion. When the Ni-Si alloy was aged at 873 K for 6.048 X 105 s, a large number of fine 7' particles formed and these particles became aligned along ( 100 ) directions, as shown in Fig. 5(a). When the alloy was subjected to the reversion at 1090 K for 5.4 X 103s after aging at 873 K, some 7' particles dissolved into the 7 matrix. Dissolution of 7' particles t o o k place so locally that some particles were left in groups, as shown in Fig. 5(b).
Fig. 4. TEM images of 7' particles in N i - 7 a t . % S i 7at.%A1 alloy aged at 1073 K (a) for 2.88 X 104 s and (b) for 5.184 X 105 s.
Fig. 5. TEM images showing the m o r p h o l o g i c a l change of 3" particles in N i - 1 2 a t . % S i alloy during reversion: (a) aged at 873 K for 6.048 x 105 s; (b) reversion treated at 1090 K for 5.4 × 103 s after the aging at 873 K for 6.048 X 105 s.
\. • \ \ -....~\
._o ¢= 0 . 5
\~,,o e~e
ee
II o\
\~.."
==
-"~.\.~
•\ .
~."
Ni-AI
%
'
o'.1 Particle
' Size
o'.2
'
o'.3
(prn)
90
Fig. 6. TEM images showing the morphological change of 7 t particles in Ni-12at.%A1 alloy during reversion: (a) aged at 1023 K for 1.80 X 104 s; (b) reversion treated at 1163 K for 1.8 X 103 s after the aging at 1023 K for 1.80 X 104 s.
The same p h e n o m e n o n was also observed in Ni-12at.%A1 alloy subjected to reversion at 1163 K for 1.8 X 103 s after the aging at 1023 K for 1.80 X 104 s, as shown in Fig. 6. Furthermore, 7' particles were sometimes left in a line along one of <100> directions by reversion. Figure 7 shows ~,' particles in Ni-8at.%A1-5at.%Ti alloy subjected to reversion at 1403 K for 6.0 X 102 s after aging at 1173 K for 7.2 × 103 s. On the contrary, in Ni-7at.%Si-7at.%A1 alloy, dissolution of 7' particles during reversion t o o k place quite uniformly throughout the ~, matrix so that some V' particles were left in a sparse and random distribution throughout the matrix (Fig. 8).
4. DISCUSSION
4.1. Shape change from a cuboid to a plate The observed morphological change is discussed here on the basis of microelasticity
Fig. 7. TEM images showing the morphological change of "),' particles in Ni-8at.%A1-5at.%Ti alloy during reversion: (a) aged at 1173 K for 7.2 × 103 s after the continuous cooling at the rate o f 10-100 K s-1 from 1523 to 1173 K; (b) reversion treated at 1407 K for 6.0 × 102 s after the aging at 1173 K for 7.2 X 103 s.
theory. Model structures are set up for calculating the energies before and after the shape change from a cuboid to a plate. Because at least the elastic interaction energies between the first-, the second- and the third-neighbour particles should be taken into consideration, we chose 27 particles which are distributed so as to construct a simple cubic lattice with an externally cubic form. With respect to the 7' plates (p = ½), there are different types of mutual configuration [5], so that it is important to choose which t y p e of morphology is best as a model structure for plates. Figure 9(a) shows a typical distribution o f 7' plates (p ~ ½) in the 7 matrix. Although this micrograph was taken from the Ni-Si alloy, this t y p e of morphology is often observed in other nickelbase alloys such as Ni-A1, Nimonic 115 etc. Figure 9(b) illustrates the distribution of r 1 7 plates (p = ~) modelled on the electron
91
Eoo,l Fig. 8. TEM images showing the morphological change of ~" particles in Ni-7at.%Si:" 7at.%A1 alloy during reversion: (a) aged at 1073 K for 8.64 X 104 s;(b) reversion treated at 1263 K for 3.6 x 103 s after the aging at 1073 K for 8.64 x 104 s.
