The Pennsylvania State University The Graduate School A COMPUTATIONAL INVESTIGATION OF THE EFFECT OF ALLOYING ELEMENTS ON THE THERMODYNAMIC AND

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1 The Pennsylvania State University The Graduate School A COMPUTATIONAL INVESTIGATION OF THE EFFECT OF ALLOYING ELEMENTS ON THE THERMODYNAMIC AND DIFFUSION PROPERTIES OF FCC NI ALLOYS, WITH APPLICATION TO THE CREEP RATE OF DILUTE NI-X ALLOYS. A Dissertation in Materials Science and Engineering by Chelsey L. Zacherl c 2012 Chelsey L. Zacherl Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2012

2 The dissertation of Chelsey L. Zacherl was reviewed and approved by the following: Zi-Kui Liu Professor of Materials Science and Engineering Dissertation Advisor, Chair of Committee Long-Qing Chen Professor of Materials Science and Engineering Paul R. Howell Professor of Metallurgy Jorge O. Sofo Professor of Physics Professor of Materials Science and Engineering Joan M. Redwing Professor of Materials Science and Engineering Chair, Intercollege Graduate Degree Program in Materials Science and Engineering Signatures are on file in the Graduate School.

3 Abstract Ni-base superalloys have become ubiquitous in the materials science community in the past half-century because of their superior ability to resist chemical and mechanical degradation at temperature upwards of 70 % of their melting temperature. Future generations of Ni-base superalloys with increased service lifetimes and higher efficiencies will require the development of more complex, multi-component alloys carefully engineered to meet specific materials properties specifications, such as high temperature creep resistance. This can only come from an intimate knowledge of the exact effects of each individual alloying element on the thermodynamic and kinetic properties of the Ni-base superalloys. In this dissertation, two computational techniques have been employed to understand the alloying effects of various transition elements in Ni and its alloys. Thermodynamics and phase stability has been investigated through use of the CALculation of PHAse Diagram(CALPHAD) modeling technique, supplemented by first-principles calculations based on density functional theory (DFT). To better understand the kinetics involved in materials transport, the self-diffusion in ferromagnetic fcc Ni is calculated by first-principles, followed by a systematic investigation of the effects of 26 alloying elements on dilute Ni-rich binary alloys. Mechanisms causing specific diffusion behavior are explored, and the usefullness of such a database of knowledge is demonstrated by applying the generated data to a secondary creep rate model to show how each alloying element affects the creep behavior of the dilute Ni alloy systems. Finally, the next frontier of diffusion coefficient calculations by first-principles is explored by extending the calculation to non-dilute impurity concentrations, employing the fourteen-frequency model applied to the Ni-Al system. To aid in the process of narrowing down the large composition space for the design of future Ni-base superalloys, a thermodynamic model using the CALPHAD approach is developed, where Gibbs energy functions of individual phases are parameterized based on fittings to experimentally measured phase equilibria or iii

4 thermochemical data and computationally predicted thermochemical data. Multicomponent Ni-base superalloys can be accurately described within the CALPHAD approach through the extrapolation of the Gibbs energy functions of the simpler sub-systems which are modeled where experimental and computational data is usually more abundant. The Re-Y and Re-Ti systems, integral binary alloy systems in the Ni-base superalloy database, are modeled in the present work. Since little thermochemical data was available for either system, first-principles calculations were used to improve the thermodynamic models of the solid solution phases and the compounds. Both phase diagrams show excellent agreement with available experimental phase equilibria data. To further demonstrate the utility of first-principles calculations, an investigation on the phase stability of the ReTi compound is performed, because it was reported to have one crystal structure through experimental measurements and different crystal structure through first-principles calculations. To demonstrate the ability of the CALPHAD method for successful extrapolation to higher order systems, the Ni-Re-Y is modeled by combining the Re-Y system with the previously modeled constituent binaries. Good agreement is found with an experimental isothermal prediction of a two- and three- phase boundary at 1000 K. In addition to studying thermodynamic and phase stability properties of Nibase superalloys, this thesis also highlights the importance of the kinetic properties of these materials through their diffusion coefficients. Vacancy mediated self-diffusion coefficients are calculated on ferromagnetic and non-magnetic fcc Ni as a function of temperature. Within Eyring s reaction rate theory, minimum energy pathways for the diffusing atom is calculated using the Nudged Elastic Band method. It is observed that ferromagnetism is necessary for reproducing both Arrhenius and thermodynamic diffusion parameters, while the corresponding non-magnetic calculations show significantly poorer agreement. It is also observed that the use of the Debye-Grünseisen model for calculating the finite temperature entropic contributions to the diffusion coefficient reproduces the experimental self-diffusion more accurately than phonon calculations based on the supercell approach. Reasons for this surprisingly result are discussed in detail. Based on the success of the first-principles calculation of self-diffusion coefficient in pure Ni, the approach using ferromagnetic spin polarization and the Debye-Grünseisen model to calculate the entropic contributions to the diffusion coefficient are employed for 26 alloying elements including: Al, Co, Cr, Cu, Fe, Hf, Ir, Mn, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Sc, Si, Ta, Tc, Ti, V, W, Y, Zn, and Zr. The five-frequency model is implemented to accommodate the various jump frequencies associated with impurity diffusion coefficients in fcc Ni. The present work demonstrates that the mid-row 5d transition row element impurities have the highest activation barriers for impurity diffusion, and subsequently are the iv

5 slowest diffusers in Ni. The fastest diffusers in Ni coupled with the lowest activation barriers for impurity diffusion are demonstrated to be at the far left of the 3d and 4d transition element rows on the periodic table. The present work also demonstrates that the primary mechanism driving the variation in the impurity diffusion coefficient from element to element is the migration barrier for impurity diffusion. In addition, the correlation of the impurity diffusion coefficients is not found to be to the size of the element as previously predicted in the literature, but rather, that the impurity diffusion coefficients have a much stronger correlation to the compressibility of the associated Ni-X dilute alloy. A charge density analysis on the transition state of six of the twenty-six systems shows how the impurity affects surrounding Ni atoms. Assessments of the validity of the five-frequency model and the relaxation techniques for the treatment of the maximum energy point along the diffusion pathway are discussed using the Ni-Al system as a model case. First, an analysis of the assumptions made for the relaxation scheme of the three saddle configurations are made. It is shown that by using a new relaxation scheme, the calculated impurity diffusion coefficient can be improved with respect to experimental data. Additionally, an alternate calculation of the correlation factor for the impurity diffusion coefficient calculation is performed that assumes interactions of the solute and vacancy go beyond the first nearest neighbor shell. The alternate method includes the jump frequency associated with the migration of the host atom in the presence of an impurity at a second nearest neighbor position, as opposed to the original method which assumes this type of jump is analogous to self-diffusion in the host system. The result shows an increased agreement with the experimental data in the case of the Ni-Al system. In order to provide quantitative measures for the improvement of future generations of Ni-base superalloys, a model from the literature for the secondary creep rate typically applied to Ni-base superalloys is selected. The impurity diffusion results as a function of temperature are combined with previous first-principles calculations of elastic and stacking fault energy properties to predict a creep rate for each of the 26 systems normalized to the creep rate of pure Ni. Results from the application of the creep model reveal similar results to the impurity diffusion coefficients in fcc Ni. Mid-row 4d and 5d transition row element impurities decrease the creep rate compared when to pure Ni. In particular, Mo, Tc, Ru, and Re impurities cause the dilute alloys to retain more creep resistance at higher temperatures. Four elements that show significantly faster impurity diffusion coefficients relative to self-diffusion in pure Ni are Hf, Ti, Nb, and Ta. The creep rates of these four elements, however, are equal to or slower than the creep rate of pure Ni, indicating greater influence from stacking fault energy and elastic properties on these dilute alloys. v

6 Finally, the first-principles methodology for calculating dilute impurity diffusion coefficients in Ni-X alloy systems is extended further to impurity diffusion in more concentrated Ni-X alloys systems. The fourteen-frequency model is a natural extension of the five-frequency model to take solute-solute interactions into account. Using the Ni-Al system as a benchmark, all of the atomic jump frequencies as a function of temperature associated with the proposed fourteen-frequency model for diffusion in non-dilute systems are calculated using the ferromagnetic spin polarization and the Debye-Grünseisen model approaches. Solute and solvent enhancement factors to show the effect of impurity concentration on the diffusion coefficient are calculated. The impurity diffusion coefficient is fit to an empirical equation to show its dependence on composition in terms of the solute and solvent enhancement factors. Trends in the migration barriers and atomic jump frequencies for all fourteen jump frequencies are examined. The goal of the present thesis is to provide a better understanding of the thermodynamic and kinetic parameters of Ni-base alloys to aid in the future development of more advanced Ni-base superalloy systems. The methodology, results, and analysis presented in this thesis provide a better understanding of the effects of alloying elements on the diffusion properties of dilute and non-dilute Ni alloys, and establish a benchmark for effects and trends of impurity diffusion in other magnetic alloy systems, such as bcc Fe as the host matrix. vi

7 Table of Contents List of Figures List of Tables Acknowledgments xi xxi xxiv Chapter 1 Introduction Motivation Overview Organization Chapter 2 Computational Methodology CALPHAD Modeling First-principles calculations based on density functional theory Density functional theory - an overview Density functional theory at 0 K Special Quasirandom Structures Finite temperature thermodynamics Phonon supercell approach Harmonic Approximation Quasiharmonic Approximation Phonon supercell method implementation Debye-Grünseisen model Thermal electronic free energy Diffusion theory vii

8 2.4.1 Diffusion overview Eyring s reaction rate theory Nudged Elastic Band (NEB) method Chapter 3 First-principles calculations and thermodynamic modeling of the Re-Y system with extension to the Ni-Re-Y system Literature Review Calculation and Modeling details First-principles calculations CALPHAD modeling Results and Discussion First-principles calculations Re-Y Thermodynamic modeling Ni-Re Thermodynamic Re-Modeling Extension to the Ni-Re-Y system Conclusions Chapter 4 Phase stability and thermodynamic modeling of the Re-Ti system supplemented by first-principles calculations Literature Review Calculation and Modeling details First-principles calculations CALPHAD modeling Results and Discussion First-principles calculations Re-Ti Thermodynamic Modeling without the ReTi phase Re-Ti Thermodynamic Modeling with the B2-ReTi phase Conclusions Chapter 5 First-principles calculations of the self-diffusion coefficients of fcc Ni Diffusion Theory Computational details Results and Discussion K results Finite temperature results Non-magnetic and other phonon results viii

9 5.4 Conclusions Note on saddle configuration relaxation Chapter 6 First-principles calculations of dilute impurity diffusion coefficients in fcc Ni Literature Review Diffusion Theory Five-frequency model System setup for dilute impurity diffusion Diffusion equations Computational details Results and Discussion Test case: Impurity diffusion coefficient of Cu in Ni Non-Transition Element Impurities d Transition Element Impurities d Transition Element Impurities d Transition Element Impurities Conclusions Chapter 7 Analysis of dilute impurity diffusion coefficient calculations: methodology and trends Discussion of first-principles methodology Analysis of the approximations of the five-frequency model Discussion of diffusivity trends Results at 0 K Finite temperature diffusivity results Diffusivity mechanisms Charge density analysis Application to a steady state creep model Selection of creep model Finite temperature results Conclusions Chapter 8 First-principles calculations of non-dilute impurity diffusion coefficients in fcc Ni Literature review Diffusion Theory ix

10 8.2.1 Fourteen-frequency model System setup for non-dilute impurity diffusion Non-dilute impurity diffusion equations Computational details Results and Discussion Conclusions Chapter 9 Conclusions and Future Work Summary Final Conclusions Directions for future work Appendix A Thermo-Calc Ni-Re-Y database 209 Appendix B Thermo-Calc Re-Ti database 216 Appendix C EOS fitting, migration properties and thermodynamic parameters of the 26 Ni-X systems 220 C.1 Non-Transition Elements C.2 3d Transition Elements C.3 4d Transition Elements C.4 5d Transition Elements C.5 0 K EOS Properties Bibliography 233 x

11 List of Figures alloying elements and their atomic number studied in the present work in their approximate location on the periodic table Energy landscape of an hcp SQS structure following the manual volume and shape relaxation method proposed in the present work, for Y-25 at. % Re hcp SQS structure Schematic illustration of a close-packed plane of atoms where(a) the diffusing atom is adjacent to a vacancy in a normal lattice position and (b) the diffusing atom is in the high-energy state between its initial position and the vacancy site, known as the transition state or saddle configuration Volume per atom versus mole fraction Y for hcp ( ) and bcc( ) solid solution structures Properties for Re from first-principles (a) Phonon dispersion curve with phonon density of states calculated frequencies (solid lines) compared to experimental data of Smith et al. [1] (transverse:, longitudinal: ) and Shitikov et al. [2] ( ), (b) calculated heat capacity (solid line) compared to experimental data by Taylor et al. [3] ( ), Jaeger et al. [4] ( ), Rudkin et al. [5] ( ), Arutyuno et al. [6] ( ), and Filippov et al. [7] ( ), (c) calculated enthalpy as a function of temperature (solid line) compared to the SGTE SSUB database [8] (dashed line), and (d) calculated entropy as a function of temperature (solid line) compared to the SGTE SSUB database [8] (dashed line) xi

12 3.3 Properties for Y from first-principles (a) Phonon dispersion curve with phonon density of states calculated frequencies (solid lines) compared to experimental data of Sinha et al. [9] ( ), (b) calculated heat capacity (solid line) compared to experimental data by Jennings et al. [10] ( ), Berg, et al. [11] ( ), and Novikov et al. [12] ( ), (c) calculated enthalpy as a function of temperature (solid line) compared to the SGTE Unary database [13] (dashed line), and (d) calculated entropy as a function of temperature (solid line) compared to the SGTE Unary database [13] (dashed line) PropertiesforRe 2 Yfromfirst-principles(a)Phonondispersioncurve with phonon density of states calculated frequencies (solid lines), (b) calculated heat capacity (solid line) compared to the SGTE SSUB database [8] (dashed line) and a Neumann-Kopp approximation from the pure element phonon calculations (dotted line), (c) calculated Gibbs energy as a function of temperature (solid line) compared to the SGTE SSUB database [8] (dashed line) and experimental work done by Rezukhina et al. [14] ( ), and (d) calculated entropy as a function of temperature (solid line) compared to the SGTE SSUB database [14] (dashed line) and a Neumann-Kopp approximation from the pure element phonon calculations (dotted line) Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re hcp solid solution structure comparing manual volume and shape relaxation, volume and shape relaxation in VASP, and full relaxation in VASP to a pure hcp structure Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re bcc solid solution structure comparing manual volume and shape relaxation, volume and shape relaxation in VASP, and full relaxation in VASP to a pure hcp structure Enthalpy of mixing for hcp( ) and bcc( ) SQS from first-principles (points) and CALPHAD modeling in the current work (lines) Calculated phase diagram of the Re-Y system with experimental data from Lundin [15] ( ) Re-calculated phase diagram of the Ni-Re system with experimental data from Savitskii et al. [16]: ( ) melting, ( ) one phase, ( ) two phase Calculated phase diagram of the Ni-Y system as modeled by Du and Lu [17] Isothermal section of the Ni-Re-Y system at 1000 K Liquidus projection of the Ni-Re-Y system xii

13 4.1 (a) Entropy, (b) enthalpy, and (c) heat capacity of hcp-ti as a function of temperature calculated in the present work using the Debye model (line), compared to SGTE Pure Elements database [13] (dashed line), and the NIST-JANAF [18] experimental data ( ) (a) Entropy and (b) enthalpy of hcp Re as a function of temperature calculated in the present work using the Debye model (solid lines), compared to quasiharmonic phonon calculations of Zacherl et al. [19] (blue dotted lines) and SGTE Unary PURE4 database [20, 8] (red dashed lines), and (c) heat capacity as a function of temperature calculated in the present work (solid line), compared to quasiharmonic phonon calculations [19] (blue dotted lines) and SGTE Unary PURE4 database [20, 8] (dashed line), and the experimental data from Taylor et al. [3] ( ), Jaeger et al. [4] ( ), Arutyuno et al. [6] ( ), and Filippov et al. [7] ( ) (a) Entropy, (b) enthalpy, and (c) heat capacity of Re 24 Ti 5 as a function of temperature from the Debye model in the present work PhonondensityofstatesoftheReTiphaseintheB2structure(solid black line) and the MoTi structure (dashed blue line) calculated by quasiharmonic phonon first-principles calculations (a) Entropy, (b) enthalpy, and (c) heat capacity of ReTi as a function of temperature for the B2 structure from the phonon supercell approach (red solid line) and the Debye model (red dotted line) and for the MoTi structure (blue dotted-dashed line) Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re hcp solid solution structure comparing shape then ions relaxed in VASP, manual volume and shape relaxation, and volume and shape relaxation in VASP, to a pure hcp structure Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re bcc solid solution structure comparing shape then ions relaxed in VASP, manual volume and shape relaxation, and volume and shape relaxation in VASP, to a pure hcp structure Volume per atom vs. mole fraction Ti for hcp ( ) and bcc ( ) solid solution structures with symbols as the SQS predictions with experimental data of Joubert et al. ( ) [21] and Wu et al. ( ) [22] Enthalpy of mixing of SQS supercells for hcp shape and volume relaxed manually, ( ), hcp shape and volume in VASP, ( ), bcc shape and volume relaxed in VASP, ( ), and bcc fully relaxed (shape, volume, and atomic position) in VASP ( ), shown with the CALPHAD modeling from the present work (lines) xiii

14 4.10 Calculated phase diagram of the Re-Ti system without the ReTi phase, with experimental data from Savitskii et al. [23] showing the melting temperature of Re 24 Ti 5 ( ), the solidus ( ), bcc-ti single-phase region ( ), and hcp-ti + bcc-ti two-phase region ( ), Savitskii et al. [24] showing hcp-re single-phase region ( ), and hcp-re + Re 24 Ti 5 two-phase region ( ) Calculated phase diagram of the Re-Ti system with the ReTi phase, with experimental data from Savitskii et al. [23] showing the meltingtemperatureofre 24 Ti 5 ( ),thesolidus( ),bcc-tisingle-phase region ( ), and hcp-ti + bcc-ti two-phase region ( ), Savitskii et al. [24] showing hcp-re single-phase region ( ), and hcp-re + Re 24 Ti 5 two-phase region ( ) Enlarged calculated phase diagram for the Re-Ti system on the Re-rich side with data from Savitskii et al. [24] for the hcp-re single-phase region ( ) and hcp-re + Re 24 Ti 5 two-phase region ( ) Enlarged calculated phase diagram of the Re-Ti system on the Tirich side with data from Savitskii et al. [23] for bcc-ti single-phase region( ), and hcp-ti + bcc-ti two-phase region ( ) Enthalpy of formation at 300 K of the Re-Ti system without the ReTi compound (solid line), and with the ReTi compound from first-principles calculations in two structures (B2:, MoTi: ) and CALPHAD modeling (dashed line) of the present work, and with the previous first-principles high throughput study ( ) presented by Levy et al. [25] calculated at 0 K Entropy of formation at 300 K of the Re-Ti system when modeled without the ReTi compound (solid line) and with the ReTi compound (dashed line) Schematic diagram of the diffusion process showing(a) the IS with a vacancy in the first nearest neighbor site of a normal lattice position and (b) the SC showing the atom at the maximum energy point along the diffusion path, and the displacement of the lattice that results from the atomic migration Phonon density of states plotted for the three configurations needed to calculate the factors entering into vacancy mediated self-diffusion, the perfect state (PS), initial vacancy configuration (IS), and the transition state, or saddle configuration (SC) xiv

15 5.3 Vacancy concentration, C, plotted as a function of 1000/T from the present work within the LDA using the quasiharmonic Debye model (solid line) and quasiharmonic phonon calculations (dashed line) for finite temperature thermodynamic properties compared to experimental results of Scholz [26] and to an embedded atom study by de Koning et al. [27] Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model (solid line) and harmonic phonon calculations (dashed line) for the finite temperature thermodynamic contribution compared to the self-diffusion mobility assessment of Zhang et al. [28] and to selected single-crystal, (Wazzan [29], Ivantsov [30], Maier [31], Feller-Kniepmeier [32], Bakker [33], and Vladimirov [34]) and poly-crystal (MacEwan [35], Hoffman [36], Monma [37], and Bronfin [38]) experimental data Entropy of a 31 atom supercell of fcc Ni with a vacancy calculated with the Debye-Grünesien model or the phonon supercell approach with the thermal-electronic contribution and compared to the NIST-JANAF thermochemical tables [18] Calculated self-diffusion coefficient of ferromagnetic (FM) fcc Ni using the Debye model (solid line) and non-magnetic (NM) fcc Ni using the Debye model (dotted line) compared to selected singlecrystal, (Wazzan [29], Ivantsov [30], Maier [31], Feller-Kniepmeier [32], Bakker [33], and Vladimirov [34]) and poly-crystal (MacEwan [35], Hoffman [36], Monma [37], and Bronfin [38]) experimental data Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model (solid line) and non-magnetic fcc Ni using the Debye model (dotted line) compared to the consensus fit (blue dashed line) and 95 % confidence band (red dashed line) from the weighted means statistics study of Campbell et al. [39] Calculated magnetization charge densities(spin up minus spin down) looking down the b-axis for (a) the perfect, defect-free supercell, PS and (b) the initial vacancy configuration (IS) with a vacancy at all four corners of the figure, and (c), the saddle configuration (SC) with the diffusing atom at its maximum energy point along the diffusion path Deformation charge density (final total charge density minus total charge density after one relaxation step) of the SC looking down the a-axis, b-axis, and c-axis Slice of the deformation charge density along the [202] plane showing the SC atom (center) and its surrounding atoms xv

16 5.11 Calculated self-diffusion coefficient of non-magnetic fcc Ni using the Debye model (dotted line) and non-magnetic fcc Ni using the phonon supercell approach (dashed line) compared to single-crystal and poly-crystal experimental data and also the ferromagnetic QHA Debye model work Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model (solid line) and non-magnetic fcc Ni using the Debye model (dotted line) compared to the consensus fit (blue dashed line) and 95 % confidence band (red dashed line) from the weighted means statistics study of Campbell et al. [39] Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model showing the difference in relaxation algorithms of the CINEB approach vs the original NEB approach used in this chapter alloying elements and their atomic number studied in the present work in their approximate location on the periodic table. Various colors indicate the properties of the different impurity elements Five-frequency model as developed by Lidiard and LeClaire [40, 41] and shown by Mehrer [42] showing the five possible jump frequencies, w i, defined in the text, where represents the solute/impurity atom, one of 26 X s in the present work, represents a vacancy, and represents the host/solvent atom, pure fcc Ni in the present work Vacancy formation energy for a 32 and 64 fcc Ni supercell, respectively Solute vacancy binding energy for a 32 and 64 fcc Ni 31 X supercell, where X=, Al, Cu, or Re Diffusion activation energy for the 26 Ni 31 X systems and the activation energy for self-diffusion in Ni plotted from the present work versus diffusion activation energies from Janotti, et al. [43]. Note, Y, Zn, Sc, Al, and Si were not studied in the previous work Impurity diffusion of Cu in Ni calculated in the present work (solid line) compared to single-crystal data of Helfmeier et al. [44], and poly-crystal data of Anand et al. [45], Monma et al. [37], and Taguchi et al. [46]. Self-diffusion of pure fcc Ni is also shown for comparison (dashed line) The magnetic moment, in Bohr magnetons, of each volume of the ps in the Ni-Cu system plotted as a function of the six volumes used in the E-V fitting xvi

17 6.8 Impurity diffusion of Al in Ni calculated in the present work (solid line) compared to single-crystal data of Gust et al. [47], and polycrystal data of Swalin et al. [48] and Allison et al. [49] Impurity diffusion of Si in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [49] and Swalin et al. [50] Impurity diffusion coefficient of Al in Ni and of Si in Ni compared to self-diffusion of fcc Ni Impurity diffusion of Sc in Ni calculated in the present work Impurity diffusion of Ti in Ni calculated in the present work (solid line) compared to poly-crystal data of Bergner [51] and Swalin et al. [48] Impurity diffusion of V in Ni calculated in the present work (solid line) compared to poly-crystal data of Murarka et al. [52] Impurity diffusion of Cr in Ni calculated in the present work (solid line) compared to poly-crystal data of Monma et al. [53]. Růžičková et al. [54], Tutunnik et al. [55], and Glinchuk et al. [56] Impurity diffusion of Mn in Ni calculated in the present work (solid line) compared to poly-crystal data of Swalin et al. [48] Impurity diffusion of Fe in Ni calculated in the present work (solid line) compared to single-crystal data of Bakker et al. [57], and to poly-crystal data of Guiarldenq [58] and Badia et al. [59] Impurity diffusion of Co in Ni calculated in the present work (solid line) compared to single-crystal data of Vladimirov et al. [60] and to poly-crystal data of Badia et al. [59], Hirano et al. [61], Hassner et al. [62]. Divya et al. [63], and McCoy et al. [64] Impurity diffusion of Zn in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [49] and Swalin et al. [50] Impurity diffusion coefficients of 3d transition metal Ni-X systems, where X=Sc, Ti, V, Cr, Mn, Fe, Co, Cu, and Zn compared to self-diffusion in pure Ni Impurity diffusion of Y in Ni calculated in the present work (solid line) Impurity diffusion of Zr in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [49] and Bergner [51] Impurity diffusion of Nb in Ni calculated in the present work (solid line) compared to poly-crystal data of Bergner [51] xvii

18 6.23 Impurity diffusion of Mo in Ni calculated in the present work (solid line) compared to poly-crystal data of Swalin et al. [50] Impurity diffusion of Tc in Ni calculated in the present work (solid line) Impurity diffusion of Ru in Ni calculated in the present work (solid line) Impurity diffusion of Rh in Ni calculated in the present work (solid line) Impurity diffusion of Pd in Ni calculated in the present work (solid line) Impurity diffusion coefficients of 4d transition metal Ni-X systems, where X=Y, Zr, Nb, Mo, Tc, Ru, Rh, and Pd compared to selfdiffusion in pure Ni Impurity diffusion of Hf in Ni calculated in the present work (solid line) compared to the poly-crystal data of Bergner [51] Impurity diffusion of Ta in Ni calculated in the present work (solid line) compared to the poly-crystal data of Bergner [51] Impurity diffusion of W in Ni calculated in the present work (solid line) compared to the single-crystal data of Vladimirov et al. [60], and the poly-crystal data of Bergner [51], Swalin et al. [48], and Monma [65] Impurity diffusion of Re in Ni calculated in the present work (solid line) Impurity diffusion of Os in Ni calculated in the present work (solid line) Impurity diffusion of Ir in Ni calculated in the present work (solid line) Impurity diffusion of Pt in Ni calculated in the present work (solid line) Impurity diffusion coefficients of 5d transition metal Ni-X systems, where X=Hf, Ta, W, Re, Os, Ir, and Pt compared to self-diffusion of pure Ni Impurity diffusion of Al in Ni calculated in the present work without fully relaxation the transition states (solid line) and with full relaxation of the transition states (dashed line) compared to singlecrystal data of Gust et al. [47], and poly-crystal data of Swalin et al. [48] and Allison et al. [49] xviii

19 7.2 w im included in conjunction with the five-frequency model in the present work to include second nearest neighbor binding effects on the impurity diffusion correlation factor Impurity diffusion coefficient of Al in Ni calculated with Manning s method for binding at second nearest neighbor site, along with the original method presented in Chapter 6 and the fully relaxed saddle configurations presented in this chapter EOS calculated equilibrium properties for the ps (Ni 31 X at 0 K without the effect of zero point vibrational energy. The (a) equilibrium volume V 0, (b) bulk modulus, B 0, (c) first derivative of bulk modulus with respect to pressure, B 0, and (d) spin magnetic moment MM are plotted as a function of atomic number along different rows in the periodic table Impurity diffusion coefficient at T=1000 K normalized to self-diffusion in fcc Ni, according to periodic table placement Impurity diffusion coefficient at T=1000 K plotted as a function of increasing atomic number along different transition element rows in the periodic table The 26 impurity diffusion coefficients at T=1000 K plotted vs increasing experimental atomic radius from Pearson [66] Diffusion coefficient of each of the 26 dilute Ni 31 X systems plotted as a function of increasing compressibility at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements Diffusion coefficient of each of the 26 dilute Ni 31 X systems plotted as a function of increasing calculated vacancy formation energy at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements Diffusion coefficient of each of the 26 dilute Ni 31 X systems plotted as a function of increasing solute-vacancy binding energy at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements Diffusion coefficient of each of the 26 dilute Ni 31 X systems plotted as a function of increasing migration barrier for impurity diffusion at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements Relative deformation charge density of Re in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane xix

20 7.13 Relative deformation charge density of Y in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane Relative deformation charge density of Cu in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane Relative deformation charge density of Fe in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane Relative deformation charge density of Ta in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane Relative deformation charge density of W in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane Relative creep rate ratio plotted vs atomic number at T=300 K for all 26 Ni 31 X alloy systems Relative creep rate ratio plotted vs atomic number at T=600 K for all 26 Ni 31 X alloy systems Relative creep rate ratio plotted vs atomic number at T=900 K for all 26 Ni 31 X alloy systems Relative creep rate ratio plotted vs atomic number at T=1200 K for all 26 Ni 31 X alloy systems Fourteen-frequency model proposed by Bocquet [67, 68] showing the fourteen possible jump frequencies, w i, as defined in the text, where represent the host/solvent atoms, represents the solute/impurity atoms, and represents the vacancies Solventenhancementfactor,b 1,plottedasafunctionoftemperature for the solvent diffusion of Ni in the presence of at. % Al Self-diffusion coefficients of pure Ni, shown in the pure host matrix, and with additions of at. % Al, and 6.25 at % Al Soluteenhancementfactor, B 1, plottedasafunctionoftemperature based on Equation 8.6 showing the effect of paired solute atoms Impurity diffusion coefficient of Al in the Ni host matrix, shown with the initial calculation using the five-frequency model, and calculated enhancements of 6.25 at. % Al and 10 at. % Al xx

21 List of Tables 3.1 First-principles lattice parameters for Re, Y, Re 2 Y, and error with respect to experiments. Calculations from Wang [69] are also compared to the present work Model parameters of the Re-Y system, given in J/mol-formula Model parameters of the Ni-Re system, given in J/mol-formula Lattice parameters calculated from first-principles compared to experimental and other calculated values in the literature Fitted equilibrium properties from the EOS of hcp-re, hcp-ti, cubic Re 24 Ti 5, and ReTi at 0 K (unless noted) including equilibrium volume, V 0, bulk modulus, B 0, and first derivative of bulk modulus with respect to pressure, B 0, compared to previous experimental and DFT studies Model parameters of the Re-Ti system from the present work, given in J/mol-formula Arrhenius diffusion parameters for Ni compared to experimental data Thermodynamic parameters from the quasiharmonic Debye model obtained from the self-diffusion calculations of fcc Ni in the present work for both ferromagnetic (FM) and non-magnetic (NM) cases, along with available experimental and other DFT calculations at 0 K: with one using the LDA and PAW potentials [70], one using the LDA with exact muffin-tin orbitals [71], and one using the generalized gradient approximation [72] Thermodynamic parameters from the non-magnetic quasiharmonic Debye model and the non-magnetic harmonic phonon supercell approach, along with available experimental and other DFT calculations at 0 K Arrhenius diffusion parameters for the impurity diffusion of Cu in Ni compared to experimental data xxi

