NORTHWESTERN UNIVERSITY
Nanoscale Studies of the Early Stages of Phase Separation in Model Ni-Al-Cr-X Superalloys
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
For the degree
DOCTOR IN PHILOSOPHY
Field of Materials Science and Engineering
By
Christopher Booth-Morrison
EVANSTON, ILLINOIS
June, 2009
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© Copyright Christopher Booth-Morrison 2009
All Rights Reserved
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Abstract
Nanoscale Studies of the Early Stages of Phase Separation in Model Ni-Al-Cr-
X Superalloys
Christopher Booth-Morrison
The phase separation of model Ni-Al-Cr-X alloys is studied at the nanoscale employing
atom-probe tomography (APT), electron microscopy and first-principles calculations.
A comparison of the kinetic pathways resulting from the formation of coherent γ’-precipitates in
two Ni-Al-Cr alloys, Ni-7.5 Al-8.5 Cr and Ni-5.2 Al-14.2 Cr at.%, with similar γ’-precipitate
volume fractions at 873 K, is performed. The morphologies of the γ’-precipitates of the alloys are
similar, though the degrees of γ’-precipitate coagulation and coalescence differ. Quantification
within the framework of classical nucleation theory reveals that differences in the chemical
driving forces for phase decomposition result in differences in the nucleation behavior of the two
alloys. The temporal evolution of the γ’-precipitate average radii and the γ-matrix
supersaturations follow the predictions of classical coarsening models. The compositional
trajectories of the γ-matrix phases of the alloys are found to follow approximately the
equilibrium tie-lines, while the trajectories of the γ’-precipitates do not, resulting in significant
differences in the partitioning ratios of the solute elements.
Phase separation in a Ni-6.5 Al-9.5 Cr at.% alloy aged at 873 K occurs in four distinct
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regimes: (i) quasi-stationary-state γ’ (L12)-precipitate nucleation; (ii) concomitant precipitate
nucleation, growth, and coagulation and coalescence; (iii) concurrent growth and coarsening,
wherein coarsening occurs via both γ’-precipitate coagulation and coalescence and by the
classical evaporation-condensation mechanism; and (iv) quasi-stationary-state coarsening of γ’-
precipitates, once the equilibrium volume fraction of precipitates is achieved. The predictions of
classical nucleation and growth models are not validated experimentally, likely due to the
complexity of the atomistic kinetic pathways involved in precipitation. During coarsening, the
temporal evolution of the γ’-precipitate average radius, number density, and the γ(FCC)-matrix
and γ’-precipitate supersaturations follow the predictions of classical models.
The γ’(L12)-precipitate morphology of a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at
1073 K is found to evolve from a bimodal distribution of spheroidal precipitates, to {001}-
faceted cuboids and parallelepipeds aligned along the elastically soft <001>-type directions. The
phase compositions and the widths of the γ’-precipitate/γ-matrix heterophase interfaces evolve
temporally as the Ni-Al-Cr-Ta alloy undergoes quasi-stationary state coarsening after 1 h of
aging. Tantalum is observed to partition preferentially to the γ’-precipitate phase, and suppresses
the mobility of Ni in the γ-matrix sufficiently to cause an accumulation of Ni on the γ-matrix side
of the γ'/γ interface. Measurements of the γ’-phase composition suggest that Al, Cr and Ta share
the Al sublattice sites of the γ’-precipitates. The calculated substitutional energies of the solute
atoms at the Ni and Al sublattice sites, from first-principles calculations, indicate that Ta has a
strong preference for the Al site, while Cr has a weak Al site preference.
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Acknowledgements
I would like to thank my advisor, Professor David N. Seidman, for his support and
encouragement, and for providing a project which was both scientifically challenging, and a
pleasure to work on. I am grateful to Dr. Ronald D. Noebe at NASA Glenn Research Center for
his technical guidance and critical evaluation of the work as it progressed, and for providing
specimens and sample preparation. I would also like to thank Professors Peter W. Voorhees and
David C. Dunand, and Dr. Georges Martin for their advice and technical guidance.
I am very grateful to a number of people who have helped me with both experimental and
analytical work, and whose contributions to the research can be found throughout this thesis,
namely:
• Yang Zhou, for his invaluable assistance with LEAPTM tomography, and for all the lively
debates.
• Dr. Zugang Mao for his computational work, and for all of his help and good cheer.
• Drs. Chantal Sudbrack and Kevin Yoon, who pioneered the work on nickel-based
superalloys in the Seidman group, and provided a great foundation of knowledge and
advice. I would also like to thank Chantal for mentoring me during my first years at NU,
and for her constant support and encouragement.
• Professor Dieter Isheim, for all of his assistance and advice, and for keeping NUCAPT in
top shape.
• Drs. Mark Anderson and Yaron Amouyal for their assistance with SEM and TEM
microscopy, respectively, and for all their other contributions to this work.
• Dr. Carelyn E. Campbell at the National Institute of Standards and Technology, who
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graciously calculated diffusivities for a wide variety of alloy compositions and aging
temperatures.
• Prof. G. B. Olson and Dr. H.-J. Jou of QuesTek LLC (Evanston, IL) for use of
PrecipiCalc.
• Dr. Kathleen Stair for her help with metallography, and her patience.
• Undergraduate research assistants Jessica Weninger and Gillian Hsieh, whose work is
included herein.
• The past and present members of the Seidman group for all their help over the years.
This research was sponsored by the National Science Foundation under grants DMR-
9728986, DMR-0241928 and DMR-0804610. I am grateful to the Fonds québécois de la
recherche sur la nature et les technologies for partial support, and to The Graduate School at
Northwestern University for providing a Walter P. Murphy graduate fellowship during my first
year of studies, as well as a B.J. Martin dissertation year fellowship in my last year.
Atom-probe tomographic measurements were performed at the Northwestern University
Center for Atom-probe Tomography (NUCAPT). The LEAPTM tomograph was purchased with
funding from the NSF-MRI (DMR 0420532, Dr. C. Bouldin, grant officer) and ONR-DURIP
(N00014-0400798, Dr. J. Christodoulou, grant officer) programs, and enhanced with a
picosecond laser with funding from the ONR-DURIP (N00014-0610539, Dr. J. Christodoulou,
grant officer). The SEM and TEM studies were performed in the EPIC facility of the NUANCE
Center at Northwestern University. The NUANCE Center is supported by NSF-NSEC, NSF-
MRSEC, the Keck Foundation, the State of Illinois, and Northwestern University.
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List of acronyms and symbols
APT atom-probe tomography
DTA differential thermal analysis
GGA generalized gradient approximation
LDA local density approximation
LEAP local-electrode atom-probe
NUCAPT Northwestern Center for Atom-Probe Tomography
PAW projector augmented-wave potentials
PLAP pulsed-laser atom-probe tomography
SEM scanning electron microscopy
TEM transmission electron microscopy
VASP Vienna ab initio simulation package
AKV coarsening rate constant for Nv(t) from the KV model
>< )(, tC ffiγ far-field γ-matrix concentration of atomic species i
)(, ∞eqiC γ equilibrium γ-matrix concentration of each atomic species i
)(,' ∞eqiC γ equilibrium γ’-precipitate concentration of each atomic species i
D diffusivity
f fraction of coagulating and coalescing γ’-precipitates γ
jiG , partial derivatives of the molar Gibbs free-energy of the γ-matrix phase
Jeff effective nucleation rate
Jst stationary-state nucleation rate
kB Boltzmann’s constant
KKV coarsening rate constant for <R(t)> from the KV model
Kiγ’/γ partitioning ratio of each atomic species i
Kγ’ bulk modulus of the γ’-precipitate phase
m/Δm mass resolving power
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m/n mass-to-charge state ratio
N0 total number of possible nucleation sites per unit volume
Nv(t) number density of γ’-precipitates
Nv(t0) precipitate number density at the onset of quasi-stationary coarsening
pi magnitude of the partitioning between two phases
R radius of the γ’-precipitate/γ’-nucleus
R* radius of the critical γ’-nucleus
<R(t)> mean γ’-precipitate radius
<R(t0)> average precipitate radius at the onset of quasi-stationary coarsening
S/N signal to noise ratio
t0 time at the onset of quasi-stationary coarsening
tc time required to reach stationary-state coarsening
T absolute temperature in degrees Kelvin γ ′
mV average atomic volume per mole of the γ’-precipitate phase
WR net reversible work required to form a γ’-nucleus *
RW net reversible work required to form a critical nucleus
Z Zeldovich factor
β kinetic coefficient describing the rate of condensation of a single atom on the
critical nucleus
δ lattice parameter misfit
)(tCiγΔ supersaturation of element i in the γ-matrix phase
)(' tCiγΔ supersaturation of element i in the γ’-precipitate phase
ΔFch chemical free energy change on forming a γ’-nucleus
ΔFel elastic free energy change on forming a γ’-nucleus
ΔFnuc bulk free energy change on forming a γ’-nucleus
φ volume fraction of the γ’-precipitate phase
φeq equilibrium volume fraction of the γ’-precipitate phase
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γκ KVi, coarsening rate constant of the atomic species i for )(tCi
γΔ from the KV model
<λe-e> mean edge-to-edge interprecipitate spacing
μi chemical potential per atom of the bulk element i
μγ shear modulus of the γ-matrix phase
σ standard deviation
σγ/γ’ γ-matrix/γ’-precipitate interfacial energy
υγ specific volume of an atom in the γ-matrix
υ specific volume of an atom in the γ’-precipitate phase
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For my wife Agatha, my parents Donald and Pamela, and my sister Gloria, for all
their love and support.
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Table of contents
Abstract………………………………………………………………………………………….. 3
Acknowledgements…………………………………………………………………………...… 5
List of acronyms and symbols………………………………………………………………….. 7
Table of contents…………………………………………………………………………….… 11
List of figures………………………………………………………………………….……..... 15
List of tables…………………………………………………………………………..……..... 21
Chapter 1: Introduction………………………………………………………………………. 23
Chapter 2: Background……………………………………………………………………….. 27
2.1 An introduction to nickel-based superalloys………………………………… 27
2.1.1 Metallurgy of modern nickel-based superalloys…………………………28
2.1.2 Strengthening and deformation mechanisms in
nickel-based superalloys……………………………………………… 31
2.2. Atom-probe tomography for characterizing nanostructures…………….... 34
Chapter 3: Effects of solute concentrations on kinetic pathways in Ni-Al-Cr alloys...…… 35
Abstract……………………………………………………………………………. 35
3.1. Introduction………………………………………………………………….. 36
3.1.1 Classical nucleation theory……………………………………………. 38
3.1.2 Coarsening theory………………………………………………………. 40
3.2. Experimental………………………………………………………………… 42
3.3. Results…………………………………………………………………………. 44
3.3.1 Morphological development………………………………………….. 44
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3.3.2 Temporal evolution of the nanostructural properties of γ’-precipitates.. 48
3.3.2.1 Nucleation and growth of γ’-precipitates…………………………... 51
3.3.2.2 Growth and coarsening of γ’-precipitates……………………………. 52
3.3.3 Temporal evolution of the compositions of the γ and γ’-phases……… 54
3.4. Discussion…………………………………………………………………… 64
3.4.1 Effects of solute concentration on nucleation behavior………………. 64
3.4.2 Effects of Solute Concentration on the Coarsening Behavior………… 71
3.5. Summary and conclusions………………………………………………….… 74
Chapter 4: On the field evaporation behavior of a model Ni-Al-Cr superalloy studied by
picosecond pulsed-laser atom-probe tomography……………………………… 78
Abstract………………………………………………………………………….…. 78
4.1 Introduction………………………………………………………………….. 79
4.2 Materials and methods……………………………………………………… 82
4.3 Results and discussion………………………………………………………… 85
4.3.1 Mass spectra……………………………………………………………. 85
4.3.2 Compositional accuracy: Effects of laser energy…………………….. 92
4.3.3 Compositional accuracy: Effects of specimen base temperature……… 94
4.3.4 Analysis of events detected at mass-to-charge state
ratios of 100-300 amu…………………………………………….….. 97
4.4 Conclusions…………………………………………………………………….. 99
Chapter 5: On the nanometer scale phase decomposition of a low-supersaturation
Ni-Al-Cr alloy…………………………………………………………………… 101
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Abstract…………………………………………………………………………… 101
5.1 Introduction………………………………………………………………… 102
5.2 Experimental…………………………………………………………………. 104
5.3 Results and discussion……………………………………………………… 106
5.3.1 Morphological evolution…………………………………………….. 106
5.3.2 Temporal evolution of the nanostructural properties
of the γ’-precipitates…………………………………………………. 108
5.3.3 Temporal evolution of the compositions of
the γ and γ’ phase compositions…………………………….………. 116
5.4 Conclusions…………………………………………………………………. 124
Chapter 6: Effects of tantalum on the temporal evolution of a model
Ni-Al-Cr superalloy during phase decomposition……………………………. 127
Abstract…………………………………………………………………………. 127
6.1 Introduction………………………………………………………………… 128
6.2 Experimental……………………………………………….………………… 129
6.3 Results………………………………………………………………………… 132
6.3.1 Morphological development……………………………………...…… 133
6.3.2 Compositional evolution…………………………………………..….. 138
6.3.3 Partitioning of elemental species………………………………….….. 140
6.4 Discussion……………………………………………………………….……. 142
6.4.1 Morphological development…………………………………………... 142
6.4.2 Compositional evolution…………………………………………….… 148
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6.4.3 Comparison with other Ni-Al-Cr-X alloys…………………………….. 153
6.5 Summary and conclusions…………………………………………………. 155
Chapter 7: Chromium and tantalum site substitution patterns
in Ni3Al (L12) γ’-precipitates…………………………………………………… 158
Abstract…………………………………………………………………………… 158
7.1 Introduction…………………………………………….…………………….. 159
7.2 Experimental…………………………………………………………………. 160
7.3 Results and discussion……………………………………………………….. 161
7.4 Conclusions………………………………………………………………….. 165
Chapter 8: Future work…………………………………………………………………….. 166
References……………………………………………………………………………………. 169
Appendix 1: Recalculation of the Ni-Al-Cr interfacial free energy from the data of
Schmuck et al. ……………………………………………………………….… 190
Appendix 2: Temporal evolution of the phase compositions of
Ni-Al-Cr alloys at 873 K……………………………………………………… 192
Vita………………………………………………………………………………………….. 195
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List of figures
Figure 2.1. Arrangements of Ni (green) and Al (red) in the (a) disordered γ-matrix and (b) the ordered γ’-Ni3Al, L12, crystal structures. (Image courtesy of Dr. Y. Amouyal, Northwestern University)……………………………………………………………………………………. 30
Figure 3.1. A partial ternary phase diagram of the Ni-Al-Cr system at 873 K calculated using the Grand Canonical Monte Carlo simulation technique [113], showing the proximity of both alloys (A) Ni-7.5 Al- 8.5 Cr and (B) Ni-5.2 Al-14.2 Cr to the (γ + γ’) / γ solvus line. The tie-lines are drawn through the nominal compositions of the alloys and between the equilibrium phase compositions determined by extrapolation of APT concentration data to infinite time. Equilibrium solvus curves determined by Thermo-Calc [114], using databases for nickel-based superalloys due to Saunders [115] and Dupin et al. [116], are superimposed on the GCMC phase diagram for comparative purposes……………………………………………………………... 37
Figure 3.2. The temporal evolution of the γ’-precipitate nanostructure in Ni-7.5 Al-8.5 Cr aged at 873 K is displayed in a series of APT parallelepipeds. The parallelepipeds are 10 x 10 x 25 nm3
subsets of the analyzed volume and contain ~ 125,000 atoms. The γ/γ’ interfaces are delineated in red with 10.5% Al isoconcentration surfaces………………………………………………. 45
Figure 3.3. A subset of an APT micrograph of Ni-7.5 Al-8.5 Cr aged at 873 K for 1024 h, containing 350,000 atoms, with Al atoms in red, and Cr atoms in blue; Ni atoms are omitted for clarity. A γ’-precipitate with a radius of ~ 9 nm is delineated by the red 10.5% Al isoconcentration surface, and shows {110}-type superlattice planes with an interplanar spacing of 0.26 ± 0.03 nm……………………………………………………………………………… 46
Figure 3.4. A centered superlattice reflection dark–field image of spheroidal Ni3(AlxCr(1-x)) γ’-precipitates for a Ni-7.5 Al-8.5 Cr sample aged for 1024 h at 873 K, recorded near the [011] zone axis……………………………………………………………………………………………. 47
Figure 3.5. The temporal evolution of the value of the fraction of γ’-precipitates interconnected by necks, f, and the average interprecipitate edge-to-edge spacing, <λe-e>. The maximum value of f is 18±4 % corresponds to the minimum value of <λe-e> of 7±2 nm at an aging time of 1 h for alloy (A) Ni-7.5 Al-8.5 Cr. For alloy (B) Ni-5.2 Al-14.2 Cr, the minimum value of <λe-e> of 5.9±0.8 nm and the maximum value of f of 30±4 % coincide at an aging time of 4 h……….. 48
Figure 3.6. The temporal evolution of the γ’-precipitate volume fraction, φ, number density, Nv(t), and mean radius, <R(t)>, for (A) Ni-7.5 Al-8.5 Cr and (B) Ni-5.2 Al-14.2 Cr alloys aged at 873 K as determined by APT microscopy. The quantity <R(t)> is proportional to t1/3 as predicted by the Umantsev and Olson (UO) and Kuehmann and Voorhees (KV) models for isothermal coarsening in ternary alloys. The temporal dependence of the diminution of the quantity Nv(t) deviates from the t-1 prediction of the UO and KV models for both alloys…….. 50
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Figure 3.7. The composition profiles on either side of the heterophase γ-matrix/ γ’-precipitate interface for (A) Ni-7.5 Al-8.5 Cr aged at 873 K for aging times of 1/4, 1, 4, 1024 h. The phase compositions evolve temporally, as the γ-matrix becomes enriched in Ni and Cr and depleted in Al. The values of <R(t)> for these aging times are 1.00 ± 0.11 nm for 1/4 h, 1.24 ± 0.12 nm for 1 h, 1.70 ± 0.25 nm for 4 h and 8.30 ± 2.93 nm for 1024 h……………………………………… 54
Figure 3.8. The compositional trajectories of the γ-matrix and γ’-precipitate phases of alloys (A) Ni-7.5 Al-8.5 Cr and (B) Ni-5.2 Al-14.2 Cr as they evolve temporally, displayed on a partial Ni-Al-Cr ternary phase diagram at 873 K. The tie-lines are drawn through the nominal compositions of the alloys (squares) and between the experimentally determined equilibrium phase compositions. The trajectories of the γ-matrix phases in alloys (A) and (B) lie approximately on the experimental tie-lines. The trajectories of the γ'-precipitate phases do not lie along the tie-line, as predicted by the Kuehmann and Voorhees model of isothermal coarsening in ternary alloys [133]……………………………………………………………………………………. 56
Figure 3.9. The partitioning ratios, Kiγ’/γ
, of Al and Cr demonstrate that both alloys exhibit partitioning of Al to the γ’-precipitates and Cr to the γ-matrix. Partitioning is more pronounced in alloy (B), where the smaller Al and larger Cr concentration results in a smaller value of the γ-matrix Al solubility, and a larger γ-matrix Cr solubility………………………………………. 59
Figure 3.10. The magnitude of the values of the supersaturations, )(tCiγΔ , of Al and Cr in the γ-
matrix are smaller for alloy (A) Ni-7.5 Al-8.5 Cr than for alloy (B) Ni-5.2 Al-14.2 Cr. The magnitude of the )(tCi
γΔ values decrease as t-1/3 in the coarsening regimes for both alloys, as predicted by the Umantsev and Olson (UO) and Kuehmann and Voorhees (KV) models for isothermal quasi-stationary state coarsening in ternary alloys. The values of )(tCCr
γΔ are expressed as an absolute value because they are negative, and reflect a flux of Cr into the γ-matrix with increasing aging time…………………………………………………………….… 62
Figure 3.11. The supersaturation values of the γ’-precipitates, )(tCiγ ′Δ , reflect the chemical
compositions of the two alloys, as the value of )(tCAlγ ′Δ is larger in alloy (A) Ni-7.5 Al- 8.5 Cr,
which contains more Al, than in alloy (B) Ni-5.2 Al-14.2 Cr, while the inverse is true for Cr. The values of )(tCi
γ ′Δ decrease as approximately t-1/3 in the coarsening regimes for both alloys… 63
Figure 4.1. The steady-state dc voltage applied to the tip to maintain a specimen evaporation rate of 0.04 ions per pulse for a Ni-6.5 Al-9.5 Cr at.% alloy is shown as a function of: (a) the laser pulse energy applied in PLAP tomography at a specimen base temperature of 40±0.3 K and a pulse repetition rate of 200 kHz; and (b) the specimen base temperature used in pulsed-voltage APT at a pulse repetition rate of 200 kHz and a pulse fraction of ~20%..................................... 85
Figure 4.2. A comparison of the mass spectra obtained by PLAP tomography with laser pulse energies of: (a) 0.2 nJ per pulse at a tip voltage of 7.40-7.70 kV; (b) 1.0 nJ per pulse at a tip voltage of 8.0-8.3 kV; and (c) 2.0 nJ per pulse at a tip voltage of 6.55-6.85 kV. The frequencies
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of detected events are normalized to the number of events detected in the most populous m/n bin. The m/n ratios of the evaporating species change by over four orders of magnitude with increasing laser pulse energy. At a laser pulse energy of 2.0 nJ per pulse, the m/n spectrum is populated by numerous hydrides, oxides and metal ion clusters peaks, which are a result of overheating and are undesirable………………………………………………………………… 87
Figure 4.3. A section of the m/n range showing the Ni2+ peaks, highlighting both the improved mass resolving power of PLAP tomography when compared to pulsed-voltage APT, and the improved mass resolving power with increasing laser pulse energy. The relative frequencies of the Ni2+ peaks from the data set collected at a laser pulse energy of 2.0 nJ per pulse are nearly two orders of magnitude smaller than the relative frequencies of the Ni2+ peaks from the set collected at 0.2 nJ per pulse……………………………………………………………….……. 88
Figure 4.4. The ratio of the ions detected in the 2+ charge state to those detected in the 1+ charge state, as a function of laser pulse energy. The Ni and Al ions appear to have similar evaporation behavior, whereas Cr atoms evaporate exclusively as Cr2+ until the laser energy reaches 1.0 nJ per pulse………………………………………………………………………………………. 89
Figure 4.5. The relative frequencies of the Ni1+ peaks at specimen base temperatures of 125, 150 and 175±0.3 K, at a constant steady-state dc tip voltage of 5.2±0.2 kV. The increase in the relative frequency of Ni1+ ions with increasing specimen base temperature suggests that the evaporation of Ni1+ ions occurs by a different thermally activated mechanism than the evaporation of Ni2+ ions………………………………………………………………………. 92
Figure 4.6. Variation in the detected composition of as-quenched Ni-6.5 Al-9.5 Cr at.% as a function of laser pulse energy (solid line) compared to the concentration measured by pulsed-voltage APT (dashed line) and the nominal composition of the alloy (dotted line). Preferential evaporation of Ni is responsible for the inaccuracy in the concentrations measured by pulsed-laser and pulsed-voltage APT………………………………………………………………….. 95
Figure 4.7. Variation of the detected composition of as-quenched Ni-6.5 Al-9.5 Cr at.% as a function of specimen base temperature by pulsed-voltage APT. At higher temperatures, preferential evaporation of Ni is less severe, while preferential evaporation of Cr is more severe.. ……………………………………………………………………………………………………96
Figure 4.8. The relative frequency of detected events between 100-300 amu, as a function of: (a) laser pulse energy in PLAP tomography; and (b) specimen base temperature in pulsed-voltage APT. The amount of noise measured by PLAP tomography is directly proportional to the tip voltage, and does not increase with increasing laser pulse energy, and thus increasing thermal energy. On the contrary, the noise measured by pulsed-voltage APT increases as the specimen base temperature increases. This result demonstrates that the Ni-Al-Cr tips are not affected by significant steady-state heating during PLAP tomography……………………………………. 98
Figure 5.1. A partial ternary phase diagram of the Ni-Al-Cr system at 873 K calculated using the Grand Canonical Monte Carlo simulation technique [113], showing the proximity of Ni-6.5 Al-
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9.5 Cr at.% to the (γ + γ’) / γ solvus line. Two other alloys that have been investigated by APT, Ni-7.5 Al- 8.5 Cr and Ni-5.2 Al-14.2 Cr at.%, are shown for comparative purposes. Equilibrium solvus curves determined by Thermo-Calc [114], using databases for nickel-based superalloys due to Saunders [115] and Dupin et al. [116], are superimposed on the GCMC phase diagram. The tie-lines are determined from the equilibrium phase compositions, determined by extrapolation of APT concentration data to infinite time……………………………………. 103
Figure 5.2. APT reconstructed 3D images of a Ni-6.5 Al-9.5 Cr at.% alloy aged at 873 K for (a) 1 h, (b) 4 h, (c) 64 h and (d) 4096 h. The nanometer-sized γ’-precipitates are delineated with red Al isoconcentration surfaces. Al, which partitions to the γ’-precipitates, is shown in red, while Cr, which partitions to the γ-matrix, is shown in blue; Ni atoms are omitted for clarity……. 107
Figure 5.3. The fraction of γ’-precipitates undergoing coagulation and coalescence, f, is significant for aging times of 0.5 to 16 h. After aging for 64 h, only 4±1 % of γ’-precipitates are undergoing coagulation and coalescence, and therefore coarsening proceeds primarily via the evaporation-condensation mechanism. No γ’-precipitate coagulation and coalescence is detected beyond an aging time of 64 h. ……………………………………………………………..…. 108
Figure 5.5. The temporal evolution of the γ’-precipitate volume fraction, φ, number density, Nv(t), and mean radius, <R(t)>, for Ni-6.5 Al-9.5 Cr at.% at 873 K. The quantity <R(t)> is approximately proportional to t1/3 during quasi-stationary state coarsening for aging times of 4 h and longer, as predicted by classical coarsening models. Once the equilibrium volume fraction is achieved after 256 h, the temporal dependence of the quantity Nv(t) achieves the t-1 prediction of the coarsening models…………………………………………………………………………. 109
Figure 5.6. The composition profiles across the γ-matrix/ γ’-precipitate interface for Ni-6.5 Al-9.5 Cr at.% at 873 K for aging times of 1, 4, 4096 h. The phase compositions evolve temporally, as the γ-matrix becomes enriched in Ni and Cr and depleted in Al. The values of <R(t)> for these aging times are 0.65 ± 0.19 nm for 1 h, 1.23 ± 0.43 nm for 4 h and 15.80 ± 3.39 nm for 4096 h………………………………………………………………………………………………. 119
Figure 5.7. The compositional trajectories of the γ-matrix and γ’-precipitate phases of Ni-6.5 Al-9.5 Cr at.%, displayed on a partial Ni-Al-Cr ternary phase diagram at 873 K. The trajectory of the γ-matrix phase lies approximately on the experimental tie-line, while the trajectory of the γ'-precipitate phases does not lie along the tie-line, as predicted by the Kuehmann-Voorhees coarsening model [133]. The tie-lines are determined from the equilibrium phase compositions, determined by extrapolation of APT concentration data to infinite time……………………. 121
Figure 5.8. The partitioning ratios, Kiγ’/γ
, quantify the partitioning of Al to the γ’-precipitates and of Ni and Cr to the γ-matrix. The values of Ki
γ’/γ are constant from 256 to 4096 h, when the alloy is undergoing quasi-stationary state coarsening………………………………………………. 122
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Figure 5.9. The magnitude of the values of the γ-matrix (left) and γ’-precipitate (right) supersaturations, )(tCi
γΔ and )t(C 'iγΔ , of Ni, Al and Cr, decrease as t-1/3 in the coarsening
regime, as predicted by classical coarsening models…………………………………………. 123
Figure 6.1. Vickers microhardness measurements for Ni-10.0 Al-8.5 Cr-2.0 Ta and Ni-10.0 Al-8.5 Cr at.% aged at 1073 K. The addition of 2.0 at.% Ta results in a 47±5% increase in the microhardness over the full range of aging times, t = 0 to 256 h, due to increases in solid-solution strengthening and to a dramatic increase in the value of φ.………………………………….. 132
Figure 6.2. APT reconstructed 3D image of a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 0 h. The solute elements that partition to the γ’-precipitates, Ta and Al, are shown in orange and red, respectively, while Cr, which partitions to the γ-matrix, is shown in blue; Ni atoms are omitted for clarity. The morphology of the γ’-precipitate phase is spheroidal in the as-quenched state, and a bimodal particle size distribution is apparent………………………….134
Figure 6.3. SEM images of a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for: (a) 0 h: (b) 0.25 h: (c) 1 h: (d) 4 h: (e) 16 h: and (f) 64 h. The primary γ’-precipitate morphology evolves from spheroidal γ’-precipitates to a cuboidal and parallelepipedic morphology with primary γ’-precipitates aligned along the elastically soft <001>-type directions……………………….. 135
Figure 6.4. APT reconstructed 3D image of a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 0.25 h. The primary γ’-precipitates in the APT reconstruction have both spheroidal and cuboidal characteristics. The radius at which the γ’-precipitates undergo the spheroidal-to- cuboidal morphological transformation is measured to be 61±7 nm. Tantalum and Al are shown in orange and red, respectively, Cr is shown in blue, and Ni atoms are omitted for clarity… 136
Figure 6.5. An APT reconstructed 3D image of cuboidal and parallelepipedic primary γ’-precipitates aligned along <001>-type directions from a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 64 h. Tantalum and Al are shown in orange and red, respectively, Cr is shown in blue, and Ni atoms are omitted for clarity………………………………………………….. 137
Figure 6.6. After aging for 256 h, the primary γ’-precipitates have cuboidal and parallelepipedic morphologies with {001}-type facets, and are aligned along the <001>-type crystallographic directions, as confirmed by a selected-area electron diffraction pattern taken along the ]101[ zone axis…………………………………………………………………………………………….. 137
Figure 6.7. The elemental concentration profiles across the primary γ’-precipitate/γ-matrix heterophase interface for a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K. The phase compositions evolve temporally, and the widths of the concentration profiles decrease with increasing aging time. An accumulation of Ni is observed to develop on the γ-matrix side of the interface, evidence of a kinetic effect associated with the addition of Ta. The concentration profiles shown here represent the average of all of the interfaces in the analyzed volume….. 139
20
Figure 6.8. The temporal evolution of the widths of the concentration profiles across the primary γ’-precipitate/γ-matrix interface for a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K. The interfacial widths are observed to decrease with increasing aging time. The species with the largest diffusivity in the γ-matrix, Al and Ta, have the smallest interfacial widths, while the opposite is true for Ni and Cr, the slower diffusing species…………………………………. 140
Figure 6.9. The temporal evolution of the partitioning ratios, γγ /′iK , of the constituent elements i
of: (a) Ni-10.0 Al-8.5 Cr-2.0 Ta; and (b) Ni-10.0 Al-8.5 Cr at.% aged at 1073 K. In both alloys, Al is observed to partition to the γ’-precipitates, while Cr and Ni partition to the γ-matrix. Ta shows a strong preference for the γ’-phase……………………………………………………. 141
Figure 6.10. The values of the γ’-precipitate number density, Nv(t ), radius, <R(t )>, and volume fraction, φ, during the water quench from 1503 K for Ni-10.0 Al-8.5 Cr-2.0 Ta at.%, as modeled by PrecipiCalc. A bimodal distribution of γ’-precipitates is predicted for the as-quenched state with primary γ’-precipitates with <R(t)> and φ values of 30.2 nm and 31.5 %, and secondary γ’-precipitates with <R(t)> and φ values of 0.8 nm and 14.2 %. The predicted values of Nv(t) for the primary and secondary γ’-precipitates are 2.7 x 1021 and 3.4 x 1025 m-3, respectively………. 145
Figure 6.11. The evolution of the values of the bulk driving force for nucleation, ΔFnuc, the effective nucleation rate, Jeff, and the critical radius for nucleation, R*, during the quench. Decreases in the magnitude of ΔFnuc are predicted at 1380 and 1190 K due to the nucleation of primary and secondary γ’-precipitates, respectively. The magnitude of ΔFnuc increases and the values of R* decrease during the quench. This figure was calculated using PrecipiCalc…….. 146
Figure 6.12. The magnitude of the values of the supersaturations, )(tCiΔ , of Ni, Al, Cr and Ta in the γ-matrix and γ’-precipitates for Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K. The magnitudes of the )(tCiΔ values decrease as approximately t-1/3 in the coarsening regimes for both phases, as predicted by the Umantsev and Olson (UO) models for quasi-stationary state coarsening………………………………………………………………………………….….. 150
Figure 7.1. APT reconstruction of a cuboidal γ’-precipitate in a model Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 256 h. The elements that partition to the γ’-precipitates, Al and Ta, are shown in orange and red, respectively, while Cr, which partitions to the γ-matrix, is shown in blue; Ni atoms are omitted for clarity…………………………………………………………. 160
Figure 7.2. The elemental concentration profiles across the γ’/γ interface for a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 256 h. Tantalum and Al partition to the γ’-precipitates, while Cr partitions to the γ-matrix…………………………………………………………… 161
21
List of tables
Table 3.1. Temporal evolution of the nanostructural properties of γ’-precipitates determined by APT for Ni-7.5 Al-8.5 Cr aged at 873 K…………………………………………..…………… 51 Table 3.2. Equilibrium γ’-precipitate and γ-matrix concentrations, as determined by atom-probe tomography (APT), Grand Canonical Monte Carlo (GCMC) simulation, and thermodynamic modeling employing Thermo-Calc for alloy (A) Ni-7.5 Al-8.5 Cr aged at 873 K……………. 58 Table 3.3. Equilibrium γ’-precipitate volume fraction, φeq, as determined by atom-probe tomography (APT), Grand Canonical Monte Carlo (GCMC) simulation, and Thermo-Calc for alloy (A) Ni-7.5 Al-8.5 Cr, and alloy (B) Ni-5.2 Al-14.2 Cr, aged at 873 K………………….. 60 Table 3.4. Curvatures in the molar Gibbs free-energy surface of the γ-matrix phase evaluated at the equilibrium composition with respect to components i and j, γ
jiG , , obtained from ideal solution theory and Thermo-Calc thermodynamic assessments for alloy (A) Ni-7.5 Al-8.5 Cr aged at 873 K…………………………………………………………………………………. 66
Table 3.5. Free-energy of the γ/γ’ interfaces, σγ/γ’, at 873 K in Ni–7.5 Al–8.5 Cr calculated from the experimental values of the Kuehmann-Voorhees coarsening rate constants for the average precipitate radius and the supersaturation of solute species i employing Equation 3.9 with solution thermodynamics described by the ideal solution and Thermo-Calc databases given in Table 3. 4……………………………………………………………………………………… 67 Table 3.6. The interfacial free energy, σγ/γ’, the chemical free energy, ΔFch, and the elastic strain energy, ΔFel, components of the driving force for nucleation used to estimate the net reversible work, *
RW , required for the formation of critical nuclei of size, R*, and the nucleation current, Jst, according to classical nucleation theory……………………………………………………… 69 Table 5.1. Temporal evolution of the nanostructural properties of γ’-precipitates a determined by APT for Ni-6.5 Al-9.5 Cr at.% aged at 873 K. The γ’-precipitate mean radius, <R(t)>, number density, Nv(t), and volume fraction, φ, are given, along with their standard errors…………… 110 Table 6.1. Equilibrium γ’-precipitate volume fraction, φeq, determined by APT, ICP chemical analysis, and thermodynamic modeling employing Thermo-Calc for Ni-10.0 Al-8.5 Cr-2.0 Ta at.% aged at 1073 K…………………………………………………………………………… 133 Table 6.2. Equilibrium compositions (at.%) of the γ’-precipitate, )(C eq,'
i ∞γ , and γ-matrix, )(C eq,
i ∞γ , determined by APT, ICP chemical analysis, and thermodynamic modeling employing Thermo-Calc for Ni-10.0 Al-8.5 Cr-2.0 Ta at.% aged at 1073 K…………………………….. 151 Table 6.3. Tracer diffusivity of element i in the γ-matrix, γ
iD , of Ni-10.0 Al-8.5 Cr and Ni-10.0
22
Al-8.5 Cr-2.0 Ta at.% calculated with Dictra, and Campbell [208] and Saunders [115] databases at 1073 K…………………………………………………………………………………….. 152
Table 7.1. Total, Etot, and substitutional, EZ→Ni, Al, energies determined by first-principles calculations. (For these calculations x = 0.042, y = 0.125, and Z = Cr, Ta.)……………….…. 164 Table 7.2. Average atomic forces and displacements at the first nearest-neighbor distance from first-principles calculations for four different substitutional structures……………………….. 165 Table A.1.1 Equilibrium γ’-precipitate and γ-matrix equilibrium concentrations, as determined by atom- probe tomography (APT) and Thermo-Calc for Ni-5.2 Al-14.8 Cr at.% aged at 873 K from the data of Schmuck et al. [28] .……………………………………………………….. 191 Table A.2.1. Temporal evolution of the phase compositions in Ni-7.5 Al-8.5 Cr at.% aged at 873 K, as measured by atom-probe tomography. The lattice parameter for this alloy is estimated to be 0.0027 ± 0.0004, and the equilibrium volume fraction is 16.4 ± 0.6 %.……………………… 192 Table A.2.2. Temporal evolution of the phase compositions in Ni-5.2 Al-14.2 Cr at.% aged at 873 K, as measured by atom-probe tomography. The lattice parameter misfit for this alloy is estimated to be 0.0006 ± 0.0004 and the equilibrium volume fraction is 15.7 ± 0.7 %.…….. 193 Table A.2.3. Temporal evolution of the phase compositions in Ni-6.5 Al-9.5 Cr at.% aged at 873 K, as measured by atom-probe tomography. The lattice parameter misfit for this alloy is estimated, from PrecipiCalc, to be 0.00284, and the equilibrium volume fraction from APT, is 9.6 ± 2.9 %.………………………………………………………………………………….. 194
23
Chapter 1
Introduction
Global warming and the volatility of fuel costs have amplified the urgency for increased
fuel efficiency in high-performance engines. Fuel efficiency is directly related to engine
operating temperature, which is tied to the high-temperature properties of the engine materials.
