Zr,Hf) Carbides and Borides Revisited and Informed by the Calculation of Defect Formation Energies in Zrc

Home , Hafnium carbide, Zirconium carbide

Thermodynamic assessments of the (Zr,Hf) carbides and borides revisited and informed by the calculation of defect formation energies in ZrC

A thesis submitted in partial fulﬁlment for the degree of Doctor of Philosophy and Diploma of Imperial College

Theresa Davey

Under the supervision of Professor M.W. Finnis and Professor W.E. Lee

Thomas Young Centre Department of Materials Royal School of Mines Faculty of Engineering Imperial College London

April 2017

Abstract

Some transition metal carbides and borides are ultra-high temperature ceramic compounds that have application in the nuclear and aerospace industries. The existing thermodynamic assessments of the systems containing the hafnium and zirconium carbides and borides were examined, along with all available experimental and theoretical data. One of the challenges of thermodynamic modelling is in accurately describing the point defects in ordered compounds. In the transition metal carbides and borides, vacancies and substitutional defects are common and play a significant role in determining the material behaviour. Hafnium and zirconium are very difficult to separate, and as such there is always some level of contamination in these compounds. Despite this, there is very limited quantitative information available about the effect that this contamination has on the properties of the material. Using first principles calculations of substitutional point defects in UHTC compounds, the CALPHAD modelling in existing thermodynamic assessments of the ternary boron-hafnium-zirconium and carbon-hafnium-zirconium systems was considered and areas for improvement were identified using the first principles insights. Hafnium and zirconium carbide have a wide range of stoichiometry facilitated by carbon vacancies. Developments to first principles techniques have allowed calculations of properties such as the vacancy formation energy that should be considered in thermodynamic assessment, but are currently not explicitly considered in the modelling. Using the carbon-zirconium system as an example, a method was developed by which the vacancy formation energy and vacancy-vacancy interaction energies can be considered directly in an existing CALPHAD optimisation without losing any of the information already encoded within it. This was applied to a thermodynamic assessment in the literature and to a new assessment of the carbon-zirconium system that was performed incorporating new experimental insights. 4 Declaration of Originality

I declare that all of the work presented in this thesis is my own or is otherwise referenced appropriately.

The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work.

5 6 Publications, Presentations, and Awards

Publications

• Improved method of calculating ab initio high-temperature thermodynamic properties with application to ZrC. Andrew Ian Duff, Theresa Davey, Do- minique Korbmacher, Albert Glensk, Blazej Grabowski, JörgNeugebauer, and Michael W. Finnis. Phys. Rev. B 91, 214311 (2015) • Using a defect-centric approach to include point defect calculations in the thermodynamic modelling of zirconium carbide. Theresa Davey, Thomas A. Mellan, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis. In preparation. • An updated thermodynamic assessment of the carbon-zirconium system with first principles insights. Theresa Davey, Thomas A. Mellan, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis. In preparation. • A review of all experimental and first principles data related to the thermodynamic description of the system boron-carbon-hafnium-zirconium. Theresa Davey, Suzana G. Fries, Michael W. Finnis. In preparation. • Chair’s update on PCSA activities and welcome to the student ACerS Bulletin. Theresa Davey. Bulletin of the American Ceramic Society, in press. • Expanding my ceramic networks to reach around the world. Theresa Davey. Bulletin of the American Ceramic Society, 95(2), 48 (2016). • United or divided? A student perspective from the United Kingdom. Theresa Davey. Bulletin of the American Ceramic Society, 93(9), 24 (2014).

7 Conference contributions

• CALPHAD assessment of the carbon-hafnium-zirconium system. Theresa Davey, Thomas A. Mellan, Suzana G. Fries, Michael W. Finnis – accepted for oral presentation at MS&T 2017, October 2017 (Pittsburgh, USA) • Thermodynamic assessment of the C-Zr system. Theresa Davey, Thomas A. Mellan, Suzana G. Fries, Michael W. Finnis – accepted for oral presentation at CALPHAD XLVI, June 2017 (Saint-Malo, France) • Phase diagram assessment of the carbon-zirconium system. Theresa Davey, Andrew I. Duff, Thomas A. Mellan, Suzana G. Fries, Michael W. Finnis – oral presentation at 1DRAC meeting, December 2016 (London, UK) • CALPHAD assessment of the carbon-zirconium system. Theresa Davey, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis – oral presentation at MS&T 2016, October 2016 (Salt Lake City, USA) • Using DFT Calculations of the Vacancy Formation Energy to Inform the As- sessment of the C-Zr Phase Diagram Theresa Davey, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis – oral presentation at MS&T 2016, October 2016 (Salt Lake City, USA) • Using DFT data to inform the CALPHAD assessment of the carbon-zirconium phase diagram Theresa Davey, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis – poster presentation at the International Workshop on Theory and Modelling of Materials In Extreme Environments, September 2016 (Oxford- shire, UK) • Using DFT calculations of the vacancy formation energy to inform the assessment of the C-Zr phase diagram. Theresa Davey, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis – oral presentation at TOFA 2016, September 2016 (Santos, Brazil) • Using DFT data to inform the assessment of the C-Zr phase diagram. Theresa Davey, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis – oral presentation at CALPHAD XLV, May 2016 (Awaji Island, Japan) • Including DFT data in phase diagram calculations for zirconium carbide. Theresa Davey, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis – oral presentation at the DPG Spring Meeting, March 2016 (Regensberg, Germany)

8 • CALPHAD examination of the system boron-carbon-hafnium-zirconium. Theresa Davey, Suzana G. Fries, Andrew I. Duff, Michael W. Finnis – oral presentation at MS&T, October 2015 (Columbus, Ohio, USA) • Fully anharmonic first principles data in the assessment of the B-C-Hf-Zr system. Theresa Davey, Andrew I. Duff, Suzana G. Fries, Michael W. Finnis – oral presentation at CALPHAD XLIV, June 2015 (Loano, Italy) • Phase diagrams in the system boron-carbon-hafnium-zirconium. Theresa Davey, Suzana G. Fries, Michael W. Finnis – oral presentation at UHTC III, April 2015 (Surfer’s Paradise, Australia) • Revisiting Boron-Carbon-Hafnium-Zirconium Thermodynamics. Theresa Davey, Suzana G. Fries, Michael W. Finnis, Alan Dinsdale – poster at TMS, February 2015 (Orlando, Florida)

Other presentations

• Phase diagrams in the boron-carbon-hafnium-zirconium system. Theresa Davey et al.– oral presentations at XMat Research Meetings, October 2013 – April 2017 • Phase stabilility in the B-C-Hf-Zr system. Theresa Davey et al. – oral presentation in Postgraduate Research Seminars, Department of Materials, Impe- rial College London, June 2015 • Student and early-stage career opportunities in the American Ceramic Society. Theresa Davey – oral presentation in I UK-ACerS, March 2017

Scholarships and awards

• Scholarship to attend CALPHAD XLVI – awarded by the Foundation of Ap- plied Thermodynamics (Stiftelsen f¨orTill¨ampadTermodynamik, STT) – May 2017 • Finalist (Sapphire) in the Graduate Excellence in Materials Science Awards – awarded by the Basic Science Division of the American Ceramic Society – October 2016

9 • Scholarship to attend MS&T 2016 – awarded by the Refractory Ceramics Di- vision of the American Ceramic Society – October 2016 • Scholarship to attend CALPHAD XLV – awarded by the Foundation of Applied Thermodynamics (Stiftelsen f¨orTill¨ampad Termodynamik, STT) – May 2016 • Scholarship to attend ICACC 2016 – awarded by the Engineering Ceramics Division of the American Ceramics Society – January 2016 • Scholarship to attend the American Ceramic Society’s Winter Workshop – awarded by the Basic Science Division of the American Ceramic Society – January 2016 • Scholarship to attend UNITECR 2015 – awarded by the Refractory Ceramics Division of the American Ceramics Society – September 2015

10 Acknowledgements

I would like to thank the Engineering and Physical Sciences Research Council for funding this work through EPSRC Project Grant EP/K008749/2 (XMat: Materials Systems for Extreme Environments). I am grateful to Mike Finnis for giving me guidance when I needed it over the last four years, while giving me the freedom to pursue the avenues that I wanted to explore, and for never losing patience at my consistently last-minute attitude. I would also like to thank Bill Lee for giving me the opportunities to expand as a professional and leader, while emphasising the importance of actually getting the research done as well. I would like to extend my sincere thanks to Suzana Fries for teaching me about CALPHAD and for all her care and support in all aspects of my life over the last three years. I would like to thank everyone at ICAMS, Ruhr-UniversitätBochum for their frequent hospitality and the Alexander von Humboldt Foundation for support during my many visits. In particular I would like to thank Silvana (Daisy) Tumminello for coming on this CALPHAD journey with me, and Mauro Palumbo for helping with my early understanding of first principles calculations and curve fitting. I wish to thank Andy Duff and Tom Mellan for their invaluable guidance in my first principles education, and for providing calculations without which this project would not have succeeded. I would also like to thank Eugenio Zapata Solvas for giving me access to his experimental results as soon as they were measured. I will be ever grateful to Darakshan Khan and Emma Warriss for going out of their way to make sure that I was always taken care of in the Department of Materials. This project was enhanced by Armando Fernandez Guillermet, Gabriele Cacciamani, and Peter Rogl who both provided helpful discussion, literature, and insight, and I am grateful for their very thorough and well documented work over the last few

11 decades that made this project significantly more pleasant to do. I would also like to thank Nathalie Dupin for providing critique and suggestions helpful to hacking the Thermo-Calc input files to make it do my bidding. I am also thankful to Bill Fahrenholtz for interesting discussion on UHTCs and for providing experimental data. My Nomad Chemists, Cassie Marker, Carl-Magnus Arvhult, and Florian Tang have made the CALPHAD community feel like family, and I will always be grateful for their advice and unwavering support both professionally and personally. I would like to thank Joseph Letts, Philippa Skett, Grace Rahman, and Lef Apos- tolakis for nurturing the over-opinionated writer within me over the last four years, and for giving me a creative outlet in which to develop my writing abilities. Mattin Mir-Tahmasebi has been invaluable in teaching me about the importance of a clear style guide, which has made this thesis much more readable. A special thanks to all the delegates in the American Ceramic Society’s President’s Council of Student Advisors, particularly to Lisa Rueschhoff for being my number one #PCSAbae and academic inspiration. I am indebted to Rebecca Davenport for being a constant source of encouragement and always knowing exactly how to help, and to Kella Kapnisi for being my Materials best friend and providing coffee and sanity-restoring pep talks over the last three and a half years. I also thank Simran Kukran for being a constant source of personal inspiration, who compensated for being the most distracting person on the planet by making sure that I was always well looked-after. I am hugely thankful Max Boleininger and Aaron Thong for listening patiently every time I ranted incoherently about phase diagram optimisations in the kitchen (and for actually providing helpful suggestions), and to Chiara Liverani for gently making my time writing up more enjoyable. I am beyond grateful to everyone at Imperial College Women’s Rugby Club who has helped shape me as a person throughout my PhD, and has given me the grit to persevere when I felt inadequate. Finally, I would like to thank Beth Sheldon and Ian Lammin for being generous beyond any reasonable expectation, and providing a home for me during the writing process. This thesis would not have been possible without their support.

12 Contents

1 Introduction and Motivation 33

2 Overview of the CALPHAD method 37 2.1 Introduction ...... 37 2.2 Classical Thermodynamics ...... 38 2.2.1 Connection to statistical thermodynamics ...... 39 2.3 Gibbs energy models ...... 41 2.3.1 General form of the Gibbs energy parameterisation ...... 41 2.3.2 Phases with ﬁxed composition ...... 42 The Kopp-Neumann rule ...... 43 2.3.3 Modelling using sublattices ...... 44 2.3.4 Composition dependence ...... 45 2.3.5 Physical phenomena ...... 48 2.3.6 Solution Models ...... 48 The Compound Energy Formalism ...... 48 Ideal substitutional solution models ...... 50 2.3.7 Excess Gibbs energy ...... 51 Binary contributions ...... 51 Ternary contributions ...... 53 Higher order contributions ...... 53 Substitutional regular solution models ...... 54 2.4 Assessment Methodology ...... 55 2.4.1 Useful experimental data ...... 57 Experimental phase diagram data ...... 57 Thermodynamic data ...... 58 Other experimental data ...... 58 Crystallographic data ...... 58 First principles calculations ...... 58

13 2.5 Overview of optimisation using the PARROT module of Thermo-Calc 59 2.6 Summary ...... 60

3 First principles calculations 62 3.0.1 Schr¨odinger’s equation ...... 62 3.0.2 Hartree-Fock Theory ...... 64 3.0.3 Density Functional Theory (DFT) ...... 65 Hohenberg-Kohn Theorems ...... 65 Kohn-Sham approach ...... 67 3.0.4 Exchange-correlation functionals ...... 68 Local Density Approximation ...... 68 Generalised Gradient Approximation GGA ...... 69 3.0.5 Plane Waves, k-points and cutoﬀ energies ...... 69 3.0.6 Pseudopotentials ...... 71

4 A review of the literature relating to thermodynamic assessments of the B-C-Hf-Zr system 72 4.1 Introduction ...... 72 4.2 The (hafnium, zirconium) diboride and carbide phases ...... 76 4.2.1 Hafnium diboride ...... 76 Phases in the boron-hafnium system ...... 76 Experimental investigations ...... 76 Thermodynamic data ...... 77 First principles investigations ...... 78 4.2.2 Zirconium diboride ...... 79 Phases in the boron-zirconium system ...... 79 Experimental investigations ...... 80 Thermodynamic data ...... 82 First principles investigations ...... 83 4.2.3 Hafnium carbide ...... 85 Phases in the carbon-hafnium system ...... 85 Experimental investigations ...... 86 Thermodynamic data ...... 86 First principles investigations ...... 87 4.2.4 Zirconium carbide ...... 88 4.3 The carbon-zirconium system ...... 89 4.3.1 Overview of phase diagram data ...... 89 Investigations of the melting temperature of zirconium carbide 89

14 Investigations of the boundaries of the homogenous region . . 92 Investigations of the solubility of carbon in zirconium . . . . . 93 Investigations of the solubility of zirconium in carbon . . . . . 93 Investigations of the ZrC-C eutectic ...... 94 4.3.2 Thermodynamic data ...... 95 Enthalpy of formation of zirconium carbides ...... 95 Heat capacity of zirconium carbides ...... 96 Heat content of zirconium carbides ...... 97 Activity as a function of composition ...... 98 Activity as a function of temperature ...... 99 4.3.3 Review of ﬁrst principles data ...... 99 Thermodynamic properties ...... 99 Point defects ...... 100

5 Examining the eﬀect of hafnium and zirconium substitutional point defects on the (Hf,Zr) carbides and borides 102 5.1 Background ...... 102 5.2 Investigations of purity ...... 103 5.2.1 The eﬀect of transition metal contamination in the existing CALPHAD models ...... 104 5.3 First principles calculations ...... 110 5.3.1 Borides ...... 110 Zirconium diboride ...... 110 Hafnium diboride ...... 112 Pure element calculations ...... 116 Lattice parameters ...... 118 Enthalpy of formation ...... 120 Improvements to the CALPHAD modelling ...... 122 5.3.2 Carbides ...... 123 Zirconium carbide ...... 125 Hafnium carbide ...... 129 Pure element calculations ...... 131 Lattice parameters ...... 133 Enthalpy of formation ...... 135 Improvements to the CALPHAD modelling ...... 139 5.4 Summary and Conclusions ...... 139

6 A defect-centric approach to modelling the vacancy formation en-

15 ergy in a thermodynamic assessment 142 6.1 Introduction ...... 143 6.1.1 The CALPHAD modelling of C-Zr ...... 144 6.1.2 The thermodynamic modelling of the γ phase ...... 145 6.2 Expressing a “reference” vacancy formation energy using Gibbs energy functions ...... 146 6.2.1 Defects in zirconium carbide ...... 146 6.2.2 The reference defect formation energy ...... 148 6.3 A defect-centric approach to the excess energy ...... 150 6.3.1 Separating defect-related quantities in the excess energy parameterisation ...... 150 6.3.2 The need for change in the excess energy parameterisation . . 151 6.4 First principles calculations of the vacancy formation energy . . . . . 152 6.4.1 Existing studies in the literature ...... 152 6.4.2 DFT calculations in this study ...... 153 Choice of graphite function in DFT VFE ...... 155 Details of the calculations ...... 158 Analysis of the DFT calculations ...... 161 Comparison with the existing CALPHAD assessment . . . . . 164 6.5 Including the vacancy formation energy explicitly in a CALPHAD assessment ...... 169 6.5.1 Existing CALPHAD assessment from the literature ...... 170 Comparison with experimental data ...... 170 6.5.2 Adding constraint without optimisation ...... 173 Comparison with experimental data ...... 173 The excess energy parameters ...... 176 6.5.3 Optimising γ, liquid, and β-Zr excess parameters ...... 178 Choice of parameters to optimise ...... 178 Parameterisation of the γ phase ...... 179 Parameterisation of the liquid phase ...... 179 Parameterisation of the β phase ...... 180 Parameters varied in the optimisation ...... 180 Data used in the optimisation ...... 180 Process of reoptimisation ...... 182 Comparison with experimental data ...... 182 Quantifying the quality of the reoptimisation ...... 187 Gibbs energies of each phase ...... 188 The lattice stability of FCC-Zr ...... 191

16 Comparing the excess energy parameters ...... 192 6.6 Summary and conclusions ...... 197 6.6.1 Further work ...... 198 6.6.2 The future of the CALPHAD approach ...... 199

7 Thermodynamic assessment of the carbon-zirconium phase diagram using the CALPHAD approach 202 7.1 Introduction ...... 202 7.2 Thermodynamic modelling ...... 203 7.2.1 Phases in the carbon-zirconium system ...... 203 7.2.2 Gibbs energy parameterisation ...... 204 7.3 Experimental information available ...... 207 7.3.1 Unary Gibbs energy functions ...... 207 7.3.2 Phase diagram data ...... 207 The γ+liquid+graphite eutectic temperature and composition 207 The γ + liquid + β eutectic temperature and composition . . 208 The α + β + γ eutectic temperature and composition . . . . . 208 The γ and graphite liquidus ...... 208 The solidus ...... 209 7.3.3 Thermodynamic data ...... 210 Heat capacity of zirconium carbides ...... 210 Heat content ...... 212 Activity ...... 212 Enthalpy of formation at 298.15 K ...... 213 7.4 Determination of parameters ...... 213 7.4.1 Choice of parameters to vary ...... 213 7.4.2 Optimisation strategy ...... 216 7.5 Results and discussion/comparison with experiment/previous assessment...... 217 7.5.1 Invariant temperatures ...... 218 7.5.2 Comparison with experimental data ...... 219 Phase diagram ...... 219 Thermodynamic data ...... 224 7.6 Vacancy formation energy ...... 231 7.7 Summary and conclusions ...... 237

8 Conclusions and future direction 240

A Assessed Gibbs energy functions 246

17 A.1 Assessed Gibbs energy functions incorporating a calculated vacancy formation energy into the CALPHAD assessment of the carbon-zirconium system by Fernandez Guillermet [66] ...... 246 A.2 Assessed Gibbs energy functions from the CALPHAD assessment of the carbon-zirconium system presented in Chapter 7 ...... 249 A.3 Assessed Gibbs energy functions incorporating a calculated vacancy formation energy into the CALPHAD assessment of the carbon-zirconium system presented in Chapter 7 ...... 252

B Constraining the VFE within Thermo-Calc input files 256 B.1 Constraining the vacancy formation energy within a TDB file . . . . 256 B.2 Constraining the vacancy formation energy within a setup file for an assessment ...... 258

18 List of Figures

2.1 A ﬂowchart from Lukas et al. [161] showing the assessment and optimisation procedure used in the CALPHAD approach to produce a thermodynamic description. 56

4.1 Phase diagram of the boron-hafnium system from the CALPHAD assessment by Bittermann and Rogl [28]...... 77 4.2 Phase diagram of the boron-zirconium system from the CALPHAD assessment by Duschanek and Rogl [55]...... 81 4.3 Phase diagram of the carbon-hafnium system from the CALPHAD assessment by Bittermann and Rogl [27]...... 85 4.4 Phase diagram of the carbon-zirconium system from the CALPHAD assessment by Fernandez Guillermet [66]...... 88 4.5 Phase diagram of the carbon-zirconium system from the CALPHAD assessment by Fernandez Guillermet [66] shown with some of the reported experimental phase diagram superimposed [226, 219, 216, 6, 242]...... 90

5.1 Experimentally measured heat capacity of zirconium diboride with varying amounts of hafnium dopant, from Lonergan et al. [159]...... 104 5.2 Phase diagram of the boron-zirconium system from the assessment by Duschanek and Rogl [55]...... 105 5.3 Phase diagram of the boron-hafnium system from the assessment by Bittermann and Rogl [28]...... 105 5.4 Isothermal slice of the boron-hafnium-zirconium system at 1000 K from the assessment by Cacciamani et al. [41]...... 106

5.5 Heat capacity of ZrB2 from the assessment by Duschanek and Rogl [55] and from the assessment by Cacciamani et al. [41]...... 107

5.6 Heat capacity of HfB2 from the assessment by Bittermann and Rogl [29]. .... 107

5.7 Heat capacity of ZrB2 from the assessment by Duschanek and Rogl [55] and from

the assessment by Cacciamani et al. [41], and of HfB2 from the assessment by Bittermann and Rogl [29]...... 108

19 5.8 Experimentally measured heat capacity data of ZrB2 with varying hafnium doping from Lonergan et al. [159] with the equivalent heat capacity values from the CALPHAD assessment of the B-Hf-Zr system by Cacciamani et al. [41]...... 109 5.9 Unit cell for the zirconium diboride phase, where the boron is shown in blue and zirconium is shown in pink, drawn using VESTA [177]...... 110 5.10 3x3x2 supercell for the zirconium diboride phase, where the boron is shown in blue and zirconium is shown in pink, drawn using VESTA [177]...... 111 5.11 3x3x2 supercell for the zirconium diboride phase with a single hafnium substitution at a zirconium site, where the boron is shown in blue, zirconium is shown in pink, and hafnium is shown in orange, drawn using VESTA [177]...... 112

5.12 The energy difference between the perfect ZrB2 supercell and the ZrB2 supercell with a single hafnium substitutional defect in a zirconium site, calculated at several different cutoff energies, using LDA and GGA exchange-correlation functionals with a 7x7x9 k-point mesh...... 113 5.13 Unit cell for the hafnium diboride phase, where the boron is shown in blue and hafnium is shown in orange, drawn using VESTA [177]...... 113 5.14 3x3x2 supercell for the hafnium diboride phase, where the boron is shown in blue and hafnium is shown in orange, drawn using VESTA [177]...... 114 5.15 3x3x2 supercell for the hafnium diboride phase with a single zirconium substitution at a zirconium site, where the boron is shown in blue, zirconium is shown in pink, and hafnium is shown in orange, drawn using VESTA [177]...... 115

5.16 The energy difference between the perfect HfB2 supercell and the HfB2 supercell with a single zirconium substitutional defect in a hafnium site, calculated at several different cutoff energies, using LDA and GGA exchange-correlation functionals with a 7x7x9 k-point mesh...... 115 5.17 Unit cell of HCP hafnium, drawn using VESTA [177]...... 116 5.18 Unit cell of HCP zirconium, drawn using VESTA [177]...... 117 5.19 Unit cell of β-rhombohedral boron, drawn using VESTA [177]...... 118

5.20 Enthalpy of formation of ZrB2 and HfB2 with a single substitutional hafnium and zirconium defect respectively calculated with DFT using LDA or GGA exchange- correlation functionals at 0 K, compared with a quasi-binary representation of the enthalpy of formation of the γ NaCl structure carbide phase from the CALPHAD assessment by Cacciamani et al. [41] as a function of hafnium content at 298.15 K. 123 5.21 Phase diagram of the carbon-zirconium system from the assessment by Fernandez Guillermet [66]...... 124 5.22 Phase diagram of the carbon-hafnium system from the assessment by Bittermann and Rogl [27]...... 124

20 5.23 Isothermal slice of the carbon-hafnium-zirconium system at 1000 K from the assessment by Bittermann and Rogl [29]...... 125 5.24 Unit cell for the zirconium carbide phase, where the carbon is shown in green and zirconium is shown in pink, drawn using VESTA [177]...... 126 5.25 2x2x2 supercell for the zirconium carbide phase, where the carbon is shown in green and zirconium is shown in pink, drawn using VESTA [177]...... 126 5.26 2x2x2 supercell for the zirconium carbide phase with a single hafnium substitutional defect at a zirconium site, where the carbon is shown in green, zirconium is shown in pink, and hafnium is shown in orange, drawn using VESTA [177]. ... 127 5.27 The energy difference between the perfect ZrC supercell and the ZrC supercell with a single hafnium substitutional defect in a zirconium site, calculated at several different cutoff energies, using an LDA exchange-correlation functional with a 5x5x5 k-point mesh...... 128 5.28 The energy difference between the perfect ZrC supercell and the ZrC supercell with a single hafnium substitutional defect in a zirconium site, calculated with 5x5x5 and 7x7x7 k-point meshes, using an LDA exchange-correlation functional and a cutoff energy of 600 eV...... 128 5.29 Unit cell for the hafnium carbide phase, where the carbon is shown in green and hafnium is shown in orange, drawn using VESTA [177]...... 129 5.30 2x2x2 supercell for the hafnium carbide phase, where the carbon is shown in green and hafnium is shown in orange, drawn using VESTA [177]...... 130 5.31 2x2x2 supercell for the hafnium carbide phase with a single zirconium substitutional defect at a hafnium site, where the carbon is shown in green, hafnium is shown in orange, and zirconium is shown in pink, drawn using VESTA [177]. .. 131 5.32 The energy difference between the perfect HfC supercell and the HfC supercell with a single hafnium substitutional defect in a zirconium site, calculated with 5x5x5 and 7x7x7 k-point meshes, using an LDA exchange-correlation functional and a cutoff energy of 600 eV...... 132 5.33 The energy difference between the perfect HfC supercell and the HfC supercell with a single hafnium substitutional defect in a zirconium site, calculated at several different cutoff energies, using an LDA exchange-correlation functional with a 5x5x5 k-point mesh...... 133 5.34 Unit cell of carbon in the diamond structure, drawn using VESTA [177]. .... 134 5.35 Enthalpy of formation of ZrC and HfC with a single substitutional hafnium and zirconium defect respectively calculated with DFT using LDA or GGA exchange- correlation functionals at 0K, compared with a quasi-binary representation of the enthalpy of formation of the γ NaCl structure carbide phase from the CALPHAD assessment by Bittermann and Rogl [29] as a function of hafnium content at 298.15 K.137