22 6.2 Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of Cu in Ni Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the Ni-Cu system Arrhenius diffusion parameters for the impurity diffusion of Al in Ni and Si in Ni compared to experimental data Arrhenius diffusion parameters for the impurity diffusion of 3d transition elements in pure Ni compared to experimental data Arrhenius diffusion parameters for the impurity diffusion of 4d transition elements in pure Ni compared to experimental data Arrhenius diffusion parameters for the impurity diffusion of 5d transition elements in pure Ni compared to experimental data Arrhenius diffusion parameters for the impurity diffusion of Al in Ni calculated by the NEB and CINEB methods compared to experimental data of [47, 48, 49] Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of Al in Ni comparing the two different relaxation schemes for the saddle configurations discussed in the present work Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the Ni-Al and Ni-Si systems Comparison of jump frequencies and jump frequency ratios for the two methods of calculating f 2 proposed by Manning [73] Comparison of impurity correlation factors, f 2, for the two methods proposed by Manning [73] Normalized creep rate for Ni-rich Ni-W alloys from the experimental work of Johnson et al. [74] compared to the normalized creep rate calculated in the present work at 1200 K Elastic [75] and stacking fault energy [76] data used for calculation of the relative creep rate ratio in Equation Atomic jump frequencies, w i, for non-dilute impurity diffusion of Al in Ni calculated using the fourteen-frequency model Migration energy barriers, G mi, for non-dilute impurity diffusion of Al in Ni calculated using the fourteen-frequency model xxii

23 C.1 Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of Al in Ni and Si in Ni C.2 Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the Ni-Al and Ni-Si systems C.3 Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of the 3d transition row impurities C.4 Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the Ni-Al and Ni-Si systems C.5 Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of the 4d transition row impurities C.6 Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the 4d transition row impurities C.7 Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of the 5d transition row impurities C.8 Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the 5d transition row impurities C.9 0 K equilibrium EOS properties calculated from the Debye-Grünesien model for all 26 dilute Ni 31 X alloy systems xxiii

24 Acknowledgments I would like to thank the following people for their support during my tenure at Penn State: My advisor, Dr. Zi-Kui Liu, for his continued support, guidance, and mentorship during my four years at Penn State. His confidence and unwavering belief in me, along with his memorable stories such as the ten pancakes story, kept me motivated and engaged in my research. I am much wiser, confident, and careful in both work and life for constantly asking myself the three whys and thinking forwards and backwards for my decisions. The committee members of my PhD dissertation, including Dr. Jorge Sofo, Dr. Paul Howell, and Dr. Long-Qing Chen, for their time devoted to reading my dissertation and for their constructive criticism and thoughtful advice. Dr. ShunLi Shang for his patience and involvement on this project for the past three years. His knowledge in first-principles and careful approach to research helped to make my project a success. The PRL members with whom I developed lasting friendships, including James, Arkapol, DongEung, Sunghoon, Brian, Alyson, Bi-Cheng, and Xuan. A big thank you to you all for always listening, discussing the thermodynamics of bacon, and making me laugh when I needed it the most. Thank you to Dr. Yi Wang for critique on my thesis. To all my friends in State College, especially Cairsty and Mike D., for seeing me through the best of times and the worst of times. My sister, who still probably thinks I am a mad scientist making crazy explosions every day but supports me 100 % nonetheless. My parents, for raising me with such determination and motivation to succeed at whatever I set out to do in life. To my fiance Mike, for loving me unconditionally and supporting me through the end. I can t wait to be Dr. Mrs.! xxiv

25 Dedication To my parents, for raising me with the determination to move my mountain, and to Mike, for giving me the strength to finish doing it. xxv

26 Chapter 1 Introduction 1.1 Motivation Ni-base superalloys, composed primarily of nickel and up to ten other alloying elements, are highly regarded in the materials science community for their ability to resist mechanical and chemical degradation at operating temperatures up to 70 % of their melting points when used in structural applications [77, 78]. The unique structure and capabilities of the Ni-base superalloys arise from what is known as the γ/γ microstructure that consists of a Ni-based fcc solid solution matrix phase (γ) with an ordered precipitate phase (γ ) that is a compound of the form Ni 3 X (X=Al, Ta, Ti) [77]. Other alloying elements such as rhenium and molybdenum are used to stabilize the γ phase or increase the high temperature creep resistance of the alloy [77]. Because of their excellent ability to withstand mechanical and chemical attacks at high temperatures, Ni-base superalloys have become the workhorse materials of the aerospace industry. They are primarily used on turbine blades for jet engines at the hottest section of the engines where the mechanical stresses are the greatest [77]. Ni-base superalloys have become the leading material in other high temperature and stress environments, including rocket components, nuclear reactor components, heat exchangers, and biomedical devices [78]. Improving the performance of Ni-base superalloys to be able to withstand higher temperatures with increased service lifetimes are two factors desirable to the materials and aerospace industries alike [79]. With the emergence of each

27 2 new generation of Ni-base superalloys, an increase in the upper limit of the operating temperature of C compared to the previous generation is usually obtained [80]. Development of the future, more advanced alloys, will require complex, multicomponent systems with carefully engineered compositions designed to meet specific materials properties specifications [81]. As the the third generation of Ni-base superalloys emerged, the element Re became a popular addition to improve the high temperature creep resistance of the superalloys [82]. Creep is a time-dependent process due to a constant load or stress that results in the permanent deformation of the material [83]. When creep occurs, it is inelastic and irrecoverable [77]. Rhenium is thought to have advantageous effects on the creep rate of Ni-base superalloys because of its low vacancy-solute exchange rates, high solute vacancy binding energy, and the high diffusion energy barrier when present in an alloy [77, 84]. There are also disadvantages caused by the addition of Re to Ni-base superalloys, which motivate understanding of the behavior of rhenium and the potential to replace Re while still maintaining high temperature creep resistance. At more than $10,000.00/kg, Re is one of the least abundant elements in the Earth s crust [85]. Also, whenreconcentrationsapproachorexceeds5wt. %oftheoverallcomposition of the superalloy, the formation of topologically packed phases (TCPs) is promoted. TCPs are detrimental to the creep strength of the alloys. Consequently, Ru or other platinum-group metals were added to the fourth generation of the Nibase superalloys to suppress the formation of the TCPs. [77] The results of the addition of both Re, Ru, or other Pt-group elements was a higher overall materials cost and a higher overall production cost of the latest generations of Ni-base superalloys. The overarching goal of this thesis is to aid in the understanding of the advantageous and deleterious effects of Re in Ni-base superalloys using various computational approaches with regards to thermodynamic and kinetic properties. Computational techniques are desirable for work on Ni-base superalloy development because of the high cost of preparing experimental Re and other rare earth containing samples. The present work uses two computational methods to develop new knowledge for the enhancement of Ni-base superalloy: the CALculation of PHAse Diagrams (CALPHAD) thermodynamic modeling technique and an ab-

28 3 initio approach for diffusion coefficient calculations. 1.2 Overview Future improvements of Ni-base superalloys for high temperature applications are going to come from the development of more complex, multi-component alloys with very specific compositions. Computational approaches are key in narrowing the composition space with which experimentalists should work to define new alloys with carefully engineered compositions. In the present thesis, thermodynamic and kinetic properties of Ni and its alloys are explored with the goal of aiding in the narrowing of this wide composition of Ni-base superalloys. Thermodynamic and phase stability of Ni and its alloys are investigated first. A key element in this process is gaining the understanding of how constituents such as rhenium, yttrium, and titanium affect the phase stability of multi-component alloys. The CALculation of PHAse Diagram (CALPHAD) [86, 87] modeling technique predicts the thermodynamic properties of a multi-component system from extrapolation of the constituent binary and ternary Gibbs energy descriptions, where experimental data is usually more plentiful. With this method, the properties of complex alloys can be efficiently and accurately predicted in a reduced amount of time compared to an equivalent experimental investigation. The Re-Y, Ni-Re, and Re-Ti systems are constituent binary systems of a Ni-base superalloy database being created within the CALPHAD community. The Gibbs energy descriptions of these three systems modeled in the present work to be added to the Ni-base superalloy database. When experimental data for the parameterization of the Gibbs energy functions is unavailable, first-principles calculations based on density functional theory (DFT) at 0 K have been shown to provide a realistic substitution to experimental data. If finite temperature thermodynamic properties are needed, the phonon supercell approach or empirical Debye model can be used to calculate the vibrational contribution to the Helmholtz free energy as a function of temperature. In the case of the Re-Ti system, first-principles calculations are used to investigate the stability of a phase reported in the literature having two different crystal structures; one from an experimental study and one from a first-principles study. Finally, to demonstrate the ability of the ability of the

29 4 CALPHAD method to extrapolate more simple systems to higher-order systems, the Ni-Re-Y system is modeled as an extension of the Re-Y modeling by combining the Re-Y thermodynamic descriptions with the Ni-Re and Ni-Y systems. In addition to thermodynamics, understanding the kinetics of materials transport is equally important in the design of new alloys. To do this, an intimate knowledge of the diffusion coefficients of the constituent subsystems with the Nibase superalloy composition space is essential, as diffusion is the primary mechanism for mass transfer in solids. In solids, diffusion coefficients are one of the key data types for predicting diffusional phase transformations, high temperature creep, and coarsening. The mechanisms of creep are considered to be dislocation glide at lower temperatures and dislocation climb at higher temperatures. The activation energy of dislocation creep is often correlated to the activation energy of self-diffusion of the host system, making the predicted diffusion coefficients in this work valuable to the Ni-base superalloy community. Particularly in close-packed metals, the diffusion process is mediated by the vacancy exchange mechanism, where host atoms or diffusion atoms are moving via hopping into an adjacent vacant lattice site. Previously, first-principles, parameter free approaches were developed to predict self- and impurity diffusion coefficients in metallic fcc systems with Al as the host matrix by the pioneering work of Mantina et al. [88, 89, 90]. The methodology of Mantina has been extended to bcc systems [91] and hcp systems [92, 93] with varying degrees of success. In systems with Ni as the host matrix, impurity diffusion data is much more scarce, particularly at higher temperature where vacancies have shorter lifetimes [94]. Previous theoretical works for calculating diffusion coefficients in Ni and its alloys have been primarily empirical [27, 95]. The present thesis adopts the methodology proposed by Mantina et al. [88, 89, 90] using Eyring s reaction rate theory [96] and the nudged elastic band method [97] to calculate the maximum energy points along the diffusion path. This dissertation explores the effect of ferromagnetism, as well as the effect of alloying elements on the self-, impurity, and non-dilute impurity diffusion coefficient of fcc Ni. DFT is used to calculate the ground state structures at 0 K necessary to represent the least energy diffusion path of all of the elementary atomic jumps involved in self-, impurity, and non-dilute impurity diffusivity. Finite temperature thermodynamic properties are then calculated using the phonon

30 5 supercell approach [98] for the sake of accuracy or the Debye-Grünseisen model [99] for the sake of simplicity and efficiency. Calculated finite temperature thermodynamic properties are used in conjunction with Eyring s reaction rate theory [96] to predict all of the components of vacancy mediated diffusion. As a benchmark calculation, self-diffusion in Ni is calculated via first-principles and shows good agreement with experimental diffusion quantities. While impurity diffusion data for more popular elements such as Al, Ti, and Ta in Ni are known, the impurity diffusion coefficient for many integral impurity elements in Ni-base superalloys such as Re, Pt, Pd, and Ru in Ni are unknown. The unavailability of this data plus the importance of values such as the activation energy for diffusion as a function of temperature are the primary motivators for the present work. The calculation of the impurity diffusion coefficient are completed using the first-principles methodology described above based on the five-frequency model of Lidiard and LeClaire [40, 41] for the 26 alloying elements shown in Figure 1.1: Sc 21 Y 39 Ti 22 Zr 40 Hf 72 V 23 Nb 41 Ta 73 Cr 24 Mo 42 W 74 Mn 25 Tc 43 Re 75 Fe 26 Ru 44 Os 76 Co 27 Rh 45 Ir 77 Ni 28 Pd 46 Pt 78 Cu 29 Zn 30 Al 13 Si 14 Figure 1.1: 26 alloying elements and their atomic number studied in the present work in their approximate location on the periodic table. For the impurity systems with available experimental data, calculated Arrhenius diffusion properties show good agreement with experimental values. Results indicate that the mid 4d and 5d transition row elements are the slowest diffusers in Ni, indicating the highest activation energy for diffusion. The slowest diffusers are Re, Os, Ir, Tc, and W. The elements with the lowest activation energy for diffusion and thus the fastest diffusion in Ni are Y, Sc, Zr, and Hf. The present work uses a charge density analysis coupled with an analysis of equilibrium properties, such as compressibility, of the 26 dilute alloy systems to analyze the trends in the diffusion behavior. To further understand diffusion behavior, an investigation into the

31 6 mechanism at 1000 K causing the effect of the change in diffusivity as a function of alloying element is observed. It is shown that the migration barrier for impurity diffusion is strongly correlated to the increase or decrease of the impurity diffusion coefficient with respect to that of pure Ni. Creep is often the failure mechanism of Ni-base superalloys. Creep is permanent, inelastic deformation, and the Ni-base superalloy in service will spend most of its lifetime in the secondary creep regime, where the creep rate is constant. To demonstrate the usefulness of the impurity diffusion coefficient calculation performed in the present work, a secondary creep rate model is applied to compare a normalized creep rate [100] as a function of temperature of each impurity system relative to the creep rate of pure Ni. It is agreed upon in the scientific community that a higher creep resistance for Ni-base superalloys can be linked to slower diffusion coefficients (and subsequent higher activation energies for diffusion), lower stacking fault energy, and smaller γ/γ misfit. The resulting trends of all 26 dilute alloy systems show similar correlation to the activation energy for diffusion, i. e., impurity systems that cause an increase in the activation barrier for diffusion in the dilute impurity system show increased creep resistance. The final goal of this thesis is to use the fourteen-frequency model of Bocquet [67], a natural extension of the five-frequency model, to demonstrate a methodology for the first-principles calculations of non-dilute impurity diffusion in a Ni-Al alloy containing 6 at. % Al. The fourteen-frequency model includes the effects of solute-solute interactions on the diffusion coefficient. Using the Ni-Al system as a demonstrative case, all factors entering into non-dilute impurity diffusion can be calculated using the first-principles methodology described above. Solute and solvent enhancement factors are calculated and can be used to describe the change in the impurity diffusion coefficient as composition of the alloy increases. 1.3 Organization The contents of this thesis are organized as follows. Chapter 2 is the methodology section. It includes a detailed methodology for the thermodynamic modeling using the CALPHAD approach, the background and details for all type of first-principles calculations used in the CALPHAD modeling and for first-principles calculations of

32 7 diffusion coefficients. It also introduces basic diffusion theory in close-packed cubic systems. Specifics for each type of calculation as well as the diffusion equations for calculating pertinent properties are given in the relevant chapters. Chapter 3 and Chapter 4 present the first-principles calculations supplemented thermodynamic modeling of the Re-Y and Ni-Re-Y systems and the Re-Ti system, respectively, showing the value of adding first-principles calculated properties to obtaining a more accurate thermodynamic description of the system. Chapter 5 validates the diffusion coefficient calculation procedure by presenting the first-principles predicted vacancy concentration and diffusion coefficient for self-diffusion in ferromagnetic fcc Ni. The effects of magnetism on the diffusion coefficient are explored and discussed in detail. Following the most successful methodology demonstrated in Chapter 5, Chapter 6 presented the results of the 26 Ni-X impurity diffusion coefficient calculations from first-principles. Basic trends among the 3d, 4d, and 5d transition row impurities are presented. Chapter 7 presents a detailed analysis of the impurity diffusion coefficient calculation methodology as well as a detailed analysis on the success of the calculations and the reasons why 5d transition elements show the slowest diffusivity and subsequent best creep resistance in dilute Ni-X alloys. Finally, Chapter 8 presents work on the non-dilute impurity diffusion of Al in an fcc Ni host matrix, following Bocquet s 14 frequency model. Solute and solvent enhancement factors are presented. Chapter 9 concludes this thesis by presenting a summary of all of the work done and recommendations for the future development of Ni-base superalloys. Possible areas for the future work on the first-principles calculations of self-, impurity, and non-dilute impurity diffusion coefficients are provided.

33 Chapter 2 Computational Methodology In this chapter, the methodology is given in order to reproduce the results obtained in this dissertation. First, thermodynamic modeling theory is presented, including an overview of the CALPHAD technique and the details of the parameterization of the Gibbs energy functions used for each phase. Then an overview of DFT and the associated finite temperature thermodynamic models used in both the CAL- PHAD modeling and the diffusion coefficient calculations is given. The chapter concludes with a review of diffusion and the relevant equations and assumptions are presented, while a more detailed procedure will be given in each respective chapter of self-, impurity, and concentrated impurity diffusion. 2.1 CALPHAD Modeling Thermodynamic modeling parameterizes the Gibbs energy of the individual phases in the system of interest as temperature, pressure, and composition dependent expressions. Thermochemical data of individual phases and phase equilibrium data between phases are fit to the expressions to determine the model parameters [86, 87]. Thermochemical data used to evaluate a single phase could be experimentally measured heat capacity, activity, or other property, or the corresponding data from first-principles calculations if the experimental data is missing or unreliable. Phase equilibria data such as phase boundaries and liquidus or solidus curves are determined primarily by experiments, and are useful to evaluating Gibbs energy functions of one phase relative to another phase under a given set of conditions.

34 9 The usefulness of thermodynamic modeling is observed once the Gibbs energy description has been evaluated for each phase in the system, because the functions can be extrapolated to other systems where experimental data does not exist to predict how new systems will behave. In the present work, the evaluation of the model parameters for each phase was performed within the PARROT module of the ThermoCalc software [101]. The general expression for Gibbs energy, G, can be expressed as follows: G = H TS (2.1) where H is enthalpy, S is entropy, and T is temperature. Both H and S are temperature-dependent. In the CALPHAD community, the Gibbs energy is often refined to be expressed in the following temperature-dependent equation: G H SER = a+bt +bt lnt +dt 2 +et 1 (2.2) where a, b, c, d, and e are model parameters evaluated in Thermo-Calc. The left side of Equation 2.2 shows that the Gibbs energy is defined with respect to a standard element reference state (SER) which is defined as the stable structure at K and 1 atm. This type of function was determined based on the analysis of the thermochemical behavior of several properties of the pure elements found in [13]. The function can be evaluated with at least three sets of experimental data. In the present work, the three sets of data chosen to fit Equation 2.2 are generally enthalpy of formation, ( f H), temperature-dependent heat capacity, (C P ) and entropy at a given temperature, (S). To fit the experimental or first-principles data according to Equation 2.2, the equation must be transformed to represent the various thermodynamic quantities. First, entropy is the negative first derivative of Gibbs energy with respect to temperature and is given as: S = dg dt = b c(1+lnt) 2dT +et 2 (2.3) Second, enthalpy can be derived by plugging Equation 2.2 and Equation?? into Equation 2.1 and then solving for H, which yields:

35 10 H = G+TS = a ct dt 2 +2eT 1 (2.4) Third and finally, heat capacity is taken as the first derivative of enthalpy, or the second derivative of Gibbs energy with respect to temperature time the negative of temperature: C P = dh dt = Td2 G dt 2 = c 2dT 2eT 2 (2.5) In order to fit the model parameters in Thermo-Calc, the heat capacity data is first optimized to the c, d, and e terms as shown in Equation 2.5. When those parameters are determined and fixed, the b term from Equation 2.3 is fit to entropy data at a given temperature. Finally, the a term from Equation 2.4 is fit based on f H at the SER of 298 K. When parameters a e have been defined, the Gibbs energy description as a function of temperature of the specific phase is complete. It should be noted, however, that at lower temperatures, the logarithmic and inverse terms of Equation 2.2 are not applicable. It is the standard in CALPHAD modeling to ignore the function below room temperature, K. To combat the loss of low temperature data, S can be used to fit the b term, being the integral of the heat capacity from 0 K to K. The fitted Gibbs energy function given in Equation 2.2 is used to describe all of the stoichiometric compounds in the present work, including Re 2 Y, Re 24 Ti 5, and ReTi. In solution phases, the compound energy formalism [102] is employed to represent the change in composition in a single phase via sublattice models. In the present work, the sublattices are necessitated by the fact that in a solution phase in a binary phase diagram such as hcp, bcc, or liquid, can have atom A or atom B sitting on any given site, based on the composition and crystal structure of the phase. The molar Gibbs energy of a solution phase of atoms A and B is given by: G ϕ m = x A G ϕ A +x B G ϕ B +RT(x Alnx A +x B lnx B )+ xs G ϕ (2.6) where x A and x B are the mole fractions of A and B, respectively, G ϕ A and Gϕ B are the Gibbs energies of pure A and pure B in the structure ϕ, respectively, and xs G ϕ is the excess Gibbs energy. The first two terms represent the mechanical mixing of the alloying elements and the third term represents the ideal mixing between the

36 11 two elements based on the configurational entropy of each sublattice. The excess Gibbs energy is modeled with a Redlich-Kister polynomial [103]: xs G ϕ = x A x k B L ϕ A,B (x A x B ) k (2.7) k=0 where L ϕ a,b represents the non-ideal interactions between A and B and is usually defined with a linear temperature dependence: k L ϕ Re,Ti = k A+ k BT (2.8) where k A and k B are model parameters to be evaluated. The Redlich-Kister polynomial is chosen for having a symmetrical contribution to the Gibbs energy. 2.2 First-principles calculations based on density functional theory Density functional theory total energy calculations are often known as ab-initio calculations, meaning from first principles because the inputs are the atomic coordinates and atomic numbers, and they do not rely on any experimental or empirical data. The total energy of the crystalline structure is then determined by using quantum mechanical electronic theory based on the electronic charge density. In this dissertation, the thermodynamic properties and ground state energies calculated in this work through the use of first-principles calculations based on density functional theory are used in several ways. In the CALPHAD modeling, thevaluesobtainedsuchasentropyandenthalpyofformationasafunctionoftemperature help to constrain the Gibbs energy functions of various phases to realistic values, which provide a more accurate extrapolation to higher order systems. In a different way, DFT is used to obtain the thermodynamic properties as a function of temperature for all of the configurations necessary to calculate the governing factors entering into vacancy mediated self- and impurity diffusion. Additionally, this approach can be extended to solve for relative energies for phases that are not thermodynamically stable.

37 Density functional theory - an overview The fundamental theory underlying density functional theory is that the ground state energy of a many electron system can be represented a functional of its electron density, ρ( r), and is obtained by minimizing the energy with respect to ρ( r) [104]. The quantum mechanical behavior of particles can be described by solving the time-independent Schrödinger equation where the wavefunctions of the particles are defined as ĤΨ = EΨ (2.9) where Ĥ is the Hamiltonian operator, E is the energy of the system, and Ψ is the eigenvalue wavefunction of the particles in the system. Since the number of atoms in a crystalline solid is on the order of 10 23, this equation cannot be solved explicitly as defined. The system can be simplified by employing periodic boundary conditions over the simplest repeating unit of a crystalline solid, which is usually the unit cell. However, the systems will still contain more than one electron and thus, this simplification does not allow for solving Equation 2.9 explicitly. Further simplification as proposed by Hohenberg, Kohn, and Sham[104, 105] states that the multi-electron wave functions, Ψ( r 1, r 2, r 3, r 4,... r n ), that are a function of of the position of every electron in the system can be replaced by a system with one electron interacting with other electrons through an effective potential. The effective potential is a functional of density of electrons, ρ( r), hence density functional theory. An additional assumption, know as the Born-Oppenheimer approximation or adiabatic decoupling, states that electrons are a set of interacting quantum mechanical points around a static nuclei [106]. With all properties of the system being described by the charge density, the total energy of the system is represented as a functional of its charge density through the following equation: E T = E[ρ( r)] (2.10) where E T is minimized iteratively with respect to the charge density to solve for the electronic ground state of the system. With the aforementioned assumptions in place, the total energy of the system can then be described as the sum of several components as:

38 13 E[ρ] = T 0 [ρ] = V ext [ρ]+v Hartree [ρ]+e xc [ρ] (2.11) where T 0 is the kinetic energy of the electrons without interactions, V ext is the external potential energy of the ions acting on the charge density, and V Hartree is the Coulombic interaction between a single electron in the system and all of the other electrons in the system, including itself. E xc is the exchange-correlation energy which accounts for all many-body physics electron interactions occurring in the system that are not Coulombic. Within the framework of DFT, the exact form of E xc is not known, and various approximations are used to define it. Two notable approximations for the calculation of the exchange-correlation functional used in the present work are the local density approximation (LDA) of Ceperley- Alder [107] under the Perdew-Zunger [108] parameterization and the generalized gradient approximation (GGA) as implemented by Perdew, Burke, and Ernzerhof [109]. In the LDA, the exchange-correlation energy is assumed to be a functional of the local electron density, depending solely upon the value of the electron density at each point in space. The LDA has had good success when applied to solids via first-principles calculations for relative total energy calculations [106], but its major drawback is having unphysical self-interaction terms and computing properties with absolute accuracy. For example, it typically underestimates the lattice parameters while cohesive energies and elastic properties are overestimated [110]. The GGA showed improvement in solid metal systems compared to the LDA because the exchange-correlation energy is assumed to be a functional of the local electron density and the gradient (rate of change) of the electron density at each point in space. The additional gradient term improves the accuracy of the DFT calculations. Both the LDA and GGA are used in this dissertation as implemented by [107, 109], with reasons for use given in the relevant chapters. In the present work, the Vienna ab-initio Simulation Package(VASP)[111, 112] is used to perform the density functional theory electronic structure calculations using the theory described above. Diffusion calculations performed with ferromagnetic Ni and other magnetic alloying elements employ the spin-polarized approximation.

39 Density functional theory at 0 K It should be noted that the total energies of a system computed using density functional theory are often not equitable to a physical situation until a reference state is put in place. Additional structure are often calculated to provide such a reference state for a compound at its SER. For example, in the thermodynamic modeling modeling portion of this dissertation, total energy calculations are converted into formation or mixing enthalpies. For a binary compound A i B j, the following relation is used to define enthalpy of formation: f H(A i B j ) = E(A i B j ) x A E(A) x B E(B) (2.12) where E s are the calculated total energies of the compound A i B j and pure elements A and B in their SER, respectively. One limitation of using DFT is that the ground state energies are computed at 0 K. Excited states, such as the vibrational or electronic excitations at finite temperatures, are not calculated directly. In the CALPHAD modeling, the assumption is often made that the enthalpy of formation has little variation with temperature and temperature effects can be incorporated into the temperaturedependent entropy and heat capacity terms. In the present work, this assumption is rarely used and finite temperature models are applied. When it comes to the diffusion calculations in the present dissertation, temperature effects from the entropic contribution must be taken into effect. Since DFT methodologies can be used to calculate any structure, the ability to input data into finite temperature models becomes important. For example, an equation of state (EOS) is a function relating the energy of the structure to one of its properties, such as volume, V. Volume dependence of any structure calculated form DFT can be used as an input into the finite temperature thermodynamic models needed to obtain values beyond 0 K. An EOS can take on several derived forms [113]. For metallic systems, one of the most widely used EOS s is the four-parameter Birch-Murnaghan [114] EOS chosen for its ability to reproduce properties of pure metallic elements [115, 113] and its non-zero second derivative of bulk modulus with respect to pressure [113]. The EOS is represented here in its linear form [113] as:

40 15 E(V) = a+bv 2/3 +cv 4/3 +dv 2 (2.13) where a, b, c, and d are model parameters. Using DFT, the EOS is determined by a series of fixed volume calculations across both negative and positive volumes, which provides an advantage over experimental results of the same EOS. Once the EOS fitting is complete, important materials properties such as the bulk modulus, B, and the first derivative of the bulk modulus with respect to pressure, B, can be obtained. The thermodynamic relations of the EOS for the obtained materials properties define pressure as: B and B are then defined as: P = E V (2.14) B = V P V = V 2 E V 2 (2.15) B = B P (2.16) The relations presented in Equations will become important in the following sections to help describe the finite temperature thermodynamic models used in the present work Special Quasirandom Structures In the case of disordered solid solution phases, special quasirandom structures (SQS) [116] are used to produce enthalpy of mixing. While there is often no reported experimental data for solid solution phases, it is important to have physically meaningful interaction parameters for binary solid solution phases for future use in higher order systems and databases. SQSs are important because they demonstrate that a limited number of atoms, suitable for first-principles calculations, can mimic the correlation functions of a completely random solid solution very closely for the first several nearest neighbor shells. These SQS structures have been successfully applied to fcc [117], bcc [118], and hcp [119] solid solution structures and are employed in the present work. The enthalpies of mixing for use

41 16 in the CALPHAD modeling can then be calculated analogously to Equation 2.12, but with the reference states of A and B in the same hcp or bcc structure at 0 K, not their SER. In the present work, hcp and bcc SQSs were used following the structures presented in the references above. Three compositions were used for each structure, A 25 B 75, A 50 B 50, and A 75 B 25, such that each solid solution phase could be represented by the composition at three points. The goal of fully relaxing the SQS structures with DFT is to obtain the 0 K ground state energy. However, local atomic relaxations are known to cause a significant distortion of the SQS, which may result in the symmetry of the parent structure being lost. In this work, the structural symmetry of the SQS supercell is kept during DFT calculations to ensure the fcc, bcc, or hcp structural symmetries are preserved. For both systems being examined, Re-Y, and Re-Ti, bcc and hcp structures are relaxed first following a scheme of volume relaxation only, then volume and shape relaxation only, then full relaxation including volume, shape and ions being simultaneously relaxed. This approach was met with varying degrees of success, and the results of the approach will be discussed in following relevant chapters. Unlike cubic structures, however, the hcp structures have an additional factor to take into consideration during the relaxation process, the c/a lattice parameter ratio. Previously, Shin et al. [119] allowed both the cell shape and volume to relax simultaneously in VASP for the hcp SQS structures. However, this direct relaxation approach could cause a loss of symmetry by changing the angles between the primitive lattice vectors of the hcp structure. In the present work, a method is proposed that manually minimizes the energy of the SQS by changing the a lattice parameter or the c/a ratio. The cell shape is fixed during a volume relaxation to ensure that the angles between the lattice vectors cannot change. With this method, the equivalent of a cell shape and cell volume relaxation that guarantees the preservation of the hcp symmetry is achieved. The energy landscape becomes a function of the hexagonal lattice vector and is calculated from a matrix of lattice parameter a and the c/a ratio. The minimum energy is determined by interpolation and a final static calculation is performed to confirm the interpolated minimum energy. An example of the energy landscape obtained through this SQS relaxation method is found in the following figure for a 25 at. % Re Re-Y hcp SQS.

