Interfacial free energies, nucleation, and precipitate morphologies in Ni-Al-Cr alloys: Calculations and atom-probe tomographic experiments

Accepted Manuscript Interfacial free energies, nucleation, and precipitate morphologies in Ni-Al-Cr alloys: Calculations and atom-probe tomographic experiments Zugang Mao, Christopher Booth-Morrison, Chantal K. Sudbrack, Ronald D. Noebe, David N. Seidman PII:

S1359-6454(19)30031-X

DOI:

https://doi.org/10.1016/j.actamat.2019.01.017

Reference:

AM 15085

To appear in:

Acta Materialia

Received Date: 13 June 2018 Revised Date:

6 January 2019

Accepted Date: 13 January 2019

Please cite this article as: Z. Mao, C. Booth-Morrison, C.K. Sudbrack, R.D. Noebe, D.N. Seidman, Interfacial free energies, nucleation, and precipitate morphologies in Ni-Al-Cr alloys: Calculations and atom-probe tomographic experiments, Acta Materialia, https://doi.org/10.1016/j.actamat.2019.01.017. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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The effects of Cr additions on the morphologies οf γ’(L12 ordered) precipitates are studied in three Ni-Al-Cr alloys utilizing atom-probe tomography (APT) and first-principles calculations. The Cr additions to Ni-Al alloys have significant effects on the critical radius and critical net reversible work of nucleation. With increasing Cr concentration, Ni3(Al,Cr)(L12) requires the smallest critical net reversible work to form but has a much larger critical size. An embryo of Ni3Cr(L12) only acts as a nucleant for mixed L12 Ni3(AlxCr1-x). The morphology of Ni3(Al,Cr)(L12) precipitates transforms from cuboidal-to-spheroidal with increasing Cr concentration, in agreement with APT observations.

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Revised for Acta Materialia, January 6, 2019

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Interfacial free energies, nucleation, and precipitate morphologies in Ni-Al-Cr alloys: Calculations and atom-probe tomographic experiments Zugang Maoa, Christopher Booth-Morrisona, Chantal K. Sudbracka,b, Ronald D. Noebeb, and David N. Seidmana,c,*

Department of Materials Science and Engineering, Northwestern University, 2220 Campus

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a

Drive, Evanston, Illinois, 60208-3108 U.S.A. b

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Structures and Materials Division, NASA Glenn Research Center, 21000 Brookpark Road, Cleveland, Ohio 44135 USA

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Northwestern University Center for Atom-Probe Tomgraphy, 2220 Campus Drive, Evanston, Illinois, 60208-3108 U.S.A.

Abstract

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The effects of Cr additions on the morphologies οf γ’(L12 ordered) precipitates are studied in three Ni-Al-Cr alloys utilizing atom-probe tomography (APT) and first-principles calculations. We find that Cr and Al share the same sublattice sites in the L12-structure: Chromium partitions from the L12 structure into the disordered f.c.c. matrix for a driving force of 0.436 eV atom-1.

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Mixing between the Ni3Al(L12) and Ni3Cr(L12) structures is energetically unfavorable based on first-principles quasi-random structure calculations. Interfacial Gibbs free energies of

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Ni/Ni3Al(L12), Ni/Ni3Cr(L12), Ni/Ni3Cr(DO22), and Nin(AlyCr1-y)/Ni3(AlxCr1-x)(L12) interfaces are calculated for the {100}, {110}, and {111} planes. The temperature dependencies of the interfacial Gibbs free energies are determined by including phonon vibrational entropies. The equilibrium interfacial Gibbs free energies are determined using concentration profiles as a function of aging time, obtained using APT. At early aging times, the initial values of the interfacial Gibbs free energies are higher with small Cr concentrations in the γ’ (L12)precipitates. With increasing aging time, the interfacial Gibbs free energies decrease with increasing Cr concentration and achieve equilibrium values after 16-h of aging at 873 K (600 oC) for the three Ni-Al-Cr alloys. From classical nucleation theory, the nucleation of metastable |Page

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Ni3Cr(L12) is easier than Ni3Al(L12) at 873 K (600 oC). Ni3Cr(DO22) is more difficult to nucleate than Ni3Cr(L12) and Ni3Al(L12) at 873 K (600 oC). The Cr additions to Ni-Al alloys have significant effects on the critical radius and critical net reversible work of nucleation. With increasing Cr concentration, Ni3(Al,Cr)(L12) requires the smallest critical net reversible work to

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form but has a much larger critical size. An embryo of Ni3Cr(L12) only acts as a nucleant for mixed L12 Ni3(AlxCr1-x).

The morphology of Ni3(Al,Cr)(L12) precipitates transforms from

(284) words

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cuboidal-to-spheroidal with increasing Cr concentration, in agreement with APT observations.

Key words: First-principles calculations; Interfacial Gibbs free-energy; phonon vibrational

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entropy; phase partitioning, atom-probe tomography.

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*[email protected]

1. Introduction Ni-Al-Cr alloys are model alloys for Ni-based superalloys, which are widely used in hightemperature aerospace jet engines and land-based gas turbine engines, due to their superior

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strength, coarsening, creep, corrosion and oxidation resistance at elevated temperatures [1, 2]. The microstructures of high-strength Ni-based superalloys consist of coherent ordered γ’(L12)precipitates in a disordered nickel-rich γ(f.c.c.)-matrix [3]. It is of great scientific interest to understand the temporal evolution of the microstructures of the model alloys as an important step

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toward understanding more complex commercial alloys. The γ(f.c.c.)/γ’(L12) heterophase interfacial structure and concentrations play a key role in determining the mechanisms of nucleation, growth, and coarsening of the γ’(L12)-phase in a disordered γ(f.c.c.)-matrix [4, 5].

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The γ(f.c.c.)/γ’(L12) heterophase interfacial Gibbs free energy, σ, for different crystallographic orientations, affects strongly the morphology of the ordered precipitates, which enters directly in all coarsening theories used to model microstructural stability, and is needed to predict the reliability of existing superalloys and to further improve them. The values of the γ(f.c.c.)/γ’(L12) heterophase interfacial Gibbs free energy in Ni-Al-Cr alloys are evaluated experimentally using the Lifshitz-Slyozov-Wagner (LSW) coarsening model and its variants, which yields average

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values of the interfacial Gibbs free energy for all the {hkl} facets of γ’(L12)-precipitates [6-8]. There are no prior reports on theoretical calculations for the γ(f.c.c.)/γ’(L12) heterophase interfacial Gibbs free energies for specific {hkl} facets of precipitates in Ni-Al-Cr alloys. The phonon vibrational entropy was calculated to determine the γ(f.c.c.)/γ’(L12) heterophase

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interfacial Gibbs free energies at an elevated temperature in Ni-Al alloys in our prior research [9]. Previous experimental studies demonstrated that the morphology changes from cuboidal-to-

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spheroidal with the addition of Cr to Ni-Al alloys [6-8, 10-14]. In this research, the morphological evolution of γ’(L12)-precipitates in Ni-Al-Cr alloys as

well as their compositional evolution as a function of aging time at 873 K were previously studied utilizing atom-probe tomography (APT) and the pertinent results are reviewed [6-8]. To understand theoretically the mechanism and its effects on the morphological changes, we employ density functional theory (DFT) first-principles calculations to obtain physically meaningful values of the interfacial free energies of the {100}, {110} and {111} heterophase interfaces, accounting for the first time for the effect of temperature on both coherency strain-energy and phonon vibrational entropy. Magnetic spin-polarized calculations are performed to evaluate the |Page

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effects of ferromagnetism, and phonon vibrational entropies are utilized to determine the temperature dependencies of the γ(f.c.c.)/γ’(L12) heterophase interfacial Gibbs free energies of the {100}, {110} and {111} interfaces. We calculated the chemical properties and elastic properties of three possible ordered phases in Ni-Al-Cr alloys: Ni3Al (L12), Al3Cr (L12), and

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Ni3Cr (DO22). The stabilities of these possible ordered phases has been studied, and mixing between the two L12 ordered phases [Ni3Al (L12) and Al3Cr (L12)] is demonstrated not to be energetically favored. The site preference and partitioning behavior of Cr atoms across γ(f.c.c.)/γ’(L12) heterophase interfaces is considered in detail. The predicted equilibrium

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morphologies of γ’(L12)-precipitates, based on Wulff constructions, are in excellent agreement with experimental atom-probe tomography (APT) results. Classical nucleation theory is utilized

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to determine the critical radius and net reversible work for all possible nuclei: Ni3Al (L12), Al3Cr (L12), Ni3Cr (DO22), and Ni3(Al,Cr)(L12). 2. Methodology 2.1.

