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A new method for the determination of the precipitate-matrix interfacial energy
Scripta
METALLURGICA
V o l . 18, pp. 6 1 1 - 6 1 6 , 1984 P r i n t e d in t h e U . S . A .
P e r g a m o n P r e s s Ltd. All r i g h t s r e s e r v e d
A NEW METHOD FOR THE DETERMINATION
OF THE PRECIPITATE-MATRIX INTERFACIAL ENERGY
S. F. Baumann and D. B. Williams Department of Metallurgy and Materials Engineering Lehigh University Bethlehem, Pennsylvania 18015
(Received February 24, 1 9 8 4 ) ( R e v i s e d M a r c h 20, 1 9 8 4 )
Introduction The energy of the interface between a precipitate and its matrix is an important thermodynamic factor in many aspects of solid state precipitation reactions. It is the primary barrier to the nucleation event, it is influential in determining particle morphologies during growth and it is the driving force for particle coarsening. While this interfacial energy ( V ) is clearly of great interest in the study of phase transformations, a direct determination of its magnitude is not possible. Experimentally, W has been determined indirectly by means such as: measuring precipitate coarsening rates (I), examining precipitate morphological development (2) and by analysis of equilibrium precipitate shapes (3). This paper describes a new procedure to determine ~ for the fully coherent interface between 6 ' (AI3Li) and its matrix ~ (A1-Li solid solution) in a binary AI-Li alloy. The method involves experimental measurement of the critical particle size for nucleation at a given temperature. This is then used to obtain ~ through the capillarity equation. The value obtained in this way is compared with previous determinations made from particle coarsening studies. Previous Studies in AI-Li Alloys On low temperature aging of dilute AI-Li binary alloys (6-14 at% Li), 6', a metastable Lip ordered phase, precipitates homogeneously and coherently. The 6' is cube/cube oriented with the ~ and the constrained lattice mismatch is less than 0.1% (4). The high compatibility of the two lattices suggests that the interfacial energy should be isotropic and this is reflected in the spherical shape of the 6' particles (see Figure i). Two previous determinations of V~/6' hav~ been made from 6' coarsening data (5,6). These experiments yielded values o~ 0.24 J/m L (5) and 0.18 J/m 2 (6). Noble and Thompson (5) noted that their value of 0.24 J/m was about an order of magnitude higher than that expected on the basis of analogous interfaces between ~' (Ni~X) in ~-Ni (where X = AI, Si, Ti). They suggested that this was due to an uncertainty in the v~lue of D (diffusivity) used in the coarsening rate equation. Williams and Edington (4) studied 6' coarsening in binary AI-Li alloys and, noting that previously reported values of ~ were high, assumed V = 0.025 J/m 2 based on existing data for similar coherent precipitates (7). The coarsening data of Williams and Edington (4) indicated clear evidence for bulk alloy composition influence on particle coarsening rates, which was interpreted as reflecting compositional effects on effective diffusivities. These effects were large enough to cast doubt on the accuracy of the previously reported values of WExperimental Procedures The material used in this work was a high purity 7.9 at% Li binary alloy in the form of 0.25 nun thick sheet obtained from Fulmer Research Institute (4). Standard 3 mm discs, suitable for transmission electron microscope (TEM) samples, were punched from the sheet and wrapped in AI foil. Solution treatment was carried out for 8 mins. at 570°C followed by a water quench to room temperature, after which various aging treatments were performed. All heat treatments were carried out in diffusion furnaces accurate to ± 2°C. Samples were then prepared for TEM investigation by conventional metallographic techniques. All samples were examined in a Philips EM400T TEM.
0036-9748/84 Copyright (c) 1 9 8 4
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Ltd.
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DETEI~IINATION OF PRECIPITATE-MATRIX
INTERFACIAL
ENERGY
Vol.
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No.