42 17 Energy, ev a, Α c/a (ratio) Figure 2.1: Energy landscape of an hcp SQS structure following the manual volume and shape relaxation method proposed in the present work, for Y-25 at. % Re hcp SQS structure. A radial distribution analysis is performed on all relaxed SQS structures to compare resulting symmetry of the SQS structures. A radial distribution analysis examines the packing of atoms around each specific atom in the structure as a function of nearest neighbor distance. The resulting radial distribution analysis is then compared to the respective pristine structure, such as hcp or bcc. It is not expected that the SQS structure will have an identical radial distribution to the pristine structure, but it is expected that the nearest neighbor distance and frequency will be similar. 2.3 Finite temperature thermodynamics As discussed previously, the total energies obtained from DFT calculations are constrained to 0 K. For both the CALPHAD modeling and diffusion projects involved in this dissertation, the work would be incomplete without obtaining finite temperature thermodynamic properties, particularly, Gibbs energy as a function of temperature. Since DFT calculations predict E 0, the ground state static free energy, one can relate Gibbs energy to this calculated free energy through the following equation: G(P,T) = F(V,T)+PV (2.17)

43 18 where F(V,T) is the temperature and volume dependent Helmholtz energy of the system. At the 0 pressure conditions adopted during the DFT calculations, Gibbs energy and Helmholtz energy are equal to each other. It is possible to split up the Helmholtz energy of the system into contributions from various phenomena and calculate these contributions separately from various models. Helmholtz energy of a system as a function of volume, V, and temperature, T, can be defined in terms of the various phenomenological contributions as [99, 69, 120]: F(V,T) = E 0K +F vib (V,T)+F t el (V,T)+... (2.18) wheree 0 isthegroundstatestaticenergydiscussedaboveandisdirectlycalculated by first-principles. F vib (V,T) is the vibrational free energy originating from the phonons exciting the atoms out of their ground state positions. F vib (V,T) can be calculated from the phonon supercell approach for the sake of accuracy or from the Debye-Grünseisen model for the sake of simplicity and efficiency. F t el (V,T) is the thermal-electronic contribution to the Helmholtz energy where thermal excitations cause the electrons to be activated to higher states. There are many other possible contributions to the Helmholtz energy, including chemical energy, van der wall forces, or defect energy. The effects that contribute significantly to the particular system vary greatly from system to system, and a disadvantage of DFT calculations liesinthefactthatthereisno apriori knowledgeofwhatwillbemostinfluential. In the present work, which is primarily transition metals, previous studies on Ni and Ni containing compounds show that the three contributions presented in Equation 2.18 will have the largest influence on the thermodynamic properties of the system [69, 120, 113], with the largest contribution to the temperature dependence coming from the vibrational free energy. The following two sections describe in detail the aforementioned phonon supercell approach and the Debye- Grünseisen model Phonon supercell approach The phonon supercell approach [121] is widely considered to be the more accurate methodology for calculating temperature dependent thermodynamic quantities in metals, and is shown in particular to work well for fcc Ni and its alloys [69, 113].

44 19 Thepremiseofthesupercellapproachisthatbasedonsymmetry, forcesactingona perturbed atom can be used to construct a three dimensional force constant matrix, calledthedynamicalmatrix,whichcanbefittoamodeltocalculatethevibrational contribution to the Helmholtz energy, F vib (V,T). This essentially involves slightly perturbing the positions of the atoms away from their equilibrium positions and calculating the resulting reaction forces [98]. The calculated forces are put into the harmonic model to solve the dynamical matrix. The name, phonon supercell approach arises from DFT being periodic in nature. In other words, the perturbed atom must be far enough from its next periodic image in order to isolate the effect and accurately calculate the force constants, so supercell that is large enough to isolate the effect must determined based on a case by case investigation Harmonic Approximation Under the standard harmonic approximation, atoms are considered to only deviate slightly from their equilibrium positions. With this approximation made, the potential energy of a system is expanded around its equilibrium value in quadratic terms based on atomic distance. [122] Thus the harmonic approximation can account for all atomic interactions with other atoms in a 3 x 3 force constant matrix. Consider a system with N atoms and Greek letter subscripts that denote the Cartesian components of a vector. Under the harmonic approximation, the vibrational energy of the system can be written as [98]: H vib = 1 u T 2 α(i)φ α,β (i,j)u β (j) (2.19) α,β The 3 x 3 matrices, Φ(i,j) are the force constant tensors that relate the displacement of atom j to the force, f, exerted on atom i through: f(i) = Φ α,β (i,j)u β (j) (2.20) Through the use of static first-principles calculations, Φ(i, j) is determined by calculating a set of individual atomic perturbations in a supercell. When the harmonic approximation is applied, the force constant tensors can be written as:

45 20 Φ α,β (i,j) = 2 E u α (i) u β (j) (2.21) When Equation 2.19 is summed over all displacement u(i) over all atoms N of atomic mass M i, the resulting vibrational frequencies are the 3N eigenvalues of the dynamical matrix [98]. The vibrational free energy is based on the vibrational entropy, which is defined as the number of thermally activated vibrational modes at a specific temperature. Maradudin et al. [123] defines the equation for Helmholtz energy of a system under the harmonic approximation based on the partition function of lattice vibrations as: F vib (T) = k B q j { ln 2sinh [ ]} hwj (q) 2k B T (2.22) where Equation 2.22 is called the vibrational free energy or kinetic energy, which gets its contributions from the vibrational degrees of freedom of the system. The eigenvalues of Φ are the vibrational frequencies, w, and q is the wave vector. From this equation, the vibrational enthalpy and entropy can be derived and detailed equations can be found in [123]. Under the harmonic approximation, temperature dependence of the vibrational free energy is described solely from the vibrational degrees of freedom, and often does not give the most complete vibrational description of the system. Another limitation of the harmonic approximation lies in the fact that in reality, a nearly infinite number of perturbations would be necessary to accurately describe all of the phonon interactions and calculate the force constant. Thus, an assumption is made that the most important interactions occur in the first several nearest neighbor shells of the atoms in question, and the further away interactions are truncated from the calculation. This is a reasonable assumption to make, and this method has shown to be quite accurate given its limitations [98] Quasiharmonic Approximation The quasiharmonic approximation, while more computationally expensive, is more completeinandofthefactthatittakesthetemperaturedependenceofvolumeinto account by performing the harmonic approximation at volumes varying from the

46 21 equilibrium volume of the system. The non-harmonic nature of the potential energy is taken into account by extrapolating these harmonic contributions at different volumes into volume dependence. With the same force constant approach as given in the previous section, the Helmholtz energy of the system is now described with an additional volume dependence as [123, 98, 69]: F vib (V,T) = k B T q { [ ]} hwj (q,v) ln 2sinh 2k B T j (2.23) where w j (q,v) represents the frequency of the jth phonon-mode at wave vector q. Equations for the enthalpic and entropic vibrational contributions to the free energy can be derived from Equation 2.23 in order to calculate thermodynamic properties. Finally, two notes should be made about the phonon supercell approach and its limitations. While accurate with respect to common thermodynamic properties such as thermal expansion, heat capacity, etc, the quasiharmonic phonon supercell approach is very computationally expensive. For systems with low symmetry, defects, impurities, instabilities, or a large number of atoms, the number of perturbations and the number of atoms in the supercell increase rapidly, as does the time needed for the calculations. The force constant fittings are an additional expense on top of the actual perturbation calculations. Additionally, dynamically unstable systems will yield imaginary phonon frequencies, adding additional complexities when evaluating the thermodynamic properties by integrating the phonon DOS. When one or both of these problems are encountered, the Debye-Grünseisen model can be implemented for the sake of simplicity and efficiency in the calculations Phonon supercell method implementation The calculations for the phonon supercell approach are implemented using the ATAT code [98, 124] in VASP through the following steps: (1) fully relax the primitive cell using first-principles code, VASP, (2) increase or decrease the volume of the primitive cell several times by straining the lattice parameters by 2% and allow to relax using VASP under a fixed volume constraint, (3) generate the perturbed supercells within each increased volume according to the nearest neighbor interaction distance to ensure isolation of the perturbation and recalculate

47 22 the forces acting on the atoms using VASP, and (4) use ATAT and the calculated forces to evaluate the force constants and the phonon (vibrational) contribution to the free energy for each of the volumes calculated. Based on the vibrational properties from the quasiharmonic approximation, the equilibrium volume at each temperature is evaluated and the corresponding thermodynamic properties, including thermal expansion, are calculated based on the vibrational contribution to the Helmholtz energy. Further details including supercell size and other pertinant information to compelte the phonon calculations are given in the relevant chapters Debye-Grünseisen model The phonon supercell approach can be greatly simplified in the Debye model, and the Debye model has the additional value of being able to calculate dynamically unstable phases without having to deal with imaginary phonon frequencies. The model assumes that the velocity of sound, v, is constant for every vibrational mode in a crystal as it would be in classical mechanics [125, 99], essentially using an average of the longitudinal and transverse vibrational modes in a crystal. The phonon dispersion through the k-vector, K, is written as: ω = vk (2.24) where ω is the vibrational frequency of the lattice. Constant sound velocity is then defined as a function of the bulk modulus B, of the materials as: v = [ ] 1/2 B (2.25) ρ where ρ is the density of the system. Following Debye theory [125, 99], lattice vibrations are allowed to occur at all frequencies until the Debye cutoff frequency which is defined as [99]: h 2π ω D = k B Θ D (2.26) where h and k B are the Plank s and Boltzmann s constants, respectively. Θ D is the characteristic Debye temperature of the system and is defined as the temperature where the vibrations reach the vibrational cutoff frequency. It is also a tem-

48 23 perature below which the system exhibits low-temperature behavior and above which the system exhibits high-temperature behavior. Θ D can be calculated experimentally and is defined through the Debye model as: Θ D = (6π 2 ) 1/3 h 2πk B [ 4π 3 ] 1/6 [ ] 1/2 rb (2.27) M where r is the atomic radius and M is the atomic mass. Equation 2.27 simplifies to: Θ D = A [ ] 1/2 rb (2.28) M where A = when B is defined in GPA and r is defined in Å. Testing the definition of the Debye temperature by using experimental bulk moduli showed that the approximation given in Equation 2.28 yielded Debye temperatures higher than the ones obtained through experiments [99]. This is due to the fact that in reality, the stiffness of s crystal is related to its longitudinal, L, and transverse, S, moduli, and is anisotropic. The Debye model assumed that the longitudinal and transverse properties could be represented through the bulk modulus. To correct this overestimation, Moruzzi et al. [99] introduced a scaling parameter, s, into the definition of the sound velocity shown in Equation To determine the value of s, Moruzzi et al. [99] did two linear fittings, B vs L and B vs S on 14 cubic non-magnetic transition elements. They confirmed a value of s = for these 14 elements. While the value of s cannot be accepted for all material types, it has been shown to be closer to 1 in oxide system, for example, it has been shown to provide an accurate representation of thermodynamic properties of interest in the present work [113, 115]. The Debye model is by nature harmonic, and leaving out anharmonic effects on the crystal such as phonon-phonon interactions, linear expansion, and temperature dependence of other properties can have significant effects on the calculated thermodynamic properties. For example, a result of harmonicity gives a constant heat capacity at high temperature. [125] In the Debye model, anharmonic effects are included through the addition of the volume dependent Grüneisen constant, γ, defined as:

49 24 γ = lnθ D lnv (2.29) By substituting the definitions of the Debye temperature and bulk modulus, Equations 2.28 and 2.15, respectively, the high-temperature limit of the Grüneisen constant where longitudinal and transverse modes are excited can be defined by: γ = 2 3 V 2 2 / V 2 P/ V (2.30) Finally, Θ D can be presented as a volume-scaled term with V 0 as the ground state volume as: Θ D = sa [ rb M ] 1/2 [ ] γ V0 (2.31) The vibrational contribution to the total Helmholtz energy using the Debye model is given as F vib (V,T) = E vib (V,T) TS vib (V,T) in terms of the Debye temperature: V F vib (V,T) = 9 8 k BΘ D (V) k B T [ D ( ) ΘD (V) T ] +3ln(1 e ΘD(V)/T ) (2.32) where 9 8 k BΘ D (V) is the zero-point energy at 0 K due to the Heisenberg uncertainty principle. The Debye function, D(x), is defined in Equation 2.33 D(x) = 3 x 3 x 0 z 3 dz e z 1 (2.33) The section shows how the Debye-Grüneisen model can effectively and more efficiently than phonon calculations predict the vibrational free energy, entropy, heat capacity, or other thermodynamic quantity, of the system, with the main inputs being the Bulk modulus, B, and the equilibrium volume of the system, V 0, both of which are determined from the 0 K EOS fitting as described above. A brief discussion on the accuracy of first-principles Debye model calculations is merited here, and will be done with regards to calculated properties of the wellstudied hcp-ti phase. When the traditional Murnaghan EOS fitting is used instead of the Birch-Murnaghan one used for the present work, both B 0 and B 0 are shown

50 25 to decrease by 2.6 and 2.8%, respectively. Additionally, the equilibrium properties ofpurehcp-tiaresensitivetothenumberofvolumesusedforthefitting, i.e., when thenumberofvolumesusedisreducedfrom11to7,changesinb 0 andb 0 areonthe order of 1 and 2%, respectively. Regardless of the EOS fitting used, the calculated values in the present work still fall within the bounds of the experimentally reported data for both properties of hcp-ti [126, 127, 128]. However, it should be noted that with all of the variables in the usage of the Debye model, one can expect the thermodynamic properties calculated from first-principles to have an error less than 5%, which is consistent with previous computational modeling studies in the literature [129, 130] Thermal electronic free energy The thermal electronic contribution to the Helmholtz energy described in Equation 2.18, F t el, describes the energetic contribution due to thermally excited electrons at high temperatures around the Fermi level [69]. This contribution can be ignored in systems where the energy of the band gap is larger than the thermal energy, k B T, such as semi-conductors and insulators. For metallic systems, this thermal electronic contribution should be considered due to the non-zero electonic density at the Fermi level. While it has been shown to be negligible in diffusion calculations due to the cancellation of the contribution between the various states [88], the thermal electronic contribution has been included in the self-diffusion coefficient calculations for the sake of completeness. It has also been used in all calculations entering into the CALPHAD modeling in Chapters 2 and 3. The thermal electronic contribution is based on Fermi-Dirac statistics and can be written as a function of volume, V, and temperature, T, [69, 113]: F t el (V,T) = E t el (V,T) TS t el (V,T) (2.34) where E t el (V,T) and S t el (V,T) represent the energy and entropy of thermal electronic excitations, respectively. The electronic entropy, S t el (V,T), is defined as the integration of the energies of the excited electrons [69, 113]:

51 26 S t el (V,T) = k B n(ǫ,v)[f(ǫ,t)lnf(ǫ,t)+(1 f(ǫ,t))ln(1 f(ǫ,t))]dǫ (2.35) where n(ǫ,v) is the electronic density of states (DOS) and f is the Fermi-Dirac distribution defined as [125]: f(ǫ,t) = exp 1 [ ǫ µ(t) k B T ] +1 (2.36) where µ(t) is the chemical potential of an electron, and should be kept equivalent to the Fermi energy, ǫ F at 0 K. To obtain the thermal electronic energy, E t el, the difference between the charge density of the excited electrons and the 0 K energies is taken: E t el (V,T) = n(ǫ,v)f(ǫ,t)ǫdǫ ǫf n(ǫ, V)ǫdǫ (2.37) The charge density and Fermi energy used in Equations 2.35 and 2.37 are calculated from DFT using the methods described in the present work. 2.4 Diffusion theory Computational techniques are becoming increasingly desirable to obtain reliable self- and impurity diffusion data, because diffusion is the primary mechanism for mass transfer in solids. Additionally, obtaining these coefficients experimentally is a lengthy and time consuming process. In the present work, self-diffusion and impurity diffusion coefficients have been calculated for ferromagnetic fcc Ni, and have been simplified through the use of Eyring s reaction rate theory [96, 90]. While the specific details regarding the formulation and other theories related to self-, dilute impurity, and concentrated impurity diffusion will be presented in the relevant chapters, this section given an overview of diffusion followed by a description of how Eyring s reaction rate theory is implemented in the present work. Finally, we discuss the implementation of the methods needed to calculate the transition state structure via first-principles calculations.

52 Diffusion overview The present work focuses on vacancy-mediated diffusion in a crystalline solid and calculates the least energy diffusion path of an atom between the initial and final states of an elementary atomic jump. Essentially, two processes are occurring. First, the defect, in this case a vacancy, is formed. Second, a thermally activated jump is occurring where the diffusion atom and the first nearest neighbor vacancy exchange lattice sites, known as vacancy migration. In the case of fcc self-diffusion, there is one type of nearest neighbor jump. In the case of dilute impurity diffusion, one solute atom is inserted into the supercell. This creates five possible types of jumps, as the impurity atom can jump into the vacancy site, or an solvent (host atom) can jump into the vacant site in varying combinations related to the solute atom. In concentrated impurity systems where the solute atoms do not have infinite separation, fourteen jumps are possible. The addition of an additional solute atom creates many possibilities of solute and solvent atoms to move in relation in relation to the vacancy and each other. All jumps are correlated based on the symmetry and number of impurities in the crystal, and calculation of the correlation factors will be discussed in following relevant chapters. In any cubic system, all jumps are of the same length and are correlated identically to jumps occurring later in time. The diffusion coefficient can be represented as [131]: D = 1 2 Γ x 2 (2.38) wherexisthelengthofajumpinaspecificdirection, andγthejumpfrequency(or rate). In a pure fcc crystal, there is one possible type of independent jump which depends on the concentration of vacancies, C, adjacent to the diffusing atom, and the atomic jump frequency of the diffusing atom, w. Total jump frequency, Γ, is defined as [131]: Γ = ZCw (2.39) where Z is the number of nearest neighbor sites available for the diffusing atom to jump to, making CZ the probability that the diffusing atom will have a vacant site in one of its nearest neighbor sites. In cubic systems, the jump distance along

53 28 a specific direction, shown as x in Equation 2.38, can be expressed in terms of the jump length, r, as x 2 = r2. Combining this definition of the jump length with 3 Equations 2.38 and 2.39, diffusion in a cubic crystal is then written as: D = 1 6 ZCw r 2 (2.40) Atomic jumps are said to be correlated when one jump depends on the direction of the previous jump. This is self-evident as only nearest neighbor atoms can diffuse into a vacant site, so which one diffuses impacts the availability of the subsequent jumps. When diffusion is correlated, the diffusing atom has a higher than random probability of diffusing back to its starting position. The effects of correlation then depend on the type of diffusion that is occurring and the symmetry of the crystal in which the atom is diffusing. This correlation factor is described at f and is related to the length of the diffusing atom s jump as [42]: r 2 = f r 2 random (2.41) which makes the final equation for diffusion with correlation in a cubic system as [131]: D = 1 6 ZCwf r 2 (2.42) Finally, as the atom is diffusing, when it is exactly between its initial site and the vacant site, it is at its highest energy, unstable site. Determining all of the factors to calculate vacancy mediated diffusion requires knowledge of the energy and thermodynamic properties at this unstable state. In the present work, this atmoic migration is calculated under the confines of Eyring s reaction rate theory [96], which will be discussed in the following section Eyring s reaction rate theory As mentioned previously, the diffusion atom passes through a high-energy unstable states when it is exactly between its initial position and the vacant lattice site it is diffusing too. This is illustrated in Figure 2.2 [42] where site A and site B are the normal lattice positions, and the saddle point is shown as the maximum energy

54 29 point between those two normal lattice positions. Energy along the reaction path can be considered as and is also shown by [42] in Figure 2.2 where the highest energy point correspond to the saddle configuration and the energy required for this jump to occur is noted by G M, or the Gibbs energy of migration. Figure 2.2: Schematic illustration of a close-packed plane of atoms where (a) the diffusing atom is adjacent to a vacancy in a normal lattice position and (b) the diffusing atom is in the high-energy state between its initial position and the vacancy site, known as the transition state or saddle configuration. The premise of Eyring s reaction rate theory [96] is to show that thermal equilibrium exists between the activated states, the saddle configuration, and the low energy points which the atom is diffusion to and from. Thus, a primary assumption of using this theory as a model is that the diffusing atom moves isothermally and reversibly along the reaction path. Since thermal equilibrium must be observed, the motion is occurring at an infinitely slow velocity [131]. Another underlying principle of the theory is that the forces on atoms cause by the electron mobility and distribution are calculated using quantum mechanics, or DFT in the present work. But the atoms themselves are moving according to classic mechanics as one atom simply jumps to the nearest neighbor vacancy position [96]. The energy of reactions are then determined by a relation of the partition functions of the ac-

55 30 tivated states and the initial normal lattice position before the migration began. Glasstone et al. [132] expanded the concept of the reaction rate theory to describe the thermal activation of solids with a rate constant of k that would represent the jump frequency, w, with a frequency of vibrations in the system in the direction of the vacancy represented by a velocity, v. Wert and Zener [133] were the first to rectify this theory in an equation relating the partition functions of the vibrations to the jump frequency as [131]: w = k BT h ( ) F F (2.43) where F is the partition function for the normal lattice and F is the partition function for the activated state (saddle configuration) in all directions except the one that leads to the decomposition of the saddle configuration, results in F having one less degree of freedom than F. To solve the problems with the degrees of freedom, Wert and Zener [133] created a ratio of partition functions that have the same degrees of freedom by separating F into two components, F = F V F 1, where F V is the partition function for a single linear oscillator of frequency v. F V represents the atomic vibrations around its equilibrium position along the direction of the diffusion jump, and F 1 is the partition function for the complete system, but with the jumping atom constrained to move in the plane normal to the direction of diffusion. By breaking down F into these two components, Wert and Zener allowed the jump frequency to be written as [131]: w = k ( ) BT Gm exp hf V k B T (2.44) where G m is the migration barrier, or the difference in the saddle configuration and the initial normal lattice position, represented by G m = H m TS m. This can be done since F and F 1 have equal degrees of freedom. By ignoring the contribution to the free energy from the imaginary vibrational frequencies of the saddle configuration, atomic jump frequency as implemented in the present work can be defined as: w = k ( BT h exp G SC G ) IS k B T (2.45)

56 31 where G SC and G IS represent the Gibbs energy of the saddle configuration and the initial lattice site of the atomic jump, respectively Nudged Elastic Band (NEB) method As mentioned earlier, the framework of transition state theory involves calculating the forces acting on the atoms at the various states along the diffusion path using quantum mechanics, before treating diffusion as classical motion. However, creating a saddle point and allowing all degrees of freedom to relax using VASP would cause the unstable atom to fall backwards down the diffusion path and end up in one of the initial or final lattice positions, shown as site A or site B in Figure 2.2. The saddle configuration is first predicted as the middle of the minimum energy path between the initial and final equilibrium vacancy configurations, and its final position and energy are computed using the nudged elastic band (NEB) method [97] within VASP. The premise of the NEB method involves using a spring-like force acting on the unstable atom while looking at intermediate steps along the diffusion path, called images. In the present work, one or three images are used to calculate the the forces acting on the saddle configuration. A 5.0 ev/å 2 spring constant was used in all NEB calculations to nudge the image to the minimum energy path between the initial and final vacancy configurations. Calculations were allowed to fully relax within the confines of NEB, and the structure was checked to ensure that local relaxations did not distort the cell or cause reversal of the diffusing atom to one of the equilibrium vacancy spots. This however, was not the fullest relaxation possible for the saddle configuration. Towards the end of the present work, a newer formulation for the NEB method was instituted in the more refined climbing-image nudged elastic band (CINEB) method [134] that allowed for a fuller relaxation of the saddle configuration. The usage of the CINEB method will be discussed in the relevant chapters.

57 Chapter 3 First-principles calculations and thermodynamic modeling of the Re-Y system with extension to the Ni-Re-Y system Improving the ability of Ni-base superalloys to withstand higher temperatures with increasing longevity is desirable to the aerospace industry. More advanced alloys will require complex, multicomponent alloys with carefully engineered compositions. To develop such alloys, it is necessary to understand how different alloying elements, such as rhenium and yttrium, affect phase stability in multi-component Ni-base systems. The CALculation of PHAse Diagram (CALPHAD) [87] modeling technique meets this challenge by predicting the thermodynamic descriptions of a multi-component system from extrapolation of the constituent binary and ternary systems, where data is more plentiful. With this method, the properties of complex alloys can be efficiently and accurately predicted reducing costly and time-consuming experimental investigations.

58 Literature Review The Re-Y system was first investigated experimentally by Lundin and Klodt [15] using metallographic and x-ray diffraction techniques. It was found to have one intermetallic compound, Re 2 Y, with a hexagonal laves structure (C14), prototype MgZn 2, and spacegroup P6 3 /mmc. Two invariant reactions were observed: a peritectic reaction involving Re and Re 2 Y at 2793 K with an unreported composition for the liquid. A eutectic at 1723 K with a composition x y = 95 at. % was also reported. No solubility of Re in hcp Y or Y in hcp Re was observed experimentally. A galvanic cell experiment done by Rezukhina and Pokarev [14] reported the enthalpy of formation of Re 2 Y to be -45 kj/mole-atom. 3.2 Calculation and Modeling details First-principles calculations The calculations in this chapter employ the projector augmented wave (PAW) method [135, 136] and the generalized gradient approximation (GGA) as implemented by Perdew, Burke, and Ernzerhof [109]. A plane wave energy cutoff of 350 ev is used for all calculations. For total energy calculations of the pure elements and Re 2 Y, all degrees of freedom for the structures are allowed to relax. The pure Re and pure Y calculations are performed with a gamma k-point mesh 16 x 16 x 10. Re 2 Y calculations are done with a 10 x 10 x 6 k-point mesh. The SQS calculations are performed with a gamma k-point mesh of 8 x 8 x 5 for hcp structures and a 8 x 8 x 6 mesh for bcc structures. Phonon calculations are performed with a gamma k-point mesh of 6 x 6 x 6 and 5 x 5 x 6 for the pure elements and Re 2 Y, respectively. Supercell sizes were with 114, 48, and 48 atoms for Re, Y, and Re 2 Y, respectively CALPHAD modeling The Gibbs energy of Re 2 Y and the solution phases follow the equations and methodology as described in Chapter 2. The Gibbs energy function of Re 2 Y takes the form of Equation 2.2, and the Gibbs energy functions of the liquid and hcp

59 34 and bcc solid solutions phases take the form of Equation 2.6. The Gibbs energy descriptions for pure Y was taken from the Scientific Group Thermodata Europe pure element (SGTE Unary) database [13]. The most recent Gibbs energy description for pure Re differs from the original SGTE Unary database and can be found in the most recent SGTE substance (SSUB) [8] database and on the SGTE Unary database webpage [20]. 3.3 Results and Discussion First-principles calculations based on density functional theory in the present work are validated for known experimental data such as the heat capacity of pure Re and pure Y. First-principles is then performed on the compound of the system, Re 2 Y, to obtain its Gibbs energy function used later in the CALPHAD modeling First-principles calculations The validity of the first-principles calculations is examined by comparing the lattice parameters of the fully relaxed structures to experimental and other first-principles results. In Table 3.1 the lattice parameters of Re, Y, and Re 2 Y are compared to experimental [137, 125] and calculated [69] data. Differences are on the order of 1%, which are typical for such DFT calculations. Excellent agreement is found with Wang et al. [69], where lattice parameters were calculated using the original GGA [138] pseudopotential, while the GGA-PBE [109] method was used in the present work. Volume per atom of each structure as a function of mole fraction Y are shown in Figure 3.1 for the solid solution phases, hcp and bcc, for the compositions x Y = 0.25, 0.50, and We see a good relationship between the mole fraction of Y and increasing volume per atom for the respective structures. The phonon dispersion curves and phone density of states (DOS) along with the heat capacity, enthalpy, and entropy as a function of temperature for hcp Re are compared to experiments and the SGTE SSUB [8] database and SGTE Unary database webpage [20] in Figure 3.2. The phonon density of states was measured in Re through inelastic incoherent neutron scattering with polycrystalline plates

60 35 Table 3.1: First-principles lattice parameters for Re, Y, Re 2 Y, and error with respect to experiments. Calculations from Wang [69] are also compared to the present work. Element Reference a, Å % Error c, Å % Error Re Expt. [125] Calc. [69] This work Y Expt. [125] Calc. [69] This work Re 2 Y Expt. [137] This work Volume (Å) 3 /atom Mole Fraction Y Figure 3.1: Volume per atom versus mole fraction Y for hcp ( ) and bcc( ) solid solution structures. by Shitikov, et al. [2] at 300 and 500 K with negligible temperature dependence. Compared with current calculations in Figure 3.2(a), good agreement of the DOS shape is observed, particularly at lower frequencies. The phonon dispersion along the Γ-A direction is compared with experiments from Smith and Wakabayashi [1], and good agreement is found, including the representation of what Smith and Wakabayashi describe to be a Khon anomaly at the Fermi surface. Smith and Wakabayashi postulated that the anomaly is from strong electron-phonon interactions. Zolyomi et al. [139] show through ab-initio calculations that this

61 36 Frequency, THz G K M G A (a) C P, J/mol-atom/K (b) Temperature, K H-H 298, kj/mol-atom (c) Temperature, K S, J/mol-atom/K (d) Temperature, K Figure 3.2: Properties for Re from first-principles (a) Phonon dispersion curve with phonon density of states calculated frequencies (solid lines) compared to experimental data of Smith et al. [1] (transverse:, longitudinal: ) and Shitikov et al. [2] ( ), (b) calculated heat capacity (solid line) compared to experimental data by Taylor et al. [3] ( ), Jaeger et al. [4] ( ), Rudkin et al. [5] ( ), Arutyuno et al. [6] ( ), and Filippov et al. [7] ( ), (c) calculated enthalpy as a function of temperature (solid line) compared to the SGTE SSUB database [8] (dashed line), and (d) calculated entropy as a function of temperature (solid line) compared to the SGTE SSUB database [8] (dashed line). anomaly exists due to a small hole pocket on the Fermi surface at the edge of the Brillouin zone. We see excellent agreement with available experimental data for heat capacity [3, 4, 5, 6, 7] and the SGTE data in ThermoCalc for all properties in Figure 3.2(b-d). The phonon dispersion curves and phonon DOS of Y as well as the heat capacity, enthalpy of formation, and entropy as a function of temperature are compared with experiments and the SGTE Unary database [13] in Figure 3.3. The phonon dispersion curve in Figure 3.3(a) is compared to measurements obtained via inelastic neutron scattering by Sinha et al. [9]. The calculated Y phonon dispersion curves also show good agreement with experiment, especially along the Γ-A direc-

62 37 Frequency, THz G K M G A C P, J/mol-atom/K (a) (b) Temperature, K H-H 298, kj/mol-atom (c) Temperature, K S, J/mol-atom/K (d) Temperature, K Figure 3.3: Properties for Y from first-principles (a) Phonon dispersion curve with phonon density of states calculated frequencies (solid lines) compared to experimental data of Sinha et al. [9] ( ), (b) calculated heat capacity (solid line) compared to experimental data by Jennings et al. [10] ( ), Berg, et al. [11] ( ), and Novikov et al. [12] ( ), (c) calculated enthalpy as a function of temperature (solid line) compared to the SGTE Unary database [13] (dashed line), and (d) calculated entropy as a function of temperature (solid line) compared to the SGTE Unary database [13] (dashed line). tion. The calculations slightly underestimate the frequencies along the Γ-K-M and M-Γ directions; a result similar to another first-principles calculation of Y lattice dynamics done by Souvatzis et al. [140]. Heat capacity shows good agreement with experiments [10, 11, 12] as well as with the SGTE Unary database [13] in Figure 3.3(b-d). From the agreement of the phonon dispersion and phonon DOS for Re and Y, it is concluded that the finite temperature properties of Re 2 Y can be accurately predicted by phonon calculations employing the supercell method. The phonon dispersion curve and density of states plot, heat capacity, Gibbs energy, and entropy as a function of temperature for Re 2 Y are shown in Figure 3.4. In Figure 3.4(b) we see the heat capacity calculated from this work compared to the Neumann-Kopp

63 38 approximation from the pure element phonon calculations and from the SGTE SSUB database [8]. Excellent agreement is found. Figure 3.4(c) shows the Gibbs energy as a function of temperature for this work based on phonon calculations, the SSUB database and experimental points calculated from galvanic cell experiments with solid calcium fluoride done by Rezukhina et al. [14] between 1010 K and 1180 K. It is clearly seen in Figure 3.4(c) that the SSUB function was created based on the experimental data from [14], and that a discrepancy exists between the experimental and first-principles calculation Gibbs energy. Several sources of error for measuring energies of solid state fluoride galvanic cells were given by Azad and Sreedharan [141]; factors which could have caused incorrect measurements of the energy of the cell involving the formation of Re 2 Y. The work by Rezukhina and Pokarev [14] suggests a room temperature enthalpy of formation around -45 kj/mol-atom which is higher than all of the values of C14 Laves compounds reported in both experimental and computational studies done on HfM 2 alloys by Levy et al. [142] and on MCr 2 alloys by Chen et al. [143]. SQS calculations are performed for hcp and bcc solid solution phases allowing volume and shape relaxation using the symmetry preservation method described above, volume and shape relaxation following the work of Shin et al. [119], and full relaxation to occur. The symmetry is analyzed by a radial distribution analysis for each structure that is compared to a pristine hcp or bcc structure. It is not expected that the SQS symmetry will have the intensity of the ideal structure, but Figure 3.5 shows excellent agreement and only minor local relaxations for both volume and shape relaxed cases in the hcp structure for the three SQS compositions calculated, x Re = 25, 50, and 75 at. %. For the present work, the process described in the methodology section involving a manual relaxation of shape using static firstprinciples is used to relax the shape and volume of the hcp structures because it achieves a slightly lower energy per atom for the structure. In the case of the x Re = 50 and 75 at. % cases, the method from the present work shows a significant increase in the retention of symmetry compared to the VASP volume and shape relaxation. The splitting of the first peak can be attributed to the different atom types (Re, Y), in the structure, and do not indicate a loss of symmetry. For the bcc structures, volume and shape are relaxed in VASP because it achieves the lowest energy per atom while still retaining local symmetry. The corresponding RDFs

64 39 Frequency, THz G K M G A (a) C P, J/mol-atom/K (b) Temperature, K D G, kj/mol-atom (c) Temperature, K S, J/mol-atom/K (d) Temperature, K Figure 3.4: Properties for Re 2 Y from first-principles (a) Phonon dispersion curve with phonon density of states calculated frequencies (solid lines), (b) calculated heat capacity (solid line) compared to the SGTE SSUB database [8] (dashed line) and a Neumann-Kopp approximation from the pure element phonon calculations (dotted line), (c) calculated Gibbs energy as a function of temperature (solid line) compared to the SGTE SSUB database [8] (dashed line) and experimental work done by Rezukhina et al. [14] ( ), and (d) calculated entropy as a function of temperature (solid line) compared to the SGTE SSUB database [14] (dashed line) and a Neumann-Kopp approximation from the pure element phonon calculations (dotted line). for x Re = 25, 50, and 75 at. % in the bcc SQS strcutre can be found in Figure 3.6. It is possible that some local distortion occures in the x Re = 25 at. % RDF as additional peaks are observed in the RDF. This is also observed in Figure 3.7, where the same point has a relatively lower energy on the enthalpy of mixing curve. Inclusion of this data point in the Thermo-Calc modeling did not have an effect on the enthalpy of mixing parameter. The full relaxation of the SQS structures results in the complete loss of symmetry for both hcp and bcc SQSs, and the the data is not used in the present work.