micrograph in Fig. 9(a); particle A in the micrograph (Fig. 9(a)) corresponds to particle A in the schematic illustration (Fig. 9(b)), and so on. The particles aligned along the [001] direction are in a mutually perpendicular configuration, i.e. in a face-edge configuration (e.g. A, D and G in the figure), and those aligned along [100] direction are in an edge-edge configuration (e.g. A, B and C in the figure). Furthermore, the same configuration is layered along the [010] direction, and the 27 plates are distributed to form a simple cubic lattice. In calculating the energies before and after the shape change from a cuboid to a plate, the following two cases are considered: (i) the 27 cuboids are distributed on the simple cubic lattice points which form a cube whose edge length is twice the first-neighbour distance and (ii) each 7' cuboid (p = 1; volume V) changes into a plate (p = _1. 2' volume V) and the 27 plates are distributed
Fig. 9. (a) TEM image of 7' plates (p ~ 12) in Ni14at.%Si alloy aged at 1023 K for 6.048 × 105 s; (b) schematic illustration of ~" plates (p -- 21) modelled on the TEM image in (a).
as illustrated in Fig. 9(b). Furthermore, the following two assumptions are made: (a) each cuboid is regarded as a sphere o f diameter D and each plate is regarded as an ellipsoid of revolution of aspect ratio ½, because the elastic interaction energies and the elastic strain energies can be calculated only for an ellipsoid of revolution, and (b) the displacement of 7' particles and hence the change in the inter-particle distances do not occur during the shape change, i.e. the distance between the first-neighbour particles (aligned along (100) directions), is 1.09D (= do)* and hence the distances between the secondneighbour particles (aligned along (110)) and the third-neighbour particles (aligned along (111)) are 21/2d0 and 31/2do respectively. *Since the elastic interaction energy between two ~' particles has a negative m i n i m u m when they are separated by a distance of 1.09D in (100) directions [5, 6 ], the first-neighbour distance is regarded as 1.09D in the present calculations.
92
The total energy o f 27 ellipsoids o f revolution is expressed b y the following equation: Z(p) = 27VEi.cl(p) + 27S(p)% + Eint(27)(p) where E~,cl(P) is the elastic strain energy of one ellipsoid, S(p) the surface area of the ellipsoid, % the surface energy density of the ellipsoid a n d Eint(27)(p) the elastic interaction energy among the 27 ellipsoids. The total energies before and after the shape change from a cuboid to a plate are given b y Ez = E (1) and El~ 2 -- E ( 1 ) , respectively. The theoretical basis for calculating Erect(P) has been explained in our previous paper [6]. Furthermore, the surface area of an ellipsoid of revolution is given as follows: S ( p ) = ~rr2p-213{2 4- F ( p ) }
where
F~)= 2
~rp = 1
and
F(p) -
(1
2 logtl + --2pp2)I/2 .
(1
--
P
p~)I12j
forp <1 Because the calculation o f E=tC27)(p) belongs to the m a n y - b o d y problem, only an approximate calculation is possible. In the present study the elastic interaction energy for every different pair taken from the 27 particles is calculated, and the total sum of the respective interaction energies is regarded as the approximate value o f Em~ZT)(p). The elastic interaction energy is calculated b y the m e t h o d given in our previous paper [5, 6]. In the present calculation the modulus effect is not taken into account. However, since its contribution is at most 10% o f the elastic interaction energy treated here, neglecting the effect has no serious influence on the final o u t c o m e of the present discussions. The energy calculations are performed for the 7' particles in the Ni-A1 alloy. The numerical values o f the eigenstrain e w* (i. e. the 7 - 7 ' lattice misfit), 7, and the elastic constants used here are as follows (for the source from which these parameters were taken, reference should be made to our previous paper [6]): e T* = 0.005 63; % = 0.0142 J m - 2 ; C l l = 11.24 X 10 a MN m -2, Cz2 = 6.27 X 104 MN m -2 and C44 = 5.69 X
104 MN m -2 for the 7 matrix; Cll* = 16.66 × 104 MN m -2, C12" = 10.65 X 104 MN m -2 and C~* = 9.92 × 104 MN m -2 for the 7' particle. The o u t c o m e of the present discussions can be applied essentially to other nickel-base alloys such as Ni-Si, Ni-A1-Ti and Nimonic 115. Figure 10 illustrates the comparison between the total energies of the aggregate o f 27 particles before and after the shape change from a cuboid (p = 1) to a plate ( p = Z ). The abscissa indicates the particle diameter of the sphere which has the same volume as one o f the 27 cuboids or plates. When the particles remain small, Ez is lower than Ez/2, and the energetically favourable morphology is an aggregate of cuboids which are closely aligned along {100> directions. When the cuboids grow larger than Dz/2*, then E l l 2 becomes lower than Ez and the cuboids can potentially change into plates. Such a shape change, however, is n o t realized immediately after the particles grow to D1/2". For example, although D l / 2 * is about 13 nm in Fig. 10, Fig. 3 clearly indicates that the size o f the plates with aspect ratio less than ~ is at least 3 0 - 4 0 nm (indicated by Dz/~** in Fig. 10). Such a difference is attributable to the kinetics of the shape change. The shape change from a cuboid to a plate is realized when a certain amount of energy has accumulated and acts as the driving force (indicated by AE in Fig. 10). As seen from Fig. 3, there are two ways o f decreasing the aspect ratio o f the 7' patti-
1.05
~ 27Gubolds
E o z
0.95 0
i
27 elate: ~ E1/2 P=I/2 i 110 20 J 310 i 40' i 50 Particle Size D (nm)
Fig. 10. Comparison between the energies of the aggregate of 27 7' particles in the Ni-AI alloy before and after the shape change from a cuboid (p = 1) to 1 . a plate (p = ~), i.e. E 1 and El~ 2 respectwely.