For example, a 200 K rise in the turbine engine entry temperature of a Boeing 777 yields a 5%
savings in fuel burn, which in turn equals a $12.5 million savings over 15 years. Due to their
excellent strength and resistance to both corrosion and creep-induced damage at temperatures up
to 1373 K, nickel-based superalloys are used for critical components of both aerospace and land-
based turbine engines.
Though the most technologically demanding applications of nickel-based superalloys
remain in the aerospace sector , these alloys are also used in land-based turbines for generating
electricity [1]. The U.S.A. consumes >4 trillion kWh of electrical power per year, of which ca.
70% comes from fossil fuels [2]. Decreasing CO2 emissions to avert global warming is crucial,
given the imminent dangers associated with climate change, and the ongoing industrial
expansion in developing nations. Nickel-based superalloys are also used in the chemical industry
because of their outstanding corrosion resistance, and in the oil and gas industry for coating pipes
and down-hole equipment to enhance resistance to hydrogen embrittlement due to H2S, and to
reduce high-temperature corrosion.
The nickel-based superalloys used in aerospace turbine blades have progressed from
equiaxed to directionally solidified grain-structures, and presently to single-crystal blades that
24
contain 12 elements [3]. The current generation of nickel-based superalloys take advantage of the
sophisticated physical metallurgy of the γ/γ’ system and of the effects of alloying on the high-
temperature microstructure and physical and mechanical properties [4]. The addition of
refractory elements such as Ta, Mo, W, Re, and recently Ru [5], increases the solvus
temperatures, thereby increasing the maximum operating temperature [6].
The high-temperature strength and creep resistance of modern nickel-based superalloys
are due to the presence of coherent, elastically hard, L12-ordered γ’-precipitates in a γ (FCC)
nickel-rich solid-solution matrix [7, 8]. Efforts to improve these properties by process
optimization, and to develop reliable life-prediction techniques, have created a demand for a
quantitative understanding of the kinetic pathways that lead to phase decomposition at service
temperatures up to 1373 K [3, 9, 10]. The technological importance of commercial nickel-based
superalloys has motivated extensive investigations of the precipitation of the γ’-phase from the
supersaturated γ-matrix of model alloys by conventional [11-15] and high–resolution [16]
transmission electron microscopy (TEM), x-ray analysis [17-19], small-angle and wide angle
neutron scattering [20-23], atom-probe field-ion microscopy (APFIM) [24-27], atom-probe
tomography (APT) [28-30], phase-field modeling [31-36], and lattice kinetic Monte Carlo
simulation (LKMC) [37-39]. Many of these techniques are limited by either, or both, their spatial
and analytical resolutions for composition [40], and thus the early stages of phase decomposition
are still not well understood. This is particularly true of concentrated multicomponent alloys due,
in part, to the complexity of the diffusion processes involved [39].
Due to their excellent mechanical and physical properties at elevated temperatures,
nickel-based superalloys based on Ni-Al-Cr compositions are technologically extremely
25
important alloys [3, 7, 41]. The addition of Cr to the binary Ni-Al system reduces the lattice
parameter misfit between the γ’-precipitates and the γ-matrix, often leading to γ’-precipitates that
are nearly misfit-free [12], thereby allowing the γ’-precipitates to remain spheroidal to fairly
large dimensions as aging progresses [17, 42]. The formation of coherent, spheroidal γ’-
precipitates with relatively stress-free precipitate/matrix heterophase interfaces in model Ni-Al-
Cr alloys makes them excellent candidates for comparison of experimental data with the
predictions of classical treatments of nucleation, growth and coarsening for ternary alloys, for
which there is only a little detailed quantitative data available.
This Ph.D. thesis focuses on the decomposition behavior of two model Ni-Al-Cr alloys,
and one model Ni-Al-Cr-Ta alloy. This work is part of a systematic investigation of the phase
decomposition of model Ni-Al-Cr alloys at 873 K [39, 42-54], and of the effects of dilute
refractory additions, such as W [43, 55, 56], Re [50, 51, 57-59], and Ta [60-63], on the γ/γ’ phase
decomposition of a model Ni-10.0 Al-8.5 Cr at.% alloy aged at 1073 K. The thesis is divided as
follows:
Chapter 2 gives a short background to nickel-based superalloys, including their basic
physical metallurgy and materials properties.
Chapter 3 presents an investigation of the effects of solute content on the kinetic
pathways that lead to phase decomposition in two Ni-Al-Cr alloys with moderate
solid-solution supersaturations (Ni-7.5 Al-8.5 Cr and Ni-5.2 Al-14.2 Cr at.%) aged at
873 K [52].
Chapter 4 describes the field evaporation behavior of a model Ni-Al-Cr superalloy
(Ni-6.5 Al-9.5 Cr at.%) studied by picosecond pulsed-laser atom-probe tomography
26
[64].
Chapter 5 provides the details of the phase decomposition of a low-supersaturation
Ni-Al-Cr alloy (Ni-6.5 Al-9.5 Cr at.%) aged at 873 K, and compares the results to the
predictions of the classical models of nucleation, growth and coarsening [53].
Chapter 6 is a description of the effects of a dilute addition of Ta to a model Ni-Al-Cr
alloy (Ni-10.0 Al-8.5 Cr at.%) aged at 1073 K [60].
Chapter 7 is an investigation of the sublattice site preference of Cr and Ta in Ni3Al
(L12) γ’-precipitates employing both atom-probe tomography and first-principles
calculations [61].
Chapter 8 gives possible directions of future research on these subjects.
27
Chapter 2
Background
2.1 An introduction to nickel-based superalloys
Considerable research and development has led to significant improvements in the
service temperatures of commercial superalloys since their first appearance in the 1940s. The
complex multicomponent alloys in service today are a result of both tremendous progress in
alloy development, and improvements in manufacturing techniques, such as the evolution of
alloys for blade applications from wrought, to polycrystalline to single crystal form.
Alloy development has evolved through four generations of superalloys. First generation
alloys contained high concentrations of the γ’-precipitate hardening elements Al, Ti and Ta, as
well as the grain boundary-strengthening additions C and B. The second and third generation
superalloys were characterized by 3 and 6 wt.% additions of Re, respectively, which provides
solid solution strengthening. Rhenium also significantly decreases the rate of γ’-precipitate
coarsening, extending the temperature capability of directionally solidified, columnar grain, and
single-crystal nickel superalloys. The fourth generation of superalloys, in use since 2000, is
characterized by additions of ruthenium, which inhibits the formation of the deleterious
topologically close-packed (TCP) phases [65-69]. These refractory-rich TCP phases evolve from
localized supersaturations of Cr, Re and W at temperatures above 1273 K [70-74]. The TCP
phases are deleterious to the mechanical properties of commercial superalloys because they
consume refractory additions, reducing both solid-solution and precipitation strengthening [71,
75]. In addition, Ruthenium improves the oxidation and creep resistance of nickel-based
28
superalloys [76].
2.1.1 Metallurgy of modern nickel-based superalloys
Nickel has emerged as the premiere solvent for high-temperature applications. As a
transition metal, Ni offers both considerable toughness and ductility due to its high cohesive
energy, related to bonding of the outer d electrons. Nickel remains stable in the FCC structure up
to its melting point [77], providing phase stability that is critical to high-temperature
applications. In addition, the small diffusivity of Ni leads to excellent resistance to diffusion
mediated processes such as γ’-precipitate coarsening, creep, and oxidation [3, 78, 79]. Nickel
also offers considerable cost and weight benefits when compared to other suitable FCC metals,
such as the platinum group metals.
While some solid-solution strengthened nickel-based superalloys, such as Hastelloy X,
are used commercially when modest strength is required, the most demanding applications
require a class of superalloys that are strengthened by intermetallic compound precipitation in
the austenitic (FCC) γ-matrix. Generally, alloying additions in nickel-based superalloys can be
characterized as:
(1) γ-matrix formers, elements that partition to the FCC matrix (Ni, Co, Cr, Fe,
Mo, W, Re, Ru)
(2) γ’-precipitate formers (Al, Ti, Ta, Nb)
(3) γ’’-precipitate formers (Nb, V)
29
(4) Carbide formers (reactive refractory additions such as Ti, Ta, Nb, Hf form
carbides that may decompose at temperature to form complex carbides with
Cr, Mo and W at γ-matrix grain boundaries)
(5) Boride formers (Cr and Mo form borides at γ-matrix grain boundaries)
(6) Grain boundary segregants (C, B, Zr)
(7) TCP phase formers (excessive additions of refractory additions such as W and
Re, as well as Cr, lead to the formation of these highly deleterious phases)
This work focuses exclusively on the γ → γ’ phase transformation in model Ni-Al-Cr-X
alloys, thus emphasis will be placed on these two phases, which provide the strengthening for
nickel alloys used in the most extreme environments, such as the hot sections of gas turbine
engines.
The binary Ni-Al phase diagram exhibits a number of solid phases, all of which show a
significant degree of directional covalent bonding, leading to precise stoichiometric
relationships, and crystal structures in which Ni-Al bonds are preferred, rather than Al-Al or Ni-
Ni bonds [3]. The Ni3Al γ’-phase has the primitive cubic L12 crystal structure, where Ni and Al
have distinct site occupancies, Figure 2.1. The L12 crystal structure can accommodate substantial
solute additions [3], and the mechanical properties of the γ’-phase depend on the sublattice site
substitution behavior of these alloying additions [80]. Considerable experimental and theoretical
work has demonstrated that Al, Ta, Ti, Nb prefer to occupy the Al sublattice sites, and Co, Mo,
Re, Ru, Pt prefer the Ni sublattice sites, while Cr, Fe and Mn have been shown to display mixed
behavior, depending on the alloy composition and temperature of interest [3].
30
(a) (b)
Figure 2.1. Arrangements of Ni (green) and Al (red) in the (a) disordered γ-matrix and (b) the ordered γ’-Ni3Al, L12, crystal structures. (Image courtesy of Dr. Y. Amouyal, Northwestern University)
The γ/γ’ interface is of critical importance in nickel-based superalloys. Interfacial
coherency is critical in order to minimize γ’-precipitate coarsening. As such, the lattice parameter
misfit between the γ-matrix and γ’-precipitate phases must be controlled to prevent a loss of
coherency. The morphology of the γ’-precipitate phase has been demonstrated to be a function of
the Gibbs interfacial free energy and the elastic energy, which is directly related to the lattice
parameter misfit. The sign of the γ/γ' lattice parameter mismatch is important for the elevated
temperature creep strength; a mismatch of zero improves creep properties by preventing the
evolution of a γ’-rafted morphology [81, 82]. Nuclei of the γ’-precipitate phase have a spherical
morphology to minimize the elastic free energy, however as particle growth occurs, {001}-
faceted cuboids develop, which then form arrays, and eventually solid-state dendrites, as
coarsening proceeds [83]. The crystallographic orientation of the two phases corresponds to the
cube-on-cube relationship, that is, {100}precipitate//{100}matrix, and ⟨010⟩precipitate//⟨010⟩matrix.
31
2.1.2 Strengthening and deformation mechanisms in nickel-based superalloys
To achieve optimal yield and creep strengths for turbine engine components, modern
commercial nickel-base superalloys contain a high volume fraction, typically 70 %, of coherent
γ'-precipitates having an average edge length of the order of 500 nm [3, 7, 84]. This leads to a
microstructure of closely spaced cuboidal γ'-precipitates aligned along the <001>-type directions,
and separated by narrow, ca. 60 nm, γ-channels. When the applied load is small, plastic
deformation is generally confined to the γ-matrix channels. At higher loads, precipitate-cutting
mechanisms, which entail deformation in both the matrix and precipitate phases, become
dominant. If the temperature is sufficiently high, dislocations can bypass the precipitates by
thermally-activated climb processes.
The high-temperature strength of nickel-based superalloys is a result of the deformation
mechanisms operating in the γ/γ’ system. The slip system of the FCC γ-matrix is a2
11 0 111{ },
and glide occurs by the passage of two Shockley partial dislocations. The dissociation reaction is
of the form:
a2
110 1 11{ }→a6
211 1 11{ }+a6
121 1 11{ } (1.1)
The magnitudes of the Burgers vectors of the Shockley partials are a / 6 , equivalent to the
displacement of neighboring planes in the FCC structure. As such, the passage of the first
Shockley partial dislocation results in an intrinsic stacking fault, which is removed by the
passage of the second Shockley partial. Slip deformation in the γ’-precipitate phase occurs along
the {111} planes, however the preferred slip system at low temperatures is a 11 0 111{ }, thus
32
the passage of a a2
11 0 111{ } dislocation from the γ-matrix into the γ’-precipitates results in the
formation of an anti-phase boundary (APB). The energy of an APB is substantial, typically on
the order of 100 mJ m-2, although the energy is anisotropic, depending on the plane on which the
APB resides [3, 85, 86]. Therefore, a2
11 0 111{ } dislocations travel in pairs, known as
superdislocations, linked by an APB [87-91]. Dislocations in the γ’-L12 structure dissociate into
partial dislocations just as they do in the γ-matrix phase, though the Burgers vectors of the
dislocations are expected to be longer as a result of ordering in the crystal structure, and the
dissociation reactions are quite complex [3, 92]. Cutting of the γ/γ’ microstructure thus requires a
pair of dislocations, and the formation of an APB, leading to significant particle strengthening,
and hence an increase in the static tensile properties.
Upon deformation, the applied stress, the anisotropy of the APB energy, and
contributions of the elastic anisotropy combine to promote cross-slip of the superpartial
dislocations from the {111} to the {010} planes of the γ’-phase, where the APB energy is lowest.
These cross-slipped segments are then locked in place, as they cannot move without producing a
trail of APBs [93]. Since a component of the cross-slip process is thermally activated, the yield
strength of many nickel-based superalloys increases with increasing temperature, up to ca. 1073
K. For alloy CMSX-10, a single-crystal alloy used in turbine engines, for example, the yield
strength increases from a value of ca. 1000 MPa at a temperature of 473 K, to a peak value of ca.
1200 MPa at 1073 K [3].
Creep strengthening in nickel-based superalloys is a result of both solid-solution
strengthening and precipitation hardening. Because of its electronic structure, nickel has a large
33
solid-solubility for many other elements, and thus there are many possibilities for solid-solution
strengthening [3, 94]. Studies of the effects of solid-solution strengtheners on the creep
resistance of nickel-based superalloys have determined a strong correlation between the
percentage difference in atomic size between the solute and solvent (Ni) atoms, and the creep-
strengthening increment [95]. Since a large-range of solid solubility is achieved only when the
atomic size of the metals are similar in size, a trade-off between solid solubility and lattice
distortion is mandated for effective solid-solution strengthening. High melting point elements,
such as Mo, Re, Ru, Nb and W, can provide strong lattice cohesion and reduce diffusion,
particularly at high temperatures, and are effective solid-solution strengtheners in nickel-base
superalloys [3, 7, 84]. Atomic clustering and short-range order can also strengthen the matrix,
but the concentration of refractory elements must be such that the deleterious TCP phases are not
formed. Substantial improvements in the creep resistance are achieved by the precipitation of the
γ’-phase, particularly when the grain size is large [96-98]. The prominent form of creep damage
in polycrystalline nickel-based alloys is creep cavitation at γ-grain boundaries, which is
minimized by increasing the grain size, hence the use of single-crystal superalloys for
applications that require creep resistance [3]. In current manufacturing processes, it is common
to cast engine components, such as nozzle guide vanes and turbine blades as single crystals, thus
eliminating the potential for grain boundary embrittlement and the need for grain boundary
strengtheners such as C, B, and Zr.
34
2.2 Atom-probe tomography for characterizing nanostructures
The temporal evolution of the γ/γ’ microstructure in this thesis is studied from its earliest
stages with subnanometer spatial resolution employing pulsed-voltage and pulsed-laser atom-
probe tomography [99-104]. The atom-probe technique yields the position and chemical identity
of individual atoms in direct lattice space with a lateral spatial resolution of 0.3 to 0.5 nm, and a
depth resolution for composition analyses that may be less than 0.1 nm [25, 105]. The detection
sensitivity is identical for all elements in the periodic table; light elements (H, B, C, N, O) are
detected with the same sensitivity as Ta, Ru, W, Re. Rectangular parallelepipeds with a
maximum cross-sectional area of ca. 200 x 200 nm2 and a length approaching several microns
are obtained and generate 3-D atom-by-atom reconstructions that can contain 100-600 million
atoms. From the 3-D reconstruction, we obtain local compositions and concentration profiles of
each element, precipitate dimensions and morphology, number density, spatial dispersion,
volume fraction, size distributions, and edge-to-edge interprecipitate spacing distributions [106].
Additionally, the Gibbsian interfacial excesses of solutes at the γ/γ’ interfaces [50, 107-111], and
the partitioning of all elements between the γ- and γ’-phases, can be measured. These results, in
concert with simulations employing advanced computational methods such as first-principles
calculations, lattice kinetic Monte-Carlo simulation, and thermodynamic simulations, elucidate
the kinetic pathways that lead to phase decomposition at high-temperatures. The development of
future generations of nickel-based superalloys that are able to withstand higher engine operating
temperatures will rely on a detailed understanding of the γ/γ' phase transformation. These
complex multi-component alloys will serve as the building blocks for advanced turbine engines
that will require less fuel, and produce fewer CO2 greenhouse gas emissions.
35
Chapter 3
Effects of solute concentrations on kinetic pathways in Ni-Al-Cr alloys
Abstract
The kinetic pathways resulting from the formation of coherent γ’-precipitates from the γ-
matrix are studied for two Ni-Al-Cr alloys with similar γ’-precipitate volume fractions at 873 K.
The details of the phase decompositions of Ni-7.5 Al-8.5 Cr at.%, and Ni-5.2 Al-14.2 Cr at.%,
for aging times from 1/6 to 1024 h are investigated by atom-probe tomography, and are found to
differ significantly from a mean-field description of coarsening. The morphologies of the γ’-
precipitates of the alloys are similar, though the degrees of γ’-precipitate coagulation and
coalescence differ. Quantification within the framework of classical nucleation theory reveals
that differences in the chemical driving forces for phase decomposition result in differences in
the nucleation behavior of the two alloys. The temporal evolution of the γ’-precipitate average
radii and the γ-matrix supersaturations follow the predictions of classical coarsening models. The
compositional trajectories of the γ-matrix phases of the alloys are found to follow approximately
the equilibrium tie-lines, while the trajectories of the γ’-precipitates do not, resulting in
significant differences in the partitioning ratios of the solute elements.
36
3.1 Introduction
Τhe research of Schmuck et al. [28, 112] and Pareige et al. [37, 38] combined atom-probe
tomography and lattice kinetic Monte Carlo (LKMC) simulation to analyze the decomposition of
a Ni-Al-Cr solid-solution. A similar approach was applied by Sudbrack et al. [42, 45, 46, 55, 56]
and Yoon et al. [51, 58, 59] for studying Ni-5.2 Al-14.2 Cr at.% aged at 873 K and Ni–10 Al-8.5
Cr at.%, Ni–10 Al-8.5 Cr-2.0 W at.% and Ni–10 Al-8.5 Cr-2.0 Re at.% aged at 1073 K, which
decompose via a first-order phase transformation to form a high number density, Nv(t), of
nanometer-sized γ’-precipitates (~1020-1025 m-3). The addition of Cr to the binary Ni-Al system
reduces the lattice parameter misfit between the γ’-Ni3(AlxCr1-x)-precipitates and the γ-matrix,
often leading to γ’-precipitates that are nearly misfit free [12], thereby allowing the γ’-
precipitates to remain spheroidal to fairly large dimensions as aging progresses [17]. At 873 K,
γ’-precipitate radii as large as ~10 nm are measured by APT for concentrated Ni-Al-Cr alloys
[42]. The formation of coherent, spheroidal, γ’-precipitates with relatively stress-free
precipitate/matrix heterophase interfaces in the model Ni-Al-Cr alloys studied herein makes
them excellent candidates for comparison of experimental data with the predictions of classical
treatments of nucleation, growth and coarsening for ternary alloys, for which there is very little
detailed quantitative data available.
The present investigation focuses on comparing the temporal evolution of the
nanostructural and compositional properties at 873 K of alloy (A), Ni-7.5 Al-8.5 Cr at.%, with
those previously reported for alloy (B), Ni-5.2 Al-14.2 Cr at.% [42, 45, 46]; all concentrations
herein are in atomic percent (at.%) unless otherwise noted. A ternary Ni-Al-Cr phase diagram
determined by the Grand Canonical Monte Carlo (GCMC) technique at 873 K, Figure 3.1 [113],
37
predicts that the values of φeq are 17.5 ± 0.5 and 15.1 ± 0.5 for alloys (A) and (B), respectively.
Since the predicted equilibrium volume fractions of the γ’-phase, φeq, for alloys (A) and (B) are
similar, it follows that any differences observed in the kinetic pathways during decomposition
are principally due to differences in solute concentrations.
Figure 3.1. A partial ternary phase diagram of the Ni-Al-Cr system at 873 K calculated using the Grand Canonical Monte Carlo simulation technique [113], showing the proximity of both alloys (A) Ni-7.5 Al- 8.5 Cr and (B) Ni-5.2 Al-14.2 Cr to the (γ + γ’) / γ solvus line. The tie-lines are drawn through the nominal compositions of the alloys and between the equilibrium phase compositions determined by extrapolation of APT concentration data to infinite time. Equilibrium solvus curves determined by Thermo-Calc [114], using databases for nickel-based superalloys due to Saunders [115] and Dupin et al. [116], are superimposed on the GCMC phase diagram for comparative purposes.
38
3.1.1 Classical nucleation theory
The early stages of phase decomposition by nucleation have been studied theoretically in
a set of models known as classical nucleation theory (CNT), which have been reviewed
extensively in the literature [40, 117-120]. According to CNT, nucleation is governed by a
balance between a bulk free energy term, which has both chemical, ΔFch, and elastic strain
energy, ΔFel, components, and an interfacial free energy term, σγ/γ’, associated with the formation
of a γ-matrix/γ’-precipitate heterophase interface; F is the Helmholtz free energy. Thus, the
expression for the net reversible work required for the formation of a spherical nucleus, WR, as a
function of nucleus radius, R, is given by:
'/23 43
4)( γγσππ RRFFW elchR +Δ+Δ= . (3.1)
According to CNT, the net reversible work acts as a nucleation barrier that nuclei must
surmount to achieve a critical nucleus radius, R*. The critical net reversible work, *RW , required
for the formation of a critical spherical nucleus is expressed as:
2
3*
)(316 '/
elchR FF
WΔ+Δ
=γγσπ ; (3.2)
and R* is given by:
)(2*
'/
elch FFR
Δ+Δ−=
γγσ . (3.3)
For nucleation to occur, (ΔFch + ΔFel) must be negative. From CNT, the stationary-state
nucleation current, Jst, which is the number of nuclei formed per unit time per unit volume, is
given by:
39
)exp(**
0 TkWNZJB
Rst −= β ; (3.4)
where Z, the Zeldovich factor, accounts for the dissolution of supercritical clusters, β∗ is a
kinetic coefficient describing the rate of condensation of single atoms on the critical nuclei, N0 is
the total number of possible nucleation sites per unit volume, taken to be the volume density of
lattice sites occupied by Al, the precipitate-forming solute [121, 122], kB is Boltzmann’s constant
and T is the absolute temperature in degrees Kelvin. The standard definitions of Z and β∗ were
employed and are given by [40, 117-120, 122]:
2/1
b2
i2
Tk21
i*)i(W
Z⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−=
π, (3.5)
40
aDC*R4* π
β = ; (3.6)
where Wi is the net reversible work required for the formation of a spherical nucleus containing i
atoms, D is the diffusion coefficient of the precipitating solute element, C0 is the initial
concentration of said solute, and a is the average of the lattice parameters of the matrix and
precipitate phase.
In their review of nucleation kinetics results for binary alloys, Aaronson and Legoues
[123] note that while there exists some experimental evidence to support the correctness of CNT,
the nucleation currents predicted by the extant theories are often several orders of magnitude
smaller than those measured experimentally, which is most likely due to the presence of
precursor clusters that form between the solutionizing, and quenching and aging treatments. Such
precursor clustering has been detected in both alloys (A) [45] and (B), and thus, it is anticipated
that our calculated values of Jst will be smaller than those measured by APT.
40
3.1.2 Coarsening theory
The first comprehensive mean-field treatment of Ostwald ripening [124], due to Lifshitz
and Slyozov [125] and Wagner [126], known as the LSW model, is limited to dilute binary
alloys with spatially-fixed spherical precipitates whose initial compositions are equal to their
equilibrium values. The LSW model for a binary alloy assumes: (i) no elastic interactions among
precipitates, thereby limiting the precipitate volume fraction to zero; (ii) precipitates have a
spherical morphology; (iii) coarsening occurs in a stress-free matrix; (iv) the precipitate diffusion
fields do not overlap; (v) dilute solid-solution theory obtains; (vi) the linearized version of the
Gibbs–Thomson equation is valid; (vii) coarsening occurs by the evaporation-condensation
mechanism; and (viii) precipitates coarsen with a fixed chemical composition, which is the
equilibrium composition. These requirements are highly restrictive and difficult to meet in
practice, and while experimental evidence exists to support the prediction of the time
dependency of the mean precipitate radius, <R(t)>, experimentalists have been unable to achieve
the exact stationary-state precipitate size distributions (PSDs) predicted by the LSW model [9,
10, 127]. Researchers have worked to remove the mean-field restrictions by developing models
based on multiparticle diffusion that are able to describe stress-free systems with finite volume
fractions [10, 128, 129].
Umantsev and Olson (UO) [130] were the first to demonstrate that the exponents of the
temporal power-laws predicted for binary alloys by LSW-type models are identical for
concentrated multi-component alloys, but that the explicit expressions for the rate constants
depend on the number of components. The UO model did not allow the composition of the
precipitates to evolve temporally, which is also true for the Morral and Purdy treatment [131].
41
Kuehmann and Voorhees (KV) [132] considered isothermal quasi-stationary state coarsening in
ternary alloys and developed a model that permits the precipitate composition to evolve, such
that the matrix and precipitate compositions can deviate locally from their equilibrium
thermodynamic values. In the quasi-stationary limit in the KV model, δCi/δ t ≈ 0, the exponents
of the power-law temporal dependencies for <R(t)>, Nv(t), and the γ-matrix
supersaturation, )(tCiγΔ , of each solute species i, are:
);()()( 03
03 ttKtRtR KV −=><−>< (3.7)
);tt(K74.4
)t(N)t(N 0eqKV1
0v1
v −≅− −−
φ and (3.8)
;)()()( 3/1,
,, −=∞−>=<Δ tCtCtC KVieq
iff
iiγγγγ κ (3.9)
where KKV, and γκKVi ,
are the coarsening rate constants for <R(t)> and )(tCiγΔ , respectively;
<R(t0)> is the average precipitate radius and Nv(t0) is the precipitate number density at the onset
of quasi-stationary coarsening, at time t0. The quantity )(tCiγΔ is denoted a supersaturation and
is the difference between the concentration in the far-field γ-matrix, >< )(, tC ffiγ , and the
equilibrium γ-matrix solute-solubility, )(, ∞eqiC γ . The quantity )(, ∞eq
iC γ needs to be calculated
or determined experimentally as it is not available for Ni-Al-Cr multicomponent alloys at 873 K.