65 Volume and Shaped Relaxed This work Volume and Shaped Relaxed This work Volume and Shaped Relaxed This work Number of Bonds Volume and Shape Relaxed VASP Volume, Shape, and Ions Relaxed Volume and Shape Relaxed VASP Volume, Shape, and Ions Relaxed Volume and Shape Relaxed VASP Volume, Shape, and Ions Relaxed Pure hcp Pure hcp Pure hcp (a) Distance (Å) (b) Distance (Å) (c) Distance (Å) Figure 3.5: Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re hcp solid solution structure comparing manual volume and shape relaxation, volume and shape relaxation in VASP, and full relaxation in VASP to a pure hcp structure. 40

66 Volume Relaxed Volume Relaxed Volume Relaxed Number of Bonds Volume and Shape Relaxed Volume, Shape, and Ions Relaxed Volume and Shape Relaxed Volume, Shape, and Ions Relaxed Volume and Shape Relaxed Volume, Shape, and Ions Relaxed Pure bcc Pure bcc Pure bcc Distance (Å) Distance (Å) Distance (Å) Figure 3.6: Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re bcc solid solution structure comparing manual volume and shape relaxation, volume and shape relaxation in VASP, and full relaxation in VASP to a pure hcp structure. 41

67 42 Enthalpy of Mixing, kj/mol Mole Fraction Y Figure 3.7: Enthalpy of mixing for hcp ( ) and bcc ( ) SQS from first-principles (points) and CALPHAD modeling in the current work (lines) Re-Y Thermodynamic modeling The calculated enthalpies of mixing from first-principles and the CALPHAD modeling as a function of Y content are shown in Figure 3.7 for both hcp and bcc solid solutions phases. The first-principles results predict a parabolic behavior for the enthalpy of mixing of both phases, indicating that a 0 L description is sufficient to describe their behavior. Since the enthalpies of mixing for both the bcc and hcp SQS s are positive, miscibility gaps are possible in the solution phases. The two solutions are modeled in ThermoCalc as regular solutions, employing only 0 L interaction parameters determined from the enthalpy of mixing predictions. With Gibbs energy descriptions for Re 2 Y, the pure elements, and the hcp and bcc solution phases, only the Gibbs energy parameters of the liquid phase remain to be determined. The liquid phase is modeled as a sub-regular solution phase, employing 0 L and 1 L terms in the Redlich-Kister polynomial of Equation 2.7. The sub-regular solution model is necessary to reproduce the eutectic and peritectic

68 43 reactions. With a regular solution, Re 2 Y melts congruently. The eutectic and peritectic data from Lundin and Klodt [15] are used for the evaluation of the liquid model parameters. The complete list of parameters for all of the phases in the Re-Y system can be found in Table 3.2. Also in Table 3.2 is a list of parameters for the Re-Y system if the modeling is completed using the function for Re 2 Y found in the SGTE SSUB database based on the experimental data of Rezukhina and Pokarev [14] and the functions for the pure elements mentioned above. As the SSUB database predicts a significantly more stable description for Re 2 Y, parameters larger by an order of magnitude are needed to compensate for this difference and to reproduce the phase equilibrium experiments. In addition to being an order of magnitude larger than the original modeling, the values are also physically unrealistic. In an article by Witusiewicz and Sommer [144] on the excess entropy of mixing in binary liquid alloys, we see that for more than 70 systems examined, the majority of the entropy of mixing terms for the liquid phases fell between -40 J/mol-K and 20 J/mol-K, with no system s term outside of a -60 J/mol-K and 60 J/mol-K range. Therefore, at J/mol-K, the magnitude of the entropy of mixing term in the liquid phase for the SSUB modeling is physically unrealistic. The description of the Re-Y system based on the results of the firstprinciples calculations for the thermochemical properties of Re 2 Y yields a more physical modeling. The phase diagram based on the assessment from this work of the Re-Y system is plotted in Figure 3.8 and compared to the experimental data points described above. The experimental eutectic at a liquid composition of at. % Y and 1723K is well reproduced by the model parameters. The peritectic reaction is also well reproduced, with the liquid composition predicted to be around 37 at. % Y Ni-Re Thermodynamic Re-Modeling In order to extend the Re-Y modeling to a prediction of the ternary Ni-Re-Y modeling, the Ni-Re and Ni-Y binaries are needed. In the Ni-Re system, the description of the pure elements is from the second version of the SGTE SSUB pure element database [13], while the phase descriptions of the pure elements of the Ni-Y and Re-Y systems are from the most recent version, version four. This

69 Table 3.2: Model parameters of the Re-Y system, given in J/mol-formula Phase (model) Parameters Modeling from first-principles Modeling from SSUB Liquid (Re, Y) 0 L liq 125, *T 508, *T 1 L liq 32, ,890 hcp (Ni, Re) 0 L hcp 320, ,000 bcc (Ni, Re) 0 L bcc 170, ,000 Re 2 Y -81, *T *Tln(T) -157, *T *Tln(T) *T 2-3,598.4*T *T 2-20,920.4*T 1 44

70 Liquid 3000 Temperature, K hcp Re 2 Y bcc hcp Mole Fraction Y Figure 3.8: Calculated phase diagram of the Re-Y system with experimental data from Lundin [15] ( ). creates an incompatibility in the ternary description of hcp Re as the function had changed between version two and three. The Ni-Re system was remodeled using the phase description for pure Re from version four of the SGTE SSUB pure element database. As described by Huang and Chang [145], interaction parameters for magnetic properties were not included because this binary system has not been investigated at temperatures below 1000 K. The liquid interaction parameter was kept as similar as possible to the previous modeling and was based on an estimate of the enthalpy of mixing at 50 at. % Re done by de Boer et al. [146]. The recalculated phase diagram is presented in Figure 3.9 along with its corresponding modeling parameters in Table 3.3. The present modeling agrees well with both the experimental data from Savitskii et al. [16] and the previous modeling done by Huang and Chang [145]. It should be noted, however, that the boundary between the fcc/fcc+hcp one- and two- phase regions is not perfectly reproduced with respect to the experimental data of Savitskii et al. [16], indicating that reproduction of the experimental peritectic is not exact.

71 Liquid Temperature, K hcp 1000 fcc Mole Fraction Re Figure 3.9: Re-calculated phase diagram of the Ni-Re system with experimental data from Savitskii et al. [16]: ( ) melting, ( ) one phase, ( ) two phase. Table 3.3: Model parameters of the Ni-Re system, given in J/mol-formula Phase (model) Parameters This work Liquid (Ni, Re) 0 L liq 16,000 hcp (Ni, Re) 0 L hcp 12, *T fcc (Ni, Re) 0 L fcc 27, *T 1 L fcc -16,906 bcc (Ni, Re) 0 L bcc 27, Extension to the Ni-Re-Y system As a further contribution to the Ni-superalloy database being created, phase equilibria in the Ni-Re-Y system is predicted. The modeling uses the Ni-Y database from Du and Lu [17] and the Ni-Re system from Huang and Chang [145]. The phase diagram of the re-modeled Ni-Re system was presented in Figure 3.9 in the previous section. The Ni-Y system as modeled by Du and Lu is reproduced in Figure In the Ni-Re-Y system there are no known ternary compounds so it was as-

72 47 Figure 3.10: Calculated phase diagram of the Ni-Y system as modeled by Du and Lu [17]. sumed that no ternary phases formed. Ternary interaction parameters in the hcp, fcc, and liquid solid solutions were not introduced. The Gibbs energy parameters for the Ni-Y systems can be found in the paper by Du and Lu [17]. After constructing the ternary description from the three binary systems, a stable bcc phase formed in the Ni-Re binary system. Instead of choosing an arbitrary positive interaction parameter for the Ni-Re bcc phase, an SQS calculation was done to predict the enthalpy of mixing at 50 at. % Re. This ensured the value would have a physical value. It is also a testament to the usefulness of SQS calculations which can predict values such as enthalpies of mixing for systems that are not attainable experimentally. Completion of the calculation led to a bcc interaction parameter of 27,558 J/mol-formula, which is reflected in Table 3.3. An isothermal section of the Ni-Re-Y system at 1000 K is shown in Figure The Ni-rich portion of the diagram agrees well with experimental data from Savitskii et al. [147], showing the boundary between the two phase fcc + Ni 17 Y 2 and the three phase fcc + hcp + Ni 17 Y 2 at 1273K. There is no known experimental data regarding ternary solubilities in the binary intermetallics. Therefore, ternary solubilities were not modeled in this work. The liquidus projection for this system is shown in Figure 3.12 with

73 48 the phases forming from the liquid during primary solidification. Ni Mole Fraction Y fcc +Ni 17 Y hcp + NiY 3 + Re 2 Y NiY 3 + Ni 2 Y 3 + Re 2 Y Ni 2 Y 3 + NiY+ Re 2 Y NiY + Ni 2 Y+ Re 2 Y Ni 2 Y + Ni 3 Y + Re 2 Y hcp + Ni 3 Y + Ni 7 Y 2 hcp + Ni 4 Y + Ni 5 Y hcp + Ni 17 Y 2 + Ni 3 Y fcc + hcp + Ni 17 Y 2 Y hcp + Ni 4 Y + Ni 7 Y Mole Fraction Re hcp + Ni 3 Y + Re 2 Y hcp + Ni 5 Y hcp + Re 2 Y Figure 3.11: Isothermal section of the Ni-Re-Y system at 1000 K. Re There are two significant areas in which experiments could enhance the present work. In the case of the modeling of Re-Y, no experimental data exists on the shape of the liquidus curves. Therefore, the liquid was modeled with the fewest parameters possible. With experimental data regarding the liquidus curve, the inflection of the liquidus curves and the exact location of the eutectic could be reproduced more accurately. Another area that would benefit from experimental data is the presence of ternary solubility of the binary intermetallics. Currently, no experimental data exists on the Ni-Re-Y system beyond the work of Savitskii et al. [147] showing the boundary between the two phase fcc + Ni 17 Y 2 and the three phase fcc + hcp + Ni 17 Y 2 in the ternary system. Experiments on the solubility of pure Re and Re 2 Y in the Ni-Y intermetallics could help to improve this modeling.

74 49 Ni 0.1 Ni 17 Y Mole Fraction Y Ni 5 Y 1700 Ni 3 Y Ni 7 Y 2 Ni4 Y fcc Y Re 2 Y 2600 hcp Mole Fraction Re Re Figure 3.12: Liquidus projection of the Ni-Re-Y system.

75 Conclusions The coupling of first-principles calculations and thermodynamic modeling is a valuable approach when designing new Ni-base superalloys, particularly with elements such as rhenium that exist in limited quantities. First-principles calculations based on density functional theory have supplemented the limited experimental thermochemical data of the Re-Y system. The enthalpy of formation for the compound Re 2 Y is calculated from first-principles, showing that the experimental value may have a substantial error. The enthalpies of mixing for the hcp and bcc solution phases are calculated through the use of special quasirandom structures and are found to be positive for both phases. Phonon calculations provided heat capacity, entropy, and enthalpy of formation at finite temperatures. A complete self-consistent thermodynamic evaluation of the Re-Y system has been obtained with first-principles calculations of solid phases and experimental phase equilibrium data. Phase equilibria in the Ni-Re-Y system was predicted based on the databases of the constituent binary systems and included a re-modeling of the Ni-Re system.

76 Chapter 4 Phase stability and thermodynamic modeling of the Re-Ti system supplemented by first-principles calculations Increasing the service lifetime of Ni-base superalloys as well as increasing the upper limit of their operating temperatures are two challenges that materials scientists and engineers are striving to meet. Future improvements of Ni-base superalloys for high temperature applications are going to come from the development of more complex, multi-component alloys with very specific compositions. A key element in this process is gaining the understanding of how constituents such as rhenium and titanium affect the phase stability of these multi-component alloys. Understanding the effects of rhenium computationally is of particular importance due to it being one of the least abundant elements in the Earth s crust yet one of the most effective alloying elements for the current generation of Ni-base superalloys. The CALculation of PHAse Diagram (CALPHAD) technique [87] meets this challenge by modeling the thermodynamic descriptions of a multi-component system through extrapolation from the binary and ternary systems that frequently have more experimental data available. With this method, the properties of complex alloys can be efficiently and accurately predicted in a significantly lesser amount

77 52 of time than an equivalent experimental investigation. There are three sets of experimental phase equilibria data published in the literature for the Re-Ti system [23, 148, 24] and no experimental thermochemical data for compounds and solutions phases. In the present work, first-principles calculations based on density functional theory (DFT) are performed to obtain enthalpies of mixing for the bcc and hcp solid solution phases using special quasirandom structures and enthalpy of formation of Re 24 Ti 5. Levy et al. [25] recently predicted four unreported compounds on the convex hull of the Re-Ti system using high-throughput calculations. In the present work, phonon calculations using the supercell approach are performed on the two reported crystal structures of the ReTi compound; a B2 phase from experiments [149, 150] and an orthorhombic superstructure predicted by first-principles [25, 151]. The values obtained from the DFT calculations are used to parameterize the Gibbs energy of the individual phases in the system using the CALPHAD technique. 4.1 Literature Review The cubic stoichiometric compound, Re 24 Ti 5, with space group I43m and prototype α-mn was first determined by Trzebiatowski et al. [152] using X-ray analysis and powder methods. Re 24 Ti 5 has a congruent melting temperature around 3023 K. Philip and Beck [150] annealed an alloy of the composition ReTi at 1473 K and quenching to room temperature, then re-annealing at 973 K or 873 K and reported the existence of an ordered B2 CsCl-type structure of composition ReTi, space group P m3m, and stable at temperatures below 873 K. X-ray diffraction results by Philip and Beck showed that two Re-Ti alloys of the composition Re-75 at. % Ti and Re-40 at. %Ti have bcc structures, indicating that the solubility of Re in bcc-ti extends greater than 50 at. % Re at 1473 K. Dwight and Beck [149] argued that the electron concentration of ReTi, i.e., the average number of electronsoutsideofclosed shells, istoolowtohaveastableb2 structure. Toprovethis argument, Dwight and Beck performed X-ray diffraction on Ti50(Re,X)50 (X=Os, Ir, Pt) buttons to show that only a slight increase in average electron concentration was needed to stabilize the B2 structure, concluding that ReTi may be a solid solution. By performing XRD on button ingots of Ti-xRe (x=4.26, 13.87, 27.78, 47.48,

78 53 and 73.88) alloys performed by an arc melting method, Wu et al. [22] reported an estimate of the lattice parameter of the ReTi compound in the B2 structure. Levy et al. [25] performed DFT calculations on Re-X binary alloy systems to supplement sparse experimental thermochemical data and predicted stable ordered structures by examining convex hulls of enthalpy of formation of a set of compounds for each system. At 0 K, they found Re 24 Ti 5 to be stable as the reported structure in the literature [152], ReTi to be stable as an orthorhombic superstructure with prototype MoTi [151], not as the bcc B2 structure reported experimentally [149, 150]. They further suggested three unreported compounds with compositions Re 5 Ti 3, ReTi 2, and ReTi 3. The phase equilibrium of the Re-Ti system was investigated experimentally by Savitskii et al. on multiple occasions [23, 148, 24] and found to have two peritectic reactions with one around 48 at. % Re at 2298 K [23] and the second around 80 at. % Re at 3023 K [148]. They [23, 148] determined the solidus of hcp-ti and bcc-ti between 973 K and 1273 K using the drop method and dilatometric studies and found that the solubility of Re in hcp-ti is less than several tenths of wt. % Re and the solubility of Re in bcc-ti at least 50 at. % Re. The bcc- Ti solidus curve was also investigated using a dilatometric method and analyzed using metallographic and X-ray techniques by Savitskii et al. [23] showing solidus temperatures to range from 1943 K to 2298 K in the composition range of 0-50 at. % Ti. Finally, Savitskii et al. [24] used microscopy and X-ray phase analysis to observe that the solubility of Ti in hcp-re does not exceed 2 at. % Re just above 2800 K. The Re-Ti system was first modeled by Kaufman and Bernstein [153] using lattice stability estimates of the pure metals. The general shape of the phase diagram was reproduced but the liquidus and solidus curves varied significantly from the observations by Savitskii et al. [23, 148, 24] because of the rough thermodynamic approximations. The binary system was also modeled by Murray [154] using an ideal solution model for hcp-re. The lack of thermodynamic data, such as the enthalpy of formation for Re 24 Ti 5, caused disagreement with experimental data, particularly with that of the bcc-ti solidus data shown by Savitskii [23]. Neither version of the previously modeled Re-Ti system included the ReTi compound.

79 Calculation and Modeling details First-principles calculations The calculations in this chapter employ the projector augmented wave (PAW) method [135, 136] and the generalized gradient approximation (GGA) as implemented by Perdew, Burke, and Ernzerhof [109]. A plane wave energy cutoff of 350 ev is used for all calculations, which is at least 1.3 times higher than the default energy cutoffs for Re and Ti. For total energy calculations of the pure elements and Re 24 Ti 5, all degrees of freedom for the structures are allowed to relax. Dense k-point meshes at of least 5000 k-points per reciprocal atom in the first Brillouin zone with Γ-centered k-point scheme are used for hexagonal lattices, and a Monkhorst-Pack scheme is used for cubic lattices. Samplings are 36 x 36 x 32 for hcp-re and hcp-ti, 32 x 32 x 32 for bcc-ti, 15 x 15 x 15 for B2-ReTi, 11 x 11 x 11 for MoTi-ReTi, and 15 x 15 x 15 for cubic Re 24 Ti 5. The energy convergence criterion for electronic self-consistency is a maximum of 10 5 ev/atom, and is generally smaller. Unless otherwise specified, all degrees of freedom of the crystal structures are allowed to relax, including cell shape, volume, and atomic positions. The shape and volume of the hcp SQS structures are relaxed manually with a Γ-centered k-point mesh of 8 x 8 x 6, and the bcc SQS structures were allowed to fully relax with an 8 x 8 x 5 k-point mesh. To investigate the ReTi compound, a supercell size of 64 atoms was used for both the B2 and MoTi structures CALPHAD modeling The Gibbs energy of Re 24 Ti 5, ReTi, and the solution phases follow the equations and methodology as described in Chapter 2. The Gibbs energy function of Re 24 Ti 5 takes the form of Equation 2.2, and the Gibbs energy functions of the liquid and hcp and bcc solid solutions phases take the form of Equation 2.6. The Gibbs energy descriptions for pure Ti and pure Re were taken from the most recent SGTE substance (SSUB) [8] database and they can also be found on the SGTE Unary database webpage [20].

80 Results and Discussion First-principles calculations To verify the results from first-principles calculations, comparisons to experimental lattice parameters of the pure elements and compounds are made. Table 4.1 shows the space groups, calculated lattice parameters, and errors between the calculated and experimental data for the phases in the system. It can be seen that errors between calculated values in this work and experimental values are usually less than 1% which is typical for such DFT calculations. Excellent agreement is also found with the work done by Wang et al. [69] in previous DFT calculations that used the GGA PW91 [138] exchange and correlation functional. Senkov et al. [155] observed the bcc-ti lattice parameter between 1173 and 1273 K using neutron diffraction. Disagreement is on the order of 2% because in the present work, calculations were done at 0 K using DFT, which does not take into account the vibrational contribution as a function of temperature, making the experimental lattice parameter higher than the calculated value. A more thorough discussion of the problems associated with calculating bcc-ti in VASP can be found in [115]. Table 4.2 presents the fitted equilibrium properties of all phases in the system calculated at 0 K from energy versus volume EOS curves, and the results are compared to experimental data or other DFT studies when available. Good agreement is seen when compared to both previous DFT studies and experimental data. Figure 4.1 shows entropy, enthalpy, and heat capacity as a function of temperature for hcp-ti compared to experimental data from the NIST-JANAF tables [18] and the SGTE Unary database [13]. Good agreement is found, especially with entropy and enthalpy, though in both properties we see that the Debye model slightly underestimates the properties at high temperatures. The same result is found for heat capacity, and in addition, the convex curvature of the heat capacity as it increases more rapidly towards its melting temperature is not reproduced. The underestimated values are consistent with a previous Debye model study performed on Ti by Mei et al. [115]. Grabowski et al. [157] pointed out that the heat capacity increases more rapidly as a function of temperature as the melting point, or transition point in the case of Ti, is approached due to increased vacancy concentration at elevated temperatures. By first-principles calculations, Grabowski

81 Table 4.1: Lattice parameters calculated from first-principles compared to experimental and other calculated values in the literature. Phase Space Group a, Å % Difference c, Å % Difference Source Re P 6/mmc This work Expt. [125] DFT [69] Ti Im3m This work 3.31 Expt. [155] DFT [69] P 6/mmc This work Expt. [125] DFT [69] Re 24 Ti 5 I43m This work 9.68 DFT [22] Expt. [152] ReTi - B2 Pm3m This work DFT [22] Expt. [149] ReTi - MoTi Imma This work 56

82 57 Table 4.2: Fitted equilibrium properties from the EOS of hcp-re, hcp-ti, cubic Re 24 Ti 5, and ReTi at 0 K (unless noted) including equilibrium volume, V 0, bulk modulus, B 0, and first derivative of bulk modulus with respect to pressure, B 0, compared to previous experimental and DFT studies. Phase Reference V 0, Å 3 /atom B 0 (GPa) B 0 hcp-re The present work DFT [156] hcp-ti The present work DFT [115] Expt. (300 K) [126] ± Expt. (300 K) [127] ± ± 0.4 Expt. (300 K) [128] Re 24 Ti 5 The present work B2 - ReTi The present work MoTo - ReTi The present work et al. demonstrate that reproducing the convex curvature of the heat capacity was dependent on including the vacancy excitation contribution to the free energy, which is beyond the scope of this study. Additionally, when compared to experiments, they state that it remains unclear if the underestimation of the heat capacity calculated by first-principles is due to errors in the experimental work or a lack of ability of the exchange-correlation functional to accurately reproduce the data. The thermodynamic properties of rhenium are represented in Figure 4.2 with the calculated entropy, enthalpy, and heat capacity compared to a previous computation study on Re using the quasiharmonic phonon approach from Zacherl et al. [19] to validate the use of the Debye model, the SGTE Unary PURE4 database [20, 8] and in the case of heat capacity, selected experimental data [3, 4, 6, 7]. The SGTE work shows a convex curvature in the heat capacity from about K. This curvature is not reflected in the Debye model calculations of the present work, the previous computational study using the phonon supercell approach, or the experimental data. It is unclear if this curvature is an effect of increased vacancy concentration at higher temperatures [157] or if it is an error associated with deriving the heat capacity of Re from the Gibbs energy function defined by the SGTE PURE4 database [20, 8]. As far as validating the Debye model compared to quasiharmonic phonon calculations, it is observed in Figure 4.2 that the entropy

83 58 80 S, J/mol-atom/K H-H 298, kj/mol-atom (a) (b) C P, J/mol-atom/K (c) Temperature, K Figure 4.1: (a) Entropy, (b) enthalpy, and (c) heat capacity of hcp-ti as a function of temperature calculated in the present work using the Debye model (line), compared to SGTE Pure Elements database [13] (dashed line), and the NIST-JANAF [18] experimental data ( ). and enthalpy as a function of temperature of pure Re vary little with respect to the method used. For heat capacity, however, the two computational methods create high (phonon supercell approach) and low (Debye model) bounds of the experimental data above 1000 K. The properties of Figure Figure 4.1 and Figure Figure 4.2 are well reproduced using the scaling parameter of s=0.617 when compared to experimental data, with a slight underestimation occurring for entropy as a function of temperature of both hcp-ti and hcp-re. From the agreement of the thermodynamic properties of hcp-re and hcp-ti

84 59 S, J/mol-atom/K H-H 298, kj/mol-atom (a) (b) C P, J/mol-atom/K (c) Temperature, K Figure 4.2: (a) Entropy and (b) enthalpy of hcp Re as a function of temperature calculated in the present work using the Debye model (solid lines), compared to quasiharmonic phonon calculations of Zacherl et al. [19] (blue dotted lines) and SGTE Unary PURE4 database [20, 8] (red dashed lines), and (c) heat capacity as a function of temperature calculated in the present work (solid line), compared to quasiharmonic phonon calculations [19] (blue dotted lines) and SGTE Unary PURE4 database [20, 8] (dashed line), and the experimental data from Taylor et al. [3] ( ), Jaeger et al. [4] ( ), Arutyuno et al. [6] ( ), and Filippov et al. [7] ( ). compared to experimental data, it is concluded that the Debye model methodology can accurately predict finite temperature thermodynamic properties of Re 24 Ti 5. Its enthalpy of formation is obtained to be kj/mol-atom at 300 K, a value which agrees with the recent work by Levy et al. [25] of kj/mol-atom for Re 24 Ti 5 that used high throughput calculations. Figure 4.3 presents the calculated

85 S, J/mol-atom/K H-H 298, kj/mol-atom (a) (b) C P, J/mol-atom/K (c) Temperature, K Figure 4.3: (a) Entropy, (b) enthalpy, and (c) heat capacity of Re 24 Ti 5 as a function of temperature from the Debye model in the present work. enthalpy, entropy, and heat capacity of Re 24 Ti 5 using the Debye model. Quasiharmonic phonon calculations using the supercell method are performed to investigate the structure of the ReTi compound, reported to be the B2 structure byphilipandbeck[150]andwuetal. [22]experimentally, andthemotistructure by Levy et al. [25] computationally. The 0 K phonon DOS are shown in Figure 4.4 for ReTi in both the B2 and MoTi structures. Neither structure shows imaginary phonon frequencies, meaning that both structures are dynamically stable. The shapes of the two DOS plots are very similar between the two structures, but a noticeable difference is the increase of modes at higher frequencies for ReTi in the MoTi structure. Figure 4.5 shows the thus obtained entropy, enthalpy, and

86 61 4 Density of States (10-12 ) Frequency (THz) Figure 4.4: Phonon density of states of the ReTi phase in the B2 structure (solid black line) and the MoTi structure (dashed blue line) calculated by quasiharmonic phonon first-principles calculations. heat capacity as a function of temperature for the ReTi compound in the two structures. Between the two structures, the values of heat capacity and enthalpy do not show a significant difference when calculated using the phonon supercell approach. The enthalpy of formation and entropy, on the other hand, shows a difference of 6-9% between the two structures, particularly at temperatures above 800 K. The predicted enthalpy of formation of the B2 phase in the present work of kj/mol-atom at 0 K agrees well with the results published by Wu et al. [22] of about -38 kj/mol-atom. The results for enthalpy of formation in the MoTi phase of kj/mol-atom at 0 K from the present work is consistent with that published by Levy et al. [25] of kj/mol-atom, though the result from the present work is less negative than what was calculated using the high throughput method as was the case with Re 24 Ti 5. To further validate our Debye model method compared to the phonon supercell approach, the same thermodynamic properties are presented as a function of temperature for the B2 phase from the Debye model and compared to the phonon supercell approach. The only noticeable difference in the thermodynamic properties between the two methodologies is a difference in the entropy of about 2 J/mol-atom/K above 600 K.