21 6.1 Several Gibbs energies for various diamond and graphite are shown as a function of temperature from 0 K to 4000 K (or 298 K to 4000 K where the range of validity is limited). The energy of diamond calculated using DFT is shown in red. All other functions (a)-(d) represent the graphite phase, determined using different methods. (a) The DFT diamond energy augmented by the graphite-diamond energy difference from SGTE [52] is shown in green. (b) The SGTE diamond Gibbs energy [52] offset by the room temperature DFT diamond energy plus the SGTE graphite-diamond difference at room temperature is shown in blue. (c) The CALPHAD Gibbs energy function from Hallstedt et al. [93] offset by the zero temperature DFT diamond energy plus the SGTE graphite-diamond difference at room temperature is shown in grey. (d) The CALPHAD Gibbs energy function from Hallstedt et al. [93] offset by the room temperature DFT diamond energy plus the SGTE graphite-diamond difference at room temperature is shown in black.156 6.2 The VFE in eV/defect from the 2x2x2, 3x3x3, and 4x4x4 unit cell supercells at 0 K against the volume of the stoichiometric zirconium carbide supercell are shown in blue. A linear fit is applied to these points, and extrapolated to represent the formation energy of one vacancy in the bulk, where the extrapolated vacancy formation is shown in green...... 160 6.3 The 0K reference VFE from the 2x2x2 DFT calculations with zero point effects are shown in grey, with finite size correction added in green, with temperature dependence from the quasiharmonic approximation added in blue, and with temperature dependent electronic contribution added in red...... 161 6.4 The vacancy formation energy extracted from the previous CALPHAD assessment [66] using equation 6.20 is shown in green. The calculated vacancy formation energy from this study, including quasiharmonic and electronic contributions, and a finite size correction is shown in red. DFT calculations for the 0K vacancy formation energy from the literature (given in Table 6.1) are also shown. .... 162 0 γ 1 γ 6.5 LZr:C,Va + LZr:C,Va is the coefficient of the linear yVa term in the excess energy, 0 γ 1 γ − LZr:C,Va − 3 LZr:C,Va is the coefficient of the quadratic yVa term in the excess 1 γ energy, and 2 LZr:C,Va is the coefficient of the cubic yVa term in the excess energy. The magnitude of each of these coefficients from the CALPHAD assessment by Fernandez Guillermet [66] is shown as a function of temperature from room temperature to 3700 K (the congruent melting temperature of zirconium carbide). 165

22 6.6 The three contributions to the excess energy from equation 6.16 are shown as a

function of temperature for diﬀerent vacancy concentrations between yVa = 0.02

and yVa = 0.10 from room temperature to 5000 K. The red curves represent the vacancy formation energy term, the green curves represent the vacancy-vacancy interaction energy terms, and the blue curves represent the higher order vacancy interaction energy terms...... 167 6.7 The three contributions to the excess energy from equation 6.16 are shown as a function of vacancy composition for diﬀerent temperatures between 300 K and

4000 K for vacancy concentrations between yVa = 0 and yVa = 0.5. The red curves represent the vacancy formation energy term, the green curves represent the vacancy-vacancy interaction energy terms, and the blue curves represent the higher order vacancy interaction energy terms...... 168 6.8 The phase diagram of the carbon-zirconium system from the CALPHAD assessment by Fernandez Guillermet [66], shown with the experimental phase diagram data used in this assessment superimposed [226, 219, 216, 242, 282, 6]...... 171 6.9 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.497487 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66], with experimental data measured by Turchanin and Fesenko [254]...... 172 6.10 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.431818 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66], with experimental data measured by Turchanin and Fesenko [254]...... 172 6.11 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.408284 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66], with experimental data measured by Turchanin and Fesenko [254]...... 172 6.12 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.494949 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66], with experimental data measured by Bolgar et al. [33]...... 172 6.13 The phase diagram of the carbon-zirconium system created from parameters from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy by modifying the ﬁrst order excess energy parameter in the γ phase, without changing any other parameters. The experimental phase diagram data used in the assessment of the original phase diagram is superimposed [226, 219, 216, 242, 282, 6]...... 174 6.14 Enthalpy of zirconium carbide for x(C) = 0.497487 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, with experimental data measured by Turchanin and Fesenko [254]...... 175

23 6.15 Enthalpy of zirconium carbide for x(C) = 0.431818 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, with experimental data measured by Turchanin and Fesenko [254]...... 175 6.16 Enthalpy of zirconium carbide for x(C) = 0.408284 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, with experimental data measured by Turchanin and Fesenko [254]...... 175 6.17 Enthalpy of zirconium carbide for x(C) = 0.494949 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, with experimental data measured by Bolgar et al. [33]...... 175 1 γ 6.18 The ﬁrst order γ excess parameter, LZr:C,Va from the CALPHAD assessment by Fernandez Guillermet [66] is in red, and the same parameter after it has been constrained such that the excess energy of the γ phase reproduces the calculated reference vacancy formation energy is shown in green, between 298 K and 3700 K (the congruent melting temperature of zirconium carbide)...... 177 6.19 The phase diagram of the carbon-zirconium system created from parameters from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy by modifying the ﬁrst order excess energy parameter in the γ phase, and reoptimising the excess parameters in the γ, liquid, and β phases. The experimental phase diagram data used in the assessment of the original phase diagram is superimposed [226, 219, 216, 242, 282, 6]. .... 183 6.20 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.497487 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, and reoptimising the excess parameters in the γ, liquid, and β phases, with experimental data measured by Turchanin and Fesenko [254]...... 185 6.21 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.431818 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, and reoptimising the excess parameters in the γ, liquid, and β phases, with experimental data measured by Turchanin and Fesenko [254]...... 185 6.22 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.408284 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, and reoptimising the excess parameters in the γ, liquid, and β phases, with experimental data measured by Turchanin and Fesenko [254]...... 186

24 6.23 Enthalpy of zirconium carbide as a function of temperature for x(C) = 0.494949 from 298 K to 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy, and reoptimising the excess parameters in the γ, liquid, and β phases, with experimental data measured by Bolgar et al. [33]...... 186 6.24 Gibbs energy of each phase in the carbon-zirconium system as a function of carbon concentration at 2000 K from the CALPHAD assessment by Fernandez Guillermet [66]...... 189 6.25 Gibbs energy of each phase in the carbon-zirconium system as a function of carbon concentration at 2000 K from the reoptimised assessment incorporating the reference vacancy formation energy function...... 189 6.26 Gibbs energy of each phase in the carbon-zirconium system as a function of carbon concentration at 2500 K from the CALPHAD assessment by Fernandez Guillermet [66]...... 189 6.27 Gibbs energy of each phase in the carbon-zirconium system as a function of carbon concentration at 2500 K from the reoptimised assessment incorporating the reference vacancy formation energy function...... 189 6.28 Gibbs energy of each phase in the carbon-zirconium system as a function of carbon concentration at 3000 K from the CALPHAD assessment by Fernandez Guillermet [66]...... 190 6.29 Gibbs energy of each phase in the carbon-zirconium system as a function of carbon concentration at 3000 K from the reoptimised assessment incorporating the reference vacancy formation energy function...... 190 0 γ 1 γ 6.30 ( LZr:C,Va + LZr:C,Va) (red) is the coefficient of the linear yVa term in the excess 0 γ 1 γ energy, (− LZr:C,Va − 3 LZr:C,Va) (green) is the coefficient of the quadratic yVa 1 γ term in the excess energy, and (2 LZr:C,Va) (blue) is the coefficient of the cubic yVa term in the excess energy. The magnitude of each of these coefficients from the CALPHAD assessment by Fernandez Guillermet [66] is shown as a function of temperature from room temperature to 3700 K...... 193

6.31 The coeﬃcients of the linear (red), quadratic (green), and cubic (blue) in yVa terms of the excess energy of the γ phase following the reoptimisation of the excess γ, liquid, and β Gibbs energy parameters from the CALPHAD assessment from Fernandez Guillermet [66] under the constraint of reproducing the calculated reference vacancy formation energy are shown as a function of temperature from room temperature to 3700 K...... 193

25 6.32 The three contributions to the excess energy from equation 6.16 are shown as a

function of temperature for diﬀerent vacancy concentrations between yVa = 0.02

and yVa = 0.10 from room temperature to 5000 K, from the CALPHAD assessment by Fernandez Guillermet [66]...... 194 6.33 The three contributions to the excess energy from equation 6.16 are shown as

a function of temperature for diﬀerent vacancy concentrations between yVa =

0.02 and yVa = 0.10 from room temperature to 5000 K, from the reoptimised assessment. The red curves represent the vacancy formation energy term, the green curves represent the vacancy-vacancy interaction energy terms, and the blue curves represent the higher order vacancy interaction energy terms. These can be compared with the curves from the original CALPHAD optimisation in Figure 6.32.194 6.34 The three contributions to the excess energy from equation 6.16 are shown as a function of vacancy concentration at diﬀerent temperatures between300 K and

4000 K, between yVa = 0 and yVa = 0.5 from room temperature to 5000 K, from the CALPHAD assessment by Fernandez Guillermet [66]...... 196 6.35 The three contributions to the excess energy from equation 6.16 are shown as a function of vacancy concentration at diﬀerent temperatures between300 K and

4000 K, between yVa = 0 and yVa = 0.5 from room temperature to 5000 K, from the reoptimised assessment. The red curves represent the vacancy formation energy term, the green curves represent the vacancy-vacancy interaction energy terms, and the blue curves represent the higher order vacancy interaction energy terms. These can be compared with the curves from the original CALPHAD optimisation in Figure 6.34...... 196

7.1 Heat capacity of stoichiometric zirconium carbide as a function of temperature from the CALPHAD assessment by Fernandez Guillermet [66], DFT calculations by Duﬀ et al. [54], and experimentally measured values [265, 257, 170, 172, 199, 134, 275]...... 211 7.2 Carbon-zirconium phase diagram from the assessment by Fernandez Guillermet [66].218 7.3 Carbon-zirconium phase diagram from the assessment shown in this work. .... 218 7.4 Phase diagram of the carbon-zirconium system as assessed in this study, with experimental data superimposed [114, 135, 96, 219, 216, 226, 6]...... 219 7.5 C-Zr phase diagram from the thermodynamic assessment by Fernandez Guillermet [66] with experimental data superimposed...... 220 7.6 C-Zr phase diagram from this thermodynamic assessment with experimental data superimposed...... 220 7.7 Heat content from Fernandez Guillermet [66] with experimental data from Bolgar et al. [33]...... 225

26 7.8 Heat content from the assessment in this study with experimental data from Bolgar et al. [33]...... 225 7.9 Heat content from Fernandez Guillermet [66] with experimental data from Tur- chanin and Fesenko [254]...... 225 7.10 Heat content from the assessment in this study with experimental data from Tur- chanin and Fesenko [254]...... 225 7.11 Heat content from Fernandez Guillermet [66] with experimental data from Tur- chanin and Fesenko [254]...... 226 7.12 Heat content from the assessment in this study with experimental data from Tur- chanin and Fesenko [254]...... 226 7.13 Heat content from Fernandez Guillermet [66] with experimental data from Tur- chanin and Fesenko [254]...... 226 7.14 Heat content from the assessment in this study with experimental data from Tur- chanin and Fesenko [254]...... 226 7.15 Heat content of stoichiometric zirconium carbide from Fernandez Guillermet [66] with experimental data from Kantor and Fomichev [119]...... 227 7.16 Activity of zirconium as a function of composition at 2103 K from the assessment by Fernandez Guillermet [66], with experimental data from Storms and Griﬃn [242].228 7.17 Activity of zirconium as a function of composition at 2103 K from the assessment in this study, with experimental data from Storms and Griﬃn [242]...... 228 7.18 Activity of zirconium as a function of composition at 2500 K from the assessment by Fernandez Guillermet [66], with experimental data from Andrievskii et al. [14]. 228 7.19 Activity of zirconium as a function of composition at 2103 K from the assessment in this study, with experimental data from Andrievskii et al. [14]...... 228

7.20 Heat capacity of ZrC0.96 as measured by Westrum and Feick [265] and from the thermodynamic assessment in this study...... 230 7.21 Molar enthalpy of formation for zirconium carbide as a function of carbon content at 298.15 K from thermodynamic assessments by Fernandez Guillermet [66] and this study, compared with experimentally measured values [16, 3, 163, 138]. ... 231 7.22 Reference vacancy formation energies (as deﬁned in Chapter 6) as calculated using DFT (red), and extracted from the CALPHAD assessment from Fernandez Guillermet [66] (blue) and from the assessment shown above (green)...... 232 7.23 Carbon-zirconium phase diagram from the assessment including recent experimental data, with experimental data superimposed [114, 135, 96, 219, 216, 226, 6]. .. 234 7.24 Carbon-zirconium phase diagram from the assessment including recent experimental data under the constraint of reproducing the calculated vacancy formation energy, with experimental data superimposed [114, 135, 96, 219, 216, 226, 6]. .. 234

27 7.25 Heat content from the assessment including recent experimental data with experimental data from Bolgar et al. [33]...... 235 7.26 Heat content from the assessment including recent experimental data under the constraint of reproducing the calculated vacancy formation energy, with experimental data from Bolgar et al. [33] ...... 235 7.27 Heat content from the assessment including recent experimental data with experimental data from Turchanin and Fesenko [254]...... 236 7.28 Heat content from the assessment including recent experimental data under the constraint of reproducing the calculated vacancy formation energy, with experimental data from Turchanin and Fesenko [254]...... 236 7.29 Heat content from the assessment including recent experimental data with experimental data from Turchanin and Fesenko [254]...... 236 7.30 Heat content from the assessment including recent experimental data under the constraint of reproducing the calculated vacancy formation energy, with experimental data from Turchanin and Fesenko [254]...... 236 7.31 Heat content from the assessment including recent experimental data with experimental data from Turchanin and Fesenko [254]...... 237 7.32 Heat content from the assessment including recent experimental data under the constraint of reproducing the calculated vacancy formation energy, with experimental data from Turchanin and Fesenko [254]...... 237

B.1 The FCC phase part of the TDB file that constrains the first order excess parameter in the γ phase to fulfil a chosen vacancy formation energy. The original parameter is shown and commented out in the TDB...... 257 B.2 Extract from the setup file for carbon-zirconium. The parameters to be optimised can be defined in terms of the variables to be optimised and the end-member parameters as functions...... 259

28 List of Tables

4.1 The CALPHAD assessments for each subsystem within the B-C-Hf-Zr quaternary system...... 73

5.1 Properties calculated using DFT with LDA PAW and GGA PAW PBE exchange- correlation functionals compares with DFT calculations from the literature, experimental data, and values from the CALPHAD assessment of the B-Hf-Zr system. The choice of significant figures indicates the accuracy to which the values were obtained, where the uncertainty would not affect the value shown...... 121 5.2 Properties calculated using DFT with LDA PAW and GGA PAW PBE exchange- correlation functionals compares with DFT calculations from the literature, experimental data, and values from the CALPHAD assessment of the C-Hf-Zr system. The choice of significant figures indicates the accuracy to which the values were obtained, where the uncertainty would not affect the value shown...... 136

6.1 T =0 K DFT calculations of the raw vacancy formation energy of zirconium carbide calculated using LDA and GGA with 2x2x2 supercells, with the energy diﬀerence between diamond and graphite added to correct the reference energies...... 153

7.1 Crystal structure data of all condensed phases in the C-Zr system...... 203 7.2 Invariant points in the carbon-zirconium system in the optimised thermodynamic assessment...... 218

29 30 List of acronyms

AM05 Armiento and Mattsson, 2005 (exchange-correlation functional) BCC Base-centred Cubic CALPHAD CALculation of PHase Diagrams CEF Compound Energy Fomalism DFE Defect Formation Energy DFT Density Functional Theory DTA Differential Thermal Analysis FCC Face-centred Cubic GGA Generalised Gradient Approximation HCP Hexagonal Close-packed ICSD Inorganic Crystal Structure Database LDA Local Density Approximation PAW Projector Augmented Wave PBE Perdew-Burke-Ernzerhof, 1996 (exchange-correlation functional) PW91 Perdew et al., 1991 (exchange-correlation functional) SER Stable Element Reference SGTE Scientific Group Thermodata Europe SQS Special Quasi-random Structures TDB Thermodynamic database (file) TRISO Tristructural-isotropic (fuel particle) UHTC Ultra-high Temperature Ceramic VASP Vienna Ab initio Simulation Package VFE Vacancy Formation Energy

31 32 Chapter 1

Introduction and Motivation

Ultra-high temperature ceramics (UHTCs) are materials that are widely used in the aerospace and nuclear industries due to their extremely high melting points (in excess of 2000 ◦C). Within the boron-carbon-hafnium-zirconium system, there are four refractory UHTC compounds: the hafnium and zirconium diboride and carbide.

UHTCs are candidate materials for the protective components of hypersonic vehicles. Hypersonic vehicles require heat protection due to their extreme speeds. Space re-entry vehicles require heat protection due to the heating that occurs upon entering the atmosphere at such high speeds. As these vehicles need to be sharp to maintain manoeuvrability, these temperatures can be in excess of 2000 ◦C at the nose of the aircraft, where plasma ablation may occur [192]. The materials used to protect such aircraft are subject to extreme heat ﬂuxes, severe oxidation, and high mechanical stresses upon manoeuvrings. These requirements limit the available materials to UHTCs, typically non-oxides with melting or decomposition temperatures above ◦ 3000 C. HfB2 and ZrB2 are well known examples of such materials [58, 117]. The development of materials for these purposes means combining the materials in order to obtain the optimal mechanical, thermodynamic, and oxidation properties. Progress in the development of these UHTCs is necessary in order to further progress in the aerospace industry, to protect hypersonic vehicles during ﬂight.

33 Within the nuclear sector, UHTCs are used in near-core fuel coating and cladding applications due to their stability at extremely high temperatures [150]. The desired nuclear applications require material functionality in environments with very high temperatures, and high oxidation, stress and irradiation levels. While there are considerable challenges in the understanding of the material response to neutron irradiation, UHTCs such as ZrC are considered good candidates, and zirconium carbide is being considered as a ﬁssion-product barrier in Tristructural-isotropic (TRISO) coated fuel micro-particles in test reactors[115]. The use of ceramics in future fusion reactors will be both structural (requiring strength and temperature resistance) and functional (ie. ceramic superconductors) [151]. The importance of the roles of these materials in this application has lead to a revived interest in their study.

This work focusses on the understanding of the hafnium and zirconium carbides and borides. In order to determine the composition and structure of these materials that will provide the most suitable properties for each application, it is necessary to fully understand the phase stability of these materials in terms of the external conditions, constituents, and defect properties.

The CALPHAD (CALculation of PHAse Diagrams) approach is a numerical method by which data from thermodynamics, phase diagrams, crystal structure, and properties such as magnetism can be combined into a single consistent model represented by Gibbs energy functions for each phase. It is a powerful tool in materials research whereby it can be used as a predictive model where there is no available data, or experimental results are conflicting. In the case of the UHTCs considered in this thesis, all of the relevant binary and ternary systems were considered using the CALPHAD approach in the late nineties and early noughties. In the decades since these assessments were conducted, there have been significant leaps in accurate measurements in extreme conditions, which have furthered the understanding of the phase stability of UHTCs. There have also been significant developments in first principles calculations that have lead to increased understanding of the properties of many materials.

Taking into account all of the available experimental and theoretical data relating to the thermodynamics and phase diagrams of these materials, the relevant phases

34 and systems are considered and examined. One area where ﬁrst principles calculations have opened the door to signiﬁcant new understanding is in the modelling of point defects and their associated properties, which are very challenging to examine experimentally.

By considering insights from the ﬁrst principles modelling of substitutional defects in the hafnium and zirconium borides and carbides, the existing CALPHAD modelling of the boron-hafnium-zirconium and carbon-hafnium-zirconium systems is examined. Without using calculated quantities as direct input in a thermodynamic assessment, the ﬁrst principles calculations provide insight that can inform the choice of thermodynamic models.

In order to have a high-quality description of the thermodynamics of a system, it is necessary to ensure that such a description is consistent with all available physical understanding of the system. Point defect related properties, such as defect formation energies and defect-defect interaction energies are now calculable, but are not generally included directly in CALPHAD assessments. In this work, it is shown how the current parameterisation of the Gibbs energies can be adapted to explicitly include ﬁrst principles calculations of properties such as defect formation energies, demonstrated for the case of vacancies in zirconium carbide.

Evolution of the CALPHAD methodolgy to keep up with developments in ﬁrst principles calculations is of vital importance in order for it to continue to be a powerful tool in understanding the properties of materials. The currently used parameterisation of the Gibbs energy is considered in the context of some defect related properties, and its appropriateness for compatibility with some ﬁrst principles calculations examined.

35 36 Chapter 2

Overview of the CALPHAD method

2.1 Introduction

One of the primary goals of engineering is to be able to control the final properties of a material, understanding the ways in which the chemical composition, processing conditions, and microstructure affect the final properies. Often during the processing or use of a material, it will undergo one or more phase transformations, which can be understood through phase diagrams. However, experimental phase diagrams are challenging and time consuming to produce, and often are not available for systems of more than two elements. The CALPHAD (CALculation of PHAse Diagrams) method allows us to calculate multi-component phase diagrams by using phase diagrams as a representation of the thermodynamic properties of a system. As such, these thermodynamic descriptions of lower-order systems can be combined to extrapolate the thermodynamic properties of higher-order systems. This can be done predictively, and fine-tuned using higher-order data where available [100].

Within the CALPHAD method the thermodynamic properties, experimental phase diagram, and crystallographic data are combined to give a consistent description

37 of each phase in the system. Within the CALPHAD approach, the Gibbs energy is chosen as a thermodynamic quantity to be modelled. Although the thermodynamic functions could be expressed in terms of any thermodynamic quantity, it is convenient to use the Gibbs energy as experiments are usually conducted at constant temperature and pressure, and as such it is possible to easily calculate other thermodynamic quantities from the Gibbs energy [161].

2.2 Classical Thermodynamics

Gibbs described a phase as referring to the composition and thermodynamic state of homogeneous body formed out of a set of component substances without any regard for its quantity or shape [78]. Diﬀerent crystal structures of a material can also be used to distinguish phases of the material, even if the bodies have the same composition and external form. The modern deﬁnition of a phase has been extended to not only distinguish whether a material is a solid, liquid, or a gas, but also to describe its crystal structure.

The components that make up a material can be any chemically distinct ingredient. In order to be called a component, an ingredient must be necessary for describing all possible compositions of the phase that could form. Components may be elemental or molecular, but the number of components must be the minimum number of independent species that when present in diﬀerent amounts are needed to describe all composition points. The components used to describe each material may be arbitrarily chosen for convenience in the description.

A system can be deﬁned as being in equilibrium if all of the system has the same temperature and the same pressure. If a system is in equilibrium, the system must be homogeneous, unless there are diﬀerent phases that coexist, each of which being homogeneous, as a consequence of the second law of thermodynamics [161].

Gibbs’ phase rule uses the number of independent state variables in a system to determine the maximum number of phases that can coexist simultanously. This is

38 expressed as f = c + 2 − p (2.1) where f is the number of degrees of freedom which is the number of independent intensive variables, c is the number of components, and p is the number of stable phases. The number two appears in the equation to account for the state variables temperature and pressure which are present in all thermodynamic systems, but this number increases if diﬀerent types of work, such as electric or magnetic work, are present [161].

If we have thermodynamic descriptions for each phase, they can be used to produce a phase diagram of the system by finding stable phases at each point in temperature- , pressure-, and composition-space. Within the CALPHAD method, experimental and calculated data is combined into a consistent model of a single thermodynamic quantity for each phase, which can be used to find the stable phase or phases. As most experimental data are measured at a given temperature, pressure, and composition, Gibbs energy is used as the modelled quantity as this is the most convenient for derivation of other quantities [161]. Therefore, to produce the phase diagram, it is necessary to find the minimum of the Gibbs energy at constant temperature, pressure, and fraction of components at each point.

2.2.1 Connection to statistical thermodynamics

Statistical thermodynamics is used as the basis for producing a model for the Gibbs energy of each phase that can be ﬁtted to experimental data. The energy terms are usually described as energies of formation of molecules of units of crystal lattices, and the entropy terms are separated into vibrational terms and entropies of mixing [161].

In the description of solid phases, the crystal structure of the phase must be considered in the modelling of the thermodynamic functions. A sublattice is deﬁned as being a set of equivalent positions in a crystal. Using several sublattices it is possible

39 to describe the positions and ordering of diﬀerent atoms in a crystal. The Com- pound Energy Formalism (CEF) [101] used in the calculations of the Gibbs energy uses the occupation of positions in the crystallographic sublattices to determine the mixing in solution models, and if the crystallographic descriptions used do not corre- spond to the physical system, the thermodynamic description will not be physically meaningful [161].

The Gibbs energy for a phase can be considered to have contributions from the bonding of the constituents, and their conﬁguration. The mixing of unlike atoms is always enhanced by the conﬁgurational part of the Gibbs energy. If there is a tendency for formation of a compound or ordering of the components, the bond energy between two unlike atoms will be more negative than the bond energy between two equal atoms. If there is a tendency towards a miscibility gap, the bond energy between two unlike atoms is more positive than the bond energy between two equal atoms.

The model for each phase is chosen in order to describe any particular chemical or physical features of the phase. Most of the commonly used models for describing the Gibbs energies of common phases are special cases of the Compound Energy Formalism (CEF). Particular features that are usually considered include the crystallography of the phase, the type of bonding, magnetic properties, and order-disorder transitions. Most models for crystalline phases account for the speciﬁc crystallography by modelling the compound as a series of sublattices, as discussed in Section 2.3.3.

Several constituents can be on each sublattice, but if there is just a single constituent on each sublattice, it is a stoichiometric compound of ﬁxed composition. The Gibbs energy of formation of such a compound is a very important part of the bond energies. If the compound is a pure element, the phase has a single set of sites and the element as constituents. For a solution phase, there are at least two such compounds, which are referred to as the end members of the solution phase, which deﬁne the limit of solubility. The model may have end members with a composition within the composition range of the phase.

40 The number of parameters in the Gibbs energy model for each phase is dependent on the phase itself, and it is necessary to adjust this for each description in order to fully describe the phase equilibria and thermodynamic properties of the system. Parameters within the CALPHAD approach can be functions of temperature, pressure or composition, and may be split into other parameters. Each parameter is a quantity that is part of the model, and is represented with several coeﬃcients, which are simply numerical values.

The modelling of the Gibbs energy for diﬀerent systems is shown in Section 2.3.

2.3 Gibbs energy models

2.3.1 General form of the Gibbs energy parameterisation

In general, the Gibbs energy of a phase ϕ is partitioned into terms as

ϕ srf ϕ phys ϕ cnf ϕ E ϕ Gm = Gm + Gm − T Sm + Gm (2.2)

srf ϕ where Gm is the “surface of reference” term, representing the Gibbs energy of an phys ϕ unreacted mixture of the constituents of the phase, Gm is the contribution to cnf ϕ the Gibbs energy by any physical models such as magnetic transitions, and Sm is the conﬁgurational entropy of the phase, based on the number of arrangements of the constituents in the phase according to statistical thermodynamics [161], and m indicates that it is a molar quantity. The energy contribution from this term is cnf ϕ E ϕ given by T Sm, where T is the temperature of the system. Gm is the excess Gibbs energy, and describes the remaining part of the real Gibbs energy. As the Gibbs energy is partitioned into terms in this way, the physical origins of the Gibbs phys ϕ energy are not modelled (apart from those modelled in Gm). Conﬁgurational, vibrational, and electronic contributions, as well as others, are contained within the srf ϕ E ϕ terms Gm and Gm.