93
cles. One way is when the 7' particles are closely distributed (aligned along (100)) because of a high volume fraction; each 7' cuboid changes into a plate under the influence of elastic interaction. The other is the case in which the 7' particles are sparsely distributed because of a low volume fraction; a single 7' cuboid splits into a pair of plates under the influence of elastic interaction. It should be noted that each of the sparsely distributed 3" particles (i. e. virtually isolated particles) never decreases its aspect ratio without splitting. This fact obviously indicates that, even when the particle is isolated, the elastic interaction energies definitely govern the shape change from a cuboid to a plate, i.e. the decrease in the aspect ratio.
4.2. Morphological change during reversion When a nickel-base alloy is subjected to reversion at a temperature higher than the aging temperature but lower than the 7' solvus line, the total number of 7' particles should decrease. Therefore, some of the 7' particles formed during aging should dissolve into the 7 matrix. In some nickelbase alloys such as Ni-A1, Ni-Si and NiA1-Ti, dissolution of 7' particles takes place very locally during reversion. For example, in the regions around A in Fig. 5(a), the 3" particles are orderly and closely aligned along (100) directions, and t h e y can survive after reversion. In the regions around B in Fig. 5(a), however, the 7' particles are not in a good order, and t h e y dissolve preferentially into the 7 matrix. We cannot see any obvious differences between the conditions around A and B except the difference in the magnitude of the elastic interaction. We infer from Fig. 5(a) that the elastic interaction between the 7' particles around A is far stronger than that around B. Therefore the precipitate morphology in regions such as A is energetically quite stable, and every 3" particle in these regions has a much greater resistance to preferential dissolution during reversion. In the Ni-Si-A1 alloy the elastic interaction energies between the 7' spheres are negligibly small because the eigenstrain and hence the elastic strain energies of the spheres are small. Therefore the 3" spheres are not aligned along a certain crystallographic direction. Further-
more, t h e y dissolve quite uniformly throughout the matrix during reversion; preferential dissolution of 7' particles never takes place during reversion and hence some 3" particles survive sparsely and not in groups, as shown in Fig. 8.
5. CONCLUSIONS
From the results obtained in the present study, we have concluded as follows. (1) The sparsely distributed 7' cuboids change into plates, i.e. their aspect ratio decreases from 1 to less than ½, only when the splitting of a single cuboid into a pair of plates has taken place. (2) Each of the closely distributed 3" cuboids changes into a plate without splitting. (3) The closely distributed 3" cuboids change into plates easily at a smaller size than those sparsely distributed. (4) During reversion, dissolution of the closely distributed 3" cuboids takes place very locally and some 3" cuboids are left in groups. (5) During reversion, dissolution of 7' spheres takes place quite uniformly throughout the 3" matrix even if the 3" particles are closely distributed, and some 3" spheres are left sparsely and randomly in the matrix. The formation of such types of precipitate morphology is a result of the elastic interaction energies. Knowledge of the elastic interaction energies between 3" precipitates is indispensable to the understanding of a ~ e a t variety of microstructures.
ACKNOWLEDGMENT
This research was financially supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan.