87 62 S, J/mol-atom/K H-H 298, kj/mol-atom (a) (b) C P, J/mol-atom/K (c) Temperature, K Figure 4.5: (a) Entropy, (b) enthalpy, and (c) heat capacity of ReTi as a function of temperature for the B2 structure from the phonon supercell approach (red solid line) and the Debye model (red dotted line) and for the MoTi structure (blue dotted-dashed line). SQS calculations for the bcc and hcp solid solution phases are performed for compositions x Re = 0.25, 0.50, and To check that local relaxation did not destroy the symmetries of the SQS, each relaxation method is compared to the radial distribution analysis of a pristine hcp, Figure 4.6, or bcc, Figure 4.7, structure. For the three structures examined for hcp SQS in Figure 4.6, it is clear that symmetry of the structure is completely lost when volume and shape relaxation is performed in VASP. If the relaxation scheme for a manual shape and volume

88 63 relaxation is performed following the method proposed by Zacherl et al. [19], the resulting structure is kept with regard to a pristine hcp structure. Thus, the resulting energies from this method is used for the hcp SQS relaxations for x Re = 0.25, 0.50, and 0.75 to calculate the enthalpy of mixing at each composition for used in the CALPHAD modeling. For the bcc SQS shown in Figure 4.7, RDFs for the three relaxation schemes including volume relaxation in VASP, volume and shape relaxation in VASP, and full relaxation (volume, shape, and ions relaxed) in VASP are presented. In all three cases, volume and volume and shape relaxations completely maintained symmetry when compared to the pristine bcc structure. When fully relaxed, tt is observed that for x Ti = 0.25 and 0.50, the symmetry of the fully relaxed bcc SQS was maintained, while for x Ti = 0.75, the full relaxation causes significant local distortion resulting in a structure that is no longer bcc in nature. Since full relaxation gives a more realistic and lower energy than volume and shape relaxation, the total energies of the fully relaxed bcc SQS for x Ti = 0.25 and 0.50 and fully relaxed x Ti = 0.75 point was eliminated. Figure 4.8 shows the thus obtained volume per atom of each SQS for both bcc and hcp solutions as a function of mole fraction Ti. It can be seen that the volume per atom increases non-linearly as a function of increasing Ti content. X- ray diffraction results used to estimate the lattice parameter of bcc-ti at room temperature by Wu et al. [22] are superimposed on the bcc SQS volume per atom, and a calculated value from an hcp alloy of 97 at. % Re from Joubert et al. [21] that reported lattice parameter by electron probe micro-analysis is superimposed on the hcp SQS volume per atom. The non-linearity of the relationship in the experimental data from Wu et al. [22] is reproduced, and disagreement between their estimated values and the present calculated values is less than 5%. For both solid solution phases, the first-principles results predict a parabolic behavior for the enthalpy of mixing, as shown in Figure 4.9. The positive enthalpy of mixing in the hcp solid solution phase indicates that a miscibility gap exists at low temperatures. In the bcc solid solution phase, the negative enthalpy of mixing indicates a tendency for intermixing between Re and Ti. The experimental work discussed earlier demonstrates that the bcc solid solution exists at least up to and possibly beyond 50 at. % Re, while Ti has almost no solubility in hcp- Re. The predictions from first-principles SQS calculations are thus consistent with

89 Volume and Shape > Ions Relaxed VASP Volume and Shape > Ions Relaxed VASP Volume and Shape > Ions Relaxed VASP Number of Bonds Volume and Shape Relaxed VASP Volume and Shape Relaxed Manually Volume and Shape Relaxed VASP Volume and Shape Relaxed Manually Volume and Shape Relaxed VASP Volume and Shape Relaxed Manually Pure hcp Pure hcp Pure hcp Distance (Å) Distance (Å) Distance (Å) Figure 4.6: Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re hcp solid solution structure comparing shape then ions relaxed in VASP, manual volume and shape relaxation, and volume and shape relaxation in VASP, to a pure hcp structure. 64

90 Volume, Shape, and Ions Relaxed VASP Volume, Shape, and Ions Relaxed VASP Volume, Shape, and Ions Relaxed VASP Number of Bonds Volume and Shape Relaxed VASP Volume Relaxed VASP Volume and Shape Relaxed VASP Volume Relaxed VASP Volume and Shape Relaxed VASP Volume Relaxed VASP Pure bcc Pure bcc Pure bcc Distance (Å) Distance (Å) Distance (Å) Figure 4.7: Radial distribution analysis for (a) 25, (b) 50, and (c) 75 at. % Re bcc solid solution structure comparing shape then ions relaxed in VASP, manual volume and shape relaxation, and volume and shape relaxation in VASP, to a pure hcp structure. 65

91 66 Volume (Å) 3 /atom hcp SQS bcc SQS Mole fraction Ti Figure 4.8: Volume per atom vs. mole fraction Ti for hcp ( ) and bcc ( ) solid solution structures with symbols as the SQS predictions with experimental data of Joubert et al. ( ) [21] and Wu et al. ( ) [22]. experimental observations Re-Ti Thermodynamic Modeling without the ReTi phase Both the hcp and bcc solid solution phases are modeled in Thermo-Calc [101] as regularsolutionsusingonlythe 0 Ltermdeterminedfromthefirst-principlesresults. The calculated enthalpies of mixing from the first-principles SQS calculations and the CALPHAD modeling as a function of mole fraction Ti are shown in Figure 4.9 for the hcp and bcc solution phases. Having determined the Gibbs energy descriptions of the solid solution phases, the remaining phases to be modeled are Re 24 Ti 5, ReTi, and liquid. For Re 24 Ti 5, and ReTi, the entropy and heat capacity as a function of temperature presented in Figure 4.3 and Figure 4.5, respectively, are used to evaluate the thermodynamic parameters b, c, d, and e in Equation 2.2 for each compound. a is modeled from the room temperature enthalpy of formation mentioned above and the reference states as shown in Equation 2.2. Since the melting temperature and subsequent phase equilibria surrounding the

92 67 10 Enthalpy of Mixing, kj/mol Mole Fraction Ti hcp bcc Figure 4.9: Enthalpy of mixing of SQS supercells for hcp shape and volume relaxed manually, ( ), hcp shape and volume in VASP, ( ), bcc shape and volume relaxed in VASP, ( ), and bcc fully relaxed (shape, volume, and atomic position) in VASP ( ), shown with the CALPHAD modeling from the present work (lines). melting of the ReTi compound is unknown, two versions of the phase diagram are presented, one with and one without for the ReTi phase. The ReTi phase was modeled according to the corresponding thermodynamic data from the B2 structure investigations. The B2 structure was chosen to be consistent with the known experimental literature since no dynamic instability of the phase was found using phonon calculations. For the version without ReTi, the liquid phase is modeled as a sub-sub-regular solution phase, using 0 L and 2 L terms in the Redlich-Kister polynomial defined in Equation 2.7. Because of the parameters of Re 24 Ti 5 being determined from firstprinciples results and the asymmetry in the liquidus temperature with increasing titanium content (largely due to the difference in melting temperatures of Re and Ti), one sub-regular and one sub-sub-regular term were the minimum number of terms needed to accurately reproduce the peritectic reaction and titanium solidus data.

93 hcp 3000 Liquid Temperature, K Re24Ti5 bcc Mole Fraction Ti bcc + hcp Figure 4.10: Calculated phase diagram of the Re-Ti system without the ReTi phase, with experimental data from Savitskii et al. [23] showing the melting temperature of Re 24 Ti 5 ( ), the solidus ( ), bcc-ti single-phase region ( ), and hcp- Ti + bcc-ti two-phase region ( ), Savitskii et al. [24] showing hcp-re single-phase region ( ), and hcp-re + Re 24 Ti 5 two-phase region ( ). The phase diagram of the Re-Ti system based on the modeling from this work is plotted in Figure 4.10 version without ReTi and Figure 4.11 version with ReTi and compared to the experimental data [23, 148, 24] described above. A complete list of thermodynamic parameters for all phases in both versions of the Re-Ti system can be found in Table 4.3. For the version without ReTi, Figure 4.10, the solidus curve agrees well with the experimental data, and both peritectic reactions are accurately reproduced. The melting temperature of the compound, Re 24 Ti 5, agrees well with its experimental value of 3023 K. The solubility data of both Ti in hcp-re and Re in bcc-ti gives a good guideline for modeling the single and two phase regions in this system as shown in the enlarged phase diagram in Figure 4.12 and Figure Both sets of experimental data for the single and two phase regions are well reproduced by this modeling.

94 hcp 3000 Liquid Temperature, K Re24Ti5 ReTi bcc Mole Fraction Ti Figure 4.11: Calculated phase diagram of the Re-Ti system with the ReTi phase, with experimental data from Savitskii et al. [23] showing the melting temperature of Re 24 Ti 5 ( ), the solidus ( ), bcc-ti single-phase region ( ), and hcp-ti + bcc- Ti two-phase region ( ), Savitskii et al. [24] showing hcp-re single-phase region ( ), and hcp-re + Re 24 Ti 5 two-phase region ( ). bcc + hcp Re-Ti Thermodynamic Modeling with the B2-ReTi phase For the version with ReTi included, no additional modeling is done after insertion of the ReTi phase. From Figure 4.11, it is clear that the ReTi compound is stable with the melting temperature around 3000 K. The results are further confirmed by a convex hull plot of the enthalpy of formation of the phases in the system shown in Figure Figure 4.14 is plotted with the 0 K enthalpy of formation data from Levy et al. [25]. It is observed that at 300 K, two of the compounds of Levy et al. [25] fall inside the convex hull due to the stability of the bcc phase at lower temperatures, making them metastable at and above 300 K. For comparison, the convex hull of the modeled phase diagram without ReTi is also plotted and shows similar results. As there is no experimental data regarding the melting temperature of ReTi, the variables from Table 4.3 remain unchanged aside from the addition

95 hcp hcp + Liquid Liquid Temperature, K hcp + Re 24 Ti Mole Fraction Ti Figure 4.12: Enlarged calculated phase diagram for the Re-Ti system on the Re-rich side with data from Savitskii et al. [24] for the hcp-re single-phase region ( ) and hcp-re + Re 24 Ti 5 two-phase region ( ) Temperature, K bcc hcp + bcc Mole Fraction Ti Figure 4.13: Enlarged calculated phase diagram of the Re-Ti system on the Tirich side with data from Savitskii et al. [23] for bcc-ti single-phase region( ), and hcp-ti + bcc-ti two-phase region ( ).

96 71 Table 4.3: Model parameters of the Re-Ti system from the present work, given in J/mol-formula. Phase(model) Parameters Value hcp (Re,Ti) 0 L hcp 22, *T bcc (Re,Ti) 0 L bcc -190, *T Liquid (Re,Ti) 1 L liq 25,000 2 L liq -15,000 Re 24 Ti 5-815, *T *Tln(T) *T 2 + 1,714,300*T 1 ReTi -90, *T *T ln(t) *T ,680*T 1 of the ReTi compound. A eutectic around x Ti = 0.41 and T=2987 K is predicted due to the addition of the ReTi compound. The uncertainly in the first-principles calculations merits a discussion with regards to the ReTi phase, and as discussed in the method, variances in the use of the Debye model can cause a ± 2% change in the outcome of the thermodynamic properties. The enthalpy of formation, or a term from Equation 2.2 will typically have an error between 1-5 kj/mole-atom when first-principles error is taken into account [130, 129], in accordance with our predicted accuracy. From examination oftheconvexhullplotinfigure4.14, itisobservedthatachangeintheenthalpyof formationofretiby1-5kjcouldmakeitmetastableat300kwhenitiscompared to the modeled version without ReTi. The entropy of formation from both versions of the modeling is presented in Figure Figure 4.5 shows a slight difference in the entropy of ReTi when calculated with the Debye model and the phonon supercell approach. At 300 K the difference is about 1 J, but increases to just over 2 J at temperatures above 600 K, which can affect the melting temperature of ReTi in the modeling by about 300 K. Further experimental investigations are desirable to check the prediction from the present work.

97 72 Enthalpy of Formation, kj/mol-atom Mole Fraction Ti Figure 4.14: Enthalpy of formation at 300 K of the Re-Ti system without the ReTi compound (solid line), and with the ReTi compound from first-principles calculations in two structures (B2:, MoTi: ) and CALPHAD modeling (dashed line) of the present work, and with the previous first-principles high throughput study ( ) presented by Levy et al. [25] calculated at 0 K

98 73 38 Entropy of Formation, J/mol-atom/K Mole Fraction Ti Figure 4.15: Entropy of formation at 300 K of the Re-Ti system when modeled without the ReTi compound (solid line) and with the ReTi compound (dashed line).

99 Conclusions In the present work first-principles calculations based on density functional theory are coupled with CALPHAD modeling to supplement limited experimental data and provide an accurate thermodynamic modeling of the Re-Ti binary system, including versions with and without the ReTi compound. By employing special quasirandom structures, the enthalpies of mixing of the hcp and bcc solid solutions are predicted to be positive and negative, respectively. The special quasirandom structure results are supported well by the literature, as there is little solubility of Ti in hcp-re and significant solubility of Re in bcc-ti. The thermodynamic model of Re 24 Ti 5 is obtained using the Debye-Gruneisen model by first-principles calculations. The possibility of including the ReTi compound is explored by comparing first-principles results performed on ReTi in the B2 structure reported by experiments, and the MoTi structure reported by first-principles high throughout calculations. The present work shows that there is no dynamic instability in the initially reported B2 structure for ReTi or the newly reported MoTi structure. A thermodynamic model of ReTi is obtained using the phonon supercell approach and a separate modeling of the system is performed to include this compound in the B2 structure. The present work shows that within the standard uncertainty of first-principles calculations, the ReTi compound exists somewhere between stable and metastable. The version of the thermodynamic modeling without the ReTi compound shows good agreement with all experimental data. The version of the thermodynamic modeling with the ReTi compound also shows good agreement with some experimental data, and poorer agreement for the bcc solidus experimental data because no additional modeling was performed on the liquid phase. The present work contributes to the Ni-base superalloy database in the CALPHAD community.

100 Chapter 5 First-principles calculations of the self-diffusion coefficients of fcc Ni Diffusion coefficients are one of the key data for predicting diffusional phase transformations, high temperature creep, and coarsening, as diffusion is the primary mechanism for mass transfer in solids. Theoretical techniques are becoming increasingly desirable for obtaining reliable self- and impurity diffusion data, since experimentation is a time consuming process. However, in order to accurately calculate diffusion coefficients, regardless of the crystal structure or concentration of the system or alloy, accurate temperature-dependent atomic jump frequencies of the diffusion species must be obtained [158]. Recently, parameter-free first-principles approaches based on density functional theory (DFT) were developed to predict self- and impurity diffusion coefficients in metallic fcc Al systems [88, 159, 89] within the framework of transition state theory. More recently, this method was applied to predicting self- and impurity diffusion coefficients in hcp systems [92, 93, 160]. It has also been used conjunction with empirical parameters to represent the change in diffusivity above and below the Curie temperature in α-fe [91]. Previous attempts at studying the self- and impurity diffusion behavior in ferromagnetic fcc Ni systems [43, 84, 72, 161] have not incorporated accurate entropic contributions to the diffusion prefactor and have performed calculations only at 0 K. Also noticeably missing from other DFT studies is the influence of ferromagnetism on the diffusion coefficient of fcc Ni when compared to the nonmagnetic case.

101 76 Mantina et al. [90] developed a straight-forward computational approach to calculate self-diffusion coefficients simplified by the use of Eyring s reaction rate theory [96] when compared to her previous work [88, 159]. This method is adopted in the present work and applied to self-diffusion in fcc ferromagnetic Ni. The finite temperature thermodynamic parameters relating to self-diffusion are calculated and compared using both the phonon supercell approach for the sake of accuracy [69,?] and the Debye model [113, 99] for the sake of efficiency. The self-diffusion coefficients are computed within the local density approximation (LDA) [107] with the saddle configuration calculated using the nudged elastic band method (NEB) [97] to ensure the diffusing atom remains at the maximum energy point along the diffusion path. Here, the choice of LDA is due to its capability to predict more accurate diffusion properties compared to the generalized gradient approximation (GGA) [89, 92], and the NEB gives almost identical results for fcc Ni with respect to the climbing-image nudged elastic band (CINEB) method [134] based on our tests. Arrhenius parameters, as well as enthalpy and entropy of formation, and entropy of migration are shown to agree well with experimental data. The present work also explores the effect of that ferromagnetism on the self-diffusion coefficient ofni, anditisshownthattheferromagneticcaseproducesabetteragreementwith experimental data than the non-magnetic case. 5.1 Diffusion Theory The present work focuses on vacancy-mediated self-diffusion in a crystalline solid and calculates the least energy diffusion path of an atom between the initial and final states of an elementary atomic jump. Essentially, two processes are occurring. First, the defect, in this case a vacancy, is formed. Second, a thermally activated jump is occurring where the diffusion atom and the first nearest neighbor vacancy exchange lattice sites, known as vacancy migration. The notations for the Gibbs energies, G PS, G IS, and G SC, represent the three supercell configurations calculated in the present work. They are defined as: The perfect state (PS) refers to a perfect N-atom fcc supercell. The initial state (IS) refers to a (N 1)-atom fcc supercell with a vacancy

102 77 IS (a) (b) SC Figure 5.1: Schematic diagram of the diffusion process showing (a) the IS with a vacancy in the first nearest neighbor site of a normal lattice position and (b) the SC showing the atom at the maximum energy point along the diffusion path, and the displacement of the lattice that results from the atomic migration. in the first nearest neighbor site of the diffusing atom. The transition state (SC) refers to the saddle point of the diffusing atom as it travels along the minimum energy path, halfway between the initial and final vacancy configurations in an N 1 supercell. A schematic of the IS and the SC during the diffusion process can be found in Figure 5.1. In the present work, the Gibbs energies as a function of temperature, G PS, G IS, and G SC, are obtained using first-principles total energy calculations with either the Debye model for the sake of efficiency or phonon calculations for the sake of accuracy, and the results are compared. In a cubic system, the diffusion coefficient can be represented as [131]: D = 1 6 r2 fγ (5.1) where r is the length of the atomic jump f the jump correlation factor given as for self-diffusion in fcc system [41], which describes the possibility of the atom jumping back into the position it initially started in, which would effectively cancel out the supposed jump. A value of f = indicates that just over

103 78 20 % of the jumps that occur during self-diffusion in an fcc lattice will immediately returntotheirinitialpositions. Γisthejumpfrequency(or rate). In anfcccrystal, there is one possible independent jump. The jump frequency is defined as [131]: Γ = ZCw (5.2) where Z is the number of nearest neighbor sites available for the diffusing atom to jump to, C the vacancy concentration, and W the successful jump frequency of the diffusing atom to its nearest neighbor vacant site. The vacancy concentration in thermal equilibrium is given as [42]: ( C = exp G ) f k B T where G f is the Gibbs energy of vacancy formation defined as: (5.3) G f = G IS ((N 1)/N)G PS (5.4) where N is the number of atoms in the fcc supercell, k B Boltzmann s constant, and T temperature. The atomic jump frequency of the diffusing atom, w is governed by Eyring s reaction rate theory [96] and described by Wert and Zener [133] as: w = k ( BT h exp G SC G ) IS k B T where G m is the Gibbs energy of vacancy migration defined as: (5.5) G m = G SC G IS (5.6) where h is Planck s constant. It should be noted that in both transition state theory and Eyring s reaction rate theory, the imaginary vibrational frequency at the unstable transition state is removed from the calculations. Substituting Γ from Equation 5.2 into the general diffusion equation in Equation 5.1 ultimately yields [131]: D = fa 2 Cw (5.7)

104 79 where a is the lattice parameter representing the length of the jump that the diffusing atom makes. For the sake of comparison to experimental data, one can represent the diffusion coefficient empirically from Equation 5.7 in Arrhenius form [88, 162, 42]. In terms of the diffusion prefactor, D 0, and the activation energy for diffusion, Q, D is represented as: where the diffusion prefactor is given as: ( D = D 0 exp Q ) k B T D 0 = k ( ) BT Sf + S m h fa2 exp and the activation enthalpy is represented by : k B (5.8) (5.9) Q = H f + H m. (5.10) 5.2 Computational details Total energy calculations are carried out using the plane wave density functional code called the Vienna ab-initio Simulation Package (VASP) [111]. All calculations use the projector augmented wave (PAW) method [136, 135] to describe electronion interactions and the local density approximation of Ceperley-Alder [107] under the Perdew-Zunger parameterization[108] to describe the exchange and correlation functional. Aplanewave energycutoffof350evisusedfor all calculations, avalue which is 1.3 times the default plane wave energy cutoff of nickel. Dense k-point meshes in the first Brillouin zone with a Monkhorst-Pack scheme is used for all calculations, with a sampling of 9 x 9 x 9 for each structure described above. For relaxation during the VASP calculations, the Methfessel-Paxton smearing method [163] is used for the calculation of forces acting on the atoms, and a final static calculation is performed after each relaxation using the linear tetrahedron method with Blöchl s [164] correction for an accurate total energy calculation. Total electronicenergyisconvergedtobeatleast10 5 ev/atom, andunlessotherwisenoted, all degrees of freedom of the crystal structures are allowed to relax, including cell

105 80 shape, cell volume, and atomic positions. Due to the ferromagnetism of nickel, calculations are performed separately and identically with and without spin polarization to isolate the effect of magnetism on the diffusion coefficient. A 32 atom (2 x 2 x 2) fcc supercell is created to represent the PS discussed above. The initial and final equilibrium vacancy configurations are identical, and created by replacing one atom by a vacancy, IS, and moving it to its first nearest neighbor position in the final equilibrium vacancy configuration. The saddle configuration, SC, is first predicted as the middle of the minimum energy path between the initial and final equilibrium vacancy configurations, and its final position and energy are computed using the nudged elastic band (NEB) method [97] within VASP. The premise of the NEB method involves using a spring-like force acting on the unstable atom while looking at intermediate steps along the diffusion path, called images. In the present work, one image is used to calculate the SC. A 5.0 ev/å 2 spring constant was used in all NEB calculations to nudge the image to the minimum energy path between the initial and final vacancy configurations. Calculations were allowed to fully relax within the confines of NEB, and the structure was checked to ensure that local relaxations did not distort the cell or cause reversal of the diffusing atom to one of the equilibrium vacancy spots. Finite temperature thermodynamic properties to describe the diffusion coefficient entering into Equations 5.7 and 5.8 were calculated using the quasiharmonic phonon supercell approach and the quasiharmonic Debye-Grünesien model as discussed in Chapter 2. F vib (V,T) was calculated for the PS, IS, and SC. F t el (V,T) has been shown to have a negligible effect on the diffusion coefficient [88] due to the cancellation effect when properties such as G f are calculated, but was considered in the present work for the sake of completeness and accuracy. 5.3 Results and Discussion K results As an initial check to ensure the proper relaxation of the PS, IS, and SC in VASP, the normal phonon frequencies were calculated for the three configurations in VASP following the procedure outlines in Chapter 2. The phonon density of states (DOS)

106 81 Density of States (10-12 ) pure Ni vacancy saddle Frequency (THz) Figure 5.2: Phonon density of states plotted for the three configurations needed to calculate the factors entering into vacancy mediated self-diffusion, the perfect state (PS), initial vacancy configuration (IS), and the transition state, or saddle configuration (SC). are obtained from these calculations, and are presented in Figure 5.2. From Figure 5.2, we see that the perfect configuration shows a phonon DOS as expected for a perfect fcc crystal. Because of the empty space in the supercell due to the vacancy in the IS, some variation in the shape of the phonon DOS is observed at mid and high frequencies. Particularly, there are fewer high frequency modes observed. A more drastic change is observed when the phonon DOS of the SC is observed. As shown in Figure 5.1, and atom in the SC will be further away from certain atoms and closer to certain atoms, which is uncharacteristic of the perfect crystal. Because it is a dynamically unstable configuration, imaginary phonon frequencies are expected and observed. Additionally, the DOS is lower at mid range frequencies due to the absence of the atom relative to the initial and final vacancy configurations. Because it is closer to some atoms when it is in the SC compared to the typical bond length, we see modes appearing at higher frequencies that for the PS or the IS.

107 Finite temperature results The introduction of a vacancy into a system creates a small amount of internal surface inside the supercell for the IS and SC configurations used in the present work. It has been noted in several self-diffusion studies [88, 92] that the LDA will underestimate this energy during first-principles calculations less severely than the generalized gradient approximation (GGA). While computational methods have been developed [165] to calculate correction terms that compensate for the LDA underestimating this surface, no such values exist for pure fcc Ni. Thus, no surface correction term is applied to the parameters calculated in the present work. Following Equation 5.3 and using the calculated Gibbs energy of vacancy formation, vacancy concentration as function of 1000/T is plotted in Figure 5.3 and compared to experimental vacancy concentration results measured using differential dilatometry on % pure Ni by Scholz [26] and a previous embedded atom computational study performed by de Koning et al. [27]. The very short lifetime of vacancies in Ni [94] make experimental vacancy concentration measurements difficult, which is probably the reason for so little experimental data on Ni vacancy concentration. Results using the Debye model for the calculation of Gibbs energy of vacancy formation show better agreement with the results of Scholz [26] and de Koning et al. [27] than the quasiharmonic phonon calculations based on the supercell approach. In fact, the difference between the calculated vacancy concentration by the Debye model and the phonon supercell approach is two orders of magnitude. Since it is expected that these two methods would give more similar results, plausible reasons for this result are given after the presentation of the diffusion coefficient data in the following paragraph. Self-diffusion coefficients of Ni, calculated in the present work, are shown in Figure 5.4 with both the quasiharmonic Debye model and the quasiharmonic phonon supercell approach used for calculating the thermodynamic parameters needed from Equation 5.7. The calculated self-diffusion coefficients are compared to select single-crystal experimental data [29, 30, 31, 32, 33, 34] and poly-crystal experimental data [35, 36, 37, 38], along with the most recent self-diffusion mobility assessment performed by Zhang et al. [28]. The Debye model method shows better agreement with experimental self-diffusion data and with the self-diffusion mobility assessment than the phonon supercell calculations. This result is also

108 83 Vacancy concentration, C Scholz, Expt., 2001 de Koning, EAM, 2003 LDA - FM QHA Debye LDA - FM QHA Phonon /T (1/K) Figure 5.3: Vacancy concentration, C, plotted as a function of 1000/T from the present work within the LDA using the quasiharmonic Debye model (solid line) and quasiharmonic phonon calculations (dashed line) for finite temperature thermodynamic properties compared to experimental results of Scholz [26] and to an embedded atom study by de Koning et al. [27]. reflected in the data shown in Table 5.1, which shows the Arrhenius diffusion parameters calculated in the present work as a function of temperature compared to selected single-crystal experimental data. In Table 5.1 it is observed that the diffusion prefactor, D 0, corresponding to the y-intercept of the D versus 1000/T plot in Figure 5.4, is better reproduced with respect to experimental data and the self-diffusion mobility assessment by the Debye model. The same trends are observed for Q, the activation energy for diffusion, which represents the slope of the D versus 1000/T plot in Figure 5.4. From Equation 5.10, we see that the diffusion prefactor depends on the entropy of formation of the vacancy and the entropy of migration of the diffusing atom. A previous computational study by Shang et al. [113] on thermodynamic properties of fcc Ni using both the quasiharmonic Debye model and the quasiharmonic phonon supercell approach shows that while there is marginally better agreement of the entropy of pure Ni as a function of temperature calculated by the high-temperature Debye model case, there is no significant difference in the enthalpy as a function of temperature between the two methods.

109 Table 5.1: Arrhenius diffusion parameters for Ni compared to experimental data. Method References D 0 (m 2 /sec) Q (ev) T (K) DFT LDA FM QHA Debye The present work 1.99E E DFT LDA NM QHA Debye The present work 2.32E E DFT LDA FM QHA Phonon The present work 4.44E E DFT LDA NM HA Phonon The present work 3.32E E Mantina (DFT) [162] 7.00E E Zhang (Mobility assessment) [28] 5.34E Assessment based on expt. data Bakker (Expt.) [33] 1.77E Vladimirov (Expt.) [34] 0.85E Maier (Expt.) [31] 1.33E

110 D (m 2 /sec) Wazzan, 1965 Ivantsov, 1966 Maier, 1976 Feller-Kniepmeier, 1976 Vladimirov, 1978 Bakker, 1968 MacEwan, 1953 Hoffman, 1965 Monma, 1964 Bronfin, 1975 Zhang et al. assessment LDA - QHA Debye FM LDA - QHA Phonon FM /T (1/K) Figure 5.4: Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model (solid line) and harmonic phonon calculations (dashed line) for the finite temperature thermodynamic contribution compared to the self-diffusion mobility assessment of Zhang et al. [28] and to selected single-crystal, (Wazzan [29], Ivantsov [30], Maier [31], Feller-Kniepmeier [32], Bakker [33], and Vladimirov [34]) and poly-crystal (MacEwan [35], Hoffman [36], Monma [37], and Bronfin [38]) experimental data. Since there is no clear difference in the values of the thermodynamic properties of pure Ni when calculated by either method proposed in the present work [113], the effect of the better agreement due to using the Debye model must be influenced by the properties of the IS and SC. As previously discussed, introducing a vacancy into the system creates a small amount of internal surface, the energy of which is underestimated during VASP calculations. Without a correction term, neither the phonon supercell approach nor the Debye model will produce perfectly accurate results for vacancy concentration or diffusivity [88, 92, 166]. In the case of ferromagnetic Ni, it is possible that the magnetic properties of the crystal are also influencing this internal surface effect in the IS and SC configurations. Additionally, the phonon calculations used in this work were simplified in and of the fact that they did not include finite temperature effects from magnetic excitations;

111 86 only the ferromagnetic spin state was included. In the literature, quantum Monte Carlo [167] and partition function approaches [168, 169] have been applied to magnetic materials to represent this contribution, but they were not considered in the present work since the contribution is shown to be relatively small for Ni [169] when compared to Fe or Co [167, 168]. There may be a bigger influence from the magnetic excitations, however, in the IS or SC, that are causing the disagreement of the results from the phonon supercell method when compared to experiments. In conclusion, since the Debye model is empirical in nature, it is likely that the empiricism in the parameters is correcting this internal surface underestimation and effects due to magnetism, thus making two errors cancel out and producing diffusion coefficient that agrees better with the experimental data. Another consideration when discussing the agreement of the calculations with experimental data is the slope of the D vs 1000/T plot, which is based on the entropic thermodynamic property. How the method for the finite temperature thermodynamic properties calculates the entropy as a function of temperature could also be influencing the diffusion coefficient. For example, Shang et al. [113] studied Ni and Ni 3 Al using both the phonon supercell method and the Debye- Grünesien model to calculate a multitude of finite temperature thermodynamic properties using the GGA PBE [109] exchange and correlation functional. Shang s work showed that in the case of fcc Ni, the calculated entropy as a function of temperature showed almost no dependence on the finite temperature model that was used, using the high-temperature parameter and traditional scaling factor as used in the present work. The same result was observed in the present work using the LDA for the exchange and correlation functional, but the calculated entropy by both methods was consistently lower than the experimental data [18] as a function of temperature. However, when the vacancy was introduced into the system, a difference in the calculated entropy is observed, as demonstrated in Figure 5.5. Figure 5.5 shows that particularly for temperatures above 500 K, the Debye model predicts a higher entropy than the phonon supercell approach. This is another factor playing into the calculation of the diffusion properties as a function of temperature, as the same results is observed for the saddle configuration. To further show the accuracy of the calculations in the present work, thermodynamic quantities including enthalpy and entropy of vacancy formation, H f

112 87 Entropy, J/mol-K Temperature, K NIST-JANAF Phonon + elec. Debye + elec Figure 5.5: Entropy of a 31 atom supercell of fcc Ni with a vacancy calculated with the Debye-Grünesien model or the phonon supercell approach with the thermal-electronic contribution and compared to the NIST-JANAF thermochemical tables [18]. and S f, respectively, and enthalpy and entropy of vacancy migration H m and S m, respectively, are given in Table 5.2 at both 700 K and 1700 K for ferromagnetic and non-magnetic cases calculated by the quasiharmonic Debye model. The values calculated in the present work are compared to most recent experimental data [26, 94, 170, 95] and to three different DFT studies performed at 0 K. It is observed from Table 5.2 that in order to accurately reproduce experimental data, thermodynamic properties must be calculated as a function of temperature, and the NEB method for the saddle configuration must be employed, in addition to employing the spin polarized approximation. Particularly good agreement is observed for H f at higher temperatures. The vacancy formation properties are observed to increase with increasing temperature, and conversely, the vacancy migration properties show a slight decrease with increasing temperature. For S f, significantly better agreement is found with recent experimental measurements[26] than the previous DFT study [72]. The EAM model used by de Koning et al. [27] predicted the value of S f to approach 5 k B at the melting temperature of Ni, a value which is also reflected in Table 5.2 from the present calculations. To better understand the effect that ferromagnetism plays on the diffusion coef-