Atom-probe tomography of the nanostructures of alloys (A) and (B) provides an in-depth
examination of the compositional and nanostructural evolution of the γ’-precipitate phase as it
evolves toward its equilibrium thermodynamic composition. In this chapter, the decomposition
of the γ-matrix phase, from the earliest stages of solute-rich γ’-nuclei formation, to the
subsequent growth and coarsening of γ’-precipitates, is accessed within the framework of
42
classical nucleation, growth, and coarsening theories. The effects of varying the solute
concentrations on the temporal evolution of the Ni-Al-Cr alloys are shown to provide a
quantitative understanding of the kinetic pathways that lead to phase separation, and the
achievement of the equilibrium compositions of both phases. We demonstrate that the kinetic
pathways to achieve thermodynamic equilibrium for the two alloys are very different, even
though the equilibrium volume fractions of the γ’-phase are approximately equal.
3.2 Experimental
High-purity constituent elements (99.97 Ni wt.%, 99.98 Al wt.%, and 99.99 Cr wt.%)
were induction-melted and chill cast in a 19 mm diameter copper mold under an Ar atmosphere.
The overall compositions of the two alloys were determined by inductively coupled plasma
(ICP) atomic-emission spectroscopy, which yielded average atomic compositions of 83.87 Ni-
7.56 Al- 8.56 Cr and 80.52 Ni-5.24 Al-14.24 Cr for alloys (A) and (B), respectively. Chemical
homogeneity of the cast ingots was achieved by annealing at 1573 K in the γ-phase field for 20 h.
Next, the ingots were held in the γ−phase field at 1123 K for 3 h to reduce the concentration of
quenched-in vacancies, and then water quenched and sectioned. Ingot sections were then aged at
873 K under flowing argon for times ranging from 1/6 to 1024 h, then water quenched, and
microtip specimens were prepared from each of the aged sections for study by APT. We
performed voltage-pulsed APT with a conventional APT [133, 134] and an Imago Scientific
Instrument’s local-electrode atom-probe (LEAPTM) tomograph [100-103]. APT data collection
was performed at a specimen temperature of 40.0 ± 0.3 K, a voltage pulse fraction (pulse
voltage/steady-state direct current voltage) of 19%, a pulse frequency of 1.5 kHz (conventional
43
APT) or 200 kHz (LEAPTM tomograph), and a background gauge pressure of < 6.7 x 10-8 Pa.
The average detection rate in the area of analysis ranged from 0.011 to 0.015 ions per pulse for
conventional APT and from 0.04 to 0.08 ions per pulse for the LEAPTM tomographic analyses.
Conventional APT data were visualized and analyzed with APEX a12 software, the successor of
ADAM [135], while LEAPTM data were analyzed employing the IVAS 3.0 software program
(Imago Scientific Instruments, Madison, Wisconsin). The γ/γ' interfaces were delineated with
10.5 at.% and 9 at.% Al isoconcentration surfaces generated with efficient sampling procedures
[136], for alloys (A) and (B), respectively, and in-depth compositional information was obtained
with the proximity histogram method [137]. The volume fraction of the γ’-precipitate phase was
defined as the ratio of the total number of atoms contained within the isoconcentration surfaces
to the total number of atoms collected. Further experimental and analytical details for alloy (B)
can be found elsewhere [42, 46], the same procedures were employed for alloy (A). It is noted
that spatial convolution effects such as ion trajectory overlap and local magnification effects
have been cited as possible sources of misleading results in APT analysis of nickel-based
superalloys [138]. Ion trajectory overlap may cause both interfacial broadening and artificially
small values of the γ’-precipitate Al composition, particularly at early aging times. A comparison
of the composition profiles across the γ/γ’ interfaces measured by APT and simulated by lattice
kinetic Monte Carlo simulations for alloy (B) [39], showed no evidence of artificial interfacial
broadening in the APT data. Additionally, our results indicate that the Al composition is largest
at the shortest aging times due to a supersaturation of Al in the γ’-precipitates, thus trajectory
overlap effects are assumed to be minimal for the APT conditions used herein. Local
magnification effects due to differences in the required evaporation fields of different phases are
44
not present in Ni-Al-Cr superalloys containing only γ-matrix and γ-precipitate phases because the
evaporation fields of the two phases are essentially identical.
The commercial software package Thermo-Calc [139] was used to calculate the values of
φeq, )(, ∞eqiC γ and the equilibrium γ’-precipitate composition of each solute species i, )(,' ∞eq
iC γ ,
for Ni-7.5 Al-8.5 Cr and Ni-5.2 Al-14.2 Cr at a pressure of 1 atmosphere, using databases for
nickel-based superalloys due to Saunders [115] and Dupin et al. [116]. For comparative
purposes, the equilibrium phase boundaries determined by Thermo-Calc using the Saunders and
Dupin et al. databases are superimposed on the GCMC phase diagram at 873 K, Figure 3.1.
While the generated γ / (γ+γ’) solvus lines show good agreement, the curvatures of the (γ+γ’) / γ’
phase lines differ for each technique. Although the Ni-Al-Cr system was studied extensively by
Taylor and Floyd [12], there is no extant experimental phase diagram at 873 K to compare to.
3.3 Results
3.3.1 Morphological development
Nanometer-sized spheroidal γ’-precipitates are detected in both alloys over the full range
of aging times, from 1/6 to 1024 h. The temporal evolution of the morphology of alloy (A) is
shown in a series of APT micrographs in Figure 3.2, which can be compared with a similar series
for alloy (B) in reference [42]. Figure 3.3 is a projection of a 15 x 15 x 25 nm3 subset of an APT
analysis of alloy (A) aged to 1024 h, showing a spheroidal γ’-precipitate of radius ~ 9 nm,
delineated by a red 10.5% Al isoconcentration surface. Atomic planes are clearly visible within
the γ’-precipitate, and the value of the interplanar spacing is 0.26 ± 0.03 nm, suggesting {110}-
type planes. This APT image demonstrates that the γ’-precipitates remain spheroidal for aging
45
times as long as 1024 h at an aging temperature of 873 K, as confirmed by the TEM micrograph
in Figure 3.4.
Figure 3.2. The temporal evolution of the γ’-precipitate nanostructure in Ni-7.5 Al-8.5 Cr aged at 873 K is displayed in a series of APT parallelepipeds. The parallelepipeds are 10 x 10 x 25 nm3
subsets of the analyzed volume and contain ~ 125,000 atoms. The γ/γ’ interfaces are delineated in red with 10.5% Al isoconcentration surfaces.
46
Figure 3.3. A subset of an APT micrograph of Ni-7.5 Al-8.5 Cr aged at 873 K for 1024 h, containing 350,000 atoms, with Al atoms in red, and Cr atoms in blue; Ni atoms are omitted for clarity. A γ’-precipitate with a radius of ~ 9 nm is delineated by the red 10.5% Al isoconcentration surface, and shows {110}-type superlattice planes with an interplanar spacing of 0.26 ± 0.03 nm.
47
Figure 3.4. A centered superlattice reflection dark–field image of spheroidal Ni3(AlxCr(1-x)) γ’-precipitates for a Ni-7.5 Al-8.5 Cr sample aged for 1024 h at 873 K, recorded near the [011] zone axis.
There is evidence of γ’-precipitate coagulation and coalescence in alloys (A) and (B),
characterized by the formation of necks that interconnect the γ’-precipitates, and exhibit L12-type
ordering. Although the first γ’-nuclei are detected by APT at an aging time of 1/6 h,
interconnected γ’-precipitates first evolve in both alloys after 1/4 h of aging, which coincides
with the end of the quasi-stationary-state nucleation regime. After 1/4 h of aging, the fraction of
coagulating and coalescing γ’-precipitates, f, is 15 ± 4% and 9 ± 3%, for alloys (A) and (B),
respectively. From Figure 3.5, the maximum value of f of 18 ± 4% for alloy (A) occurs at an
aging time of 1 h, while the maximum value for alloy (B) of 30 ± 4% occurs at an aging time of
4 h.
48
Figure 3.5. The temporal evolution of the value of the fraction of γ’-precipitates interconnected by necks, f, and the average interprecipitate edge-to-edge spacing, <λe-e>. The maximum value of f of 18 ± 4 % corresponds to the minimum value of <λe-e> of 7 ± 2 nm at an aging time of 1 h for alloy (A) Ni-7.5 Al-8.5 Cr. For alloy (B) Ni-5.2 Al-14.2 Cr, the minimum value of <λe-e> of 5.9 ± 0.8 nm and the maximum value of f of 30 ± 4 % coincide at an aging time of 4 h.
3.3.2 Temporal evolution of the nanostructural properties of γ’-precipitates
Figure 3.6 provides a quantitative description of the temporal evolution of the γ’-
precipitate volume fraction, φ, and the quantities <R(t)> and Nv(t), for alloys (A) and (B). The γ’-
49
precipitate nanostructural properties determined by APT analysis for alloy (A) are summarized in
Table 3.1, while reference [42] contains details for alloy (B). The standard errors for all
quantities, σ, are calculated based on counting statistics and reconstruction scaling errors using
standard error propagation methods [140], and represent one standard deviation from the mean.
The two Ni-Al-Cr alloys were designed to have similar values of φeq of approximately 16%, and
the measured values of φ for alloy (A) and (B) are statistically indistinguishable over the full
range of aging times, from 1/6 to 1024 h. It is worth noting that over the range of aging times
studied, the values of <R(t)> of alloy (A) are, on average, 32 ± 6% larger than those of alloy (B).
From Figure 3.6, the temporal evolution of alloys (A) and (B) can be divided into three regimes:
(i) quasi-stationary-state γ’-precipitate nucleation at early aging times; followed by (ii)
concomitant precipitate nucleation and growth, and finally (iii) concurrent growth and
coarsening once the maximum value of Nv(t) is achieved. The characteristic microstructural
evolution associated with these three regimes is described in detail below.
50
Figure 3.6. The temporal evolution of the γ’-precipitate volume fraction, φ, number density, Nv(t), and mean radius, <R(t)>, for (A) Ni-7.5 Al-8.5 Cr and (B) Ni-5.2 Al-14.2 Cr alloys aged at 873 K as determined by APT microscopy. The quantity <R(t)> is proportional to t1/3 as predicted by the Umantsev and Olson (UO) and Kuehmann and Voorhees (KV) models for isothermal coarsening in ternary alloys. The temporal dependence of the diminution of the quantity Nv(t) deviates from the t-1 prediction of the UO and KV models for both alloys.
51
Table 3.1. Temporal evolution of the nanostructural properties of γ’-precipitates a determined by APT for Ni-7.5 Al-8.5 Cr aged at 873 K.
Aging time (h)
Npptb <R(t)> ± σ
(nm) Nv(t) ± σ
(x 1024 m-3) φ ± σ (%)
f ± σ (%)
<λe-e> ± σ (nm)
1/6 8 0.90 ± 0.32 0.26 ± 0.14 0.31 ± 0.11 ND c 17.7 ± 6.3
1/4 101 1.00 ± 0.11 1.89 ± 0.59 1.36 ± 0.14 15 ± 4 7.7 ± 2.7
1 70.5 1.24 ± 0.12 2.21 ± 0.64 2.48 ± 0.25 18 ± 4 7.0 ± 2.5
4 46 1.70 ± 0.25 1.02 ± 0.31 5.98 ± 0.88 16 ± 3 8.9 ± 3.2
16 42 2.80 ± 0.43 0.60 ± 0.18 9.12 ± 1.4 13 ± 3 9.1 ± 3.2
64 76.5 3.59 ± 0.41 0.34 ± 0.10 11.8 ± 1.4 7.4 ± 1.9 10.6 ± 3.7
256 15 5.54 ± 1.43 0.20 ± 0.05 14.6 ± 3.8 2.5 ± 0.9 10.1 ± 3.6
1024 8 8.30 ± 2.93 0.13 ± 0.05 16.0 ± 5.7 ND c 7.9 ± 2.8
a Mean radius of γ’-precipitates, <R(t)>; the number density, Nv(t); precipitated volume fraction, φ; fraction of γ’-precipitates interconnected by necks, f; average edge-to-edge interprecipitate spacing, <λe-e>; and their standard errors, σ, one standard deviation is reported.
b The number of γ’-precipitates analyzed, Nppt, is smaller than the total number of γ’-precipitates detected by APT. Precipitates that intersect the sample volume contribute 0.5 to the quantity Nppt, and are included in the estimates of Nv(t), φ, f, and the phase compositions, and not in the measurement of <R(t)>.
c ND = not detected.
3.3.2.1 Nucleation and growth of γ’-precipitates
Phase decomposition in alloys (A) and (B) begins with the formation of solute-rich nuclei
that grow to form stable γ’-precipitates. Precipitates are first detected by APT in alloys (A) and
(B) after 1/6 h of aging. Alloy (A) forms (2.6 ± 1.4) x 1023 γ’-precipitates m-3 with an <R(t = 1/6
h)> value of 0.90 ± 0.32 nm, corresponding to a φ value of 0.31±0.11 %, after 1/6 h of aging. For
the same aging time, Alloy (B) forms (3.6 ± 1.3) x 1023 γ’-precipitates m-3 with an <R(t = 1/6
52
h)> value of 0.74 ± 0.24 nm, accounting for a φ value of 0.11±0.04 %. The nucleation of stable
γ’-precipitates in alloy (A) for aging times between 1/6 and 1/4 h results in a sharp linear slope of
the Nv(t) profile of (5.4 ± 1.5) x 1021 m-3 s-1, while the values of <R(t)> of 0.90 ± 0.32 nm and
1.00 ± 0.11 nm, remain constant within statistical error. This slope for Nv(t), although based on
two experimental data points, is taken to be an estimate of the quasi-stationary-state nucleation
current of γ’-precipitates, Jst. In contrast, alloy (B) undergoes nucleation of stable γ’-precipitates
with a constant <R(t)> value of 0.75 ± 0.24 nm for aging times less than 1/4 h. The value of Jst
for alloy (B) is estimated to be (5.9 ± 1.7) x 1021 m-3 s-1, which is statistically indistinguishable
from the value of Jst of (5.4 ± 1.5) x 1021 m-3 s-1
measured for alloy (A).
At an aging time of 1/4 h, alloys (A) and (B) enter a regime of concomitant γ’-precipitate
nucleation and growth, which results in steadily increasing φ and <R(t)> values, and a maximum
value for Nv(t). The peak value in Nv(t) of (2.21 ± 0.64) x 1024 m-3 for alloy (A) occurs at 1 h,
while for alloy (B), the peak value in Nv(t) of (3.2 ± 0.6) x 1024 m-3 is achieved at 4 h. We note
that the prediction of the temporal dependence of <R(t)> of t1/2 for interface reaction controlled
growth [141-143], is observed in neither of the Ni-Al-Cr alloys studied, contrary to the results
for an earlier APT study of an alloy whose composition is essentially identical to that of alloy
(B) [28].
3.3.2.2 Growth and coarsening of γ’-precipitates
Beyond the peak in Nv(t), alloys (A) and (B) enter a quasi-stationary-state of growth and
coarsening characterized by a steady diminution of Nv(t) and increasing values of φ and <R(t)>.
Beyond 1 h, the quantity Nv(t) for alloy (A) displays a temporal dependence of t-0.42 ± 0.03, which
53
differs significantly from the predicted value of t-1 from the UO and KV models. This is not
surprising, since the quantity φ continues to evolve temporally in this regime, implying that the
system has not achieved a stationary-state [144]. During coarsening in alloy (B), the quantity
Nv(t) displays a temporal dependence of t-0.67 ± 0.01 beyond an aging time of 4 h, which also differs
from the predicted value of t-1, though to a lesser extent. The values of φ for the two alloys
increase steadily in this regime, reaching values of φ of 16.0 ± 5.7 % and 15.6 ± 6.4 %, for alloys
(A) and (B), respectively, which are statistically indistinguishable. In the growth and coarsening
regime, the quantity <R(t)> displays a temporal dependence of t0.29 ± 0.03 for alloy (A) and t0.29 ±
0.05 for alloy (B), which both agree approximately, but not exactly, with the predicted value of
t1/3, indicating that the phase transformation is primarily diffusion-limited.
From Figure 3.5, the maximum value for f, after 1 h of aging, of 18 ± 4% for alloy (A)
coincides with the minimum value of the average edge-to-edge interprecipitate spacing, <λe-e>,
of 7.0 ± 2.5 nm and the peak value of Nv(t) of (2.21 ± 0.64) x 1024 m-3. For alloy (B), the peak
value of Nv(t), at an aging time of 4 h, is (3.2 ± 0.6) x 1024 m-3, corresponding to a minimum
value of <λe-e> of 5.9 ± 0.8 nm at a maximum value of f of 30 ± 4%. The quantity <λe-e> is
calculated using Equation 3.10, which assumes (i) a regular simple cubic array of γ’-precipitates;
and (ii) the unit volume is equivalent to that of a lattice enclosed sphere in order to preserve the
radial symmetry of the 3D structure [42]:
⎥⎥⎦
⎤
⎢⎢⎣
⎡><−⎟
⎠⎞
⎜⎝⎛ ⋅>=<
−
− )()(342
3/1
tRtNvee πλ . (3.10)
54
3.3.3 Temporal evolution of the compositions of the γ and γ’-phases
The compositions of the γ-matrix and the γ’-precipitate phases of alloys (A) and (B)
evolve temporally as the γ-matrix becomes enriched in Ni and Cr and depleted in Al, as shown in
Figure 3.7 for alloy (A).
Figure 3.7. The composition profiles on either side of the heterophase γ-matrix / γ’-precipitate interface for (A) Ni-7.5 Al-8.5 Cr aged at 873 K for aging times of 1/4, 1, 4, 1024 h. The phase compositions evolve temporally, as the γ-matrix becomes enriched in Ni and Cr and depleted in Al. The values of <R(t)> for these aging times are 1.00 ± 0.11 nm for 1/4 h, 1.24 ± 0.12 nm for 1 h, 1.70 ± 0.25 nm for 4 h and 8.30 ± 2.93 nm for 1024 h.
55
The compositional trajectories of the γ-matrix and γ’-precipitate phases are shown on a
partial Ni-Al-Cr ternary phase diagram at 873 K in Figure 3.8. The mean-field KV model of
quasi-stationary state coarsening predicts that the slope of the trajectory of the γ-matrix phase
during coarsening lies along the equilibrium tie-line and has a value of pAl/pcr, where pi is the
magnitude of the partitioning of solute species i, defined as pi = ,[ ( )eqiCγ ′ ∞ - , ( )]eq
iCγ ∞ . For a binary
alloy, the compositions at the interface are given by the assumption of local equilibrium,
whereas for a ternary alloy, the interfacial compositions must be further defined by the condition
of flux balance at the interface [128]. As such, according to the KV model, the addition of a third
alloying element alters both the form of the Gibbs-Thompson equations and the predictions of
the temporal evolution of the phase compositions. The compositional trajectory of the γ’-
precipitate phase is predicted to lie on a straight line that is not necessarily parallel to the
equilibrium tie-line, which is the case for both alloys (A) and (B). In order to quantify deviations
from the equilibrium tie-line, the quantities )(/)( tCtC CrAlγγ δδ and )(/)( '' tCtC CrAl
γγ δδ for aging times
of 4-1024 h are compared to the values of pAl/pcr for both alloys, where δCpi is the slope of the
concentration of element i in phase p with aging time. The trajectories of the γ-matrix phase and
the γ’-precipitate phase compositions of alloy (A) have slopes of – 2.33 ± 0.06 and 5.71 ± 1.42,
respectively, while the slope of the equilibrium tie-line is estimated to be -3.52 ± 0.09. Slopes of
-1.36 ± 0.24 and 0.85 ± 0.20 are estimated for the γ-matrix and γ’-precipitate phases,
respectively, from the APT data for alloy (B), while the slope of the equilibrium tie-line is -1.53
± 0.07. From these results, it is absolutely clear that the trajectories of the composition of the γ’-
precipitate phases in alloys (A) and (B) do not lie along the equilibrium tie line of the respective
alloys, which is contrary to all coarsening models except the KV model. Hence, our results
56
indicate that the temporal evolution of ternary alloys is considerably more complicated than for
binary alloys, and certainly warrants future research.
Figure 3.8. The compositional trajectories of the γ-matrix and γ’-precipitate phases of alloys (A) Ni-7.5 Al-8.5 Cr and (B) Ni-5.2 Al-14.2 Cr as they evolve temporally, displayed on a partial Ni-Al-Cr ternary phase diagram at 873 K. The tie-lines are drawn through the nominal compositions of the alloys (squares) and between the experimentally determined equilibrium phase compositions. The trajectories of the γ-matrix phases in alloys (A) and (B) lie approximately on the experimental tie-lines. The trajectories of the γ'-precipitate phases do not lie along the tie-line, as predicted by the Kuehmann and Voorhees model of isothermal coarsening in ternary alloys [132].
During nucleation, solute-rich γ’-nuclei form with large values for the Al and Cr
supersaturations. The first γ’-nuclei detected by APT for alloy (A) have solute-supersaturated
compositions of 70.9 ± 1.4 Ni, 23.3 ± 1.5 Al and 5.8 ± 0.9 Cr at an <R(t = 1/6 h)> value of 0.90
± 0.32 nm for alloy (A). The γ’-nuclei in alloy (B) have a composition of 71.3 ± 1.6 Ni, 19.1 ±
1.4 Al and 9.7 ± 1.1 Cr, and an <R(t = 1/6 h)> value of 0.74 ± 0.24 nm. As phase decomposition
57
progresses beyond nucleation, the magnitude of the values of )(tCiγΔ decrease asymptotically
toward a value of zero as the equilibrium γ- and γ’-phase compositions are approached. The
equilibrium γ-matrix and γ’-precipitate compositions are extrapolated by fitting the measured
concentrations from the quasi-stationary coarsening regime to Equation 3.9 for aging times
beyond 1 h for alloy (A) and 4 h for alloy (B). The equilibrium γ’-precipitate composition of
alloy (A) is estimated to be 76.33 ± 0.12 Ni, 17.82 ± 0.15 Al and 5.85 ± 0.12 Cr, while the
equilibrium γ-matrix has a composition of 85.19 ± 0.08 Ni, 5.42 ± 0.09 Al and 9.39 ± 0.09 Cr at
infinite time. The equilibrium γ’-precipitate phase composition of alloy (B) is estimated to be
76.53 ± 0.25 Ni, 16.69 ± 0.22 Al and 6.77 ± 0.15 Cr, while the equilibrium composition of the γ-
matrix phase is 81.26 ± 0.09 Ni, 3.13 ± 0.04 Al and 15.61 ± 0.09 Cr. Summaries of the
equilibrium phase compositions of alloy (A) determined by APT are presented in Table 3.2, and
agree with results obtained from Thermo-Calc and GCMC simulation. Reference [42] contains a
similar table for alloy (B).
58
Table 3.2. Equilibrium γ’-precipitate and γ-matrix concentrations, as determined by atom-probe tomography (APT), Grand Canonical Monte Carlo (GCMC) simulation, and thermodynamic modeling employing Thermo-Calc for alloy (A) Ni-7.5 Al-8.5 Cr aged at 873 K.
Equilibrium composition of γ'-precipitates Ni (at.%) Al (at.%) Cr (at.%)
Measured by APT at 1024 h: 76.11 ± 0.09 18.02 ± 0.09 5.87 ± 0.05
Extrapolated from APT data: 76.33 ± 0.12 17.82 ± 0.15 5.85 ± 0.12
Modeled by GCMC simulation [113]: 76.3 ± 0.5 17.8 ± 0.5 5.9 ± 0.5
Calculated with Thermo-Calc and Saunders database [115]:
76.40 17.79 5.81
Calculated with Thermo-Calc and Dupin et al. database [116]:
75.44 17.87 6.99
Equilibrium composition of γ-matrix Ni (at.%) Al (at.%) Cr (at.%)
Measured by APT at 1024 h: 85.13 ± 0.06 5.53 ± 0.07 9.34 ± 0.04
Extrapolated from APT data: 85.19 ± 0.08 5.42 ± 0.09 9.39 ± 0.09
Modeled by GCMC simulation [113]: 85.8 ± 0.5 5.2 ± 0.5 9.0 ±0 .5
Calculated with Thermo-Calc and Saunders database [115]:
85.71 5.42 8.86
Calculated with Thermo-Calc and Dupin et al. database [116]:
85.3 5.99 8.70
The partitioning behavior of the elements can be determined quantitatively by calculating
the partitioning ratio, Kiγ’/γ
, defined as ratio of the concentration of an element i in the γ’-
precipitates to the concentration of the same element in the γ-matrix. Figure 3.9 demonstrates
that alloys (A) and (B) both exhibit partitioning of Al to the γ’-precipitates and of Ni and Cr to
the γ-matrix. Partitioning is more pronounced in alloy (B) than (A), as the smaller Al and the
larger Cr concentrations of this alloy result in a smaller solubility of Al, and a larger solubility of
Cr in the γ-matrix, respectively. Nickel partitions to the γ-matrix with a Kiγ’/γ of ~ 0.9 for both
alloys, and is only slightly time dependent.
59
Figure 3.9. The partitioning ratios, Kiγ’/γ
, of Al and Cr demonstrate that both alloys exhibit partitioning of Al to the γ’-precipitates and Cr to the γ-matrix. Partitioning is more pronounced in alloy (B), where the smaller Al and larger Cr concentration results in a smaller value of the γ-matrix Al solubility, and a larger γ-matrix Cr solubility.
The lever rule is applied to the equilibrium phase compositions for both alloys to estimate
a φeq value of 16.4 ± 0.6 % for alloy (A), and 15.7 ± 0.7 % for alloy (B). The φ values of 16.0 ±
5.7 % and 15.6 ± 6.4 % determined by APT at an aging time of 1024 h for alloy (A) and (B), are
within experimental error of the φeq values estimated by the lever rule. The Thermo-Calc
software package yields φeq values of 16.7 % and 14.9 % for alloy (A) according to the Saunders
60
and Dupin et al. databases, respectively, thus the Saunders database yields the value closest to
the experimental value of 16.4 ± 0.6%. Alloy (B) is predicted to achieve φeq values of 12.83%
and 12.34% for the same databases, which are both smaller than the experimentally determined
value of 15.7 ± 0.7%. Summaries of the φeq values determined by APT are presented in Table
3.3, and agree with results obtained from Thermo-Calc and GCMC modeling to different
degrees.
Table 3.3. Equilibrium γ’-precipitate volume fraction, φeq, as determined by atom-probe tomography (APT), Grand Canonical Monte Carlo (GCMC) simulation, and Thermo-Calc for alloy (A) Ni-7.5 Al-8.5 Cr, and alloy (B) Ni-5.2 Al-14.2 Cr, aged at 873 K.
Technique used to estimate φeq (A) Ni-7.5 Al-8.5 Cr (B) Ni-5.2 Al-14.2 Cr
Determined by lever rule calculation with phase compositions measured by APT:
16.4 ± 0.6 15.7 ± 0.7
Measured by APT at 1024 h: 16.0 ± 5.7 15.6 ± 6.4
Modeled by GCMC simulation [113]: 17.5 ± 0.5 15.1 ± 0.5
Calculated with Thermo-Calc and Saunders database [115]:
16.69 12.83
Calculated with Thermo-Calc and Dupin et al. database [116]:
14.90 12.34
The values of the solid-solution supersaturations of alloys (A) and (B) are calculated
based on the equilibrium phase compositions from APT data. The initial solid-solution
supersaturation values are estimated to be 2.08 ± 0.02 Al and -0.89 ± 0.01 Cr, for alloy (A), and
2.09 ± 0.04 Al and -1.46 ± 0.02 Cr, for alloy (B). Figure 3.10 shows the temporal evolution of
the Al and Cr γ-matrix supersaturation values for both alloys. The formation of γ’-nuclei during
the early stages of aging results in a decrease in the magnitude of the values of )(tCiγΔ , which in
61
turn causes the slowing and eventual termination of γ’-precipitate nucleation. Beyond the aging
time corresponding to the peak γ’-precipitate number density, 1 h for alloy (A), and 4 h for alloy
(B), the diminution of the )(tCiγΔ values approximately follow the t-1/3 prediction of the KV
model. From Figure 3.10, alloy (A) demonstrates a temporal dependence of t-0.32 ± 0.03 for
)(tCAlγΔ and t-0.29 ± 0.04 for )(tCCr
γΔ , and alloy (B) exhibits a dependence of t-0.33±0.04 for )(tCAlγΔ
and t-0.34±0.07 for )(tCCrγΔ . The supersaturation values of the γ’-precipitate phases, )(tCi
γ ′Δ , of
alloys (A) and (B) are a reflection of their alloy composition, as the magnitude of )(tCiγ ′Δ is
greater in alloy (A), Ni-7.5 Al-8.5 Cr, which contains more Al, than in alloy (B), Ni-5.2 Al-14.2
Cr, while the inverse is true for Cr. From Figure 3.11, the quantities )(tCAlγ ′Δ and )(tCCr
γ ′Δ for
alloy (A) exhibit temporal dependencies of t-0.34 ± 0.04 and t-0.32 ± 0.05 respectively, which are close
to the predicted value of t-1/3. A dependence of t-0.30 ± 0.05 for )(tCAlγ ′Δ , and t-0.29 ± 0.07 for )(tCCr
γ ′Δ ,
are demonstrated for alloy (B).
62
Figure 3.10. The magnitude of the values of the supersaturations, )(tCiγΔ , of Al and Cr in the γ-
matrix are smaller for alloy (A) Ni-7.5 Al-8.5 Cr than for alloy (B) Ni-5.2 Al-14.2 Cr. The magnitude of the )(tCi
γΔ values decrease as t-1/3 in the coarsening regimes for both alloys, as predicted by the Umantsev and Olson (UO) and Kuehmann and Voorhees (KV) models for isothermal quasi-stationary state coarsening in ternary alloys. The values of )(tCCr
γΔ are expressed as an absolute value because they are negative, and reflect a flux of Cr into the γ-matrix with increasing aging time.
63
Figure 3.11. The supersaturation values of the γ’-precipitates, )(tCiγ ′Δ , reflect the chemical
compositions of the two alloys, as the value of )(tCAlγ ′Δ is larger in alloy (A) Ni-7.5 Al- 8.5 Cr,
which contains more Al, than in alloy (B) Ni-5.2 Al-14.2 Cr, while the inverse is true for Cr. The values of )(tCi
γ ′Δ decrease as approximately t-1/3 in the coarsening regimes for both alloys.
64
3.4. Discussion
While the temporal evolution of the morphology and the volume fraction of the γ’-
precipitates in alloy (A) Ni-7.5 Al-8.5 Cr and alloy (B) Ni-5.2 Al-14.2 Cr are similar, the
nanostructural and compositional results for the two alloys exhibit significant differences. These
differences can be attributed to the effects of solute concentrations on the kinetic pathways
involved in phase decomposition.
3.4.1 Effects of solute concentration on nucleation behavior
The nucleation behavior observed by APT for alloys (A) and (B) can be compared with
the predictions of CNT using Equations 3.2-3.4, given the values of σγ/γ’, ΔFch, and ΔFel for both
alloys. The values of σγ/γ’ are estimated as first shown by Ardell for a binary alloy [145, 146],
and later for a ternary alloy by Marquis and Seidman [147], and as applied to alloy (B) by
Sudbrack et al. [46]. The relationship for σγ/γ’ in a nonideal, nondilute ternary alloy consisting of
a γ-matrix and a γ’-precipitate phase with a finite volume fraction of the γ’-phase is given by
[128]:
)GpGpp2Gp(pV2
)K(Cr,Cr
2CrAl,CrCrAlAl,Al
2Al
im
KVi3/1
KV/ γγγγ
γγγ κ
σ ++= ′′ ; (3.11)
where KKV and γκ KVi, are the rate constants for the quantities <R(t)> and )(tCiγΔ , respectively,
from the KV coarsening model, pi is the magnitude of the partitioning as previously defined,
γ ′mV is the molar volume of the γ’-precipitate phase, calculated to be 6.7584 x 10-6 m3 mol-1,
and γjiG , is shorthand notation for the partial derivatives of the molar Gibbs free-energy of the γ-
65
matrix phase with respect to the solute species i and j. The quantities KKV and γκ KVi, are
determined by fitting the experimental APT data to Equations 3.7 and 3.9 from the KV
coarsening model. For alloy (A), KKV is (1.84 ± 0.43) x 10-31 m3s-1 and γκ KVAl , and γκ KVCr , are 0.18
± 0.05 at. fr. s1/3 and -0.06 ± 0.01 at. fr. s1/3, respectively, while a value of KKV of (8.8 ± 3.3) x 10-
32 m3s-1, and values of γκ KVAl , and γκ KVCr , of 0.19 ± 0.02 at. fr. s1/3 and -0.14 ± 0.05 at. fr. s1/3, are
found for alloy (B). For the general case described by nonideal and nondilute solution theory, the
values of γjiG , may be calculated using Thermo-Calc employing the extant databases for nickel-
based superalloys. As noted by Sudbrack et al. [46], the calculations of γjiG , from Thermo-Calc
predict a more highly-curved free energy surface than ideal solution theory with respect to all
solute species combinations. The Thermo-Calc assessments take into account the excess free-
energies of mixing and the magnitudes of γjiG , are 1.5 to 13 times larger than those for ideal
solution theory for alloys (A) and (B) as shown in Table 3.4 for alloy (A). See Table 3.4 in
Sudbrack et al. [46] for the values of γjiG , for alloy (B). Note that two values of σγ/γ’, '/ γγσ Al and
'/Cr
γγσ , are obtained from the coarsening data, depending on the choice of the solute element used
to estimate γκ KVi, and pi.