41 2.3.2 Phases with ﬁxed composition

A phase has ﬁxed composition if it is a pure element, a stoichiometric compound, or if it is a solution phase with an externally maintained composition. End-members of solutions are phases in this class, and have Gibbs energies that are only temperature and pressure dependent, with no composition dependence. For all of the work in this study, experiments and calculations were conducted at atmospheric pressure, and thus any pressure dependence of the Gibbs energy and other thermodynamic quantities can be ignored.

For a stable end member of a phase ϕ, the temperature dependence of the molar Gibbs is often expressed as a power series as

ϕ X SER 2 −1 3 Gm − biHi = a0 +a1T +a2T ln T +a3T +a4T +a5T +..., T1 < T < T2 (2.3) i where bi is the stoichiometric factor of element i in phase ϕ.

P SER biHi represents the sum of the enthalpies of the elements in their Stable Element i Referece (SER) states, at 298.15 K and atmospheric pressure. As the enthalpy of a system has no absolute value, some reference state must be chosen. This power series expression can only be valid within a limited temperature range, T1 < T < T2, above the Debye temperature.

From the relationships between thermodynamic quantities, it is possible to obtain the following expressions for the enthalpy, entropy, and heat capacity of a phase ϕ.

ϕ X SER 2 −1 3 Hm − biHi = a0 − a2T − a3T + 2a4T − 2a5T ... (2.4) i

∂G Sϕ = − = −a − a (1 + ln T ) − 2a T + a T −2 − 3a T 2... (2.5) m ∂T 1 2 3 4 5

42 ∂H Cϕ = − = −a − 2a T − 2a T −2 − 6a T 2... (2.6) p ∂T 2 3 4 5

From the expression for the heat capacity, it can be seen that the T ln T term in Equation 2.3 comes from the temperature independent heat capacity coeﬃcient. By relating certain parameters to measurable terms that can be ﬁtted to experimental data, we can determine the values for certain parameters in the Gibbs energy functions.

It is desireable to use as few coefficients as possible to describe the Gibbs energy function to avoid overfitting and prevent erroneous behaviour of the function. Some- times the Gibbs energy can be expressed using several different temperature ranges, in order to reduce the number of coefficients needed in the description. It is necessary that the Gibbs energy and its first derivative are continuous through such a breakpoint in the description, otherwise the behaviour would be like that of a phase transition.

When there is solubility of one compound within another, a solution phase is formed, which may have a limited solubility range that varies with temperature. This means that it is necessary that the Gibbs energy of the solution phase must have an endpoint value representing the compound at temperatures outside its range of stability. In cases such as this, the Gibbs energy is extrapolated to the relevant value. It is also sometimes necessary to use the Gibbs energy of a compound that is never stable in the model, using data extrapolated from experimental measurements or from ﬁrst principles calculations.

The Kopp-Neumann rule

Where the heat capacity of a compound has not been measured, the Kopp-Neumann rule can be applied, which states that the heat capacity of a compound is equal to the stoichiometric average of the heat capacities of the constituent pure elements in

43 their reference state [136]. This means that the Gibbs energy will be

ϕ X SER Gm − biGi = a0 + a1T (2.7) i

Here, the expression has only two coeﬃcients which may be determined from experimental information, as the heat capacity will be taken from the descriptions of the pure elements. This equation is also applied when modelling the end members of solution phases which have a composition within the composition range of the phase.

2.3.3 Modelling using sublattices

Sublattices represent long range ordering of the system, by taking into account different types of nested lattices that may be occupied by atoms, with diﬀerent coordination numbers. This long range ordering aﬀects the entropy expression and the excess Gibbs energy.

Two sublattices with two constituents in each sublattice can be expressed with the notation

(A,B)m(C,D)n (2.8) where m and n give the ratio of the number of sites on each sublattice. The constituents A, B, C, and D may be atoms, ions, or vacancies [211]. The most general formalism for describing the thermodynamic properties of phases with more than one sublattice is the CEF [101], described in Section 2.3.6. Where there is a single component on each sublattice, it is a stoichiometric phase, identical to having a single species as a constituent. Where there is mixing on one sublattice and a single constituent on another, it is identical to mixing species in the substitutional solution model.

When there are several constituents in each sublattice, the model can be written

(A,B,...)m(U,V,...)n (2.9)

44 where a constituent may or may not exist in both sublattices. The Gibbs energy for this model is given by ! X X 0 00 o phys X 0 0 X 00 00 E Gm = yiyj Gi:j + Gm+RT m yi ln yi + n yj ln yj + Gm (2.10) i j i j

0 00 where yi and yj are the constituent fractions on sublattices 1 and 2 respectively o [101]. The Gibbs energy of formation of these compounds, Gi:j, is multiplied by the fractions of one constituent from each sublattice. A colon is used to separate constituents on diﬀerent sublattices, where two constituents on the same sublattice are separated by a comma.

The two-sublattice model is often used to model interstitial solutions of carbon, boron or nitrogen in metals. As carbon atoms are relatively small, they have a tendency to occupy interstitial sites in metallic sublattices, as is the case for the hafnium and zirconium carbides and borides. In an interstitial solution, one of the constituents is the vacancy.

2.3.4 Composition dependence

The Gibbs energy of a phase is dependent on the amount of each constituent in that phase. When modelling the composition-dependence of the Gibbs energy, G, it is convenient to use the molar Gibbs energy, Gm, and deﬁne the size of the system as the total amount of components N.

G = NGm (2.11)

X N = Ni (2.12) i

The composition dependence of the molar Gibbs energy is usually described in terms

45 of the mole fractions of the components i, xi, which is deﬁned as

N x = i (2.13) i N

For a closed system, the amounts of each component is an external variable, controlled from outside the system. Within each phase, each component may form different species or ions, and can be a constituent in different types of sites in a crystalline lattice. The composition of a phase in equilibrium is defined as the amounts or fractions of all the components in the phase, usually expressed using the mole fractions defined above.

The number of components in a system is always equal to the number of elements, but it may be convenient to use components other than the elements to describe the system [161].

At equilibrium, the chemical potential of a component j can be written as the weighted sum of the chemical potentials of the elements i as

X µi = bijµij (2.14) i where bij is the stoichiometric factor of component i in constituent j, and µj is the chemical potential of constituent j.

The species making up a phase are referred to as constituents. In a crystalline phase, made up of several sublattices, the constituent fraction can also be called the site fraction. On each sublattice, the constituent fractions sum to one, and the mole fractions can be calculated from the constituent fractions yi as P bijyj j xi = P P (2.15) bkjyj k j where bij are the stoichiometric factors of the element in each component, as deﬁned

46 above.

For phases with sublattices, labelled by (s), the constituent fractions are deﬁned as

N (s) y(s) = i (2.16) i N (s)

(s) (s) Ni is the number of sites occupied by constituent i on sublattice (s), N is the total number of sites on sublattice (s).

(s) Where there are empty sites, a vacancy fraction yVa can be deﬁned as

(s) P (s) N − Ni (s) i X y = = 1 − y (2.17) Va N (s) i i6=Va

Vacancies can be treated as a real component, with the restriction that their chemical potential is always equal to zero. Deﬁned this way, these are structural, or constitutional vacanies, not thermal vacancies.

If a phase is modelled using sublattices, the molar Gibbs energy can be easily expressed per mole of formula unit rather than per mole of real components. The formula unit of a phase with sublattices is given by the sum of the site ratios P a(s), s where a(s) are the smallest integers which give the correct ratio between the num- (s) bers of sites N on each sublattice. The mole fraction xi in a crystalline phase is therefore P bijyj X j x = (2.18) i a(s) P P b y s kj j k j where vacancies are excluded from summations.

47 2.3.5 Physical phenomena

The contributions to the Gibbs energy arising from various physical phenomena can be modelled within the CALPHAD approach. These include the contribution from lattice vibrations and ferromagnetism. Other physical contributions such as electronic heat capacity, size mismatch, and short range order are not usually modelled separately as they are relatively small and cannot be modelled accurately [161]. However, these are not used in the CALPHAD modelling of any systems within the boron-carbon-hafnium-zirconium system, and so will not be described further here.

2.3.6 Solution Models

The term solution applies to all kinds of phases with variable composition. The composition of all phases varies to some extent, but for practical purposes it is possible to ignore variations in composition for certain compounds. When modelling a solution phase, it is necessary to describe the properties of the end points of the solutions, be they pure elements or compounds. This may require properties of compounds that cannot be measured separately and must be determined by extrapolation, such as in the case where a compound or structure is not stable at a given stoichiometry or temperature. To describe the Gibbs energy of real solutions, each term contributing to the Gibbs energy must be modelled with parameters ﬁtted to experimental data on the properties of the solution.

The Compound Energy Formalism

The Compound Energy Formalism (CEF) [101] is based on the two sublattice model formulated by Hillert and Staffansson (1970) [102], which was extended to an arbitrary number of sublattices and constituents on each sublattice by Sundman and Agren˚ (1981) [243]. A constituent array was defined, which is denoted I and specifies one or more constituents i on each sublattice (s). A zeroth order constituent array

48 contains just one constituent on each sublattice. The constituent array denotes a compound.

The Gibbs energy expression for the CEF is

srf X o Gm = PI0 (Y ) GI0 (2.19)

n ns cnf X X (s) (s) Sm = −R as yi ln yi (2.20) s=1 i=1

E X X Gm = PI1 (Y )LI1 + PI2 (Y )LI2 + ... (2.21)

I1 I2 where In is a constituent arrray of order n, PIn (Y ) is the product of constituent fractions specified by In [161], where Y is a matrix of the site fractions of each o component on each sublattice. GI0 is the Gibbs energy of formation of the compound (s) I0. as is the number of sites on each sublattice s, where yi is the constituent fraction of i on s. The first sum in the configurational entropy term is over all sublattices, E the second over all constituents on each sublattice. The excess Gibbs energy Gm is the sum over all interaction parameters L by component arrays of each order.

The partial Gibbs energy for a constituent array of zeroth order is given as

n n X ∂Gm X X (s) ∂Gm GI0 = Gm + (s) − yi (s) (2.22) s=1 ∂yi s=1 j ∂yj

n where the ﬁrst sum P is over the constituent i on each sublattice (as this is zeroth s=1 order there is just one constituent on each sublattice), and the second sum, P is j over all constituents on each sublattice.

Within the CEF, each compound or end member has an independent Gibbs energy of formation, which is a function of temperature and pressure, but is independent of

49 composition. If there is no experimental data for this end member, for instance when such an end member is purely ﬁcticious, the energy of the compound is sometimes estimated as the sum of the bond energies between all the atoms in the compound [161].

First principles calculations are often used to calculate the energies of ordered compounds where not all states are stable and measurable. While this method is good at providing extrapolations to higher order systems, it is necessary to compare with experimental data wherever possible to ensure the best possible physical description.

The temperature and pressure dependent Gibbs energy of formation of the constituent array I in phase ϕ with reference to the stable states of the elements included o ϕ in the constituents, GI , is given as

o ϕ X SER GI − bijHj = f(T, p) (2.23) k where bij is the stoichiometry factor of component j in the constitent array I, and HSER is the enthalpy of component j in its SER state.

Ideal substitutional solution models

An ideal substitutional solution is one where there is random mixing between two non-interacting components. When the constituents are the same as the components, a simple ideal substitutional solution model, the Gibbs energy is

n n X o X Gm = xi Gi + RT xi ln xi (2.24) i=1 i=1

Where there are more constituents than components, the Gibbs energy is given by

n n X o X Gm = yi Gi + RT yi ln yi (2.25) i=1 i=1

50 where the constituent fractions, yi are used instead of the mole fractions, xi. The term ideal model is used to describe models where the excess contribution of the Gibbs energy is zero, meaning that the Gibbs energy comes purely from the Gibbs energies of the end members and random mixing between constituents.

2.3.7 Excess Gibbs energy

The excess Gibbs energy is the remainder of the total Gibbs energy after the surface, conﬁgurational and physical terms have been summed. It is the diﬀerence between the Gibbs energy of the real phase and the Gibbs energy of the ideal model that has been selected. Each order of interaction within the excess Gibbs energy is modelled separately, so for a phase ϕ

total ϕ srf ϕ phys ϕ cnf ϕ E ϕ Gm = Gm + Gm − T Sm + Gm (2.26) where the binary, ternary, and higher order terms in the excess Gibbs energy are separated as E ϕ bin.E ϕ tern.E ϕ high.E ϕ Gm = Gm + Gm + Gm (2.27) which is known as the Muggianu method.

Binary contributions

The binary contributions of the excess Gibbs energy in a multicomponent substitutional solution are parameterised as

n−1 n bin.E X X Gm = xixjLij (2.28) i=1 j=i+1 which is a sum over all binary interactions in all i-j systems. This model can include temperatures and composition dependences of the interaction parameter Lij, and to consider interactions between constituents on diﬀerent sublattices. As the interaction

51 parameter is dependent on the bond energies, which remain approximately the same regardless of the structure of the phase, it is expected that the interaction parameters are of approximately the same order of magnitude in all phases modelled with these regular solution parameters.

The composition dependence of the interaction parameters can be expressed in terms of the difference in the fractions of constituents i and j with a Redlich-Kister power series: k X ν ν Lij = (xi − xj) Lij (2.29) ν=0 This parameterisation preserves the shape of the excess Gibbs energy in the binary system in the multicomponent system, with only changes to the magnitude. The number of coefficients in a Redlich-Kister power series expansion of the interaction parameters should be kept as small as possible to avoid overfitting of the functions and to reduce the number of degrees of freedom.

ν The parameters Lij often have a temperature dependence, usually of the form

ν ν ν Lij = aij + bijT (2.30)

ν where the composition dependence of the excess enthalpy is given by aij and the ν excess entropy by bij. Normally only a linear expansion of these parameters is necessary, but a higher order expansion in T may be used where excess heat capacity data is available for the system.

The Redlich-Kister expansion can only be used when the enthalpy varies smoothly with composition. Where it has sharp variations, a diﬀerent model must be used. There are many other excess models for binary systems although the Redlich-Kister expansion is the most common.

52 Ternary contributions

In the same way as the binary interaction contributions were expressed, the ternary interaction contribution to the excess Gibbs energy is expressed as

n−2 n−1 n tern.E X X X Gm = xixjxkLijk (2.31) i=1 j=i+1 k=j+1 where the ternary interaction parameter, Lijk can be composition dependent. The composition dependence can be modelled as

i j k Lijk = νi Lijk + νj Lijk + νk Lijk (2.32) where (1 − x − x − x ) ν = x + i j k (2.33) i i 3 (1 − x − x − x ) ν = x + i j k (2.34) j j 3 (1 − x − x − x ) ν = x + i j k (2.35) k k 3

By introducting the νi fractions, which sum to 1 even in multicomponent systems, the ternary parameters will behave symmetrically when extrapolated to a higher order system.

Higher order contributions

More summations can be added to the excess Gibbs energy in the same was as for the binary and ternary contributions. Hence the higher order contributions to the excess Gibbs energy can be written as

n=3 n−2 n−1 n high.E X X X X Gm = xixjxkxlLijkl + ... (2.36) i=1 j=i+1 k=j+1 l=k+1

53 The composition dependence of higher order interaction parameters has not been considered in existing CALPHAD assessments due to limitations of experimental data for accurate modelling.

Substitutional regular solution models

In a substitutional solution, each constituent has the same probability of occupying each site in the unit cell. The Gibbs energy is described by

n n X o X E Gm = xi Gi + RT xi ln xi + Gm (2.37) i=1 i=1 where the excess Gibbs energy for binary interactions is expressed as

E X X Gm = xixjLij (2.38) i j>i where the binary interaction parameter Lij is given as z L = (2.39) ij 2 ij where z is the number of bonds and is related to the bond energies [102]. This substitutional model can be used for liquids and solids where the constituents mix on the same sites.

Models based on the CEF do not have any short range ordering contributions explicitly. Long range ordering, such as that arising from sublattices with diﬀerent numbers of nearest neighbours, and preference for certain sublattices by the constituents must be included explicitly in the modelling.

54 2.4 Assessment Methodology

The basis of a CALPHAD is a thorough survey and review of all of the available literature, seeking all the information that has been calculated or experimentally measured that relates to the thermodynamic assessment, be it phase diagram data, crystallographic information, or thermodynamic quantities. Once all of the information has been gathered, decisions must be made to remove conﬂicting sets of data from the collection of data to be used in the assessment. These decisions can be made on the basis of the reported experimental uncertainties and the reliability of the techniques used.

Once the set of data has been chosen and formatted appropriately for the assessment, accounting for all errors in the measurements, the model parameters for the system can be ﬁtted to this data. The models used should be chosen such that they can be appropriately ﬁtted to the available data. For instance, if there is no available data on a ternary system, ternary parameters cannot be modelled from the experimental data relating to the binary sub-systems.

The chosen model parameters are then fitted simultaneously to all of the available data. This process is referred to as an optimisation. The fitting of the model parameters is usually done using a least squares fitting, and is done as an iterative process whereby the various experimental points can be weighted differently according to their experimental uncertaintly and quantity, and compared with all of the existing data until a description that reproduces all of the data satisfactorily is obtained.

The methodolgy of a CALPHAD assessment is summarised in Figure 2.1 from Lukas et al. [161].

For systems with multiple elements, this is done using a bottom-up approach, whereby the unary systems are assessed ﬁrst and used as the end members of a binary assessment, and so on. One of the beneﬁts of the CALPHAD approach is that descriptions of sub-systems can be used to extrapolate the thermodynamic properties of higher order systems. Although simple extrapolation of, for instance, three binary sys-

55 Figure 2.1: A ﬂowchart from Lukas et al. [161] showing the assessment and optimisation procedure used in the CALPHAD approach to produce a thermodynamic description. tems to a ternary system, will be unlikely to completely describe all of the detailed behaviour, it can be used as an approximation.

The extrapolation of a higher order system can also be used to guide the assessment

56 of its subsystem. The nature of CALPHAD is that the thermodynamic description is not entirely unique, and it is possible to fit the parameters differently to the available data and obtain a thermodynamic description of the same quality. If the extrapolation of a higher order system proves to be significantly different to the expectation of that system, this can suggest that the sub-system assessment in use may be unsuitable and may need to be reevaluated.

2.4.1 Useful experimental data

The experimental data used in a CALPHAD assessment can be broken down into several categories. It is useful to separate the data like this as it may be used at diﬀerent stages of the assessment procedure. For all data, the sources of error must be considered, and in the case of systematic errors, corrected for where possible (such as in the case of using historic temperature scale standards). Where sets of data are contradictory, the uncertainties associated with the data must be considered and a single set of data chosen for use in the optimisation. It is possible to use data that is not used in the optimisation for comparison with the quantities described by the optimised parameters.

Experimental phase diagram data

Experimental phase diagram data can be phase boundaries, solidus and liquidus lines, the phases present at a specific point, invariant points, or other data that can be represented on an experimental phase diagram. This information can be found in a variety of ways including metallography, X-ray diffraction, thermal analysis, and diffusion-probe experiments, among others [161].

57 Thermodynamic data

Thermodynamic data can be any diﬀerent thermodynamic quantities measured as a function of temperature or composition. These can include enthalpies of formation, mixing, or other enthalpies, and data related to chemical potential measurements. These are often measured using calorimetry, measurements of the electromotive force, or vapour-pressure methods [161, 1]

Other experimental data

Other physical measurements that can be related to thermodynamic quantities can be considered in the assessment. These can include elastic moduli and constants, and thermal expansion properties, amongst others. These properties can be derived from the Gibbs energy, and thus can be included in the assessment process where available.

Crystallographic data

Information about the crystal structure of the material, densities and ordering of point defects within it, and properties such as the resistivity can be used to inform the Gibbs energy models used in the CALPHAD assessment.

First principles calculations

In recent years, the quality of ﬁrst principles calculations of thermodynamic and elastic properties has become comparable with experiment [179] and theoretical studies have become abundant due to their comparative cheapness. These calculations and their associated error can be used in the same way as experimentally measured thermodynamic data in the ﬁtting process of the assessment. Structure stability calculations can also be used to inform the models used, and to give insight where there is limited experimental data or there are contradictory or ambiguous data.

58 2.5 Overview of optimisation using the PARROT module of Thermo-Calc

Unary end member descriptions are usually kept as the SGTE (Scientific Group Thermodata Europe) unary descriptions of the condensed phases of the pure elements [52] to allow consistency between different assessments. This approach was adopted by the CALPHAD community in 1991 to prevent the same systems being considered by others, and to allow easy extrapolation of higher order systems from standard ’building block’ assessments. For some applications, the SGTE functions are not sufficient and it is necessary to consider the Gibbs energy modelling of the pure elements, but for all the work in this study, SGTE unary end members were maintained.

During the assessment it is common to optimise each phase to its associated data, before optimising all of the parameters to the data simultanously. When using the PARROT optimisation module in Thermo-Calc [13, 245], it is recommended to ﬁt the parameters describing the heat capacity of each phase separately, and not to allow them to vary in the optimisation [1, 161]. Where no experimental or calculated heat capacity data is available, the Kopp-Neumann rule is used to give an estimate for the heat capacity of that material.

During the Gibbs energy modelling used to descibe each phase, it is necessary to choose certain parameters which are to be made adjustable during the optimisation process. It is desirable to use the minimum number of parameters possible to accurately describe the thermodynamics of the system to reduce the amount of error introduced and the possibility of overﬁtting [161]. If it is not possible to ﬁt the parameters such that the Gibbs energy reproduces certain key experiments well, it can be necessary to reconsider the choice of models.

In a CALPHAD assessment of a system, it is desirable to guess parameters that can be used as a starting point in the optimisation, in order to guide the ﬁtting. If a CALPHAD assessment of the system has been conducted before, this can be used

59 to inform the assessment, otherwise it is necessary to use experience and trial and error to make physically appropriate estimates of the parameters.

The least squares minimisation is guided by assigning different weights to each dat- apoint. When using PARROT, the data are automatically weighted according to their assigned uncertainty, but it is also possible to assign different weighting to each set of data, or even to each data point. This is particularly useful if there are many measurements of a single property over a small range compared to the rest of the available data. To avoid an uneven fitting of data over this range, it is possible to reduce the weighting on these data points.

Once a converged set of parameters has been reached during the optimisation, it is necessary to check the thermodynamic description against all available experimental and calculated data to ensure a good ﬁt. Data that was not chosen for use in the optimisation can be used for comparison. The properties described by the thermodynamic description should be physically realistic in all regions, even where no data is present, and should reproduce the experimental data well where it is available. In order to check the quality of an assessment, it is sometimes possible to use a higher order extrapolation to give an indication of whether the description is physically meaningful. The PARROT module in Thermo-Calc provides a sum of squares of errors that can be used to indicate whether or not a set of parameters is converged and ﬁtting the data well.

2.6 Summary

The CALPHAD approach is a highly powerful tool for combining all the known phase diagram, thermodynamic, and crystallographic information into a single consistent description, that can be used to concisely describe a system, or can be used to predictively describe higher order systems by extrapolation. In this chapter a background to the modelling and ﬁtting of parameters used in a CALPHAD method has been given.

60 In order to produce a high-quality consistent description of the thermodynamics and phase diagram of a system, it is necessary to use insights from experimental data and first principles calculations to choose a parameterisation of the Gibbs energy that is appropriate and can be fitted to selected experimental and first principles data.

61 62 Chapter 3

First principles calculations

Within this thesis, ground state energy structures and various thermodynamic quantities were calculated using Density Functional Theory (DFT), a computational quantum mechanical modelling method which calculates these quantities from ﬁrst principles, with various approximations.

Density functional theory uses quantum mechanical electronic theory based on electronic charge density to compute the interactions between atoms in a periodically repeating supercell. It has been used as the dominant method of simulation of quantum mechanical systems for decades [94]. Within this chapter, an overview of the theoretical background to DFT calculations is given.

3.0.1 Schr¨odinger’sequation

The many-body problem of calculating the interactions between positively charged nuclei and negatively charged electrons can be solved by considering Schr¨odinger’s time-independent non-relativistic equation and the Born-Oppenheimer approximation (in which it is assumed that the motion of the atomic nuclei and electrons can be treated separately) [35]. This is

ˆ HΨ({ri}) = EΨ({ri}) (3.1)

63 where Ψ is the wavefunction of electrons, E is the energy of the system, and Hˆ is the

Hamiltonian operator, and ri is the position of the ith electron. The Hamiltonian operator can be expressed as a sum of three terms: the kinetic energy, the interaction with an external potential Vext, and an electron-electron interaction Vee, where the Born-Oppenheimer approximation lets us assume that the nuclei are stationary on the electronic timescale, and hence ignore nuclei-nuclei interactions. This can be written as " N # X 1 X − ∇2 + V (r ) + U(r , r ) Ψ = EΨ (3.2) 2 i ext i i j i=1 i

The electron-electron interaction potential can be written in atomic units as

N X 1 V = (3.3) ee |r − r | i

The external potential that we are interested in is the interaction of the electrons with the atomic nuclei, which can be written as

Natoms X Zα V = − (3.4) ext |r − R | α i α where Zα is the charge on the nucleus at the coordinate Rα.

This time-independent Schr¨odinger equation can be solved for a set of Ψ assuming that the Ψ are anti-symmetric. The lowest energy eigenvalue, E0, is the ground state energy of the system, and the electrons exist with a probability density at any ri 2 being |Ψ0| .

The energy is a functional of the wavefunction, and the variational theorem states

64 that the energy must be higher than that of the ground state unless Ψ is Ψ0 as

E[Ψ] ≥ E0 (3.5)

3.0.2 Hartree-Fock Theory

The ground state is the one that minimises the total energy when searching over all possible wavefunctions. Hartree-Fock theory assumes Ψ to be an antisymmetric product of functions φi, each being dependent on the coordinates of a single electron. This gives a Hartree-Fock wavefunction as

1 ΨHF = √ det [φ1φ2φ3...φN ] (3.6) N! which leads to a Hartree-Fock energy that can be written as

N ! Z 1 X E = φ *(r) − ∇2 + V φ (r)dr HF i 2 i ext i i N Z 1 X φi*(r1)φi(r1)φj*(r2)φj(r2) (3.7) + dr dr 2 |r − r | 1 2 i,j i j N Z 1 X φi*(r1)φj(r1)φi(r2)φj*(r2) − dr dr 2 |r − r | 1 2 i,j i j

The ﬁrst term is the kinetic energy and external potential energy, the second is the classical Coulomb energy in terms of orbitals, and the third term is the exchange energy. By applying the variational theorem in equation 3.5 under the constraint that the orbitals are orthonormal, the ground state orbitals can be determined, yielding the Hartree-Fock equations. These equations describe non-interacting electrons in the presence of a mean ﬁeld potential that consists of a classical Coulomb potential

65 and a non-local exchange potential, written as

1 Z ρ(r0) Z − ∇2 + ν (r) + dr0 φ (r) + ν (r, r0)φ (r0)dr0 = ε φ (r) (3.8) 2 ext |r − r0| i X i i i where νX is the non-local exchange potential that fulﬁls

Z N Z 0 X φj(r)φ*j(r ) ν (r, r0)φ (r0)dr0 = φ (r0)dr0 (3.9) X i |r − r0| i j

The computational eﬀort required to solve the Hartree-Fock equations scales badly with the number of electrons [91], and although correlated methods can be used to obtain approximations for Ψ and E0, the computational cost scales prohibitively with the number of electrons [94]. These equations have 3N degrees of freedom, which makes them computationally intensive due to the number of variables, but it can be shown that the ground state energy is determined by the electron density, which simpliﬁes the problem.