6
The quality of the results is dependent upon the accurate imaging and size measurement of very small particles (particle d i a m e t e r s N 3 nm). In this work the 6' was imaged by centered dark field (CDF) techniques using the superlattice reflections. To allow measurement of individual particles, it was necessary to work in thin regions of the foil to avoid excessive particle image overlap. All foils were less than I00 nm thick in the regions where this work was done. The practical limitation to image resolution and, therefore, small particle detection in this work is chromatic aberration since some electrons will experience energy losses on passing through the foil. Calculations show that in the worst case (i00 nm thick) chromatic aberration will limit the smallest resolvable particle size to approximately 1.5 nm radius, although in most specimens smaller particles were resolvable. Particle size measurements in this work were made from enlargements of high magnification 6' CDF images. Results
and Discussion
After a solution treatment, the alloy showed a fully recrystallized microstructure. The diffraction pattern contained well-defined LIp superlattice reflections which, when imaged, revealed a homogeneous distribution of ordere~ 6' particles, approximately 1.5 nm radius. The uniformity of their distribution and small particle sizes suggest that nucleation occurred at or near room temperature as proposed by previous investigators (4,8). A close examination of the regions adjacent to grain boundaries revealed a gradation in the 6' particle size from 1.5 nm radius in the bulk to less than i nm near the boundary (see Figure 2). This phenomenon is evident in previous publications (e.g., Figure i, reference (4)) but has not been commented on to date. It is proposed that this gradient in particle size arises from a gradient in excess quenched-in vacancies which will affect diffusivity at room temperature (9). During the quench from the solution treatment temperatur~ excess vacancies will diffuse to grain boundaries and dislocations. At room temperatures, adjacent to these defects, there will therefore be a gradient of excess vacancies trapped in the structure. Since excess vacancies will contribute to low temperature diffusivity and, therefore increase both the nucleation and growth rate of precipitates, the 6' particles near grain boundaries should be smaller than those within the bulk of the grain as observed in Figure 2. There is no evidence to suggest that this particle size gradient results from a solute gradient since the structure is fully recrystallized and no precipitate phases are present at the grain boundary. Having established a particle size gradient near the grain boundary the material is then given a short, high temperature aging treatment near the solvus line for the phase. In this case, the solution-treated foils were aged for 2 minutes at 200°C, which is approximately 50°C below the 6' solvus for AI 7.9 at% Li (4). The sample takes about 1 minute to reach temperature and is at 200°C for i minute. (This was observed by simulating a heat treatment with a thermocouple included in the foil.) During the "heat up" period the particles coarsen slightly When the sample reaches 200°C all those particles whose size is below the critical size necessary for survival at that temperature will dissolve, the others will grow and coarsen. Examination of the sample aged at 200°C shows a precipitate free zone (PFZ) adjacent to the grain boundary where the smaller particles had dissolved (see Figure 3 and compare with Figure 2). If we ignore the small amount of particle coarsening that occurred at 200°C, then the smallest particle size observed at the edge of this 6' PFZ approximates to the critical 6' nucleus size for this alloy at 200°C. This was measured to be approximately 2 nm in radius. This measurement was straightforward because there was a well-defined decrease in image contrast between these stable 6 ' nuclei (~ 2 nm radius) and much smaller ill-defined particles, which were not present at 200°C but nucleated on the quench back to room temperature (see Figure 3b). The solute content of the matrix in equilibrium with a precipitate of a given size is defined by the capillarity equation (Gibbs-Thompson equation). For the case where the precipitate phase is spherical and is a compound this is given by (i0):
[i x Vm)]
w h e r e X an d X~ are the mole fractions r curvatures r and ~ respectively. Xp i s V i s t h e m o l a r v o l u m e and RT i s t h e r m a l m~tastable equilibrium with the matrix of
of solute in the matrix at interfaces with radii of the mole fraction of solute in the precipitate phase, energy. By d e f i n i t i o n , the critical nucleus is in bulk alloy composition. T h e r e f o r e by s e t t i n g X =
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DETERMINATION OF PRECIPITATE-MATRIX
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0.079, X= =0.066 (~/~ + 6' phase boundary at 200°C), X. = 0.25, V = 10 -5 m 3, r = 2 x 10 -9 m m 2 and T = 473 K equation (i) can be solved for ~. This yields a value of V = 0.014 J/m . This value is an order of magnitude smaller than those previously obtained from 6' particle coarsening experiments (6). Figure 4 shows how ? will vary as a function of the measured value of r. It is of interest to find the reason for the discrepancy in the value of N obtained here and those previously determined. There is some uncertainty in the value of .