113 Table 5.2: Thermodynamic parameters from the quasiharmonic Debye model obtained from the self-diffusion calculations of fcc Ni in the present work for both ferromagnetic (FM) and non-magnetic (NM) cases, along with available experimental and other DFT calculations at 0 K: with one using the LDA and PAW potentials [70], one using the LDA with exact muffin-tin orbitals [71], and one using the generalized gradient approximation [72] Thermodynamic The present work, The present work, Other DFT Experiment property FM NM study Temperature 700 K 1700 K 700 K 1700 K 0 K K H f (ev) [70] 1.73 [94] 1.67 [71] 1.79 [170] S f (k B ) [72] 3.3 ± 0.05 [26] H m (ev) [70] 1.04 [94] S m (k B )

114 D (m 2 /sec) Wazzan, 1965 Ivantsov, 1966 Maier, 1976 Feller-Kniepmeier, 1976 Vladimirov, 1978 Bakker, 1968 MacEwan, 1953 Hoffman, 1965 Monma, 1964 Bronfin, 1975 LDA - QHA Debye FM LDA - QHA Debye NM /T (1/K) Figure 5.6: Calculated self-diffusion coefficient of ferromagnetic (FM) fcc Ni using the Debye model (solid line) and non-magnetic (NM) fcc Ni using the Debye model (dotted line) compared to selected single-crystal, (Wazzan [29], Ivantsov [30], Maier[31], Feller-Kniepmeier[32], Bakker[33], and Vladimirov[34]) and polycrystal (MacEwan [35], Hoffman [36], Monma [37], and Bronfin [38]) experimental data. ficient, Figure 5.6 plots the calculated diffusion coefficient using the quasiharmonic Debye model for the finite temperature thermodynamic contributions for the ferromagnetic case and the non-magnetic case compared to experimental data. (For tabulated data, see Table 5.2.) A qualitative analysis shows that the ferromagnetic case reproduces the slope of the diffusion coefficient more accurately than the non-magnetic case, and that the ferromagnetic case also shows a slight improvement in the intercept of the diffusion coefficient, particularly when compared to single-crystal experimental data [29, 30, 31, 32, 33, 34]. Table 5.1 shows that the Arrhenius parameters from the ferromagnetic case agree better with experimental parameters than the non-magnetic case. To verify the calculated Ni self-diffusion coefficient quantitatively compared to experimental data, the self-diffusion coefficients as a function of temperature are plotted with the confidence band and consensus analysis from a statistical

115 90 analysis performed using weighted mean statics by Campbell et al. [39] on all known experimental self-diffusion data for Ni. The work of Campbell et al. uses a Gaussian distribution for experimental error, and creates consensus fit and a 95 % confidence band for all experimental data considered. Figure 5.7 plots the analysis of Campbell et al. with the ferromagnetic and non-magnetic diffusion coefficient calculated in the present work. It is observed from Figure 5.7 that the non-magnetic calculated diffusion coefficient is consistently outside of upper limit of the 95 % confidence band and gets increasingly further from the confidence band as temperature increases. For the ferromagnetic calculated diffusion coefficient, it is within the confidence band at lower temperatures and higher temperature, and falls out of the confidence band where it tightens between K. The ferromagnetic case also coincides with the consensus fit at higher temperatures indicating an accurate intercept based on the diffusion prefactor, and shows a more accurate slope, which is based on the activation energy for diffusion, than the non-magnetic case. These results from the present work indicate the importance of completing self-diffusion coefficient calculations for Ni under the spin-polarized approximation. In a further effort to understand the spin-polarized ferromagnetic effect on the slope of the self-diffusion coefficient of fcc Ni, the magnetization charge densities (spin up minus spin down) are plotted for the PS, IS, and SC. The magnetization charge densities of the 3-D 32 or 31 atom supercells are shown in Figure 5.8 as 2-D projections looking down the b-axis where the colored atoms are cut by the edge of the supercell and display the magnetization charge density. Figure 5.8(a) shows that for pure, defect-free fcc Ni, the PS, there is a nearly cubic magnetization charge density surrounding each atomic position. In Figure 5.8(b), a periodic vacancy is present at all four corners of the supercell and represents the IS. For the IS, a change in the shape of the magnetization charge density from nearly cubic to more spherical is observed for the atoms surrounding the vacancy in the first nearest neighbor positions. Figure 5.8(c) represents the SC where there are two vacant sites and in the middle is the SC atom following the least energy diffusion path. A completely different shape of the magnetization charge density for the SC atom is observed; a shift from the nearly cubic shape to a diamond shape is found. Also of note in Figure 5.8(c) is that the effects on the magnetization charge

116 91-30 ln D Ni (m 2 /sec) LDA - QHA FM Debye LDA - QHA NM Debye Consensus Analysis, Campbell et al. 95% confidence band, Campbell et al /T x 10 4 (K -1 ) Figure 5.7: Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model (solid line) and non-magnetic fcc Ni using the Debye model (dotted line) compared to the consensus fit (blue dashed line) and 95 % confidence band (red dashed line) from the weighted means statistics study of Campbell et al. [39]. density are much further reaching when the atom is at the SC. All of the atoms in this plane see a change from a nearly cubic shape to a more spherical, and in some cases, ellipsoid shape. This is also observed for atoms in the plane behind in the SC atom. These observations may be an indication that the magnetic effect is not isolated from other factors in the system. For example, the magnetic contribution to the Gibbs (and Helmholtz) energy may be influenced by changes in bonding occurring as the SC atom is at its maximum energy point along the diffusion path. The shape of charge density is directly linked to the supercell s anisotropic properties, and the change of shape and magnitude of the magnetization charge density results in a change of properties. In conclusion, the shape and magnitude changes of the magnetization charged density throughout the various configurations used to calculate the diffusion coefficient signal that magnetism greatly influences the thermodynamic properties, and subsequently, the diffusion coefficient. To continue elaboration on the fact that the effects from the SC atom is far reaching throughout the supercell, the total deformation charge density, the dif-

117 92 Figure 5.8: Calculated magnetization charge densities (spin up minus spin down) looking down the b-axis for (a) the perfect, defect-free supercell, PS and (b) the initial vacancy configuration (IS) with a vacancy at all four corners of the figure, and (c), the saddle configuration (SC) with the diffusing atom at its maximum energy point along the diffusion path. ference between the total electronic charge density and the initial charge density from one electronic step, is plotted in Figure Figure 5.11 shows a 2-D projection of the deformation charge density of the SC looking down the a, b, and c axes. Several important observations can be made. In the b-axis projection, it is observed that the majority of the atoms in the same plane as the SC atom show significant changes in their shape, going from the nearly cubic shape as observed in Figure 5.8 to a more ellipse shape with different orientations due to the strain imposed on the lattice by the SC. the atoms in the perpendicular a-axis and c-axis projections have a greater tendency to retain their nearly cubic shapes. Also interesting is the shape of the SC atom itself, shown in the bottom left corner of the b-axis projection, as it takes on a diamond shape to fill the empty space as it is at the maximum energy path between the initial and final vacancy configurations it is diffusing between. Finally, we can observe from the b-axis projection that the atoms in the plane behind the SC remain largely unaffected in their nearly cubic shape as the atoms of the PS where in Figure 5.8. In Figure 5.12, a slice of the deformation charge density along the[202] direction is shown to show the effects that the SC atom has on its first and second nearest

118 93 neighbor atoms. It is observed that the four first nearest neighbors shown in Figure 5.12 must distort locally in order to accommodate the strain induced by the diffusing atom. Figure 5.9: Deformation charge density (final total charge density minus total charge density after one relaxation step) of the SC looking down the a-axis, b-axis, and c-axis.

119 94 Figure 5.10: Slice of the deformation charge density along the[202] plane showing the SC atom (center) and its surrounding atoms Non-magnetic and other phonon results In the previous section, it is clearly shown that the vibrational contributions produced by the use of the QHA Debye model under the ferromagnetic spin polarization approximation produce the best results for the self-diffusion coefficient of fcc Ni. Possible reasons for the better agreement with experimental data of the QHA Debye model technique over the QHA phonon supercell technique were discussed. Another argument that surfaces regarding the self-diffusion coefficient of fcc Ni is the fact that all of the experimental data is in the paramagnetic regime, i.e., it is above the Curie temperature and the spin states are now disordered. The purpose of this section is to show that non-magnetic results, whether produced with the Debye model or HA phonon method for vibrational properties, do not agree with the experimental data, and that the QHA FM Debye model shown in the previous section is the best way to calculate the self-diffusion coefficient of fcc Ni. Full QHA phonon calculations are not completed for the non-magnetic case since it is computationally inefficient and there is very little difference in the resulting diffusion coefficient, which was previously discussed. Figure 5.11 plots the selfdiffusion coefficient of fcc Ni for the non-magnetic QHA Debye and non-magnetic HA phonon compared to the QHA Debye FM results shown to agree very well with experimental data in the previous section. While at first glance, the observation

120 95 may be that one of the non-magnetic calculated diffusion coefficients shows better agreement with the low temperature data, it can clearly be seen that the slope of the entire diffusion coefficient deviates substantially from the experimental data and the FM case. The intercept of the diffusion coefficient is also incorrect when compared to experiments. These results show that magnetic contributions to the entropy and enthalpy are important and are necessary to produce an accurate diffusion coefficient. The non-magnetic data for the Arrhenius diffusion parameters is also presented in Table 5.1, and the non-magnetic thermodynamic parameters are presented in Table D (m 2 /sec) Wazzan, 1965 Ivantsov, 1966 Maier, 1976 Feller-Kniepmeier, 1976 Vladimirov, 1978 Bakker, 1968 MacEwan, 1953 Hoffman, 1965 Monma, 1964 Bronfin, 1975 LDA - QHA Debye FM LDA - QHA Debye NM LDA - HA Phonon NM /T (1/K) Figure 5.11: Calculated self-diffusion coefficient of non-magnetic fcc Ni using the Debye model (dotted line) and non-magnetic fcc Ni using the phonon supercell approach (dashed line) compared to single-crystal and poly-crystal experimental data and also the ferromagnetic QHA Debye model work. Finally, to validate the work compared to the consensus analysis of Campbell et al. [39], Figure 5.7 is reproduced using the ferromagnetic and non-magnetic selfdiffusion coefficient when the vibrational properties are calculated using the phonon supercell approach. The results are plotted in Figure It is clearly observed that when compared to the weight-mean statistical analysis, neither ferromagnetic nor non-magnetic phonon results show improvement of either Debye model case.

121 Table 5.3: Thermodynamic parameters from the non-magnetic quasiharmonic Debye model and the non-magnetic harmonic phonon supercell approach, along with available experimental and other DFT calculations at 0 K. Thermodynamic The present work, The present work, Other DFT Experiment property NM Debye model NM phonon supercell study Temperature 700 K 1700 K 700 K 1700 K 0 K K H f (ev) [70] 1.73 [94] 1.67 [71] 1.79 [170] S f (k B ) [72] 3.3 ± 0.05 [26] H m (ev) [70] 1.04 [94] 96

122 97 Infact, theresultsshowninfigure5.12aremuchworsethanthoseshowninfigure 5.7. The same reasons for this disparity in the phonon results discussed above can be attributed to the discrepancies involving the weighted-means statistics data. -30 ln D Ni (m 2 /sec) LDA FM HA Phonon LDA NM HA Phonon Consensus Analysis, Campbell et al. 95% confidence band, Campbell et al /T x 10 4 (K -1 ) Figure 5.12: Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model (solid line) and non-magnetic fcc Ni using the Debye model (dotted line) compared to the consensus fit (blue dashed line) and 95 % confidence band (red dashed line) from the weighted means statistics study of Campbell et al. [39]. 5.4 Conclusions Vacancy-mediated self-diffusion coefficients of fcc ferromagnetic and non-magnetic Ni were calculated in the present work using the LDA and the NEB method for calculating the minimum energy pathway of the diffusion atom. It should be noted that this method is limited to calculation of the ferromagnetic phase or the paramagnetic phase; no account for the energy change at the magnetic transition can be achieved. Therefore, even though the majority of the experimental data exists in the paramagnetic regime, the results presented are ferromagnetic. Two methods for calculating the vibrational contribution to the Helmholtz energy were

123 98 employed: the Debye-Grünesien model and the phonon supercell approach. Excellent agreement with experimental data for the self-diffusion coefficient and related thermodynamic parameters is found when the Debye model method is used. Based on previous diffusion studies, it was expected that the phonon supercell approach would yield more accurate results, but it is clear that the magnetic effects are far reaching and that the empiricism in the Debye-Grünesien model is making up for the effect of magnetism and the effect of internal surface created within the supercell due to vacancies. It is shown through quantitative and qualitative analyses that inducing ferromagnetic spin on the atoms calculates a more accurate self-diffusion coefficient and Arrhenius parameters than the non-magnetic case at temperatures both above and below the ferromagnetic to paramagnetic transition. For future calculation of diffusion coefficients in magnetic systems, performing ferromagnetic spin-polarized calculations using the Debye-Grünesien model for the self-diffusion coefficient of fcc Ni based on the quantitative and qualitative analysis will yield accurate results. Finally, to support the final recommendation, it is shown that contributions from the magnetization charge density are different for the structure with the diffusing atom at the SC, indicating bonding may affect the magnetic contribution to the thermodynamic properties energy and subsequently, the calculated diffusion parameters. 5.5 Note on saddle configuration relaxation It should be pointed out that a new relaxation algorithm for the saddle configurationinvasp5usingthecinebmethodbecameavailableatpennstatemorethan one year after completion of this work. It is described more fully in the methodology section Chapter 7. The advantage to what Henkelman calls the ss-cineb method is that it allows for a more fully relaxed saddle configuration without having to deal with the potential for local atomic relaxations significantly distorting the supercell. The present calculation uses five images along the band to calculate the energy of the saddle configuration, and then used the Debye model to calculate the finite temperature contribution. The results from the more full relaxation are plotted below in Figure 5.4 and compared to the original results presented in Figure Examination of Figure 5.13 shows that the new relaxation algorithm

124 99 allows for a slightly lower energy of the SC relative to the previous calculation. This will decrease the migration barrier for the diffusing atom, and thus, slightly increase the slope of the D vs 1000/T plot. It provides slightly better agreement with experimental data D (m 2 /sec) Wazzan, 1965 Ivantsov, 1966 Maier, 1976 Feller-Kniepmeier, 1976 Vladimirov, 1978 Bakker, 1968 MacEwan, 1953 Hoffman, 1965 Monma, 1964 Bronfin, 1975 LDA - QHA Debye FM LDA - QHA Debye FM - Full SC relaxation /T (1/K) Figure 5.13: Calculated self-diffusion coefficient of ferromagnetic fcc Ni using the Debye model showing the difference in relaxation algorithms of the CINEB approach vs the original NEB approach used in this chapter.

125 Chapter 6 First-principles calculations of dilute impurity diffusion coefficients in fcc Ni The importance of Ni and its alloys was discussed in the Introduction of Chapter 1 and has been highlighted in each subsequent chapter. Diffusion coefficients are key data for diffusional phase transformations, high temperature creep rates, and coarsening rates, among other properties. One consistent challenge facing materials scientists and engineers while developing new Ni-base superalloys is creep deformation that occurs at high temperatures. Creep is a time-dependent process and when it occurs, it is inelastic and irrecoverable [77]. Rhenium is thought to have advantageous effects on the creep rate of Ni-base superalloys because of its low vacancy-solute exchange rates and the high diffusion energy barrier when present in an alloy [77, 84]. However, the availability of Re in the Earth s crust is diminishing rapidly [171]. In order to replicate the creep-dampening effects of having Re in the future development of new Ni-base superalloys, insight into why it is so advantageous and what can be done to replicate it is needed. In the present work, the dilute impurity diffusion coefficient for 26 Ni-X systems are calculated and analyzed with the intention of understanding why the 5d rare earth elements possess such a slow diffusivity in Ni, and what materials property can correlate to diffusion behavior in fcc Ni. X can be represented by the 26 elements shown in their approximate placement on the periodic table of the elements shown in

126 101 Figure 6.1: 26 alloying elements and their atomic number studied in the present work in their approximate location on the periodic table. Various colors indicate the properties of the different impurity elements. Figure 6.1. The various shaded colors indicate the crystal structure of the various impurity elements. The four elements in blue font, Sc, Y, Zr, and Hf, are almost insoluble in fcc Ni as based on their binary phase diagrams. The six elements in red font, Ni, Cu, Rh, Pd, Ir, and Pt, are intermiscible in fcc Ni. pv or sv after the atomic name indicates that the p or s states are treated as valence electrons for the first-principles calculations, respectively. In this Chapter, investigations of the impurity diffusion coefficients in the 26 systems begin with basic convergence tests and proof of concept for the fivefrequency model using the well-studied Ni-Cu system as a test case. The all of the results of the impurity diffusion coefficients for the 26 elements shown in Figure 6.1 will be presented. Following in Chapter 7, a discussion on the accuracy of the calculation method, evaluation of the assumptions made in the five-frequency model, trends in the diffusion coefficients as a function of alloying element, and analyses of all of the diffusion work will be presented, and implications for tailoring the creep rate in light of the secondary creep rate model will be discussed. 6.1 Literature Review Before the pioneering work of Mantina [162], dilute impurity diffusion coefficients in any system had not been calculated using first-principles calculations based on density functional theory. Mantina et al. [162, 89, 159] adopted the fivefrequency model as described by Lidiard and LeClaire [40, 41] and through the use of transition state theory simplified by Eyring s reaction rate theory[96], calculated

127 102 all of the factors entering into vacancy mediated dilute impurity diffusion in fcc Al-X alloys. The work has since been adopted to hcp systems with an eightfrequency model [93], and used in bcc Fe host systems with empirical magnetic parameters [91]. Little work has been done on impurity diffusion in fcc Ni. Two studies by Tucker et al. [72] and Choudhury et al. [161] looked at Ni-Cr and Ni-Fe systems using the GGA at 0 K, but did not explicitly calculate finite temperature contributions, rather they were fitted to experimental data. Studies by Janotti et al. [43] and Krčmar et al. [84] calculated activation energy for diffusion using firstprinciples at 0 K, but assumed a constant vibrational contribution for all elements considered. Results in the present work will be compared to the work of Janotti et al. [43] to validate the methodology in the present work. The present work attempts to get a broader understanding of the effect of impurity elements on the dilute impurity diffusion coefficient of fcc Ni by using the Debye model for the vibrational contribution for the sake of simplicity and efficiency due to the number of structures needing to be calculated increasing from 3 for self-diffusion to 9 for impurity diffusion. 6.2 Diffusion Theory Five-frequency model In this chapter, the impurity diffusion coefficient is described as the diffusion coefficient of the impurity/solute atom as it diffuses through a host lattice. The host/solvent is fcc Ni, and the impurity exists in dilute concentrations, which are considered to be less than 3 at. % impurity. It is accepted that dilute concentrations of impurity elements diffuse in fcc crystals via the vacancy mechanism [41, 42], so the present work demonstrates a method to calculate all factors entering into the equation for dilute impurity diffusion coefficients in an analogous manner to the self-diffusion work presented in Chapter 5. The impurity diffusion coefficient is determined from the jump frequency of the impurity atom as it moves into a first neighbor vacant site. However, with the presence of a solute atom, there are several possibilities for the host atom to jump to that are different from that of self-diffusion based on the placement of the impurity atom. LeClaire developed

128 103 a model to represent the possible jumps of the host atom, and defined the relationship between self-diffusion and impurity diffusion in the five-frequency model [40, 41] as: D 2 D 0 = f 2 f 0 w 4 w 0 w 1 w 3 w 2 w 1 (6.1) whered 0 istheself-diffusioncoefficientofthehostlattice,purefccniinthepresent work (shown in Chapter 5), D 2 the diffusion coefficient of the impurity atom, X, as it diffuses through the Ni host lattice, f 2 the correlation factor for impurity diffusion in the Ni-X system, and f 0 the correlation factor for self-diffusion in fcc Ni. The remaining terms represent the five possible jump frequencies occurring in the system and can be visualized as (note that there is only one vacancy that is present in the system, but two are shown here for visualization purposes) [42]. Figure 6.2: Five-frequency model as developed by Lidiard and LeClaire [40, 41] and shown by Mehrer [42] showing the five possible jump frequencies, w i, defined in the text, where represents the solute/impurity atom, one of 26 X s in the present work, represents a vacancy, and represents the host/solvent atom, pure fcc Ni in the present work. w 0 is the host atom jump frequency in the absence of an impurity, w 1 is the jump frequency of a host atom jumping around the vacancy that always remains its first nearest neighbor, i.e., not dissociating the impurity from the vacancy, w 2 is the impurity atom exchanging places with the vacancy, w 3 is the jump of the host atom dissociating the impurity and vacancy, and w 4 is the reverse of w 3, where the jump of the host atom associates the impurity and vacancy. Thus, w 3 and w 4

129 104 share the same transition states, and w 1, w 2, and w 4 share the same initial state for the various jumps. One obvious assumption of this model is that only the first neighbor shell of host atoms are considered to be influenced by the presence of this impurity. The validity of this assumption will be discussed later in this chapter, because the effects of the magnetism of Ni may be further reaching than the first neighbor shell System setup for dilute impurity diffusion To aid in the understanding of the following section regarding the equations needed to calculate the impurity diffusion coefficient of Ni-X alloys, the definition of all of the configurations that needed to be considered must be made. From Chapter 5, three configurations already exist and can be used in the calculation of w 0 in the five-frequency model. The are the N-atom Ni supercell with no defects (PS), the (N 1)-atom Ni supercell with a vacancy (IS) that is the initial state of the atomic jump of interest, and the transition state (SC) when the diffusing atom is between the IS and the corresponding final vacancy configuration that is one nearest neighbor from the IS. In the five-frequency model, additional configurations must be included, including three additional transition states. They are: ps: The N-atom supercell with with (N 1) Ni atoms and 1 impurity atom, X is the perfect impurity configuration, ps. is: The configuration with 1 impurity atom interacting (first nearest neighbor) to 1 vacancy atom with (N 2) Ni atoms is the interacting solutevacancy state, is. ts: The transition state to define the vacancy-impurity exchange, w 2 is referred to as the ts. ts2: The transition state for the w 1 jump is called ts2 and indicates the host atom jump always keeping the vacancy as a first nearest neighbor. ts3: The transition state which is the same for the association and dissociation jumps, w 3 and w 4, is called ts3, which either associates or dissociates the impurity-vacancy pair.

130 105 2NN: The initial configuration of the associate jump, w 4 is not the is as is the case with the three other jump frequencies because it associated the impurity and vacancy. The diffusing solvent atom starts separates the solute and the impurity, so the vacancy is at a second nearest neighbor position to the impurity atom. This is known as the second nearest neighbor impurity configuration, 2NN. This nomenclature will be used in the remainder of this thesis Diffusion equations The correlation factor for impurity diffusion, f 2, is related to the probability that the impurity atom will make the reverse jump back to its previous position in subsequent jumps. Unlike the self-diffusion in Chapter 5, this value is not a constant because five different atomic jumps are possible, and the properties of the impurity will also affect the value of f 2. LeClaire and Lidiard [41] defined this value as: f 2 = 1+3.5(w 3 /w 1 )F(w 4 /w 0 ) 1+(w 2 /w 1 )+3.5(w 3 /w 1 )F(w 4 /w 0 ) (6.2) where F is the escape probability defining the chance that after a dissociation jump, the vacancy will not return to a first nearest neighbor site of the impurity. Manning [73] derived this function to be: F(x) = 1 10x x x (2x x x x+435) (6.3) where x = w 4 /w 0. Since the atomic jump frequencies under the five-frequency model used to calculation f 2 are temperature dependent, f 2 also becomes temperature dependent. The temperature dependence of f 2 is defined by LeClaire [158] and can be written as: C = k B lnf 2 (1/T) (6.4) C is calculated from the relationship between the self- and impurity diffusion coefficient equations and the experimental activation energy which is defined as Q = k lnd (1/T) [158]. C is usually a very small value when compared to the factors

131 106 needed to calculate the activation energy for impurity diffusion, and if C varies with temperature, small contributions to curvature in the Arrhenius diffusion plot may be observed. In light of the five frequency model, one must take into account the difference in energy that it will take to form a vacancy when it is in the pure host matrix, G 0 f and surrounded by only host atoms, and when the vacancy forms adjacent to the impurity atom [42]. In the latter case, a solute-vacancy pair is formed, and its associated energy, known as the solute-vacancy binding energy, G b, describes the energy it takes to bring an infinitely separated solute and vacancy into a solutevacancy pair. In terms of the vacancy formation and vacancy binding energies, the vacancy concentration, C 2, when it is adjacent to the impurity is described as: ( C 2 = exp G0 f G ) b k B T (6.5) where G 0 f can be taken from the work completed in Chapter 5, and the solutevacancy binding energy, G b can be defined as [172]: G b = (G is +G PS G IS G ps ) (6.6) Anegativesignisusedontherighthandsideoftheequationtokeepthecalcualted values consistent with experiemntal values; where a positive binding energy indicates favorable interaction and a negative binding energy indicates repulsion. Note that the definition for solute-vacancy binding energy in Equation 6.6 differs from other used previously in diffusion coefficient calculations [90, 89, 93] because it includes four components. Previous works have assumed that within a 32 or 36 atom supercell, a vacancy and solute atom can be considered non-interacting if separated by a few nearest neighbor shells. LeClaire [173] states the the region of influence of the solute atom is assumed to extend to the fourth nearest neighbor position. Thus, the author of the present work felt that the definition in Equation 6.6 was more complete, and therefore calculated an additional configuration, ps, to make this more accurate. At thermal equilibrium, the association (solute-vacancy pair) and dissociation (separated solute and vacancy) frequencies can be related to the solute-vacancy binding energy through the following equation [42, 158]:

132 107 ( ) w 4 Gb = exp w 3 k B T where G b is the solute-vacancy binding energy defined in Equation 6.6. (6.7) To represent Equation 6.2 in a more recognizable form and in a form analogous to Equation 5.7, Equation 6.7 is substituted into Equation 6.1. Additionally, the equations describing the self-diffusion coefficient in the pure host system, D 0 = f 0 w 0 C 0 a 2, and the vacancy ( concentration ) in the absence of impurities in the pure host system, C 0 = exp are inserted into Equation 6.2. The new relation G0 f k B T can be rearranged to solve for the impurity diffusion coefficient, f 2 which takes the form [42, 158]: D 2 = f 2 w 2 C 2 a 2 (6.8) where f 2 and C 2 are described above. The jump frequencies for impurity diffusion are calculated within the confines of Eyring s reaction rate theory [96] as described in Chapter 5. To describe the impurity jump frequency, w 2, the enthalpy of migration, or the difference in enthalpy between the atom in its transition state between the two vacant sites and the initial state of that jump, must be calculated. In Equation 6.8, the impurity jump frequency is described in terms of the enthalpy of migration of the impurity atom as follows: w 2 = k ( BT h exp G ) ts G is k B T (6.9) where G ts G is = G m, known as the Gibbs energy of migration for the impurity atom. The remaining jump frequencies that are present in Equation 6.2 needed to complete the description f 2 are calculated analogously to Equation 6.8, where the transition state and initial configuration for each jump frequency is described in the previous section. To calculate the Arrhenius form of the diffusion coefficient, consider the free energy of vacancy formation as the difference between the vacancy formation energy in the host system and the solute vacancy binding energy in order to account for the effects on the bonding due to the solute being present in the system [42]. Free energy of vacancy formation can then be written as G f = G 0 f G b =

133 108 H f T S f. Plugging G f and the equation for w 2 into Equation 6.8, the Arrhenius form of the impurity diffusion coefficient in terms of Eyring s reaction rate theory is expressed as [158]: D 2 = f 2 a 2k BT h exp ( Sf + S m +C k B ) ( exp H ) f + H m +C k B T (6.10) where the activation energy is defined as: and the diffusion prefactor can be represented as: D 0 = f 2 a 2k BT h exp 6.3 Computational details Q = H f + H m +C (6.11) ( Sf + S m+c/(k B T) k B ). (6.12) First-principles calculations are performed on a 32-atom 2 x 2 x 2 fcc Ni supercell including 31 Ni atoms and one alloying element, X. Total energy calculations are carried out using the plane wave density functional code called the Vienna ab-initio Simulation Package (VASP) [111]. All calculations use the projector augmented wave (PAW) method [136, 135] to describe electron-ion interactions and the local density approximation of Ceperley-Alder [107] under the Perdew-Zunger parameterization [108] to describe the exchange and correlation functional. A plane wave energycutoffof350evisusedforallcalculations, avaluewhichis1.3timesthedefault plane wave energy cutoff of nickel. Dense k-point meshes in the first Brillouin zone with a Monkhorst-Pack scheme is used for all calculations, with a sampling of 8 x 8 x 8 for each structure described above. For relaxation during the VASP calculations, the Methfessel-Paxton smearing method [163] is used for the calculation of forces acting on the atoms, and a final static calculation is performed after each relaxation using the linear tetrahedron method with Blöchl s [164] correction for an accurate total energy calculation. Total electronic energy is converged to be at least 10 5 ev/atom, and unless otherwise noted, all degrees of freedom of the crystal structures are allowed to relax, including cell shape, cell volume, and

134 109 atomic positions. Due to the ferromagnetism of nickel, calculations are performed separately and identically with and without spin polarization to isolate the effect of magnetism on the diffusion coefficient. A 32 atom (2 x 2 x 2) fcc supercell is created. Results from Chapter 5 are used for the PS, IS, and SC. The ps, is, and 2NN configurations described above are allowed to fully relax within VASP, and their shape and atomic positions were allowed to relax during the E-V curve calculation. The three saddle configurations, ts, ts2, and ts3, are first predicted as the middle of the minimum energy path between the initial and final configurations for that particular jump. The final position and energy for each of the three saddle configurations are computed using the nudged elastic band (NEB) method [97] within VASP. The premise of the NEB method involves using a spring-like force acting on the unstable atom while looking at intermediate steps along the diffusion path, called images. In the present work, one image is used to calculate the TS. A 5.0 ev/å 2 spring constant was used in all NEB calculations to nudge the image to the minimum energy path between the initial and final vacancy configurations. Because of the more intricate nature of the supercell when a vacancy and impurity are present, full relaxation of the saddle configurations within VASP is not appropriate because it cannot be guaranteed that local distortions would not distort the supercell and cause discrepancies in the results. Therefore, the lattice vectors were taken from the fully relaxed ps and inserted with the atomic positions of the respective saddle configuration, and only atomic positions were allowed to relax. Subsequently, the six-volumes used for the E-V fittings were completed using only static calculations. Finite temperature thermodynamic properties to describe the diffusion coefficient entering into Equations 6.8 and 6.10 were calculated using the quasiharmonic Debye-Grünesien model for the sake of simplicity and efficiency as discussed in Chapter 2. F vib (V,T) results for the PS, IS, and TS are used from the selfdiffusion study in Chapter 5. F vib (V,T) results are calculated for the ps, is, 2NN, ts, ts2, and ts3 configurations discussed above. F t el (V,T) has been shown to have a negligible effect on the diffusion coefficient [88] due to the cancellation effect when properties such as G f are calculated, and it is not considered in the impurity diffusion coefficient calculations in this chapter.