3.0.3 Density Functional Theory (DFT)

Hohenberg-Kohn Theorems

Hohenberg and Kohn proved two theorems which simplify this problem [104]. The ﬁrst is that the external potential is determined to within an additive constant by the electron density, from which it follows that the electron density uniquely determines the Hamiltonian operator, as the Hamiltonian is speciﬁed by the total number of electrons and the external potential, which can be found from the electron density by integrating over all space. Hence, from the charge density, the Hamiltonian operator and thus the wavefunctions of all states and the material properties can be calculated. The proof of this theorem by Hohenburg and Kohn was generalised to include systems with degenerate states by Levy [156].

66 The second theorem is a variational principle, which states that the electron density that minimises the total energy is the exact ground state density, thus allowing the ground state to be obtained variationally. It can be given formally as saying that for R any positive deﬁnite trial density ρt such that ρt(r)dr = N, then E[ρt] ≥ E0 [94]. This variational theorem restricts density functional theory to ground state studies only.

These two theorems lead to the fundamental statement of density functional theory,

Z δ E[ρ] − µ ρ(r)dr − N = 0 (3.10) where the ground state energy and density are given by the minimum of a functional E[ρ] where the density must contain the correct number of electrons. µ is the electronic chemical potential. From this we know that there is a universal functional E[ρ] that does not depend on the external potential of the system that can be minimised in the above equation to determine the exact ground state density and energy [94].

From the Schr¨odingerequation, we can see that the energy functional will have three terms: a kinetic energy term, an interaction with an external potential, and the electron-electron interaction.

E[ρ] = T [ρ] + Vext[ρ] + Vee[ρ] (3.11) where the interaction with the external potential is Z ˆ Vext[ρ] = Vextρ(r)dr (3.12) and the kinetic and electron-electron functionals are unknown [94], but can be approximated using the Kohn-Sham approach [133].

67 Kohn-Sham approach

Kohn and Sham proved showed that the problem can be solved as if the electrons were non-interacting and in an eﬀective local orbital and the same ground state density can be obtained as if they were considered to be interacting. They show that the energy functional can be written as

E[ρ] = Ts[ρ] + Vext[ρ] + VHartree[ρ] + Exc[ρ] (3.13)

where Exc[ρ] is the exchange-correlation function,

Exc[ρ] = (T [ρ] − Ts[ρ]) + (Vee[ρ] − VHartree[ρ]) (3.14) where Z 1 ρ(r1)ρ(r2) VHartree[ρ] = dr1dr2 (3.15) 2 |r1 − r2|

The Kohn-Sham equations

1 Z ρ(r0) − ∇2 + ν (r) + dr0 + ν (r) φ (r) = ε φ (r) (3.16) 2 ext |r − r0| xc i i i where a local potential that is the functional derivative of the exchange-correlation energy with respect to density has been introduced,

∂E [ρ] ν (r) = xc (3.17) xc ∂ρ and the true ground state density is given by

N X 2 ρ(r) = |φi| (3.18) i are then satisﬁed, and the energy of the system can be computed.

68 3.0.4 Exchange-correlation functionals

However, as there is no exact solution for the exchange-correlation energy that can be calculated, there are many different approximations. Different functional forms exist that make approximations to account for different factors, and within DFT it is necessary to choose which approximation is suitable for a particular study. These choices are usually done by comparing results from several different functionals with experimental data or more accurate theory, but the derivations of the approximations can also inform the choice.

Local Density Approximation

The Local Density Approximation (LDA) was developed by Perdew and Zunger [198]. For a homogenous electron gas, the system can be speciﬁed by the value of the electron density ρ = N/V , where the electron charge density is constant and the system is subject to a constant external potential. The orbitals of this system are plane waves due to symmetry [61, 246]. Approximating the electron-electron interaction with the classical Hartree potential (neglecting exchange-correlation eﬀects) the total energy function was computed by Thomas and Fermi [246, 61] and the dependence of the kinetic and exchange energy on the density of the electron gas can be extracted and written as a local function of the density. This suggests that for an inhomogenous system it is possible to approximate the functional as an integral of a local function of the charge density [94].

This gives the local exchange-correlation energy per electron as a function of the local charge density εxc(ρ) Z Exc[ρ] = ρ(t)εxc(ρ(r))dr (3.19)

The local density approximation (LDA) is then to take εxc(ρ) to be the exchange and correlation energy density of the uniform electron gas with density ρ. εxc can be

69 separated into its exchange and correlation contributions,

εxc(ρ) = εx(ρ) + εc(ρ) (3.20)

The LDA reliably describes many systems, giving properties such as crystal structure, mechanical properties such as elastic moduli, and phase stability that are consistent with experimentally measured quantities.

Generalised Gradient Approximation GGA

The ﬁrst Generalised Gradient Approximation (GGA) was developed by Perdew et al. [194, 197] (PW 91) and extended by Perdew, Burke, and Ernzerhof [196, 195] (PBE).

The GGA is a variation on the Gradient Expansion Approximation (GEA), which is natural next step to the LDA, where the ﬁrst order gradient terms of the density are also considered. The GGA has a typical functional form Z Exc = ρ(r)εxc(ρ, ∇ρ)dr (3.21)

Including this ﬁrst order gradient term improves the description of the binding energy of the molecules [94]. The GGA gives better results for the total energy when compared to other approximations such as the LDA, particularly for inhomogenous densities.

3.0.5 Plane Waves, k-points and cutoﬀ energies

Because of the periodic nature of a repeating crystal structure, a plane-wave solution can be applied, where the solution can be expressed as a 3D Fourier series summed over all spatial frequencies with the right periodicity [98].

70 As plane waves are used to solve the Kohn-Sham equations, periodic boundary conditions are introduced. The interactions between repeated images must be accounted for by use of a suﬃciently large supercell and by choice of appropriate k-points.

Translational invariance of the system implies that a quantum number exists, the Bloch wave vector k, which can be used to index all electronic states. k is usually constrained to lie within the ﬁrst Brillouin zone in reciprocal space [166]. In order to evaluate key quantities such as charge density and total energy, it is necessary to integrate over the ﬁrst Brillouin zone.

The wave function at k-points that are close together will be almost identical, and so the integration over k can be approximated as a weighted sum over a discrete set of points. Therefore the calculation can be reduced to calculating the wavefunction at a discrete set of k-points in the ﬁrst Brillouin zone for a number of bands that is the order of the number of electrons per unit cell.

Meshes can either be centred on Γ, where Γ belongs to the mesh, or can be centred around Γ, which can break the symmetry of the cell geometry. A Monkhorst-Pack mesh is one of equally spaced k-points in a Brillouin zone, and is widely used. For certain cell geometries, such as hexagonal cells, evenly spaced meshes break the symmetry. By symmetrising the k-points to the cell geometry, and using a Γ point centred mesh, the symmetry of the cell can be preserved.

The kinetic energy cutoff controls the completeness of the basis set used, as at each k-point the plane waves that are included are controlled by the cutoff energy [140]. The kinetic energy cutoff must kept consistent between calculations to ensure that different numbers of plane waves are not used at each k-point. VASP applies an automatic convergence correction based on the cutoff energy used, and due to this correction, which can be only partial for bulk calculations, the energy might increase. When the same cutoff energy is used for all calculations, there is some cancellations of errors. The kinetic cutoff energy should be chosen according to the pseudopotential [147], where a default cutoff energy that will meet a certain uncertainty criteria will be given. However, in some circumstances such as surface or slab calculations,

71 amongst others, it is necessary to change the cutoff energy to ensure that sufficient plane waves are included to have convergence of the calculation at that cutoff energy.

3.0.6 Pseudopotentials

For most elements, the number of plane waves needed would be beyond practical limits, and so pseudopotentials are used instead of exact potentials [141]. This means that a lower cutoﬀ energy can be used and the calculation is much less computationally intensive [98]. The use of pseudopotentials also allows the calculation to concentrate on the valance electrons, which reduces the number of states that need to be included.

There are three types of psuedopotentials implemented in VASP (Vienna Ab initio Simulation Package) [144, 145, 142, 143], the software used for all the DFT calculations in this thesis. These are norm-conserving pseudopotentials, ultra-soft pseudopotentials, and projector augmented wave (PAW) potentials. All three of these methods are frozen-core methods, meaning that the core electrons are pre-calculated in an atomic environment and kept frozen throughout the subsequent calculations [141].

As the resulting Hamiltonian depends on the electron density which in turn depends on the wave functions, the equations must be solved iteratively to find the ground state [98]. This is done until the value has converged to a defined convergence tolerance. The forces may also be calculated and can also be given a convergence tolerance, which means performing a relaxation of the material to find the minimum in the forces. This is done in two steps: volume relaxation and ionic relaxation. Volume relaxation minimises the energy with respect to volume, and ionic relaxation moves the atoms of the material and finds the positions that yield the lowest energy.

The ﬁnal convergence must be considered for the k-point sampling and kinetic cutoﬀ energy to ensure that a true minimum energy and set of ionic positions has been found.

72 Chapter 4

A review of the literature relating to thermodynamic assessments of the B-C-Hf-Zr system

4.1 Introduction

Within the boron-carbon-hafnium-zirconium system there are four ultra-high temperature ceramic (UHTC) compounds that are widely used in the nuclear and aeronautical industries in applications relating to their high melting points, and strength and hardness at temperature.

Each of the binary phase diagrams containing the compounds hafnium diboride, zirconium diboride, hafnium carbide, and zirconium carbide has been the subject of extensive experimental study, but due to the nature of these materials accurate information at high temperatures is very limited as the experiments are expensive and challenging. The boron-carbon-hafnium-zirconium system contains a total of four unary systems, six binary subsystems (B-C, B-Hf, B-Zr, C-Hf, C-Zr, Hf-Zr) and four ternary subsystems (B-C-Hf, B-C-Zr, B-Hf-Zr, C-Hf-Zr), all of which have been assessed experimentally and using the CALPHAD approach with the limited data available. There is no existing CALPHAD assessment of the quaternary system in

73 System Author Year B Gurvich et al. [87] 1982 C Gustafson [90] 1986 Hf Saunders and Dinsdale [52] 1991 Zr Fernandez Guillermet [64] and Saunders et al. [230] 1987, 1988 B-C Kasper and Lukas [122], Rogl et al. [214] 1996, 2014 B-Hf Rogl and Potter [188], Bittermann and Rogl [28] 1988, 1997 Cacciamani et al. [41] 2011 B-Zr Rogl and Potter [213], Duschanek [55], Tokunaga et al. [248] 1988, 1998, 2008 Chen et al. [45], Cacciamani et al. [41] 2009, 2011 C-Hf Bittermann and Rogl [27] 1997 C-Zr Fernandez-Guillermet [66] 1995 Hf-Zr Bittermann and Rogl [29] 2002 B-C-Hf Rogl and Bittermann [212], Rogl et al. [214] 2000, 2014 B-C-Zr Duschanek and Rogl [55] 1998 B-Hf-Zr Cacciamani et al. [41] 2011 C-Hf-Zr Bittermann and Rogl [29] 2002 B-C-Hf-Zr No existing assessment

Table 4.1: The CALPHAD assessments for each subsystem within the B-C-Hf-Zr quaternary system. the open literature.

The published CALPHAD assessments in the open literature are listed in Table 4.1.

Recent studies show that transition metal contamination can have a signiﬁcant effect on the properties of the borides, and such materials are often manufactured in graphite crucibles, leading to some carbon contamination. As such, the existing studies on the binary subsystems can be treated as ternary or quaternary studies with very low levels of the third and fourth components, and experimental studies using ultra-pure samples and ﬁrst principles calculations become crucial in understanding the true binary systems.

The increasing use of ceramic matrix composites (CMCs) drives the need to understand the interaction between all elements in this system, as UHTCs such as zirconium diboride are being combined with other compounds such as carbides in order to improve the structural and oxidation properties. In order to fully understand

74 the behaviour in these systems, a consistent description of the quaternary B-C-Hf-Zr system is desirable.

The CALPHAD approach allows higher order systems to be built easily from assessments of their subsystems, with interaction parameters modelled between each set of elements. In order for the CALPHAD assessments to be combined in this way, it is necessary that the modelling of a phase is the same in all assessments. In the existing assessments of the ternary subsystems of the B-C-Hf-Zr system, the only existing B- Hf-Zr assessment uses diﬀerent crystal structure modelling for the diboride phases, and the B-Zr binary system used within it has diﬀerent stable phases present than the binary assessment than in the B-C-Hf and B-C-Zr ternary systems. As such, the existing binary and ternary assessments cannot be combined into a quaternary database, and in order to produce a consistent description of this system, certain systems need to be reassessed with compatible modelling.

Improvements to ﬁrst principles techniques in recent decades have enabled insight into the stability and properties of phases where experimental data may not be available. Consideration of point defects within the CALPHAD approach has become a topic of interest in recent years since the ﬁrst principles calculations of some defect properties can be shown to have an accuracy comparable to the uncertainty in experimentally measured properties [179, 73].

Within this thesis, the eﬀect of point defects in the form of vacancies and substitutional defects on the phase properties and phase diagram are considered. Within the CALPHAD approach, point defects are treated as an additional constituent on the sublattice, and have associated energies of interaction between components, although the interaction energy between vacancies is typically given a zero value. Calculations of defective structures using ﬁrst principles techniques provides data that can inform the assessment of the system.

Hafnium and zirconium diboride are both AlB2-type phases that exhibit similar properties as a result of hafnium and zirconium both being group IV transition metals with the same number of valence electrons. Similarly, hafnium and zirconium carbide

75 are known to exhibit similar properties. However, as a result of the mining process of these transition metals, these two metals are often mutually contaminated, which can be seen as substitutional defects at the metal sites. In Chapter 5 ﬁrst principles calculations of the hafnium and zirconium dibories and carbides with and without substitutional defects are presented and compared to experimental and theoretical data in the literature. In this literature review, the experimental data relevant to these phases is considered.

The carbon-zirconium system has been assessed once using the CALPHAD approach by Fernandez Guillermet [66] with very limited experimental data available. Since this assessment in 1995, the carbon-zirconium system has been the subject of several studies that obtain experimental data where it was previously limited, conﬂicting, or missing, and ﬁrst principles studies that examine the stability of the metastable phases and ordering of vacancies in the substoichiometric carbide phase. Taking this data into account, the carbon-zirconium system is reassessed in Chapter 7 using a method outlined in Chapter 6 to incorporate calculations of vacancy-related quantities into the assessment. The experimental data relating to the carbon-zirconium system is considered and reviewed in this literature review.

In conducting a CALPHAD assessment, one of the greatest challenges is assembling and considering all available experimental data. One thing that all of the subsystems in the boron-carbon-hafnium-zirconium system have in common is that the experimental data is limited beyond the stoichiometric or near-stoichiometic UHTC compounds of interest, and the available data has a signiﬁcant scatter due to large uncertainties associated with the measurements, inpurities in the samples, and the use of diﬀerent methods. A CALPHAD assessment must be conducted while considering all of this data and its associated uncertainty and reliability. This review aims to summarise the available data that is considered in the work in this thesis, while demonstrating the scarcity of data in certain areas, and the spread of data in others.

76 4.2 The (hafnium, zirconium) diboride and carbide phases

4.2.1 Hafnium diboride

Phases in the boron-hafnium system

Within the boron-hafnium system, the condensed phases are known to be a liquid phase, α (HCP) and β (BCC) hafnium that has some boron solubility within it, β-rhombohedral boron that has limited or no solubility of hafnium reported, and two intermediate phases, HfB and HfB2 that have limited to no solubility of either hafnium or boron.

The boron-hafnium phase diagram as assessed by Bittermann and Rogl [28] is shown in Figure 4.1, where the intermediate phases are modelled as line compounds, and the solubility of hafnium in boron is assumed to be negligible.

The intermediate phase that we are interested in, hafnium diboride, has a hexagonal hP3 structure with space group P6/mmm and AlB2 prototype.

Experimental investigations

Congruent melting temperature of hafnium diboride

There are very few studies of the solidus and liquidus in the boron-hafnium system. Rudy and Windisch [221] used metallographic methods, X-ray methods, and the

Pirani method to record a melting temperature of 3653±20 K for HfB2. Portnoi et al. [206] determined the melting temperature of HfB2 as 3643 K using X-ray and metallographic methods.

77 Figure 4.1: Phase diagram of the boron-hafnium system from the CALPHAD assessment by Bittermann and Rogl [28].

Homogeneity range of hafnium diboride

Numerous experimental investigations report hafnium diboride as having a small boron solubility of between 1 % and 2 % [188]. Kaufman and Clougherty [126] determined the phase boundaries of the HfB2, ﬁnding that both sub and superstoichiom- etry was present, with the homogeneity range extending from 65.5-67.7 at. % B at 2000 K.

Thermodynamic data

Enthalpy of formation

Paderno et al. [190] detemined the enthalpy of formation to be -105.02 kJ/mol of atoms by reducing HfO2 with boron and B4C at temperatures between 1673-1973 K.

Kaufman and Clougherty [126] report an enthalpy of formation of -96.2 kJ/mol of atoms.

78 Johnson et al. [118] measured the enthalpy of formation of HfB2 at 298.15 K to be -109.46±2.98 kJ/mol of atoms using ﬂourine bomb calorimetry.

Holcombe et al. [105] used measurements of the vapour pressure between 2400-2647 K to report the enthalpy of formation at 298.15 K as -106.55 kJ/mol of atoms.

Kirpichev et al. [132] reports the enthalpy of formation of HfB1.93 to be -85.1±2.0 kJ/mol of atoms using calorimetric methods in an argon atmosphere.

Heat capacity

The only heat capacity measurements for hafnium diboride are from Westrum and

Feick [266], who report measurements of the heat capacity of HfB2.035 between 5 K and 350 K using adiabatic calorimetry.

Heat content

Mezaki et al. [174] reports measurements of the heat content for 486-1150 K, and Pears et al. [193] report the heat content for 599.8-2813.7 K.

Bolgar et al. [34] measure the enthalpy of HfB2 using drop calorimetry for 1181- 2224 K.

Bolgar et al. [32] report measurements of the enthalpy of HfB2 from 447 K to 2295 K.

First principles investigations

Vajeeston et al. [256] report calculations of the enthalpy of formation at 0 K of several AlB2-type transition metal diborides, including HfB2.

Zhang et al. [278] report 0 K properties of the diborides of hafnium and zirconium including the enthalpy of formation using a variety of diﬀerent exchange-correlation functionals.

79 Zhang and Cheng [277] report calculations of the properties of stoichiometric HfB2 at 0 K from zero pressure to high pressure at 0 K. They report the structural properties and enthalpy of formation.

Lawson et al. [149] report the defect enthalpies of formation at 0 K of ZrB2 and

HfB2, where the defects are transition metal or boron vacancies and antisites, using several exchange-correlation functionals.

Xu et al. [271] report the structural and elastic properties and enthalpy of formation at 0 K for several transition metal diborides including ZrB2 and HfB2.

Pan et al. [191] report the enthalpy of formation for several intermediate compounds of different structures in the boron-hafnium system at 0 K, finding that the HfB2 phase is significantly more stable than the other intermediate compounds.

Dahlqvist et al. [50] use the special quasi-random structure (SQS) method to examine the eﬀect of boron vacancies on the stability and structure of HfB2 and other transition metal diborides.

4.2.2 Zirconium diboride

Phases in the boron-zirconium system

There is some debate about the stable condensed phases in the boron-zirconium system. There is agreement amongst all studies on the presence of a liquid phase, α (HCP) and β (BCC) zirconium in which there is some boron solubility, β-rhombohedral boron in which there is limited zirconium solubility, and intermediate phases ZrB2

(AlB12-type) and ZrB12 (UB12-type).

Amongst the experimental literature there are some reports of the ZrB phase (NaCl- type such as reported by Schedler [231] and Glaser and Post [81], who report it as being stable in the temperature range 1073-1523 K with a very narrow range of stoichiometry. Other experimental investigations, such as by Rudy and Windisch [222] conclude that there is no monoboride phase, and the phase observed in other

80 studies was stabilised by the presence of carbon, oxygen, and nitrogen impurities, because of the lattice parameters of the reported ZrB phases and the lack of ZrB presence in studies using high purity starting materials [213].

The extensive studies by Portnoi and Romashov [204] or Portnoi et al. also do not show presence of the ZrB phase, but Nowotny et al. [184], Shulishova and Shcherbak [236], and Haggerty et al. [92] report the presence of the phase, and Champion and Hagege [43, 44, 44] observe the formation of the ZrB phase by peritectoid formation below 1573 K.

First principles calculations have been conducted in studies such as by Tokunaga et al. [248] to determine the enthalpy of formation of a ZrB phase of diﬀerent stoichiometries. While the recent CALPHAD phase diagrams from Tokunaga et al. [248], Chen et al. [45], and Cacciamani et al. [41] all include ZrB as a stable phase, there is no conclusive agreement in the literature, and zirconium monoboride remains a controversial topic.

The phase diagram of the boron-zirconium system as assessed by Duschanek and Rogl [55] is shown in Figure 4.2, where the zirconium monoboride phase does not stabilise at equilibrium, and the solubility of zirconium in boron is assumed to be negligible.

The intermediate phase of interest is zirconium diboride that, like hafnium diboride has a hexagonal hP3 structure with space group P6/mmm and AlB2 prototype.

Experimental investigations

Congruent melting temperature of zirconium diboride

Agte and Moers [9] report the congruent melting temperature of zirconium diboride to be 3265 K

Glaser and Post [82, 207] report the melting temperature of the diboride to be 3323 K.

81 Figure 4.2: Phase diagram of the boron-zirconium system from the CALPHAD assessment by Duschanek and Rogl [55].

In the study by Rudy and Windisch [222] the melting point of the alloys were determined using the Pirani technique [218]. They report a congruent melting temperature of 3518 K. They attribute the diﬀerence between to values mesasured by Agte and Moers, and Glaser and Post to the variation of melting temperature at diﬀerent stable stoichiometries of the diboride phase, as the narrow stoichiometry range has a steep phase boundary.

Portnoi et al. [204] report a congruent melting temperature of 3518±25 K from their experimentally derived phase diagram.

Homogeneity range of zirconium diboride

Experimental studies (primarily X-ray methods) from van Arkel [259], McKenna [171], Kiessling [130], Norton et al. [183], Brewer et al. [37], Glaser and Post [82], Baroch and Evans [17], Nowotny et al. [184], Rudy and Benesovsky [217], Holleck et al. [106], Bernstein [23], Gebhardt and Cree [76], Portnoi et al. [204, 203], and

Kennard and Davis [129] report the formation of a ZrB2 phase with a solubility of

82 less then 1 at. % B.

Kaufman [125] used electron microprobe analysis to determine the width of the ZrB2 phase, ﬁnding the phase to extend on both sides of the line compound.

Kaufman and Clougherty [127] prepared various boride compounds by high pressure- hot pressing powders and using X-ray, metallographic, pyrometric, and chemical analysis to determine the phases and phase boundaries in the boron-zirconium system. Kaufman and Clougherty [127] reported the formation of the non-stoichiometric diboride phase ZrB1.89 at 2273-2473 K, and ZrB1.7 at 2073 K.

Thermodynamic data

Enthalpy of formation of ZrB2

Leitnaker et al. [152] used measurements of the vapor pressure to determine the enthalpy of formation of the diboride phase to be -97.6±9.8 kJ/mol of atoms at 298.15 K

Huber et al. [107] used oxygen bomb calorimetry to measure the enthalpy of formation of the diboride phase as -107.7±1.7 kJ/mol of atoms at 298.15 K

Trulson and Goldstein [250] used measurements of the vapor pressure to determine the enthalpy of formation of the diboride phase to be -103.3 kJ/mol of atoms at 298.15 K

Johnson et al. [118] used ﬂuorine bomb calorimetry at Zr:B 1:1.993±0.006 to give an enthlpy of formation of =-108.9±2.1 kJ/mol of atoms at 298.15 K

Kirpichev et al. [132] reported the enthalpy of formation of ZrB1.93 to be -92.7±1.9 kJ/mol of atoms using calorimetric methods in an argon atmosphere.

83 Heat capacity of ZrB2

Westrum and Feick [56] measure the heat capacity of ZrB1.97 from 5 to 350 K by adiabatic calorimetry.

Lonergan et al. [160] report measurements of the heat capacity of zirconium diboride containing very low amounts of hafnium as a contaminant, ﬁnding that the heat capacity decreases with added hafnium content.

Heat content of ZrB2

Valentine et al. [258] report measurements of the heat content of ZrB1.97 from 410 to 1125 K using drop calorimetry, where the samples were made in the study by Westrum and Feick [56].

Matthieu et al. [168] report measurements of the high temperature heat content of

ZrB2.

Schick [103] also reports heat content data expressed as thermodynamic functions.

Vapor pressure

Leitnaker et al. [152], Trulson and Goldstein [250], Wolﬀ and Alcock [268], and Lyon and Linersky [162] all report measurements of the vapor pressure across the Zr +

ZrB2 two phase region, but the Gibbs energy of formation of the ZrB2 phase derived from these results does not agree across the studies.

First principles investigations

Vajeeston et al. [256] report calculations of the enthalpy of formation at 0 K of several AlB2-type transition metal diborides, including ZrB2.

84 Mihalkovic et al. [176] present ﬁrst principles enthalpies of formation for diﬀerent structures in the boron-zirconium system at 0 K, and determine that the only stable intermediate phase is hP3 ZrB2 , with cF52 ZrB12 being less than 100 meV above the convex hull.

Tokunaga et al. [248] calculated the enthalpies of formation of several intermediate compounds, including diﬀerent ZrB structures. They report the 0 K enthalpy of formation of ZrB2 as -100.3 kJ/mol of atoms.

Zhang et al. [278] report 0 K properties of the diborides of hafnium and zirconium including the enthalpy of formation using a variety of diﬀerent exchange-correlation functionals.

Middleburgh et al. [175] uses atomic scale modelling to look at point defect structures within zirconium diboride up to high temperatures, ﬁnding that non-stoichiometry can be attributed to Schottky and Frenkel defect processes.

Lawson et al. [149] report the defect enthalpies of formation at 0 K of ZrB2 and

HfB2, where the defects are transition metal or boron vacancies and antisites, using several exchange-correlation functionals.

Xu et al. [271] report the structural and elastic properties and enthalpy of formation at 0 K for several transition metal diborides including ZrB2 and HfB2.

Li et al. [158] examine diﬀerent structures across stoichiometries and pressures to determine the convex hull and thus stable structures. At zero pressure, they ﬁnd that the only stable phases at 0 K are ZrB2 and ZrB12, reporting the enthalpies of formation of each phase.