~ for 6 '. Cocco et al. (ii) report X_ = 0.22 at 200°C. However, this discrepancy would yield only a slightly z lower value of V from equation (i) ( V = 0.012 J/m ). There is also an uncertainty in the measurement of the particle size since at these small particle sizes the interface is often not clearly defined. Measurements were made from ~ 40 of the thousands of particles visible in the enlarged prints of Figure 3b. Errors w e r e ~ ± 50%. However, even if the error were ± 100% the value of ~ would only change by a factor of 2 (see Figure 4). Furthermore if we insert the values of V due to Noble and Thompson (5) and Tamura et al. (6) into equation (I), the predicted critical nucleus radii at 200°C are 35 nm and 26 nm respectively, ~ 10X more than the measured data from Figure 3. Therefore uncertainty in Xp and measurement errors cannot explain the 10X discrepancy in ~ between the present an~ previous values. It is suggested here that the difference arises from a failure to account for volume fraction effects on 6' coarsening rates when ~ is calculated from coarsening experiments. In the Lifshitz-Slyozov and Wagner theories (12,13) of particle coarsening, the average particle radius 7 at time t is related to the average particle radius ~ at time t by the o t coarsening rate constant, k,°through: -
r
3
t
with
- r
3
o
= k (t - t ) o
D V 2 m 9 RT
(2)
8 V C k =
e
(3)
where C is the equilibrium matrix solute content, D is diffusivity, and V is the molar volume e of the precipitate phase. In the determination of N through coarsening data, k is measured experimentally by application of equation (2) to the data, and equation (3) defines N (assuming all other variables are known). Equation (3) was originally derived for the condition where the volume fraction ( ~ ) of the precipitate is very small. Many coarsening rate studies, however, involve alloys where ~ is in the range of 0.05 to 0.2 or larger. Several theoretical treatments of the effect of ~ on particle coarsening have been reported in the literature (14-17). The distinguishing assumptions regarding particle distributions and diffusion geometry result in different predictions for each of the models. While there is no quantitative agreement amongst the various models, they all predict an increase in the value of k as increases due to reduced diffusion distances between particles. The 6' coarsening data of Williams and Edington (4) and Williams (18) clearly indicates an effect of bulk Li content on k, with k increasing as Li increases. Figure 5 is from the work of Williams (18) and illustrates the effect of bulk Li content on 6' coarsening at 200°C for 3 binary a l ~ y s . ^ From these data the following values Qf k were measured: For AI-7.9 at% Li, k=2.08 x2½0=~3cmb/sec ; for AI-10.7 at% Li, k=l.40 x i0 -zz cmJ/sec; for AI-12.9 at% Li, k=6.24 x I0cm /sec. For each of these cases, the variables on the rlght-hand side of equation (3) should be the same and, therefore, k should be constant. Experimentally, this is not found to be true. If potential effects of ~ on k are not accounted for, it is obvious that the three sets of k values reported above would yield three different values of N which would vary by more than one order of magnitude from the high to low value. Quantitative metallography was performed on a 7.9 at% binary alloy aged at 200°C. This showed ~ 0.098. We have selected five available models for the effects of ~ on k to backcalculate k (0) (i.e., k when ~ = 0) for the 7.9 at% coarsening data of Williams (18). These were then used to determine N through equation (3). Using a diffusivity calculated from the work of Costas (19) we get the results shown in Table i. The values of ~ in Table 1 are in reasonable agreement with the val~e of 0.014 J/m 2 calculated in this work. Moreover, they are in the range of 0.01 to 0.25 J/m reported for similar coherent interfaces when the lattice misfit is less than I% (7). It is suggested therefore that previous determinations of N from 6' particle coarsening data, in making no correction for the effect of ~ on k, produced erroneously high values of ~ These prior studies were
614
DETERMINATION OF PRECIPITATE-MATRIX INTERFACIAL ENERGY
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performed on alloys of 10.6 and 13.8 at% Li which would contain large volume fractions of 8 ' ( ~ _ 0 . 3 and 0.55 respectively) and thus experience enhanced coarsening. No corrections for effects on k were made so the analyses would have overestimated the value of V. TABLE i Calculated Values of k(0) and V from Different Models for Effects on k when applied to 7.9 at% coarsening rate data in Figure 5 Model
k(O) (cm2/sec)
Ardell (15) Asimow (14) Brailsford and Wynblatt (16) Tsumuraya and Miyata (Modell III) (17) Tsumuraya and Miyata (Model Vl) (17)
3.69 9.72 1.08 9.45 3.45
x x x x x
10-24 -24 10_23 i0 10-24 10-24
~(J/m 2) .008 .022 .025 .021 .008
Conclusions The experiment described in this paper yields a reasonable value of N~/6,which is also in agreement with values calculated from coarsening data when corrections are made for volume fraction effects. This experimental procedure should be applicable to any system where nucleation is coherent, transformation strains are small and a particle size distribution can be established near grain boundaries while the solute concentration remains constant. It might be possible to extend this to more general situations. For example, when interfaclal energies are highly anisotropic, it might be possible to get an approximation of N for the high energy interface by assuming it determines the thermodynamic stability at temperature. Equation (I) should then be adjusted to reflect the mean radius of curvature for that interface. Finally, due to the need to image rather small particle sizes accurately in the TEM, the technique would be restricted to precipitates that are ordered, exhibit discrete diffraction spots or are of much larger atomic number than the matrix. Alternatively, lattice imaging techniques could be used rather than conventional CDF images if smaller precipitates have to be detected. Acknowledgment The authors wish to acknowledge the National Science Foundation for their financial support under Grant #DMR-81-08308. References
i)