Atom-probe tomography

The temporal evolution of γ’(L12)-precipitates was studied experimentally in three different NiAl-Cr alloys using APT. The compositions in at.% of the three alloys studied are: Ni-7.5Al-8.5Cr

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(alloy A); Ni-5.2Al-14.2Cr (alloy B); and Ni-6.5Al-9.5Cr (alloy C). Solution-treated ingotsections of alloy samples were aged at 873 K (600 oC) under flowing argon for times ranging from 1/4 to 1024 h. We performed voltage-pulsed APT on microtip specimens utilizing a conventionl APT [15, 16], and additionally a Cameca Instruments (formerly Imago Scientific

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Instruments, Madison, WI) picosecond green laser assisted local-electrode atom-probe (LEAP) tomograph [17-22] in the Northwestern University Center for Atom-Probe Tomography

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(NUCAPT). Pulsed-voltage APT was performed for alloys A and B at a specimen temperature of 40.0 ± 0.3 K, a voltage pulse-fraction (pulse voltage/steady-state dc voltage) of 19%, a pulse repetition rate of 1.5 kHz (a conventional APT) or a 200 kHz (LEAP4000X Si tomograph), and at a background gauge pressure of < 6.7x10-8 Pa (< 5x10-10 Torr). The average detection rates in the areas of analysis ranged from 0.011 to 0.015 ion pulse-1 for conventional APT and from 0.04 to 0.08 ion pulse-1 for 3-D-LEAP tomographic analyses. Pulsed-laser APT experiments were performed for alloy C with a LEAP4000X Si tomograph at a target evaporation rate of 0.04 ion per pulse, a specimen temperature of 40.0±0.3 K, a laser pulse energy of 0.6 nJ per pulse, a pulse repetition rate of 200 kHz, and an ambient gauge pressure of <6.7 x 10-8 Pa, employing a green |Page

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laser (wavelength = 535 nm). These evaporation conditions were optimized to provide the highest compositional accuracy for this alloy [23]. The detailed experimental and analytical information, including data regarding the nanostructural properties of the γ’(L12)-precipitates; comparisons with classical nucleation theory, growth and coarsening models can be found

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elsewhere. [6, 7, 24-30]. Concentration profiles from APT data discussed herein are generated utilizing the proximity historgram methodology [31, 32] by averaging across the γ(f.c.c.)/γ’(L12)-heterophase interfaces of tens to hundreds of γ’(L12)-precipitates for the different aging conditions employed. All concentrations are reported in at.%, unless otherwise

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specified.

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2.2. Computational details

In this study, the plane-wave total-energy methodology with the Perdew-Burke-Ernzerhof parameterization of the generalized gradient approximation (GGA) is employed [33, 34] for exchange-correlation in our density functional theory (DFT) first-principles calculations, as implemented in the Vienna ab initio simulation package (VASP) [35-39]. We use the projector augmented wave (PAW) potentials in our calculations [40]. All the structures considered are fully relaxed with respect to the atomic coordinates as well as the volume inside the supercell.

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We considered and tested the convergence of results carefully with respect to a range of energy cutoffs from 200 to 700 eV and 4x4x4 to 16x16x16 k-points. A plane-wave basis set with spinpolarized method was used with an energy cutoff of 400 eV to represent the Kohn-Sham wave functions. The summation over the Brillouin zone for the bulk structures is performed on an calculations.

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optimized 12×12×12 spacing with a Monkhorst-pack k-point mesh per f.c.c unit cell for all

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We calculated the phonon vibrational entropies to determine the temperature

dependencies of interfacial Gibbs free energies and nucleation energies of L12 and DO22 ordered precipitates in the Ni-Al-Cr alloys. The harmonic vibrational frequency approximation was used for all vibrational calculations. To determine the phonon dispersions for all the related structures, either for ordered structures or the disordered solid-solution phase, we utilized the frozen-phonon methodology [41-43] to compute the vibrational thermodynamics for ordered Ni3Al (L12), Ni3Cr (L12), Ni3Cr (DO22), Ni3(AlxCr1-x)( L12) and solid-solutions, and for Ni (f.c.c.)/Ni3Al (L12), Ni (f.c.c.)/Ni3Cr(L12), Ni(f.c.c.)/Ni3Cr (DO22), and Ni (f.c.c.) /Ni3(AlxCr1-x) (L12) interfaces. We

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calculated the vibrational free energies of these supercells from the phonon density of states (DOSs) using standard thermodynamic expressions. For the solid-solution Nin(Al,Cr), we utilized a 64-atom cubic cell (2 × 2 × 4 conventional f.c.c.) of Al with a Cr atom replacing a portion of the Ni atoms on the central sites.

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For Ni3Al (L12), Ni3Cr (L12), and Ni3(AlxCr1-x) phases, we employed supercells of 3 × 3 × 3 cubic units cells, comprising 108 total atoms. For the Ni3Cr (DO22) phase, we used supercells with 2 × 2 × 4 units cells, comprising 64 total atoms.

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3. Results and discussion 3.1 Atom-probe tomography

Three different Ni-Al-Cr alloys are studied: Ni-7.5Al-8.5Cr (alloy A); Ni-5.2Al-14.2Cr (alloy

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B); and Ni-6.5Al-9.5Cr (alloy C). Nanometer-sized γ’(L12)-precipitates are experimentally observed and studied over the full range of aging times from 1/6 to 1024 h at 873 K (600 oC). The temporal evolution of the morphology of the precipitate phase in the three alloys is discussed in great detail in our prior research [6-8, 24, 29]. The compositions of the γ(f.c.c.)matrix and the γ’(L12)-precipitate phases of these alloys evolve temporally as the γ(f.c.c.)-matrix becomes enriched in Ni and Cr and depleted in Al, while the γ’(L12)-precipitates become

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enriched in Al and depleted in Cr as function of aging time, Tables 1, 2, and 3. During nucleation, solute-rich γ’(L12)-nuclei form with large values of the Al and Cr supersaturations, which decrease with increasing aging time. The first γ’(L12)-nuclei detected by APT for alloy A have solute-supersaturated compositions of 72 ± 3 Ni, 21 ± 3 Al, and 6 ± 1 Cr; 71 ± 3 Ni, 19 ± 3

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Al and 10 ± 2 Cr for alloy B; and 70 ± 3 Ni, 21 ± 5 Al and 9 ± 5 Cr for alloy C. The final equilibrium compositions of the L12 ordered precipitates are 76.3 ± 0.1 Ni, 17.8 ± 0.2 Al and 5.9

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± 0.1 Cr for alloy A; 76.5 ± 3.0 Ni, 16.2 ± 0.4 Al and 6.8 ± 0.3 Cr for alloy B; and 76.4 ± 0.2 Ni, 17.5 ± 0.3 Al and 6.1 ± 0.6 Cr for alloy C. In these three alloys there is a depletion in the Al concentration in the γ(f.c.c.)-matrix (0.4, -0.4 and -0.1 at.% in a layer

≈ 2 nm thick),

respectively, together with an accumulation of Cr ( +0.4, +0.4 and +0.2 at.% over the same distance, respectively). A small and more localized retention of Ni occurs as a result of the local imbalance between the Al-depletion and Cr-accumulation. Such solute concentration changes as a function of aging time affect the interfacial Gibbs free energy, Section 3.5. The morphology of the precipitating phase becomes more spheroidal with increasing Cr concentration in all three alloys, whereas the precipitates in Ni-Al binary alloys have a cuboidal morphology. |Page

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Experimentally, the interfacial free energy, σ γ / γ ' , can be determined using the Kuehmann Voorhees (KV) coarsening model [44]. The relationship for σ γ /γ ' for a nonideal, nondilute ternary alloy consisting of a γ-matrix and a γ’-precipitate phase with a finite volume fraction of

σ

γ /γ '

=

2 γ ( K KV )1/3 κ iγ ( p Al2 GAlγ , Al + pAl pCr GAlγ ,Cr + pCr GCr ,Cr )

2 piVmγ '

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the γ’-phase is given by:

(1)

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where K KV and κ iγ are the rate constants for the mean radius, , of the γ’-precipitates and supersaturation, respectively, Vmγ ' is the molar volume of the γ’-precipitate phase,

pi is the

magnitude of the partitioning coefficient as defined by Ciγ ',eq (∞) − Ciγ ,eq (∞) , and Giγ, j is a

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shorthand notation for the partial derivatives of the molar Gibbs free-energy of the γ-matrix phase with respect to the solute species i and j. The experimentally determined average interfacial Gibbs free energy from coarsening data is 23.0 ±1.2 mJ m-2 for alloy A; 24.0 ±1.6 mJ m-2 for alloy B; and 21.0 ±1.0 mJ m-2 for alloy C using the Kuehmann-Voorhees coarsening model for a ternary alloy at 873 K (600 oC) [6-8].

3.2 Bulk chemical properties and elastic properties of the possible ordered phases in Ni-Al-Cr alloys

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In this article, we present both experimental determinations and calculations of the interfacial Gibbs free-energies, nucleation barriers, and morphologies of precipitates in three Ni-Al-Cr alloys. We calculated the energetic and thermodynamic properties of the bulk phases, as well as the interfacial Gibbs free energies. We first discuss the first-principles results for the bulk

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properties, including formation energies of the ordered structures, lattice parameters, elastic constants and strain energies, and discuss subsequently the calculated interfacial energies. From

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these results, we obtain a more complete physical picture of the energetics of nucleation of L12 structures in Ni-AL-Cr alloys. These energies permit an understanding of the morphology of L12 precipitates.