Dahlqvist et al. [50] use the SQS method to examine the eﬀect of boron vacancies on the stability and structure of ZrB2 and other transition metal diborides.

85 4.2.3 Hafnium carbide

Phases in the carbon-hafnium system

Within the carbon-hafnium system, the condensed phases are known to be a liquid phase, α (HCP) and β (BCC) hafnium that dissolve the other phases within them, carbon in the graphite phase that has limited or no solubility of hafnium reported, and a single intermediate phase, HfCx, that is stable over a wide range of stoichiometry, facilitated by the presence of carbon vacancies.

The carbon-hafnium phase diagram as assessed by Bittermann and Rogl [27] is shown in Figure 4.3, where it is assumed that there is no solubility of hafnium in graphite.

The intermediate phase, HfCx has a cubic cF8 structure, with space group Fm3m, and NaCl prototype. The carbon atoms can be considered to ﬁll the octahedral interstitial sites in FCC hafnium, where the HfC phase is stable with these sites partially or fully ﬁlled giving a wide single phase region in the phase diagram.

Figure 4.3: Phase diagram of the carbon-hafnium system from the CALPHAD assessment by Bittermann and Rogl [27].

86 Experimental investigations

The carbon-hafnium phase diagram, include the congruent melting temperature and phase boundaries have been measured experimentally in numerous studies [8, 49, 20, 123, 205, 15, 30, 48, 124, 4, 215, 225, 5, 51, 57] giving measurements with numerous inconsistencies that suggest signiﬁcant experimental uncertainty. Experimental studies by Rudy et al. [215, 216] and Sara [225] report systematic and comprehen- sive studies of the width and melting temperatures of the HfCx phase, using X-ray analysis, metallography, themal analysis, and chemical analysis.

Thermodynamic data

Enthalpy of formation

There are limited measurements of the enthalpy of formation for stoichiometric HfC in the literature. Coﬀman et al. [47], and Fesenko et al. [72], who measured it it be -131.35 kJ/mol of atoms, and -105.25±10.45 kJ/mol of atoms at 298 K respectively, both using the Langmuir method. Zhelankin and Kutsev [281] and Kornilov et al. [139] use combustion calorimetry to measure -115.55±1.05 kJ/mol of atoms and - 104.7 kJ/mol of atoms at 298 K respectively.

Mah [3], McClaine and Little [169], Zhelankin and Kutsev [281], Kornilov et al. [139], and Maslov et al. [167] report enthalpies of formation of substoichiometric

HfCx compounds at 298 K, and Berkane [21] reports the enthalpy of formation of several stoichiometries in the homogeneity region at 1870 K.

Heat capacity

The heat capacity of stoichiometric HfC was measured by McDonald et al. [170] from 1300-2100 K using adiabatic calorimetry. McClaine [169] and Westrum and Feick [266] report measurements of diﬀerent near-stoichiometric compounds from 5-298 K

87 and 5-350 K respectively using adiabatic calorimetry, and Buravoi and Taubin [40] report measurements of the heat capacity of HfC0.99 taken using pulse calorimetry.

Heat content

There are no experimental heat content measurements available in the literature for stoichiometric HfC. Neel et al. [180] and Coﬀman et al. [47] report measurements of from 540-3016 K and 440-1367 K of substoichiometric HfCx respectively. Levin- son [154] and Bolgar et al. report measurements of near-stoichiometric HfC using drop calorimetry and the mixing method. Guseva et al. [89] reports drop calorimetry measurements between 1300 K and 2500 K of several stoichiometries across the homogeneity range.

First principles investigations

Iikubo et al. [111] calculate the heat capacity of HfC up to 3000 K using the quasiharmonic approximation of DFT with LDA exchange-correlation functionals. They found that below 2000 K there was good agreement with the experimental data in the literature, and above 2000 K there was a sharp increase in the calculated value that diverged from the trend of the experimental values in the literature.

Zeng et al. [276] use a first principles evolutionary algorithm to search for stable hafnium carbide phases with different structures and stoichiometries, finding that NaCl-type HfC was by far the most stable compound, but that other compounds may also be stable, facilitated by ordering of vacancies as suggested by Gusev and Rempel [88]. Enthalpies of formation are reported with and without zero point energy (ZPE) corrections.

Razumovskiy et al. [208] examined the properties and interactions of point defects in transition metal carbides, reporting the vacancy formation energies and vacancy- vacancy pair interaction energies for the diﬀerent nearest neighbour sites.

88 Yu et al. [274] also used a ﬁrst principles evolutionary algorithm to determine the stable structures in the carbon-hafnium system, examining the ordering of vacancies and stacking faults within the structures and reporting the enthalpy of formation.

4.2.4 Zirconium carbide

As the carbon-zirconium system is treated more thoroughly in Chapters 6 and 7, the full carbon-zirconium sysem is considered in Section 4.3, which also contains the information relevant to the calculations of zirconium carbide and zirconium carbide with hafnium substitutional defects at the zirconium sites in Section 5.3.2.

The phase diagram of the carbon-zirconium system from the CALPHAD assessment by Fernandez Guillermet [66] is shown in Figure 4.4 for reference.

Figure 4.4: Phase diagram of the carbon-zirconium system from the CALPHAD assessment by Fernandez Guillermet [66].

89 4.3 The carbon-zirconium system

The carbon-zirconium system has been the subject of experimental and theoretical investigations for more than 150 years [240]. A review of the experimental studies in the literature is given here, with key experimental information and an assessment of the uncertainty associated with the measurement given where suﬃcient information is available.

All the experimental studies conﬁrm that within the carbon-zirconium system, there is only one intermediate carbide phase, ZrC, which is stable over a range of stoichiometry in a NaCl-type structure. This phase coexists with the three other stable phases and liquid. The other stable phases in the system are carbon in the graphite structure, and two allotropes of zirconium: hexagonal close packed at low temperature, and base centred cubic at higher temperatures, which each have some reported solubility of carbon.

The only CALPHAD assessment of the carbon-zirconium system in the open literature is from 1995, by Fernandez Guillermet [66].

4.3.1 Overview of phase diagram data

The carbon-zirconium phase diagram is shown with a selection of the reported experimental phase diagram data in Figure 4.5.

Investigations of the melting temperature of zirconium carbide

Friederich and Sittig [74] fabricated ZrC from ZrO2 and carbon powders and reduced at at least 1900K to produce a solid sample. The samples were heated in a furnace in a hydrogen atmosphere and the melting point was measured using a Holborn- Kurlbaum pyrometer. Frederic and Sittig, 1925 [74] observed melt at 3300 K and 3500 K in a sample that was reported to contain 7-10 % graphite. The exact melting

90 Figure 4.5: Phase diagram of the carbon-zirconium system from the CALPHAD assessment by Fernandez Guillermet [66] shown with some of the reported experimental phase diagram superimposed [226, 219, 216, 6, 242]. temperature could not be determined as a white smoke obscured the sample, so an uncertainty of 200 K is estimated.

Agte and Altherthum [8] mixed ZrO2 and carbon powders and heated and hot- pressed. The melting point was determined using a method developed by Pirani and Alterthum [200]: a borehole was treated as a blackbody, whose temperature was determined with a Siemens and Halske micropyrometer calibrated to 3775 K. The sample was heated in argon (15 % nitrogen). The melting temperature was measured to be 3805±125 K.

Agte and Moers [9] investigated diﬀerent methods of fabricating zirconium carbide

(sintering and producing from ZrCl4). They report the melting temperature of ZrC to be 3805 K. measured using the method developed by Pirani and Alterthum [200]. By adding carbon, the melting temperature was reported to be lowered to 2703 K.

91 Brownlee [38] references measurements by F.W. Glaser [80] of the melting temperature of ZrC as 3448±50 K. Brownlee [38] made zirconium carbide by carburising zirconium foil (containing up to 0.5 wt. % hafnium) giving a powder with 1.84 wt. % free carbon which was then sintered. The melting point was measured as 3808 K by melting in an argon-arc furnace and using a disappearing-ﬁlament optical pyrometer. The accuracy of these measurements is reported as “not very high”.

Farr, in unpublished work (cited by [229]) reports the highest melting point of zirconium carbide to be 3673±50 K at 45 at. %C.

Sara et al. [227, 228, 229] produced samples of zirconium carbide with carbon content between 0 and 90 at. % from zirconium hydride and graphite powders, before sintering. They were heated in a resistance furnace in vacuum, and DTA, high temperature x-rays, and optical pyrometers calibrated to 3773 K were used using blackbody holes in the specimens as described in [227, 228]. Sara et al. [229] report a series of measurements at various compositions and temperatures. From this data, it is suggested that the highest melting point is 3693 K at approximately 46 at. %C. The solidus is reported to be relatively ﬂat with composition.

Sara [226] used DTA, x-ray, and metallographic methods to investigate zirconium- carbon samples at a variety of compositions, produced from spectroscopic graphite powder and zirconium hydride powder. From the melting point investigations, it was observed that the solidus of the zirconium carbide phase is relatively ﬂat with composition, with the highest melting point at 3693 K at approximately 46 at. %C. Sara et al. report the uncertainty on their temperatures to be within ±1 %.

Rudy et al. [219] performed experiments to explore the areas of the phase diagram that remained uncertain following investigations by Sara et al. [227, 228, 229, 226]. Samples were prepared using zirconium and carbon powders, as well as pre-made

ZrH2 and ZrC powders. Melting point determinations, DTA, and metallographic analysis was performed. The chemical analysis of the carbon content of the samples is believed to be accurate within ±0.05 wt. %.

Rudy et al. [219] record a maximum melting temperature of 3713±20 K at a carbon

92 concentration of approximately 45 at. %C.

Adelsberg et al. [7] determined the liquidus of the Zr+ZrC region for 2273-3073 K. This was done using the reaction between liquid zirconium (99.9 % purity) and graphite (99.9 % purity) in argon or helium gas. Temperatures were measured with a Milletron two-colour pyrometer. Temperatures in the crucible were partially calibrated using a Zr-ZrC eutectic temperature of 2093 K as determined by [229, 226]. Adelsberg et al. report a maximum error in temperature of around ±50 K, corre- sponding to a deviation in composition of ±1 at. %.

Rudy and Progulski [220] use a Pirani furnace to measure the congruent melting point of ZrC to be 3713±25 K at 45±1 at. %C.

Rudy et al. [216] summarise previous data reported in [219] as well as reproducing some new data (not previously reported).

Investigations of the boundaries of the homogenous region

Umanski reports a width of the homogeneity region of 27-50 at. %C (reported by [19] and [229]). Kovalski and Makarenko report a width of 36-50 at. %C (reported by [19] and [229]). Samsonov and Rozinova report a width of 21-50 at. %C (reported by [19] and [229]).

Benesovsky and Rudy [19] investigated the width of the homogeneity range of zirconium carbide using x-rays to investigate hot pressed samples prepared from carbon and zirconium powders. At 1673 K, it is reported that zirconium carbide is homogenous for 35-50 at. %C.

Farr [59] (reported by [229]) reports boundaries of the homogeneity region at 35.4 at. %C (no temperature dependence given) and at 49.4 at. %C at 2673 K. The ZrC/ZrC+C boundary was measured as 49.1 at. %C at a eutectic temperature 3123±50 K.

Wallace et al. [262] report the lower boundary of the zirconium carbide region to be at Zr0.64C0.36 at 1773 K. At 2373 K the homogenous phase is reported to extend

93 from around 35-50 at. %C. The samples were fabricated from 99 % purity zirconium sponge and spectrographic grade carbon rod. Chemical analysis and x-ray techniques were used to investigate the phase boundaries.

Sara et al. [228] report the ZrC phase boundary is reported as being at 38.5 at. %C at 2173 K from metallographic evidence, Sara et al. [229] found this to be invariant with temperature to 3573 K. The high-carbon boundary is reported at 48.9 at. %C, and invariant between 3123 K and 3573 K. Sara [226] reports again that the high- carbon boundary at 3573 K is 48.9 at. %C, and the low-carbon boundary is at 38.5 at. %C for 2173-3303 K, with a “small amount” of liquid forming by 3573 K.

From metallographic studies, Rudy et al. [219] report the boundaries of the homogenous zirconium carbide phase to be 37.5±0.5 at. %C and approximately 50 at. %C for samples quenched from above 3073 K.

Adelsberg et al. [7] reported an upper stoichiometry for ZrC as C/Zr=0.975 for room temperature x-ray studies. Results from this study appeared consistent with the relatively temperature independent phase range as reported by [229, 226, 219].

Investigations of the solubility of carbon in zirconium

Anderson et al. [11] report the solubility of carbon in zirconium to be 0.35-0.38 wt. %C (reported by Benesovsky et al. [19] and Sara et al. [228]).

Rudy et al. [219] report a very slight carbon solubility in zirconium at the α-β zirconium phase transition. The carbon solubility in the bcc-structure zirconium is also reported as being small.

Investigations of the solubility of zirconium in carbon

Godin et al. [84] used pure carbon rods and 99.8 purity zirconium pieces to produce powders. By x-ray and spectral analysis, they report that the solubility of zirconium in carbon, if it occurs, is below 0.01 %.

94 Investigations of the Zr-ZrC eutectic

Benesovsky and Rudy [19] report a eutectic at around 5 at. %C at 2103 K.

Farr [59] (reported by [229]) measured a eutectic temperature at 2083 K.

Sara et al. [228] observed the Zr/ZrC solidus to be 2133±20 K based on incipient melting experiments and DTA on heating and cooling. Sara et al. [229] report the invariant point of the Zr-ZrC eutectic at less than 2 at. %C. This is reported again by Sara [226].

Rudy et al. [219] report the Zr-ZrC eutectic temperature to be 2108±15 K, with the eutectic composition at less than 5 at. %C.

Rudy and Progulski [220] use a Pirani furnace to measure the Zr-ZrC eutectic point as 2108±20 K at 3±1.5 at. %C.

Bhatt et al. [24] use the optical pyrometric method known as the spot technique to measure the Zr-Zr+C eutectic temperature, by placing pure zirconium metal in- side a graphite eﬀusion cell, and heating it under a dynamic vacuum in an electron bombardment furnace. It was measured to be 2127±5 K.

Investigations of the ZrC-C eutectic

Portnoi et al. [202] report a measurement of the ZrC-C eutectic temperature as 3193±50 K using optical determination of the appearance of liquid.

Farr [59] (reported by [229]) measured a eutectic temperature at 3123±50 K at 49.1 at. %C in equilibrium with graphite.

Wallace et al. [262] measured the melting temperature of Zr0.28C0.72 (with some free carbon) to be 3143±30 K.

Andseron et al. (Bureau of Mines report) [11] report the eutectic temperature as 3072 K with the eutectic composition at 19.15 wt. %C (64.5 at. %C) [228].

95 Sara et al. [228] reports the invariant point of the eutectic at 3123 K and 65±3 at. %C. This is reported again by Sara [226].

Rudy et al. [219] report the ZrC-C eutectic at 3184±12 K and 64.5±0.5 at. %C.

Adelsberg et al. [7] reported the ZrC-C eutectic temperature to be 3253±50 K. This is reported again in [6].

Rudy and Progulski [220] use a Pirani furnace to measure the ZrC-C eutectic point as 3184±19 K at 64.5±1 at. %C.

Zotov and Kotelnikov [282] report the eutectic temperature as 3200±50 K.

4.3.2 Thermodynamic data

Enthalpy of formation of zirconium carbides

Mah and Boyle [163] report the heat of formation of zirconium carbide using combustion calorimetry. The zirconium carbide was prepared using graphite and hafnium- free zirconium sponge prepared by [110]. The energy of combustion results were adjusted for the presence of zirconium nitride, zirconium monoxide, and free car- o bon. The raw data is provided, but using the corrected values give ∆H298.16 = −651.7kJ/mol of atoms as the standard heat of combustion of zirconium carbide. Using the heat of formation for zirconium dioxide from [110], the heat of formation o of ZrC from the elements is found of be ∆Hf,298.16 = −92.3 ± 3.1kJ/mol of atoms.

Mah [163, 3] (reported by [66, 16]) measured the enthalpy of combustion of ZrC0.93 o and ZrC0.99 as ∆H298 = −645.6 ± 1.2kJ/mol of atoms, where the impurities were well deﬁned. For a sample with composition ZrC0.71, the enthalpy of combustion was o measured as ∆H298 = −722.21 ± 1.7kJ/mol of atoms.

Baker et al. [16] produced samples of ZrCx by arc-melting zirconium metal with spectroscopic graphite rods, before grinding and sintering. The enthalpy of formation was determined using a Lund rotating bomb calorimeter, modiﬁed for use without

96 rotation. Six diﬀerent samples of ZrCx were tested, and the energies of combustion at 298 K reported. By using a least-squares ﬁt to these values, the enthalpy of formation o of stoichiometric ZrC was given as ∆Hf,298 = −103.5 ± 1.3kJ/mol of atoms. Kornilov et al. [137] determined the heats of formation for seven compositions of

ZrCx for 0.716 < x < 0.99 which they represent by the equation

1 o 2 −2 2 ∆Hf (ZrCx) = −(13.3+36.1x)±1.2[0.164+(x−0.845) /9.4×10 ] kJ/mol of atoms (4.1) which is consistent with results from [16].

Maslov et al. [167] use high temperature synthesis to measure the heat of formation o of ZrC0.92 as ∆Hf,298 = −94.8±2.9kJ/mol of atoms, and from this calculate the heat o of formation of stoichiometric ZrC as ∆Hf,298 = −103.3 ± 2.0kJ/mol of atoms.

Heat capacity of zirconium carbides

Westrum and Feick [265] use an adiabatic vacuum cryostat and a copper calorimeter to measure the heat capacity of zirconium carbide at temperatures between 5 K and 350 K. The samples were prepared from commercial hafnium-free zirconium carbide powder with some free carbon powder added to correct the carbon deﬁciency. The mass per formula unit of ZrC was reported as 103.231, which corresponds to a stoichiometric compound within the reported accuracy of measurement. The error in the measured heat capacity is reported as decreasing from around 5 % at 5 K to around 1 % at 10 K, and less than 0.1 % above 50 K. The temperatures were measured using a platinum-resistance thermometer, and are reported to be consistent with the thermodynamic temperature scale to within 0.04 K for 90-350 K and within 0.03 K for 10-90 K. Based on these measurements, Westrum and Feick produced thermodynamic functions for the heat capacity, entropy, enthalpy, and Gibbs energy of the compound between 5 K and 350 K.

97 Heat content of zirconium carbides

Levinson [155] reports measurements of the heat content of ZrC measured at various temperatures between 1275 and 2788 K using a drop calorimeter (with reference to the heat content at 310 K). The samples contained 88.24 wt. % Zr, 11.13 wt. % combined C, 0.24 wt. % combined C, and small quantities of oxygen and hafnium. The reported uncertainty on the temperature measurements is ±0.5 %, and the estimated error on the heat content measurements is reported as ±1.4 %. The experimental results were used to ﬁt an equation for heat content valid between 1275 and 2788K.

Kantor and Fomichev [119] report measurements of the enthalpy for ZrC over a temperature range of 500 K to 2400 K, with temperatures measured by thermocouple and optical pyrometer, and enthalpies measured with an electric calorimeter.

Bolgar et al. [33] determined the enthalpy of ZrC0.98 in the temperature range 298-2500 K using a high-temperature, high-vacuum calorimeter. It is reported that error in the temperature measurement is less than 0.8 % and the relative error in the measured heat content is less than 1.5 %. The data were summarised in the following equation:

o o −3 2 HT − H298 = 6.018T + 21.92 × 10 T − 381.0 kJ/mol of atoms (4.2)

From this data, thermodynamic functions were also produced.

Turchanin and Fesenko [254] investigate the enthalpy of various stoichiometries of zirconium carbide in the temperature range 1300-2500 K. Samples were fabricated from the elements, producing stoichiometries of ZrC0.69, ZrC0.76, and ZrC0.99, and the heat content was measured with a relative error of less than 1.1 % using a high temperature vacuum calorimeter, The temperatures were measured with an uncertainty of 0.8 % using an optical pyrometer. The experimental results are ﬁtted into equations for the enthalpy and heat capacity for each composition.

98 For ZrC0.99:

o o −3 2 HT − H298 = 7.329T + 1.663 × 10 T − 7749 kJ/mol of atoms (4.3)

For ZrC0.76:

19.957 × 103 Ho −Ho = 7.762T +1.174×10−3T 2+1.659×106 exp − −1595 kJ/mol of atoms T 298 T (4.4)

For ZrC0.69:

31.397 × 103 Ho −Ho = 8.057T +1.928×10−3T 2+7623×109T −2 exp − kJ/mol of atoms T 298 T (4.5)

Activity as a function of composition

Storms and Griffin [242] report measurements of the partial enthalpy of vaporisation and activity in ZrC as a function of composition (between ZrC0.58 and ZrC+C) and as a function of temperature (between 1700 K and 2200 K) using a high temperature mass spectrometer by Knudsen effusion. The samples were prepared by arc-melting reactor-grade zirconium metal and spectroscopic graphite rods. The activity coeffi- B cients A and B in the form ln(a) = A + T were determined from the measurements by a least-squares fit. The partial enthalpy of vaporisation was determined using a Gibbs-Duhem integration.

Based on measurements of partial enthalpy and activity as a function of composition in ZrC, Storms and Griﬃn [242] place the lower phase boundary at 2100 K at ZrC0.565, moving slightly higher as the temperature is lowered.

As well as their measurements, Storms and Griﬃn [242] provide thermodynamic functions for ZrC0.96. Temperature measurements were made using an optical pyrometer looking through a hole in the side of the eﬀusion cell, and was reported to be accurate to within ±4 K.

99 Andrievskii et al. [14] determine the partial pressure of carbon vapour within the homogeneity region. Samples were made by hot-pressing zirconium and carbon black to create the following compositions: ZrC0.63, ZrC0.67, ZrC0.84, ZrC0.89, ZrC0.92, and

ZrC0.95. The temperature measurement accuracy is reported as ±15-20 K, with a ±30 % error in determining the vapour pressure. The partial vapour pressure, ac- B tivity, and heat of sublimation activity are reported, in the form ln(p) = A − T and B ln(a) = A− T for both carbon and zirconium activity, where the experimental results have been put into this form using a least-squares ﬁtting. The partial enthalpy was determined using a Gibbs-Duhem graphical integration.

Activity as a function of temperature

Storms and Griﬃn [242] report measurements of the activity and free energy of formation as a function of temperature for a C/Zr ratio of 1.97 using the same methods used to determine the activity as a function of composition.

4.3.3 Review of ﬁrst principles data

The ﬁrst principles data relating to the carbon-zirconium system is divided into thermodynamic properties, which can be used directly in a CALPHAD assessment in place of experimental data, and point defect investigations, which take many forms, and can be considered explicitly in the assessment (in the case of compound formation energies) or used to inform the modelling (such as in the case of defect formation energies or interaction energies).

Thermodynamic properties

Iikubo et al. [111] calculate the heat capacity of ZrC up to 3000 K using the quasiharmonic approximation of DFT and LDA exchange-correlation functionals. They

100 found that above 2000 K, there was a sharp increase in the calculated heat capacity that was divergent from the experimental data in the literature.

Abdollahi [2] report calculations of the pressure and temperature dependent thermodynamic properties of ZrC using the quasiharmonic approximation of DFT with GGA exchange-correlation functionals.

Duff et al. [54] report thermodynamic properties of stoichiometric ZrC up to the melting point using DFT to obtain the fully anharmonic vibrational contribution and electron excitation contribution to the free energy. It was found that above the Debye temperature, the anharmonic effects had a significant contribution to the thermodynamic properties. This was done using both LDA and GGA exchange-correlation functionals, where it was found that the LDA calculations better reproduced the experimental data available.

Point defects

Kim et al. [131] consider the point defect structures within ZrC, reporting carbon and zirconium vacancy formation energy and carbon and zirconium antisite formation energy at 0 K.

Zhang et al. [280] consider the phase stability in the presence of native point defects in ZrC, finding that carbon vacancies and carbon interstitial defects are the dominant defects, existing primarily in certain configurations. The formation energy of these defects in different configurations is reported.

Zhang et al. [279] use the cluster expansion and first principles methods to examine the self-assembly of carbon vacancies in sub-stoichiometric ZrC, finding a significant tendency towards vacancy ordering at 0 K.

Razumovskiy et al. [208] examine the properties and interactions of point defects in transition metal carbides, reporting the vacancy formation energies and vacancy- vacancy pair interaction energies for the diﬀerent nearest neighbour sites.

101 Yu et al. [274] use a ﬁrst principles evolutionary algorithm to determine the stable structures in the carbon-zirconium system, examining the ordering of vacancies and fault stacking within the structures and reporting the enthalpy of formation.

Xie et al. [270] use first principles methods to examine the effect of the ordering of vacancies in different structural configurations on the stability of non-stoichiometric ZrC. The convex hull produced shows several vacancy-ordered configurations that are stable at 0 K.

102 Chapter 5

Examining the eﬀect of hafnium and zirconium substitutional point defects on the (Hf,Zr) carbides and borides

5.1 Background

Zirconium and hafnium are both group IV transition metals from period 5 and period 6 respectively. Having the same number of valence electrons, zirconium and hafnium form many of the same compounds with other elements. Chemically, zirconium and hafnium are very similar, both having hexagonal close-packed (hcp) strutures at low temperatures, and allotropic phase transformations to a body-centred cubic structure (bcc) at 1136 K and 2016 K respectively.

Zirconium and hafnium have largely similar mechanical properties, although they have signiﬁcantly diﬀerent densities (at room temperature being 6.52 and 13.31g/cm3 respectively), melting points (2128 and 2506K), and behaviour under neutron irradiation (where zirconium is almost neutron transparent, and hafnium readily absorbs neutrons, with a neutron absorption cross-section 600 times than of zirconium).

103 These properties make zirconium, hafnium, and their compounds ideal materials for use in diﬀerent applications in nuclear reactors. Hafnium compounds such as hafnium diboride are used as reactor control rods thanks to the high absorption of neutrons, while zirconium alloys are used to create cladding for nuclear fuels thanks to their low neutron capture.

Zirconium and hafnium occur together in the Earth’s upper crust in a solid solution with zirconium compounds such as zircon, ZrSiO4, from which it is mined. Due to their similar chemical properties, hafnium and zirconium can be difficult to separate, with commercial zirconium metal containing between 1 % and 2.5 % hafnium typically. For some applications, this level of impurity is not a problem because of the similar mechanical properties of the two transition metals. However, for use in nuclear applications, even small amounts of the other metal as a contaminant can have a significant effect on the properties.

5.2 Investigations of purity

The experimental studies of the properties of zirconium and hafnium diboride have signiﬁcant scatter amongst the results, as examined in Chapter 4.

Lonergan et al. suggest that the differences in the properties measured for ZrB2 could be attributed to variations in the hafnium content of the studied materials [159]. They noted that hafnium additions in the compound affect its thermal properties, and in a later study, Lonergan et al. systematically investigate the effect as a function of the impurity concentration, through experimental measurements of the thermal conductivity and heat capacity at room temperature [160]. The experimentally measured heat capacity as a function of atomic % hafnium in zirconium diboride can be seen in Figure 5.1.