A. J. Ardell, Met. Trans., !, 525 (1970).
2)
J. A. Malcolm and G. R. Purdy, Trans. Met. Soe. AIME, 239, 1391 (1967). G. R. Spelch and R. A. Oriani, Trans. Met. Soc. AIME, 233, 623 (1965). D. B. Williams and J. W. Edington, Metal Sci., 9, 529 (1975). B. Noble and G. E. Thompson, Metal Sci. J., ~, 114 (1971). M. Tamura, T. Mori and T. Nakamura, J. Jap. Inst. Met., 34, 919 (1970). B. A. Parker and D. R. F. West, J. Aust. Inst. Met. 14, 102 (1969). S. Ceresara, A. Giarda and A. Sanchez, Phil. Mag., 35, 97 (1977). M. H. Jacobs and D. W. Pashley, in The Mechanism of Phase Transformations in Crystalline Solids, Institute of Metals, 1969, pp. 43-48. R. K. Trivedl, in Lectures on the Theory of Phase Transformations, H.I. Aaronson, ed., pp. 79-80, AIME, New York, 1975. G. Coeco, G. Fagherazzl and L. Schiffini, J. AppI. Cryst., iO, 325 (1977). I . M . Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids, 19, 35 (1961). C. Wagner, Z. Electrochem., 65, 581 (1961). R. Asimow, Acta Met., Ii, 72 (1963). A. J. Ardell, Aeta Met., 20, 61 (1972). A. D. Brailsford and P. Wynblatt, Aeta Met., 27, 489 (1979). K. Tsumuraya and Y. Miyata, Acta Met., 31, 437 (1983). D. B. Williams, Ph.D. Thesis, 1974, Cambridge University. L. P. Costas, U.S. Atomic Energy Comm. Rep. (DP-813), 1963.
3) 4) 5) 6) 7) 8) 9) io) 11) 12) 13) 14) 15) 16) 17) 18) 19)
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Figure i: Centered dark field (CDF) image from a 6' s uperlattice reflection showing spherical 6' particles in a AI-7.9 at% Li binary alloy aged 170 hrs. at 220°C.
Figure 2: a) Low magnification and b) higher magnification 6' CDF images taken near a grain boundary in a solution treated and water quenched AI-7.9 at% Li alloy. A gradient in 6' particle size is present adjacent to the boundary. Tilting of the foil showed this was not just a consequence of local diffracting conditions or thickness variations.
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Figure 3: a) Low magnification 6' CDF image showing a 6' precipitate-free zone (PFZ) adjacent to the grain boundary; b) a higher magnification image of the border of the 6' PFZ, in AI 7.9 at% Li solution-treated, water-quenched and aged 2 min. at 200°C. The arrows in b) indicate some typical high contrast 6' particles which were considered to be stable nuclei at 200°C. Fainter 6' particles in the original PFZ nucleated on the quench back to room temperature.
TEMPERATURE
0.04
30
v
2OO'C 3 /
- 20 >.-
laJ z
15
0.02 ._1 <_ ~o
E n,
,o
w _z 0.01
5
Oo
I MINIMUM
:3 PARTICAL
4 RAD(US (rim)
6
Figure 4: The ~/6' interfacial energy, ~, as a function of the critical 5' particle radius in a 7.9 at% b i n a r y a l l o y a t 200°C as calculated from e q u a t i o n ( 1 ) .
O
I
8
I
16 TIME
I
I
*
24
|
32
I
I
40
113 (stlc.I/3)
Figure 5: Steady state 6' coarsening data for three different binary alloys at 200°C (after Williams (18)).