REFERENCES I A. J. Ardell, R. B. Nicholson and J. D. Eshelby, Acta Metall., 14 (1966) 1295. 2 E. Seitz and D. de Fontaine, Acta MetaU., 26 (1978) 1671.
94 3 W. C. Johnson and J. K. Lee, Metall. Trans. A, 10 (1979) 1149. 4 H. Yamauchi and D. de Fontaine, Acta MetaU., 27 (1979) 763. 5 T. Miyazaki, H. Imamura, H. Mori and T. Kozakai, J. Mater. Sci., 16 (1981) 1197. 6 T. Miyazaki, H. Imamura and T. Kozakai, Mater. Sci. Eng., 54 (1982) 9.
7 M. Doi and T. Miyazaki, Proc. 5th Int. Syrup. on Superalloys, Seven Springs, PA, October 7-11, 1984, Metallurgical Society of AIME, Warrendale, PA, 1984, p. 543. 8 M. Doi, T. Miyazaki and T. Wakatsuki, Mater. Sc£ Eng., 67 (1984) 247. 9 M. Doi, T. Miyazaki and T. Wakatsuki, Mater. Sci. Eng., 74 (1985) 139.
~,' P r e c i p i t a t e M o r p h o l o g y F o r m e d Energies in Nickel-base Alloys
87
under the Influence
o f E l a s t i c Interaction
MINORU DOI and TORU MIYAZAKI Department of Materials Science and Engineering, Metals Section, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466 (Japan) (Received July 12, 1985; in revised form August 6, 1985)
ABSTRACT
The morphological change o f ~" precipitates in some nickel-base alloys was investigated by means o f transmission electron microscopy. In Ni-Si, N i - A l and Nimonic 115 alloys, each o f the ~[' cuboids which were closely aligned along (100> directions changed into a plate during coarsening. In an N i - S i Al alloy, however, the ~,' particles were not aligned along a certain crystallographic direction, and they remained spherical during coarsening. The ~,' spheres in an N i - S i - A l alloy had dissolved uniformly into the "y matrix during reversion, and some spheres were left in a sparse and random distribution throughout the matrix. In Ni-Si, N i - A l and N i - A l - T i alloys, however, the cuboidal or plate-like "y' particles dissolved very locally into the ~[ matrix during reversion, and some particles were left in groups in the matrix. The elastic interaction energies between the "y' precipitates played an essential role in the development o f such types o f precipitate morphology as the above. 1. INTRODUCTION It is widely recognized nowadays that the 7' precipitate morphology (i. e. the shape and distribution of 7' precipitate particles) greatly influences the properties of nickel-base superalloys. Therefore, one subject which is important to the development of new types of superalloy is h o w to produce the desirable ~,' precipitate morphology. The morphology of a coherent 7' particle is determined by minimizing the sum o f the elastic strain energy of the particle, the surface energy o f the particle and the elastic interaction energy between particles. 0025-5416/86/$3.50
The elastic interaction arises from the overlap of the strain fields of the individual 7' particles. Knowledge of the effect of elastic interaction energies on the precipitate morphology has been rapidly advancing [1-9]. It used to be thought that the elastic interaction energy does n o t affect the shape of the individual precipitate particle b u t has a marked influence only on the distribution of precipitate particles in the matrix. However, our group has recently found the extraordinary phenomenon that a single ~/' cuboid in some nickel-base alloys splits into a pair of parallel plates (i.e. a doublet) or into eight small cuboids (i.e. an ogdoad) during coarsening [ 6 - 9 ] . This is a typical example which clearly indicates that the elastic interaction energy also has an essential effect on the shape of the ~/' precipitates. From our previous results [ 6 - 9 ] , we expect that the elastic interaction energy has a marked influence on the formation and stability of a particular variety of precipitate morphology in addition to the influence on the splitting. In the present study the morphological changes of ~' precipitates during aging at a temperature considerably lower than the 7' solvus line and those during reversion are investigated b y means of transmission electron microscopy (TEM). Such morphological changes are discussed b y means of energy calculations based on microelasticity theory.