It can be seen that the measured heat capacity decreases by over 14% over up to 0.4 at. % added hafnium. Furthermore, experimental studies of the heat capacity of hafnium diboride and zirconium diboride show that the heat capacity of hafnium

104 Figure 5.1: Experimentally measured heat capacity of zirconium diboride with varying amounts of hafnium dopant, from Lonergan et al. [159]. diboride is higher than that of zirconium diboride, meaning that the experimental data from Lonergan et al. is showing the opposite trend than we would expect.

5.2.1 The eﬀect of transition metal contamination in the existing CALPHAD models

There is only one existing CALPHAD assessment of the boron-hafnium-zirconium system in the literature, which is Cacciamani et al., 2011 [41]. This assessment uses the boron-hafnium CALPHAD assessment from Bittermann and Rogl [28] and the boron-zirconium assessment from Duschanek and Rogl [55], modifying the solubility ranges of HfB2 and ZrB2 respectively by allowing the formation of vacancies on the metal sublattice. The experimental data used in these CALPHAD assessments are considered in Chapter 4.

The phase diagrams for the boron-zirconium and boron-hafnium systems are shown

105 Figure 5.2: Phase diagram of the boron- Figure 5.3: Phase diagram of the boron- zirconium system from the assessment by hafnium system from the assessment by Bit- Duschanek and Rogl [55]. termann and Rogl [28]. in Figures 5.2 and 5.3 respectively. The phase diagrams are broadly similar, as a result of the similarities between hafnium and zirconium, but the boron-zirconium system has a stable zirconium dodecaboride phase, ZrB12, and the boron-hafnium phase has a stable hafnium monoboride phase, HfB. It can also be seen that the allotropic transformation of pure zirconium occurs at a much lower temperature compared to its solidus than hafnium.

An isothermal slice from the ternary phase diagram from the assessment by Caccia- mani et al. is shown in Figure 5.4. It can be seen that the AlB2-structure diboride phase extends across the phase diagram from ZrB2 to HfB2 indicating that the metal atoms are substituted on the metal lattice to fulﬁl the diﬀerent compositions of this phase. In the CALPHAD modelling in this assessment, this phase is modelled as an ideal solution between the two diborides, where there is no enthalpy of mixing and thus no interaction parameters (excess Gibbs energy) parameters are modelled.

The Gibbs energy functions for this assessment are not published with the paper, but were obtained by private communication with the author. While this assessment reproduces experimental binary and ternary phase diagrams from the published paper, the heat capacity of zirconium diboride can be seen to be non-physical at high tem-

106 Figure 5.4: Isothermal slice of the boron-hafnium-zirconium system at 1000 K from the assessment by Cacciamani et al. [41]. peratures in the assessment by Cacciamani et al., and to diﬀer from the assessment by Duschanek and Rogl at low temperatures. This can be seen in Figure 5.5.

For hafnium diboride, the heat capacity from the modiﬁed assessment by Cacciamani et al. and the original assessment by Bittermann and Rogl are shown in Figure 5.6 and are very similar.

As the assessment by Cacciamani et al. does not show physical behaviour at high temperature, the assessments by Duschanek and Rogl and Bittermann and Rogl can be used to compare the heat capacity of zirconium diboride and hafnium diboride. From Figure 5.7 we can see that in these CALPHAD assessments, at low temperature, the heat capacity of ZrB2 is less than that of HfB2, and remains lower until very high temperatures (around 3800K) where the heat capacity of ZrB2 becomes higher.

As it is the only CALPHAD assessment of the B-Hf-Zr system in the open literature,

107 Figure 5.5: Heat capacity of ZrB2 from the assessment by Duschanek and Rogl [55] and from the assessment by Cacciamani et al. [41].

Figure 5.6: Heat capacity of HfB2 from the assessment by Bittermann and Rogl [29].

108 Figure 5.7: Heat capacity of ZrB2 from the assessment by Duschanek and Rogl [55] and from the assessment by Cacciamani et al. [41], and of HfB2 from the assessment by Bittermann and Rogl [29]. we have no choice but to use the heat capacity from the assessment by Cacciamani et al. to compare with the experimental data by Lonergan et al. [159]. At low temperatures (below 2000K) the heat capacity from Cacciamani et al. is consistent with the earlier assessment by Duschanek and Rogl [55] and experimental data. As the experimental heat capacities measured by Lonergan et al. are at room temperature, we can use this assessment as a point of comparison, but it must be noted that it cannot be taken as completely physically representative of the system.

From the assessment by Cacciamani et al. [41], the heat capacity of (Hf,Zr)B2 at room temperature increases linearly as a function of hafnium content between the end member heat capacity values at this temperature. In Figure 5.8 the experimental heat capacity is shown along with the value from the assessment at each composition.

As the experimentally measured heat capacity of the ultra-pure samples are higher than the heat capacity in the CALPHAD assessment, decreasing with hafnium content, it is likely that the experimentally measured heat capacity data used in the

109 Figure 5.8: Experimentally measured heat capacity data of ZrB2 with varying hafnium doping from Lonergan et al. [159] with the equivalent heat capacity values from the CALPHAD assessment of the B-Hf-Zr system by Cacciamani et al. [41].

CALPHAD assessment of the boron-zirconium system by Duschanek and Rogl [55] had some level of contamination above 0.4 at. % Hf, which is the highest hafnium content reported by Lonergan et al. [159].

110 5.3 First principles calculations

DFT calculations were performed to investigate the eﬀect of zirconium-hafnium substitution on the zirconium and hafnium carbides and borides. The calculations were all done using the Vienna ab initio simulation package (VASP) [144, 145, 142, 143], with PAW [31, 146], using both the Local Density Approximation (LDA) exchange- correlation functional [198] and the GGA-PBE exchange-correlation functional [196, 195]. The quasiharmonic approximation was used throughout.

Electronic and ionic relaxation was performed with geometry optimisations having a cut-oﬀ criterion of 10−6 eV/A,˚ and electronic optimisations having a cut-oﬀ criterion of 10−8 eV. The calculations of the Helmholtz energy were performed using Methfessel-Paxton smearing at 0.1 eV.

5.3.1 Borides

Zirconium diboride

Zirconium diboride is a hexagonal hp3 structure with AlB2 prototype, with space group P6/mmm. The zirconium diboride unit cell has three atoms, which can be seen in Figure 5.9, produced using 3D visualisation software VESTA [177].

Figure 5.9: Unit cell for the zirconium diboride phase, where the boron is shown in blue and zirconium is shown in pink, drawn using VESTA [177].

111 Using a 3x3x2 supercell with 54 atoms, DFT calculations of stoichiometric zirconium diboride were performed using both LDA PAW and GGA PAW PBE exchange- correlation functionals. The atomic positions that were used for the starting point of the calculations were downloaded from the Inorganic Crystal Structure Database (ICSD) [18]. The crystal structure used as the starting values for the calculation were from neutron diﬀraction studies by Li et al. [157]. A visualisation of the 3x3x2 supercell can be seen in Figure 5.10.

Figure 5.10: 3x3x2 supercell for the zirconium diboride phase, where the boron is shown in blue and zirconium is shown in pink, drawn using VESTA [177].

Calculations were performed with 500 eV, 550 eV, 600 eV, and 700 eV cutoff energies, and with 7x7x9 k-point mesh, that was chosen to give a k-point density equivalent to the 7x7x7 k-point mesh in the carbide calculations in Section 5.3.2 for consistency. For hexagonal lattices, such as zirconium and hafnium diboride, it is necessary to use Gamma centred grids [147] instead of standard Monkhorst-Pack grids, in order to maintain full hexagonal symmetry. Different k-point densities were also calculated at the different cutoff energies to check convergence in the same way as in Section 5.3.2. The convergence of the calculations was examined by considering the difference in the energies of the perfect supercell and the supercell with a point defect present calculated in exactly the same way, to allow cancellation of errors.

A single substitutional point defect was made on the zirconium lattice to replace a single zirconium atom with a hafnium atom. This gave a composition of 1.85 at.%Hf, 31.48 at.%Zr, 66.67 at.%B. It was not necessary to repeat the calculations with the

112 Figure 5.11: 3x3x2 supercell for the zirconium diboride phase with a single hafnium substitution at a zirconium site, where the boron is shown in blue, zirconium is shown in pink, and hafnium is shown in orange, drawn using VESTA [177]. defect at diﬀerent positions in the supercell as all metal positions in the unit cell are equivalent by symmetry. The structure of the defective crystal can be seen in Figure 5.11.

Figure 5.12 shows the energy diﬀerence between the perfect supercell and the supercell with a substitutional defect, calculated using LDA-PAW and LDA-PAW-PBE exchange-correlation functional in VASP with a 7x7x9 k-point mesh, using several kinetic cutoﬀ energies.

For the LDA values, there is a 0.0015 eV difference between the highest and the lowest energy difference calculated with the different cutoff energies used. For the GGA calculations, there is a 0.0038 eV difference. This indicates that the calculations have converged for both LDA and GGA at any kinetic cutoff energy above 500 eV.

The results of the calculations using 700 eV cutoﬀ energy and 7x7x9 k-points are summarised in Table 5.1.

Hafnium diboride

Hafnium diboride has the same crystal structure type as zirconium diboride, having a hexagonal hp3 structure with AlB2 prototype and space group P6/mmm. As for zirconium diboride, the unit cell has three atoms. The unit cell can be been seen in

113 Figure 5.12: The energy difference between the perfect ZrB2 supercell and the ZrB2 supercell with a single hafnium substitutional defect in a zirconium site, calculated at several different cutoff energies, using LDA and GGA exchange-correlation functionals with a 7x7x9 k-point mesh.

Figure 5.13.

Figure 5.13: Unit cell for the hafnium diboride phase, where the boron is shown in blue and hafnium is shown in orange, drawn using VESTA [177].

Using a 3x3x2 supercell with 54 atoms, DFT calculations of stoichiometric hafnium diboride were performed using both LDA PAW and GGA PAW PBE exchange- correlation functionals. The atomic positions that were used for the starting point of the calculations were downloaded from the ICSD [18]. The crystal structure used as the starting values for the calculation were from neutron diﬀraction studies by

114 Zhang and Cheng [277]. A visualisation of the 3x3x2 supercell can be seen in Figure 5.14.

Figure 5.14: 3x3x2 supercell for the hafnium diboride phase, where the boron is shown in blue and hafnium is shown in orange, drawn using VESTA [177].

As for ZrB2, calculations were performed using a kinetic cutoff energy of 700 eV and 7x7x9 k-point mesh, where different cutoff energies and k-point densitites were considered to check the convergence of the calculations. As in the case of the zirconium diboride calculations, convergence was examined by considering the difference between the energies of the perfect supercell and the supercell with a single substitutional defect, calculated in the same way.

A single substitution was made on the hafnium lattice to replace a single hafnium atom with a zirconium. This gave a composition of 1.85 at.% Zr, 31.48 at.% Hf,

66.67 at.% B. As for the ZrB2 calculations, it was not necessary to repeat the calculations with the defect at diﬀerent positions in the supercell as all metal positions are positionally equivalent. The structure of the defective supercell can be seen in Figure 5.15.

The energy difference between the perfect supercell and the supercell with a single zirconium substitution is shown for several different kinetic cutoff energies in Figure 5.16.

There is a 0.0125 eV difference between the highest and lowest energy values calculated with the LDA exchange-correlation functions at difference cutoff energies, and

115 Figure 5.15: 3x3x2 supercell for the hafnium diboride phase with a single zirconium substitution at a zirconium site, where the boron is shown in blue, zirconium is shown in pink, and hafnium is shown in orange, drawn using VESTA [177]. a 0.0410 eV diﬀerence between the highest and lowest energy values calculated using GGA. This indicates that the values calculated have converged for both LDA and GGA exchange-correlation functionals at kinetic cutoﬀ energies above 500 eV.

The results of the calculations using 700 eV cutoﬀ energy and 7x7x9 k-points are summarised in Table 5.1.

Figure 5.16: The energy difference between the perfect HfB2 supercell and the HfB2 supercell with a single zirconium substitutional defect in a hafnium site, calculated at several different cutoff energies, using LDA and GGA exchange-correlation functionals with a 7x7x9 k-point mesh.

116 Pure element calculations

To compare the energies calculated, they were taken with reference to the energies of the pure elements. Calculations were done of the energy of the pure elements using the same settings and cutoﬀ energies for each of the considered exchange-correlation functionals used in the ZrB2 and HfB2 calculations.

Hafnium

At low temperatures, hafnium is stable in a HCP structure. DFT calculations of a 1x1x1 supercell of HCP hafnium were performed using the same LDA PAW and

LDA PAW PBE functionals used in the ZrB2 and HfB2 calculations. A 700 eV cutoﬀ energy was used for consistency, and a k-point density matching the density of 7x7x9 k-point mesh used in the ZrB2 and HfB2 calculations were used. For 1x1x1 hafnium, this was a 20x20x12 k-point density. As HCP hafnium is a hexagonal structure, a Gamma centered grid of k-points was used instead of Monkhorst-Pack k-points as Monkhorst-Pack grids do not have full hexagonal symmetry [147]. The unit cell positions for HCP hafnium were obtained from the ICSD [18] where values from Lejaeghere et al. [153] were used as the starting values. The HCP hafnium unit cell can be seen in Figure 5.17.

Figure 5.17: Unit cell of HCP hafnium, drawn using VESTA [177].

117 Zirconium

Below 1159 K, zirconium is stable in a HCP structure. DFT calculations of a 1x1x1 supercell of HCP zirconium were performed using the same LDA PAW and LDA PAW PBE functionals used in the ZrC and HfC calculations. A 700 eV cutoﬀ energy was used for consistency and a 20x20x12 Gamma centered k-point mesh was used to give the same k-point density as the ZrB2 and HfB2 supercells. The unit cell positions for HCP zirconium were obtained from the ICSD [18] where values from Srivastava et al. [239] were used as the starting values. The HCP zirconium unit cell can be seen in Figure 5.18.

Figure 5.18: Unit cell of HCP zirconium, drawn using VESTA [177].

Boron

DFT calculations of a 1x1x1 supercell of β-rhomobohedral boron were performed using the same LDA PAW and LDA PAW PBE functionals used in the ZrB2 and

HfB2 calculations. A 700 eV cutoﬀ energy was used for consistency with k-point densities matching k-point mesh used in the ZrB2 and HfB2 calculations, meaning a 6x6x6 Monkhorst-Pack k-point mesh was used. The unit cell positions for β- rhomobohedral boron were obtained from the ICSD [18] where atomic positions of a 105 atom unit cell from Geist et al. [77] were used as the starting values. The β-rhomobohedral boron unit cell can be seen in Figure 5.19.

118 Figure 5.19: Unit cell of β-rhombohedral boron, drawn using VESTA [177].

Lattice parameters

Pure elements

The a and c lattice parameters for HCP hafnium have been measured by Russell [223] as 3.1946 A˚ and 5.0510 A˚ respectively. The DFT calculations using the LDA PAW exchange-correlation functional give lattice parameters 3.1206 A˚ and 4.9426 A,˚ and the GGA PAW PBE functional gives 3.2021 A˚ and 5.0593 A.˚

Fast [60] reports the a and c lattice parameters of HCP zirconium as 3.2224 A˚ and 5.134 A˚ respectively. The DFT calculations using the LDA PAW exchange- correlation functional give lattice parameters 3.1464 A˚ and 5.0878 A˚ , and the GGA PAW PBE functional gives 3.2317 A˚ and 5.1702 A.˚

As expected, there is better agreement with the experimentally measured lattice parameters of hafnium and zirconium and the values calculated using the GGA PAW PBE exchange-correlation functionals than with the LDA PAW functionals.

119 The structure of β-rhombohedral boron can be described with a single lattice parameter, which has been measured in several studies between 1963 [108] and 1994 [264] as being 10.145 A˚ at 300 K. The DFT calculations using the LDA PAW exchange- correlation functional give a lattice parameter of 9.9937 A˚ , and using the GGA PAW PBE functional gives a lattice parameter of 10.1117 A˚ . The results calculated using the GGA PAW PBE functional give a lattice parameter that is closer to the experimental value than the values calculated using the LDA PAW functional.

ZrB2 and HfB2

The lattice parameters of zirconium and hafnium diboride have been reported in the literature by numerous studies, but were reported by Fahrenholtz et al. [58] in a review of the properties of hafnium and zirconium diboride as the values in Table 5.1.

The lattice parameters for ZrB2 and HfB2 calculated using the GGA PAW PBE exchange-correlation functionals are closer to the experimental values than the lattice parameters from the calculations using LDA PAW exchange-correlation functionals.

Eﬀect of the defect on the lattice parameter

The relaxation of the crystal structures in the DFT calculations allows comparison of the lattice parameters of the perfect crystal and the crystal with a single substitutional defect per 18 metal atoms. Using the LDA PAW exchange-correlation functional, the a and c lattice parameters for the perfect ZrB2 crystal were calculated as 3.1338 A˚ and 3.4908 A˚ respectively, and with a single defect, these became

3.1322 A˚ and 3.4876 A˚ . The c/a ratio in the perfect ZrB2 crystal was calculated as 1.1139, and for the crystal with a hafnium defect it was 1.1134.

Using the GGA PAW PBE exchange-correlation functional, the a and c lattice parameters for the perfect ZrB2 crystal were calculated as 3.1711 A˚ and 3.5418 A˚ , and

120 with the defect they were 3.1697 A˚ and 3.5390 A˚ , where the c/a ratio was 1.1169 for the perfect crystal and 1.1165 for the crystal with the defect.

For the calculations with LDA and GGA, the presence of the hafnium defect caused the lattice parameters to decrease, with the c lattice parameter decreasing twice as much as the a parameter, causing a distortion in the lattice.

Using the LDA PAW exchange-correlation functionals, the a and c lattice parameters for the perfect HfB2 crystal were calculated as 3.1056 A˚ and 3.4331 A˚ , with c/a ratio 1.1054. For the defective crystal, the a and c lattice parameters were calculated as 3.1072 A˚ and 3.4362 A˚ respectively, with c/a ratio 1.1059.

Using the GGA PAW PBE exchange-correlation functional, the a and c lattice parameters were calculated as 3.1453 A˚ and 3.4920 A˚ for the perfect crystal and 3.1468 A˚ and 3.4948 A˚ for the crystal with the substitutional defect. The c/a ratio was 1.1102 for the perfect crystal and 1.1106 for the defective crystal.

For the calculations with both LDA and GGA functionals, the presence of the zirconium defect caused the lattice parameters to increase, with the c lattice parameter increasing twice as much as the a parameter as in the case for the hafnium substitutional defect in zirconium diboride, causing a distortion in the lattice.

Enthalpy of formation

The energies calculated with DFT for the boride compounds at 0 K are summarised in Table 5.1. It can be seen that there is some diﬀerence between the energies calculated using the LDA and GGA exchange-correlation functionals,

The calculated enthalpies of formation at 0 K for the diﬀerent compositions are shown compared with the enthalpy of formation as a function of hafnium content at 298.15 K from the CALPHAD assessment by Cacciamani et al. [41] in Figure 5.20.

In the assessment from Cacciamani et al. [41], there are no ternary mixing parameters used in modelling the diboride phase in the thermodynamic assessment, and so the

121 Lattice Enthalpy of c/a Volume Compound Composition Method constants formation ratio (A˚3) (A)˚ (kJ/mol) at.% Hf at. % Zr a c Zr18B36 0 33.33 3.1338 3.4908 1.1139 29.690 -107.67 Hf1Zr17B36 1.85 31.48 3.1322 3.4876 1.1134 29.632 -107.69 LDA Zr1Hf17B36 31.48 1.85 3.1072 3.4362 1.1059 28.731 -108.16 Hf18B36 33.33 0 DFT (0 K) 3.1056 3.4331 1.1054 29.648 -108.20 Zr18B36 0 33.33 (this work) 3.1711 3.5418 1.1169 30.845 -99.97 Hf1Zr17B36 1.85 31.48 3.1697 3.5390 1.1165 30.792 -99.91 GGA Zr1Hf17B36 31.48 1.85 3.1468 3.4948 1.1106 29.971 -98.87 Hf18B36 33.33 0 3.1454 3.4920 1.1102 29.919 -98.80 LDA-CA-PZ [278] 3.1510 3.4941 1.1089 30.044 -106.04 ZrB 0 33.33 2 GGA-PBE [278] 3.1681 3.5355 1.1160 30.731 -98.25 DFT (0 K) LDA-CA-PZ [278] 3.1246 3.4507 1.1044 29.176 -98.67 HfB 33.33 0 2 GGA-PBE [278] 3.1416 3.4976 1.1133 29.895 -88.99 ZrB 0 33.33 3.17 [58] 3.53 [58] 1.114 -107.4 [232] 2 Experiment (298 K) HfB2 33.33 0 3.139 [58] 3.473 [58] 1.106 -111.9 [232] ZrB 0 33.33 [55] -100.56 2 CALPHAD (298 K) HfB2 33.33 0 [29] -106.49

Table 5.1: Properties calculated using DFT with LDA PAW and GGA PAW PBE exchange- correlation functionals compares with DFT calculations from the literature, experimental data, and values from the CALPHAD assessment of the B-Hf-Zr system. The choice of significant figures indicates the accuracy to which the values were obtained, where the uncertainty would not affect the value shown.

mixing of ZrB2 and HfB2 is modelled with the ideal solution model in which the enthalpy of mixing is zero. As a result of this, the enthalpy of formation of the diboride phase is linear with composition as the hafnium or zirconium content is varied.

Neither the LDA or GGA calculations give enthalpies of formation that are consistent with the CALPHAD description, but the results are very consistent with calculations of ZrB2 and HfB2 from Zhang et al. [278]. As the DFT calculations are at 0 K and the values from the CALPHAD assessment shown are at 298.15 K, it would be expected that the enthalpies of formation from the DFT would be lower than the CALPHAD values. The experimental values from Schneider [232] show that the enthalpy of formation of hafnium diboride is lower than that of zirconium diboride, a trend that is reproduced by the DFT calculations using the LDA PAW exchange-correlation functionals, although the diﬀerence between the calculated enthalpies of formation of the end member single metal diborides is not consistent with the experimental data.

122 The calculated enthalpy of formation of the defective cells is seen to be very similar to the enthalpy of formation of the perfect supercell of the fraction of substitutional defect considered. The calculations done with the LDA functionals show that the defective ZrB2 is slightly more stable than the perfect crystal, and the defective HfB2 is slightly less stable than the perfect crystal, where there is suggestion of a linear trend between the ZrB2 and HfB2 end members, which is decreasing with hafnium content. The GGA calculations show that the defective ZrB2 crystal is less stable than the perfect crystal, and the defective HfB2 crystal is more stable than the perfect crystal, again being consistent with a linear trend across the range of composition but increasing, contrary to experimental observations.

We can deﬁne a reference defect formation energy for the case of a metal (M2) substitutional defect in metal (M1) diboride in a 2x2x3 supercell as

f 1 1 E = E(M21M117B36) − E(M118B36) + E(M12) − E(M22) (5.1) M2 substitution 2 2

The reference defect formation energy for a hafnium substitutional defect at a zirconium site in zirconium diboride was calculated as -0.011 eV (-1.08 kJ) using the LDA PAW exchange-correlation functional, and 0.034 ev (3.24 kJ) using the GGA PAW PBE functional. The reference defect formation energy for a zirconium substitutional defect at a hafnium site in hafnium diboride was calculated as 0.023 eV (2.24 kJ) using the LDA PAW functional, and -0.037 eV (-3.61 kJ) using the GGA PAW PBE functional.

Improvements to the CALPHAD modelling

The experimental data from Lonergan et al. indicate that the mixing of the hafnium diboride and zirconium diboride phases is not ideal at low defect contents, where the defect is a metal substitution on the metal lattice. The calculations of the defective supercells for the diborides are not inconsistent with the ideal solution model used in

123 Figure 5.20: Enthalpy of formation of ZrB2 and HfB2 with a single substitutional hafnium and zirconium defect respectively calculated with DFT using LDA or GGA exchange-correlation functionals at 0 K, compared with a quasi-binary representation of the enthalpy of formation of the γ NaCl structure carbide phase from the CALPHAD assessment by Cacciamani et al. [41] as a function of hafnium content at 298.15 K. the CALPHAD modelling by Cacciamani et al., but the calculations of the perfect hafnium and zirconium diboride supercells are in disagreement with the enthalpies of formation as modelled in the CALPHAD assessment, while being consistent with other ﬁrst principles calculations. This indicates a need for reassessment of the ternary boron-hafnium-zirconium phase diagram while considering the calculated enthalpies of formation which were not available when the previous assessments were published.

5.3.2 Carbides

As well as considering the eﬀect that hafnium and zirconium substitutional defects have on the transition metal borides in Section 5.3.1, ﬁrst principles calculations of the properties of zirconium and hafnium carbides were performed, with and without the presence of substitutional defects in the same way as was considered for the

124 borides.

Figure 5.21: Phase diagram of the carbon- Figure 5.22: Phase diagram of the carbon- zirconium system from the assessment by hafnium system from the assessment by Bit- Fernandez Guillermet [66]. termann and Rogl [27].

The phase diagrams for the carbon-zirconium and carbon-hafnium systems as assessed by Fernandez Guillermet [66] and Bittermann and Rogl [27] are in Figures 5.21 and 5.22 respectively. It can be seen that both systems have a wide single phase region of the metal monocarbides that are stable over a wide range of stoichiometries. As hafnium and zirconium have such similar atomic structures and properties, the phase diagrams are broadly similar, each having the same phases present. The allotropic transformation of hafnium is at a higher temperature relative to its melting temperature than the equivalent transformation in zirconium, and the solubility of carbon in hafnium is greater than the solubility of carbon in zirconium.

The carbon-hafnium-zirconium ternary system has been assessed using the CAL- PHAD approach by Bittermann and Rogl [29]. An isothermal slice from this assessment at 1000 K is shown in Figure 5.23. It can be seen that there is a single phase NaCl-structure γ phase region that extends across the ternary phase diagram from ZrC to HfC, where is it stable at a range of stoichiometries. As in the B-Hf-Zr modelling, this indicates that the metal atoms are substituted on the metal lattice to fulﬁl the diﬀerent compositions of this phase. In the CALPHAD modelling in this

125 assessment, there are no interaction parameters modelled, as it is assumed that the two end members, HfC and ZrC show ideal mixing with no enthalpy of mixing.

Figure 5.23: Isothermal slice of the carbon-hafnium-zirconium system at 1000 K from the assessment by Bittermann and Rogl [29].

Zirconium carbide

Zirconium carbide is a cubic compound, with an NaCl structure with space group Fm3¯m, and eight atoms in the unit cell. The atomic positions that were used for the starting point of the calculations were downloaded from the ICSD [18]. The crystal structure used as the starting values for the calculation were from neutron diﬀraction studies by Nakamura et al. [178]. The ZrC unit cell was drawn using 3D visualisation software VESTA [177] and can be seen in Figure 5.24.