135 Results and Discussion To begin the study of dilute impurity diffusion in fcc Ni-X alloys, a convergence test for the vacancy formation energy, G f is compared to the solute vacancy binding energy, G b, as reflected in Equation 6.6. Three systems were chosen to be tested for the solute vacancy binding energy, Re, for its slow diffusion in Ni, Al, for its fast diffusion in Ni, and Cu for its similar diffusivity in Ni with respect to self-diffusion in Ni. From Figure 6.3, it is observed that the vacancy formation energy converges within 0.01 ev, indicating that the 32 atom supercell is a sufficient size for the diffusion calculations. For the solute vacancy binding energy, energy convergence is within 0.03 ev. The value for the Ni-Cu solute vacancy binding energy agrees withapreviousstudyon3dsolutesinnipublishedbykrčmaretal. [84]. Sincethe solute vacancy binding energy is less than 5 % of the contribution to the vacancy concentration given in Equation 6.5, it was concluded that 32 atoms would be an acceptable size to get reasonable results in a convenient amount of time. A study by Janotti et al. [43] using ultra-soft psedopotentials also showed 1 % difference between solute-vacancy binding energies for 32-atom and 64-atom supercells in a 0 K study. Relaxation time increased significantly for the 64 atom supercells, and likewise, the time for calculating the vibrational contribution would have been immense to acheive a 0.01 ev convergence in the solute vacancy binding energy. Or particular interest is the fact that the solute vacancy binding energy of Re in Ni went from being negative to positive as a results of convergence testing, indicating a change from unfavorable binding (negative) to favorable binding (positive). This trend was also observed in transition metal solutes in a first-principles study on Al-X solute vacancy binding energies by Wolverton [172]. The work of Wolverton defines the uncertainty of the calculations to be the difference between the two supercell sizes, in this case, the 0.03 ev in the change in binding energy between the 32 and 64 atom supercell size. As a second check to validate the 0 K methodology, comparison of the activation energyfordiffusionq, asgiveninequation6.11, ismadetoastudyofjanottietal. [43] that predicted diffusion rates of transition metal solutes in Ni by calculating Q and estimated the finite temperature entropic contribution to the free energy to be constant for all transition state elements. While there are slight differences in

136 pure Ni 1.75 G f, ev # of atoms Figure 6.3: Vacancy formation energy for a 32 and 64 fcc Ni supercell, respectively Ni-Al Ni-Re Ni-Cu G b, ev # of atoms Figure 6.4: Solute vacancy binding energy for a 32 and 64 fcc Ni 31 X supercell, where X=, Al, Cu, or Re. the methodology between the work of Janotti et al. at the present work, including the use of the ultra-soft pseudopotentials in the case of Janotti et al. as opposed to the projector-augmented wave method used in the present work. However, as demonstrated in Figure 6.5, excellent correlation between the previous study and the present work is shown, particularly for 3d solutes. Slightly lower values are shown in the present work for the 4d and 5d solutes, which can be attributed to the

137 112 different in pseudopotential method. X=Y, Sc, Al, and Si are denoted as other because they were not studied by Janotti et al. The good correlation in Figure 6.5 indicates a sound method for calculating 0 K contributions to the diffusivity of the 26 Ni 31 X systems in the present work. Q, ev, Janotti et al., d 4d 5d Other Zr Sc Hf Re Ru Ir Os Cr Ni Pt Tc W Cu V Zn Al Si Fe Mo Co Pd Rh Ta Ti Mn Nb 0.5 Y Q, ev, The present work Figure 6.5: Diffusion activation energy for the 26 Ni 31 X systems and the activation energy for self-diffusion in Ni plotted from the present work versus diffusion activation energies from Janotti, et al. [43]. Note, Y, Zn, Sc, Al, and Si were not studied in the previous work.

138 Test case: Impurity diffusion coefficient of Cu in Ni The first Ni 31 X system examined in the present study is Ni-Cu. All of the factors entering into vacancy mediated dilute impurity diffusion are calculated using first-principles calculations as described above with the Debye-Grünesien model for the finite temperature vibrational contributions to the Helmholtz energy. The Arrhenius parameters are calculated following Equation 6.10 for comparison to experimental data. Four sets of experimental diffusivity data are used for comparison to the calculations in the present work: single-crystal data of Helfmeier et al. [44], and poly-crystal data of Anand et al. [45], Monma et al. [37], and Taguchi et al. [46]. Figure 6.6 plots D vs 1000/T for Cu diffusing in Ni. Very good agreement with the experimental data is observed, particularly with the single-crystal data. In general, there is slight underestimation of the diffusivity from the calculated present work compared to experiments. This is most likely due to the method used to relaxed the transition states as discussed in the methodology section in order to ensure local relaxations did not distort the fcc structure of the supercell. Arrhenius parameters calculated in the present work compared to the experimental data are presented in Table 6.1. In general, the error between the calculated activation energy and experimental activation energy is less than 5%. D 0 is highly dependent on temperature but shows acceptable agreement with the experimental values in the reported temperature range. Compared to self-diffusion in pure fcc Ni, Cu diffuses very similarly, which is expected as they are neighbors on the periodic table with similar coordination shells and atomic radii. Table 6.1: Arrhenius diffusion parameters for the impurity diffusion of Cu in Ni compared to experimental data. Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) LDA Debye The present work Expt. Monma [37] Expt. Anand [45] Expt. Helfmeier [44] Expt. Taguchi [46] Since all of the calculations are spin polarized, it is interesting to note the

139 D (m 2 /sec) Helfmeier, 1970 Anand, 1965 Taguchi, 1984 Monma, 1964 Pure Ni LDA Debye /T (1/K) Figure 6.6: Impurity diffusion of Cu in Ni calculated in the present work (solid line) compared to single-crystal data of Helfmeier et al. [44], and poly-crystal data of Anand et al. [45], Monma et al. [37], and Taguchi et al. [46]. Self-diffusion of pure fcc Ni is also shown for comparison (dashed line). magnetic moments as a function of volume from the E-V curves used to enter the Deby-Grünesien model to calculate the vibrational contribution to the Gibbs energy. In Figure 6.7, the total magnetic moment (in Bohr magnetons) of the unit cell of the ps is plotted as a function of its six necessary volumes required for the E- V fitting. It is expected that the total magnetization will increase approximately linearly with increasing volume. It is not expected that from system to system the magnetic moments as a function of volume will be the same, because the magnetic interactions will vary greatly from system to system. It is expected, however, that within each system, the six structures calculated for the impurity diffusion coefficient have similar behavior. This property is presented here for Cu impurity diffusion in Ni as a baseline for future discussion on the behavior of other impurities in Ni. Finally, in Table 6.2 the migration barriers, G m and atomic jump frequencies, w i are presented at 700 K and 1700 K for all five jump frequencies in the Ni-Cu system. While there are no known experimental values to compare to, all of the migration properties are presented in Table 6.2 for the sake of completeness. The same is true for the thermodynamic parameters of the Ni-cu system, there is no known experimental data to compare to, but the

140 115 Table 6.2: Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of Cu in Ni. w 0 w 1 w 2 w 3 w 4 G m (ev) T=700 K T=1700 K w i (Hz) T=700 K 1.25x x x x x10 4 T=1700 K 2.37x x x x x10 9 results are tabulated in Table 6.3. H f and S f are presented as defined in the methodology to include the effects of the solute vacancy binding energy where G f = G 0 f G b = H f T S f. Magnetic moment, µ B Volume, Å 3 /atom Figure 6.7: The magnetic moment, in Bohr magnetons, of each volume of the ps in the Ni-Cu system plotted as a function of the six volumes used in the E-V fitting. In conclusion, using the impurity diffusion of Cu in Ni as a case study, it is demonstrated that the five-frequency model of Lidiard and LeClaire [40, 41] using the Debye-Grünesien model for the finite temperature contribution to the Gibbs energy produces results that are in good agreement with experimental data. For the remainder of the chapter, the diffusion plot of D vs. 1000/T will be presented for each of the 26 Ni 31 X systems along with the Arrhenius diffusion

141 116 Table 6.3: Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the Ni-Cu system. f 2 H f H m S f S m (ev) (ev) (k B ) (k B ) C (ev) T=700 K T=1700 K parameters, split up by grouping on the periodic table. The migration properties and thermodynamic properties of each system will be presented in Appendix C.

142 Non-Transition Element Impurities In the present work, two non-transition element systems were examined, Al in Ni and Si in Ni. Both systems have experimental data and the calculated impurity diffusion coefficient is in acceptable agreement with experimental data. The calculated diffusion coefficients slightly underestimate the experimental data on the order of one-half to one order of magnitude. Figure 6.8 and Figure 6.9 plot D vs. 1000/T for Ni-Al and Ni-Si, respectively. Arrhenius parameters for both systems are presented in Table 6.4, and show good agreement with experiments. The underestimation of the Arrhenius parameters compared to the experimental data is consistent with other fcc studies of impurity diffusion in Al [89, 159] and is better than impurity diffusion studies on hcp Mg [93]. The migration properties and thermodynamic properties of each system are presented in Appendix C, as there is no known experimental data available for comparison Al in Ni D (m 2 /sec) Gust, 1981 Swalin, 1956 Allison, 1959 LDA Debye /T (1/K) Figure 6.8: Impurity diffusion of Al in Ni calculated in the present work (solid line) compared to single-crystal data of Gust et al. [47], and poly-crystal data of Swalin et al. [48] and Allison et al. [49]. In Figure 6.10, both the impurity diffusion of Al in Ni and Si in Ni are plotted with the self-diffusion coefficient of fcc Ni. It is observed that Al diffuses at about the same speed in Ni as self-diffusion occurs in Ni and becomes slightly faster than Ni at higher temperatures. Si is a slightly slower diffuser in Ni, and the difference

143 Si in Ni D (m 2 /sec) Swalin, 1957 Allison, 1959 LDA Debye /T (1/K) Figure 6.9: Impurity diffusion of Si in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [49] and Swalin et al. [50]. becomes greater as temperature decreases. With only two non-transition metal impurity elements it is hard to make a conclusion about the trend of the behavior in Ni, but it appears that the non-transition row impurity diffusion coefficient in Ni have similar diffusion behavior to that of self-diffusion in pure Ni. Reasons for the variation of the calculated diffusion coefficient with respect to the known experimental data along with an analysis of trends are discussed in Chapter 7.

144 119 Table 6.4: Arrhenius diffusion parameters for the impurity diffusion of Al in Ni and Si in Ni compared to experimental data. Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) Al in Ni LDA Debye The present work Expt. Gust [47] Expt. Swalin [48] Expt. Allison [49] Si in Ni LDA Debye The present work Expt. Swalin [50] Expt. Allison [49] Ni-Al Ni-Si Pure Ni D (m 2 /sec) /T (1/K) Figure 6.10: Impurity diffusion coefficient of Al in Ni and of Si in Ni compared to self-diffusion of fcc Ni.

145 d Transition Element Impurities Nine 3d transition element systems were examined and they are presented from left-to-right across the periodic table of the elements: Ni-Sc, Ni-Ti, Ni-V, Ni-Cr, Ni-Mn, Ni-Fe, Ni-Co, Ni-Cu, Ni-Zn, Ni-Cu. An in-depth analysis of Ni-Cu was already presented, so it is only included when all of the 3d transition elements are plotted compared to self-diffusion in pure Ni. Experimental data is known and presented along with the D vs. 1000/T plots for all remaining eight systems with the exception of Ni-Sc and Ni-Zn in Figures The Ni-Ti, Ni-V, and Ni-Fe systems show very good agreement with the experimental data; agreement is less than one order of magnitude across all ranges of experimental data. Agreement in the Ni-Cr and Ni-Co systems are good at higher temperatures, but become only satisfactory as temperature decreases to the Curie temperature of Ni. This disagreement most likely has to do with the magnetic effects of the impurity atom, which will be discussed in Chapter 7. In the case of Ni-Mn, high temperature agreement is excellent when compared to the experimental data. This may be fortuitous however, because the the magnetic moment of manganese in all six of the structures was inconsistent with its sign, ie, sometimes spin up and sometimes spin down, which caused severe fluctuations in the plotted E-V curve. Another indicationisthefluctuationofqandd 0 valuesasafunctionoftemperatureforthe Ni-Mn system presented in Table 6.5, which is not consistent with the Arrhenius values for the other systems. The Arrhenius data for all nine 3d transition metal impurity Ni-X systems is plotted in Table 6.5, again in left-to-right order across the periodic table. Table 6.5: Arrhenius diffusion parameters for the impurity diffusion of 3d transition elements in pure Ni compared to experimental data. Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) Sc in Ni LDA Debye The present work Ti in Ni LDA Debye The present work 1.83x Continued on next page...

146 121 Table 6.5 Continued from previous page... Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) 1.25x Expt. Swalin [48] Expt. Bergner [51] V in Ni LDA Debye The present work Expt. Murarka [52] Cr in Ni LDA Debye The present work Expt. Růžičková [54] Expt. Monma [53] Expt. Tutunnik [55] Expt. Glinchuk [56] Mn in Ni LDA Debye The present work Expt. Swalin [48] Fe in Ni LDA Debye The present work Expt. Bakker [57] Expt. Badia [59] Expt. Guiarldenq [58] Co in Ni LDA Debye The present work Expt. Vladimirov [60] Expt. Badia [59] Expt. Hirano [61] Continued on next page...

147 122 Table 6.5 Continued from previous page... Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) Expt. Hassner [62] Expt. Divya [63] Expt. McCoy [64] Zn in Ni LDA Debye The present work 6.14x Finally, the D vs. 1000/T plots are shown for all nine 3d transition metal impurity Ni-X systems compared to self-diffusion of pure Ni in Figure Sc is clearly the fastest diffusing element of all 3d transition metals. With the exception of Mn, the 3d mid-row elements are the slowest diffusers in fcc Ni, including V, Cr, Fe, and Co. The remaining 3d impurities, Ti, Mn, Cu, and Zn, diffuse similarly to the self-diffusion of Ni and are barely distinguishable on the D vs. 1000/T plot. From examination of Table 6.5, It is observed that the general trend of the calculated impurity diffusion coefficients underestimates the activation barrier for diffusion, Q, and the diffusion prefactor, D 0. The three exceptions to this general trend are the 3d transition elements with higher magnetic moments, including the Ni-Cr, Ni-Co, and Ni-Fe systems, where Q is actually overestimated. Reasons for the variation of the calculated diffusion coefficient with respect to the known experimental data along with an in-depth analysis of diffusivity trends are discussed in Chapter 7.

148 Sc in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.11: Impurity diffusion of Sc in Ni calculated in the present work Ti in Ni D (m 2 /sec) Swalin, 1956 Bergner, 1977 LDA Debye /T (1/K) Figure 6.12: Impurity diffusion of Ti in Ni calculated in the present work (solid line) compared to poly-crystal data of Bergner [51] and Swalin et al. [48].

149 V in Ni D (m 2 /sec) Murarka, 1968 LDA Debye /T (1/K) Figure 6.13: Impurity diffusion of V in Ni calculated in the present work (solid line) compared to poly-crystal data of Murarka et al. [52] Cr in Ni D (m 2 /sec) Ruzickova, 1981 Monma, 1964 Tutunnik, 1957 Glinchuk, 1960 LDA Debye /T (1/K) Figure 6.14: Impurity diffusion of Cr in Ni calculated in the present work (solid line) compared to poly-crystal data of Monma et al. [53]. Růžičková et al. [54], Tutunnik et al. [55], and Glinchuk et al. [56].

150 Mn in Ni D (m 2 /sec) Swalin, 1956 LDA Debye /T (1/K) Figure 6.15: Impurity diffusion of Mn in Ni calculated in the present work (solid line) compared to poly-crystal data of Swalin et al. [48] Fe in Ni D (m 2 /sec) Badia, 1969 Bakker, 1971 Guiarldenq, 1962 LDA Debye /T (1/K) Figure 6.16: Impurity diffusion of Fe in Ni calculated in the present work (solid line) compared to single-crystal data of Bakker et al. [57], and to poly-crystal data of Guiarldenq [58] and Badia et al. [59].

151 Co in Ni D (m 2 /sec) Vladimirov, 1978 Badia, 1969 Hirano, 1962 Hassner, 1965 Divya, 2011 McCoy, 1963 LDA Debye /T (1/K) Figure 6.17: Impurity diffusion of Co in Ni calculated in the present work (solid line) compared to single-crystal data of Vladimirov et al. [60] and to poly-crystal data of Badia et al. [59], Hirano et al. [61], Hassner et al. [62]. Divya et al. [63], and McCoy et al. [64] Zn in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.18: Impurity diffusion of Zn in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [49] and Swalin et al. [50].

152 d impurities D (m 2 /sec) Ni-Sc Ni-Ti Ni-V Ni-Cr Ni-Mn Ni-Fe Ni-Co Ni-Cu Ni-Zn Pure Ni /T (1/K) Figure 6.19: Impurity diffusion coefficients of 3d transition metal Ni-X systems, where X=Sc, Ti, V, Cr, Mn, Fe, Co, Cu, and Zn compared to self-diffusion in pure Ni.

153 d Transition Element Impurities The impurity diffusion coefficient of eight 4d transition element systems were examined and they are presented from left-to-right across the periodic table of the elements: Ni-Y, Ni-Zr, Ni-Nb, Ni-Mo, Ni-Tc, Ni-Ru, Ni-Rh, and Ni-Pd. Of these eight systems in the 4d transition element row, the only experimental impurity diffusion coefficients that exist are for Ni-Zr, Ni-Nb, and Ni-Mo. The calculated impurity diffusion coefficients for the Ni-Mo and Ni-Nb systems show good agreement with the experimental data; on the order of less than one magnitude. For the Ni-Zr system, there is a huge discrepancy in the two sets of experimental data. The calculated impurity diffusion coefficient from the present work is essentially an average of the two sets of experimental data which differ by about 4 orders of magnitude. The work presented by Allison et al. [49] states that the grain size was too small for the experimental observations of the diffusion coefficient and that effects of grain boundary diffusion played a role in the presented diffusion coefficient. The diffusivity plots of D vs. 1000/T are plotted in Figure Figure 6.27 for all eight 4d transition metal impurity systems and the Arrhenius data is tabulated at 700 K and 1700 K and compared to experimental data where available in Table Y in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.20: Impurity diffusion of Y in Ni calculated in the present work (solid line).

154 Zr in Ni D (m 2 /sec) Bergner, 1977 Allison, 1959 LDA Debye /T (1/K) Figure 6.21: Impurity diffusion of Zr in Ni calculated in the present work (solid line) compared to poly-crystal data of Allison et al. [49] and Bergner [51]. Table 6.6: Arrhenius diffusion parameters for the impurity diffusion of 4d transition elements in pure Ni compared to experimental data. Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) Y in Ni LDA Debye The present work Zr in Ni LDA Debye The present work Expt. Allison [49] 1x Expt. Bergner [51] Nb in Ni LDA Debye The present work 2.35x Expt. Bergner [51] Mo in Ni LDA Debye The present work Continued on next page...

155 130 Table 6.6 Continued from previous page... Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) Expt. Swalin [50] Tc in Ni LDA Debye The present work Ru in Ni LDA Debye The present work Rh in Ni LDA Debye The present work Pd in Ni LDA Debye The present work Finally, the D vs. 1000/T plots are shown for all eight 4d transition metal impurity Ni-X systems compared to self-diffusion of pure Ni in Figure Y is the fastest diffusing element of the 4d transition metals examined and is two order of magnitude faster than the next element, Zr. Y and Zr are from the far left of the 4d transition element row, and show very different diffusion behavior than those 4d elements more towards the center of the periodic table. Ni-Nb and Ni-Pd diffuse slightly faster than the self-diffusion in pure Ni, but show relatively similar behavior to pure fcc Ni. The remaining four mid-row elements, Mo, Rh, Ru, and Tc show slower diffusion in pure Ni than the self-diffusion of pure Ni, following the trend of the 3d elements that the mid-row transition elements diffuse the slowest in fcc Ni. Another interesting trend is that the diffusivities of the 4d transition row impurity elements covers a much wider span than the 3d impurity transition elements, the latter of which tended to stay closer to the self-diffusion values of pure Ni. Examination of the Arrhenius parameters presented in Table 6.6 compared to the experimental data show less severe underestimation of Q in the case of Nb

156 Nb in Ni D (m 2 /sec) Bergner, 1977 LDA Debye /T (1/K) Figure 6.22: Impurity diffusion of Nb in Ni calculated in the present work (solid line) compared to poly-crystal data of Bergner [51]. and Mo diffusion in Ni compared to the 3d transition element impurity behavior. Just like the plotted experimental data of Figure 6.21, the calculated values for the Ni-Zr system fall in between the two sets of experimental data. A more in depth analysis on the diffusivity behavior of the transition elements in Ni will be presented in the following Chapter 7, along with a discussion on possible reasons for disagreement between the calculated and experimental diffusion coefficients.

157 Mo in Ni D (m 2 /sec) Swalin, 1957 LDA Debye /T (1/K) Figure 6.23: Impurity diffusion of Mo in Ni calculated in the present work (solid line) compared to poly-crystal data of Swalin et al. [50] Tc in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.24: Impurity diffusion of Tc in Ni calculated in the present work (solid line).

158 Ru in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.25: Impurity diffusion of Ru in Ni calculated in the present work (solid line) Rh in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.26: Impurity diffusion of Rh in Ni calculated in the present work (solid line).

159 Pd in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.27: Impurity diffusion of Pd in Ni calculated in the present work (solid line) d impurities D (m 2 /sec) Ni-Y Ni-Zr Ni-Nb Ni-Mo Ni-Tc Ni-Ru Ni-Rh Ni-Pd Pure Ni /T (1/K) Figure 6.28: Impurity diffusion coefficients of 4d transition metal Ni-X systems, where X=Y, Zr, Nb, Mo, Tc, Ru, Rh, and Pd compared to self-diffusion in pure Ni.

160 d Transition Element Impurities Seven 5d transition metal impurity diffusion coefficients were calculated in the present work. Previous DFT calculations have shown that the 5d elements are typically the slowest diffusers in Ni, with elements such as Re known to enhance creep resistance at high temperatures. The seven 5d transition metals system examined are Ni-Hf, Ni-Ta, Ni-W, Ni-Re, Ni-Os, Ni-Ir, and Ni-Pt. Three of these systems have experimental data: Ni-Hf, Ni-Ta, and Ni-W. Within these 5d transition elements, varying agreement with the experimental data is observed. Diffusion of Hf in Ni shows agreement with the experimental data within one order of magnitude at high temperatures, though there is considerable spread in the experimental data. Like Mo and Cr in the same column above it, impurity diffusion of Ta in Ni shows excellent agreement at higher temperatures. W diffusion in Ni also shows good agreement with experimental data and unlike many systems, agreement gets better as temperature decreases. The diffusivity plots of D vs. 1000/t are plotted in Figure Figure 6.35 for all seven 5d transition elements listed above, in order of left-to-right going across the 5d transition element row in on the periodic table. Arrhenius parameters for the same system is plotted in Table Hf in Ni D (m 2 /sec) Bergner, 1972 LDA Debye /T (1/K) Figure 6.29: Impurity diffusion of Hf in Ni calculated in the present work (solid line) compared to the poly-crystal data of Bergner [51].

161 Ta in Ni D (m 2 /sec) Bergner, 1977 LDA Debye /T (1/K) Figure 6.30: Impurity diffusion of Ta in Ni calculated in the present work (solid line) compared to the poly-crystal data of Bergner [51]. D (m 2 /sec) Vladimirov, 1978 Swalin, 1956 Monma, 1964 Bergner, 1977 LDA Debye /T (1/K) W in Ni Figure 6.31: Impurity diffusion of W in Ni calculated in the present work (solid line) compared to the single-crystal data of Vladimirov et al. [60], and the polycrystal data of Bergner [51], Swalin et al. [48], and Monma [65].

162 Re in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.32: Impurity diffusion of Re in Ni calculated in the present work (solid line). Table 6.7: Arrhenius diffusion parameters for the impurity diffusion of 5d transition elements in pure Ni compared to experimental data. Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) Hf in Ni LDA Debye The present work 5.77x Expt. Bergner [51] Ta in Ni LDA Debye The present work Expt. Bergner [51] W in Ni LDA Debye The present work Expt. Vladimirov [34] Expt. Monma [65] Expt. Swalin [48] Continued on next page...

163 138 Table 6.7 Continued from previous page... Method References D 0 (10 4 m 2 /sec) Q (ev) T (K) Expt. Bergner [51] Re in Ni LDA Debye The present work Os in Ni LDA Debye The present work Ir in Ni LDA Debye The present work Pt in Ni LDA Debye The present work Finally, the D vs 1000/T plots are shown for all seven 5d transition metal impurity Ni-X systems compared to the self-diffusion of pure Ni in Figure The 5d transition row elements exhibit very different behavior than the 3d and 4d elements. Only Hf and Ta diffusion in Ni is faster than the self-diffusion of pure Ni, and they are the two left most elements of all 5d transition row elements examined. Ta in Ni diffuses almost identically to the self-diffusion of pure Ni. There are no fast diffusers like Sc and Y in Ni present in the 5d impurity elements examined. The remaining five elements, W, Re, Os, Ir, and Pt, diffuse more slowly in Ni than the self-diffusion coefficient of pure Ni. At high temperatures, Re and Os diffusing in Ni are almost 5 orders ofmagnitudeslower than theself-diffusion ofni, and they are the two slowest metals of all 26 Ni-X systems examined. Re and Os are also both centered on the periodic table of the elements in the 5d row, supporting the trend observed in the previous two sets of data that the slowest diffusing elements in each row tend toward the center of the row, not on the outsides of the row. This is also reflected in the tabulated Arrhenius data from Table 6.7, where Re and Os have the highest activation barriers for diffusion in Ni. As a whole, there is less

164 Os in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.33: Impurity diffusion of Os in Ni calculated in the present work (solid line). spread in the magnitude of the impurity diffusivities in Ni compared to the 3d and 4d transition row impurity elements. Reasons for the variation of the calculated diffusion coefficient with respect to the known experimental data along with an in-depth analysis of diffusivity trends are discussed in Chapter 7.

165 Ir in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.34: Impurity diffusion of Ir in Ni calculated in the present work (solid line) Pt in Ni D (m 2 /sec) LDA Debye /T (1/K) Figure 6.35: Impurity diffusion of Pt in Ni calculated in the present work (solid line).

166 d impurities D (m 2 /sec) Ni-Hf Ni-Ta Ni-W Ni-Re Ni-Os Ni-Ir Ni-Pt Pure Ni /T (1/K) Figure 6.36: Impurity diffusion coefficients of 5d transition metal Ni-X systems, where X=Hf, Ta, W, Re, Os, Ir, and Pt compared to self-diffusion of pure Ni.

167 Conclusions In this Chapter, the calculation of dilute impurity diffusion coefficients is explained in light of Eyring s reaction rate theory and the five-frequency model for impurity diffusion as developed by Lidiard and LeClaire. Convergence tests for the Gibbs energy of vacancy formation and the solute-vacancy binding energy are presented for 32 and 64 atom supercells. While the Gibbs energy of vacancy formation shows little change between supercell sizes, more variation is observed in the solute-vacancy binding energy. Based on the complexity of the ferromagnetic nature of the fcc Ni supercell, 32 atoms is chosen for the remainder of the calculations. The overall impurity diffusion coefficient methodology is validated by comparing the 0 K activation energy for diffusion to the previous work of Janotti et al. As a test case, dilute Cu diffusion in Ni is calculated and all Arrhenius and thermodynamic parameters including jump frequencies and migration barriers are presented and compared to experimental values when available. After showing the robustness of the calculations, impurity diffusion coefficients and Arrhenius parameters are presented for the remaining 25 Ni-X dilute binary systems. In general, the 3d solutes are the fastest diffusers in fcc Ni while the 5d transition metal solutes are the slowest. The 4d transition metal solutes show the largest range of speed of diffusivity in Ni. The 5d elements and vanadium group elements show very good agreement with experimental data. The magnetic elements such as Co and Cr, show the worst agreement with experimental data. The significance of the results, along with an in-depth analysis of the cause for the variation of the impurity diffusion coefficient based on alloying element will be discussed in Chapter 7. Further analysis of the accuracy of the calculation method and assumptions of the five-frequency model are also presented in Chapter 7.

168 Chapter 7 Analysis of dilute impurity diffusion coefficient calculations: methodology and trends InChapter6,aproofofconceptfortheimpuritydiffusionofCuinNiandallresults from the 26 dilute impurity diffusion coefficient calculations were presented, where X = Al, Co, Cr, Cu, Fe, Hf, Ir, Mn, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Sc, Si, Ta, Tc, Ti, V, W, Y, Zn, and Zr. Basic trends were discussed, such as the 5d transition metal solutes being the slowest diffuser in Ni. This chapter seeks to analyze the data and the calculation method to better understand the trends and effects of each solute impurity diffusion coefficient in fcc Ni. Particular questions that are under consideration are: How is the calculation method influencing the results and how can the calculation method be improved; How good of an approximation is the five-frequency model being used and what are its deficiencies; and What is causing the advantageous slow diffusion observed as the alloying element approach the lower center regions of the periodic table of the elements? Finally, the chapter concludes with implications for the effects of the various solute elements on the creep rate of dilute binary Ni-rich alloys by employing a secondary creep rate model.