There are three possible basic ordered structures in Ni-Al-Cr alloys: Ni3Al (L12), Al3Cr

(L12), and Ni3Cr (DO22). The calculated bulk properties and elastic properties of the ordered phases are listed in table 4. From our calculations, the formation energies of Al3Cr (L12) and Al3Cr (DO22) have small negative values (-0.26 kJ·mol-1) for Al3Cr (L12) and -1.31 kJ·mol-1 for Al3Cr (DO22) ), which are significantly greater than for Ni3Al (L12) (-41.88 kJ·mol-1). The driving force for the transformation of Al3Cr from L12 to DO22 is 1.05 kJ mol-1. There is a large |Page

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driving force for Al to substitute for Cr to form Ni3(AlxCr1-x) ( L12). For the purposes of predicting the morphology of the precipitate phases in Ni-Al-Cr alloys we mainly consider Ni3Al (L12), Al3Cr (L12), and mixed Ni3(AlxCr1-x)(L12) as Ni3Al is the experimentally observed phase in concentrated Ni-Al-Cr alloys [45].

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The calculated lattice parameters at 0 K are 3.53 Å for f.c.c. Ni (3.54 Å experimentally at o

20 C), 3.57 Å for Ni3Al (L12) (3.56 Å experimentally at 20 oC), 3.55 Å for Al3Cr (L12) at 0 K. The calculated lattice parameter misfit is 1.2% for Ni(f.c.c.)/Ni3Al(L12) and 0.6% for Ni(f.c.c.)/Al3Cr(L12) at 0 K. These lattice parameter misfits are in agreement with the

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experimental value of 1.1 % at room temperature for Ni(f.c.c.)/Ni3Al(L12) [46].

We now discuss the bulk elastic properties for the ordered phases considered: Ni3Al

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(L12), Al3Cr (L12), and Ni3Cr (DO22). First, we calculated the three elastic constants C11, C12, and C44 of a cubic crystal by either tetragonal or trigonal lattice distortions [47]. The bulk modulus and the Voigt averaged shear modulus are determinedfrom the bulk elastic constants by considering a hydrostatic elastic deformation. The bulk modulus, B, and the Voigt averaged shear modulus, , are given by: C11 + 2 C12 3 3 C 44 + C 11 − C 12 < G >V = 5

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B=

(2) (3)

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The calculated elastic constants, C11, C12, and C44, Voigt averaged shear modulus < G > V , and bulk modulus, B, of Ni, Ni3Al(L12), Ni3Cr(L12), Ni3Cr(DO22) and Al are listed in Table 4. From our calculations, the two Al3Cr crystal structures have the highest bulk moduli,

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196 GPa for L12 and 201 GPa for DO22. The ordered phases all have larger shear moduli than Ni, and Ni3Al(L12) has larger values of the elastic constants than Ni3Cr(L12). We next computed the coherency strain energies for different strain orientations across

the following three different interfaces: Ni/Ni3Al(L12), Ni/Ni3Cr(L12) and Ni/Ni3Cr(DO22). The strain energies are used as one contribution of the energetic quantities utilized to determine the driving force for nucleation, the radius of the critical nucleus, and the eventual stable precipitate structure in Ni-Al-Cr alloys. We have reported the values for the Ni/Ni3Al(L12) heterophase interface [9]. For the sake of completeness, we describe briefly the methodology for the computation of coherency strain. The coherency strain can be visualized in terms of the |Page

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formation energy of coherent long-period ApBq superlattices (A and B represent the different phases considered, such as disordered Ni-rich phase and ordered L12 phase), which are nonzero and depend on the superlattice direction,

) ) G. The formation energy per atom, ∆ECSeq ( x, G) , can be

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defined as the equilibrium value of the composition-weighted sum of the energies to deform both bulk disordered Ni-rich phase A and ordered L12 phase B to the epitaxial geometry with lattice )

constant as perpendicular to superlattice direction G as following [48]:

aS

,

(4)

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) ) ) eq ∆ECS ( x, G) = min  x∆EAepi (aS , G) + (1 − x)∆EBepi (aS , G) 

where x is the mole fraction of Ni-rich phase A, ∆EA is the epitaxial deformation energies of epi

Ni-rich phase A, and ∆EB is the epitaxial deformation energies of ordered L12 phase B. The

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epi

coherency strain energies for the [100], [110], and [111] directions for the Ni/Ni3Al(L12), Ni/Al3Cr(L12) and Ni/Al3Cr(DO22) interfaces are displayed in Fig. 1. We find that the Ni/Ni3Cr(L12) interface has the smallest strains of the three possible interfaces with f.c.c. Ni. The [100] direction has smallest strain and the [111] direction has the largest strain for both the Ni/Ni3Al(L12) and Ni/Al3Cr(L12) interfaces, whereas the [100]

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direction is the elastically softest, and the [111] direction is the hardest. Alternatively, the Ni/Al3Cr (DO22) interface has the largest elastic strain energy among the three systems, which is consistent with the largest lattice parameter mismatch of 3.48%. The Ni/Ni3Al(L12) interface has the largest difference (0.021 eV·atom-1) between the [100] and [111] directions, which means the

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precipitates tend toward a cuboidal morphology, whereas the Ni/Al3Cr(L12) interface has a very small difference (0.003 eV·atom-1) between the [100] and [111] directions, which implies that

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the precipitates tend toward a spheroidal morphology. The strain for the Ni/Al3Cr(DO22) interface is different from that of the Ni/Ni3Al(L12) and Ni/Al3Cr(L12) interfaces, where the [111] direction is the elastically softest and the [100] direction is the elastically hardest. 3.3 Solute site preference and partitioning pattern of Cr addition across Ni(γ)/Ni3Al(γ’) interface The site preference of Cr and the energetic driving force for the partitioning of Cr were determined by first-principles calculations for a system of 12 × 2 × 2 unit cells (192 atoms) constructed along the [100] direction. The supercell was divided by a (100) interface, and the |Page

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two halves of the supercell were occupied by the γ-Ni(f.c.c.) and the γ’-Ni3Al(L12) phases, respectively. Every Ni atom on the γ-Ni(f.c.c.) of the interface was treated as a potential substitutional site for the Cr atoms, while on the Ni3Al side, the Ni and Al sublattice sites were treated as separate substitutional sites. To ensure coherency of the (100) γ(f.c.c.)/γ’-Ni3Al(L12)

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interface, the structures on either side of the interface were relaxed within the constraints of the Ni3Al crystal structure. The substitutional formation energies of Cr, as a function of distance from the (100) interface, were calculated as:

ECr → Ni = ( EZtot, matrix + nNi µ Ni ) − ( E tot + nCr µCr )

(5)

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tot tot ECr → M = ( ECr + nCr µCr ) , precipitate + nM µ M ) − ( E

(6)

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tot tot where M is Ni or Al, E tot is the total energy prior to substitution, ECr , precipitate and ECr , matrix are the

total energies when Cr partitions to the matrix or precipitate phases, respectively.The site substitutional energies of Cr are smaller at the Al sublattice-sites than at the Ni sublattice-sites, Table 5, which confirms that Cr atoms prefer to occupy the Al sublattice-sites of the Ni3Al(L12) structure. The calculated substitutional formation energies of Cr are smaller in the γ-Ni(f.c.c.) matrix than in the γ’-Ni3Al(L12)-precipitate, Fig. 2, providing the energetic driving force, 0.436

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eV atom-1, for the partitioning of Cr atoms to the γ-Ni(f.c.c.) matrix phase. The addition of Cr leads to an increase in the total energy of the L12 crystal-structure. The average atomic forces and displacements associated with the local stresses and strains resulting from the substitution of Cr on the Ni and Al sublattice sites are displayed in

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Table 5. The substitution of Cr in the γ-matrix phase results in a smaller value of the atomic force and average atomic displacement than substitution of Cr at either Ni or Al sublattice sites in

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Ni3Al(L12) precipitate phase, thus the partitioning and substitution of Cr in the γ-matrix phase is preferred.

3.4 Mixing between L12 Ni3Al(L12), Ni3Cr(L12) ordered phase As discussed above, for both the experimental results, Tables 1, 2 and 3, and the substitutional pathway calculations, the solute element, Cr, in concentrated Ni-Al-Cr alloys has a tendency to partition to the γ-matrix solid-solution phase, whereas Cr and Al share the solute sublattice in the Ni3Al(L12) precipitate phase. To determine the stability of the mixed L12-ordered phases, we | P a g e 10

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calculate the mixing energy between the ordered structures. To mimic adequately the mixing statistics of a random distribution of ordered structures in an alloy, with a supercell size that is computationally feasible for DFT, we use the special quasi-random structure (SQS) approach developed by Zunger et al. [2,3]. SQSs are specifically designed small-unit-cell periodic

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structures that mimic the most relevant nearest-neighbor pair and multisite correlation functions of random substitutional solute positions in alloys. The concept of SQS originated from the wellestablished cluster expansion (CE) methodology [5–12], and SQS structures have been developed for mixing on several important lattice types: f.c.c. [49], b.c.c. [50], and h.c.p. [51].