Using a 2x2x2 supercell with 64 atoms, DFT calculations of stoichiometric zirconium carbide were performed using both the LDA PAW and GGA PAW PBE functionals

126 Figure 5.24: Unit cell for the zirconium carbide phase, where the carbon is shown in green and zirconium is shown in pink, drawn using VESTA [177]. from VASP. The same functionals for zirconium and hafnium were used as in the

ZrB2 and HfB2 calculations. Duﬀ et al. [54] found that LDA functionals better reproduced experimental values for zirconium carbide, but many other studies in the literature (e.g. Giorgi [79] and Kim et al. [131]) use GGA functionals, so both LDA PAW and GGA PAW PBE were used here for comparison. The 2x2x2 supercell is shown in Figure 5.25.

Figure 5.25: 2x2x2 supercell for the zirconium carbide phase, where the carbon is shown in green and zirconium is shown in pink, drawn using VESTA [177].

In order to be consistent with the earlier zirconium carbide calculations by Duff et al. [54], the highest kinetic cutoff-energy used was 700 eV, which was also used as the highest cutoff energy in the calculations of the diborides. Calculations were also performed at 500 eV, 550 eV, and 600 eV to check that the values had converged at

127 these kinetic cutoff energies. Calculations were also performed for different densities of k-point Monkhorst-Pack meshes at each energy to ensure convergence. These were 5x5x5 and 7x7x7 k-point meshes. As was done for the borides, the convergence of the calculations was examined by taking the difference between the energy of the perfect supercell and the supercell with a single defect calculated in the same way in order to ensure cancellation of errors.

A single substitutional defect was made on the zirconium lattice, whereby a single zirconium atom in the supercell was replaced with a hafnium atom. This gave a composition of 1.56 at.% Hf, 48.44 at.% Zr, 50 at.% C. It was not necessary to repeat the calculations with the substituted atom in diﬀerent positions in the supercell as all metal atoms are spatially equivalent due to the symmetry. The defective supercell can be seen in Figure 5.26.

Figure 5.26: 2x2x2 supercell for the zirconium carbide phase with a single hafnium substitutional defect at a zirconium site, where the carbon is shown in green, zirconium is shown in pink, and hafnium is shown in orange, drawn using VESTA [177].

Figure 5.27 shows the energy difference between the 2x2x2 ZrC supercell and the 2x2x2 ZrC supercell with a single hafnium substitutional defect on the metal lattice for different cutoff energies, where the same LDA PAW functional is used throughout. Figure 5.28 shows the energy difference using different k-point densities, where the x-axis shows the number of k-points in each direction.

It can be seen that for the cutoff energies used, there is a 0.00022 eV difference between the highest and lowest energies calculated, indicating that the calculations have good convergence at any kinetic cutoff energy above 500 eV.

128 Figure 5.27: The energy difference between the perfect ZrC supercell and the ZrC supercell with a single hafnium substitutional defect in a zirconium site, calculated at several different cutoff energies, using an LDA exchange-correlation functional with a 5x5x5 k-point mesh.

Figure 5.28: The energy diﬀerence between the perfect ZrC supercell and the ZrC supercell with a single hafnium substitutional defect in a zirconium site, calculated with 5x5x5 and 7x7x7 k-point meshes, using an LDA exchange-correlation functional and a cutoﬀ energy of 600 eV.

It can be seen that for these different k-point densities, there is no difference in energy between the calculations with the two different k-point meshes within the range of the significant figures output by the VASP calculations.

129 A summary of the converged values using LDA PAW and GGA PAW PBE exchange- correlation functionals with a 7x7x7 Monkhorst-Pack k-point mesh and a 700 eV kinetic cutoﬀ energy is given in Table 5.2.

Hafnium carbide

The same process for zirconium carbide was repreated for hafnium carbide with a single zirconium substitutional defect at a hafnium site.

Like zirconium carbide, hafnium carbide is a cubic compound, with an NaCl structure with space group Fm3¯m, and eight atoms in the unit cell. The atomic positions that were used for the starting point of the calculations were downloaded from the ICSD [18]. The crystal structure used as the starting values for the calculation were from neutron diﬀraction studies by Nakamura et al. [178]. The unit cell can be seen in Figure 5.29.

Figure 5.29: Unit cell for the hafnium carbide phase, where the carbon is shown in green and hafnium is shown in orange, drawn using VESTA [177].

As the structure and lattice parameters for zirconium carbide and hafnium carbide are so similar, the same kinetic cutoﬀ energies and k-point densities as were used in the zirconium carbide investigation were used, and convergence was examined in the same way. Calculations were performed using both LDA PAW and GGA PAW PBE exchange-correlation functionals. A 2x2x2 supercell was used, meaning that

130 the impurity density was the same as in the zirconium carbide calculations. The HfC supercell that was used is shown in Figure 5.30.

Figure 5.30: 2x2x2 supercell for the hafnium carbide phase, where the carbon is shown in green and hafnium is shown in orange, drawn using VESTA [177].

The convergence of the calculations was examined in the same way as for the ZrC calculations, where the diﬀerence between the energies calculated for the perfect supercell and a supercell where a single zirconium substitutional is made at a hafnium site was considered in order to cancel any errors. Calculations were performed for the perfect and defective supercells using cutoﬀ energies at 500 eV, 550 eV, 600 eV, and 700 eV, and 5x5x5 and 7x7x7 k-point meshes.

A single substitutional defect was made on the hafnium lattice, where a single hafnium atom in the 64 atom supercell was replaced with a zirconium atom. This gave a composition of 1.56 at.% Zr, 48.44 at.% Hf, 50 at.% C. Due to the positional equivalence of all metal sites due to the symmetry of the NaCl-type structure, the calculations did not need to be repeated with the defect in diﬀerent positions in the lattice. The defective supercell can be seen in Figure 5.31.

Figure 5.32 shows the energy difference using different k-point densities, and Figure 5.33 shows the energy difference for different cutoff energies, where the same LDA PAW functional is used throughout. It can be seen that for these different k-point densities, the energy difference using the 7x7x7 k-point mesh was 0.00009 eV higher than that calculated with the 5x5x5 k-point mesh, indicating that within the error of the DFT calculations, the calculations have converged.

131 Figure 5.31: 2x2x2 supercell for the hafnium carbide phase with a single zirconium substitutional defect at a hafnium site, where the carbon is shown in green, hafnium is shown in orange, and zirconium is shown in pink, drawn using VESTA [177].

It can be seen that for the cutoff energies used, there is a 0.06588 eV difference between the highest and lowest energies calculated, indicating that there is an adequate convergence for the purposes of examining the trends when adding impurities at any cutoff energy above 500 eV. We note however that the convergence is not as good as the convergence reached by the ZrC calculations.

Pure element calculations

As was done for the borides in Section 5.3.1, the calculations were taken with reference to calculations of the pure elements. The hafnium and zirconium calculations that were used in Section 5.3.1 are used again here, and the properties of the carbon unit cell in the diamond crystal structure were calculated and corrected in order to obtain the energy of graphite, the stable state of carbon.

Carbon

DFT calculations of a 1x1x1 supercell of carbon in the diamond structure were performed using the same LDA PAW and LDA PAW PBE functionals used in the ZrC and HfC calculations. A 700 eV cutoﬀ energy was used for consistency with k-point densities matching the highest density 7x7x7 k-point mesh used in the ZrC and HfC calculations, meaning a 6x6x6 Monkhorst-PAck k-point mesh was used.

132 Figure 5.32: The energy diﬀerence between the perfect HfC supercell and the HfC supercell with a single hafnium substitutional defect in a zirconium site, calculated with 5x5x5 and 7x7x7 k-point meshes, using an LDA exchange-correlation functional and a cutoﬀ energy of 600 eV.

The unit cell positions for diamond were obtained from the ICSD [18] where values from Bindzus et al. [25] were used as the starting values. The diamond unit cell can be seen in Figure 5.34.

As the enthalpy of formation is defined with respect to the enthalpies of the pure component elements in their stable state, the enthalpy of carbon in the graphite phase is required. As the DFT calculations are for carbon in the diamond phase, it is necessary to augment this value to reflect the difference in stability between graphite and diamond at 0 K. At 0 K, there is data from both experiment extrapolated to 0 K [36], CALPHAD extrapolation [93] and first principles calculations of diamond [235] that report the zero temperature energy difference to be 27 meV, 25 meV, and 28 meV respectively, where graphite is more stable than diamond. As such, the first principles value from Shin et al. of 28 meV per atom is used to modify the energy of the diamond calculation to be used in the calculation of the enthalpy of formation.

133 Figure 5.33: The energy difference between the perfect HfC supercell and the HfC supercell with a single hafnium substitutional defect in a zirconium site, calculated at several different cutoff energies, using an LDA exchange-correlation functional with a 5x5x5 k-point mesh.

Lattice parameters

Pure elements

Calculations were performed of carbon in the diamond structure. Renninger [210] reports an experimentally lattice parameter of diamond as 3.5595 A˚ . The DFT calculations using the LDA PAW exchange-correlation functional give a lattice parameter 3.5322 A˚ and the GGA PAW PBE functional gives 3.5697 A˚ . Neither the DFT calculations using either the LDA nor the GGA functional give a lattice parameter that is very consistent with the experimental value, but the value calculated using the GGA PAW PBE functional are closer to the experimentally measured value.

ZrC and HfC

The lattice parameters of zirconium carbide, hafnium carbide, and the intermediate mixture between them have been measured in numerous studies, including by Rude et al. [219], Storms [241], [39]. The lattice parameters from the calculations us-

134 Figure 5.34: Unit cell of carbon in the diamond structure, drawn using VESTA [177]. ing the GGA PAW PBE exchange-correlation functional are seen to provide lattice parameters for ZrC and HfC that are closest to experiment.

Eﬀect of the defect on the lattice parameter

As was done for the diboride calculations, the relaxation of the crystal structures in the DFT calculations allows comparison of the lattice parameters of the perfect crystal and the crystal with a single substitutional defect per 32 metal atoms.

It was found that by comparing the DFT calculations of the perfect ZrC and HfC crystals, and the same crystals with a single metal substitutional defect per supercell, the lattice parameter of the crystal changed when the defect was added, but the symmetry of the crystal was not lost, with the lattice parameter changing by the same amount in all directions. This can be attributed to the symmetry of the repeating supercell, shown in Figures 5.26 and 5.31 where the distance between defects is the same in each of the x, y, and z directions.

Using the LDA PAW exchange-correlation functionals, the lattice parameter for the perfect ZrC crystal was calculated as 4.6436 A˚ and with a single defect, this became 4.6417 A˚ . Using the GGA PAW PBE exchange-correlation functionals, the lattice parameter for the perfect ZrC crystal was calculated as 4.7081 A˚ , and with the defect it was 4.7064 A˚ . The presence of the hafnium defect caused the lattice parameters

135 to decrease by 0.0019 A˚ and 0.0017 A˚ for the LDA PAW and GGA PAW PBE exchange-correlation functionals respectively.

Using the LDA PAW exchange-correlation functionals, the lattice parameter for the perfect HfC crystal was calculated as 4.5814 A˚ and with a single defect, this became 4.5835 A˚ . Using the GGA PAW PBE exchange-correlation functionals, the lattice parameter for the perfect HfC crystal was calculated as 4.6528 A˚ , and with the defect it was 4.6546 A˚ . The presence of the zirconium defect caused the lattice parameters to increase by 0.0021 A˚ and 0.0018 A˚ for the calculations using LDA PAW and GGA PAW PBE exchange-correlation functionals respectively.

As the lattice parameter of hafnium carbide is smaller than the lattice parameter of zirconium carbide, it is expected that the lattice parameter of the (Zr, Hf)-carbide would decrease with increasing hafnium content, as these calculations show. The lattice parameter changes more for the introduction of zirconium substitutional defects into hafnium sites in hafnium carbide than for the introduction of hafnium substitutional defects into zirconium carbide at the same concentration, indicating that the variation of lattice parameter of the monocarbide phase with composition between zirconium carbide and hafnium carbide is not linear, in agreement with results from Brukl and Harmon [39].

Enthalpy of formation

The most recent currently accepted CALPHAD assessment of the carbon-hafnum- zirconium system in the literature is by Bitterman and Rogl [29]. In order to compare the DFT calculations with the CALPHAD assessment, it is convenient to compare the enthalpy of formation of the compound at the diﬀerent compositions calculated. In the DFT calculations, we calculated the energy at 0 K, which we can use to calculate the enthalpy of formation at 0 K by combining the calculations described above with DFT calculations of the energy of the pure elements in the compounds. We compare these with calculations of the enthalpy of formation at 298.15 K from the CALPHAD assessment, as the assessment is only valid above room temperature.

136 The results of the calculation are summarised and compared with other values in the literature in Table 5.2.

Lattice Enthalpy of Volume Compound Composition Method constants formation (A˚3) (A)˚ (kJ/mol) at.% Hf at. % Zr a Zr32C32 0 50 4.6436 100.130 -88.28 Hf1Zr31C32 1.56 48.44 4.6417 100.006 -84.52 LDA Zr1Hf31C32 48.44 1.56 4.5835 96.295 -99.37 Hf32C32 50 0 DFT (0 K) 4.5814 96.161 -91.60 Zr32C32 0 50 (this work) 4.7081 104.364 -87.54 Hf1Zr31C32 1.56 48.44 4.7064 104.246 -87.69 GGA Zr1Hf31C32 48.44 1.56 4.6546 100.844 -97.01 Hf32C32 50 0 4.6528 100.729 -89.24 GGA-PBEsol [224] 4.685 -83.5 ZrC 0 50 GGA-PBE [224] -84.0 DFT (0 K) GGA-PBEsol [224] 4.608 -98.0 HfC 50 0 GGA-PBE [224] -96.6 ZrC 0 50 4.6983 [241] Experiment (298 K) HfC 50 0 4.6385 [219] ZrC 0 50 [66] -103.46 CALPHAD (298 K) HfC 50 0 [27] -104.66

Table 5.2: Properties calculated using DFT with LDA PAW and GGA PAW PBE exchange- correlation functionals compares with DFT calculations from the literature, experimental data, and values from the CALPHAD assessment of the C-Hf-Zr system. The choice of significant figures indicates the accuracy to which the values were obtained, where the uncertainty would not affect the value shown.

Figure 5.35 shows the enthalpy of formation of the various carbides calculates using LDA PAW and GGA PAW PBE exchange-correlation DFT functionals at 0 K, and the enthalpy of formation of the NaCl structure carbide (γ) phase as a function of hafnium content from the carbon-hafnium-zirconium ternary CALPHAD assessment by Bittermann and Rogl [29]. The CALPHAD assessment does not give any ternary interaction parameters for the γ phase, and so the enthalpy of formation is linear with composition between the enthalpies of formation of the end member carbides as calculated by Fernandez Guillermet [66] and Bittermann and Rogl [27] for zirconium carbide and hafnium carbide respectively. The enthalpy of formation from the CALPHAD assessment is consistently lower than the enthalpy of formation from the DFT calculations for both exchange-correlation functionals used. The enthalpies of formation from the DFT calculations and from the CALPHAD description are at

137 Figure 5.35: Enthalpy of formation of ZrC and HfC with a single substitutional hafnium and zirconium defect respectively calculated with DFT using LDA or GGA exchange-correlation functionals at 0K, compared with a quasi-binary representation of the enthalpy of formation of the γ NaCl structure carbide phase from the CALPHAD assessment by Bittermann and Rogl [29] as a function of hafnium content at 298.15 K. diﬀerent temperatures (0 K and 298.15 K respectively), but as we would expect the enthalpy of formation to increase with temperature, this does not account for this disagreement.

Experimental data shows that the enthalpy of formation of hafnium carbide is lower than the enthalpy of formation of zirconium carbide, which is also shown from the DFT calculations of the enthalpy of formation using both LDA PAW and GGA PAW PBE exchange-correlation functionals. The experimentally measured enthalpies of formation of zirconium carbide are significantly lower the the calculated values. Al- though the presence of hafnium impurities in the experimental samples might result in a lower enthalpy of formation as the enthalpy of formation of hafnium carbide is lower, we would not expect this to be significant enough to account for the difference between the calculated and measured values shown in Table 5.2.

The DFT calculations of the enthalpy of formation of the perfect ZrC crystal are very similar using both LDA and GGA exchange-correlation functionals, with only

138 0.7 kJ difference. However, the calculations of the ZrC compound with a single substitutional defect show different trends for the LDA and GGA functionals. The results using the LDA functional indicate that the defective compound is less stable than the perfect compound, with the difference in energy being 3.75 kJ, whereas the results using the GGA functional suggest that the defective compound is more stable than the perfect by 0.14 kJ.

The calculations using both the LDA and GGA functionals display a similar trend for the perfect and defective hafnium carbide, showing that the defective compound is 7.76 kJ or 7.77 kJ more stable than the perfect compound for LDA and GGA respectively. However, there is a 2.74 kJ diﬀerence between the enthalpy of formation calculated with the two functionals, with the enthalpy of formation calculated with the LDA functional being lower.

The calculations using the LDA PAW exchange-correlation functionals suggest that zirconium carbide will oppose the presence of hafnium carbide, meaning that the defective crystal will be less likely to form than the pure crystal, while it is more likely that the hafnium carbide crystal will form with a defect than without. The calculations using the GGA PAW PBE exchange-correlation functional indicate that both hafnium and zirconium carbide are more likely to form with substitutional defects than without. This can be used to explain why hafnium and zirconium carbide are so frequently found with substitutional zirconium and hafnium impurities.

We can deﬁne a reference defect formation energy for a metal substitution in same was as for the hafnium and zirconium diborides in Section 5.3.1. The reference defect formation energy for a hafnium substitutional defect at a zirconium site in zirconium carbide was calculated as 2.49 eV (240.3 kJ) using the LDA PAW exchange- correlation functional, and -0.10 ev (-9.8 kJ) using the GGA PAW PBE functional. The reference defect formation energy for a zirconium substitutional defect at a hafnium site in hafnium carbide was calculated as -5.13 eV (-497.3 kJ) using the LDA PAW functional, and -5.15 eV (-497.0 kJ) using the GGA PAW PBE functional. The defect formation energies for a metal substitution in 64 atoms of zirconium and hafnium carbide are of a much greater magnitude than the equivalent energies

139 calculated for a metal substitution in 54 atoms of zirconium and hafnium diboride.

From the calculations performed here, which have no or very small quantities of impurity, the full variation of the enthalpy of formation with defect concentration is unclear, and further calculations using diﬀerent defect fractions could help elucidate the stability of the defective structures and give an indication of whether the LDA or GGA exchange-correlation functionals are best describing the properties of the crystals.

Improvements to the CALPHAD modelling

The DFT calculations using both the LDA and GGA exchange-correlation functions suggest that the enthalpy of formation of the NaCl-type carbide phase does not vary linearly with composition between the enthalpy of formation of zirconium carbide and hafnium carbide, as is modelled in the CALPHAD assessment by Bittermann and Rogl [29]. As such, these calculations have given us insight that suggests that ternary interaction parameters (excess Gibbs energy parameters) are needed to full describe the behaviour, as it deviates from the ideal system. In order to model a convex or concave hull, at least a zeroth order interaction parameter is needed, but to describe the behaviour shown by the calculations using the LDA functionals, where the convacity changes sign, at least a zeroth and ﬁrst order interaction parameters will be needed.

5.4 Summary and Conclusions

Zirconium and hafnium are both group IV transition metals that form very similar compounds with carbon and boron, and are often found together as a result of their mining process. The carbon-hafnum-zirconium and boron-hafnium-zirconium systems have been assessed using the CALPHAD approach in the past, but certain experimental data suggests that these phase diagrams do not suﬃciently describe

140 the mixing of the zirconium and hafnium carbide or diboride phases.

In this chapter, first principles calculations were used to give some insight into the the effects of introducing low levels of substitutional defects on the hafnium and zirconium lattices. By considering 0 K DFT calculations, the enthalpy of formation as a function of defect concentration was considered for both the diboride and carbide compounds of hafnium and zirconium, as well as the effect on the structure of the crystals. The results were compared with other DFT investigations, experimental data, and the properties described by the CALPHAD assessments in the literature.

It was found that in the case of the hafnium and zirconium diborides, the enthalpy of formation may follow a linear trend with metal composition between zirconium diboride and hafnium diboride, as described by the ideal mixing model used in the existing CALPHAD assessment by Cacciamani et al. [41]. However, experimental data from Lonergan et al. [159] suggests that the trend displayed by the heat capacity as a function of metal defect concentration is not linear, and so to enable consistency with this, interaction parameters are needed in the thermodynamic assessment. The results also provided some evidence that the samples used in measurements of the properties of zirconium or hafnium diboride may have been contaminated with hafnium or zirconium, altering the properties.

The investigation was repeated for the zirconium and hafnium monocarbides, and the DFT calculations showed that the enthalpy of formation was not linear with metal concentration as was also modelled in the CALPHAD assessment by Bittermann and Rogl [29] where ideal mixing was again assumed. The calculations showed that the presence of the defect may serve to stabilise the compound further, which could help explain why removal of the contaminant transition metal can be so challenging. In order to incorporate this information into the CALPHAD assessment, interaction parameters describing the non ideality of the mixing between zirconium and hafnium carbide are needed.

This chapter demonstrates some ways in which ﬁrst principles calculations of defect- related properties can be used to give insight into the CALPHAD modelling of a

141 system without being implicitly included as a thermodynamic quantity or phase boundary. The modelling of point defects such as substitutional defects is often simpliﬁed in CALPHAD modelling where the interactions are often assumed to be ideal, and deviations from the ideal case are ignored. By considering properties such as the enthalpies of formation or thermodynamic quantities of structures with small amount of defects, important trends can be uncovered and included in the modelling for a more physically accurate, precise description.

142 Chapter 6

A defect-centric approach to modelling the vacancy formation energy in a thermodynamic assessment

143 6.1 Introduction

One of the challenges of thermodynamic modelling is in accurately describing the vacancies in ordered compounds. Such compounds are often stable over a range of composition, the width of which can vary as a function of temperature. Until now, neither experimental nor calculated vacancy formation energies have been explicitly included in thermodynamic assessments, although it is implicitly included in other quantities used in the assessments.

The formation energies of defects such as vacancies are diﬃcult to measure experimentally, and for many systems, there is no reliable experimental data available. Developments in ﬁrst principles calculations mean that it is now possible to use density functional theory (DFT) to calculate the formation energy of vacancies and other defects, with accuracy comparable to that of experiments [179].

In order to create a high quality thermodynamic assessment, it is important to ensure that it is consistent with all available data. To do this, it is necessary to compare all calculable or measurable quantities with experimental or theoretical data, whether or not the data is included in the optimisation. Some quantities, such as the vacancy formation energy, are implicitly represented in the thermodynamic assessment, but have not been considered explicitly in the ﬁtting of parameters describing the Gibbs energy of a phase, despite being a quantity that is often calculated. A consequence of this is the empirical Gibbs energy functions from a CALPHAD assessment are not guaranteed to reproduce a physical vacancy formation energy.

Here, it is shown how the Gibbs energy functions from a CALPHAD assessment can be used to represent the vacancy formation energy, and then how the parameters in these functions can be constrained to ensure consistency with this quantity both for existing parameters (by adding a constraint in a thermodynamic database (TDB) ﬁle) and in a setup ﬁle for an optimisation.

144 6.1.1 The CALPHAD modelling of C-Zr

Within the CALPHAD approach, the Gibbs energy of each phase is parametrised as a function of temperature, composition and pressure [161]. Model parameters of these functions are fitted to experimental and first principles data of phase boundaries and thermodynamic properties. The total Gibbs energy is minimised with respect to the many variables that describe the proportions of the possible phases and the variations of composition within each of them, under the constraint that the overall composition is fixed. This procedure is carried out at sample points in the space of overall composition and temperature, usually at a single pressure. From the solution, the chemical potentials of each component and the phase diagram in temperature- composition space can be generated. A more thorough background to the CALPHAD approach is described in Chapter 2.

θ For each phase, θ, the molar Gibbs energy, Gm is described using parameterised expressions of the ideal solution model and excess energy terms which describe the diﬀerence between the ideal solution model and the experimental (or calculated) values [161]. θ srf θ cnf θ E θ Gm = Gm − T · Sm + Gm (6.1)

This is from the generalised form in equation 2.2 where there are no additional physical contributions to the Gibbs energy.

Strictly speaking, there are no expressions to parameterise in the ideal solution model except those describing the Gibbs energies of the stoichiometric end members as a function of temperature, which are sometimes ﬁctional substances for the composition that they represent. In conducting an assessment between end members, it is then only the excess Gibbs energies that need to be parameterised.

Within the thermodynamic modelling in the widely used CALPHAD assessment of the carbon-zirconium system [66], the Gibbs energy of the zirconium carbide (γ) phase is parameterised as a function of temperature, T , and composition of components, xi, where i =C, Zr, using an ideal solution model plus excess energy

145 terms.

γ X SER X ◦ γ X E γ Gm − xiHi = yi GZr:i + RT yi ln(yi) + Gm (6.2) i =C,Zr i=C,Va i=C,Va

SER where Hi is the enthalpy of the element i in its reference state (the stable state at 298.15 K and atmospheric pressure).

6.1.2 The thermodynamic modelling of the γ phase

The zirconium carbide (γ) phase, has a NaCl-type B1 structure, which exists over a range of stoichiometry, facilitated by carbon vacancies. In the currently accepted assessment of the carbon-zirconium phase diagram as a function of temperature and composition, by Fernandez-Guillermet (1995) [66], this phase is described with a two sublattice model, (Zr)1(C,Va)1, where the zirconium atoms have an face-centred cubic (FCC) structure that is stabilised by carbon atoms occupying the octahedral interstitial sites.

Although the CALPHAD modelling from Fernandez Guillermet does not consider vacancies on the zirconium sublattice or other interstitials, there is recent evidence from simulations that Frenkel pairs will be formed at temperatures close to the melting point [53].

Zirconium carbide has been experimentally observed at a wide range of stoichiometries, up to a composition where around 20 % of the octahedral interstitial sites (the carbon sublattice) are vacant.

The Gibbs energy functions of the end members are ﬁtted to experimental data using a power law expansion with coeﬃcients an where n = [1, 6].

◦ γ 2 3 −1 −3 Gi = a0 + a1T + a2T ln(T ) + a3T + a4T + a5T + a6T (6.3)

In most Redlich-Kister binary excess models, the excess parameters are given only a linear temperature dependence [161]. In the assessment by Fernandez Guillermet

146 [66], the T ln(T ) and T 2 terms are included as there are experimental measurements of the heat capacity of various compositions as a function of temperature, meaning that these terms can be ﬁtted directly from experiment.

E γ The excess energy in the assessment by Fernandez Guillermet, Gm, is described 0 γ using a Redlich-Kister binary excess model where excess parameters LZr:C,Va and 1 γ LZr:C,Va are polynomials in temperature with coeﬃcients an where n = [7, 14] as expressed generally in equation 2.29.