6
METALLURGICA
V o l . 18, pp. 6 1 1 - 6 1 6 , 1984 P r i n t e d in t h e U . S . A .
P e r g a m o n P r e s s Ltd. All r i g h t s r e s e r v e d
A NEW METHOD FOR THE DETERMINATION
OF THE PRECIPITATE-MATRIX INTERFACIAL ENERGY
S. F. Baumann and D. B. Williams Department of Metallurgy and Materials Engineering Lehigh University Bethlehem, Pennsylvania 18015
(Received February 24, 1 9 8 4 ) ( R e v i s e d M a r c h 20, 1 9 8 4 )
Introduction The energy of the interface between a precipitate and its matrix is an important thermodynamic factor in many aspects of solid state precipitation reactions. It is the primary barrier to the nucleation event, it is influential in determining particle morphologies during growth and it is the driving force for particle coarsening. While this interfacial energy ( V ) is clearly of great interest in the study of phase transformations, a direct determination of its magnitude is not possible. Experimentally, W has been determined indirectly by means such as: measuring precipitate coarsening rates (I), examining precipitate morphological development (2) and by analysis of equilibrium precipitate shapes (3). This paper describes a new procedure to determine ~ for the fully coherent interface between 6 ' (AI3Li) and its matrix ~ (A1-Li solid solution) in a binary AI-Li alloy. The method involves experimental measurement of the critical particle size for nucleation at a given temperature. This is then used to obtain ~ through the capillarity equation. The value obtained in this way is compared with previous determinations made from particle coarsening studies. Previous Studies in AI-Li Alloys On low temperature aging of dilute AI-Li binary alloys (6-14 at% Li), 6', a metastable Lip ordered phase, precipitates homogeneously and coherently. The 6' is cube/cube oriented with the ~ and the constrained lattice mismatch is less than 0.1% (4). The high compatibility of the two lattices suggests that the interfacial energy should be isotropic and this is reflected in the spherical shape of the 6' particles (see Figure i). Two previous determinations of V~/6' hav~ been made from 6' coarsening data (5,6). These experiments yielded values o~ 0.24 J/m L (5) and 0.18 J/m 2 (6). Noble and Thompson (5) noted that their value of 0.24 J/m was about an order of magnitude higher than that expected on the basis of analogous interfaces between ~' (Ni~X) in ~-Ni (where X = AI, Si, Ti). They suggested that this was due to an uncertainty in the v~lue of D (diffusivity) used in the coarsening rate equation. Williams and Edington (4) studied 6' coarsening in binary AI-Li alloys and, noting that previously reported values of ~ were high, assumed V = 0.025 J/m 2 based on existing data for similar coherent precipitates (7). The coarsening data of Williams and Edington (4) indicated clear evidence for bulk alloy composition influence on particle coarsening rates, which was interpreted as reflecting compositional effects on effective diffusivities. These effects were large enough to cast doubt on the accuracy of the previously reported values of WExperimental Procedures The material used in this work was a high purity 7.9 at% Li binary alloy in the form of 0.25 nun thick sheet obtained from Fulmer Research Institute (4). Standard 3 mm discs, suitable for transmission electron microscope (TEM) samples, were punched from the sheet and wrapped in AI foil. Solution treatment was carried out for 8 mins. at 570°C followed by a water quench to room temperature, after which various aging treatments were performed. All heat treatments were carried out in diffusion furnaces accurate to ± 2°C. Samples were then prepared for TEM investigation by conventional metallographic techniques. All samples were examined in a Philips EM400T TEM.
0036-9748/84 Copyright (c) 1 9 8 4
611 $ 3 . 0 0 + .00 Pergamon Press
Ltd.
612
DETEI~IINATION OF PRECIPITATE-MATRIX
INTERFACIAL
ENERGY
Vol.
18,
No.