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While the CE method is more powerful and is capable of describing both ordered and disordered states, within a single unified framework, the SQS approach is computationally more efficient if

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the sole purpose is to obtain reliably the properties of random alloys. As a consequence of such atomic arrangements, local environmentally dependent effects, such as charge transfer and local atomic relaxations can be accurately captured by the SQS approach [52]. The most straightforward way to generate the SQSs is to enumerate exhaustively several possible mixed L12 structures and then calculate their pair and multisite correlation functions, as is implemented by the gensqs code in the Alloy-Theoretic Automated Toolkit (ATAT) [21]. Since both Al and Cr atoms prefer the solute sublattice, for creating a Ni3(Al1-nCrn)

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substitutional alloy, the substitutional arrangement of Al and Cr atoms share the solute sublattice- type lattice sites. In the present study, we generated different SQS-N structures (with N = 108, 32, and 108 atoms per unit cell) for a random fcc lattice, with the compositions x = 0.33, 0.5 or 0.67. For Ni3(Al0.5Cr0.5), we generate the SQS-32 structure with 32 atoms of a 2 × 2

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× 2 supercell. For Ni3(Al0.33Cr0.67) and Ni3(Al0.67Cr0.33), we generate the SQS108 structure with

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108 atoms of a 3 x 3 x 3 supercell to achieve convergence. The mixing enthalpy of the randomly distributed solute elements, Al and Cr, with compositions xNi3Al and xNi3Cr can be calculated from:

∆H mix = E (SQS ) − xNi3 Al E ( Ni3 Al ) − xNi3Cr E ( Ni3Cr )

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xNi3Al + xNi3Cr =1

where E ( N i3 Al ) , E ( N i 3 C r ) , and E ( SQ S ) are the calculated total energies of fully relaxed L12 ordered Ni3Al, Ni3Cr, and the SQS Ni3(Al,Cr), respectively. The mixed systems have positive mixing energies, Fig. 3, indicating that mixing between Ni3Al(L12) and Ni3Cr(L12) to | P a g e 11

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form Ni3(Al,Cr)(L12) is energetically unfavorable. The calculated positive mixing energies confirm Cr partitioning to the γ-matrix phase and agree with the positive substitutional energies, where Al and Cr atoms share the same sublattice sites of the L12 structure.

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3.5 Interfacial free energies of Ni/Ni3Al(L12), Ni/Ni3Cr(L12), Ni/Ni3Cr(DO22) and Nin(AlyCr1-y)/ Ni3(AlxCr1-x) (L12) interfaces

The interfacial Gibbs free energies between disordered γ and ordered γ’ phase are critical

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for determining the mechanisms that lead to the nucleation of γ’ precipitates with an ordered structure from a disordered f.c.c. solid-solution. We calculate the energies of the three possible

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interfacial systems in the Ni-Al-Cr system: Ni/Ni3Al(L12), Ni/Ni3Cr(L12), Ni/Ni3Cr(DO22) and Nin(AlyCr1-y)/Ni3(AlxCr1-x) (L12). For each system, the (100), (110), and (111) interfacies are considered. The interfacial configurations were constructed with the initial positions on an ideal and uniform f.c.c. lattice with a lattice parameter of 3.57 Å for both phases. We fully relaxed all atomic positions, cell vectors, and the total cell volume for all three interfacial systems. Firstly, we determine three basic interfaces: Ni/Ni3Al(L12), Ni/Ni3Cr(L12), and Ni/Ni3Cr(DO22). We calculate the interfacial energy including strain effects [11], where the

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interfacial internal energies are calculated by subtracting the total energy of the phases on either side of the interface from the total energy of a two-phase system containing an interface: 1  Eαtot/ β − ( Eαtot + E βtot ) − ∆ E cs  − T ∆ S vib ; 2A 

(8)

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σ α /β =

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where A is the interfacial area, the subscripts α and β denote the two phases across the interface tot plane: the possible phases are Ni(f.c.c.), Ni3Al(L12), Ni3Cr(L12) and Ni3Cr(DO22)); Eα / β is the tot

tot total internal energy of the relaxed α/β system containing an interface; Eα and Eβ are the total

internal energies of the α- and β-phases, respectively, each relaxed to their own equilibrium geometry; ∆ is the coherency strain energy between the α- and β-phases as discussed above; ∆Svib is the phonon vibrational entropy calculated by the first-principles method as discussed below. The technique of calculating an interfacial energy, including strain effects, was described by Wolverton and Zunger [11]. The resulting calculated interfacial internal energies at 0 K for Ni/Ni3Al(L12), Ni/Ni3Cr(L12), and Ni/Ni3Cr(DO22) are listed in Table 6. Generally, the | P a g e 12

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Ni/Ni3Cr(DO22) interface has the largest calculated interfacial internal energy for all (100), (110), and (111) planes, while the Ni/Ni3Cr(L12) interfacial internal energy has the smallest value for all (100), (110), and (111) planes. The (100) interface is found to have the smallest interfacial internal energy for Ni/Ni3Al(L12) and Ni/Ni3Cr(L12) interfaces. The large energy difference

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between Ni/Ni3Al(L12)(100) and Ni/Ni3Al(L12)(111), 11 mJ·m-2, affects the morphology of Ni3Al(L12) precipitates, which tends to form cuboidal structures. This finding is in agreement with prior calculations [53, 54] and with high resolution electron microscopy (HREM) results that show that Ni3Al(L12) precipitates develop large (100) facets and small (110) and (111)

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facets as they nucleate, grow and coarsen [45]. The energy difference between the (100) and (111) facets is small for Ni/Ni3Cr(L12), Table 6. Ni3Cr(L12) precipitates tend to form spheroidal

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ordered precipitates. The Ni/Ni3Cr(DO22) system has a larger interfacial internal energy than do the Ni/Ni3Al(L12) and Ni/Ni3Cr(L12) interfaces and the (111) plane has a lower value than other two planes (110), (111) at 0 K.

Secondly, we consider the effect of Cr on Nin(AlyCr1-y)/Ni3(AlxCr1-x) interfaces using the three alloy compositions. First, we calculate the interfacial internal energy at 0 K, Fig.4, according to the experimental concentration profiles as a function of aging time displayed in Table 1-3. The

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phase compositions for calculated interfacial structures of both γ− and γ’-phases are then varied to match the experimentally determined values listed in Tables 1, 2 and 3. The substitutional process is random and five configurations are generated for each aging time and the total tot

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tot energies Eαtot/ β , Eα and Eβ are calculated to obtain the average interfacial internal energy. The

individual phases are fully relaxed in our calculations. The calculated interfacial internal energy at 0 K is plotted as function of aging time, Fig. 4, for the three alloys. The (100) interface has the

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smallest interfacial internal energy of the three alloys, which exhibit the same trends as the Ni/Ni3Al(L12) and Ni/Ni3Cr(L12) interfacial systems. As the aging time increases, the calculated interfacial internal energies decrease with increasing Cr concentration and they achieve equilibrium plateaus after 16 h of aging for the three alloys. We take the interfacial internal energies at 128 h as the equilibrium values displayed in Table 6 for the vibrational entropy calculations to determine the temperature effects discussed below. We find that the calculated equilibrium interfacial free energies for the three alloys are in good agreement with the values determined experimentally at 873 K (600 oC): 23.0 ±1.2 mJ m-2 for alloy A; 24.0 ±1.6 mJ m-2 for alloy B; and 21.0 ±1.0 mJ m-2 for alloy C. | P a g e 13

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Now we consider the effect of temperature on the phonon vibrational entropy using the MedeA package with PHONON software developed by Krzysztof Parlinski [55]. The magnetic spin polarization calculations were utilized for the phonon calculations. The effects of

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temperature on the phonon vibrational entropy are based on the sharp interface model btween two phases. The vibrational entropy is calculated by: matrix / precipitates precipitates matrix ∆Svib = ∆Svib − ∆Svib − ∆Svib

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(9)

To calculate the phonon dispersions for the γ-matrix phase (pure Ni and solid-solution x)),

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Nin(AlyCr1-y)) and γ’(L12)-precipitates, Ni3Al(L12), Ni3Cr(L12), Ni3Cr(DO22) and L12 Ni3(AlxCr1we used the frozen-phonon method for 3 × 3 × 3 supercells for bulk supercells for the

γ(f.c.c.)/γ’ interfacial structures, and 2 × 2 × 2 supercells for the different (hkl) planes. Using this methodology we calculated a series of total energy differences between atomic configurations with and without a single displaced atom. The harmonic approximation was employed for all our vibrational calculations. The energy for the harmonic approximation [42, 43, 56] is calculated as:

1 1 2 mi [u& (i )] + ∑ uT (i )Φ(i, j )u ( j ) ; ∑ 2 i 2 i, j

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H=

(10)

where mi is the mass of atom i; u (i ) is the displacement away from its equilibrium position; and

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Φ (i , j ) is the tensor of the force constants from the displacement of atom j to the force on atom i.