2. EXPERIMENTAL PROCEDURES Ni-7at.%Si-7at.%A1, Ni-12at.%Si, Ni-14at.%Si, Ni-8at.%A1-5at.%Ti, Ni-12at.%A1 and Nimonic 115 alloys were © Elsevier Sequoia/Printed in The Netherlands
88
used in the present study. Each alloy was quenched into iced water after homogenizing and then aged at a temperature considerably lower than the ~/' solvus line to ensure that a large n u m b e r of 7' particles precipitated close together. Furthermore, some of the aged alloys were subjected to a reversion treatment at a temperature higher than the aging temperature but lower than the ~/' solvus line. Thin foil specimens for TEM were prepared by electropolishing the samples. TEM observations of the 7' precipitate morphology were performed at an accelerating voltage of 200 kV. The energetically stable morphology of 7' precipitates was numerically calculated with a computer on the basis of microelasticity theory. "
3. EXPERIMENTAL RESULTS
3.1. Morphological change o f closely distributed 7' particles Figure 1 shows the morphological change of 7' precipitate particles in Nimonic 115
Fig. 1. TEM images of closely distributed T' particles in Nimonic 115 superalloy aged at 1273 K (a) for 3.0 X 10 2 s and (b) for 2.4 × 103 s.
superaUoy during aging at 1273 K. A large number of fine 7' particles were formed by aging for only 3.0 X 102 s, as shown in Fig. l(a). During coarsening by the further aging at 1273 K, the 7' particles became cuboidal and aligned along <100 > directions. Finally, t h e y changed their shapes to plates (see Fig. l(b)); the aspect ratio p ( - c / a ) 1 Such a decreases from 1 to less than 5" morphological change was also observed in Ni-14at.%Si alloy, as shown in Fig. 2. Figure 3 illustrates the relationship between the size and the aspect ratio p of 7' particles in Ni-12at.%A1 alloy. As a result of aging at 1133 K, a sparse distribution formed (indicated by open circles in the figure) and splitting t o o k place; after aging at 1063 K, closely distributed precipitates formed (indicated by full circles in the figure). This figure clearly indicates that, when the ~/' particles are closely distributed in the T matrix, each particle can exist as a cuboid (i.e. p = 1) with a largest size o f about 0.05 pm. When the particle size exceeded about 0.1 pm, each of the closely distributed
Fig. 2. TEM images of closely distributed ~,' particles in Ni-14at.%Si alloy aged at 973 K (a) for 2.500 x 105 s and (b) for 5.000 x 105 s.
89
particles changed into plates (i.e. p < ½), as indicated by the open arrow in the figure. When the 7' particles were sparsely distributed, however, they retained a cuboidal shape up to a b o u t 0.3 #m. In this case the aspect ratio decreases from 1 to less than z
1.0
--O CO00 ~
000-
\ ¢9
&
- 00- -O-CO- -0 --0 O- O- -0 ....
0 .....
0 ......
because o f the splitting of a single cuboid into a pair of plates, as indicated by the broken arrow in the figure. In Ni-7at.%Si-7at.%A1 alloy the 7' particles are not aligned along a certain crystallographic direction even when they are closely distributed, as shown in Fig. 4. Furthermore, they can grow increasingly larger as spheres.
I
\
Ill
SplIQIn|
[]
3.2. Morphological change during reversion
Fig. 3. Relationship b e t w e e n the size and the aspect ratio p (-~c/a) o f 7' particles in Ni-12at.%A1 alloy: o, sparsely distributed 7' particles ( 1133 K) ; 0, closely distributed 3/ particles (1063 K).
Figure 5 shows the morphological change of 7' particles in Ni-12at.%Si alloy before and after reversion. When the Ni-Si alloy was aged at 873 K for 6.048 X 105 s, a large number of fine 7' particles formed and these particles became aligned along ( 100 ) directions, as shown in Fig. 5(a). When the alloy was subjected to the reversion at 1090 K for 5.4 X 103s after aging at 873 K, some 7' particles dissolved into the 7 matrix. Dissolution of 7' particles t o o k place so locally that some particles were left in groups, as shown in Fig. 5(b).
Fig. 4. TEM images of 7' particles in N i - 7 a t . % S i 7at.%A1 alloy aged at 1073 K (a) for 2.88 X 104 s and (b) for 5.184 X 105 s.
Fig. 5. TEM images showing the m o r p h o l o g i c a l change of 3" particles in N i - 1 2 a t . % S i alloy during reversion: (a) aged at 873 K for 6.048 x 105 s; (b) reversion treated at 1090 K for 5.4 × 103 s after the aging at 873 K for 6.048 X 105 s.