6
The quality of the results is dependent upon the accurate imaging and size measurement of very small particles (particle d i a m e t e r s N 3 nm). In this work the 6' was imaged by centered dark field (CDF) techniques using the superlattice reflections. To allow measurement of individual particles, it was necessary to work in thin regions of the foil to avoid excessive particle image overlap. All foils were less than I00 nm thick in the regions where this work was done. The practical limitation to image resolution and, therefore, small particle detection in this work is chromatic aberration since some electrons will experience energy losses on passing through the foil. Calculations show that in the worst case (i00 nm thick) chromatic aberration will limit the smallest resolvable particle size to approximately 1.5 nm radius, although in most specimens smaller particles were resolvable. Particle size measurements in this work were made from enlargements of high magnification 6' CDF images. Results
and Discussion
After a solution treatment, the alloy showed a fully recrystallized microstructure. The diffraction pattern contained well-defined LIp superlattice reflections which, when imaged, revealed a homogeneous distribution of ordere~ 6' particles, approximately 1.5 nm radius. The uniformity of their distribution and small particle sizes suggest that nucleation occurred at or near room temperature as proposed by previous investigators (4,8). A close examination of the regions adjacent to grain boundaries revealed a gradation in the 6' particle size from 1.5 nm radius in the bulk to less than i nm near the boundary (see Figure 2). This phenomenon is evident in previous publications (e.g., Figure i, reference (4)) but has not been commented on to date. It is proposed that this gradient in particle size arises from a gradient in excess quenched-in vacancies which will affect diffusivity at room temperature (9). During the quench from the solution treatment temperatur~ excess vacancies will diffuse to grain boundaries and dislocations. At room temperatures, adjacent to these defects, there will therefore be a gradient of excess vacancies trapped in the structure. Since excess vacancies will contribute to low temperature diffusivity and, therefore increase both the nucleation and growth rate of precipitates, the 6' particles near grain boundaries should be smaller than those within the bulk of the grain as observed in Figure 2. There is no evidence to suggest that this particle size gradient results from a solute gradient since the structure is fully recrystallized and no precipitate phases are present at the grain boundary. Having established a particle size gradient near the grain boundary the material is then given a short, high temperature aging treatment near the solvus line for the phase. In this case, the solution-treated foils were aged for 2 minutes at 200°C, which is approximately 50°C below the 6' solvus for AI 7.9 at% Li (4). The sample takes about 1 minute to reach temperature and is at 200°C for i minute. (This was observed by simulating a heat treatment with a thermocouple included in the foil.) During the "heat up" period the particles coarsen slightly When the sample reaches 200°C all those particles whose size is below the critical size necessary for survival at that temperature will dissolve, the others will grow and coarsen. Examination of the sample aged at 200°C shows a precipitate free zone (PFZ) adjacent to the grain boundary where the smaller particles had dissolved (see Figure 3 and compare with Figure 2). If we ignore the small amount of particle coarsening that occurred at 200°C, then the smallest particle size observed at the edge of this 6' PFZ approximates to the critical 6' nucleus size for this alloy at 200°C. This was measured to be approximately 2 nm in radius. This measurement was straightforward because there was a well-defined decrease in image contrast between these stable 6 ' nuclei (~ 2 nm radius) and much smaller ill-defined particles, which were not present at 200°C but nucleated on the quench back to room temperature (see Figure 3b). The solute content of the matrix in equilibrium with a precipitate of a given size is defined by the capillarity equation (Gibbs-Thompson equation). For the case where the precipitate phase is spherical and is a compound this is given by (i0):
[i x Vm)]
w h e r e X an d X~ are the mole fractions r curvatures r and ~ respectively. Xp i s V i s t h e m o l a r v o l u m e and RT i s t h e r m a l m~tastable equilibrium with the matrix of
of solute in the matrix at interfaces with radii of the mole fraction of solute in the precipitate phase, energy. By d e f i n i t i o n , the critical nucleus is in bulk alloy composition. T h e r e f o r e by s e t t i n g X =
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0.079, X= =0.066 (~/~ + 6' phase boundary at 200°C), X. = 0.25, V = 10 -5 m 3, r = 2 x 10 -9 m m 2 and T = 473 K equation (i) can be solved for ~. This yields a value of V = 0.014 J/m . This value is an order of magnitude smaller than those previously obtained from 6' particle coarsening experiments (6). Figure 4 shows how ? will vary as a function of the measured value of r. It is of interest to find the reason for the discrepancy in the value of N obtained here and those previously determined. There is some uncertainty in the value of .