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The Helmholtz free energy of the system is calcuated as [57]:  ∞  hv F = E * + Nk BT ∫ ln  2 sinh  0  2 k BT 

  d (v ) dv ; 

(11)

where E* is the total internal energy of the supercell at T = 0 K, at its equilibrium position; kB is Boltzmann’s constant; h is Planck’s constant, N is the number of atoms in the supercell; and d ( v ) is the phonon density-of-states (DOS). The DOS of the phonon vibrations for bulk pure Ni (solid-solution), Ni3Al(L12), Ni3Cr(L12) and Ni3Cr(DO22) are displayed in Fig. 5(a), and the phonon dispersion spectra of these three ordered structures are displayed in Fig. 5 (b). There are no imaginary frequencies in the phonon dispersion curves, and the three structures studied are | P a g e 14

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considered dynamically stable. The ordered phases, Ni3Al(L12), Ni3Cr(L12) and Ni3Cr(DO22), have higher average phonon frequencies, with less weight in the low-frequency region compared to the γ-(f.c.c.)-phase. The vibrational modes move to higher frequencies when Ni atoms are replaced by either Al or Cr atoms. Structural differences in the vibrational entropy are due to

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changes in the number of Ni-Ni, Ni-Al, Ni-Cr nearest-neighbor (NN) bonds. In the strongly intermetallic-forming Ni-Al system Ni-Al bonds are stiffer than Ni-Ni, Al-Al, Cr-Cr, or NiCrbonds, leading to higher phonon frequencies and smallest vibrational entropies [9]. The DOS of the high-frequency modes in γ’-Ni3Al is at ~10–11 THz, corresponding to Ni-Al vibrations,

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resulting in the DOS peaks moving to higher frequencies due to fewer NN Ni-Ni bonds. The DOS of the high-frequency mode in Ni3Cr(L12) is at 9.3 THz, corresponding to Ni-Cr vibrations,

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but such a mode is much weaker than γ’-Ni3Al. Interestingly, Ni3Cr(DO22) has strong lower frequency peaks at 0.4 to 0.5 THz than do Ni3Al(L12) and Ni3Cr(L12). A low frequency mode is related to the formation of magnetic domains, which make Ni3Cr(DO22) more stable than Ni3Cr(L12). The calculated vibrational entropies are: -17.13 J mol-1 K-1 for γ’-Ni3Al; -11.45 J mol-1 K-1 for Ni3Cr(L12); and -14.32 J mol-1 K-1 for γ’- Ni3Cr(DO22). The DOS of the (100) γNi/ordered-phase interface is a combination of γ-Ni and ordered phases: γ’-Ni3Al, Ni3Cr(L12), and Ni3Cr(DO22). The lower frequency phonon modes are very similar to those of γ-Ni, and the

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higher frequency modes decrease from 10-11 THz to 9-10 THz due to the distortion of the ordered phases. We observe that the higher frequency modes around 9-10 THz move to lower frequencies for the L12 phase, which indicates that the Ni-Al and Ni-Cr bonding structures and ferromagnetic fields across the denser (111) interface are more disrupted than for the (100)

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interface because of the absence of atomic symmetry across this interface.

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By accounting for vibrational entropy, the equilibrium interfacial free energies are calculated as a function of temperature. Interfacial free energies of Ni/Ni3Al(L12), Ni/Ni3Cr(L12) and Ni/Ni3Cr(DO22) decrease as the temperature increases, Fig. 6. As we discussed in our prior article [9], the calculated free energies 22, 28, and 29 mJ·m-2 for the (100), (110) and (111) Ni/Ni3Al(L12) interfaces, respectively, at 873 K are in agreement with the average values over all facets, calculated by Ardell, of 22.33 ± 1.31 and 31.08 ± 1.45 mJ m-2, obtained by fitting the existing and our previously reported experimental data to his trans-interface diffusion-controlled (TIDC) [46] and LSW-type coarsening models, respectively [58]; the latter values are, however, in better agreement with our calculations than Ardell’s TDIC values. Metastable Ni/Ni3Cr(L12) | P a g e 15

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interfacial structures have very similar interfacial free energy values: 20.5, 21.3, and 21.8 mJ m-2 for the (100), (110) and (111) planes at 873 K (600 oC), respectively. Theoretically, the preferred morphology of Ni3Cr(L12) is spherical at 873 K (600 oC), unlike the morphology of the Ni3Al(L12) phase, which is cubic. The Ni/Ni3Cr(DO22) interfacial free energy at 873 K (600 oC)

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is larger than that of the two L12 ordered phases (Ni/Ni3Al(L12) and Ni/Ni3Cr(L12)), with the values, 73.9, 69.1, and 58.9 mJ m-2 for the (100), (110) and (111) planes, which yields a cuboidal-shaped precipitate dominated by {111} planes due to the (111) interface having the

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lowest interfacial free energy.

We next investigated mixing between Ni3Al(L12) and Ni3Cr(L12). The Nin(AlyCr1-y)/Ni3(AlxCr1x)

(L12) interfacial free energy as a function of Cr concentration, Fig. 7, follows a similar trend as

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Ni3Al(L12) and Ni3Cr(L12); with the (100) having a lower interfacial free energy than the (110) or (111) interfaces. The calculated interfacial free energies at 873 K (600 oC), are: 25.5 mJ m-2 for the (100); 28.7 mJ m-2 for the (110); and 29.8 mJ m-2 for the (111) for alloy A (Ni-7.5Al8.5Cr at.% ). Then 26.6 mJ m-2 for the (100), 27.6 mJ m-2 for the (110), and 28.7 mJ m-2 for the (111) for alloy B (Ni-5.2Al-14.2Cr at.% ),

and 24.7 mJ m-2 for the (100), 29.6 mJ m-2 for the

(110), and 31.9 mJ m-2 for the (111), for alloy C (Ni-6.5Al-9.5Cr at.% ). The differece between

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the (100) and (111) interfacial free energies, calculated at 873 K (600 oC), decreases as the Cr concentration increases: 7.2 mJ m-2 for alloy C (Ni-6.5Al-9.5Cr at.% ); 4.3 mJ m-2 for alloy A (Ni-7.5Al-8.5Cr at.% ); and 2.1 mJ m-2 for alloy B (Ni-5.2Al-14.2Cr at.% ). We, therefore,

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conclude that the addition of Cr decreases the differences of the interfacial energies between different {hkl} planes, which results in the precipitate morphology transforming from cubic-to-

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spherical.

3.6. Nucleation Properties: First-Principles nucleation-barriers and critical radii for Ni3Al(L12) Ni3Cr(L12) , Ni3Cr(DO22) and Ni3(AlxCr1-x)(L12) Classic nucleation theory (CNT) can be used to determine the nucleation barrier and critical size of all the possible ordered phases in Ni-Al-Cr alloys studed: Ni3Al(L12), Ni3Cr(L12), Ni3Cr(DO22) and Ni3(AlxCr1-x)(L12). According to CNT, the thermodynamic energy barrier to

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nucleation is controlled by a three-factor competition[59, 60]: (1) a negative bulk free energy contribution, which is the chemical formation Helmholtz free energy change, ∆Fchem ; (2) a

energy contribution, σ.

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positive elastic strain energy contribution, ∆Eelastic ; and (3) a positive interfacial Gibbs free Using CNT, we can understand and explain the thermodynamic

mechanism for nucleation of Ni3Al(L12), Ni3Cr(L12), Ni3Cr(DO22) and Ni3(AlxCr1-x)(L12) in a

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Ni(f.c.c.) matrix. We described our first-principles calculations above for the bulk internal energy, elastic strain energy, and interfacial Gibbs free energy of these systems, which are for calculating the energy barrier to nucleation.

Now we assemble these three

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required

quantities into a first-principles prediction of the nucleation thermodynamics of putative Ni3Al(L12), Ni3Cr(L12) and Ni3Cr(DO22) ordered precipitates. We consider homogeneous nucleation of spherical precipitates.