\. • \ \ -....~\
._o ¢= 0 . 5
\~,,o e~e
ee
II o\
\~.."
==
-"~.\.~
•\ .
~."
Ni-AI
%
'
o'.1 Particle
' Size
o'.2
'
o'.3
(prn)
90
Fig. 6. TEM images showing the morphological change of 7 t particles in Ni-12at.%A1 alloy during reversion: (a) aged at 1023 K for 1.80 X 104 s; (b) reversion treated at 1163 K for 1.8 X 103 s after the aging at 1023 K for 1.80 X 104 s.
The same p h e n o m e n o n was also observed in Ni-12at.%A1 alloy subjected to reversion at 1163 K for 1.8 X 103 s after the aging at 1023 K for 1.80 X 104 s, as shown in Fig. 6. Furthermore, 7' particles were sometimes left in a line along one of <100> directions by reversion. Figure 7 shows ~,' particles in Ni-8at.%A1-5at.%Ti alloy subjected to reversion at 1403 K for 6.0 X 102 s after aging at 1173 K for 7.2 × 103 s. On the contrary, in Ni-7at.%Si-7at.%A1 alloy, dissolution of 7' particles during reversion t o o k place quite uniformly throughout the ~, matrix so that some V' particles were left in a sparse and random distribution throughout the matrix (Fig. 8).
4. DISCUSSION
4.1. Shape change from a cuboid to a plate The observed morphological change is discussed here on the basis of microelasticity
Fig. 7. TEM images showing the morphological change of "),' particles in Ni-8at.%A1-5at.%Ti alloy during reversion: (a) aged at 1173 K for 7.2 × 103 s after the continuous cooling at the rate o f 10-100 K s-1 from 1523 to 1173 K; (b) reversion treated at 1407 K for 6.0 × 102 s after the aging at 1173 K for 7.2 X 103 s.
theory. Model structures are set up for calculating the energies before and after the shape change from a cuboid to a plate. Because at least the elastic interaction energies between the first-, the second- and the third-neighbour particles should be taken into consideration, we chose 27 particles which are distributed so as to construct a simple cubic lattice with an externally cubic form. With respect to the 7' plates (p = ½), there are different types of mutual configuration [5], so that it is important to choose which t y p e of morphology is best as a model structure for plates. Figure 9(a) shows a typical distribution o f 7' plates (p ~ ½) in the 7 matrix. Although this micrograph was taken from the Ni-Si alloy, this t y p e of morphology is often observed in other nickelbase alloys such as Ni-A1, Nimonic 115 etc. Figure 9(b) illustrates the distribution of r 1 7 plates (p = ~) modelled on the electron
91
Eoo,l Fig. 8. TEM images showing the morphological change of ~" particles in Ni-7at.%Si:" 7at.%A1 alloy during reversion: (a) aged at 1073 K for 8.64 X 104 s;(b) reversion treated at 1263 K for 3.6 x 103 s after the aging at 1073 K for 8.64 x 104 s.
micrograph in Fig. 9(a); particle A in the micrograph (Fig. 9(a)) corresponds to particle A in the schematic illustration (Fig. 9(b)), and so on. The particles aligned along the [001] direction are in a mutually perpendicular configuration, i.e. in a face-edge configuration (e.g. A, D and G in the figure), and those aligned along [100] direction are in an edge-edge configuration (e.g. A, B and C in the figure). Furthermore, the same configuration is layered along the [010] direction, and the 27 plates are distributed to form a simple cubic lattice. In calculating the energies before and after the shape change from a cuboid to a plate, the following two cases are considered: (i) the 27 cuboids are distributed on the simple cubic lattice points which form a cube whose edge length is twice the first-neighbour distance and (ii) each 7' cuboid (p = 1; volume V) changes into a plate (p = _1. 2' volume V) and the 27 plates are distributed
Fig. 9. (a) TEM image of 7' plates (p ~ 12) in Ni14at.%Si alloy aged at 1023 K for 6.048 × 105 s; (b) schematic illustration of ~" plates (p -- 21) modelled on the TEM image in (a).