66
Table 3.4. Curvatures in the molar Gibbs free-energy surface of the γ-matrix phase evaluated at the equilibrium composition with respect to components i and j, γ
jiG , , obtained from ideal solution theory and Thermo-Calc thermodynamic assessments for alloy (A) Ni-7.5 Al-8.5 Cr aged at 873 K.
γjiG , Ideal Solution Theory
( J mol-1) Saunders database [115]
(J mol-1)Dupin et al. database
[116] (J mol-1)γ
AlAlG 142,441.7 271,392.5 306,697.8 γ
CrCrG ,85,821.1 166,601.4 166,708.5
γCrAlG ,
8,521.4 112,567.3 140,244.9
Fortuitously, the Thermo-Calc and ideal solution theory assessments of γjiG , yield
approximately the same value for σγ/γ’ for alloy (B) of 22-23 ± 7 mJ m-2 [46]. For alloy (A), the
values for σγ/γ’, calculated from the ideal solution theory of 14-16 ± 3 mJ m-2 are smaller than
those from the Thermo-Calc assessment of γjiG , of 23-25 ± 6 mJ m-2, Table 3.5. For the purposes
of CNT, values of σγ/γ’ of 24 ± 6 mJ m-2, for alloy (A), and 22.5 ± 7 mJ m-2, for alloy (B) are
used, the averages of the values generated from the Thermo-Calc assessments of γjiG , .
Additionally, estimates of 36.3 ± 3.8 mJ m-2 and 35.7 ± 1.7 mJ m-2 are obtained for alloys (A)
and (B) at 0 K, from first principles calculations, in the framework of density functional theory
(DFT) and the local density approximation (LDA), employing ultrasoft Vanderbilt potentials
(US-PP), using the VASP (Vienna Ab-initio simulation package) code [148-151]. It is noted that
the values of σγ/γ’ from the first principles calculations are larger than those determined
experimentally because the first principles calculations assume a sharp γ/γ’ interface and are
performed at 0 K, and thus do not include entropic effects. The experimental estimates of σγ/γ’ are
for diffuse γ/γ’ interfaces at 873 K, and are free energies because they include entropic effects.
67
Table 3.5. Free-energy of the γ/γ’ interfaces, σγ/γ’, at 873 K in Ni–7.5 Al–8.5 Cr calculated from the experimental values of the Kuehmann-Voorhees coarsening rate constants for the average precipitate radius and the supersaturation of solute species i employing Equation 3.11 with solution thermodynamics described by the ideal solution and Thermo-Calc databases given in Table 3.4.
Thermodynamic models '/ γγσ Al
(mJ m-2)
'/ γγσ Cr (mJ m-2)
σγ/γ’ (mJ m-2)
Ideal solution theory 14.2 ± 3.1 16.0 ± 3.6 15.1 ± 3.4
Saunders database [115] 21.6 ± 4.0 24.4 ± 6.1 23.0 ± 5.1
Dupin et al. database [116] 23.6 ± 4.9 26.6 ± 6.2 25.1 ± 5.5
A value of σγ/γ’ of 12.5 mJ m-2 was previously determined for a ternary Ni-5.2 Al-14.8 Cr
alloy aged at 873 K by Schmuck et al. [28]. The value of σγ/γ’ for this alloy, which has an overall
composition that is very close to that of alloy (B) Ni-5.2 Al-14.2 Cr, was determined assuming
the LSW model for binary alloys. We reanalyzed their data according to the method developed
by Marquis and Seidman for ternary alloys, and values of 20-21 ± 5 mJ m-2 are estimated
for σγ/γ’, in good agreement with the values of σγ/γ’ estimated for alloys (A) and (B); additional
details are presented in Appendix 1. Gleiter and Hornbogen [152] estimated a value of 13.5 mJ
m-2 for a Ni-5.4 Al-18.7 Cr alloy aged at 750 ºC, though they also applied binary LSW theory to
a ternary alloy, and their results did not include concentration data, making recalculation of their
value of σγ/γ impossible by the method developed by Marquis and Seidman for ternary alloys.
Baldan [9] provides a review of literature values of σγ/γ’ for several different Ni-Al and Ni-Al-Cr
systems over a range of aging temperatures, which need to be evaluated in detail in light of our
work.
The values of the chemical driving force of alloys (A) and (B) are estimated as proposed
68
by Lupis [153] and applied by Schmuck et al. [28], from thermodynamic data taken from
Thermo-Calc [139] and the extant nickel-based superalloy databases [115, 116]. The values of
ΔFch for alloys (A) and (B) are estimated to be -6.6 x 107 J m-3 and -8.2 x 107 J m-3, respectively.
The values of ΔFel for the two alloys are estimated using [154]:
'
2'
)43()(2
γγγ
γγγγ
a
aael VSB
VVBSF+
−=Δ ′
′
; (3.12)
where Sγ is the shear modulus of the γ-matrix phase, Bγ’ is the bulk modulus of the γ’-precipitate
phase, and γaV and 'γ
aV are the atomic volumes of the γ-matrix and γ’-precipitate phases,
respectively. No elastic constants are available for this alloy, therefore the value of Sγ of 100.9
GPa, of a similar alloy, Ni-12.69 Al at 873 K [155] is employed, while the value of Bγ’ is taken
to be 175 GPa [156]. The lattice parameters for the equilibrium phases in alloy (A) at 873 K, are
estimated to be 0.3554 ± 0.0001 nm and 0.3544 ± 0.0001 nm for the γ' and γ-phases,
respectively, based on room-temperature x-ray diffraction measurements on similar Ni-Al-Cr
alloys [12, 17]. These lattice parameter values result in a near-zero estimate of the lattice misfit
of 0.0027 ± 0.0004 for alloy (A). The lattice parameter misfit for alloy (B) is estimated to be
0.0006 ± 0.0004 [42]. Substituting these values into Equation 3.12 yields values of ΔFel of 2.5 x
106 J m-3 for alloy (A) and 1.1 x 105 J m-3 for alloy (B). The larger Cr concentration in alloy (B)
is responsible for a smaller value of the lattice parameter misfit and therefore a smaller elastic
strain energy. The quantity ΔFel is also estimated by a simpler technique due to Eshelby [157],
which yields values of 2.80 x 106 J m-3 and 1.38 x 105 J m-3 for alloys (A) and (B), respectively.
The high degree of coherency of the γ’-precipitates in these alloys is such that the bulk
component of the driving force for nucleation is dominated by the ΔFch term, as ΔFel is only
69
3.9% of the value of ΔFch for alloy (A) and 0.1% for alloy (B). As such, experimentally
determined differences in the nucleation behavior may be described as due primarily, but not
exclusively, to differing values of ΔFch. The tracer diffusivity of Al in the γ-matrix phase of these
Ni-Al-Cr alloys is significantly larger than the diffusivity of Cr. For alloy (A), fccAlD is 9.6 x 10-21
m2s-1 and fccCrD has a value of 2.6 x 10-21 m2s-1, while for alloy (B), fcc
AlD is 11 x 10-21 m2s-1 and
fccCrD is 3.0 x 10-21 m2s-1.
The value of *RW for alloy (A) is estimated from Equation 3.2 to be 35.2 kJ mol-1 or 0.365
eV atom-1, while R* is estimated from Equation 3.3 to be 0.76 nm. The value of *RW for alloy (B)
is found to be 17.0 kJ mol-1 or 0.177 eV atom-1, and *R is calculated to be 0.55 nm. The
nucleation currents for the two alloys are estimated from Equation 3.4 to be 4.0 x 1022 m-3 s-1 for
alloy (A) and 3.2 x 1023 m-3 s-1 for alloy (B). The calculated values of σγ/γ’, ΔFch, ΔFel, *RW , R*,
and Jst for both alloys are summarized in Table 3.6.
Table 3.6. The interfacial free energy, σγ/γ’, the chemical free energy, ΔFch, and the elastic strain energy, ΔFel, components of the driving force for nucleation used to estimate the net reversible work, *
RW , required for the formation of critical nuclei of size, R*, and the nucleation current, Jst, according to classical nucleation theory.
Alloy σγ/γ’ (mJ m-2)
ΔFch (107 J m-3)
ΔFel (106 J m-3)
*RW
(kJ mol-1) R*
(nm) Jst
(m-3 s-1)
(A) Ni-7.5 Al-8.5 Cr 24.0 ± 6 -6.6 2.5 35.2 0.76 4.0 x 1022
(B) Ni-5.2 Al-14.2 Cr 22.5 ± 7 -8.2 0.11 17.0 0.55 3.2 x 1023
70
The predictions of the value of R* from CNT are verified experimentally for both alloys,
as the first nucleating precipitates detected consistently by APT at an aging time of 1/6 h have
<R(t)> values of 0.9 ± 0.32 nm for alloy (A) and 0.74 ± 0.24 nm for alloy (B), which are both
slightly greater than the calculated R* estimates of 0.76 and 0.55 nm, respectively. The predicted
value of Jst of 4.0 x 1022 m-3 s-1 for alloy (A) is nearly an order of magnitude greater than the
experimental value of (5.4 ± 1.5) x 1021 m-3 s-1. The calculated value of Jst for alloy (B) of 3.2 x
1023 m-3s-1 is 50 times greater than the experimentally measured value of Jst of (5.9 ± 1.7) x 1021
m-3 s-1. Given the evidence of precursor clustering in these Ni-Al-Cr alloys [45], it is surprising
that the experimentally determined values of Jst are significantly less than the predicted values.
Xiao and Haasen [16] performed a similar comparison of experimentally determined nucleation
currents with those predicted by CNT for a binary Ni-12.0 Al at.% alloy aged at 773 K and found
that the predicted value of Jst was a factor of 500 larger than the measured value of Jst. They
attributed this discrepancy to the sensitivity of the predicted value of Jst to the value of R*. Xiao
and Haasen also pointed out that predicted nucleation currents are likely an overestimate due to
the assumption that the value of N0 is equal to the volume density of lattice sites, which is a
commonly made assumption that is not necessarily correct. By assuming that the value of N0 is
equal to the volume density of Al atoms, we seem to have mitigated this error to some extent.
Given that the experimentally determined values of Jst are measured from two experimental data
points, and that the detailed kinetics involved in the formation of γ’-nuclei in these ternary
systems are not completely understood, further analysis is presently not instructive. The
nucleation behavior of Ni-Al-Cr alloys certainly warrants future research, given the
technological importance of these systems.
71
The formation of stable, growing nuclei at early aging times causes a decrease in the
values of )(tCiγΔ , and results in a reduction in, and the eventual termination of, nucleation. By
an aging time of 1 h, the values of )(tCiγΔ for alloy (A) decrease from the solid-solution values
of 2.08±0.02 Al and -0.89±0.01 Cr to 1.21±0.02 Al and -0.52±0.04 Cr. This decrease in the
magnitude of the )(tCiγΔ values is reflected in a 50% decrease in the quantity ΔFch to a value of
-3.3 x 107 J m-3, and therefore a decrease in the predicted value of Jst of seven orders of
magnitude, after only 1 h of aging. After 4 h of aging in alloy (B), the values of )(tCiγΔ decrease
from 2.09 ± 0.04 Al and -1.46 ± 0.02 Cr, to 0.84±0.04 Al and -0.59 ± 0.09 Cr, while the value of
ΔFch decreases 51% to -4.01 x 107 J m-3. This decrease in the quantity ΔFch results in a decrease
in the predicted value of Jst of three orders of magnitude for alloy (B). From these estimates, and
our APT nanostructural results, it appears that the larger initial value of the chemical driving
force in alloy (B) sustains a significant nucleation current for aging times as long as 4 h, whereas
the nucleation current decreases significantly in alloy (A) after 1 h.
3.4.2 Effects of solute concentration on the coarsening behavior
As alloys (A) and (B) coarsen, the quantities )(tCiγΔ and )(tCi
γ ′Δ evolve temporally as
approximately t-1/3, and <R(t)> grows as approximately t1/3, as predicted by the UO and KV
models. The experimentally measured temporal dependencies of Nv(t) of t-0.42 ± 0.03. and t-0.67 ± 0.01,
for alloys (A) and (B), respectively, differ significantly, however, from the t-1 prediction of both
models. The magnitudes of the quantities )(tCiγΔ for both alloys remain non-zero at an aging
time of 1024 h, thus neither alloy has achieved a true stationary-state, which is a basic
72
assumption of classical LSW Ostwald ripening behavior. The UO and KV models, however,
assume a quasi-stationary state, which may have been achieved for alloys (A) and (B). At 1024
h, the value of φ, Figure 3.6, is still evolving temporally.
When Ostwald ripening is considered to be a diffusion-limited process, it is limited by
the characteristic length over which diffusion can occur, taken as the average edge-to-edge
interprecipitate spacing in these systems. It is important to note that as Nv(t) decreases with
increasing time, <R(t)> increases and the value of <λe-e> increases concomitantly. The time
required to reach stationary-state coarsening, tc, may be estimated employing [158]:
3
266
2
3 1172964
DKt KV
c ⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛≈
φωω ; (3.13)
where KKV is the coarsening rate constant for <R(t)> as defined in section 1.2, ωi is the ith
moment of the precipitate size distribution, ω2 = 1.046 and ω3 = 1.130 for the LSW distribution
[159], and D is the diffusivity of the least mobile atomic species in the alloy, taken to be the
diffusivity of Cr in the γ’-precipitate phase, which has a value of 1.44 x 10-23 m2 s-1 at 873 K
[160]. The estimate of tc for alloy (A) from Equation 3.13 is (8 ± 5) x 106 h, while the predicted
value of tc for alloy (B) is (2 ± 1) x 106 h, both of which are greater than the longest aging time
studied, 1024 h. Our inability to achieve stationary-state coarsening explains the continuously
increasing values of φ and the non-zero values of )(tCiγΔ and )(tCi
γ ′Δ at an aging time of 1024
h, which may explain the deviation from the t-1 prediction for the temporal evolution of Nv(t).
When D is taken to be the diffusivity of Cr in the γ-matrix phase, 3.0 x 10-21 m2s-1, for alloy (A),
and 2.6 x 10-21 m2s-1 for alloy (B), the estimates of tc (1.0 ± 0.1) h and (0.3 ± 0.1) h are generated
for alloy (A) and (B), respectively, are three orders of magnitude less then the longest aging time
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studied experimentally, and indicate that the diffusivity inside the γ’-precipitates may be the rate
limiting step in γ’-precipitate coarsening. Additionally, the values of tc for alloy (A) are longer
than those estimated for alloy (B), an indication that the coarsening kinetics are slower in alloy
(A). Thus the rate of the diminution of Nv(t) is significantly slower in alloy (A) (Nv(t) ∝ t -0.42 ±
0.03) than in alloy (B) (Nv(t) ∝ t -0.67 ± 0.01).
A recent study by Mao et. al [39] combined APT and LKMC to study the role of the
precipitation diffusion mechanism on the early-stage precipitate morphology of alloy (B). They
showed that the long-range solute-vacancy binding energies (out to fourth nearest-neighbor
distance) strongly affect the γ’-precipitate coagulation and coalescence process, which occurs
abundantly at early stages, from 0.25 to 64 h. Coagulation and coalescence are shown to result
from the overlap of nonequilibrium concentration profiles surrounding γ’-precipitates that give
rise to nonequilibrium diffuse interfaces. The concentration profiles associated with the
interfacial regions between γ’-precipitates are spread over distances significantly larger than that
of the equilibrium interfacial thickness. This is due to specific couplings between the diffusion
fluxes of the constituent elements toward and away from γ’-precipitates, which are a result of the
finite vacancy-solute binding energies. From this analysis, the coagulation and coalescence of γ’-
precipitates is more likely when <λe-e> has a minimum value, as evidenced experimentally for
alloys (A) and (B), Figure 3.5. Additionally, the larger Al concentration in alloy (A) leads to the
formation of more highly mobile Al clusters than in alloy (B), which explains why the quantities
Nv(t) and f achieve their maximum values, and the quantity <λe-e> reaches a minimum value,
after aging for only 1 h in alloy (A), while this condition is reached at 4 h in alloy (B).
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3.5. Summary and Conclusions
We present a detailed comparison of the nanostructural and compositional evolution of
alloy (A) Ni-7.5 Al-8.5 Cr and alloy (B) Ni-5.2 Al-14.2 Cr, during phase separation at 873 K for
aging times ranging from 1/6 to 1024 h, employing atom-probe topography (APT). These ternary
alloys have similar equilibrium γ’-precipitate volume fractions, φeq, of 16.4±0.6% for alloy (A)
and 15.7±0.7% for alloy (B), and were designed to study the effects of solute concentration on
the kinetic pathways in model Ni-Al-Cr alloys, leading to the following results:
• The morphology of the γ’-precipitate phase in both alloys is found by both APT and TEM
to be spheroidal for aging times as long as 1024 h, as a result of a near-zero lattice
parameter misfit between the γ-matrix and γ’-precipitate phases. This high degree of
coherency makes these alloys amenable to a comparison of APT results to predictions of
classical theories of nucleation, growth and coarsening where the chemical free energy
term is dominant. Coagulation and coalescence of γ’-precipitates is observed and is
argued to be a result of the overlap of the nonequilibrium concentration profiles
associated with adjacent γ’-precipitates [39].
• After aging to 1/6 h, precipitation of the γ’-phase is evident for both alloys, as alloy (A)
forms nuclei with a composition of 70.9 ± 1.4 Ni, 23.3 ± 1.5 Al and 5.8 ± 0.9 Cr, at a γ’-
precipitate number density, Nv(t = 1/6 h), of (2.6 ± 1.4) x 1023 m-3, a mean radius, <R(t =
1/6 h)>, of 0.90 ± 0.32 nm, and a volume fraction, φ, of 0.31 ± 0.11 %. Alloy (B) forms
71.3 ± 1.6 Ni, 19.1 ± 1.4 Al and 9.7 ± 1.1 Cr nuclei with a Nv(t = 1/6 h) value of (3.6 ±
1.3) x 1023 m-3 , an <R(t = 1/6 h)> value of 0.74 ± 0.24 nm, and a φ value of 0.11 ±
0.04%.
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• Classical nucleation theory (CNT) is applied to demonstrate that the chemical free energy
changes for forming a nucleus, ΔFch, of -6.6 x 107 J m-3 for alloy (A) and -8.2 x 107 J m-3
for alloy (B), provide the primary driving force for nucleation. As such, the differing
solute concentrations of the two alloys are responsible for differences in the nucleation
behavior and in the nanostructural and compositional evolution of the alloys as they
decompose. The high degree of γ’-precipitate coherency with the γ-matrix in these alloys
is such that the elastic strain energy values, ΔFel, estimated to be -2.5 x 106 J m-3 for alloy
(A), and -1.1 x 105 J m-3 for alloy (B), are only a small fraction of the values of ΔFch, for
both alloys.
• Estimates of the γ/γ’ interfacial free energy values, σγ/γ’, from the coarsening data
obtained by APT yield values of 23-25 ± 6 mJ m-2 for alloy (A) and 22-23 ± 7 mJ m-2 for
alloy (B). These values are in good agreement with our recalculated values of σγ/γ’ of 20-
21 ± 5 mJ m-2 for the coarsening data of Schmuck et al. [28] for a ternary Ni-5.2 Al-14.8
Cr alloy aged at 873 K.
• The predictions of the critical radii for nucleation, R*, of 0.76 nm and 0.55 nm for alloys
(A) and (B), show reasonable agreement with the average radii of the first γ’-precipitates
detected by APT at 1/6 h, of 0.90 ± 0.32 nm and 0.74 ± 0.24 nm, respectively. The
predicted value of Jst of 4.0 x 1022 m-3 s-1 for alloy (A) is nearly one order of magnitude
greater than the experimental value of (5.4 ± 1.5) x 1021 m-3 s-1, and the calculated value
of Jst for alloy (B) of 3.2 x 1023 m-3s-1 is 50 times greater than the experimentally
measured value of Jst of (5.9 ± 1.7) x 1021 m-3 s-1. This discrepancy may be due to the
sensitivity of the predicted value of Jst to the value of R*, and due to the assumption that
76
the number of nucleation sites per volume is equal to the volume density of lattice points
occupied by Al, the precipitate-forming solute element, which may be an overestimate.
Further research is required to measure the value of Jst more accurately for ternary Ni-Al-
Cr alloys to better understand the nucleation kinetics in these concentrated
multicomponent alloys. To date, there is not a generally accepted theory of nucleation in
concentrated multicomponent alloys.
• After 1 h of aging, alloy (A) achieves a peak value of Nv(t) of (2.21 ± 0.64) x 1024 m-3,
which coincides with a minimum value of the average interprecipitate edge-to-edge
spacing, <λe-e>, of 7.0 ± 2.5 nm and a maximum in the fraction of interconnected
precipitates, f, of 18 ± 4 %. At an aging time of 4 h, alloy (B) achieves a peak value of
Nv(t) of (3.2 ± 0.6) x 1024 m-3 at a minimum value of <λe-e> of 5.9±0.8 nm and a
maximum value of f of 30 ± 4 %. The larger peak value of Nv(t) for alloy (B) is a result of
the larger initial solid-solution value of ΔFch in alloy (B), which sustains nucleation for
longer aging times.
• In the growth and coarsening regime that follows the peak in the value of Nv(t), the
quantity <R(t)> displays a temporal dependence of t0.29 ± 0.03 for alloy (A) and t0.29 ± 0.05
for alloy (B), in approximate agreement with the t1/3 prediction of the Kuehmann and
Voorhees (KV) model for isothermal coarsening in concentrated ternary alloys. The
supersaturation of the γ-matrix phase in alloy (A) exhibits a temporal dependence of t-0.32
± 0.03 for Al and t-0.29 ± 0.04 for Cr, while alloy (B) demonstrates a temporal dependence of t-
0.33 ± 0.04 for Al and t-0.34 ± 0.07 for Cr, in good agreement with the t1/3 prediction of the UO
and KV models. The supersaturation values of Al and Cr in the γ’-precipitate phase of
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Alloy (A) display a temporal dependence of t-0.34 ± 0.04 for Al and t-0.32 ± 0.05 for Cr, while
the supersaturation values in alloy (B) demonstrate a temporal dependence of t-0.30 ± 0.05
for Al and t-0.29 ± 0.07 for Cr.
• During growth and coarsening, the diminution of the quantity Nv(t) in alloy (A) exhibits a
temporal dependence of t-0.42 ± 0.03, while alloy (B) exhibits a temporal dependence of t-0.67
± 0.01. These temporal exponents deviate from the t-1 prediction of the KV coarsening
model because neither alloy has achieved stationary-state coarsening, characterized by a
constant value of φ and a zero value of the γ-matrix supersaturation. The estimate of the
critical time required for stationary-state coarsening, tc, for alloy (A) is (8 ± 5) x 106 h,
while for alloy (B) the value is (2 ± 1) x 106 h. The estimates of tc for both alloys are well
beyond the longest aging time we studied, 1024 h, and this is most likely the reason that
stationary-state coarsening is not achieved.
• The compositional trajectories of the γ-matrix during phase decomposition lie
approximately along the tie-lines, while the trajectories of the γ’-precipitate phase do not,
as predicted by the KV model for quasi-stationary state isothermal coarsening in ternary
alloys. The addition of a third alloying element alters the Gibbs-Thompson equations
significantly, and as such the KV model predicts that the compositional trajectory of the
γ’-precipitate phase will lie on a straight line that is not necessarily parallel to the
equilibrium tie-line.
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Chapter 4
On the field evaporation behavior of a model Ni-Al-Cr superalloy studied by
picosecond pulsed-laser atom-probe tomography
Abstract
The effects of varying the pulse energy of a picosecond laser used in the pulsed-laser atom-probe
(PLAP) tomography of an as-quenched Ni-6.5 Al-9.5 Cr at.% alloy are assessed based on the
quality of the mass spectra and the compositional accuracy of the technique. Compared to
pulsed-voltage atom-probe tomography (APT), PLAP tomography improves mass resolving
power, decreases noise levels, and improves compositional accuracy. Experimental evidence
suggests that Ni2+, Al2+ and Cr2+ ions are formed primarily by a thermally activated evaporation
process, and not by post-ionization of the ions in the 1+ charge state. An analysis of the detected
noise levels reveals that for properly chosen instrument parameters, there is no significant
steady-state heating of the Ni-6.5 Al-9.5 Cr at.% tips during PLAP tomography.
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4.1 Introduction
Pulsed-laser atom-probe (PLAP) tomography permits atom-probe tomography (APT)
with increased mass resolving power and pulse repetition rate, and allows the study of materials
with poor electrical conductivity [161-163]. PLAP tomography extends the range of study of the
APT technique, though there are some inherent drawbacks, such as inaccurate quantitative
results which can arise due to the complexity of the evaporation mechanism [164]. Additionally,
surface diffusion can occur if heating of samples is excessive under laser illumination. PLAP
tomography has become an increasingly popular technique for the study of metals and
semiconductors [165-169], thus a quantitative understanding of the effects of pico- or
femtosecond laser pulsing on the evaporation behavior of investigated materials is essential to
obtain physically meaningful results.
The extant theories of field evaporation, the image hump theory [170] and the charge
exchange theory [171-173], were successful in predicting both the correct charge state of
evaporating atomic species, and the fields required for evaporation, though refinement was
required [163, 174]. The ionic model upon which these theories are based treats the temperature
dependence of the evaporation field to be exclusively from the heat of sublimation, which from
Kirchhoff’s equation, displays very little temperature dependence from absolute zero to the
melting point of metals [175]. Hence, the ionic model is incapable of explaining the observed
experimental temperature dependencies of the evaporation field, and the calculated values of the
evaporation field in the literature are essentially 0 K values. While no fully satisfactory
theoretical treatment of field evaporation is currently available [176, 177], modification of the
image hump theory of field evaporation led to an expression for the temperature sensitivity of the
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evaporation field at constant evaporation rate [174, 178, 179], which predicted an increase in the
rate of evaporation with increasing temperature and was validated experimentally [174, 176,
180-185]. Currently, the mechanism by which laser pulsing induces field evaporation remains
somewhat controversial, particularly for femtosecond laser pulsing [104, 186-190]. We note that
there is convincing evidence of heating of the tip under laser illumination with picosecond
pulsing in other fields, such as scanning tunneling electron microscopy [191, 192]. The
theoretical and experimental work performed to measure the temperature rise due to laser
illumination in field-ion microscopy has been reviewed in detail [193], though more work
remains to be done, particularly for the new generation of atom-probes using pico- and
femtosecond laser pulsing.
In pulsed-voltage APT, variables such as sample preparation, instrument specifications,
and the instrumentation parameters selected to evaporate ions, contribute to the quality of the
data collected. Material- and specimen-dependent parameters, such as the evaporation behavior
of the phases being studied, and the crystallographic direction being examined, are also of
critical importance and must be considered on a sample-by-sample basis. With the continued
improvement in both instrument capability and sample preparation techniques [99, 100, 104,
165, 166], optimization of data quality must be performed by fine tuning the instrumentation
parameters, specifically: (1) specimen base temperature; (2) pulse fraction, the ratio of the pulse
voltage to the steady-state voltage; (3) the specimen evaporation rate, the number of ions
evaporated per pulse, and (4) the pulse repetition rate. This has been performed extensively for
pulsed-voltage atom-probe microscopy; see for example [194-196] who used a time-of-flight
atom-probe field-ion microscope [197]. Little, however, has been published on the quantification
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of the effects of the controllable experimental parameters in PLAP tomography [164],
particularly for the new generation of high-performance instruments employing ultrafast pico-
and femtosecond lasers. The instrumentation parameters for PLAP tomography are the same as
those for pulsed-voltage APT, though the pulse fraction is replaced by the effective pulse
fraction, which is not a directly tunable instrumentation parameter. The effective pulse fraction is
defined as one minus the ratio of the steady-state tip voltage during laser-assisted data
acquisition to the steady-state dc tip voltage required to field evaporate ions from a specimen at a
rate of 103 ions per second [104]. The laser adds a further degree of complexity by introducing
instrumentation variables such as the laser pulse energy, spot diameter, wavelength, and pulse
widths. In the case of the LEAP 3000X Si™ instrument used in this study, these variables have
been set by the instrument manufacturers [104], with the exception of the laser energy, which
remains a variable instrumentation parameter under the control of the experimentalist.
It has been demonstrated that decreasing the laser pulse repetition rate and increasing the
shank angle of specimen tips improves mass resolving power, m/Δm, the signal-to-noise ratio,
S/N, and the compositional accuracy of PLAP tomography for materials with small values of the
thermal diffusivity [104, 186]; the thermal diffusivity is equal to the ratio of the thermal
conductivity to the product of the density and the specific heat at constant pressure. Little work
has been performed to model the field evaporation of atoms in concentrated alloys because the
relevant thermodynamic quantities are generally unknown [176], though some early work was
performed to model the field evaporation of solute atoms in dilute alloys of iron [198]. The
present study determines the effects of the laser pulse energy on the evaporation behavior of a
concentrated model Ni-Al-Cr superalloy. The quality of the mass spectra and the compositional
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accuracy obtained for the same specimen with varying picosecond pulse energies are compared
at constant base specimen temperature. These results are compared with those of a test of the
effect of specimen base temperature on the evaporation behavior of the Ni-Al-Cr alloy,
conducted by performing pulsed-voltage APT at different tip temperatures.
4.2 Materials and methods
Ni-6.5 Al-9.5 Cr at.% solid-solutions were homogenized at 1573 K for 20 hours,
solutionized at 1123 K for 3 hours then water quenched to room temperature. The as-quenched
solid-solutions did not contain γ’-precipitates, and the chemical composition of the as-quenched
samples was measured by inductively coupled plasma (ICP) atomic-emission spectroscopy to be
Ni-6.24 Al-9.64 Cr at.%. Pulsed-laser and pulsed-voltage APT were performed employing an
Imago Scientific Instrument’s local-electrode atom-probe (LEAP™) tomograph [101-103], and
APT data were analyzed employing the IVAS® 3.0 software program (Imago Scientific
Instruments, Madison, Wi).
The data presented herein are obtained from two Ni-6.5 Al-9.5 Cr at.% tips; the first was
analyzed by PLAP tomography to study the effects of the laser pulse energy, while the second tip
was analyzed by pulsed-voltage APT to study the effects of specimen base temperature on the
evaporation behavior of the Ni-Al-Cr alloy. Datasets of ~5 million atoms were collected for each
laser pulse energy and specimen base temperature studied, and the first 0.5 million atoms were
discarded due to the initial instability of evaporation upon application of the pulsed-laser or
pulsed-voltage source. The depth of analysis of the datasets collected by pulsed-laser APT was
constant at 21±3 nm, with cross-sectional areas of 3,800±300 nm2, as determined by calculations
83
based on the theoretical density of this alloy, with the tip radius estimated from the evaporation
field of pure Ni at 73 K. PLAP tomography was performed at a constant specimen base
temperature of 40±0.3 K and ambient pressure of < 6.7 x 10-8 Pa, a pulse repetition rate of 200
kHz, a specimen evaporation rate of 0.04 ions per pulse, and laser energies of 0.2, 0.4, 0.6, 0.8,
1.0, 1.2 or 2.0 nJ. A laser with a wavelength of 532 nm, a pulse duration of ~ 12 ps, and a 5 μm
spot size was used. Pulsed-voltage APT at a specimen base temperature of 40±0.3 K was
performed after the PLAP tomography runs at laser pulse energies of 0.6, 1.0 and 2.0 nJ in order
to study the cumulative long-range effects of laser pulsing on the alloy composition.