The reversible work for the formation of a spherical

WR =

4π 3 R ∆FV + 4π R 2σ ; 3

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Where

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nucleus, W R , as a function of nucleus radius, R, is given by [61, 62]:

(13)

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∆FV = ∆Fchem + ∆Eelastic

(12)

The critical radius, R*, for nucleation is given by:

R* =

−2σ ∆Fchem + ∆Eelastic

(14)

And the critical net reversible work required for the formation of a critical spherical nucleus, W R ( R * ) , with radius R*, is given by: | P a g e 17

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WR ( R* ) =

16π σ3 3 (∆Fchem + ∆Eelastic )2

(15)

and n = 31 in 2x2x2 supercell. The equation for ∆Fchem is given by: ∆Fchem = ∆F ( Ni3 M ) + ( n − 3) ∆F ( Ni ) − ∆F ( Nin M )

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= [ ∆H ( Ni3 M ) − ∆H ( Nin M ) ] − T [ ∆S ( Ni3 M ) − ∆S ( Nin M ) ]

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The chemical formation energies were calculated based on the nucleation of Ni3Al (L12), Ni3Cr (L12), or Ni3Cr (DO22) from a dilute solid-solution, Nin M → Ni3M + Nin−3 , where M = Al or Cr,

(16)

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The entropy terms ∆S in Eq. 15 are from the phonon vibrational entropies of the ordered Ni3M and NinM cells in eq. (9) and (10). We consider the enthalpy, H, to be equal to the internal energy, E, because the pressure-volume term in H is negligible compared to E in the solid-state. In a dilute NinZ solution, we did not consider its contribution because the configurational entropy is much smaller than the vibrational entropy. Using Eq. 15 the calculated values of ∆Fchem at 873

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K are -38.5 meV·atom-1 for Ni3Al(L12), -9.78 meV·atom-1 for Ni3Cr(L12), and -13.78 meV·atom1 for Ni3Al(DO22). The values of ∆Eelastic are calculated from the coherency strains as explained above. We use the elastic strains in the corresponding softest directions, (100) for Ni3Al(L12) and Ni3Cr(L12), and (111) for Ni3Al(DO22). The elastic strains are 5.49 meV·atom-1 for

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Ni/Ni3Al(L12), 1.12 meV·atom-1 for Ni/Ni3Cr(L12) and 4.89 meV·atom-1 for Ni/Ni3Cr(DO22), which enables us to calculate R* and W R ( R * ) , Eqs. 13 and 14. These elastic internal strain

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energies represent an 11% contribution to the total nucleation free energies of Ni3Al(L12), a 10% contribution for Ni3Cr(L12), and a 28% contribution for Ni3Cr(DO22). The quantity ∆FV for Ni3Al(L12), Ni3Cr(L12) and Ni3Cr(DO22) is a function of temperature because of the entropic term, and the values of ∆FV were determined using Eq. 12. The calculated values at 873 K of R* (Table 7) are: 7.36 Å for Ni3Al(L12); 14.87 Å for Ni3Cr(L12); and 17.14 Å for Ni3Cr(DO22). And the values of WR ( R * ) are 2.7 and 1.62 meV (4.31x10-22 and 2.59x10-22 J) for Ni3Al(L12) and Ni3Cr(L12), respectively, 4.4 meV (7.03x10-22 J) for Ni3Cr(DO22), respectively. These results demonstrate that the formation of Ni3Cr(L12) requires a smaller ∆FV than does Ni3Al(L12) at 873 K (600 oC), while Ni3Al(L12) has the smallest critical radius. Ni3Cr(DO22) has a significantly larger ∆FV and has a very large critical | P a g e 18

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radius, which makes Ni3Cr(DO22) more difficult to nucleate than either Ni3Al(L12) or Ni3Cr(L12). This result is consistent with our experimental results as Ni3Cr(DO22) isn’t experimentally observed. Experimentally, we only observe Ni3Al(L12) and not Ni3Cr(L12). The reason is that even though Ni3Cr(L12) has the smallest, but it has a much larger critical radius

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than Ni3Al(L12). An embryo of Ni3Cr(L12) has difficulty to stabilize due to its larger critical size. Also, the Cr diffusivity is much smaller than Al’s diffusivity. Ni-Cr short-range order was only oberserved experimentally in the high Cr concentration alloy, B, in the early stage of nucleation [63]. An embryo of Ni3Cr(L12) only acts as the nucleant for mixed L12 Ni3(AlxCr1-x). Similar

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calculations were also performed for the mixed L12 Ni3(AlxCr1-x) phase using the interfacial free energies and formation energies for the three alloys studied, Table 5. The critical precipitate

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radii are: 9.23 Å for alloy B (Ni-5.2Al-14.2Cr at.% ); 8.45 Å for alloy A (Ni-7.5Al-8.5Cr at.% ); and 8.04 Å for alloy C (Ni-6.5Al-9.5Cr at.% ). The values of W R ( R * ) are: 1.12 meV (1.79x10-22 J) for alloy B (Ni-5.2Al-14.2Cr at.% ); 1.75 meV (2.81x10-22 J) for alloy A (Ni7.5Al-8.5Cr at.% ); and 2.04 meV (3.27x10-22 J) for alloy C (Ni-6.5Al-9.5Cr at.% ). We therefore conclude that the addition of Cr decreases significantly W R ( R * ) for nucleating Ni3(AlxCr1-x)(L12), but it has only a small effect on R* , ranging from 9.23 to 8.04 Å.

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3.7. The effects of Cr additions on precipitation morphology The equilibrium morphology of the γ’(L12)-precipitates transforms from cubic-to-spherical, which may be faceted, when adding Cr to the Ni-Al alloys, studied experimentally utilizing 3-D APT reconstructions for specimens aged at 873 K (600 o); Figs. 8(a) and (b) [6, 25, 64]. The

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morphology of γ’(L12)-precipitates derived from interfacial free energies calculated using firstprinciples methods involves the {100}, {110} and {111} facets. Figs. 8(c), (d), and (f) were

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determined from Wulff constructions using our first-principles interfacial free energies at 873 K (600 oC) employing the NIST program Wulffman (http://www.ctcms.nist.gov/wulffman/ ) [45, 65, 66]. If the anisotropic interfacial free energies are known, the Wulff construction yields the equilibrium morphology of a small precipitate. The smaller is the interfacial free energy the larger is the area of an {hkl}-facet. Employing our calculated interfacial free energies we determined the equilibrium Wulff morphologies at 873 K (600 oC) for Ni3Al(L12), Ni3(AlxCr1x)(L12)

and Ni3Cr(L12), which all exhibit the morphological transformation from a cube-to-a-

sphere, which may be faceted. The result for alloy A (Ni-7.5Al-8.5Cr at. %) is displayed in Fig. 8(b). For Ni3Al(L12) it is a great rhombicuboctahedron (or a truncated cuboctahedron), while | P a g e 19

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Ni3Cr(L12) has the morphology of a perfect sphere. The morphology of γ’-Ni3(AlxCr1-x)(L12)precipitates is spherical covered with smaller {100} facets than the bigger {100} facets that occur for binary Ni-Al alloys[9, 67-69]. Hence, the addition of Cr changes significantly the

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morphology of γ’(L12)-precipitates. 4. Summary and conclusions

We have studied systematically the stability and nucleation of putative ordered-precipitates in Ni-Al-Cr alloys based on bulk thermodynamics (static total energies and vibrational free

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energies) and interfacial Gibbs free energies. Additionally, the effects of Cr additions on the morphology of the ordered-precipitate phases has been studied experimentally employing 3-D

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atom-probe tomography (APT) and first-principles calculations. From these studies we reach the following conclusions: •

The formation energy and elastic constants for three possible ordered-precipitate phases, Ni3Al(L12), Ni3Cr(L12) and Ni3Cr(DO22), are calculated utilizing first-principles calculations. Both Ni3Al(L12) and Ni3Cr(DO22) are thermally stable structures, while Ni3Cr(L12) is a metastable structure. The formation energies for Ni3Cr(L12) and Ni3Cr(DO22) are significantly smaller than for Ni3Al(L12). The driving force for the transformation of ordered Al3Cr from the

•

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L12 structure to the DO22 structure is 1.05 kJ mol-1 (0.011eV atom-1). Chromium and Al share the same sublattice site in the L12 ordered-structure. Chromium partitions from the L12 ordered-structure to the disordered f.c.c. matrix with a driving force of 0.436 eV atom-1.

It is demonstrated that phase mixing between Ni3Al(L12) and Ni3Cr(L12) is not energetically

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•

preferred, utilizing first-principles quasi-random structure (SQS) calculations. From experimental 3-D APT reconstructions we observed that the precipitate morphology with

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•

Cr additions to the Ni-Al alloy changes from cubic-to-spherical at a temperature of 873 K (600 C). The compositions of the γ(f.c.c.)-matrix and the ordered γ’(L12)-precipitate phases of these

o

alloys evolve temporally as the γ(f.c.c.)-matrix becomes enriched in Ni and Cr and depleted in Al, while the γ’(L12)-precipitates become enriched in Al and depleted in Cr, Tables 1, 2, and 3. •

The interfacial Gibbs free energies of binary Ni/Ni3Al(L12), Ni/Ni3Cr(L12), and Ni/Ni3Cr(DO22) interfaces were calculated for the {100}, {110}, and {111} planes. The temperature dependencies of the interfacial Gibbs free energies is determined utilizing the phonon vibrational entropy. The calculated interfacial free energies at 873 K (600 oC) are: 25.5 mJ m-2 {100}, 28.7 | P a g e 20

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mJ m-2{110}, and 29.8 mJ m-2 {111} for alloy A (Ni-7.5Al-8.5Cr at.% ); 26.6 mJ m-2 {100}, 27.6 mJ m-2 {110}, and 28.7 mJ m-2 {111} for alloy B (Ni-5.2Al-14.2Cr at.% ); and 24.7 mJ m-2 {100}, 29.6 mJ m-2 {110}, and 31.9 mJ m-2 {111} for alloy C (Ni-6.5Al-9.5Cr at.% ). The Ni/Ni3Cr(DO22) interface has the largest calculated interfacial Gibbs free energy, 73.9 mJ m-2 for

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{100} plane, while the Ni/Ni3Cr(L12) interfacial Gibbs free energy has the smallest value 20.5 mJ m-2 for {100} plane. The {100} interface is found to have the smallest interfacial Gibbs free energy for the Ni/Ni3Al(L12) and Ni/Ni3Cr(L12) interfaces. The {111} interface is found to have the smallest interfacial energy for the Ni/Ni3Cr(DO22) interface. The large interfacial free energy

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difference between the {100} and {111} interfaces in Ni/Ni3Al(L12) affects the morphology of Ni3Al(L12) precipitates, which tends to be cubic. The interfacial Gibbs free energy difference

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between the {100} and {111} interfaces of Ni/Ni3Cr(L12) is very small and therefore Ni3Cr(L12) precipitates tend to form spherical precipitates. •

Chromium additions to Ni-Al alloys affect significantly the interfacial Gibbs free energies differences between the {100} and {111} interfaces. The interfacial Gibbs free energy differences are calculated at 873 K (600 oC), and become smaller with increasing Cr concentration: 7.1 mJ m-2 for alloy C (Ni-6.5Al-9.5Cr at.% ), 5.3 mJ m-2 for alloy A (Ni-7.5Al8.5Cr at.% ), and 1.9 mJ m-2 for alloy B (Ni-5.2Al-14.2Cr at.% ).