~ for 6 '. Cocco et al. (ii) report X_ = 0.22 at 200°C. However, this discrepancy would yield only a slightly z lower value of V from equation (i) ( V = 0.012 J/m ). There is also an uncertainty in the measurement of the particle size since at these small particle sizes the interface is often not clearly defined. Measurements were made from ~ 40 of the thousands of particles visible in the enlarged prints of Figure 3b. Errors w e r e ~ ± 50%. However, even if the error were ± 100% the value of ~ would only change by a factor of 2 (see Figure 4). Furthermore if we insert the values of V due to Noble and Thompson (5) and Tamura et al. (6) into equation (I), the predicted critical nucleus radii at 200°C are 35 nm and 26 nm respectively, ~ 10X more than the measured data from Figure 3. Therefore uncertainty in Xp and measurement errors cannot explain the 10X discrepancy in ~ between the present an~ previous values. It is suggested here that the difference arises from a failure to account for volume fraction effects on 6' coarsening rates when ~ is calculated from coarsening experiments. In the Lifshitz-Slyozov and Wagner theories (12,13) of particle coarsening, the average particle radius 7 at time t is related to the average particle radius ~ at time t by the o t coarsening rate constant, k,°through: -
r
3
t
with
- r
3
o
= k (t - t ) o
D V 2 m 9 RT
(2)
8 V C k =
e
(3)
where C is the equilibrium matrix solute content, D is diffusivity, and V is the molar volume e of the precipitate phase. In the determination of N through coarsening data, k is measured experimentally by application of equation (2) to the data, and equation (3) defines N (assuming all other variables are known). Equation (3) was originally derived for the condition where the volume fraction ( ~ ) of the precipitate is very small. Many coarsening rate studies, however, involve alloys where ~ is in the range of 0.05 to 0.2 or larger. Several theoretical treatments of the effect of ~ on particle coarsening have been reported in the literature (14-17). The distinguishing assumptions regarding particle distributions and diffusion geometry result in different predictions for each of the models. While there is no quantitative agreement amongst the various models, they all predict an increase in the value of k as increases due to reduced diffusion distances between particles. The 6' coarsening data of Williams and Edington (4) and Williams (18) clearly indicates an effect of bulk Li content on k, with k increasing as Li increases. Figure 5 is from the work of Williams (18) and illustrates the effect of bulk Li content on 6' coarsening at 200°C for 3 binary a l ~ y s . ^ From these data the following values Qf k were measured: For AI-7.9 at% Li, k=2.08 x2½0=~3cmb/sec ; for AI-10.7 at% Li, k=l.40 x i0 -zz cmJ/sec; for AI-12.9 at% Li, k=6.24 x I0cm /sec. For each of these cases, the variables on the rlght-hand side of equation (3) should be the same and, therefore, k should be constant. Experimentally, this is not found to be true. If potential effects of ~ on k are not accounted for, it is obvious that the three sets of k values reported above would yield three different values of N which would vary by more than one order of magnitude from the high to low value. Quantitative metallography was performed on a 7.9 at% binary alloy aged at 200°C. This showed ~ 0.098. We have selected five available models for the effects of ~ on k to backcalculate k (0) (i.e., k when ~ = 0) for the 7.9 at% coarsening data of Williams (18). These were then used to determine N through equation (3). Using a diffusivity calculated from the work of Costas (19) we get the results shown in Table i. The values of ~ in Table 1 are in reasonable agreement with the val~e of 0.014 J/m 2 calculated in this work. Moreover, they are in the range of 0.01 to 0.25 J/m reported for similar coherent interfaces when the lattice misfit is less than I% (7). It is suggested therefore that previous determinations of N from 6' particle coarsening data, in making no correction for the effect of ~ on k, produced erroneously high values of ~ These prior studies were
614
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performed on alloys of 10.6 and 13.8 at% Li which would contain large volume fractions of 8 ' ( ~ _ 0 . 3 and 0.55 respectively) and thus experience enhanced coarsening. No corrections for effects on k were made so the analyses would have overestimated the value of V. TABLE i Calculated Values of k(0) and V from Different Models for Effects on k when applied to 7.9 at% coarsening rate data in Figure 5 Model
k(O) (cm2/sec)
Ardell (15) Asimow (14) Brailsford and Wynblatt (16) Tsumuraya and Miyata (Modell III) (17) Tsumuraya and Miyata (Model Vl) (17)
3.69 9.72 1.08 9.45 3.45
x x x x x
10-24 -24 10_23 i0 10-24 10-24
~(J/m 2) .008 .022 .025 .021 .008
Conclusions The experiment described in this paper yields a reasonable value of N~/6,which is also in agreement with values calculated from coarsening data when corrections are made for volume fraction effects. This experimental procedure should be applicable to any system where nucleation is coherent, transformation strains are small and a particle size distribution can be established near grain boundaries while the solute concentration remains constant. It might be possible to extend this to more general situations. For example, when interfaclal energies are highly anisotropic, it might be possible to get an approximation of N for the high energy interface by assuming it determines the thermodynamic stability at temperature. Equation (I) should then be adjusted to reflect the mean radius of curvature for that interface. Finally, due to the need to image rather small particle sizes accurately in the TEM, the technique would be restricted to precipitates that are ordered, exhibit discrete diffraction spots or are of much larger atomic number than the matrix. Alternatively, lattice imaging techniques could be used rather than conventional CDF images if smaller precipitates have to be detected. Acknowledgment The authors wish to acknowledge the National Science Foundation for their financial support under Grant #DMR-81-08308. References
i)