169 Discussion of first-principles methodology In Chapter 6, the computational details of the first-principles relaxation process for the various system configurations necessary to calculate the impurity diffusion coefficients were given. In particular, it is mentioned that for the three transition state, ts, ts2, and ts3, the nudged-elastic band (NEB) [97] method was used within VASP. It is also mentioned in both Chapter 5 and Chapter 6 that full relaxation, including relaxation of the atomic positions, ions, and shape, can often greatly distort the supercell and even cause the unstable impurity atom to fall down the minimum energy pathway into one of the adjacent nearest neighbor vacancies. For the self-diffusion of Ni, this was not a problem and the structure was checked to make sure local relaxations did not distort the supercell. To combat this problem during the 26 impurity diffusion coefficient calculations, the lattice vectors were fixed as the relaxed lattice vectors of the ps, the 32 atom supercell with 31 Ni atoms and 1 impurity atom, and only the atomic positions were allowed to relax. This method showed promising results for the case study system of Cu diffusion in Ni. The results of all 25 impurity diffusion coefficient from Chapter 6 however, showed a consistent underestimation of the both the diffusion prefactor, D 0, and the activation energy for diffusion, Q, for the majority of the systems with the exception of three 3d transition row impurities, Cr, Fe, and Co, which overestimated D 0. In theory, a lower migration barrier for all of the transition states should increase the overall impurity diffusion coefficient and make better agreement with experimental data. Alternatively, a higher diffusion prefactor, D 0 would also provide better agreement with experimental data. To test the possibility of this underestimation of the Arrhenius parameters being a result of the computational technique, a new version of the climbing-image nudged elastic band (CINEB) [134] method became available at Penn State and was tested for the three transition states in the Ni-Al system. The CINEB method is an improvement on the NEB method by allowing the highest energy image to climb up the elastic band of forces holding the unstable atom in place. The maximum energy image is not affected by the spring forces. Essentially, the highest energy image tries to maximize its energy while minimizing the all other energies along the band. Typically in fcc structures,

170 145 there is no difference between the NEB and CINEB methods, as was found by tests in the present work for the calculation of both self- and impurity diffusion coefficients. The advantage of the new version of CINEB, usually called ss-cineb, performed in VASP 5 allows for a more complete relaxation of the saddle without the possibility of local distortions affecting the overall structure using a different optimization scheme [174]. To test this new methodology, tests were first performed to ensure that versions 4 and 5 of VASP gave the same result independent of relaxation scheme, so that the ps, is, and 2NN configurations did not have to be re-relaxed. Tests showed the same result was obtained whether the configuration was relaxed in VASP 4 or VASP 5. Then, the three transition configurations, ts, ts2, and ts3, were relaxed in VASP 5 using full relaxation with the CINEB method, and the Debye model was performed on the resulting structure in the same manner described in Chapter 2. A more fully relaxed saddle configuration should correspond to lower energy barriers (and higher atomic jump frequencies) for all jumps involved in the five frequency model. The new results are plotted in Figure 7.1 as a dashed line in comparison with the solid line of the methodology and results presented for the diffusion of Al in Ni in Chapter 6. The Arrhenius parameters are also given in Table 7.1. Table 7.1: Arrhenius diffusion parameters for the impurity diffusion of Al in Ni calculated by the NEB and CINEB methods compared to experimental data of [47, 48, 49]. Method References D 0 (10E-4 m 2 /sec) Q (ev) T (K) Al in Ni LDA Debye - The present CINEB work LDA Debye - The present NEB work Expt. Gust [47] Expt. Swalin [48] Expt. Allison [49] Figure 7.1 makes it clear that there is better agreement with the experimental data when full relaxation in the new CINEB approach of all three saddle con-

171 Al in Ni D (m 2 /sec) Gust, 1981 Swalin, 1956 Allison, 1959 LDA Debye LDA Debye - Full SC relax /T (1/K) Figure 7.1: Impurity diffusion of Al in Ni calculated in the present work without fully relaxation the transition states (solid line) and with full relaxation of the transition states (dashed line) compared to single-crystal data of Gust et al. [47], and poly-crystal data of Swalin et al. [48] and Allison et al. [49]. figurations (ts, ts2, and ts3) are used. Particularly at lower temperatures, the improvement of the impurity diffusion coefficient with respect to the extrapolated experimental data is nearly an order of magnitude difference from the original relaxation scheme. However, from looking at the Arrhenius parameters, it is observed that Q, the activation energy for diffusion, actually decreases and shows worse agreement with experiments when this relaxation scheme is employed. The same results is observed for the diffusion prefactor, D 0. To understand why the plot of D vs 1000/T improves with this relaxation scheme, the thermodynamic parameters and migration properties should be examined. They are presented in Table 7.2 and Table 7.4. It is observed that the migration barriers are reduced and the jump frequencies subsequently increased increased. From examination of Equation 6.2 and Equation 6.3, it is observed that the higher jump frequencies will result in a higher value of the impurity correlation factor, f 2. In conclusion, the effect of full relaxation of the saddle configurations increases the overall agreement of the plot of D vs 1000/T when compared to experiments. But by analyzing the diffusion parameters, Q and D 0 actually both show slightly worse agreement with experiments, but the effect of the low activation energy for diffusion is over com-

172 147 Table 7.2: Gibbs energy of migration, G m and atomic jump frequencies, w i for the five jump frequencies for impurity diffusion of Al in Ni comparing the two different relaxation schemes for the saddle configurations discussed in the present work. w 0 w 1 w 2 w 3 w 4 Ni-Al (CINEB) G m (ev) T=700 K T=1700 K w i (Hz) T=700 K 1.23E5 9.03E3 7.40E7 2.87E3 6.01E5 T=1700 K 2.37E E9 1.25E E9 1.68E10 Ni-Al (NEB) G m (ev) T=700 K T=1700 K w i (Hz) T=700 K 1.23E5 2.25E5 3.27E7 7.34E3 1.53E5 T=1700 K 2.37E E9 1.01E E9 1.08E10 Table 7.3: Thermodynamic parameters at 700 K and 1700 K given for all factors entering into vacancy mediated impurity diffusion in the Ni-Al and Ni-Si systems. f 2 H f H m S f S m (ev) (ev) (k B ) (k B ) C (ev) Ni-Al (CINEB) T=700 K T=1700 K Ni-Al (NEB) T=700 K T=1700 K pensated for by a diffusion prefactor. The ultimate conclusion from this analysis is that there are clear limitations with the calculation of the Arrhenius parameters for diffusion when using the Debye-Grünesien model for the calculation of the vibrational properties. Full relaxation of the saddle configurations has a direct influence on the impurity correlation factor, f 2 via the atomic jump frequencies. Better agreement with experiments is obtained though a cancellation of errors as both Arrhenius parameters show poorer agreement with experiments.

173 Analysis of the approximations of the fivefrequency model The five-frequency model of Lidiard and LeClaire [40, 41] is known to have several approximations associated with it, for example (i) only including monovacancy effects and ignoring affects from divacancies, which can be prominent in fcc metals [158], (ii) solute atoms are non-interacting because they are so far apart from oneanother, and (iii) perhaps the most severe assumption, is that solute effects are not felt beyond the first neighbor adjacent solvent atoms. In other words, this means that all atomic jump frequencies that are not w 1, w 2, w 3, or w 4 are all assumed to be equal to w 0, which is equivalent to the jump frequency of the self-diffusion in Ni. In Al, Wolverton [172] that while solute-vacancy binding energies when the solute is at a 2NN site from the impurity are smaller than the binding energies when the solute-vacancy pair is at the first nearest neighbor site, there are still many cases of attractive solute-vacancy binding effects. This is especially true for transition metal solutes, and it can be expected that this will be amplified in systems where the solvent matrix is a magnetic transition metal such as Ni. It has already been demonstrated in the present work that the magnetic effects of Ni are far reaching throughout the cell and can greatly influence the thermodynamic properties and subsequently, the diffusion coefficient. Because of the magnetic effect, ignoring any solute-vacancy binding effects at the second nearest neighbor sites may not be the proper assumption. When Manning [73] defined the exactly correlation factor dependencies for the bcc, diamond, and fcc structures, he also developed an equation for the correlation factor, f 2 for fcc systems that includes the effects of solute-vacancy binding at second nearest neighbor sites. As in the case of the bcc cell where the second nearest neighbors are much closer to the impurity atom than in the fcc structures, w 0 will include jump frequencies from third or higher nearest neighbors, and jumps from the second nearest neighbor site will be included in the impurity correlation factor correlation. The following jump frequencies should be considered: w 1 : same rotational jump around impurity atom defined in Figure 6.2. w 2 : same solute-vacancy exchange jump defined in Figure 6.2.

174 149 w ib : jumps from second to first nearest neighbor site of the impurity (associate jump, same as w 4 in Figure 6.2. w im : jumpsfromsecondnearestneighborsitetoasitethatisnotfirstnearest neighbor to the impurity. w bm : jumpsfromthefirstnearestneighbortothirdorfourthnearestneighbor site of the impurity. Unfortunately, it is outside of the bounds of the first-principles calculations to be able to calculate a jump that is not between first nearest neighbor sites. This means that calculation of the w bm jump cannot be explicitly done and must be approximated as the dissociate w 3 jump as previously used in the five-frequency model of Figure 6.2. The w im jump can easily be calculated by first-principles calculations and can be viewed in Figure 7.2. The w im is usually not considered in the five-frequency model [40, 41] or the calculation of the correlation factor f 2 [73]. It was not considered when previous first-principles studies were performed [162, 89, 159] for calculation of diffusion coefficients. The hypothesis from the inclusion of the additional jump frequencies to take into account effects from second nearest neighbor binding effects is that the resulting diffusion coefficient will have better agreement with experiments. To test this hypothesis, all configurations necessary were relaxed using the full relaxation CINEB method defined in the previous section in order to more fully understand the effects from the various contributions on the diffusion coefficient. The impurity correlation factor, f 2, was calculated based on the description by Manning [73] that includes the effects from the second nearest neighbor solutevacancy binding frequencies and can be found in the following equation: where f 2 = 2w 1 +7Fw bm 2w 2 +2w 1 +7Fw bm (7.1) 7F = (w ib/w im )+19.22(w ib /w im ) (w ib /w im ) (7.2) Since we have to assume that w bm = w 3, the value for f 2 in Equation 7.1 does not change compared to its original definite in Equation 6.2 that was used for the calculation of f 2 for all 26 impurity diffusion coefficients. The difference lies in

175 150 Figure 7.2: w im included in conjunction with the five-frequency model in the present work to include second nearest neighbor binding effects on the impurity diffusion correlation factor. the value of F in Equation 7.2, where F now depends on the jump from second to first nearest neighbor and from jumps from the second nearest neighbor to the third nearest neighbor position. To gain an understanding of the effect of including the w im on the impurity diffusion correlation factor, let us first examine the jump frequencies. In the original definition of F, F was dependent on a ratio of the solute-vacancy association jump, w 4 and the jump frequency of the self-diffusion jump, w 0, assumed to be equal to all jump frequencies beyond the second nearest neighbor of the vacancy. In the newer definition of F, F depends on the ratio between the association, w 4 jump, and the new w im jump, which we can define as the w 0 jump in the present of the impurity. Table 7.4 displays the jump frequencies and the ratios as a function of temperature to examine the effect of the impurity on the w 0 /w im jumps. Table 7.4 shows interesting results. Of the three jump frequencies presented, it is observed that w im has the lowest jump frequency, which will mean it will have the highest barrier for diffusion. This can be expected, especially when compared to the w 0 jump because there are less opportunities for this jump to be occurring than the self-diffusion w 0 jump. Particularly at lower temperatures, this will cause ahigherw 4 /w im ratiothanthew 4 /w 0 ratio, andasubsequentincreaseinthevalues

176 151 Table 7.4: Comparison of jump frequencies and jump frequency ratios for the two methods of calculating f 2 proposed by Manning [73]. T=700 K T=1000 K T=1700 K w E E E+10 w E E E+10 w im 4.98E E E+09 w 4 /w w 4 /w im of F and f 2 as a function of temperature. This is confirmed in Table 7.5 and also in the D vs. 1000/T plot shown in Figure 7.3. It is observed that the use of the new method for the calculation of f 2 results in an impurity diffusion coefficient of Al in Ni that significantly affects the agreement with experimental data. Also, the effect is more pronounced at higher temperatures, causing an increase in the diffusion coefficient of two orders of magnitude at 500 K. The conclusion from this investigation is that there is a contribution coming from the second nearest neighbor solute-vacancy binding energy on the diffusion coefficient. This can be due to the far-reaching effects of the magnetic moment of Ni throughout the supercell that has been previously demonstrated in Chapter 5. Table 7.5: Comparison of impurity correlation factors, f 2, for the two methods proposed by Manning [73]. T=700 K T=1000 K T=1700 K f 2 - original f 2 - with w im

177 Al in Ni D (m 2 /sec) Gust, 1981 Swalin, 1956 Allison, 1959 LDA Debye - ALT f 2 LDA Debye - CINEB LDA Debye /T (1/K) Figure 7.3: Impurity diffusion coefficient of Al in Ni calculated with Manning s method for binding at second nearest neighbor site, along with the original method presented in Chapter 6 and the fully relaxed saddle configurations presented in this chapter.

178 Discussion of diffusivity trends The goal of this section is to explain the behavior of the various 26 impurity diffusion coefficient in Ni, and to understand which elements are slower diffusers and can have advantageous effects on the future development of Ni-base superalloys Results at 0 K To better understand the effects of each dilute impurity element, X, on the supercell of fcc Ni, the results from the EOS fitting based on Equation 2.13 are examined. Equilibrium properties are calculated from the 32 atom supercell with 1 impurity atom and 31 Ni atoms (ps), creating a dilute Ni alloy. In Figure 7.4, the (a) equilibrium volume V 0, (b) bulk modulus, B 0, (c) first derivative of bulk modulus with respect to pressure, B 0, and (d) spin magnetic moment MM, are plotted without the contributions from zero point energy for each of the 26 systems as a function of increasing atomic number along different rows in the periodic table. Note that for pure Ni, the computed V 0 is slightly underestimated when compared to experiments ( Å 3 /atom calc. vs Å 3 /atom expt. [113]), and B 0 is overestimated compared to experiments (252.8 GPa calc. vs GPa expt. [113]), which is due to the underestimation of lattice parameters resulting from the use of the LDA exchange and correlation functional. The calculated magnetic moment, µ B, agrees well with the calculated value of 0.62 µ B /atom [175]. Several trends can be observed from the analysis of Figure 7.4. In Figure 7.4(a) it can be seen that the addition of an impurity element increases the equilibrium volume, V 0, of the dilute Ni 31 X alloy when compared to the equilibrium volume of pure fcc Ni. The largest volume effects come from the 4d and 5d transition row elements, and the general trends shows that the greatest increase in volume of the dilute Ni alloy begins at the far left of the periodic table and becomes less as the impurity element approaches the center of the periodic table, then becomes more pronounced as the far right alloying elements on the periodic table are reached. It should be noted that the impurity effects on the volume of the dilute Ni 31 X alloy given Figure 7.4(a) shows nearly the exact shape the experimental volumes of the transition row elements plotted as a function of atomic number across atomic rows as given by Pearson [66]. Pearson [66] also addresses the anomalous size behavior

179 Volume, Å 3 /atom Sc Y Hf Ti Zr Nb W Os Ir Pt Ta Re Mo Pd Tc Ru Rh V Cr Mn Fe Co (a) 3d elements 4d elements 5d elements other elements Zn Cu Ni Al Si B 0 (GPa) Sc Ti Cr (b) Mo Re Ru Ta W Rh Os V Ir Pt Tc Ni Si Nb Fe Co Pd Cu Hf Zn Al Zr Mn Y B 0 / Re W Os Rh PtPd V Ta Tc Ti Mo Ir Cu Ni Y Ru Zr Nb Cr Fe Co Sc Hf Mn (c) Zn Al Atomic number along different rows Si Magnetic moment, (µ B /atom) Fe Co (d) Rh Ni Pd Ru Pt Cu Ir Zn Sc Al Ti Y Mn Os Zr V Cr Tc Si NbMo Hf Ta W Re Atomic number along different rows Figure 7.4: EOS calculated equilibrium properties for the ps (Ni 31 X at 0 K without the effect of zero point vibrational energy. The (a) equilibrium volume V 0, (b) bulk modulus, B 0, (c) first derivative of bulk modulus with respect to pressure, B 0, and (d) spin magnetic moment MM are plotted as a function of atomic number along different rows in the periodic table. 154

180 155 of Mn, Fe, and Co, which do not show the minimum in atomic volume at the center of the transition row elements on the periodic table as the corresponding 4d and 5d elements do. Pearson postulates that this is a results of the presence of non-bonding d electrons that decreases cohesion within the atoms. For the bulk modulus, B 0 presented in Figure 7.4(b), the trend observed shows the opposite results from the addition of impurity element, X, on the dilute Ni 31 X alloy. The 5d mid row transition elements (X = W, Re, Os, Ir, Pt) show the cause the greatest increase in the bulk modulus of the dilute Ni 31 X alloys, followed by the mid row 4d transition row elements (X=Mo, Tc, Ru, Rh). The far right and far left transition row elements of all three transition rows examined decrease the bulk modulus of the Ni 31 X alloys with respect to the calculated bulk modulus of pure Ni. Another interesting result is that except in the Ti and V group elements, the 5d transition row element has the highest bulk modulus of all other elements in its column (X = W, Re, Os, Ir, Pt). The same three elements that show anomalous effects on the volume (X=Mn, Fe, and Co), show a local minimum in the effect of X on the volume of the Ni 31 X alloys from the 3d transition row elements. Figure 7.4(c) shows that the first derivative of bulk modulus with respect to pressure, B 0, do not follow an apparent trend in a manner similar to the V 0 and B 0. In both the 4d and 5d transition row elements, the Re group (X=Re and Tc) show the highest B 0 for their respective d transition row. In the 3d elements, where there is usually a local minima or maxima among the properties around (X = Mn, Fe, and Co), the local minimum encompasses (X = Cr, Mn, Fe, Co, and Ni); making any analysis of trends difficult. Two other observations are that the W and Re group elements in the 4d and 5d transition rows show the highest values of B 0, and the values for all 26 dilute Ni 31 X alloys lie between 4.75 (X=Mn) (X=Re). The resulting magnetic moment per atom, µ B /atom presented in Figure 7.4(d) show very interesting results. Within each transition element row, the MM/atom takes on a nearly complete S shape with the 3d transition row elements having a higher MM/atom than the 4d row, which has a higher MM/atom than the 5d elements, within each column of elements. The greatest effects on the MM/atom come from the 3d transition row elements (X=Fe and Co). One might expect other 3d transition row elements such as (X=V, Cr, and Mn) to have a larger influence on the MM/atom, but examination of the VASP structure files show a

181 156 preference for each of those atoms to be antiparallel (negative spin) to the positive host magnetic atoms of the ferromagnetic fcc Ni supercell. Al and Si show similar trends as the 3d transition row elements when compared to each other s effects on the dilute Ni 31 X alloys. For example, Al increases the overall volume of the dilute Ni 31 X alloy while Si decreases the volume per atom. The reverse is true for the bulk modulus, Si increases B 0 more than Al in the dilute Ni 31 X alloys. For the MM/atom, Al and Si basically become an extension of the S of the 3d transition row elements. Tabulated data for all 0 K properties presented in Figure 7.4 is presented in Appendix C in Table Finite temperature diffusivity results To get a broad understanding of the effect of each alloying element X on the impurity diffusion coefficient in fcc Ni, the data can be viewed as a function of atomic number along different rows in the periodic table, or in a normalized version to more easily compare the numerical values to self-diffusion in fcc Ni. A normalized version of D 2 is computed at 1000 K such that self-diffusion in pure fcc Ni has a value of 0.00 can be calculated using the following equation: D normalized = log 10 (D Ni31 X) log 10 (D Ni31 Ni) (7.3) where D Ni31 X is the impurity diffusion coefficient of element X in fcc Ni, and D Ni31 Ni represents self-diffusion in pure fcc Ni. This normalization is chosen because faster diffusers in Ni will have a value greater than 0.00, and slower diffusers in Ni will have a value less than In graphical form, this normalization is shown along with a gradient color scheme where self-diffusion in fcc Ni is neutral, represented by yellow, the fastest diffusers are given a red color, and the slowest diffusers are given a green color. The normalized diffusion coefficient at 1000 K according to periodic placement is shown in Figure 7.5. To treat the data numerically, the impurity diffusivity at T=1000 K is given for each of the 26 impurity elements examined in Figure 7.6. There are several conclusions regarding the impurity diffusion coefficient in Ni can be made from the analysis of Figure 7.5 and Figure 7.6. The fastest diffusers in Ni are located at the far left of the periodic table, and are X = Sc, Y, Hf,

182 Figure 7.5: Impurity diffusion coefficient at T=1000 K normalized to self-diffusion in fcc Ni, according to periodic table placement. 157

183 158 log D T=1000, m 2 /sec Y Sc Zr Ti Hf Ta V Nb Mo Cr W Mn Fe Tc Ru Rh Co Ir Pd Ni Pt 3d elements 4d elements 5d elements other Cu Zn Al Si -24 Re Os Atomic Number Figure 7.6: Impurity diffusion coefficient at T=1000 K plotted as a function of increasing atomic number along different transition element rows in the periodic table. and Zr. Regarding the binary phase diagrams of Ni-X, where (X = Sc, Y, Hf, and Zr), there four elements show little to no solubility in fcc Ni. These three are significantly faster diffusers than all 23 other elements examined, with Y being the fastest diffuser diffusion ten times faster than self-diffusion in fcc Ni, indicating that phase stability may have an effect on the impurity diffusivity. As observations move towards the center of the periodic table of the elements, the slowest diffusers are located at the lower center of the periodic table, where X = Tc, W, Re, Os, and Ir. Re is the slowest diffusing 5d element and Tc is the slowest diffusing 4d element, and they are located in the same group on the periodic table. Mn is showing an exception to the rule and is the only element in groups VIB, VIIB, and VIII that diffuses faster than self-diffusion occurs in pure Ni, which can be expected after seeing the anomalous behavior of Mn in Ni as shown in Figure 7.4. As the observation goes to the far right rows of transition elements, it is observed that diffusion of the impurity elements speeds up again, but not to the degree on the left side on the table. X = Cu, Zn, Al, and Si all exhibit diffusivities slightly faster than self-diffusion in Ni, but very close. The remainder of this section seeks to answer the questions generated from analysis of this normalized diffusivity at 1000 K, such as:

184 159 Why are the 5d transition element solutes the slowest diffusers in fcc Ni? What makes the VIIB group elements the slowest diffusers in their group with the exception of Mn? Why does the difference of one electron, ie, between X=Ta and X=W for example, cause diffusivity to go from being faster than self-diffusion in Ni (X=Nb) to slower than self-diffusion in Ni (X=Mo). Janotti et al. [43] and Krčmar et al. [84] performed two studies that calculated the 0 K activation barrier for diffusion without any entropic contributions for 22/26 of the solutes examined in the present work. The present work used the results at 0 K to validate the methodology in Figure 6.5. Their estimation of the entropic contributions that come out in the diffusion prefactor led to the conclusion that solutes with similar atomic radii to Ni move the slowest as impurity diffusers in Ni. This disproved the commonly held belief that the smaller atoms could move faster when diffusing in a solvent because the lattice misfit strain would be at its lowest. Their conclusion was that larger atoms could move faster because they were the most compressible due to the formation of solute-host directional bonds. However, examination of the calculated impurity diffusion coefficients, D, at 1000 K plotted as a function of experimental ionic radius [66] Figure 7.7 shows no direct correlation between diffusion speed and ionic radii of the atom aside from the four fastest diffusers in Ni (X = Hf, Zr, Sc, and Y) as shown in Figure 7.5. When a linear fitting is applied, a correlation of r 2 =0.64 is observed, showing the weak dependence of diffusivity on experimental ionic radius. The four fastest diffusers are definitely the largest impurity atoms, but even within those four it is not a direct correlation. In fact, from observation of Figure 7.7, the two slowest diffusers that stand out, (X = Os, Re), have larger atomic radii than only about half of the impurity diffusers considered in the present work. Also, impurity elements with diffusivity approximately equal to that of self-diffusion in Ni (X = Cu, Si, Pd, Zn, andal), showaspreadofatomicradiiacrosstheentirespectrumofradiiexamined. An additional observation that can be made that disproves this hypothesis entirely is that Re and Pd have identical ionic radii and exhibit one of the fastest (Pd) and slowest (Re) impurity diffusion coefficient in Ni.

185 Y log D T=1000, m 2 /sec Hf 16 Mn 18 Zn Ti Nb Cu Si Ta Pd Ni Al Fe V Pt 20 Cr Rh Mo Co Ru Tc W 22 Ir Os Re Experimental ionic radius, Å Figure 7.7: The 26 impurity diffusion coefficients at T=1000 K plotted vs increasing experimental atomic radius from Pearson [66]. Zr Sc Krčmar et al. [84] went on to suggest that larger atoms could move faster because they were more compressible because the larger atoms form more compressible solute-host directional bonds. Compressibility can be defined as the inverse of bulk modulus. Bulk modulus is defined as the resistance that a material has to uniform compression, and is shown to be the derivative of pressure, P, with respect to volume, V and was defined in Equation By taking the inverse of Equation 2.15 the compressibility, κ, can be defined as: κ = 1 V V P (7.4) In the present work compressibility, κ, was calculated directly from the Bulk modulus as part of the EOS fitting done at 0 K for each of the 26 dilute Ni 31 X alloy systems. The value for κ that is essentially calculated in this work is the effect of each impurity element, X, has on the dilute Ni 31 X systems. It is expected that the elements that decrease the compressibility the most compared to that of pure fcc Ni will diffuse the slowest in fcc Ni. At 1000 K, the plot of log D vs compressibility in Figure 7.8 shows exactly that, that Re with the lowest compressibility is the slowest impurity diffuser in Ni. Note that the impurity atom labels are colored according to their placement on the periodic table, where 3d=blue, 4d=red, 5d-

186 161 black, and other=green. A linear fitting is also presented along with the calculated data to show the nearly linear trend of the data, where slower diffusers have a lower compressibility in their respective alloy and faster diffusers have a higher compressibility. The plot increases nearly linearly as compressibility of the Ni 31 X alloysincreasestothefastestdiffuserinfccni, Yfollowingacorrelationofr 2 =0.88. The use of the fittings show that there is a much stronger dependence of diffusivity on compressibility than on ionic radius. The tabulated form of Figure 7.8 is also presented in Appendix C in Table C.9. The present work confirms the theory presented by Krčmar et al. [84] that the most compressible solute atoms diffuse the fastest, thought there is no direct dependence on ionic radius as they had previously predicted. log D, m 2 /sec Nb Ta Ti Si V Ni Pd Cu Cr Pt Mo Co Fe W Tc Ru Os Ir Rh Re Zn Al Hf Mn T=1000 K Increasing calculated compressibility, 1/GPa Zr Sc Y Figure 7.8: Diffusion coefficient of each of the 26 dilute Ni 31 X systems plotted as a function of increasing compressibility at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements. This plot shown in Figure 7.8 is also useful in showing the effect of the placement in the periodic table of the various transition row elements as a whole on the impurity diffusion in fcc Ni. Cluster grouping of elements are observed, for example, the four slowest diffusers are from the center of the 5d transition row (X=Re, W, Os, and Ir). The next five slowest diffusers (X=Mo, Cr, Tc, V, and Rh) are from the mid row 4d and 3d transition row elements, indicating a correlation to the relative impurity diffusion coefficient of Figure 7.5. Two previous

187 162 works semi-empirical works linked bulk modulus to activation energy, one by Patil et al. [176], and one by Toth et al. [177]. Conclusions from this work, which support the computational study of four impurity elements in hcp Mg of Ganeshan et al. [93], show that the elastic properties of the elements are directly related to the activation energy and inversely related to the impurity diffusion coefficients. Srikrishnan et al. [178] see fairly similar trends when the activation energy for self-diffusion was correlated to the number of d electrons, indicating that future study of the diffusion trends in this chapter could indicate how the diffusion is linked to the average number of bonds affected by the formation and migration of a vacancy. While bonding and electron configurations are beyond the scope of the present work, it is possible that studies on this topic would further define the behavior of transition metal solutes as impurity diffusers in any host system Diffusivity mechanisms While the previous section discussed the relationship between physical properties such as ionic radius, elastic properties, and diffusivity, a discussion on the diffusion mechanism that is a direct cause for impurity elements to be fast or slow diffusers in fcc Ni is desirable. To do this, diffusivity at T=1000 K is plotted as a function of the vacancy formation energy adjacent to the impurity, G f, the solute vacancy binding energy, G b, and the migration barrier for the diffusing impurity atom, G m, for each of the dilute Ni 31 X systems. It can be seen in Figure 7.9 that a higher vacancy formation energy adjacent to the solute atom is correlated to slower diffusers in fcc Ni. The relationship between vacancy formation energy and diffusivity does not appear to be linear, rather the slope of the correlation increases as vacancy formation energy adjacent to the solute increases. Figure 7.10 shows that lower solute-vacancy binding energy is also correlated to slower diffusers diffusers. The trend in the data is nearly a direct inverse of the relationship of diffusivity to vacancy formation energy, which makes sense since solute-vacancy binding energy enters in to the vacancy formation energy adjacent to the impurity. Finally in Figure 7.11, a higher migration barrier for impurity diffusion of the solute results in slower diffusers in fcc Ni. With the exception of the two fastest diffusers, Y and Zr, and Sc, a nearly decreasing linear relationship between diffusivity and the

188 163 migration barrier of the impurity atom is observed. It has the strongest correlation to diffusivity of the three properties of interest. This is important insight as it shows that the impurity diffusion coefficient in fcc Ni is most affected by the free energydifferencebetweenthetransitionstateofthew 2 jumpanditsinitialposition before the jump occurs. It also may be applicable to the impurity diffusion in other host systems. D 1000 (m 2 /sec) Y Sc 3d elements 4d elements 5d elements other Zr Vacancy formation energy, ev Cu Si Hf Pd TaMo Nb Zn Al V Mn Ti Fe Pt Cr Rh W Co Ru Ir Tc Os Re Figure 7.9: Diffusion coefficient of each of the 26 dilute Ni 31 X systems plotted as a function of increasing calculated vacancy formation energy at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements.

189 164 D 1000 (m 2 /sec) MoTa SiNbZn Hf V Ti Pd Fe Al CuMn Co Cr Pt W Tc Ru Rh Ir Re Os Zr 3d elements 4d elements 5d elements other Sc Solute-vacancy binding energy, ev Y Figure 7.10: Diffusion coefficient of each of the 26 dilute Ni 31 X systems plotted as a function of increasing solute-vacancy binding energy at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements. D 1000 (m 2 /sec) Y Zr Hf Ti Sc Solute migration energy, ev 3d elements 4d elements 5d elements other Nb Ta Si Mn Pd Al Zn Cu Mo Pt V FeCr Rh Ru Co W Tc Ir Os Re Figure 7.11: Diffusioncoefficientofeachofthe26diluteNi 31 Xsystemsplottedas a function of increasing migration barrier for impurity diffusion at 1000 K, colored by 3d (blue), 4d (red), and 5d (black) transition row elements.

190 Charge density analysis To understand how the presence of the impurity atom affects the diffusivity of each dilute Ni 31 X system, the electronic structure is examined for each w 2 jump when the impurity atom is at its maximum energy path. Six representative systems are chosen for this analysis, Re for being the slowest diffuser in Ni, Y for being the fastest diffuser in Ni, cu for diffusing at nearly the same speed as self-diffusion in Ni, Fe for being a magnetic impurity, and Ta and W to compare the effects of being next to each other on the periodic table but diffusing faster in Ni (Ta) and slower than Ni (W). Relative deformation charge density is plotted as the total charge density of the relaxed structure minus the initial charge density, which can be defined as the charge density of the same structure after only one electronic step, along the (a) b-axis and (b) c-axis, where the impurity atom is in the bottom left corner of the 2-D b-axis projection. This is very useful in showing how each impurity affects the structure when it is compared to Figure 5.11 showing the relative deformation charge density of self-diffusion in Ni. Also plotted is (c) a slice of the deformation charge density on the [202] plane, with the impurity atom in the center of the figure. The slice of the relative charge density is very useful to show how the four first nearest neighbors are affected by the presence of the impurity. The three versions of relative charge density described above are plotted in Figure 7.12 to Figure Figure 7.12: Relative deformation charge density of Re in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane. These six figures provide a significant amount of useful information about the

191 166 Figure 7.13: Relative deformation charge density of Y in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane. Figure 7.14: Relative deformation charge density of Cu in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane. Figure 7.15: Relative deformation charge density of Fe in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane.

192 167 Figure 7.16: Relative deformation charge density of Ta in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane. Figure 7.17: Relative deformation charge density of W in Ni shown as (a) 2-D projection down the b-axis, (b) 2-D projection down the c-axis, and (c) slice on the [202] plane. trends observed in the 26 impurity diffusion coefficients calculated in the present work. It is commonly believed that a denser charge density is indicative of stronger bonding between atoms [75], and that non-spherical distribution of charge density can hinder shear deformation [179], which could directly be linked to the elastic properties and correlation of the impurity diffusion coefficient to the compressibility of each dilute Ni 31 X system. To begin the analysis, it should be known that the total charge density of pure fcc Ni is nearly spherical surrounding each atomic position [75], while the magnetization charge density is shown in the present work to be a rounded cubic shape as shown in Figure The charge densities of the Cu impurity, which is an impurity similar in both size and compressibility when compared to Ni, is shown in Figure 7.14(a-c). It is observed that in the plane of