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The equilibrium interfacial Gibbs free energies in three Ni-Al-Cr ternary alloys are determined by utilizing the experimental concentration profiles for both the disordered γ(f.c.c.)-matrix and ordered γ’(L12)-precipitates as function of aging time from atom-probe tomography experimental three-dimensional reconstructions. The calculated Gibbs interfacial free energies for Ni3(AlxCr1are in very good agreement with experimentally determined average interfacial Gibbs

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x)(L12)

free energies at 873 K 600 oC). We find that the γ(f.c.c.)-matrix/{100}-interface has the smallest interfacial Gibbs free energy for the three alloys studied. As the aging time increases, the

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•

interfacial Gibbs free energies decrease with increasing Cr concentrations in the L12-ordered precipitates iand achieve equilibrium plateaus after 16 h aging time. Classical nucleation theory demonstrates that nucleation of Ni3Al(L12) is easier to stabilize than that of metastable Ni3Cr(L12) at 873 K (600 oC). Ni3Cr(L12) has the smallest W R ( R * ) , but it has a much larger critical radius than Ni3Al(L12). The embryo of Ni3Cr(L12) is dffucult to stabilize due to its larger critical radius and it only acts as an nucleant for mixed L12 Ni3(AlxCr1-x). Ni3Cr(DO22) is more difficult to nucleate than Ni3Cr(L12) and Ni3Al(L12). The calculated critical radius of Ni3Al(L12) nuclei (7.36 Å ) is smaller than the values for Ni3Cr(L12) (14.87 Å) and | P a g e 21

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Ni3Cr(DO22) (17.14 Å).

The critical net reversible work required for nucleation is 4.31x10-22,

2.59x10-22 and 7.03x10-22 J for Ni3Al(L12), Ni3Cr(L12) and Ni3Cr(DO22), respectively. Chromium additions to Ni-Al alloys have significant effects on the critical radius and critical net reversible work. The critical radius of Ni3(Al,Cr)(L12) is 9.23 Å for alloy B (Ni-5.2Al-14.2Cr

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at.% ); 8.45 Å for alloy A (Ni-7.5Al-8.5Cr at.% ); and 8.04 Å for alloy C (Ni-6.5Al-9.5Cr at.% ). The critical net reversible work required for Ni3(Al,Cr)(L12) is 1.12 meV (1.79x10-22 J) for alloy B (Ni-5.2Al-14.2Cr at.% ); 1.75 meV (2.81x10-22 J) for alloy A (Ni-7.5Al-8.5Cr at.% ); and 2.04 meV (3.27x10-22 J) for alloy C (Ni-6.5Al-9.5Cr at.% ). As the Cr concentration The predicted morphologies of L12-ordered precipitates is based on Wulff constructions using

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the NIST program Wulffman. The γ’-Ni3Al(L12) precipitate has the morphology of a great rhombicuboctahedron (truncated cuboctahedron), while Ni3Cr(L12)’s morphology is spherical: the {hkl}-interfacial Gibbs free energies were calculated based on first-principles calculations. The morphology of the Ni3(Al,Cr)(L12) precipitates transforms from cuboidal-to-spheroidal as a result of Cr additions to Ni-Al alloys. These results are in agreement with our three-dimensional

Acknowledgements

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atom-probe tomographic observations of the three Ni-Al-Cr alloys studied[68, 70].

This research was supported by the National Science Foundation, Division of Materials Research (DMR) grant number DMR-1610367 001 Profs. Diana Farkas and Gary Shiflet, grant officers. The alloys were processed at the NASA Glenn Research Center by Dr. Ronald D. Noebe. APT

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measurements were performed at the Northwestern University Center for Atom Probe Tomography (NUCAPT) by Drs. Chantal K. Sudbrack, Kevin E. Yoon, Chris Booth-Moorison, and Yang Zhou during their Ph.D. thesis studies and Ms. Jessica Weninger during her

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increases, from 7.5 to 14.2 at. %, Ni3(Al,Cr)(L12) becomes easier to nucleate.

undergraduate research studies. We thank Prof. C. Wolverton for helpful discussions and suggestions. The LEAP tomograph was purchased with initial funding from the NSF-MRI program (DMR 0420532, Dr. Charles Bouldin, grant monitor) and ONR-DURIP program (N00014-0400798, Dr. Julie Christodoulou, grant monitor) programs. Additionally, the LEAP tomograph was enhanced with a picosecond green laser with funding from the ONR-DURIP program (N00014-0610539, Dr. Julie Christodoulou, grant officer). We gratefully acknowledge the Initiative for Sustainability and Energy at Northwestern (ISEN) for grants to upgrade the

| P a g e 22

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capabilities of NUCAPT. We wish to thank research associate Prof. Dieter Isheim for managing NUCAPT and discussions.

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References

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Table 1. Temporal evolution of the γ(f.c.c.) and γ’(L12) phases’s compositions as a function of aging time, at 873 K (600 oC), as measured by atom-probe tomography for alloy A: Ni-7.5 Al8.5 Cr at.%. γ(f.c.c.)-matrix composition (at.%)

γ’(L12)-precipitate composition (at.%)

RI PT

Aging time (h)

Al

Cr

Ni

0.167

83.96 ± 0.05

7.37 ± 0.03

8.67 ± 0.03

72 ± 3

0.25

84.04 ± 0.05

7.09 ± 0.09

8.9 ± 0.1

73± 1

1

84.50 ± 0.04

6.63 ± 0.04

8.87 ± 0.04

73.9 ± 0.4

Al

Cr

21 ± 3

6±1

21 ± 1

6.2 ± 0.7

SC

Ni

6.2

± 0.4

± 0.2

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20.0

84.75 ± 0.04

6.20 ± 0.03

9.05 ± 0.03

74.8 ± 0.2

19.1 ± 0.2

6.1 ± 0.1

16

84.92 ± 0.06

5.88 ± 0.04

9.20 ± 0.04

75.4 ± 0.2

18.6 ± 0.2

6.1 ± 0.1

64

84.99 ± 0.05

5.76 ± 0.03

9.25 ± 0.04

75.7 ± 0.2

18.3 ± 0.2

6.0 ± 0.1

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

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Table 2. Temporal evolution of the γ(f.c.c.) and γ’(L12) phases’s compositions as a function of aging time, at 873 K (600 oC), as measured by atom-probe tomography for alloy B: Ni-5.2 Al14.2 Cr at.%.

γ(f.c.c.) -matrix composition (at.%)

γ’(L12)-precipitate composition (at.%)

RI PT

Aging time

Ni

Al

Cr

Ni

Al

Cr

0.167

80.59 ± 0.09

5.19 ± 0.05

14.22 ± 0.08

71 ± 3

19 ± 3

10 ± 2

0.25

80.73 ± 0.09

5.07 ± 0.05

14.20 ±

73 ± 1

18.2 ± 0.9

9.2 ± 0.7

SC

(h)

0.08 80.9 ± 0.1

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

75.5 ± 0.3

17.2 ± 0.2

7.3 ± 0.2

64

81.22 ± 0.07

3.45 ± 0.04

15.33 ± 0.07

75.7 ± 0.3

17.2 ± 0.3

7.2 ± 0.2

256

81.22 ± 0.07

3.30 ± 0.03

15.47 ± 0.07

76.0 ± 0.2

17.0 ± 0.2

7.1 ± 0.1

1024

81.16 ± 0.09

3.27 ± 0.04

15.57 ± 0.09

76.4 ± 0.3

16.7 ± 0.3

6.9 ± 0.2

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Table 3. Temporal evolution of the γ(f.c.c.) and γ’(L12) phases’s compositions as a function of aging time, at 873 K (600 oC) , as measured by atom-probe tomography for alloy C: Ni-6.5 Al-9.5 Cr at.%. γ(f.c.c.)-matrix composition (at.%)

γ’(L12)-precipitate composition (at.%)

(h)