A. J. Ardell, Met. Trans., !, 525 (1970).
2)
J. A. Malcolm and G. R. Purdy, Trans. Met. Soe. AIME, 239, 1391 (1967). G. R. Spelch and R. A. Oriani, Trans. Met. Soc. AIME, 233, 623 (1965). D. B. Williams and J. W. Edington, Metal Sci., 9, 529 (1975). B. Noble and G. E. Thompson, Metal Sci. J., ~, 114 (1971). M. Tamura, T. Mori and T. Nakamura, J. Jap. Inst. Met., 34, 919 (1970). B. A. Parker and D. R. F. West, J. Aust. Inst. Met. 14, 102 (1969). S. Ceresara, A. Giarda and A. Sanchez, Phil. Mag., 35, 97 (1977). M. H. Jacobs and D. W. Pashley, in The Mechanism of Phase Transformations in Crystalline Solids, Institute of Metals, 1969, pp. 43-48. R. K. Trivedl, in Lectures on the Theory of Phase Transformations, H.I. Aaronson, ed., pp. 79-80, AIME, New York, 1975. G. Coeco, G. Fagherazzl and L. Schiffini, J. AppI. Cryst., iO, 325 (1977). I . M . Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids, 19, 35 (1961). C. Wagner, Z. Electrochem., 65, 581 (1961). R. Asimow, Acta Met., Ii, 72 (1963). A. J. Ardell, Aeta Met., 20, 61 (1972). A. D. Brailsford and P. Wynblatt, Aeta Met., 27, 489 (1979). K. Tsumuraya and Y. Miyata, Acta Met., 31, 437 (1983). D. B. Williams, Ph.D. Thesis, 1974, Cambridge University. L. P. Costas, U.S. Atomic Energy Comm. Rep. (DP-813), 1963.
3) 4) 5) 6) 7) 8) 9) io) 11) 12) 13) 14) 15) 16) 17) 18) 19)
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Figure i: Centered dark field (CDF) image from a 6' s uperlattice reflection showing spherical 6' particles in a AI-7.9 at% Li binary alloy aged 170 hrs. at 220°C.
Figure 2: a) Low magnification and b) higher magnification 6' CDF images taken near a grain boundary in a solution treated and water quenched AI-7.9 at% Li alloy. A gradient in 6' particle size is present adjacent to the boundary. Tilting of the foil showed this was not just a consequence of local diffracting conditions or thickness variations.
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Figure 3: a) Low magnification 6' CDF image showing a 6' precipitate-free zone (PFZ) adjacent to the grain boundary; b) a higher magnification image of the border of the 6' PFZ, in AI 7.9 at% Li solution-treated, water-quenched and aged 2 min. at 200°C. The arrows in b) indicate some typical high contrast 6' particles which were considered to be stable nuclei at 200°C. Fainter 6' particles in the original PFZ nucleated on the quench back to room temperature.
TEMPERATURE
0.04
30
v
2OO'C 3 /
- 20 >.-
laJ z
15
0.02 ._1 <_ ~o
E n,
,o
w _z 0.01
5
Oo
I MINIMUM
:3 PARTICAL
4 RAD(US (rim)
6
Figure 4: The ~/6' interfacial energy, ~, as a function of the critical 5' particle radius in a 7.9 at% b i n a r y a l l o y a t 200°C as calculated from e q u a t i o n ( 1 ) .
O
I
8
I
16 TIME
I
I
*
24
|
32
I
I
40
113 (stlc.I/3)
Figure 5: Steady state 6' coarsening data for three different binary alloys at 200°C (after Williams (18)).
6