Pulsed-voltage APT of the second tip was performed at a constant background pressure
of < 6.7 x 10-8 Pa, a pulse repetition rate of 200 kHz, a voltage pulse fraction (pulse
voltage/steady-state DC voltage) of ~19%, a constant evaporation rate of 0.04 ions per pulse, and
specimen base temperatures of 40, 60, 80, 100, 125, 150 and 175±0.3 K. The depth of analysis of
the datasets collected by pulsed-voltage APT was constant at 27±2 nm, with cross-sectional areas
of 2,200±200 nm2, as determined above. The steady-state tip voltages required to achieve a
specimen evaporation rate of 0.04 ions per pulse for both pulsed-laser and pulsed-voltage APT
are shown in Figure 4.1. We note that the tips had different initial radii of curvature and taper
angles, and hence, under the same field evaporation conditions, the specimens required different
steady-state voltages to evaporate atoms at the same rate. As such, results from the two Ni-Al-Cr
tips studied herein are not comparable in a quantitative manner, though trends in the evaporation
behavior provide insights into the effects of the picosecond laser pulse energy. The differences in
the tip shapes and dimensions were such that the evaporation rates of the two tips differed when
normalized to the effective area subtended by the detector. Normalized evaporation rates of
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1.1±0.1 x 10-5 ions pulse-1 nm-2 and 1.8±0.2 x 10-5 ions pulse-1 nm-2 were achieved for pulsed-
laser and pulsed-voltage APT, respectively.
Ni-Al-Cr samples in the as-quenched state were studied in order to avoid γ’(L12)-
precipitation due to the problems associated with differences in the evaporation fields of the
atomic species in the γ(FCC)-matrix and γ’-precipitate phases. No γ’-precipitates are detected in
the APT reconstructions, though partial radial distribution functions (RDFs) [45] of the as-
quenched samples indicate small Ni-Al ordered domains, which evolve into γ’-precipitates upon
subsequent aging. We note that deconvolution of the overlapping 27Al1+ and 54Cr2+ peaks was
performed based on the relative abundances of Cr isotopes. The standard errors, 2σ, for all
quantities are calculated based on counting statistics and reconstruction scaling errors, using
standard error propagation methods [140], and represent two standard deviations from the mean.
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Figure 4.1. The steady-state dc voltage applied to the tip to maintain a specimen evaporation rate of 0.04 ions per pulse for a Ni-6.5 Al-9.5 Cr at.% alloy is shown as a function of: (a) the laser pulse energy applied in PLAP tomography at a specimen base temperature of 40±0.3 K and a pulse repetition rate of 200 kHz; and (b) the specimen base temperature used in pulsed-voltage APT at a pulse repetition rate of 200 kHz and a pulse fraction of ~20%.
4.3 Results and discussion
4.3.1 Mass spectra
Figure 4.2 shows a comparison of the mass spectra obtained by PLAP tomography with
laser energies of: (a) 0.2 nJ per pulse and a steady-state tip voltage of 7.40-7.70 kV; (b) 1.0 nJ
per pulse and a tip voltage of 8.0-8.3 kV; and (c) 2.0 nJ per pulse and a tip voltage of 6.55-6.85
86
kV. The amount of noise increases with increasing laser pulse energy; the percentage of
evaporation events that are detected in expected ranges decreases from 97.24 % at 0.2 nJ, to
95.15 % at 1.0 nJ, to 93.96 % at 2.0 nJ. We note that at a laser pulse energy of 2.0 nJ, the mass-
to-charge state, m/n, spectrum is populated by numerous metal hydride peaks, particularly Ni1+
hydrides, and unexpected peaks at m/n ratios that correspond to hydrides, oxides and metal ion
clusters, which are probably a result of overheating of the tip. Because these peaks affect the
quality of the compositional data obtained by PLAP tomography, laser energies greater than 1.0
nJ per pulse should be avoided for this alloy to minimize background noise and undesirable
peaks. A section of the m/n range displaying the Ni2+ peaks is presented in Figure 4.3, to
highlight both the improved m/Δm values of PLAP tomography when compared to pulsed-
voltage APT, and the improved m/Δm values at higher laser pulse energies.
87
Figure 4.2. A comparison of the mass spectra obtained by PLAP tomography with laser pulse energies of: (a) 0.2 nJ per pulse at a tip voltage of 7.40-7.70 kV; (b) 1.0 nJ per pulse at a tip voltage of 8.0-8.3 kV; and (c) 2.0 nJ per pulse at a tip voltage of 6.55-6.85 kV. The frequencies of detected events are normalized to the number of events detected in the most populous m/n bin. The m/n ratios of the evaporating species change by over four orders of magnitude with increasing laser pulse energy. At a laser pulse energy of 2.0 nJ per pulse, the m/n spectrum is populated by numerous hydrides, oxides and metal ion clusters peaks, which are a result of overheating and are undesirable.
88
Figure 4.3. A section of the m/n range showing the Ni2+ peaks, highlighting both the improved mass resolving power of PLAP tomography when compared to pulsed-voltage APT, and the improved mass resolving power with increasing laser pulse energy. The relative frequencies of the Ni2+ peaks from the data set collected at a laser pulse energy of 2.0 nJ per pulse are nearly two orders of magnitude smaller than the relative frequencies of the Ni2+ peaks from the set collected at 0.2 nJ per pulse.
The most striking feature of Figures 4.2 and 4.3 is the dramatic change in the m/n ratios
of the evaporating species with increasing laser pulse energy. The relative frequencies of the Ni2+
peaks from the data set collected at a laser energy of 2.0 nJ per pulse, Figure 4.3, are nearly two
orders of magnitude less than the relative frequencies of the Ni2+ peaks at 0.2 nJ per pulse. The
Ni atoms are observed to evaporate as Ni1+ ions with increasing regularity as the laser pulse
energy is increased. This trend is observed to hold for all the atomic species to varying degrees,
Figure 4.4. While Ni and Al have similar evaporation behavior in terms of the m/n ratio, Cr
evaporates exclusively as Cr2+ until the laser energy reaches 1.0 nJ per pulse, at which point the
trend of the Cr2+/Cr1+ ratio with increasing laser pulse energy is similar to that of Ni2+/Ni1+ and
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Al2+/Al1+. Similar trends have been observed previously for other materials such as pure W, Si,
Ni, Rh, Mo, Re, Ir and Pt [162, 183, 186, 199]. High charge states on the order of 5+ and 6+
were detected for 5d transition metals evaporated at very high electric fields [200].
Figure 4.4. The ratio of the ions detected in the 2+ charge state to those detected in the 1+ charge state, as a function of laser pulse energy. The Ni and Al ions appear to have similar evaporation behavior, whereas Cr atoms evaporate exclusively as Cr2+ until the laser energy reaches 1.0 nJ per pulse.
It was previously suggested that field evaporated ions with lower than predicted m/n
ratios were a result of post-ionization of desorbed ions by an electron tunneling mechanism
similar to that involved in field ionization of gas atoms above the surface of the tip [201-205]. It
was reasoned that the increased electric field required to maintain a constant evaporation rate at
smaller laser pulse-energies led to progressively more post-ionization of atoms, and to the
appearance of lower m/n ratios. In Kellogg’s PLAP tomography experiment with tungsten, for
example, the ions field evaporated predominantly as W3+ at voltages greater than 3.6 kV, while
90
they evaporated as predominantly W2+ at voltages less than 2.9 kV [199]. For the Ni-Al-Cr alloy
studied here, when the laser pulse energy is varied between 0.4 and 1.0 nJ, the Ni2+/Ni1+ and
Al2+/Al1+ ratios both decrease by more than two orders of magnitude. Over this range of laser
pulse energies, the tip voltage required to maintain an evaporation rate of 0.04 ions per pulse
only varies between 8.15 and 8.30 kV. Given that the applied electric field varies by no more
than 2% over this range of laser energies, it is difficult to ascribe the dramatic changes in the
charge state ratios exclusively to post-ionization. Additionally, if post-ionization is responsible
for the higher charge states, the conditions required to post-ionize Ni1+ and an Al1+ ions differ
significantly from those required to post-ionize Cr1+. Thus, it may be that post-ionization is not
solely responsible for the change in the charge state of the evaporated species in the Ni-Al-Cr
alloy. We note that the average depth of the datasets was 21±3 nm, thus the change in the radius
of curvature, which affects the local electric field at the tip surface, is not responsible for a
decrease in the ratio of the Ni2+/Ni1+ and Al2+/Al1+ of more than four orders of magnitude over
the range of pulse energies studied herein.
Previous work with PLAP tomography at constant steady-state tip voltage and varying
specimen base temperatures showed that temperature had no effect on the m/n ratio of
evaporating species [183, 199]. These results were interpreted to mean that the activation
energies of desorption for ions of different charge states are identical, and thus provided
evidence for the post-ionization model. To quantify the importance of post-ionization in our
model ternary alloy, a test at a constant specimen evaporation rate of 0.04 ions per pulse and
varying specimen base temperature was performed for a second Ni-6.5 Al-9.5 Cr at.% tip to
understand the effect of specimen base temperature on the evaporation behavior of this model
91
Ni-Al-Cr alloy. The mass spectra from pulsed-voltage APT for specimen base temperatures
ranging from 40-125 ±0.3 K do not exhibit any evidence of evaporation of any of the atomic
species in the 1+ charge state. At specimen base temperatures of 150 and 175±0.3 K, however,
when the tip voltage is a constant 5.2±0.2 kV, Ni1+ ions are detected, and increase in relative
frequency of occurrence with increasing temperature, Figure 4.5. Thus, Ni atoms at specimen
base temperatures below 150 K seem to field evaporate as Ni2+ ions, and do not to result from
post-ionization of Ni1+ ions. For the evaporation conditions employed herein, Ni1+ ions field
evaporate by a thermally activated process at temperatures of ~150 K and greater. Neither Al1+
nor Cr1+ ions are detected by pulsed-voltage APT, suggesting that, for the given evaporation
conditions, the base specimen temperatures chosen herein are insufficient for evaporation of
these ionic charge states. From these results, the change in the abundances of the charge states
with increasing laser pulse energy in PLAP tomography, Figure 4.4, may be partly or completely
due to thermally activated field evaporation of ions in the 1+ charge state at the higher tip
temperatures that result from laser illumination.
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Figure 4.5. The relative frequencies of the Ni1+ peaks at specimen base temperatures of 125, 150 and 175±0.3 K, at a constant steady-state dc tip voltage of 5.2±0.2 kV. The increase in the relative frequency of Ni1+ ions with increasing specimen base temperature suggests that the evaporation of Ni1+ ions occurs by a different thermally activated mechanism than the evaporation of Ni2+ ions.
4.3.2 Compositional accuracy: Effects of laser energy
The concentrations of Ni, Al and Cr measured by PLAP tomography as a function of
laser pulse energy are presented in Figure 4.6, and clearly demonstrate that the measured
composition of the alloy differs with increasing laser pulse energy. Interestingly, the alloy
composition differs from both the nominal composition obtained by ICP chemical analysis, and
the values measured by pulsed-voltage APT after the PLAP tomography runs at laser pulse
energies of 0.6, 1.0 and 2.0 nJ. The compositions measured by pulsed-voltage APT were found
to be 82.94±0.08 Ni, 6.98±0.02 Al, and 10.07±0.03 Cr at.% and did not vary beyond
experimental error. Thus even at the largest laser pulse energy, 2.0 nJ, the laser power has no
cumulative, long-range effect on the tip composition.
The compositions measured by PLAP tomography are closer to the nominal composition
93
of the alloy than those measured by pulsed-voltage APT of the same tip at 40±0.3 K, Figure 4.6.
Previous studies of Ni-Al-Cr alloys by pulsed-voltage APT showed evidence of preferential field
evaporation of Ni atoms resulting in differences between the expected and measured Ni
concentration of 2.7 at.% for Ni-5.23 Al-14.77 Cr at.% [112], and 3.0 at.% for Ni-5.2 Al-14.2 Cr
at.% [42, 43, 46, 50]. These differences are greater than the 1.18±0.08 at.% Ni measured here by
pulsed-voltage APT. At a pulse energy of 0.6 nJ, the difference between the nominal Ni
concentration and the value measured by PLAP tomography is 0.22±0.14 at.%, suggesting that
for the experimental conditions used herein, preferential evaporation is less severe in PLAP
tomography than in voltage-pulsed APT, leading to improved compositional accuracy. The value
of the effective pulse fraction at a laser pulse energy of 0.6 nJ per pulse is ~20%, which is close
to the pulse fraction of ~19% that was determined to minimize preferential evaporation in Ni-Al-
Cr systems in a pulsed-voltage APT [43, 50, 112]. In both voltage and PLAP tomography, the
pulse fraction, or effective pulse fraction, respectively, have been recognized as the most
important instrumentation parameter for controlling preferential evaporation; values of 15-20%
are often cited as giving the most accurate compositional results [104]. In the case of PLAP
tomography, an excessive effective pulse fraction can lead to overheating of the specimen tip,
causing diffusion on the tip surface and resulting in a loss of spatial resolution [161, 186, 193].
The most striking feature of Figure 4.6 is the variation of the concentrations of the
constituent elements with laser pulse energy during PLAP tomography. At laser pulse energies of
0.2-0.6 nJ, the measured Ni concentration increases, and the Al and Cr concentrations decrease,
with increasing pulse energy. This suggests that the severity of Ni preferential evaporation
decreases with increasing laser pulse energy, leading to a decrease in the detected concentration
94
of solute atoms. At laser energies greater than 0.6 nJ per pulse, the Ni and Al concentrations do
not vary beyond the experimental error. The measured Cr concentration increases with
increasing pulse energy at energies of 1.0 nJ per pulse and greater. At a laser pulse energy of 2.0
nJ, the Cr concentration has increased by a statistically significant 0.11±0.08 at.% Cr value over
the value obtained for 0.6 nJ per pulse. The increase in the Cr concentration at 1.0 nJ per pulse
and greater, coincides with the evaporation of Cr1+, Figure 4.4. This effect cannot be ascribed to
post-ionization, but may be a result of a thermally activated evaporation mechanism for Cr1+.
4.3.3 Compositional accuracy- Effects of specimen base temperature
The effect of varying the specimen base temperature on the alloy composition measured
by pulsed-voltage APT is displayed in Figure 4.7. The measured Ni concentration increases with
increasing temperature, reaching the nominal composition at a temperature of 175±0.3 K. Over
the specimen base temperature range of 40–175 ±0.3 K, the measured Al concentration remains
constant, while the Cr concentration decreases by 1.43±0.04 at.% Cr. As the specimen base
temperature increases, and the tip voltage reaches a plateau value, Figure 4.1a, and Ni
preferential evaporation is diminished, while the preferential evaporation of Cr increases. This is
a result of differences in the required evaporation fields of Ni and Cr at different temperatures.
95
Figure 4.6. Variation in the detected composition of as-quenched Ni-6.5 Al-9.5 Cr at.% as a function of laser pulse energy (solid line) compared to the concentration measured by pulsed-voltage APT (dashed line) and the nominal composition of the alloy (dotted line). Preferential evaporation of Ni is responsible for the inaccuracy in the concentrations measured by pulsed-laser and pulsed-voltage APT.
96
Figure 4.7. Variation of the detected composition of as-quenched Ni-6.5 Al-9.5 Cr at.% as a function of specimen base temperature by pulsed-voltage APT. At higher temperatures, preferential evaporation of Ni is less severe, while preferential evaporation of Cr is more severe.
97
4.3.4 Analysis of events detected at mass-to-charge state ratios of 100-300 amu
An analysis of the events detected at m/n ratios between 100-300 amu, ranges where
peaks are neither expected nor observed for this Ni-Al-Cr alloy, provides some insight into
random field evaporation due exclusively to the steady-state dc tip voltage. Figure 4.8a shows
that for PLAP tomography, the extent of random evaporation is directly proportional to the
steady-state dc tip voltage, and does not increase with increasing pulse laser energy, and
concomitantly increasing input thermal energy. The events detected in this same mass range for
pulsed-voltage APT increase in frequency with increasing specimen base temperature, Figure
4.8b, as a result of the decrease in the evaporation fields required for evaporation of the atomic
species. This direct experimental result demonstrates clearly that results from PLAP tomography
with picosecond laser pulsing for this Ni-Al-Cr alloy are not affected by significant steady-state
heating of the tip. This is consistent with previous calculated estimates of the increase in tip
temperature on the order of 200-300 K due to laser illumination [182, 188], and estimates of tip
cooling times that are on the order of hundreds of nanoseconds [206].
The relative frequency of detected evaporation events with m/n ratios of 100-300 amu in
the three pulsed-voltage APT measurements made between pulsed-laser analyses, averaged
1.3±0.1 % of the total spectrum at an average tip voltage of 10.5±1.0 kV. This should be
compared to a value of 1.0±0.1 % of the total spectrum measured for the same tip, run by PLAP
tomography at a steady-state dc voltage of 8.0-8.3 kV, for a laser energy of 0.6 nJ per pulse, and
an equivalent pulse fraction of ~20%. Evaporation between pulses is less severe in PLAP
tomography due to the lower steady-state dc voltage required to achieve a field evaporation rate
of 0.04 ions per pulse, minimizing the preferential evaporation of Ni, as previously discussed.
98
Figure 4.8. The relative frequency of detected events between 100-300 amu, as a function of: (a) laser pulse energy in PLAP tomography; and (b) specimen base temperature in pulsed-voltage APT. The amount of noise measured by PLAP tomography is directly proportional to the tip voltage, and does not increase with increasing laser pulse energy, and thus increasing thermal energy. On the contrary, the noise measured by pulsed-voltage APT increases as the specimen base temperature increases. This result demonstrates that the Ni-Al-Cr tips are not affected by significant steady-state heating during PLAP tomography.
99
4.4 Conclusions
The effects of picosecond laser pulse energy on the compositional accuracy and the
quality of the mass spectra obtained for as-quenched Ni-6.5 Al-9.5 Cr at.% solid-solutions
analyzed by PLAP tomography are assessed, leading to the following conclusions:
• For the conditions used herein, it is found that when compared to pulsed-voltage APT,
PLAP tomography yields improved mass resolving power and compositional accuracy
and decreased noise levels.
• The most accurate compositional results are found at a laser pulse energy of 0.6 nJ per
pulse, a specimen base temperature of 40±0.3 K, a pulse repetition rate of 200 kHz, and a
specimen evaporation rate of 0.04 ions per pulse; these conditions correspond to an
effective pulse fraction of ~20%.
• An analysis of the events detected at m/n ratios of 100-300 amu by PLAP tomography
with varying laser pulse energies shows no evidence for steady-state heating of the tips
due to laser illumination.
• The laser pulse energy is found to have an important impact on the observed m/n values
of the evaporated species. The Ni1+, Al1+ and Cr1+ ions are observed to evaporate
predominantly at higher laser energies, while at lower laser pulse energies, Ni2+, Al2+ and
Cr2+ ions are predominantly detected, an effect which had previously been ascribed to
post-ionization. The post-ionization model, however, is unable to account for: (i) the
decrease in the Ni2+/Ni1+ and Al2+/Al1+ ratios by more than two orders of magnitude when
the laser energy is varied between 0.4-1.0 nJ per pulse at steady-state tip voltages ranging
from 8.15-8.30 kV; (ii) the detection of Ni1+ ions in pulsed-voltage APT exclusively at
100
specimen base temperatures above 150±0.3 K with a constant steady-state dc tip voltage.
The latter indicates that the evaporation of Ni1+ occurs by a thermally activated process at
higher temperatures, and thus it is unlikely that the evaporation of Ni2+, Al2+ and Cr2+
ions is due primarily to post-ionization of ions in the 1+ charge state.
• This study demonstrates the importance of selecting the correct instrumentation
parameters to achieve compositional accuracy for a Ni-Al-Cr alloy, analyzed by both
pulsed-laser and pulsed-voltage APT. A similar investigation is recommended for all
materials systems, particularly those with small thermal diffusivities, prior to in-depth
APT analysis, to achieve optimal evaporation conditions and the highest degree of
compositional accuracy.
101
Chapter 5
On the nanometer scale phase decomposition of a low-supersaturation Ni-Al-
Cr alloy
Abstract
The phase decomposition of a Ni-6.5 Al-9.5 Cr at.% alloy aged at 873 K is studied by atom-
probe tomography and compared to the predictions of classical precipitation models. Phase
separation in this alloy occurs in four distinct regimes: (i) quasi-stationary-state γ’ (L12)-
precipitate nucleation; (ii) concomitant precipitate nucleation, growth, and coagulation and
coalescence; (iii) concurrent growth and coarsening, wherein coarsening occurs via both γ’-
precipitate coagulation and coalescence and by the classical evaporation-condensation
mechanism; and (iv) quasi-stationary-state coarsening of γ’-precipitates, once the equilibrium
volume fraction of precipitates is achieved. The predictions of classical nucleation and growth
models are not validated experimentally, likely due to the complexity of the atomistic kinetic
pathways involved in precipitation. During coarsening, the temporal evolution of the γ’-
precipitate average radius, number density, and the γ(FCC)-matrix and γ’-precipitate
supersaturations follow the predictions of classical models.
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5.1 Introduction
Efforts to improve the fuel efficiency of aviation fuel and natural gas burning turbine
engines have led to the development of complex concentrated multicomponent nickel-based
superalloys that can withstand the extreme environments inside these engines [3, 7]. Nickel-
based superalloys are used in sections of turbine engines where operating temperatures exceed
1073 K, and resistance to chemical and mechanical degradation is critical. The high-temperature
mechanical properties of these alloys depend primarily on the presence of coherent, elastically-
hard, L12-ordered γ’-precipitates that result from decomposition of the γ(FCC) matrix.
Recent investigations of γ’-precipitation in model Ni-Al-Cr superalloys employing atom-
probe tomography (APT) have elucidated the kinetic pathways of phase decomposition for two
alloys, Ni-7.5 Al-8.5 Cr and Ni-5.2 Al-14.2 Cr at.%, both of which have a γ’-phase volume
fraction of about 16% [28, 37, 38, 42, 45, 46, 52, 112]. These studies have, however, been
limited by both the number of γ’-precipitates investigated, and by the number of data points
collected during the γ’-precipitate nucleation and growth regimes. Recent advances in atom-
probe tomography have improved the data collection rate and the field-of-view (200x200 nm2 at
the lowest magnification) of this technique [99, 104, 165], thereby eliminating the problems
associated with an insufficient γ’-precipitate sample size.
We present details of the phase decomposition of a low-supersaturation Ni-6.5 Al-9.5 Cr
at.% alloy aged at 873 K. This alloy was chosen to employ a small chemical driving force for
nucleation, permitting a detailed study of precipitate nucleation, and subsequently growth and
coarsening. The reduced γ-matrix supersaturation also results in a smaller γ’-precipitate volume
fraction, making this alloy more amenable to comparison with the predictions of classical models
103
of phase decomposition. A ternary Ni-Al-Cr phase diagram determined by the Grand Canonical
Monte Carlo (GCMC) technique at 873 K, Figure 5.1, predicts that the value of the equilibrium
volume fraction, φeq, for Ni-6.5 Al-9.5 Cr at.% aged at 873 K is 10.2±0.5 %. Ni-7.5 Al-8.5 Cr
and Ni-5.2 Al-14.2 Cr at.%, the other Ni-Al-Cr alloys that have been investigated in detail by
APT, are also plotted in Figure 5.1, and are predicted to have values of φeq of 17.5±0.5 and
15.1±0.5 %, respectively [52, 113].
Figure 5.1. A partial ternary phase diagram of the Ni-Al-Cr system at 873 K calculated using the Grand Canonical Monte Carlo simulation technique [113], showing the proximity of Ni-6.5 Al-9.5 Cr at.% to the (γ + γ’) / γ solvus line. Two other alloys that have been investigated by APT, Ni-7.5 Al- 8.5 Cr and Ni-5.2 Al-14.2 Cr at.%, are shown for comparative purposes. Equilibrium solvus curves determined by Thermo-Calc [139], using databases for nickel-based superalloys due to Saunders [115] and Dupin et al. [116], are superimposed on the GCMC phase diagram. The tie-lines are determined from the equilibrium phase compositions, determined by extrapolation of APT concentration data to infinite time.
104
5.2 Experimental
High-purity Ni, Al and Cr were induction-melted under a partial pressure of Ar and chill
cast in a 19 mm diameter copper mold to form a polycrystalline master ingot. The overall
composition of the master ingot was determined by inductively coupled plasma atomic-emission
spectroscopy to be 84.12 Ni-6.24 Al-9.64 Cr at.%, and was indistinguishable, within
experimental error, from the targeted composition of Ni-6.5 Al-9.5 Cr at.%. Samples from the
ingot were subjected to a three-stage heat-treatment: (1) homogenization at 1573 K in the γ-
phase field for 20 h; (2) a vacancy anneal in the γ−phase field at 1123 K for 3 hours followed by
a water quench; and (3) an aging anneal at 873 K under flowing argon for times ranging from
0.25 to 4096 h, followed by a water quench. Pulsed-laser APT was performed with a 3-D
LEAPTM tomograph [100-104] at a target evaporation rate of 0.04 ions per pulse, a specimen
temperature of 40.0±0.3 K, a pulse energy of 0.6 nJ per pulse, a pulse repetition rate of 200 kHz,
and an ambient gauge pressure of less than 6.7 x 10-8 Pa. These evaporation conditions have been
optimized to provide the highest compositional accuracy for this alloy [64]. APT data were
analyzed with the software program IVAS 3.0 (Imago Scientific Instruments). The γ’-
precipitate/γ-matrix heterophase interfaces were delineated with Al isoconcentration surfaces
employing the inflection point method [55], and compositional information was obtained using
the proximity histogram methodology [136, 137]. The standard errors for all quantities are
calculated based on counting statistics, and represent two standard deviations from the mean
[140]. We note that spatial convolution effects such as ion-trajectory overlap and local
magnification effects, which have been cited as sources of misleading results in APT analysis of
nickel-based superalloys [138], are not significant for low-supersaturation Ni-Al-Cr alloys aged
105
at 873 K [52]. Further experimental details are provided in references [42, 52]; the same
procedures were employed herein.
For comparative purposes, the equilibrium phase boundaries determined by the software
program Thermo-Calc [139], using the thermodynamic databases due to Saunders [115] and
Dupin et al. [116], are superimposed on the phase diagram at 873 K, Figure 5.1 [48]. While the
generated γ / (γ+γ’) solvus lines show good agreement, the curvatures of the (γ+γ’) / γ’ phase
lines differ for each technique. We note a discrepancy between the (γ+γ’) / γ solvus lines
predicted by Thermo-Calc and the GCMC technique as they approach the binary Ni-Al axis.
Given that GCMC employs changes in chemical potentials to determine the equilibrium tie-line
compositions, it is unlikely that this discrepancy is an artifact of the simulation, and may be due
to the fitting involved in the design of Thermo-Calc databases.
All thermodynamic quantities used herein were obtained from the commercial software
package Thermo-Calc [114] using thermodynamic databases due to Saunders [115] and Dupin et
al. [116]. The tracer diffusivities of the atomic species in the γ-matrix phase were calculated
employing Dictra [207] with the mobility database due to Campbell [208] and employing the
Saunders thermodynamic database.
106
5.3 Results and discussion
5.3.1 Morphological evolution
The temporal evolution of the γ’-precipitate morphology of the model Ni-Al-Cr alloy is
displayed in a series of 25 x 25 x 200 nm3 3D-APT reconstructions, each containing ca. 5.5
million atoms, Figure 5.2. Nanometer-sized spheroidal γ’-precipitates are detected in Ni-6.5 Al-
9.5 Cr at.% for aging times ranging from 0.5 to 1024 h. The γ’-precipitates at an aging time of
4096 h have commenced a spheroidal-to-cuboidal morphological transformation to minimize
their elastic strain energy as they increase in size [83, 209, 210]. For aging times of 1.5 to 64 h,
γ’-precipitate coagulation and coalescence is evident, characterized by the formation of L12-
ordered necks that interconnect γ’-precipitates. From Figure 5.3, the fraction of γ’-precipitates
undergoing coagulation and coalescence, f, is 25±3% after 4 h of aging, and 12±2% after 16 h of
aging; thus coarsening via this mechanism is significant at these times. After aging for 64 h, only
4±1 % of γ’-precipitates are undergoing coagulation and coalescence, and therefore coarsening
proceeds primarily via the evaporation-condensation mechanism. Precipitate coagulation and
coalescence has been detected previously in two different Ni-Al-Cr alloys [42, 52], and is due to
the overlap of the nonequilibrium γ’-precipitate diffusion fields caused by the specific coupling
among diffusion fluxes of Ni, Al, and Cr toward and away from γ’-precipitates [39].
107
Figure 5.2. APT reconstructed 3D images of a Ni-6.5 Al-9.5 Cr at.% alloy aged at 873 K for (a) 1 h, (b) 4 h, (c) 64 h and (d) 4096 h. The nanometer-sized γ’-precipitates are delineated with red Al isoconcentration surfaces. Aluminum, which partitions to the γ’-precipitates, is shown in red, while Cr, which partitions to the γ-matrix, is shown in blue; Ni atoms are omitted for clarity.
108
Figure 5.3. The fraction of γ’-precipitates undergoing coagulation and coalescence, f, is significant for aging times of 0.5 to 16 h. After aging for 64 h, only 4±1 % of γ’-precipitates are undergoing coagulation and coalescence, and therefore coarsening proceeds primarily via the evaporation-condensation mechanism. No γ’-precipitate coagulation and coalescence is detected beyond an aging time of 64 h.
5.3.2 Temporal evolution of the nanostructural properties of the γ’-precipitates
The temporal evolution of Ni-6.5 Al-9.5 Cr at.% aged at 873 K is complex and is divided
into four regimes: (i) quasi-stationary-state γ’-precipitate nucleation from 0.5 to 1.5 h; followed
by (ii) concomitant precipitate nucleation, growth, and coagulation and coalescence from 1.5 to 4
h; (iii) concurrent growth and coarsening between 4 and 256 h; and finally (iv) quasi-stationary
state coarsening of γ’-precipitates from 256 to 4096 h. Figure 5.4 displays the temporal evolution
of the γ’-precipitate volume fraction, φ, average radius, <R(t)>, and number density, Nv(t), and
clearly shows these four stages of phase decomposition in this model alloy. The γ’-precipitate
nanostructural properties determined by APT analysis are summarized in Table 5.1.
109
Figure 5.4. The temporal evolution of the γ’-precipitate volume fraction, φ, number density, Nv(t), and mean radius, <R(t)>, for Ni-6.5 Al-9.5 Cr at.% at 873 K. The quantity <R(t)> is approximately proportional to t1/3 during quasi-stationary state coarsening for aging times of 4 h and longer, as predicted by classical coarsening models. Once the equilibrium volume fraction is achieved after 256 h, the temporal dependence of the quantity Nv(t) achieves the t-1 prediction of the coarsening models.
110
Table 5.1. Temporal evolution of the nanostructural properties of γ’-precipitates a determined by APT for Ni-6.5 Al-9.5 Cr at.% aged at 873 K. The γ’-precipitate mean radius, <R(t)>, number density, Nv(t), and volume fraction, φ, are given, along with their standard errors.
Aging time (h)
Npptb <R(t)> ± 2σ
(nm) Nv(t) ± 2σ
(x 1023 m-3) φ ± 2σ
(%)
0.5 4 0.57 ± 0.10 0.11 ± 0.05 0.002 ± 0.001
0.75 175 0.67 ± 0.21 0.68 ± 0.05 0.02 ± 0
1 328 0.65 ± 0.19 1.27 ± 0.07 0.05 ± 0
1.5 1599 0.90 ± 0.30 5.53 ± 0.14 0.46 ± 0.01
2 755.5 0.99 ± 0.36 6.84 ± 0.25 0.54 ± 0.02
3.5 109.5 1.16 ± 0.53 8.13 ± 0.78 2.14 ± 0.20
4 905 1.23 ± 0.43 9.84 ± 0.33 2.32 ± 0.08
16 314.5 1.64 ± 0.26 2.09 ± 0.12 6.06 ± 0.34
64 47.5 2.82 ± 0.79 1.21 ± 0.08 8.17 ± 1.19
256 40.5 3.58 ± 0.93 0.45 ± 0.07 8.29 ± 1.30
1024 25 7.10 ± 1.86 0.14 ± 0.08 8.53 ± 1.71
4096 13.5 15.80 ± 3.39 0.03 ± 0.01 8.74 ± 2.38
a The number of γ’-precipitates analyzed, Nppt, is smaller than the total number of γ’-precipitates detected by APT. Precipitates that intersect the sample volume contribute 0.5 to the quantity Nppt, and are included in the estimates of Nv(t) and φ, and the phase compositions, and not in the measurement of <R(t)>.