RI PT

Aging time

Al

Cr

Ni

Al

Cr

0.5

83.05 ± 0.03

7.13 ± 0.08

9.82 ± 0.07

70 ± 3

21 ± 5

9±5

0.75

82.99 ± 0.03

7.22 ± 0.07

9.79 ± 0.07

71.3 ± 0.1

20.4 ± 0.2

8.4 ± 0.2

1

83.73 ± 0.01

6.46 ± 0.03

9.81 ± 0.03

71.5 ± 0.5

20.5 ± 0.8

8.1 ± 0.9

1.5

83.74 ± 0.03

6.38 ± 0.07

9.88 ± 0.06

72.3 ± 0.2

19.6 ± 0.3

8.1 ± 0.4

2

83.74 ± 0.02

6.38 ± 0.04

9.87 ± 0.04

72.8 ± 0.5

19.4 ± 0.8

7.8 ± 0.8

3.5

83.83 ± 0.04

6.27 ± 0.08

10.08 ± 0.08

74.1 ± 0.4

18.7 ± 0.8

7.2 ± 0.9

4

83.72 ± 0.01

6.29 ± 0.03

9.99 ± 0.03

74.53 ± 0.09

18.4 ± 0.2

7.1 ± 0.2

16

83.90 ± 0.01

5.90 ± 0.03

10.20 ± 0.03

75.09 ± 0.08

11.0 ± 0.1

6.9 ± 0.2

64

83.99 ± 0.02

5.80 ± 0.05

10.21 ± 0.05

75.55 ± 0.07

17.9 ± 0.1

6.6 ± 0.2

256

84.06 ± 0.02

5.71 ± 0.05

10.23 ± 0.05

75.82 ± 0.09

17.8 ± 0.2

6.4 ± 0.2

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

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Ni

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Table 4. The formation energy, lattice parameter, ao, bulk modulus, B, elastic constants, Cij, anisotropy ratio, A, and the Voigt averaged shear modulus, V, from first-principles calculations at 0 K (generalized gradient approximation (GGA)).

Formation energy 0

Ni3Cr (L12)

Ni3Cr (DO22)

-41.88

-0.26

-1.31

0

0.357

0.349

a=0.342

4.032

(kJ·mol-1) Lattice constant, a 0.352

c=0.678

Bulk module B 189

175.8

196

C11(GPa)

239

228

C12(GPa)

165

149

C44(GPa)

102

115

66

71

79

184

192

109

147

156

65

96

105

32

73

76

28

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< GV > (GPa)

201

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(GPa)

SC

(nm)

Al(f.c.c.)

RI PT

Ni(f.c.c.) Ni3Al(L12)

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Table 5. Average atomic forces and displacements at the first nearest-neighbor distance, from first-principles magnetic calculations for three different structures: solid-solution NinCr, Ni3(AlyCr1-y), with Cr substituting on the Al sublattice site, and (NixCr1-x)3Al with Cr substituting on the Ni sublattice site. Average Atomic

-1

-1

Force (eV Ǻ )

NinCr

0.129

0.0091

Ni3(AlyCr1-y)

0.565

0.0108

(NixCr1-x)3Al

0.648

0.0149

Displacement

(Ǻ)

0.0289

0.0371

0.0307

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(eV atom )

Average Atomic

RI PT

ECr→Ni, Al

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Table 6.

Density functional theory (DFT) first-principles calculated γ-Ni/Ni3Al (L12), γ-

Ni/Ni3Cr (L12),

γ-Ni/Ni3Cr (DO22) and three Nin(AlyCr1-y)/Ni3(AlxCr1-x) (L12) interfacial

energies using nonmagnetic spin-polarized methods for three different {hkl} planes at 0 K (mJ m-2). {110}

{111}

γ-Ni/Ni3Al (L12)

25

33

36

γ-Ni/Ni3Cr (L12)

28

31

33

γ-Ni/Ni3Cr (DO22)

RI PT

{100}

81

66

31

36

38

B

32

34

36

C

30

37

40

Nin(AlyCr1-y)/ Ni3(AlxCr1-x) (L12)

Nin(AlyCr1-y)/ Ni3(AlxCr1-x) (L12)

M AN U

Nin(AlyCr1-y)/ Ni3(AlxCr1-x) (L12)

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89 A

Alloy A: Ni-7.5Al-8.5Cr at.%

Alloy B: Ni-5.2Al-14.2Cr at.%

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Alloy C: Ni-6.5Al-9.5Cr at.%

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Table 7. The calculated nucleation parameters (critical radius, R* and net reversible work, W R ( R * ) ) for classical nucleation theory for four possible nuclei: Ni3Al (L12); Ni3Cr(L12); Ni3Cr(DO22); and Ni3(Al,Cr)(L12). W R ( R * ) (meV)

Ni3Al (L12)

7.36

2.70

Ni3Cr(L12)

14.87

1.62

Ni3Cr(DO22)

17.14

4.40

Ni3(Al,Cr)(L12)A

8.45

1.75

Ni3(Al,Cr)(L12)B

9.23

1.12

1.79x10-22

Ni3(Al,Cr)(L12)C

8.04

2.04

3.27x10-22

Alloy B: Ni-5.2Al-14.2Cr at.%

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2.59x10-22 7.03x10-22 2.81x10-22

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Alloy C: Ni-6.5Al-9.5Cr at.%

4.31x10-22

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Alloy A: Ni-7.5Al-8.5Cr at.%

W R ( R * ) (J)

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R* (Å)

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Figure captions: Figure 1. Density-functional first-principles theory calculated equilibrium coherency strain(L12); (b) Ni(f.c.c.)/Ni3Cr (L12); and (c) Ni(f.c.c.)/Ni3Cr (DO22).

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energies at 0 K for the {100}, {110}, and {111} interfacial planes for: (a) Ni(f.c.c.)/γ’-Ni3Al

Figure 2. The substitutional formation energies of Cr atoms as a function of distance from the Ni(f.c.c.)/γ’-Ni3Al (L12) heterophase interface from density-functional first-principles

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calculations at 0 K: both Ni and Al sublattice sites are considered.

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Figure 3. The mixing energies for Ni3Al (L12)/Ni3Cr (L12) calculated using the densityfunctional first-principles quasi-random structure (SQS) method.

Figure 4. The temporal evolution of the Nin(AlyCr1-y)/ Ni3(AlxCr1-x) (L12) interfacial energy from the density-functional first-principles calculations as function of aging time for three Ni-AlCr alloys at 0 K: (a) Alloy A: Ni-7.5Al-8.5Cr at.%; (b) Alloy B: Ni-5.2Al-14.2Cr at.%; (c)

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Alloy C: Ni-6.5Al-9.5Cr at.%.

Figure 5. (a) A comparison of density-functional theory calculated phonon vibrational density of states (DOS) for the following structures: γ-Ni(f.c.c.); ordered γ’-Ni3Al (L12) and γ’-Ni3Cr(L12), and the Ni3Al (DO22). (b) the phonon dispersion spectra of three ordered γ’-Ni3Al (L12) and γ’-

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Ni3Cr(L12), and the Ni3Al (DO22).

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Figure 6. Density functional theory calculated interfacial Gibbs free energies for the {100}, {110}, and {111} interfacial planes as a function of temperature: (a) γ-Ni(f.c.c.)/γ’-Ni3Al (L12); (b) γ-Ni(f.c.c.)/Ni3Cr (L12); and (c) γ-Ni(f.c.c.)/γ/Ni3Cr (DO22). These calculations include coherency strain energies and phonon vibrational entropies. Figure 7. Density functional theory calculated Nin(AlyCr1-y)/ Ni3(AlxCr1-x) (L12) interfacial free energies for the {100}, {110}, and {111} interfacial planes as a function of temperature for three different Ni-Al-Cr systems: (a) Alloy A: Ni-7.5Al-8.5Cr at.%; (b) Alloy B: Ni-5.2Al-14.2Cr

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at.%; and (c) Alloy C: Ni-6.5Al-9.5Cr at.%. These calculations include coherency strain energies and phonon vibrational entropies. Figure 8. Comparisons of the morphologies of the precipitate phases between experimental 3-D

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atom-probe tomography reconstructions and calculated Wulff constructions based on first-

principles calculated values of the {100}, {110} and {111} interfacial Gibbs free energies at 873 K (600 oC): (a) experimental γ’(L12) phase in a Ni-13 at.% Al alloy. The specimens were aged in bulk for 4096 h at 823 K (600 oC) prior to being studied by atom-probe tomography. The [100]-

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direction is perpendicular to a cube-face of a γ’(L12)-precipitate. (b) experimental γ’(L12) phase in Alloy A (Ni-7.5 Al -8.5Cr at.%). The alloy samples were aged at 873 K (600 oC) under flowing argon for 1024 h. (c) calculated γ’-Ni3Al (L12) precipitate phase; (d) calculated γ’-

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Ni3(Al,Cr)(L12)- precipitate phase; and (f) calculated γ’-Ni3Cr (L12)-precipitate phase.

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Figure 1.

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(a)

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(b)

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(c)

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Figure 2.

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Figure 3

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Figure 4.

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Figure 5. (a)

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Figure 5. (b).

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Figure 6.

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Figure 7.

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Figure 8.

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