as illustrated in Fig. 9(b). Furthermore, the following two assumptions are made: (a) each cuboid is regarded as a sphere o f diameter D and each plate is regarded as an ellipsoid of revolution of aspect ratio ½, because the elastic interaction energies and the elastic strain energies can be calculated only for an ellipsoid of revolution, and (b) the displacement of 7' particles and hence the change in the inter-particle distances do not occur during the shape change, i.e. the distance between the first-neighbour particles (aligned along (100) directions), is 1.09D (= do)* and hence the distances between the secondneighbour particles (aligned along (110)) and the third-neighbour particles (aligned along (111)) are 21/2d0 and 31/2do respectively. *Since the elastic interaction energy between two ~' particles has a negative m i n i m u m when they are separated by a distance of 1.09D in (100) directions [5, 6 ], the first-neighbour distance is regarded as 1.09D in the present calculations.
92
The total energy o f 27 ellipsoids o f revolution is expressed b y the following equation: Z(p) = 27VEi.cl(p) + 27S(p)% + Eint(27)(p) where E~,cl(P) is the elastic strain energy of one ellipsoid, S(p) the surface area of the ellipsoid, % the surface energy density of the ellipsoid a n d Eint(27)(p) the elastic interaction energy among the 27 ellipsoids. The total energies before and after the shape change from a cuboid to a plate are given b y Ez = E (1) and El~ 2 -- E ( 1 ) , respectively. The theoretical basis for calculating Erect(P) has been explained in our previous paper [6]. Furthermore, the surface area of an ellipsoid of revolution is given as follows: S ( p ) = ~rr2p-213{2 4- F ( p ) }
where
F~)= 2
~rp = 1
and
F(p) -
(1
2 logtl + --2pp2)I/2 .
(1
--
P
p~)I12j
forp <1 Because the calculation o f E=tC27)(p) belongs to the m a n y - b o d y problem, only an approximate calculation is possible. In the present study the elastic interaction energy for every different pair taken from the 27 particles is calculated, and the total sum of the respective interaction energies is regarded as the approximate value o f Em~ZT)(p). The elastic interaction energy is calculated b y the m e t h o d given in our previous paper [5, 6]. In the present calculation the modulus effect is not taken into account. However, since its contribution is at most 10% o f the elastic interaction energy treated here, neglecting the effect has no serious influence on the final o u t c o m e of the present discussions. The energy calculations are performed for the 7' particles in the Ni-A1 alloy. The numerical values o f the eigenstrain e w* (i. e. the 7 - 7 ' lattice misfit), 7, and the elastic constants used here are as follows (for the source from which these parameters were taken, reference should be made to our previous paper [6]): e T* = 0.005 63; % = 0.0142 J m - 2 ; C l l = 11.24 X 10 a MN m -2, Cz2 = 6.27 X 104 MN m -2 and C44 = 5.69 X
104 MN m -2 for the 7 matrix; Cll* = 16.66 × 104 MN m -2, C12" = 10.65 X 104 MN m -2 and C~* = 9.92 × 104 MN m -2 for the 7' particle. The o u t c o m e of the present discussions can be applied essentially to other nickel-base alloys such as Ni-Si, Ni-A1-Ti and Nimonic 115. Figure 10 illustrates the comparison between the total energies of the aggregate o f 27 particles before and after the shape change from a cuboid (p = 1) to a plate ( p = Z ). The abscissa indicates the particle diameter of the sphere which has the same volume as one o f the 27 cuboids or plates. When the particles remain small, Ez is lower than Ez/2, and the energetically favourable morphology is an aggregate of cuboids which are closely aligned along {100> directions. When the cuboids grow larger than Dz/2*, then E l l 2 becomes lower than Ez and the cuboids can potentially change into plates. Such a shape change, however, is n o t realized immediately after the particles grow to D1/2". For example, although D l / 2 * is about 13 nm in Fig. 10, Fig. 3 clearly indicates that the size o f the plates with aspect ratio less than ~ is at least 3 0 - 4 0 nm (indicated by Dz/~** in Fig. 10). Such a difference is attributable to the kinetics of the shape change. The shape change from a cuboid to a plate is realized when a certain amount of energy has accumulated and acts as the driving force (indicated by AE in Fig. 10). As seen from Fig. 3, there are two ways o f decreasing the aspect ratio o f the 7' patti-
1.05
~ 27Gubolds
E o z
0.95 0
i
27 elate: ~ E1/2 P=I/2 i 110 20 J 310 i 40' i 50 Particle Size D (nm)
Fig. 10. Comparison between the energies of the aggregate of 27 7' particles in the Ni-AI alloy before and after the shape change from a cuboid (p = 1) to 1 . a plate (p = ~), i.e. E 1 and El~ 2 respectwely.