Phase decomposition by nucleation has been studied theoretically for binary alloys in a
set of models known as classical nucleation theory (CNT), which have been reviewed
extensively in the literature [40, 117-120, 122]. According to CNT, nucleation is governed by a
balance between a bulk free energy term, which has both chemical, ΔFch, and elastic strain
111
energy, ΔFel, components, and an interfacial free energy term, σγ/γ’; where F is the Helmholtz
free energy. Experimental validation of CNT is quite difficult for solid solutions, due to the short
time and length scales involved in precipitation of second phases, which typically contain less
than 100 atoms [40, 211]. Control of the quenching conditions is important to minimize the
vacancy concentration, while simultaneously insuring that phase decomposition does not occur
during the quench. Additionally, the quantities involved in CNT, such as the interfacial energy,
the driving force, and the solute diffusion coefficient, are difficult to measure, and the
predictions of the models are highly sensitive to uncertainties in these values. As a result, the
discrepancies between the measured and predicted nucleation rates are typically between 3 and 5
orders of magnitude [40, 212].
From CNT, the expression for the net reversible work required for the formation of a
spherical nucleus, WR, as a function of nucleus radius, R, is given by:
'/23 43
4)( γγσππ RRFFW elchR +Δ+Δ= . (5.1)
The critical net reversible work, *RW , required for the formation of a critical spherical
nucleus is given by:
2elch
3*R )FF(3
16W'/
ΔΔσπ γγ
+= ; (5.2)
The critical net reversible work for nucleation acts as a barrier that nuclei must overcome
to achieve a critical nucleus radius, R*, given by:
)(
2*'/
elch FFR
Δ+Δ−=
γγσ . (5.3)
From CNT, the stationary-state nucleation current, Jst, the number of nuclei formed per
112
unit volume per unit time (m-3 s-1), is given by:
)Tk
Wexp(N*ZJB
*Rst −
= β ; (5.4)
where Z, the Zeldovich factor, accounts for the dissolution of supercritical clusters, β∗ is a
kinetic coefficient describing the rate of condensation of single atoms on the critical nuclei, N is
the total number of possible nucleation sites per unit volume, taken to be the volume density of
lattice sites occupied by Al, the precipitating solute element [122, 213], kB is Boltzmann’s
constant, and T is the absolute temperature. The standard definitions of Z and β∗ were employed
and are given by [40, 117-120, 122]:
2/1
b2
i2
Tk21
i*)i(WZ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−=
π, (5.5)
40
aDC*R4* π
β = ; (5.6)
where Wi is the net reversible work required for the formation of a spherical nucleus containing i
atoms, D is the diffusion coefficient of the precipitating solute element, C0 is the initial
concentration of said solute, and a is the average of the lattice parameters of the matrix and
precipitate phase, equal, in this case, to 0.3568 nm. Given that the extant theories of nucleation
are for binary alloys, the calculations of the stationary-state nucleation rate were performed for
both Al and Cr, thereby providing an upper and lower bound for the predicted nucleation rate.
Diffusion coefficients of 9.47 and 2.61 x 10-21 m2 s-1 were employed for Al and Cr, respectively,
along with C0 values of 6.5 and 9.5 at.%.
The value of ΔFch of -6.25 x 107 J m-3 is determined from Thermo-Calc [139] and the
Saunders thermodynamic database [145], and is calculated employing the equilibrium phase
113
compositions, an assumption which is valid for this alloy, given the small values of the solute
supersaturations [40, 214-216]. The values of ΔFel for the two alloys are estimated using [154]:
'a
2a
'a
el V)S4B3()VV(BS2F γγγ
γγγγ
Δ+
−= ′
′
; (5.7)
where Sγ is the shear modulus of the γ-matrix phase, Bγ’ is the bulk modulus of the γ’-precipitate
phase, and γaV and 'γ
aV are the atomic volumes of the γ-matrix and γ’-precipitate phases,
respectively. No elastic constants are available for this alloy, therefore the value of Sγ of 100.9
GPa, of a similar alloy, Ni-12.69 Al at 873 K [155] is employed, while the value of Bγ’ is taken to
be 175 GPa [156]. The lattice parameters of the γ'- and γ-phases at 873 K are calculated to be
0.3573 nm and 0.3563 nm, respectively, employing the precipitation simulation software
program PrecipiCalc [217, 218]. These lattice parameter values result in a near-zero value of the
lattice parameter misfit of 0.00284 for Ni-6.5 Al-9.5 Cr at.%. Substituting these values into
Equation 5.4 yields a value of ΔFel of 2.67 x 106 J m-3. The high degree of coherency of the γ’-
precipitates in this alloy is such that the bulk component of the driving force for nucleation is
dominated by the ΔFch term, as ΔFel is only 4.27% of the value of ΔFch. A value of σγ/γ’ of 22±1
mJ m-2 is estimated by a technique that relies on thermodynamic information and experimental
coarsening data, and was first used by Ardell for a binary alloy [145, 146], and extended to the
ternary Al-Sc-Mg and Ni-Al-Cr alloys [42, 52, 147], and is described in detail therein. The value
of σγ/γ’ of 22±1 mJ m-2 estimated for Ni-6.5 Al-9.5 Cr at.% is in excellent agreement with the
values of 20-25 mJ m-2 determined by the same technique for Ni-Al-Cr alloys aged at 873 K
[52]. We note that the value of σγ/γ’ during nucleation may differ significantly from this value,
since the latter was obtained from the coarsening regime, after the γ’-precipitates have undergone
114
a significant amount of temporal evolution. Unfortunately, there is no extant technique to
estimate the value of σγ/γ’ during nucleation for this ternary alloy aged at 873 K.
Employing the values of ΔFch, ΔFel, and σγ/γ’ given above, CNT predicts nucleation
currents of 1.06 x 1023 and 6.24 x 1022 m-3 s-1, employing the diffusion data for Al and Cr,
respectively. These values are 700 and 285 times greater than the measured rate of 1.5±0.7 x 1020
m-3 s-1, respectively. Given the evidence of precursor clustering in these Ni-Al-Cr alloys [45], it
is surprising that the experimentally determined value of Jst is significantly less than the
predicted value. Predictions of the nucleation rate by CNT have been found to be 50-500 times
greater than the experimental values for other Ni-Al and Ni-Al-Cr alloys [16, 52]. Xiao and
Haasen attributed this discrepancy to the sensitivity of the predicted value of Jst to the value of R*
[16]. They also noted that predicted nucleation currents are likely overestimated due to the
assumption that the value of N0 is equal to the volume density of lattice sites, which is a
commonly made assumption that is not necessarily correct. By assuming that the value of N0 is
equal to the volume density of Al atoms, we have mitigated this error to some extent. The
prediction of R* from CNT, which depends only on thermodynamic quantities, yields a value of
0.74 nm, in agreement with the value of <R(t)> of 0.62±0.17 we measured for aging times of
0.5-1 h.
To date, there is not a generally accepted theory of nucleation in concentrated
multicomponent alloys. Given that the detailed atomistic kinetics involved in the formation of γ’-
nuclei in these ternary systems are not completely understood, further analysis herein is not
instructive. The earliest stages of precipitate formation in Ni-6.5 Al-9.5 Cr at.% are described in
greater detail elsewhere [219], and complemented with results of lattice kinetic Monte Carlo
115
simulation to develop a deeper understanding of the kinetic pathways of nucleation in this alloy.
Nucleation and growth of γ’-precipitates occurs concurrently for aging times of 1.5 to 4
h, resulting in a nucleation current of 4.1±1.6 x 1019 m-3 s-1, and an increase in the value of φ
from 0.46±0.01 % to 2.32 ±0.08 %. As such, we do not observe a pure growth regime [141-143],
which is not a surprise given the evidence of concomitant γ’-precipitate nucleation, growth and
coagulation and coalescence between 1.5 and 4 h.
For aging times between 4 and 256 h, the values of φ and <R(t)> increase, and the values
of Nv(t) decrease, as expected for quasi-stationary-state coarsening. Previous investigations of
the phase decomposition of Ni-Al-Cr alloys have shown that the predictions of the temporal
evolution of the quantity <R(t)> from classical coarsening models are validated experimentally
during quasi-stationary-state coarsening [28, 37, 38, 42, 45, 46, 52, 112]. The predicted temporal
evolutions of the quantity Nv(t) and of the particle size distributions have not been validated
experimentally. This is because experimental alloys have failed to achieve genuine stationary-
state coarsening at the equilibrium volume fraction of the precipitated phase [9, 10, 127, 144].
The first comprehensive mean-field treatment of coarsening, due to Lifshitz and Slyozov [125]
and Wagner [126], known as the LSW model, is limited to dilute binary alloys with spatially-
fixed spherical precipitates whose initial compositions are equal to their equilibrium values, as
determined by the equilibrium binary phase diagram. These stringent requirements, and other
assumptions inherent to the LSW model, are highly restrictive, and difficult to meet in practice.
Researchers have worked to remove the mean-field restrictions by developing models based on
multiparticle diffusion that are able to describe stress-free systems with finite volume fractions
[10, 128, 129, 131]. Umantsev and Olson (UO) [130] demonstrated that the exponents of the
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temporal power-laws for concentrated multi-component alloys are identical with those for dilute
binary alloys, but that the explicit expressions for the rate constants depend on the number of
components. Kuehmann and Voorhees (KV) [132] considered isothermal coarsening in ternary
alloys and developed a detailed model that includes the effects of capillarity on the precipitate
composition, such that the phase compositions can deviate from their equilibrium values. In the
quasi-stationary-state limit of the KV model, the expressions for the temporal dependencies for
<R(t)>, Nv(t), and the γ-matrix supersaturation, )(tCiγΔ , of each solute species i, are:
)()()( 0KV3
03 ttKtRtR −=><−>< ; (5.8)
)()()( 0eqKV1
0v1
v ttK74.4
tNtN −≅− −−
φ; and (5.9)
3/1KV,i
eq,i
ff,ii tCtCtC −=∞−>=< γγγγ κΔ )()()( ; (5.10)
where KKV, and γκKV,i
are the coarsening rate constants for <R(t)> and )(tCiγΔ , respectively;
<R(t0)> is the average precipitate radius and Nv(t0) is the precipitate number density at the onset
of quasi-stationary-state coarsening, at time t0. The quantity )(tCiγΔ is denoted a supersaturation
and is the difference between the concentration in the far-field γ-matrix, >< )(, tC ffiγ , and the
equilibrium γ-matrix solute-solubility, )(, ∞eqiC γ . The temporal dependence of the quantity
<R(t)> for the alloy studied herein is measured to be t0.36±0.06 from the onset of coarsening at 4 h
to 4096 h. As noted previously, the prediction of the temporal evolution of the quantity <R(t)>
has been validated for systems where the equilibrium volume fraction has not been achieved, as
is the case for Ni-6.5 Al-9.5 Cr at.% for aging times of 4-256 h. Thus, its measurement alone is
not a particularly strong test of the LSW model. For this alloy, the temporal dependence of the
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quantity Nv(t) is t -0.80±0.06 for aging times of 4 h to 4096 h, and does not achieve the value of t-1
predicted by classical models of coarsening because the equilibrium volume fraction of
precipitates was not achieved for all data points considered.
During aging from 4 to 16 h, the value of Nv(t) drops sharply from the maximum value of
9.84±0.33 x 1023 m-3 at 4 h, to 2.09±0.12 x 1023 m-3 at 16 h as a result of coarsening via both the
classical evaporation-condensation mechanism and the coagulation and coalescence of γ’-
precipitates. The fraction of coagulating and coalescing γ’-precipitates is 25±3% after 4 h of
aging, and 12±2% after 16 h of aging; thus coarsening via this mechanism is significant at these
times. Note that the value of φ increases from 2.31±0.08 to 6.06±0.34 % over this range of time,
thus γ’-precipitate growth is also ongoing. After aging for 64 h, only 4±1 % of γ’-precipitates are
undergoing coagulation and coalescence, and therefore coarsening proceeds primarily via the
evaporation-condensation mechanism. For aging times of 256 to 4096 hours, where no
coagulation and coalescence of γ’-precipitates is detected, and the volume fraction of γ’-
precipitates is approximately constant, the temporal dependence of the quantity Nv(t) is t -1.04±0.10,
as predicted by extant models of coarsening, including the detailed KV model for ternary alloys
[125, 126, 132, 220, 221]. We note that the temporal dependence of the quantity <R(t)> is
measured to be t0.54±0.15 for aging times of 256 to 4096 hours, when volume fraction of γ’-
precipitates is approximately constant.
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5.3.3 Temporal evolution of the compositions of the γ and γ’ phase compositions
The compositions of the γ-matrix and γ’-precipitate phases in Ni-6.5 Al-9.5 Cr at.%
evolve temporally, as the γ-matrix is enriched in Ni and Cr and depleted of Al, as shown in
Figure 5.5. The first γ’-nuclei, detected at an aging time of 0.5 h, have a solute supersaturated
composition of 70.1 ± 3.0 Ni, 21.3 ± 4.9 Al and 8.7 ± 5.3 Cr at.%, with a value of <R(t=0.5 h)>
of 0.57 ± 0.10 nm. As phase decomposition proceeds, the values of )(tCiγΔ and the γ’-precipitate
supersaturation, )t(C 'iγΔ , decrease asymptotically as the γ- and γ’-phase compositions approach
their equilibrium values. The equilibrium compositions of the γ-matrix and γ’-precipitate phases
are determined from Equation 5.10 to be 84.19±0.02 Ni-5.49±0.05 Al-10.30±0.05 Cr and
76.40±0.19 Ni-17.53±0.33 Al-6.06±0.36 Cr at.%, respectively. These values are used to estimate
an equilibrium volume fraction of γ’-precipitates, φeq, of 9.6±2.9 %, according to the lever rule.
The values of φ of 8.29±1.30, 8.53±1.71 and 8.74±2.38 % determined by APT for aging times of
256, 1024 and 4096 h, respectively, are all within the experimental error of the φeq value of
9.6±2.9 %. Thus, by an aging time of 256 h, growth of γ’-precipitates is complete, and phase
decomposition proceeds only by quasi-stationary-state coarsening.
119
Figure 5.5. The composition profiles across the γ-matrix/ γ’-precipitate interface for Ni-6.5 Al-9.5 Cr at.% at 873 K for aging times of 1, 4, 4096 h. The phase compositions evolve temporally, as the γ-matrix becomes enriched in Ni and Cr and depleted in Al. The values of <R(t)> for these aging times are 0.65 ± 0.19 nm for 1 h, 1.23 ± 0.43 nm for 4 h and 15.80 ± 3.39 nm for 4096 h.
The temporal evolution of the phase compositions is shown on a partial ternary Ni-Al-Cr
phase diagram determined by the GCMC technique at 873 K, Figure 5.6 [130]. For comparative
purposes, the compositional trajectories of Ni-7.5 Al-8.5 Cr and Ni-5.2 Al-14.2 Cr at.%, two
alloys that have also been investigated by APT at Northwestern University, are superimposed on
the phase diagram. According to the KV coarsening model, the addition of a third alloying
element alters both the form of the Gibbs-Thompson equations and the predictions of the
120
temporal evolution of the phase compositions [132]. The slope of the trajectory of the γ-matrix
phase during coarsening is predicted by the KV model to lie along the equilibrium tie-line and
have a value of pAl/pCr, where pi is the magnitude of the partitioning of solute species i, as
defined by the expression pi = ,[ ( )eqiCγ ′ ∞ - , ( )]eq
iCγ ∞ . Alternatively, the compositional trajectory of
the γ’-precipitate phase is predicted by the KV model to lie on a straight line that is not
necessarily parallel to the equilibrium tie-line. To verify these predictions for the ternary alloy
studied, the quantities )(/)( tCtC CrAlγγ δδ and )(/)( '' tCtC CrAl
γγ δδ for aging times of 4-4094 h are
compared to the value of pAl/pcr, where δCpi is the slope of the concentration of element i in
phase p with aging time. The trajectories of the γ-matrix phase and the γ’-precipitate phases have
slopes of -4.29±0.52 and 0.56±0.24, respectively, while the slope of the equilibrium tie-line is
estimated to be -2.84±0.06. From these results, it is absolutely clear that the trajectory of the γ’-
precipitate composition does not lie along the equilibrium tie line. This result is contrary to all
coarsening models, except the KV model for ternary alloys. We note that the KV mean-field
model is unable to predict the correct value of the slopes, which is most likely due to the
omission of the off-diagonal terms in the diffusion matrix in this model. The omission of the off-
diagonal terms, in turn, generates inaccurate predictions of the diffusion fluxes into and out of
the γ'-precipitate phase by suppressing flux coupling [39].
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Figure 5.6. The compositional trajectories of the γ-matrix and γ’-precipitate phases of Ni-6.5 Al-9.5 Cr at.%, displayed on a partial Ni-Al-Cr ternary phase diagram at 873 K. The trajectory of the γ-matrix phase lies approximately on the experimental tie-line, while the trajectory of the γ'-precipitate phases does not lie along the tie-line, as predicted by the Kuehmann-Voorhees coarsening model [132]. The tie-lines are determined from the equilibrium phase compositions, determined by extrapolation of APT concentration data to infinite time.
The values plotted on the phase diagram in Figure 5.6 are the far-field plateau
compositions of the two phases, and do not include atoms that lie within the interfacial γ/γ’
region. As such, the rule of mixtures, which dictates that the nominal γ’-precipitate and γ-matrix
compositions at a given aging time lie along a straight line, is not satisfied. Classical models of
phase decomposition often ignore the interfacial region, thus we have chosen a similar approach
for comparative purposes. The inclusion of the interfacial region in the phase compositions (not
displayed) does not alter the compositional trajectories significantly, and does satisfy the rule of
mixtures. Advanced models of phase decomposition are needed to account for the interfacial
region between phases. The interfacial region, which has a definite width on the order of ~ 2 nm
122
for this alloy, Figure 5.5, has been measured to be on the order of several nanometers for other
model nickel-based superalloys [42, 52, 56, 58, 62, 113, 222-224].
The partitioning behavior of the elements can be determined quantitatively by calculating
the partitioning ratio, Kiγ’/γ
, defined as the ratio of the concentration of an element i in the γ’-
precipitates to the concentration of the same element in the γ-matrix. Figure 5.7 demonstrates
partitioning of Al to the γ’-precipitates, and of Ni and Cr to the γ-matrix. The values of Kiγ’/γ are
constant from 256-4096 h, when the alloy is undergoing quasi-stationary state coarsening.
Figure 5.7. The partitioning ratios, Kiγ’/γ
, quantify the partitioning of Al to the γ’-precipitates and of Ni and Cr to the γ-matrix. The values of Ki
γ’/γ are constant from 256 to 4096 h, when the alloy is undergoing quasi-stationary state coarsening.
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The diminution of the γ-matrix supersaturation values approximately follow the t-1/3
prediction of the KV model. From Figure 5.8, the alloy displays a temporal dependence of t-
0.32±0.03 for )t(CNiγΔ , t-0.32±0.04 for )(tCAl
γΔ , and t-0.32±0.04 for )(tCCrγΔ . The temporal dependencies
of the γ’-precipitate supersaturation values also follow approximately the t-1/3 prediction, with
measured dependencies of t-0.32±0.03 for )t(C 'NiγΔ , t-0.29±0.04 for )t(C '
AlγΔ , and t-0.34±0.03 for )t(C '
CrγΔ .
Figure 5.8. The magnitude of the values of the γ-matrix (left) and γ’-precipitate (right) supersaturations, )(tCi
γΔ and )t(C 'iγΔ , of Ni, Al and Cr, decrease as t-1/3 in the coarsening
regime, as predicted by classical coarsening models.
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5.4 Conclusions
We present a detailed comparison of the nanostructural and compositional evolution of
Ni-6.5 Al-9.5 Cr at.% during phase separation at 873 K for aging times ranging from 0.5 to 4096
h, employing atom-probe topography (APT). This ternary alloy with a small equilibrium γ’-
precipitate volume fraction of 9.6±2.9 % was designed to study precipitate nucleation, and
subsequently growth and coarsening, leading to the following results:
• The phase decomposition of the model alloy occurs in four distinct regimes: (i) quasi-
stationary-state γ’-precipitate nucleation from 0.5 to 1.5 h; (ii) followed by concomitant
precipitate nucleation, growth and coagulation and coalescence from 1.5 to 4 h; (iii)
concurrent growth and coarsening from 4 to 256 h, wherein coarsening occurs via both
γ’-precipitate coagulation and coalescence and by the classical evaporation-condensation
mechanism; and (iv) finally quasi-stationary-state coarsening of γ’-precipitates from 256
to 4096 h.
• The morphology of the γ’-precipitate phase is found to be spheroidal for aging times as
long as 1024 h, as a result of a near-zero lattice parameter misfit between the γ-matrix
and γ’-precipitate phases. Coagulation and coalescence of γ’-precipitates is observed for
aging times of 1.5 to 64 h, as a result of the overlap of the non-equilibrium concentration
profiles associated with adjacent γ’-precipitates [39]. The γ’-precipitates that have formed
by an aging time of 4096 h have commenced a spheroidal-to-cuboidal morphological
transformation to minimize their elastic strain energy as they increase in size [83, 209,
210].
• It is found that classical nucleation theory accurately predicts the critical radius for
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nucleation for the model Ni-Al-Cr alloy, although the prediction of the stationary-state
nucleation current is at least 285 times greater than the value measured experimentally.
These findings indicate that a better understanding of the kinetic pathways of nucleation
in this concentrated ternary alloy is required.
• The predictions of the Umantsev-Olson [130] and Kuehmann-Voorhees [132] models of
coarsening are verified experimentally for this model Ni-Al-Cr alloy. Beyond an aging
time of 4 h, the temporal evolutions of the values of the average precipitate radii and γ-
matrix and γ’-precipitate supersaturations follow the predictions of t1/3 and t-1/3,
respectively. For aging times beyond 256 h, the volume fraction of γ’-precipitates is
constant and the temporal dependence of the quantity Nv(t) is t-1.0±0.1.
• The solute solubility in the γ-matrix phase of Ni-6.5 Al-9.5 Cr at.% is determined by APT
to be 5.49±0.05 Al-10.30±0.05 Cr, while the equilibrium γ’-precipitate composition is
76.40±0.19 Ni-17.53±0.33 Al-6.06 ±0.36 Cr at.%. These values are used to estimate an
equilibrium volume fraction of γ’-precipitates, φeq, of 9.6±2.9 %, according to the lever
rule. The values of φ of 8.29±1.30, 8.53±1.71 and 8.74±2.38 % determined by APT for
aging times of 256, 1024 and 4096 h, respectively, are all within the experimental error of
the φeq value of 9.6±2.9 %. Thus, by an aging time of 256 h, growth of γ’-precipitates is
complete, and phase decomposition proceeds only by quasi-stationary-state coarsening.
• The compositional trajectory of the γ-matrix during phase decomposition lies
approximately along the tie-line, while the trajectory of the γ’-precipitate phase does not,
as predicted by the KV mean-field model for quasi-stationary state coarsening in ternary
alloys. The addition of a third alloying element alters the Gibbs-Thompson equations
126
significantly, and as such, the KV model predicts that the compositional trajectory of the
γ’-precipitate phase will lie on a straight line that is not necessarily parallel to the
equilibrium tie-line. The KV model, however, is unable to predict the exact value of the
slopes, which is most likely due to the omission of the off-diagonal terms in the diffusion
matrix in this model, which generates inaccurate predictions of the diffusion fluxes into
and out of the γ'-precipitate phase by suppressing flux coupling [39].
• An estimate of the γ/γ’ interfacial free energy, σγ/γ’, from the coarsening data obtained by
APT yields a value of 22±1 mJ m-2. This value for Ni-6.5 Al-9.5 Cr at.% is in excellent
agreement with the values of 20-25 mJ m-2 measured by the same technique for other Ni-
Al-Cr alloys aged at 873 K [52].
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Chapter 6
Effects of tantalum on the temporal evolution of a model Ni-Al-Cr superalloy
during phase decomposition
Abstract
The effects of a 2.0 at.% addition of Ta to a model Ni-10.0 Al-8.5 Cr at.% superalloy aged at
1073 K are assessed using scanning electron microscopy and atom-probe tomography. The
γ’(L12)-precipitate morphology that develops as a result of γ(FCC)-matrix phase decomposition
is found to evolve from a bimodal distribution of spheroidal precipitates, to {001}-faceted
cuboids and parallelepipeds aligned along the elastically soft <001>-type directions. The phase
compositions and the widths of the γ’-precipitate/γ-matrix heterophase interfaces evolve
temporally as the Ni-Al-Cr-Ta alloy undergoes quasi-stationary state coarsening after 1 h of
aging. Tantalum is observed to partition preferentially to the γ’-precipitate phase, and suppresses
the mobility of Ni in the γ-matrix sufficiently to cause an accumulation of Ni on the γ-matrix side
of the γ'/γ interface. Additionally, computational modeling, employing Thermo-Calc, Dictra and
PrecipiCalc, is employed to elucidate the kinetic pathways that lead to phase decomposition in
this concentrated Ni-Al-Cr-Ta alloy.
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6.1 Introduction
Tantalum is an important refractory addition to commercial nickel-based superalloys as
both a solid-solution strengthener and a precipitate former. Tantalum has been shown to increase
the high-temperature strength and ductility, and to improve the resistance to creep, fatigue, and
corrosion of these high-performance materials used in land-based and aerospace turbine engines
at operating temperatures up to 1373 K [3, 7, 78, 79, 225, 226]. The effects of Ta on the
microstructure and mechanical properties of nickel-based superalloys have been investigated
[226-231], however, little has been done to characterize the morphological and compositional
changes due to the addition of Ta.
Phase decomposition in model Ni-Al-Cr alloys has been investigated in detail by atom-
probe tomography (APT) [138]. Τhe research of Schmuck et al. [28, 112] and Pareige et al. [37,
38] combined APT and lattice kinetic Monte Carlo (LKMC) simulation to analyze the
decomposition of a Ni-Al-Cr solid-solution at 873 K. A similar approach was applied by
Sudbrack et al. [42, 45, 46, 55, 56], Yoon et al. [58, 59], Mao et al. [39] and Booth-Morrison et
al. [52] for studying Ni-5.2 Al-14.2 Cr and Ni-7.5 Al-8.5 Cr at.% aged at 873 K, and Ni-10 Al-
8.5 Cr at.%, Ni-10 Al-8.5 Cr-2.0 W at.% and Ni-10 Al-8.5 Cr-2.0 Re at.% aged at 1073 K, which
decompose via a first-order phase transformation to form a high number density (~1020-1025 m-3)
of nanometer-sized γ’-precipitates. We report on the temporal evolution of a model Ni-10.0 Al-
8.5 Cr-2.0 Ta at.% alloy aged at 1073 K that decomposes to form a microstructure consisting of
γ’(L12)-precipitates in a γ(FCC)-matrix. Chromium is added to the binary Ni-Al system to reduce
the lattice parameter misfit between the γ’-precipitates and the γ-matrix. The addition of Ta
increases the volume fraction of the γ’-precipitate phase, providing significant strengthening.
129
Tantalum also decreases overall alloy diffusivity, thereby improving phase stability and service
life by retarding diffusion mediated processes such as γ’-precipitate coarsening, creep and
oxidation [78, 79]. The effect of a 2.0 at.% addition of Ta to a model Ni-Al-Cr superalloy is
studied using scanning electron microscopy (SEM) and APT. The experimental results are
complemented by computational modeling employing the commercial software packages
Thermo-Calc, Dictra and PrecipiCalc, to elucidate the thermodynamic and kinetic pathways that
lead to phase decomposition in a concentrated Ni-Al-Cr-Ta alloy.
6.2 Experimental
High-purity constituent elements were induction-melted under a partial pressure of Ar
and chill cast in a 19 mm diameter copper mold to form a polycrystalline master ingot. Samples
from the cast ingot then underwent a three-stage heat-treatment: (1) homogenization at 1573 K in
the γ-phase field for 20 h; (2) a vacancy anneal in the γ−phase field at 1503 K for 3 hours
followed by a water quench; and (3) an aging anneal at 1073 K under flowing argon for times
ranging from 0.25 to 256 h, followed by a water quench. Microtip specimens and metallographic
samples were prepared from each of the aged sections for study by APT and SEM.
Vickers microhardness was measured using a Buehler Micromet™ instrument on
samples polished to 1 μm, with an applied load of 500 g sustained for 5 s, using the average
value of fifteen independent measurements made on several grains. SEM was performed on
samples polished to an 0.02 μm finish and etched in a 100 ml HCl/100 ml deionized H2O/1g
K2S205 mixture, employing a LEO Gemini 1525™ field-emission SEM operating at 5 kV with a
20-30 μm aperture and a working distance of 6 mm. APT microtips were prepared using
130
standard procedures [176, 232], and analyzed with a local-electrode atom-probe (LEAPTM)
tomograph [99, 101-103, 165] at the Northwestern University Center for Atom-Probe
Tomography (NUCAPT). Pulsed-laser APT data collection was performed at a target
evaporation rate of 0.04 ions per pulse, a specimen temperature of 40.0±0.3 K, a pulse energy of
0.6 nJ, a pulse repetition rate of 200 kHz, and a background gauge pressure of less than 6.7 x 10-8
Pa. Pulsed-laser atom-probe was employed to improve the compositional accuracy of the APT
technique by limiting preferential evaporation [54], and to increase the tip specimen life by
reducing the dc voltage required for evaporation. APT data were analyzed with the IVAS® 3.0
software program (Imago Scientific Instruments). The γ’-precipitate/γ-matrix heterophase
interfaces were delineated with Al isoconcentration surfaces generated by efficient sampling
procedures [136], and detailed compositional information was obtained with the proximity
histogram method [137]. The equilibrium volume fraction of the γ’-precipitate phase, φeq, was
estimated by averaging the values obtained from the lever rule of the extrapolated equilibrium
concentrations of Ni, Al, Cr and Ta.
Spatial convolution effects such as ion trajectory overlap and local magnification effects
have been cited as possible sources of misleading results in the APT analysis of nickel-based
superalloys [138]. A comparison of the composition profiles across the γ/γ’ interfaces measured
by APT to those simulated by the lattice kinetic Monte Carlo technique for a Ni-5.2 Al-14.2 Cr
at.% alloy [39], showed no evidence of artificial interfacial broadening in the APT data due to
ion trajectory overlap. Local magnification effects due to differences in the required evaporation
fields of different phases are unlikely in Ni-Al-Cr superalloys containing only the γ-matrix and
γ'-precipitate phases because the evaporation fields of the phases are essentially identical.
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The overall composition of the master ingot was determined by inductively coupled
plasma (ICP) atomic-emission spectroscopy to be 80.01 Ni-9.75 Al-8.21 Cr-2.02 Ta at.%, and
was indistinguishable, within experimental error, from the targeted composition of Ni-10.0 Al-
8.5 Cr-2.0 Ta at.%. ICP chemical analysis was also used to determine the compositions of the γ-
matrix and γ’-precipitate phases of a sample aged at 1073 K for 256 h after phase extraction by
anodic dissolution of the γ-matrix phase with a 1:1 aqueous solution of citric acid and
ammonium nitrate at constant current density. We note that the standard errors for all quantities
are calculated based on counting statistics and reconstruction scaling errors using standard error
propagation methods [140], and represent two standard deviations from the mean.
The commercial software package Thermo-Calc [114] was used to estimate the values of
φeq and the equilibrium compositions of the γ-matrix, )(, ∞eqiC γ , and γ’-precipitate phases,
)(,' ∞eqiC γ , at a pressure of 1 atmosphere, using a database for nickel-based superalloys developed
by Saunders [115]. The tracer diffusivities of the atomic species in the γ-matrix phase were
calculated employing Dictra [207] with the mobility database due to Campbell [208] and
employing the Saunders thermodynamic database. Additionally, precipitation modeling was
performed with the commercial software PrecipiCalc [217, 233], employing the Saunders
thermodynamic database and the mobility database due to Campbell. Precipicalc applies
thermodynamic and kinetic data from Thermo-Calc and Dictra to continuum models of
precipitation for multicomponent, multiphase alloys to provide a unified treatment of nucleation,
growth and coarsening.
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6.3 Results
A 2.0 at.% addition of Ta to a model Ni-10.0 Al-8.5 Cr at.% alloy aged at 1073 K results
in a 47±5% increase in the microhardness, Figure 6.1, over the full range of aging times, t = 0 to
256 h. The microhardness of Ni-Al-Cr-Ta varies from 3.3 to 3.6 GPa, with peak microhardness
occurring between t = 1 and 16 h. For Ni-10.0 Al-8.5 Cr at.% aged at 1073 K, the microhardness
varies over a narrow range, 2.15 GPa to 2.5 GPa, with peak microhardness occurring at 4 h [55,
56]. The increase in the microhardness due to the addition of Ta is a result of both the added
solid-solution strengthening provided by Ta, and to a dramatic increase in the value of the γ’-
precipitate volume fraction, φ. At 256 h, for example, the value of φ from the phase extraction
results is 18.9% for Ni-Al-Cr and 36.4% for Ni-Al-Cr-Ta. The values of the equilibrium volume
fractions, φeq, estimated by APT and from Thermo-Calc for Ni-Al-Cr-Ta are presented in Table
6.1, and compared to the values of φ at 256 h, as measured by both APT and phase extraction.
Figure 6.1. Vickers microhardness measurements for Ni-10.0 Al-8.5 Cr-2.0 Ta and Ni-10.0 Al-8.5 Cr at.% aged at 1073 K. The addition of 2.0 at.% Ta results in a 47±5% increase in the microhardness over the full range of aging times, t = 0 to 256 h, due to increases in solid-solution strengthening and to a dramatic increase in the value of φ.
133
Table 6.1. Equilibrium γ’-precipitate volume fraction, φeq, determined by APT, ICP chemical analysis, and thermodynamic modeling employing Thermo-Calc for Ni-10.0 Al-8.5 Cr-2.0 Ta at.% aged at 1073 K.
Technique used to estimate φeq: φeq (%)
Lever rule with APT compositions, 256 h: 37.2 ± 8.9
Lever rule with APT equilibrium compositions: 37.0 ± 9.0
ICP chemical analysis, 256 h: 36.4
Thermo-Calc using Saunders database [115]: 38.17
6.3.1 Morphological development
The microstructure of the material as-quenched from 1503 K consists of a bimodal
distribution of γ’-precipitates, with primary γ’-precipitates of radii, R(t = 0 h), on the order of 30-
40 nm, and smaller secondary γ’-precipitates with an average radius, <R(t = 0 h)>, of 6.4±1.1
nm. From APT, the number density, Nv(t = 0 h), of the primary γ’-precipitates is 3±2 x 1021 m-3,
occupying a volume fraction of 28±13%. The value of Nv(t = 0 h) of the secondary γ’-
precipitates is 1.3±0.1 x 1023 m-3, with a volume fraction of 11±1%. The average edge-to-edge
distance, λe-e, between secondary γ’-precipitates is measured to be 4.3±1.3 nm, while the value of
λe-e for the primary γ’-precipitates is 109±63 nm, as measured by a method developed by
Karnesky et al. [106]. A solute-depleted, precipitate-free shell with a thickness of 5.7±1.1 nm
surrounds the primary γ’-precipitates. The bimodal distribution is only observed in the as-
quenched state. With aging, the secondary γ’-precipitates are likely consumed by an Ostwald
ripening process to decrease the total γ/γ’ interfacial free energy of the system.
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Figure 6.2. APT reconstructed 3D image of a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 0 h. The solute elements that partition to the γ’-precipitates, Ta and Al, are shown in orange and red, respectively, while Cr, which partitions to the γ-matrix, is shown in blue; Ni atoms are omitted for clarity. The morphology of the γ’-precipitate phase is spheroidal in the as-quenched state, and a bimodal particle size distribution is apparent.
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The morphology of the Ni-Al-Cr-Ta alloy evolves from spheroidal to cuboidal γ’-
precipitates with significantly rounded edges, and finally to cuboidal and parallelepipedic γ’-
precipitates aligned along the elastically soft <001>-type directions, Figure 6.3. For aging times
beyond the as-quenched state, the primary γ’-precipitates develop an {001}-faceted cuboidal
morphology with rounded corners, as seen by both SEM, Figure 6.3b, and APT, Figure 6.4, for a
sample aged for 0.25 h. We note that the γ’-precipitates in Figure 6.3b and 4 exhibit both
spheroidal and cuboidal characteristics. The value of R(t = 0.25 h) at which primary γ’-
precipitates in this Ni-Al-Cr-Ta alloy undergo the transformation from a spheroidal-to-cuboidal
morphology is estimated from APT to be 61±7 nm.
Figure 6.3. SEM images of a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for: (a) 0 h: (b) 0.25 h: (c) 1 h: (d) 4 h: (e) 16 h: and (f) 64 h. The primary γ’-precipitate morphology evolves from spheroidal γ’-precipitates to a cuboidal and parallelepipedic morphology with primary γ’-precipitates aligned along the elastically soft <001>-type directions.
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Figure 6.4. APT reconstructed 3D image of a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 0.25 h. The primary γ’-precipitates in the APT reconstruction have both spheroidal and cuboidal characteristics. The radius at which the γ’-precipitates undergo the spheroidal-to- cuboidal morphological transformation is measured to be 61±7 nm. Tantalum and Al are shown in orange and red, respectively, Cr is shown in blue, and Ni atoms are omitted for clarity.
The cuboidal morphology of the γ’-precipitates persists with aging, and for aging times
beyond of 4 h, the γ’-precipitates align along orthogonal <001>-type directions, Figures 6.2 and
6.5. For aging times of 64 and 256 h, the primary γ’-precipitates have a non-equiaxed,
rectangular parallelepipedic morphology with {001}-type facets. We note that the direction of γ’-
precipitate alignment was verified employing selected-area electron-diffraction (SAED) patterns
using conventional transmission electron microscopy, with an Hitachi HF-8100 at an
accelerating voltage of 200 kV, for a sample aged for 256 h at 1073 K, Figure 6.6.
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Figure 6.5. An APT reconstructed 3D image of cuboidal and parallelepipedic primary γ’-precipitates aligned along <001>-type directions from a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 64 h. Tantalum and Al are shown in orange and red, respectively, Cr is shown in blue, and Ni atoms are omitted for clarity.
Figure 6.6. After aging for 256 h, the primary γ’-precipitates have cuboidal and parallelepipedic morphologies with {001}-type facets, and are aligned along the <001>-type crystallographic directions, as confirmed by a selected-area electron diffraction pattern taken along the ]101[ zone axis.
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6.3.2 Compositional evolution
The compositional information generated by APT permits the study of the temporal
evolution of the phase compositions, and of the concentration profiles at the γ’-precipitate/γ-
matrix interface, Figure 6.7. The γ’/γ interfaces are defined by employing the inflection-point
approach to determine a threshold isoconcentration surface that represents the average of the far-
field plateau concentrations of the γ’-precipitate and γ-matrix phases [55]. The phase
compositions at, and away from, the interface evolve temporally, and the interfacial widths,
defined as the distance between the plateau concentrations of the two phases, decrease with
aging time, Figure 6.8. Interestingly, an accumulation of Ni is observed ~ 2 nm into the γ-matrix.
We note that there is no evidence of non-monotonic (confined) segregation of Ta at the γ’-
precipitate/γ-matrix heterophase interface.
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Figure 6.7. The elemental concentration profiles across the primary γ’-precipitate/γ-matrix heterophase interface for a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K. The phase compositions evolve temporally, and the widths of the concentration profiles decrease with increasing aging time. An accumulation of Ni is observed to develop on the γ-matrix side of the interface, evidence of a kinetic effect associated with the addition of Ta. The concentration profiles shown here represent the average of all of the interfaces in the analyzed volume.
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Figure 6.8. The temporal evolution of the widths of the concentration profiles across the primary γ’-precipitate/γ-matrix interface for a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K. The interfacial widths are observed to decrease with increasing aging time. The species with the largest diffusivity in the γ-matrix, Al and Ta, have the smallest interfacial widths, while the opposite is true for Ni and Cr, the slower diffusing species.
6.3.3 Partitioning of elemental species
The partitioning behavior of the elements can be determined quantitatively by calculating
the partitioning ratio, Kiγ’/γ; the ratio of the concentration of an element i in the γ’-precipitates, to
the concentration of the same element i in the γ-matrix, Figure 6.9. Aluminum and Ta are
observed to partition to the γ’-precipitates, while Cr and Ni partition to the γ-matrix. The strong
partitioning of Ta to the γ’-precipitates increases the partitioning of Cr to the γ-matrix and of Al
to the γ’-precipitates, when compared to the results obtained for a ternary Ni-10.0 Al-8.5 Cr at.%
alloy [55, 56]. For example, for a sample aged at 1073 K for 256 h, the value of KCrγ’/γ decreases
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from 0.63±0.01 to 0.26±0.03 and the value of KAlγ’/γ increases from 2.20±0.02 to 3.17±0.01, due
to the formation of a significantly higher volume fraction of γ’-precipitates. The value of KTaγ’/γ
for the Ni-Al-Cr-Ta alloy at 256 h is 10.25±0.07, and the partitioning of Ni is unaffected by the
addition of Ta.
Figure 6.9. The temporal evolution of the partitioning ratios, Kiγ’/γ, of the constituent elements i
of: (a) Ni-10.0 Al-8.5 Cr-2.0 Ta; and (b) Ni-10.0 Al-8.5 Cr at.% aged at 1073 K. In both alloys, Al is observed to partition to the γ’-precipitates, while Cr and Ni partition to the γ-matrix. Ta shows a strong preference for the γ’-phase.
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6.4 Discussion
6.4.1 Morphological development
The morphology of the γ’-precipitates is observed to evolve from a bimodal distribution
of spheroidal γ’-precipitates in the as-quenched state, to cuboids with significantly rounded
edges, and finally to cuboids and rectangular parallelepipeds aligned along orthogonal <001>-
type directions. The γ’-precipitates that form the bimodal distribution in the as-quenched state
nucleate rapidly during the quench due to the small barrier height to nucleation caused by the
large supersaturations of alloying elements. The larger γ’-precipitates nucleate first, and
subsequently undergo high-temperature growth. As the alloy cools, however, the diffusion fields
associated with the primary γ’-precipitates shrink, and growth slows [234]. With additional
cooling, a significant solute supersaturation develops in the interprecipitate spaces, resulting in a
secondary burst of precipitation, Figure 6.2. In a recent study of multimodal precipitation in the
commercial nickel-based superalloy Udimet 720 Li, Radis et al. [234] found that multimodal
precipitation is highly dependent on alloy cooling rate, and noted that this phenomenon has been
observed to occur in other nickel-based alloys during continuous cooling.
The nucleation of γ’-precipitates during the quench from 1503 K was modeled with
PrecipiCalc, which incorporates the results of the classical models of nucleation, growth and
coarsening. These calculations were performed assuming a continuous quench rate of 20 K s-1,
and a γ’-precipitate/γ-matrix interfacial free energy of 25 mJ m-2, based on results from
concentrated Ni-Al-Cr alloys [42, 52] aged at 873 K. The γ/γ’ solvus point was determined
experimentally by differential thermal analysis (DTA) to be 1393 K, which agrees well with the
value of 1412 K predicted by Thermo-Calc and employed in the PrecipiCalc modeling. The
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results of the PrecipiCalc modeling are shown in Figure 6.10, and predict a bimodal distribution
of γ’-precipitates. Primary γ’-precipitate nucleation is predicted to begin at a temperature of 1380
K due to a bulk free energy driving force for nucleation, ΔFnuc, of -123 J mol-1, and continues to
a temperature of 1357 K. Primary γ’-precipitate growth follows nucleation, and values of <R(t)>
of 28.2 nm, Nv(t) of 2.7 x 1021 m-3, and φ of 25.7 % are predicted for a temperature of 1194 K.
According to the simulation results, significant secondary γ’-precipitate nucleation begins at a
temperature of 1190 K due to a value of ΔFnuc of -132 J mol-1, and continues throughout the
remainder of the quench. The final as-quenched microstructure is predicted to be a bimodal
distribution of γ’-precipitates, with primary γ’-precipitates with <R(t)> and φ values of 30.2 nm
and 31.5 %, respectively, and secondary γ’-precipitates with <R(t)> and φ values of 0.8 nm and
14.2 %, respectively. The predicted values of Nv(t) for the primary and secondary γ’-precipitates
are 2.7 x 1021 and 3.4 x 1025 m-3, respectively.
The PrecipiCalc predictions of the values <R(t)>, φ and Nv(t) for the primary γ’-
precipitates of 30.2 nm, 31.5 % and 2.7 x 1021 m-3 are in good agreement with the measured
values of 30-40 nm, 28±13% and 3±2 x 1021 m-3. The PrecipiCalc predictions for the secondary
γ’-precipitate properties of 0.8 nm, 14.2 % and 3.4 x 1025 m-3 differ, however, from the
experimental values of <R>, φ and Nv(t) of 6.4±1.1 nm, 11±1% and 1.3±0.1 x 1023 m-3,
respectively. The accuracy of the predictions for the primary γ’-precipitates indicates that the
thermodynamics and kinetics of the γ/γ’ phase transformation are modeled correctly.
Inaccuracies in the predictions of the properties of the secondary γ’-precipitates may be due to
the limitations in of the extant models of nucleation and growth upon which PrecipiCalc is based.
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These models are best suited for dilute, misfit-free, binary alloys with near-zero volume fractions
of the secondary phase, and hence very small elemental supersaturations. These conditions are
not satisfied for a concentrated quaternary Ni-Al-Cr-Ta alloy with an equilibrium precipitated
volume fraction of 37.0 ± 9.0 %, and a complex bimodal microstructure in the as-quenched state.
We note that efforts to improve the predictive power of this modeling technique, and to
incorporate the effects of variables such as lattice misfit, are ongoing [217]. The PrecipiCalc
predictions are highly dependent on the choice of the alloy cooling rate, the shape of the cooling
curve, the γ/γ’-interfacial energy, and the kinetics and thermodynamics of the alloy during the
quench. Thus, the modeling predictions can be modified by adjusting any of these parameters.
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Figure 6.10. The values of the γ’-precipitate number density, Nv(t ), radius, <R(t )>, and volume fraction, φ, during the water quench from 1503 K for Ni-10.0 Al-8.5 Cr-2.0 Ta at.%, as modeled by PrecipiCalc. A bimodal distribution of γ’-precipitates is predicted for the as-quenched state with primary γ’-precipitates with <R(t)> and φ values of 30.2 nm and 31.5 %, and secondary γ’-precipitates with <R(t)> and φ values of 0.8 nm and 14.2 %. The predicted values of Nv(t) for the primary and secondary γ’-precipitates are 2.7 x 1021 and 3.4 x 1025 m-3, respectively.
Figure 6.11 shows the PrecipiCalc predictions of the values of ΔFnuc, the effective
nucleation rate, Jeff, and the critical radius for nucleation, R*, during the quench. Decreases in the
magnitude of ΔFnuc are predicted at 1380 and 1190 K due to the nucleation of primary and
secondary γ’-precipitates, respectively. For the remainder of the quench, the driving force for
phase separation increases rapidly. This is due to the increasing γ-matrix supersaturation values
that are not sufficiently consumed by primary γ’-precipitation growth, due, in turn, to the
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exponentially decreasing diffusivities with decreasing temperature. As expected, the predicted
values of R* decrease rapidly as the magnitude of ΔFnuc increases.
Figure 6.11. The evolution of the values of the bulk driving force for nucleation, ΔFnuc, the effective nucleation rate, Jeff, and the critical radius for nucleation, R*, during the quench. Decreases in the magnitude of ΔFnuc are predicted at 1380 and 1190 K due to the nucleation of primary and secondary γ’-precipitates, respectively. The magnitude of ΔFnuc increases and the values of R* decrease during the quench. This figure was calculated using PrecipiCalc.
The composition of the primary γ’-precipitates in the as-quenched state is measured to be
76.04±0.04 Ni-17.02±0.02 Al-3.37±0.02 Cr-3.57±0.03 Ta at.%, while the secondary γ’-
precipitates have a composition of 76.05±0.05 Ni-16.82±0.03 Al-4.60±0.03 Cr-2.53±0.03 Ta
at.%. We note that the composition of the larger γ’-precipitates is closer to the equilibrium
composition of 75.24±0.04 Ni-16.48±0.07 Al- 3.20±0.06 Cr-5.03±0.07 Ta at.% estimated from
APT data. The primary γ’-precipitates reject Cr to the γ-matrix, and preferentially consume the
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supersaturation of Ta, which partitions strongly to the γ’-phase. As such, the secondary γ’-
precipitates nucleate from a Cr-enriched and Ta-depleted γ-matrix, resulting in Cr and Ta
concentrations that are 36% larger and 29% smaller, respectively, than those measured for the
primary γ’-precipitates.
The morphological development of nickel-based superalloys has been shown to proceed
from spheroidal to cuboidal to cuboidal arrays and finally to solid-state dendrites [83]. The γ’-
precipitates of the Ni-Al-Cr-Ta alloy studied herein undergo the first two transitions,
transforming from spheroids-to-cuboids at an aging time of 0.25 h, and forming cuboidal arrays
by an aging time of 4 h. The sizes at which these transitions occur are a function of the
magnitude of the lattice parameter misfit, and are partially diffusion controlled [83]. The elastic
self-energy of a precipitate increases as R3, while the heterophase interfacial free energy
increases as R2. Thus, as growth and coarsening proceed, and the average γ’-precipitate radius
increases, the elastic energy ultimately determines the precipitate morphology, and is decreased
by the formation of cuboidal γ’-precipitates. The evolution of the spheroidal γ’-precipitates into
cuboids in the Ni-Al-Cr-Ta alloy commences at an aging time of 0.25 h, when the γ’-precipitates
have an average radius of 61±7 nm. In contrast, the radius at which the microstructure becomes
cuboidal for the Ni-Al-Cr alloy was estimated to be ~ 88 nm at an aging time of 64 h at 1073 K
[56]. The lattice parameters of the γ-matrix and the γ’-precipitates of the Ni-Al-Cr-Ta alloy are
estimated from PrecipiCalc to be 0.3589 and 0.3603 nm, respectively, resulting in a lattice
parameter misfit, δ, of 0.0039. The value of δ of the Ni-Al-Cr alloy was estimated to be 0.0022 ±
0.0007 [56]. The addition of 2.0 at.% Ta increases the lattice parameter misfit by 78 %, resulting
in a spheroid to cuboid transition that occurs at a smaller γ’-precipitate size, and at an earlier
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aging time. For aging times of 64 and 256 h, the γ’-precipitates have a non-equiaxed, rectangular
parallelepiped morphology with {001}-type facets. The parallelepipedic morphology of the γ’-
precipitates further minimizes the elastic energy [209, 210].
The alignment of the γ’-precipitates results from the minimization of the elastic
interactions between the γ'-precipitates, where the elastic interaction energy depends on the
elastic anisotropy, the difference in the elastic constants of the two phases, and the sign and
magnitude of the misfit strain [235-237]. The alignment of γ’-precipitates into cuboidal and
parallelepipedic arrays was found to occur after 64 h of aging at 1073 K for the base Ni-Al-Cr
alloy [56], while the formation of arrays begins at 4 h for the Ni-Al-Cr-Ta alloy. The increased
lattice parameter misfit due to the addition of Ta leads to γ’-precipitate alignment at smaller
precipitate radii and earlier aging times in the Ni-Al-Cr-Ta alloy. The cuboidal γ’-precipitates in
both alloys have a coherent cube-on-cube relationship with the γ-matrix, with clearly defined
{001}-type facets, and are aligned along the elastically soft <001>-type directions, Figure 6.6.
6.4.2 Compositional evolution
From the APT results, Al and Ta partition strongly to the γ’-precipitate phase, while Ni
and Cr partition to the γ-matrix phase, in agreement with past experimental results [226, 229,
230, 238-241]. The equilibrium composition of the γ’-precipitates suggests that Al, Ta and Cr
occupy the Al sublattice sites of the L12-ordered γ’-phase of the Ni-Al-Cr-Ta alloy at 1073 K. A
first-principles study, discussed in detail in Chapter 7, confirms that Ta and Cr substitute
preferentially at the Al sublattice sites of the Ni3Al-L12 structure [61], in agreement with
previous work on this subject [138, 240, 242-257].
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The measured γ’-precipitate volume fractions, Table 6.1, are close to their equilibrium
values, suggesting that the Ni-Al-Cr-Ta alloy may be undergoing quasi-stationary state
coarsening. The first comprehensive mean-field treatment of precipitate coarsening, due to
Lifshitz and Slyozov [125] and Wagner [126], is limited to very dilute binary alloys with
spatially-fixed spherical precipitates whose initial compositions are equal to their equilibrium
values. Umantsev and Olson (UO) demonstrated that the exponents of the temporal power-laws
predicted for binary alloys by LSW-type models are identical for concentrated multi-component
alloys, but that the explicit expressions for the rate constants depend on the number of
components [130]. In the UO model, the asymptotic solution for the supersaturation of solute
element i, ΔCi(t), the difference between the concentration in the far-field γ-matrix, >< )(, tC ffiγ ,
and the equilibrium γ-matrix solute-solubility, )(C eq,i ∞γ , is:
, , 1/3,( ) ( ) ( ) ;ff eq
i i i i UOC t C t C tγ γγ κ −Δ =< > − ∞ = (6.1)
where UO,iκ is the coarsening rate constant for )t(CiγΔ and is related to the partial derivatives of
the chemical potentials, the γ/γ’ interfacial free energy, the partitioning ratios of the alloying
elements, and the diffusivities of the elements [130]. The temporal evolution of the
supersaturations of the alloying elements in the γ-matrix and γ’-precipitates are observed to
follow approximately the t-1/3 prediction of the UO model, Figure 6.12. The supersaturations of
Ni, Al, Cr and Ta in the γ-matrix evolve as t-0.33±0.04, t-0.35±0.05, t-0.31±0.04 and t-0.30±0.04, respectively,
while the supersaturations of the same elements in the γ'-precipitates evolve as t-0.33±0.03, t-0.36±0.05,
t-0.28±0.04 and t-0.36±0.03, respectively. Quasi-stationary state coarsening has begun by an aging time
of 1 h, and continues throughout the aging times studied here.
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Figure 6.12. The magnitude of the values of the supersaturations, )(tCiΔ , of Ni, Al, Cr and Ta in the γ-matrix and γ’-precipitates for Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K. The magnitudes of the )(tCiΔ values decrease as approximately t-1/3 in the coarsening regimes for both phases, as predicted by the Umantsev and Olson (UO) models for quasi-stationary state coarsening.
The values of )(, ∞eqiC γ and )(,' ∞eq
iC γ estimated from APT data employing Equation 6.1
are listed in Table 6.2, and compared to the equilibrium values predicted by Thermo-Calc, and
the concentrations measured by APT and ICP analysis at 256 h. The phase compositions after
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256 h of aging are near their equilibrium values, suggesting that the magnitudes of the γ-matrix
supersaturations are small and that the γ’-precipitate growth and coarsening regime is nearly
complete.
Table 6.2. Equilibrium compositions (at.%) of the γ’-precipitate, )(C eq,'i ∞γ , and γ-matrix,
)(C eq,i ∞γ , determined by APT, ICP chemical analysis, and thermodynamic modeling employing
Thermo-Calc for Ni-10.0 Al-8.5 Cr-2.0 Ta at.% aged at 1073 K.
Technique: )(C eq,'Ni ∞γ )(,' ∞eq
AlCγ )(,' ∞eqCrCγ )(,' ∞eq
TaCγ
APT, 256 h: 75.37±0.05 16.56±0.08 3.25±0.09 4.82±0.09
APT, extrapolated: 75.24±0.08 16.48±0.07 3.20±0.06 5.03±0.07
ICP, 256 h: 75.56 16.57 3.28 4.58
Thermo-Calc: 76.47 17.00 2.66 3.88
Technique: )(, ∞eqNiC γ )(, ∞eq
AlC γ )(, ∞eqCrC γ )(, ∞eq
TaC γ
APT, 256 h: 81.65±0.01 5.23±0.03 12.66±0.03 0.47±0.03
APT, extrapolated: 81.68±0.04 5.18±0.08 12.70±0.02 0.44±0.06
ICP, 256 h: 83.17 5.49 10.74 0.60
Thermo-Calc: 81.37 5.68 12.11 0.84
The most striking feature of the concentration profiles across the γ’-precipitate/γ-matrix
heterophase interface, Figure 6.7, is the accumulation of Ni in the γ-matrix, roughly 2 nm from
the interface. This effect was observed at all γ/γ’ interfaces, regardless of direction, and is likely
kinetic in origin, resulting from a decrease in the diffusivity of Ni in the γ-matrix. Estimates of
the tracer diffusivities from Dictra predict a threefold decrease in the calculated γ-matrix tracer
diffusivity of Ni, γNiD , from 6.06 x 10-18 m2 s-1 to 2.02 x 10-18 m2 s-1 due to a 2.0 at.% addition of
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Ta, Table 6.3. This decrease in Ni diffusivity is significant because Ni becomes the least mobile
species in the γ-matrix phase. Hence, as phase decomposition proceeds, Ni partitions to the γ-
matrix and accumulates on the γ-matrix side of the γ'/γ interface. The diffusivities of all of the
atomic species decrease due to the addition of Ta, thus the γ’-precipitate coarsening kinetics are
expected to decrease. We note that the calculations of the γ-matrix diffusivities used herein are
dependent on the thermodynamic predictions from the Saunders database. Since the Saunders
database predicts an equilibrium partitioning ratio for Ta of 4.61, while a value of 11.4 is
measured herein, the predictions of the database are not absolutely accurate. As such, the
predictions of the γ-matrix kinetics likely suffer from some inaccuracy, as previously observed in
comparisons of microanalysis results of diffusion couples to Dictra predictions [217].
Table 6.3. Tracer diffusivity of element i in the γ-matrix, γ
iD , of Ni-10.0 Al-8.5 Cr and Ni-10.0 Al-8.5 Cr-2.0 Ta at.% calculated with Dictra, and Campbell [208] and Saunders [115] databases at 1073 K.
Alloy (at.%) γNiD
(10-18 m2s-1)
γAlD
(10-18 m2s-1)
γCrD
(10-18 m2s-1)
γTaD
(10-18 m2s-1)Ni-10.0 Al-8.5 Cr 6.06 13.9 5.13 -
Ni-10.0 Al-8.5 Cr-2.0 Ta 2.02 12.6 4.80 5.14
The widths of the concentration profiles across the γ’-precipitate/γ-matrix interface are
observed to decrease with increasing aging time, Figure 6.8, which is a kinetic effect. As aging
progresses, the two species with the largest predicted diffusivities in the γ-matrix, Al and Ta,
Table 6.3, have the smallest interfacial widths, while the opposite is true for Ni and Cr, the
slower species. This is expected, though the diffusivities of the atomic species in the γ’-
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precipitate, for which there are no extant kinetic databases that include Ta, also play an important
role in determining the interfacial widths. The interfacial widths of model nickel-based
superalloys have been measured to range from a single atomic layer to up to 5 nm, depending on
the alloy system and temperature studied [39, 222, 258].
6.4.3 Comparison with other Ni-Al-Cr-X alloys
This work is part of a systematic investigation of the effects of dilute refractory additions
on the γ/γ’ phase decomposition of a model Ni-10.0 Al-8.5 Cr at.% alloy aged at 1073 K. A 2.0
at.% addition of W [55, 56] was previously found to increase the microhardness and decrease the
coarsening kinetics of the γ’-phase. Tungsten was found to partition preferentially to the γ’-
precipitates, with no evidence of non-monotonic (confined) segregation of W at the heterophase
interface. The addition of W led to stronger partitioning of Al to the γ’-precipitates, and of Cr to
the γ-matrix, than was measured for the base Ni-Al-Cr alloy. The Ni-Al-Cr-W alloy, like Ni-Al-
Cr-Ta and the base Ni-Al-Cr alloys, undergoes a γ’-precipitate morphological evolution from
spheroids-to-cuboids, as well as alignment of the cuboids along the elastically-soft <001>-type
directions.
A 2.0 at.% addition of Re [58, 59] was found to result in the strong partitioning of Re to
the γ-matrix, and increased partitioning of Al to the γ′-precipitates, and of Cr to the γ-matrix.
Rhenium was found to decrease the coarsening kinetics, and to stabilize the spheroidal
morphology of the γ’-precipitates, allowing the γ’-phase to remain spheroidal after aging to 256
h. A high degree of γ’-precipitate coagulation and coalescence was detected, in contrast with the
Ni-Al-Cr-Ta and Ni-Al-Cr-W alloys. Additionally, no evidence of confined (non-monotonic)
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interfacial Re segregation was detected for Ni-Al-Cr-Re, unlike commercial nickel-based
superalloys, such as René N6, which exhibits significant confined interfacial segregation of Re
of 2.41 atoms nm-2 [110, 111]. The effects of Re and Ta in modern nickel-based superalloys are
dramatically different, though both decrease the kinetics of γ-phase decomposition, leading to
phase stability at high temperature.
The accumulation of Ni on the γ-matrix side of the γ’/γ heterophase interface is unique to
the addition of Ta, due to a decrease in the value of γNiD from 6.06 to 2.02 x 10-18 m2 s-1 for a 2.0
at.% addition of Ta. The value of γNiD was calculated to decrease to 4.92 x 10-18 m2 s-1 [56] and
4.05 x 10-18 m2 s-1 [59] due to the additions of W and Re, respectively, leading to a deceleration
of the coarsening kinetics. The values of γNiD are, however, apparently still sufficient to transport
Ni atoms away from the γ’/γ interface during phase decomposition of the Ni-Al-Cr-W and Ni-Al-
Cr-Re alloys, avoiding Ni accumulation. The bimodal distribution of γ’-precipitates in the as-
quenched state is also unique to Ni-Al-Cr-Ta among the Ni-based alloys we have studied to date.
The bimodal distribution is due to the initially high value of the Ta supersaturation in the γ-
matrix, and to the relatively large diffusivity of Ta in the γ-matrix. The value of γTaD of 5.14 x 10-
18 m2 s-1 is significantly greater than the values of γWD and γ
ReD of 4.93 x 10-19 m2 s-1 and 1.07 x
10-19 m2 s-1, respectively, in the model Ni-Al-Cr alloys.
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6.5 Summary and conclusions
The effects of a 2.0 at.% addition of tantalum to a model Ni-10.0 Al-8.5 Cr at.% superalloy aged
at 1073 K for 0 to 256 h are assessed using mainly scanning electron microscopy and atom-probe
tomography (APT), leading to the following conclusions:
• The addition of Ta results in a 47±5% increase in microhardness due to increases in
solid-solution strengthening and to a dramatic increase in the volume fraction of the γ’-
phase. After 256 h of aging, the value of φ from phase extraction results is 18.9% for Ni-
Al-Cr and 36.4% for the Ni-Al-Cr-Ta alloy. The equilibrium volume fraction of the γ’-
phase for the Ni-Al-Cr-Ta alloy is estimated by APT to be 37.0 ± 9.0 %.
• The morphology of the γ’-precipitates evolves from a bimodal distribution of spheroidal
precipitates in the as-quenched state, to primary γ’-precipitate cuboids with significantly
rounded edges, and finally to cuboidal and parallelepipedic γ’-precipitates with clearly
defined {001} facets, aligned along the elastically soft <001> directions.
• The morphology of the γ’-precipitate phase is spheroidal in the as-quenched state, and a
bimodal particle size distribution is apparent. The smaller secondary γ’-precipitates are
observed to have an average radius of 6.4±1.1 nm, and the larger primary γ’-precipitates
have radii on the order of 30-40 nm. A solute-depleted, precipitate-free shell with a
thickness of 5.7±1.1 nm surrounds the primary γ’-precipitates. Precipitation modeling
employing the commercial software program PrecipiCalc predicts that primary and
secondary γ’-precipitate nucleation begin at 1380 and 1190 K, respectively. The smaller
secondary γ’-precipitates are only observed in the as-quenched state and disappear
between 0 and 0.25 h of aging at 1073 K due to coarsening of the primary γ’-precipitates.
156
• The transition from a spheroidal-to-cuboidal morphology commences at an aging time of
0.25 h, when the primary γ’-precipitates have an average radius of 61±7 nm. In contrast,
the radius at which the microstructure becomes cuboidal for the base Ni-Al-Cr alloy is
estimated to be ~ 88 nm at an aging time of 64 h at 1073 K [56]. The addition of 2.0 at%
Ta increases the lattice parameter misfit, δ, by 78 %, from 0.0022 ± 0.0007 [56] to
0.0039. This increase in δ results in a spheroid-to-cuboid transition that occurs at a
smaller γ’-precipitate radius, and at an earlier aging time, than for the base ternary Ni-Al-
Cr alloys, to minimize the free energy associated with elastic misfit.