93
cles. One way is when the 7' particles are closely distributed (aligned along (100)) because of a high volume fraction; each 7' cuboid changes into a plate under the influence of elastic interaction. The other is the case in which the 7' particles are sparsely distributed because of a low volume fraction; a single 7' cuboid splits into a pair of plates under the influence of elastic interaction. It should be noted that each of the sparsely distributed 3" particles (i. e. virtually isolated particles) never decreases its aspect ratio without splitting. This fact obviously indicates that, even when the particle is isolated, the elastic interaction energies definitely govern the shape change from a cuboid to a plate, i.e. the decrease in the aspect ratio.
4.2. Morphological change during reversion When a nickel-base alloy is subjected to reversion at a temperature higher than the aging temperature but lower than the 7' solvus line, the total number of 7' particles should decrease. Therefore, some of the 7' particles formed during aging should dissolve into the 7 matrix. In some nickelbase alloys such as Ni-A1, Ni-Si and NiA1-Ti, dissolution of 7' particles takes place very locally during reversion. For example, in the regions around A in Fig. 5(a), the 3" particles are orderly and closely aligned along (100) directions, and t h e y can survive after reversion. In the regions around B in Fig. 5(a), however, the 7' particles are not in a good order, and t h e y dissolve preferentially into the 7 matrix. We cannot see any obvious differences between the conditions around A and B except the difference in the magnitude of the elastic interaction. We infer from Fig. 5(a) that the elastic interaction between the 7' particles around A is far stronger than that around B. Therefore the precipitate morphology in regions such as A is energetically quite stable, and every 3" particle in these regions has a much greater resistance to preferential dissolution during reversion. In the Ni-Si-A1 alloy the elastic interaction energies between the 7' spheres are negligibly small because the eigenstrain and hence the elastic strain energies of the spheres are small. Therefore the 3" spheres are not aligned along a certain crystallographic direction. Further-
more, t h e y dissolve quite uniformly throughout the matrix during reversion; preferential dissolution of 7' particles never takes place during reversion and hence some 3" particles survive sparsely and not in groups, as shown in Fig. 8.
5. CONCLUSIONS
From the results obtained in the present study, we have concluded as follows. (1) The sparsely distributed 7' cuboids change into plates, i.e. their aspect ratio decreases from 1 to less than ½, only when the splitting of a single cuboid into a pair of plates has taken place. (2) Each of the closely distributed 3" cuboids changes into a plate without splitting. (3) The closely distributed 3" cuboids change into plates easily at a smaller size than those sparsely distributed. (4) During reversion, dissolution of the closely distributed 3" cuboids takes place very locally and some 3" cuboids are left in groups. (5) During reversion, dissolution of 7' spheres takes place quite uniformly throughout the 3" matrix even if the 3" particles are closely distributed, and some 3" spheres are left sparsely and randomly in the matrix. The formation of such types of precipitate morphology is a result of the elastic interaction energies. Knowledge of the elastic interaction energies between 3" precipitates is indispensable to the understanding of a ~ e a t variety of microstructures.
ACKNOWLEDGMENT
This research was financially supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan.
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94 3 W. C. Johnson and J. K. Lee, Metall. Trans. A, 10 (1979) 1149. 4 H. Yamauchi and D. de Fontaine, Acta MetaU., 27 (1979) 763. 5 T. Miyazaki, H. Imamura, H. Mori and T. Kozakai, J. Mater. Sci., 16 (1981) 1197. 6 T. Miyazaki, H. Imamura and T. Kozakai, Mater. Sci. Eng., 54 (1982) 9.
7 M. Doi and T. Miyazaki, Proc. 5th Int. Syrup. on Superalloys, Seven Springs, PA, October 7-11, 1984, Metallurgical Society of AIME, Warrendale, PA, 1984, p. 543. 8 M. Doi, T. Miyazaki and T. Wakatsuki, Mater. Sc£ Eng., 67 (1984) 247. 9 M. Doi, T. Miyazaki and T. Wakatsuki, Mater. Sci. Eng., 74 (1985) 139.