• Alignment of primary γ’-precipitates into cuboidal and parallelepipedic arrays occurs
after 4 h of aging at 1073 K for the model Ni-Al-Cr-Ta alloy, while the formation of
arrays was observed to begin after 64 h for the base Ni-Al-Cr alloy. The increased lattice
parameter misfit due to the addition of Ta leads to alignment at smaller primary γ’-
precipitate radii and earlier aging times in the Ni-Al-Cr-Ta alloy.
• Aluminum and Ta partition to the γ’-precipitates, while Cr and Ni partition to the γ-
matrix. The strong partitioning of Ta to the γ’-precipitates increases the partitioning of Cr
to the γ-matrix and of Al to the γ’-precipitates, when compared to the results of the base
Ni-Al-Cr alloy.
• The concentration profiles of the constituent elements of the Ni-Al-Cr-Ta alloy across all
γ’-precipitate/γ-matrix interfaces exhibit an accumulation of Ni on the γ-matrix side of the
interface. This effect is likely due to the diminution of the mobility of Ni in the γ-matrix
caused by the addition of Ta, and was not previously observed in alloys with 2.0 at.%
additions of W or Re.
157
• The widths of the concentration profiles across the γ’-precipitate/γ-matrix interface are
observed to decrease with increasing aging time. Upon aging, the atomic species with the
largest diffusivities in the γ-matrix, Al and Ta, have the smallest interfacial widths, while
the opposite is true for Ni and Cr, the slower species.
• For aging times of 1 h and longer, the temporal evolutions of the supersaturations of the
alloying elements in the γ-matrix and γ’-precipitates are observed to follow
approximately the t-1/3 prediction of the Umantsev-Olson quasi-stationary state
coarsening model.
158
Chapter 7
Chromium and tantalum site substitution patterns in Ni3Al (L12) γ’-
precipitates
Abstract
The site substitution behavior of Cr and Ta in the Ni3Al (L12)-type γ’-precipitates of a Ni-Al-Cr-
Ta alloy is investigated by atom-probe tomography (APT) and first-principles calculations.
Measurements of the γ’-phase composition by APT suggest that Al, Cr and Ta share the Al
sublattice sites of the γ’-precipitates. The calculated substitutional energies of the solute atoms at
the Ni and Al sublattice sites indicate that Ta has a strong preference for the Al site, while Cr has
a weak Al site preference. Furthermore, Ta is shown to replace Cr at the Al sublattice sites of the
γ’-precipitates, altering the elemental phase partitioning behavior of the Ni-Al-Cr-Ta alloy.
159
7.1 Introduction
Modern nickel-based superalloys are used in land-based and aerospace turbine engines at
operating temperatures up to 1373 K. The high-temperature strength and creep resistance of
these alloys are due primarily to strengthening by the precipitated γ’-(L12) phase, which can
accommodate substantial solute additions [3]. The mechanical properties of the γ’-phase depend
on the sublattice site substitution behavior of these alloying additions [80], motivating an
investigation of the site occupancy of Cr and Ta in a model Ni-Al-Cr-Ta alloy that has been
studied by atom-probe tomography (APT) and scanning electron microscopy [60].
The addition of Cr to Ni-Al alloys reduces the lattice parameter misfit between the γ’-
phase and the γ-matrix, often leading to spheroidal, nearly misfit-free γ’-precipitates [12].
Tantalum provides solid-solution strengthening and increases the volume fraction of the γ’-
precipitate phase, thereby providing significant strengthening while improving γ’-phase stability
and service life [3]. Previous APT [240], atom-probe field-ion microscope [242, 243], and x-ray
analysis experiments [244-246], as well as results of first-principles calculations [247-251], the
cluster variation method [252-254], and other techniques [255-257], have indicated that Ta
occupies the Al sublattice sites in ordered Ni3Al. These findings contradict experimental results
from ion-channeling and nuclear-reaction analysis studies [259] and research based on short-
range ordering employing pseudopotential approximations [260], which claim that Ta occupies
the Ni sublattice sites in Ni3Al. The site preference of Cr has generally been found to depend on
the alloy composition and aging treatment [244, 247, 249, 253, 257, 260, 261], though some
studies concluded that Cr occupies the Al sites, regardless of composition [246], or temperature
[262].
160
7.2 Experimental
We present APT results and first-principles calculations that elucidate the site
substitution behavior of Cr and Ta in the γ’-phase of a Ni-Al-Cr-Ta superalloy aged at 1073 K. A
Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy was homogenized at 1573 K for 20 h, then annealed in the
γ−phase field at 1503 K for 3 h, and water-quenched. Ingot sections were aged at 1073 K under
flowing argon for times ranging from 0.25 to 256 h, then water quenched, and microtip
specimens were prepared for study by APT. Pulsed-laser APT data collection was performed at
an evaporation rate of 0.04 ions pulse-1, a specimen temperature of 40.0±0.3 K, a pulse energy of
0.6 nJ pulse-1, a pulse repetition rate of 200 kHz, and a gauge pressure of < 6.7 x 10-8 Pa. APT
data were visualized and analyzed with IVAS™ 3.0 (Imago Scientific Instruments). Figure 7.1
shows an APT reconstruction that intersected a cuboidal γ’-precipitate in a sample aged for 256
h. The strong partitioning of Al and Ta, shown in orange and red, respectively, to the γ’-phase,
and Cr, in blue, to the γ-phase is demonstrated.
Figure 7.1. APT reconstruction of a cuboidal γ’-precipitate in a model Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 256 h. The elements that partition to the γ’-precipitates, Al and Ta, are shown in red and orange, respectively, while Cr, which partitions to the γ-matrix, is shown in blue; Ni atoms are omitted for clarity.
161
7.3 Results and discussion
The composition profiles across the γ’/γ interface in Figure 7.1 are displayed in Figure
7.2. The measured composition of the γ’-precipitates is 75.37±0.05 Ni- 16.56±0.08 Al-
3.25±0.09 Cr- 4.82±0.09 Ta at.%, suggesting that Cr and Ta occupy the Al sites of the Ni3Al γ’-
precipitates. The relatively Cr rich γ’-precipitate composition of 76.49±0.13 Ni- 17.47±0.12 Al-
6.04±0.07 Cr at.% measured for the reference Ni-Al-Cr alloy [55, 56], indicates that the addition
of 2.0 at.% Ta leads to the preferential replacement of Cr by Ta in the γ’-precipitates.
Figure 7.2. The elemental concentration profiles across the γ’/γ interface for a Ni-10.0 Al-8.5 Cr-2.0 Ta at.% alloy aged at 1073 K for 256 h. Tantalum and Al partition to the γ’-precipitates, while Cr partitions to the γ-matrix.
162
First-principles calculations were performed to confirm the site substitution preferences
of the solute atoms in the γ’-precipitates and to determine the thermodynamic driving force for
the replacement of Cr by Ta atoms in the γ’-phase. The calculations employed the plane-wave
pseudopotential total energy method with generalized gradient approximations [263], as
implemented in the Vienna ab initio simulation package (VASP) [149-151], using the projector
augmented-wave (PAW) potentials [264]. A plane wave cutoff energy of 300 eV and 8×8×8
Monkhorst-Pack k-point grids were used, and found to be sufficient to give fully converged
results. A three-dimensional periodic supercell with 2×2×2 unit cells (32 atoms) was employed
to determine the total energies of the cells, which converge to 2 x 10−5 eV atom−1, while residual
forces converge to 0.005 eV nm−1. The Ni3Al (L12) structure was fully relaxed, and the lattice
parameter was determined to be 0.3566 nm, in good agreement with the experimental room
temperature value of 0.3570 nm [3].
The antisite formation energies associated with the Ni3(Al1-yNiy) and (Ni1-xAlx)3Al
structures were calculated and defined as:
)()(313 )( Ni
totAlNiAl
totNiAlNiAlNi EEE
yyμμ +−+=
−→ ; (7.1)
)()(331 )( Al
totAlNiNi
totAlAlNiNiAl EEE
xxμμ +−+=
−→ ; (7.2)
where μi is the chemical potential per atom of the bulk element i assuming same cell symmetry,
and are -5.432 and -3.697 eV atom-1 for Ni and Al, respectively. The antisite formation energies
are ENi→Al = 0.986 eV atom-1 and EAl→Ni = 0.742 eV atom-1, and are energetically unfavorable.
Other first-principles studies have found that the antisite formation energies are similar and range
from 0.51 to 0.99 eV [265-267]. Another investigation calculated the Al and Ni antisite energies
to be -0.92 and 2.04 eV, respectively [251].
163
Four substitutional structures, (Ni1-xCrx)3Al, Ni3(Al1-yCry), (Ni1-xTax)3Al and Ni3(Al1-
yTay), were modeled by substituting one Cr or Ta atom at one of the Ni or Al sublattice sites.
Table 7.1 summarizes the calculated total energies of the relaxed structures, Etot, and the
substitutional energies, EZ→Ni and EZ→Al, (Z = Cr, Ta), which are defined as follows:
)()(331 )( Z
totAlNiNi
totAlZNiNiZ EEE
xxμμ +−+=
−→ ; (7.3)
)()(313 )( Z
totAlNiAl
totZAlNiAlZ EEE
yyμμ +−+=
−→ ; (7.4)
where the chemical potentials of Cr and Ta are -9.103 and -11.478 eV atom-1, respectively. The
calculated results demonstrate that Ta atoms strongly prefer to occupy the Al sublattice sites of
Ni3Al, as ETa→Al is significantly smaller than ETa→Ni.. Chromium weakly prefers to substitute at
the Al sites, as ECr→Al is only slightly smaller than ECr→Ni. The substitutional energy associated
with the replacement of Cr atoms by Ta atoms on the Al sublattice sites of the Ni3Al structure is
calculated as follows:
)()( )()( 1313 Tatot
CrAlNiCrtot
TaAlNiCrTa yyyyEEE μμ +−+=
−−→ ; (7.5)
and is -1.038 eV atom-1. Thus the replacement of Cr by Ta in the γ’-precipitates is energetically
favorable, and is observed by APT to occur.
To verify that the substitutional patterns of Cr modeled herein result in the lowest energy
configurations, two other substitutional patterns (Ni1-xAlx)3(Al1-yCry) and (Ni1-xCrx)3(Al1-y-
Niy) were modeled. These structures may form by the occupation of a Ni or Al sublattice by a Cr
atom, and the displacement of the Ni or Al atom to an antisite. These structures are also found to
164
be energetically unfavorable, yielding formation energies of 1.138 and 1.337 eV atom-1 for
(Ni1-xAlx)3(Al1-yCry) and (Ni1-xCrx)3(Al1-yNiy), respectively.
Table 7.1. Total, Etot, and substitutional, EZ→Ni, Al, energies determined by first-principles calculations. (For these calculations x = 0.042, y = 0.125, and Z = Cr, Ta.)
Etot (eV) EZ→Ni, Al (eV atom-1)
Ni3Al -174.944 -
(Ni1-xCrx)3Al -177.966 0.648
Ni3(Al1-yCry) -179.748 0.565
(Ni1-xTax)3Al -180.957 0.033
Ni3(Al1-yTay) -183.198 -0.473
The local stresses and strains due to site substitution result in the average atomic forces
and displacements at the first nearest-neighbor distance displayed in Table 7.2. We note that no
atomic force was measured for relaxed Ni3Al prior to site substitution. The substitution of Ta at
the Al sublattice sites results in smaller values of the atomic force and displacement than
substitution of Ta at the Ni sites. The atomic forces and displacements associated with Cr
substitution at the Ni and Al sites are comparable, providing supporting evidence that Cr only
weakly prefers Al sublattice sites.
165
Table 7.2. Average atomic forces and displacements at the first nearest-neighbor distance from first-principles calculations for four different substitutional structures.
Average Atomic Force (eV Ǻ-1) Average Atomic Displacement (Ǻ)
(NixCr1-x)3Al 0.0150 0.0294
Ni3(AlyCr1-y) 0.0112 0.0325
(NixTa1-x)3Al 0.0154 0.0479
Ni3(AlyTa1-y) 0.0095 0.0215
From APT results, the addition of Ta to a Ni-Al-Cr alloy leads to the rejection of Cr from
the γ’-precipitates.. The elemental phase partitioning behavior is quantified by the partitioning
ratio, Kiγ’/γ, the ratio of the concentration of an element i in the γ’-precipitates to the
concentration of the same element in the γ-matrix. For example, for a sample aged at 1073 K for
256 h, the value of /CrK γ γ′ decreases from 0.63±0.01 to 0.26±0.03 and the value of /
AlK γ γ′
increases from 2.20±0.02 to 3.17±0.01, due to the formation of a significantly higher volume
fraction of Al-rich γ’-precipitates. The value of /TaK γ γ′ for the Ni-Al-Cr-Ta alloy at 256 h is
10.25±0.07, and the partitioning of Ni is unaffected by the addition of Ta.
7.4 Conclusions
To conclude, Ta and Cr substitute preferentially at the Al sublattice sites in Ni3Al (L12).
The substitution of Ta for Al is energetically favorable, while the substitution of Cr for Al is less
so. This results in the replacement of Cr by Ta at the Al sublattice sites and in the γ’-precipitates,
and alters the partitioning behavior of the elements in the Ni-Al-Cr-Ta alloy.
166
Chapter 8
Future Work
There are multiple possible extensions of the research presented herein. Given the
success of the atom-probe technique in analyzing the phase decomposition of Ni-Al-Cr-X alloys,
a natural continuation of this work would be more detailed simulation of these results,
employing, for example, molecular dynamics, phase-field modeling, first-principles calculations,
and lattice kinetic Monte-Carlo simulation. These detailed analyses would provide further insight
into the kinetic pathways that lead to phase decomposition in these alloys, and would provide
quantitative data to improve the extant theories of precipitate nucleation, growth and coarsening.
These experimental and simulated results also provide insight into precipitate coagulation and
coalescence, a mechanism that is critical to phase decomposition in these alloys, and has only
recently been examined in detail [39].
The γ/γ’ interfacial region is of critical importance in the phase decomposition of both
model and commercial nickel-based superalloys. This region has been shown to have a width on
the order of several nanometers for other model nickel-based superalloys [42, 52, 56, 58, 62,
222-224, 268]. Classical models of phase decomposition often ignore the interfacial region,
however, given the experimental and simulated evidence of the diffuseness of the γ/γ’ interface,
advanced models of phase separation will need to account for the interfacial region [269].
167
The energy of the γ/γ’ interface is of critical importance to the properties and structures of
nickel-based superalloys; it remains, however, very difficult to determine the energy directly. In
order to estimate the interfacial energy, one must resort to either indirect measurements, from the
results of the classical models of nucleation or coarsening, or directly through simulation or fist-
principles calculations [222]. Ardell [13, 145] first demonstrated that the value of σγ/γ’ can be
estimated by comparison of independent measurements of the kinetics of γ’-particle growth
during coarsening, and of the γ-matrix supersaturation to the predictions of classical LSW-type
models of Ostwald ripening [125, 126]. Attempts to remove many of the restrictions of the LSW
model have been made, such as efforts to incorporate the effects of finite volume fractions in
these models [128]. Calderon et al. developed a model for binary alloys that incorporates a finite
precipitate volume fraction, and allows for nonzero solubilities and nonideal solution
thermodynamics [270]. The results of the work by Calderon et al. were applied by Ardell to data
from Ni-3.5 and Ni-6.0 wt.% Al alloys aged at 898 and 988 K [145], and a Ni-12.5 at.% alloy
aged at 823-973 K [271], to yield values of σγ/γ’ ranging from 8 to.0 8.7 mJ m-2, assuming non-
ideal solution thermodynamics. Wang et al. estimated values of σγ/γ’ of ~8.5 to 13 mJ m-2 by the
cluster variation method for a Ni-18 Al at.% alloy at temperatures of 873 to 1573 K [272].
Mishin employed an embedded-atom-potential to calculate Ni/Ni3Al interfacial energies of 46,
28 and 12 mJ m-2 for the (100), (110) and (111) planes, respectively [268]. Recent atom-probe
tomographic investigations of Ni-Al-Cr alloys [28, 42, 46, 52] aged at 873 K have led to
estimates of σγ/γ’ of 20-25 mJ m-2, by employing the coarsening model developed by Kuehmann
and Voorhees for ternary alloys [132]. A thorough study of the effect of ternary additions on the
γ/γ’ interfacial energy of the binary Ni-Al system, employing both experimental and simulation
168
techniques, would provide a valuable tool for the design of complex, concentrated
multicomponent superalloys.
The work on Ni-Al-Cr-Ta is part of a systematic investigation of the effects of dilute
refractory additions, such as W [43, 55, 56], Re [50, 51, 57-59], and Ta [60-62], on the γ/γ’ phase
decomposition of a model Ni-10.0 Al-8.5 Cr at.% alloy aged at 1073 K. A very interesting
extension of this work would be a more detailed study of the interactions between refractory
additions in more complex model alloys containing more than one refractory addition. For
example, the interplay between Ta and W has recently been shown to be significant [63, 273],
thus the next logical step in this work would be a detailed study of a Ni-Al-Cr-Ta-W alloy,
employing both atom-probe tomography and first-principles simulations.
169
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Appendix 1
Recalculation of the Ni-Al-Cr Interfacial Free Energy from the Data of
Schmuck et al.
Schmuck et al. [28] determined a value for the interfacial free energy, σγ/γ’, of 12.5 mJ m-
2 for a ternary Ni-5.2 Al-14.8 Cr at.% alloy aged at 873 K. This value is approximately one half
the value determined by Sudbrack et al. [46] for a similar alloy Ni-5.2 Al-14.2 Cr at.% aged at
873 K. The Schmuck et al. approach assumes: (i) the LSW model for binary alloys applies to
their ternary system; (ii) the rate constant for the average γ’-precipitate radius from LSW is
correct, which it is not [270]; and (iii) that the γ-matrix and γ’-precipitate phase compositions
achieved their equilibrium compositions after aging for 64 h. We recalculate the values of σγ/γ’
from the data by Schmuck et al. by applying a relationship for σγ/γ’ in a nonideal, nondilute
ternary alloy proposed by Marquis and Seidman [147], based on the results of Calderon et al.
[270] and the Kuehmann-Voorhees coarsening model [132], and as applied to Ni-5.2 Al-14.8 Cr
at.% by Sudbrack et al. [46]. Schmuck et al. found that the γ’-precipitates had compositions very
close to their equilibrium values after only 1 h, and had achieved their equilibrium compositions
after 64 h. Sudbrack et al. [46], however, found that the γ’-precipitate phase compositions of a
similar alloy, Ni-5.2 Al-14.2 Cr at.% aged at 873 K, continued to evolve temporally at an aging
time of 1024 h. As a result, we use the equilibrium compositions calculated by Thermo-Calc for
Ni-5.2 Al-14.8 Cr at.%, employing the Saunders database [115], Table A.1.1, in our estimates of
the value of σγ/γ’ for the data of Schmuck et al. A coarsening rate constant, KKV, of (2.30 ± 0.13) x
191
10-31 m3 s-1 and values of γκ KVAl , and γκ KVCr , of 0.13 ± 0.01 at. fr. s1/3 and -0.08 ± 0.02 at. fr. s1/3,
respectively, are found from the data of Schmuck et al. These values should be compared to
values of KKV of (8.8 ± 3.3) x 10-31 m3 s-1 and of 0.19 ± 0.02 at. fr. s1/3 and -0.14 ± 0.05 at. fr. s1/3,
for γκ KVAl , and γκ KVCr , respectively, for Ni-5.2 Al-14.2 Cr at.%. The value of σγ/γ’ calculated from
the data of Schmuck et al. using Equation 3.9 is 20-21 ± 5 mJ m-2, which is in good agreement
with values of 22-23 ± 7 mJ m-2 for Ni-5.2 Al-14.2 Cr at.% and 23-25 ± 6 mJ m-2 for Ni-7.5 Al-
8.5 Cr at.%.
Table A.1.1 Equilibrium γ’-precipitate and γ-matrix equilibrium concentrations, as determined by atom- probe tomography (APT) and Thermo-Calc for Ni-5.2 Al-14.8 Cr at.% aged at 873 K from the data of Schmuck et al. [28].
Equilibrium composition of γ'-precipitates Ni (at.%) Al (at.%) Cr (at.%)
Measured by APT at 64 h [28]: 74 ± 2 18 ± 1 7.6 ± 0.8
Extrapolated from APT data [28]: 74.2 ± 0.9 18.4 ± 0.8 7.4 ± 0.5
Calculated with Thermo-Calc and Saunders database [115]: 74.91 16.08 9.01
Calculated with Thermo-Calc and Dupin database [116]: 75.79 14.44 9.76
Equilibrium composition of γ-matrix Ni (at.%) Al (at.%) Cr (at.%)
Measured by APT at 64 h [28]: 80.2 ± 0.3 4 ± 0.2 15.8 ± 0.2
Extrapolated from APT data [28]: 80.3 ± 0.2 3.94 ± 0.08 15.8 ± 0.2
Calculated with Thermo-Calc and Saunders database [115] : 80.80 3.50 15.70
Calculated with Thermo-Calc and Dupin database [116]: 80.65 3.77 15.58
192
Appendix 2
Temporal evolution of the phase compositions of Ni-Al-Cr alloys at 873 K
Table A.2.1. Temporal evolution of the phase compositions in Ni-7.5 Al-8.5 Cr at.% aged at 873 K, as measured by atom-probe tomography. The lattice parameter for this alloy is estimated to be 0.0027 ± 0.0004, and the equilibrium volume fraction is 16.4 ± 0.6 %.
Aging time (h) γ-matrix composition (at.%) γ’-precipitate composition (at.%)
Ni Al Cr Ni Al Cr
0.167 83.96 ± 0.05
7.37 ± 0.03
8.67 ± 0.03
72.14 ± 3.16
21.40 ± 3.01 6.46 ± 1.44
0.25 84.04 ± 0.05
7.09 ± 0.09
8.87 ± 0.10
73.07 ± 1.20
20.70 ± 1.09 6.23 ± 0.65
1 84.50 ± 0.04
6.63 ± 0.04
8.87 ± 0.04
73.86 ± 0.42
19.95 ± 0.38 6.19 ± 0.23
4 84.75 ± 0.04
6.20 ± 0.03
9.05 ± 0.03
74.79 ± 0.19
19.13 ± 0.17 6.08 ± 0.11
16 84.92 ± 0.06
5.88 ± 0.04
9.20 ± 0.04
75.38 ± 0.19
18.57 ± 0.17 6.05 ± 0.11
64 84.99 ± 0.05
5.76 ± 0.03
9.25 ± 0.04
75.73 ± 0.16
18.30 ± 0.16 5.97 ± 0.10
256 85.09 ± 0.02
5.61 ± 0.02
9.30 ± 0.02
75.92 ± 0.06
18.18 ± 0.06 5.90 ± 0.04
1024 85.13 ± 0.06
5.53 ± 0.07
9.34 ± 0.04
76.11 ± 0.09
18.02 ± 0.09 5.87 ± 0.05
∞ 85.19 ± 0.08
5.42 ± 0.09
9.39 ± 0.09
76.33 ± 0.12
17.82 ± 0.15 5.85 ± 0.12
193
Table A.2.2. Temporal evolution of the phase compositions in Ni-5.2 Al-14.2 Cr at.% aged at 873 K, as measured by atom-probe tomography. The lattice parameter misfit for this alloy is estimated to be 0.0006 ± 0.0004 and the equilibrium volume fraction is 15.7 ± 0.7 %.
Aging time (h) γ-matrix composition (at.%) γ’-precipitate composition (at.%)
Ni Al Cr Ni Al Cr
0.167 80.59 ± 0.09
5.19 ± 0.05
14.22 ± 0.08 71.3 ± 3.1 19.1 ± 2.7 9.7 ± 2.1
0.25 80.73 ± 0.09
5.07 ± 0.05
14.20 ± 0.08 72.6 ± 1.1 18.2 ± 0.9 9.2 ± 0.7
1 80.88 ± 0.10
4.75 ± 0.06
14.36 ± 0.09 73.4 ± 0.8 17.8 ± 0.6 8.8 ± 0.5
4 81.01 ± 0.15
3.97 ± 0.08
15.02 ± 0.14 74.3 ± 0.5 17.7 ± 0.4 8.0 ± 0.3
16 81.10 ± 0.06
3.61 ± 0.03
15.28 ± 0.06
75.48 ± 0.26
17.19 ± 0.23
7.33 ± 0.16
64 81.22 ± 0.07
3.45 ± 0.04
15.33 ± 0.07 75.7 ± 0.3
17.17 ± 0.28
7.16 ± 0.19
256 81.22 ± 0.07
3.30 ± 0.03
15.47 ± 0.07
75.95 ± 0.20
16.96 ± 0.17
7.08 ± 0.12
1024 81.16 ± 0.09
3.27 ± 0.04
15.57 ± 0.09 76.4 ±0.3
16.70 ±0.29
6.91 ± 0.20
∞ 81.26 ± 0.18
3.13 ± 0.08
15.61 ± 0.18 76.5 ± 0.5 16.7 ± 0.4
6.77 ± 0.30
194
Table A.2.3. Temporal evolution of the phase compositions in Ni-6.5 Al-9.5 Cr at.% aged at 873 K, as measured by atom-probe tomography. The lattice parameter misfit for this alloy is estimated, from PrecipiCalc, to be 0.00284, and the equilibrium volume fraction from APT, is 9.6 ± 2.9 %.
Aging time (h) γ-matrix composition (at.%) γ’-precipitate composition (at.%)
Ni Al Cr Ni Al Cr
0.5 83.05 ± 0.03
7.13 ± 0.08
9.82 ± 0.07
70.06 ± 3.02
21.28 ± 4.89
8.66 ± 5.27
0.75 82.99 ± 0.03
7.22 ± 0.07
9.79 ± 0.07
71.25 ± 0.13
20.40 ± 0.21
8.35 ± 0.23
1 83.73 ± 0.01
6.46 ± 0.03
9.81 ± 0.03
71.45 ± 0.49
20.46 ± 0.81
8.08 ± 0.88
1.5 83.74 ± 0.03
6.38 ± 0.07
9.88 ± 0.06
72.28 ± 0.20
19.57 ± 0.33
8.14 ± 0.36
2 83.74 ± 0.02
6.38 ± 0.04
9.87 ± 0.04
72.81 ± 0.46
19.35 ± 0.79
7.84 ± 0.84
3.5 83.83 ± 0.04
6.27 ± 0.08
10.08 ± 0.08
74.06 ± 0.44
18.74 ± 0.81
7.19 ± 0.86
4 83.72 ± 0.01
6.29 ± 0.03
9.99 ± 0.03
74.53 ± 0.09
18.36 ± 0.16
7.11 ± 0.17
16 83.90 ± 0.01
5.90 ± 0.03
10.20 ± 0.03
75.09 ± 0.08
17.98 ± 0.14
6.94 ± 0.15
64 83.99 ± 0.02
5.80 ± 0.05
10.21 ± 0.05
75.55 ± 0.07
17.89 ± 0.13
6.56 ± 0.17
256 84.06 ± 0.02
5.71 ± 0.05
10.23 ± 0.05
75.82 ± 0.09
17.81 ± 0.17
6.37 ± 0.18
1024 84.11 ± 0.01
5.64 ± 0.04
10.25 ± 0.03
76.05 ± 0.05
17.65 ± 0.09
6.30 ± 0.07
4096 84.16 ± 0.01
5.57 ± 0.03
10.27 ± 0.02
76.24 ± 0.03
17.54 ± 0.06
6.22 ± 0.07
∞ 84.19 ± 0.02
5.49 ± 0.05
10.30 ± 0.05
76.40 ± 0.19
17.53 ± 0.33
6.06 ± 0.36
195
Vita
Christopher Booth-Morrison
Education 2006 Arizona State University, Tempe, AZ
High Resolution Electron Microscopy Winter School (January)
2004 B. Eng., Metallurgical Engineering McGill University, Montreal, Canada
Journal Publications
1. C. Booth-Morrison, Y. Zhou, R. D. Noebe, and D. N. Seidman, “On the nanometer scale phase separation of a low-supersaturation Ni-Al-Cr alloy”. (To appear in Phil. Mag. 2009).
2. Y. Amouyal, Z. Mao, C. Booth-Morrison, and D. N. Seidman, “On the interplay between tungsten and tantalum in Ni-based superalloys: An atom-probe tomographic and first-principles study”. Applied Physics Letters. 2009;94;041917.
3. C. Booth-Morrison, R. D. Noebe, and D. N. Seidman, “Effects of tantalum on the temporal evolution of a model Ni-Al-Cr superalloy during phase decomposition”. Acta Materialia. 2009;57;909-920.
4. Y. Zhou, C. Booth-Morrison, and D. N. Seidman, “On the field-evaporation behavior of a model Ni-Al-Cr superalloy studied by picosecond pulsed-laser atom probe tomography” Microscopy & Microanalysis. 2008;14;571-580.
5. Y. Zhou, Z. Mao, C. Booth-Morrison, and D. N. Seidman, “The partitioning and site preference of rhenium or ruthenium in model Ni-based superalloys: An atom-probe tomographic and first-principles study”. Applied Physics Letters. 2008;93;171905.
6. C. Booth-Morrison, Z. Mao, R. D. Noebe, and D. N. Seidman, “Chromium and tantalum site substitution patterns in Ni3Al (L12) γ’-precipitates”. Applied Physics Letters. 2008;93;033103.
7. C. Booth-Morrison, J. Weninger, C. K. Sudbrack, Z. Mao, R. D. Noebe, and D. N. Seidman, “Effects of solute concentrations on the kinetic pathways in Ni-Al-Cr alloys”. Acta Materialia. 2008;56;3422-3438.
Conference Proceedings
1. C. Booth-Morrison, R. D. Noebe, and D. N. Seidman, “Effects of a Tantalum Addition on the Morphological and Compositional Evolution of a Model Ni-Al-Cr Superalloy”. Proceedings of 11th International Symposium on Superalloys. The Minerals, Metals and Materials Society, Warrendale, PA. (2008).
196
Conference Presentations
1. C. Booth-Morrison, R. D. Noebe, and D. N. Seidman, “Effects of Tantalum on the Phase Decomposition of a Model Ni-Al-Cr Superalloy on a Nanoscale”. The Minerals, Metals and Materials Society 2009 Annual Meeting. San Francisco, CA. (2009).
2. C. Booth-Morrison, Y. Zhou, Z. Mao, R. D. Noebe, and D. N. Seidman, “The Temporal Evolution of the Nanostructures of a Low-Supersaturation Ni-Al-Cr Superalloy”. The Minerals, Metals and Materials Society 2009 Annual Meeting. San Francisco, CA. (2009).
3. D. N. Seidman, C. Booth-Morrison, Z. Mao, C.K. Sudbrack, “Effects of Solute Concentrations on the Kinetic Pathways in Ni-Al-Cr Alloys: Experiments and Simulations”. The Minerals, Metals and Materials Society 2009 Annual Meeting. San Francisco, CA. (2009).
4. C. Booth-Morrison, R. D. Noebe, and D. N. Seidman, “Effects of a Tantalum Addition on the Morphological and Compositional Evolution of a Model Ni-Al-Cr Superalloy”. 11th International Symposium on Superalloys, The Minerals, Metals and Materials Society, Champion, PA. (2008).
5. C. Booth-Morrison, R. D. Noebe, and D. N. Seidman, “Nanoscale Studies of the Early Stages of Phase Decomposition in Model Ni-Al-Cr Superalloys”. High Throughput Analysis of Multicomponent Multiphase Diffusion Data Conference, National Institute of Standards and Technologies, Gaithersburg, MD. (2007).
6. C. K. Sudbrack, J. A. Wenninger, C. Booth-Morrison, R. D. Noebe, D. N. Seidman, “A Comparison of the Early-Stages of γ’-Precipitation in Two Ni-Al-Cr Superalloys”. The Minerals, Metals and Materials Society 2006 Annual Meeting. San Antonio, TX. (2006).
7. P. Wanjara, C. Booth-Morrison, E. Hsu, Jahazi M., “Process Optimization for Linear Friction Welding of Ti6Al4V”, The 7th International Conference on Trends in Welding Research, Pine Mountain, GA. (2005).
Awards and Honors
2008-2009 B.J. Martin Terminal Year Fellowship (Northwestern University)
2006-2007 Materials Science and Engineering Teaching Assistant of the Year (Northwestern
University)
2005-2006 Northwestern University Graduate Fellowship
2004-2006 Fonds de recherche sur la nature et les technologies Fellowship
2004-2005 Walter P. Murphy Fellowship (Northwestern University)
2000-2004 McGill University Academic Achievement Scholarships
2001-2004 Canadian Mineral Industry Education Foundation Scholarship
2002-2003 ASM International Student Chapter Award