E γ 0 γ 1 γ Gm = yCyVa LZr:C,Va + LZr:C,Va (yC − yVa) (6.4) where 0 γ 2 LZr:C,Va = a7 + a8T + a9T ln(T ) + a10T (6.5)

1 γ 2 LZr:C,Va = a11 + a12T + a13T ln(T ) + a14T (6.6)

6.2 Expressing a “reference” vacancy formation energy using Gibbs energy functions

6.2.1 Defects in zirconium carbide

When a defect is present in a crystal, there is an associated lattice deformation and formation energy. For a given number of defects in a certain arrangement in a DFT supercell, a certain stoichiometry is produced and its associated properties can be calculated. For a single point defect such as a vacancy in the carbon lattice in zirconium carbide, its possible locations are positionally equivalent by symmetry because of the FCC (A1) structure. However, when multiple defects are present, they can be arranged in several different configurations. Gusev and Rempel [88] suggest that below 1216 K there is some ordering of vacancies to form structures of certain fixed stoichiometries. Some first principles investigations [279] also suggest that there is some ordering of defects in zirconium carbide at low temperatures. When there

147 are degenerate configurations of the defects in a system, whether from a single defect or an ordered structure, there is a configurational entropy change associated with that composition and internal energy. This configurational entropy change is given from statistical thermodynamics as

∆S = kB ln(Ω) (6.7)

where ∆S is the change in entropy, kB is Boltzmann’s constant, and Ω is the number of microstates associated with the system.

Introducing a defect into a system has an associated defect formation energy (DFE). A grand canonical ensemble can be used to describe this system, whereby the system is maintained in equilibrium with a reservoir such that it can exchange total particle number and total energy, while the volume and shape of the system are kept the same. Upon removing a particle from the system, the energy of the system will change by an amount equal to the chemical potential (µ) associated with the removed particle. We deﬁne the DFE at 0 K as

X ∆Gf = ∆Ef = Edefective − Eperfect + (∆niµi) (6.8) i where ∆Gf is the Gibbs energy of formation, ∆Ef is the energy of formation, Edefective is the total Gibbs energy of the system with the defect, Eperfect is the total Gibbs energy of the perfect system, ∆ni is the change in the number of particles of species i, and µi is the chemical potential of species i. The chemical potential depends on the environment, and hence there is a need for a reference defect formation energy if it is to be useful for predicting Gibbs energies as part of a CALPHAD assessment.

In order to combine these DFT calculations with the CALPHAD assessment of the carbon-zirconium system, we deﬁne a related quantity, the reference defect formation energy that can be more easily compared with and incorporated into the thermodynamic assessment.

148 6.2.2 The reference defect formation energy

The DFT calculations of the free energy of stoichiometric zirconium carbide, defective zirconium carbide, and diamond were combined to yield a reference defect formation energy, which uses the perfect stoichiometric crystal and the graphite at the same temperature and pressure as the reference state.

For the case of creating a single carbon vacancy, we deﬁne the reference vacancy formation energy as the change in energy when we remove a single carbon atom from bulk, stoichiometric ZrC and add it to pure graphite at the same temperature and pressure, without allowing the vacant site to wander. In this way we can include just the vibrational part of the Gibbs energy in the deﬁnition. The choice of graphite at the same temperature and pressure as a reference state is arbitrary. Rogal et al. [211] referred to a “raw” defect formation energy for which the reference state of an atom of the removed element was taken to be the isolated free atom. By taking the atom as being removed to pure graphite, the value we obtain is convenient for comparing with the calculated vacancy formation energies implicit in the CALPHAD database, as we thereby avoid having to calculate the energy of a free atom with DFT.

We deﬁne the energy of ZrCx to be GZrCx (nZr, nC) where nZr is the number of atoms on the Zr sublattice, and nC is the number of atoms on the carbon sublattice. We shall f then use the notation EVa = GZrCx (n, n−1)+Egraphite(1)−GZrC(n, n) to represent the reference vacancy formation energy, where the limiting value for large n is required.

This is not the physical quantity that corresponds to the activation energy in an expression for the concentration of thermal vacancies, because it depends on an arbitrary reference energy for the species of atom removed. The ‘physical’ vacancy formation energy in this sense would be the reference vacancy formation energy augmented by the chemical potential of the species of atom removed, in this case carbon. The reference vacancy formation energy is by construction independent of the reference energy. Furthermore it renders the physical defect formation energy dependent on the activity of the species of atom removed, which is notionally added to the environment of the crystal.

149 For this reason, we use the reference vacancy formation energy as the quantity that we compare with the thermodynamic assessment. It is only necessary to ensure that the same reference states are used in the CALPHAD parameterisations and the DFT calculations.

Using the compound energy formalism modelling from [66], the total expression for the Gibbs molar energy of the γ phase as a function of site fraction of each compontent, yi, is given from equations 6.2 and 6.4 by

◦ γ ◦ γ ◦ γ Gm = yC GZr:C + yVa GZr:Va + RT (yC ln yC + yVa ln yVa) (6.9) 0 γ 1 γ + yCyVa LZr:C,Va + LZr:C,Va(yC − yVa)

In order to compare the reference vacancy formation energy with the equivalent quantity from the thermodynamic assessment, it is necessary to determine the reference vacancy formation energy expressed using the parameters from the thermodynamic modelling. This can then be directly compared with the DFT values, either at 0 K or as a function of temperature.

When lattice vibrations are well-defined harmonic or even anharmonic degrees of freedom of the system, which is plausible in crystals at all temperatures except per- haps very close to the melting point, it is customary to separate the vibrational and configurational contributions to the entropy. The latter refer to the number or degenerate positions or orientations of the defect. They are dealt with by the ideal solution expression represented in equation 6.2 by the yi ln yi term. The deviations from the ideal solution are due to defect-defect interactions and are implicitly accounted for within the excess Gibbs energy. Hence in our definition of the Gibbs energy of formation of a vacancy, we only need to account for the vibrational contribution to the Gibbs energies, since the configurational part will be dealt with in the ideal solution expression.

We deﬁne the molar Gibbs energies of the end-member constituents with reference to Stable Element Reference (SER) states [161, 52, 66] as

o γ SER SER o γ GZr:C − HC − HZr = ∆GZr:C (6.10)

150 o γ SER o γ GZr:va − HZr = ∆GZr:Va (6.11)

For n molecules of stoichiometric ZrC, yVa = 0, yC = 1, yZr = 1, therefore

n o γ GZrCx (n, n) = ( GZr:C) (6.12) NA

1 n−1 For n molecules of ZrC with one vacancy, yVa = n , yC = n , yZr = 1, therefore

n n − 1 o γ 1 o γ GZrCx (n, n − 1) = GZr:C + GZr:Va NA n n (6.13) n − 1 1 n − 1 1 + 0Lγ + 1Lγ − n n Zr:C,Va Zr:C,Va n n

This then gives the expression for the reference vacancy formation energy

f 1 ◦ γ ◦ γ 0 γ 1 γ ◦ graphite EVa = − GZr:C + GZr:Va + LZr:C,Va + LZr:C,Va + GC NA (6.14) 1 1 + −0Lγ − 3 · 1Lγ + 2 · 1Lγ n Zr:C,Va Zr:C,Va n2 Zr:C,Va

1 For a large n, we can neglect terms dependent on O( n ), which leaves

f 1 ◦ γ ◦ γ ◦ graphite 0 γ 1 γ EVa = − GZr:C + GZr:Va + GC + LZr:C,Va + LZr:C,Va (6.15) NA

6.3 A defect-centric approach to the excess energy

6.3.1 Separating defect-related quantities in the excess energy parameterisation

Another way to consider the defect formation energies, defect-defect interaction energies, and higher order defect-related terms is to consider what Rogal et al. referred

151 to as a “defect-centric” approach to the excess energy. As described in Chapter E γ 2, the excess energy, Gm, represents the diﬀerence between the energy from the ideal solution model and the real behaviour as in equation 6.2. This is expressed as a Redlich-Kister polynomial as in equation 6.4 where the parameters are ﬁtted to experimental and calculated data.

If we take a defect-centric approach to the excess energy, meaning make the defect concentration the subject of the equation, it is possible to separate the contributions to the excess energy according to the order of the defect concentration.

By using conservation of sites on the carbon sublattice, we can make the substitution yC = 1 − yVa which allows us to rewrite the excess energy as

E γ 0 γ 1 γ 0 γ 1 γ 2 1 γ 3 Gm = LZr:C,Va + LZr:C,Va yVa + − LZr:C,Va − 3 LZr:C,Va yVa + 2 LZr:C,Va yVa (6.16)

The coefficients of the different orders of the defect concentration represent different physical quantities. The term that is linear in vacancy concentration is part of what we refer to as vacancy formation energy. We can see that this quantity is the same as the excess part of the reference vacancy formation energy for a single vacancy in a large system that we extracted earlier in Equation 6.2.2. The higher order terms in vacancy concentration can be interpreted in physical terms as representing the vacancy-vacancy interactions, although this is not a concept that has been explicitly modelled in the CALPHAD community.

6.3.2 The need for change in the excess energy parameterisation

By writing the excess energy this way, it is possible to use the existing framework to parameterise the excess energy in terms of physical quantities, where it has previously consisted of polynomials ﬁtted for convenience to account for the diﬀerence between the ideal solution model and experimental data.

152 The Redlich-Kister polynomial expression is often used by default in CALPHAD assessments, with few widely used alternatives [121]. It is clear that from the Redlich- Kister expression of the excess energy, the defect-defect interaction energy will hence be dependent on the defect formation energy and higher order defect interaction terms. While it would be possible within the existing parameterisation to together constrain the diﬀerent defect related quantities with some measured or calculated input, it is clearly not physical to have each of these terms being coupled together in this way, and indeed with a cubic dependence of the vacancy concentration.

As a result of this, the Redlich-Kister excess energy expression is unsuitable for adding such types of data to assessments, as any constraint becomes a compromise between ﬁtting each of the available quantites. As it is becoming more computationally feasible to calculate values for these quantities and add them to assessments of defective structures, we have an increasing need to have a structure for the excess energy that is more appropriate to add this data to. This could be done by introducing a simple polynomial in terms of vacancy concentration with coeﬃcients of each order of yVa being independent of each other.

6.4 First principles calculations of the vacancy formation energy

Density functional theory (DFT) calculations have been performed [244] to give additional insight into zirconium carbide by calculating its electronic structure and properties. A background to DFT is given in Chapter 3.

6.4.1 Existing studies in the literature

A review of the ﬁrst principles studies relating to vacancies in zirconium carbide is given in Chapter 4 (Section 4.3.3). Calculated formation energies of a vacancy in

153 zirconium carbide at 0 K in the structure considered in this work exist within studies by Kim et al. (2010) [131] and Giorgi (2015) [79]. The effect of the different exchange-correlation functionals on the calculations of zirconium carbide thermodynamic properties was considered by Duff et al. [54], who made fully anharmonic DFT calculations of the thermodynamic properties of stoichiometric zirconium carbide from T = 0 K up to the melting point.

DFT calculations of the reference vacancy formation energy were made for temperatures up to the melting point [244]. The 0 K values from the available studies are summarised in Table 6.1.

The values calculated both by Giorgi [79] and in this study, used the diamond structure as the reference state for carbon, and so a correction from diamond to graphite needed to be applied. This is the diﬀerence between the two carbon phases, where graphite is reported experimentally to be more stable than diamond by 27 meV [272, 36], 24 ± 10 meV when the calorimetric data for graphite and diamond is extrapolated to 0 K [93] [22], and theoretically to be 28 meV [235].

Reference DFT VFE functional (eV/defect) Mellan, 2017 [244] LDA 0.760 Giorgi, 2015 [79] GGA 1.00 Kim et al., 2010 [131] GGA 0.93

Table 6.1: T = 0 K DFT calculations of the raw vacancy formation energy of zirconium carbide calculated using LDA and GGA with 2x2x2 supercells, with the energy diﬀerence between diamond and graphite added to correct the reference energies.

6.4.2 DFT calculations in this study

Separate DFT calculations were made of the energies of the stoichiometric zirconium carbide supercell, and the same supercell with a single carbon atom removed. As zirconium carbide has a sodium chloride structure, it was not necessary to repeat the calculations for diﬀerent positions of the defect, as all positions are equivalent

154 by symmetry and the energy of the supercell is not dependent on the position of the defect.

In order to account for the carbon atom being removed to the graphite phase, the energy of one atom of carbon in the graphite phase was added to the energy of zirconium carbide with a single carbon vacancy. For consistency, this energy must be equivalent to one calculated using the same DFT methodology.

Graphite has two diﬀerent crystal phases: hexagonal and rhombohedral, and it is the hexagonal phase that is abundant in nature and considered here. This phase consists of hexagonally structured carbon atoms arranged in layers (of graphene), where each carbon atom is bonded to three others. There is purely covalent in-plane bonding, and between layers there are van der Waals bonds, that are signiﬁcantly weaker than the covalent bonds within the layer [187]. This leads to graphite’s use as a lubricant, as it is easy for the layers to slide over one another.

One known weakness of the LDA and GGA exchange-correlation functionals is their poor description of van der Waals bonding in solids [26]. In the case of graphite, this means that while the atomic distances within the graphene layers are well predicted, the distances between the planes are not [26, 187]. Although there is agreement in the literature that the LDA exchange-correlation functional better simulates graphite than the GGA exchange-correlation functional, calculations tend to require specialised exchange-correlation functionals and signiﬁcant care with the parameters used. Even given this, there are very limited ways to get satisfactory energies of bulk graphite. For this reason, and for the reason that the same exchange-correlation functionals need to be used throughout all the calculations for consistency, calculations were performed for bulk diamond, and a correction was applied to account for the diﬀerence in stability between graphite and diamond in order to use the calculated energy as a reference energy.

155 Choice of graphite function in DFT VFE

As the DFT energy that was calculated was for carbon in the diamond phase, it was necessary to account for the energy diﬀerence between the graphite and diamond phases in order to represent the reference VFE as referring to an atom of carbon displaced to the graphite phase.

Several ways to account for this diﬀerence were considered in order to best represent the energy of graphite. Figure 6.1 shows the various energy functions considered.

The Gibbs energy of diamond was calculated from 0 K to 4000 K, and is shown as the red curve in Figure 6.1. One option (a) to account for the energy difference between graphite and diamond is to take the difference between the SGTE Gibbs energy functions for graphite and diamond [52] as a function of temperature and add this to the raw DFT values. This is shown in green. It can be seen that there is a significant difference between the Gibbs energy of graphite and diamond, that increases with temperature. However, as the SGTE Gibbs energy functions for graphite and diamond are only valid above 298 K, this will give us a VFE function that is only valid above room temperature.

The SGTE graphite Gibbs energy function [52] describes the energy of graphite as a function of temperature well above 298 K, and so another option (b) is to use this function and take the calculated Gibbs energy of diamond at some temperature as an oﬀset to the SGTE graphite function so that the reference energies to this value are equivalent to the other DFT quantities in the VFE calculation. This is shown in Figure 6.1 in blue. However, this again restricts the VFE to being valid only above 298 K.

The graphite Gibbs energy function is well described from 0.01 K to high temperature in an assessment by Hallstedt et al. [93]. This Gibbs energy function is in agreement with the SGTE Gibbs energy function for graphite above 298 K, but also describes T < 298 K. Another option for the energy used in the VFE calculation is to take the graphite Gibbs energy function from Hallstedt et al.. and use the calculated diamond

156 Figure 6.1: Several Gibbs energies for various diamond and graphite are shown as a function of temperature from 0 K to 4000 K (or 298 K to 4000 K where the range of validity is limited). The energy of diamond calculated using DFT is shown in red. All other functions (a)-(d) represent the graphite phase, determined using different methods. (a) The DFT diamond energy augmented by the graphite-diamond energy difference from SGTE [52] is shown in green. (b) The SGTE diamond Gibbs energy [52] offset by the room temperature DFT diamond energy plus the SGTE graphite-diamond difference at room temperature is shown in blue. (c) The CALPHAD Gibbs energy function from Hallstedt et al. [93] offset by the zero temperature DFT diamond energy plus the SGTE graphite-diamond difference at room temperature is shown in grey. (d) The CALPHAD Gibbs energy function from Hallstedt et al. [93] offset by the room temperature DFT diamond energy plus the SGTE graphite-diamond difference at room temperature is shown in black.

Gibbs energy at some temperature to give a value that has the same reference energies as the other DFT quantities. This is shown for two diﬀerent energy points ((c) 0 K and (d) 298 K) in grey and black respectively.

157 The only approach that uses the full temperature range of the DFT energies is (a), and it could be argued that this has the most physical behaviour, but this approach has the disadvantage of only being valid above 298 K. However, as the true quantity that is being calculated is the energy of diamond, there is not more physical value in using the full temperature range of calculated energies instead of the parameterised Gibbs energy functions from either SGTE or Hallstedt et al.. [93], which have been ﬁtted to experimental and ﬁrst principles data.

It can also be seen that the diﬀerence between the energy from (a) and (b), (c), and (d) is much smaller than the error in the calculations themselves. As such, it seems worthwhile to use an approach that gives a graphite Gibbs energy function from 0 K to high temperatures, instead of just above room temperature.

It can be seen that the Gibbs energy of graphite from (b), (c), and (d) are very similar, and so it does not seem necessary to restrict the Gibbs energy functions used to SGTE functions in this case. It can also be seen that (c) and (d) are very similar, so while the choice of calculated DFT temperature offset will make some difference, it is much smaller than the error on the calculations, and is not of the same order of magnitude as the calculated energies. The temperature that was chosen for this offset was 0 K, there is data from both experiment [36], CALPHAD extrapolation [93] and first principles calculations of diamond [235] that allow us to validate the different parts of this quantity.

Hence it was decided that in order to best represent the behaviour of the graphite phase, the Gibbs energy function used to represent the energy of this carbon atom was

◦Ggraphite = Ggraphite(T ) − Ggraphite(0.01 K) + Gdiamond(0 K) + ∆Ggraphite-diamond(0 K) C [93] [93] DFT [93] (6.17)

158 Details of the calculations

DFT calculations were done in the quasi-harmonic approximation using the Vienna ab initio simulation package (VASP) [144, 145, 142, 143] using the Local Density Ap- proximation (LDA) exchange-correlation functional [198]. It was found in an earlier study by Duﬀ et al. [54] that the results of the quasiharmonic approximation were dependent on the choice of functional, where calculations made using the General- ized Gradient Approximation (GGA) [194, 197] diverged from those using the LDA and experimental data strongly at high temperatures. As such, only LDA was used in these calculations.

The finite temperature calculations included the 0 K energy (including zero point effects), corrected for finite size effects, with quasiharmonic and electronic contributions included.

The Helmholtz energy was found as

F (V,T ) = Fqh(V,T ) + Fel(V,T ) (6.18) then Gibbs free energies were found by minimising the total function as

G(T ) = minV [E0(V ) + ∆FFS + F (V,T )] (6.19)

where ∆FFS is the energy difference attributed to finite size effects in the calculation due to the use to repeating supercells, and E0(V ) is the zero point energy.

The calculations were done using the LDA exchange-correlation functional. The projector-augmented wave (PAW) method was used [31, 146], where the 4s− and 4p−Zr electrons were included as valence states.

A kinetic energy cut-off of 700 eV was used throughout the calculations. For calculations of ZrCx, the k-point Monkhorst-Pack mesh density used in calculating the Helmholtz energy was equivalent to 6x6x6 for a 2x2x2 supercell, and for diamond the mesh density was 9x9x9 for a 2x2x2 supercell. Geometry optimisations were performed to a cut-off criterion of 10−6 eV/A,˚ and electronic optimisations had a cut-off

159 criterion of 10−8 eV. The calculations of the Helmholtz energy were performed using

Methfessel-Paxton smearing at 0.1 eV for ZrCx, and Gaussian smearing at 0.01 eV for diamond.

The ground state energies and volumes, E0, were found by ﬁtting a Vinet equation of state [261] for a minimum of 10 volumes.

To obtain the quasiharmonic contribution, Fqh(V,T ), a mesh of 10 volumes was used with the 64/63 atom supercells for ZrCx and for diamond. Phonons were obtained using the small displacement method to get force constants using phonopy [247]. Due to the high symmetry of ZrC with a vacancy, to obtain quasiharmonic phonons DFT calculations were required at only 18 displacements per volume.

For the 2x2x2 64 atom supercell for stoichiometric ZrC, a q−point sampling density of 25x25x25 was used, with commensurate q−point sampling used elsewhere.

The electronic contribution to the Helmholtz energy, Fel(V,T ), was calculated using Mermin’s ﬁnite-temperature formulation with electronic structure optimised to self-consistency with temperature-dependent Fermi-Dirac occupation. A structure equivalent in size to a 2x2x2 supercell was used, with a sampling density of 12x12x12. In order to obtain the volume dependence of this term, a mesh of 11 volumes was used, covering the range of quasiharmonic volume expansion. The temperature dependence was determined using a mesh of 11 temperatures, ranging from 0.01 eV to 0.33 eV.

160 Figure 6.2: The VFE in eV/defect from the 2x2x2, 3x3x3, and 4x4x4 unit cell supercells at 0 K against the volume of the stoichiometric zirconium carbide supercell are shown in blue. A linear ﬁt is applied to these points, and extrapolated to represent the formation energy of one vacancy in the bulk, where the extrapolated vacancy formation is shown in green.

DFT finite-size errors were corrected by considering 2x2x2, 3x3x3 and 4x4x4 supercells of the perfect ZrC unitcell with a single vacancy. The bulk-limit VFE was determined by adding a relative finite-size correction to the 2x2x2 zero-temperature VFE value, which takes into account the effect of zero point energy and geometry relaxation. The bulk VFE of 0.760 eV corrects DFT finite-size effects by 0.305 eV. This value was extracted from a linear fit (R2 > 0.999) of the VFE against inverse supercell volume, by extending the trend to where the reciprocal of the volume is zero, as illustrated in Figure 6.2.

Fel and Fqh were ﬁtted with a fourth order polynomial in terms of V and T . In order to get the Gibbs free energies used to calculate the vacancy formation energy, the Helmholtz energy was minimised with respect to volume at 1 K intervals up to 4000 K.

The diﬀerent contributions to the calculated vacancy formation energy are shown in

161 Figure 6.3.

Figure 6.3: The 0K reference VFE from the 2x2x2 DFT calculations with zero point eﬀects are shown in grey, with ﬁnite size correction added in green, with temperature dependence from the quasiharmonic approximation added in blue, and with temperature dependent electronic contribution added in red.

Analysis of the DFT calculations

The vibrational and electronic contributions to the free energy both stabilise the non- stoichiometric zirconium carbide, the quasiharmonic contribution decreasing such that the reference vacancy formation energy becomes negative above a certain temperature.

Including all calculated contributions to the vacancy formation energy, it can be seen that at 0K, the reference formation energy of a single carbon vacancy in stoichiometric zirconium carbide is 0.760 eV, which decreases with temperature to -1.137 eV at 3700 K (the congruent melting temperature of zirconium carbide). This decrease can be ﬁtted to a quadratic function, −3.5653109 × 10−8T 2 − 3.9345158 × 10−4T +

162 8.0750829, with R2 = 0.99991. Above 1765 K the vacancy formation energy is negative, meaning that is is more favourable for the material to have a carbon vacancy than to remain stoichiometric.

Figure 6.4: The vacancy formation energy extracted from the previous CALPHAD assessment [66] using equation 6.20 is shown in green. The calculated vacancy formation energy from this study, including quasiharmonic and electronic contributions, and a ﬁnite size correction is shown in red. DFT calculations for the 0K vacancy formation energy from the literature (given in Table 6.1) are also shown.

This result is consistent with the trend shown in other theoretical studies of the vacancy formation energy as a function of temperature. Glensk et al. [83] show calculations of the vacancy formation energy in copper and aluminium as a function of temperature, including anharmonic eﬀects, and demonstrate how the Arrhenius law describing vacancy formation energies does not fully describe the behaviour of the vacancy formation energy as a function of temperature. For both of these materials, a decrease with temperature is shown.

Glensk et al.. show the calculated vacancy formation energy both with electronic and quashiharmonic contributions, and with electronic, quasiharmonic, and anhar-

163 monic contributions, for a variety of exchange-correlation functions (GGA-PW91, GGA-PBE, LDA, and AM05). Their ﬁndings were that for both aluminium and copper, calculations using GGA-PBE and including anharmonic eﬀects were consistent with experimental data - decreasing with temperature (at a rate increasing with temperature) from around 1 eV at 0K.

The electronic plus quasiharmonic calculations allow us a point of comparison for the calculations shown here. They are shown not to exactly follow the same trend as the experimental data, indicating that in aluminium and copper, anharmonicity plays a signiﬁcant role at temperatures above the Debye temperature. However, they still show a vacancy formation energy that decreases with temperature, giving an indication of the behaviour of this property at temperature. For the vacancy formation energy including the quasiharmonic and electronic contributions from Glensk et al., the decrease with temperature appears to be close to linear with temperature. This is consistent with the calculations of the vacancy formation energy in zirconium carbide shown here.

The chemical potential of carbon augments this quantity to become a physical vacancy formation energy. If zirconium carbide is to have a negative reference vacancy formation energy such as this, the chemical potential contribution would not be enough to prevent the vacancy formation energy from also becoming negative. It would then be expected that the material would not be stable with the carbon atoms in place, and instead there would be no stable zirconium carbide phase in equilibrium. However, we know this not to be the case.

Although ﬁnding a negative vacancy formation energy is counterintuitive for a solid that does not fully decompose, it is possible to have a negative reference vacancy formation energy if there are other terms, such as defect interaction terms, that serve to oppose the vacancy formation energy and stabilise the substoichiometric carbide. Although we do not have ﬁrst principles calculated energies for these quantities, this is explored in Section 6.4.2 for the case of the CALPHAD assessment from Fernandez Guillermet [66] and in Section 6.5.3 for this study.

164 Comparison with the existing CALPHAD assessment

The calculated raw vacancy formation energy can be compared with the value extracted from the CALPHAD assessment as shown in equation 6.2.2. The expression extracted from the existing thermodynamic assessment by Fernandez Guillermet [66] is only valid above 298.15 K. The raw vacancy formation energy as given by the parameters within the previous assessment [66], extracted using the parameters in the assessment as shown in Section 6.2, is shown by the green line in Figure 6.4. Based on these parameters, the room temperature raw vacancy formation energy is 0.91 eV, which increases to around 1 eV at 2000 K, and begins to decrease slightly at higher temperatures. At low temperatures, there is agreement within errors of the DFT and CALPHAD values. There is also agreement between the values calculated in this study and other ﬁrst principles calculations, which are listed in Table 6.1.

Although we know by comparing with the DFT calculations that the reference vacancy formation energy encoded in the thermodynamic assessment by Fernandez Guillermet [66] is not physically correct, it is useful to examine the diﬀerent vacancy- related terms from this assessment in order to learn how the diﬀerent components of the excess energy complement one another.

As the energy needed to form a vacancy in zirconium carbide is around 1 eV at all temperatures, it is likely that thermal eﬀects would cause the carbon lattice to destabilise and many vacancies to form. We know from experiment that this does not happen, and so we expect the vacancy-vacancy and higher order interaction energies to act to stabilise the structure.

Using the defect-centric approach to the excess energy described in Section 6.3, we 0 γ 1 γ can extract the diﬀerent parts of the excess energy, with LZr:C,Va + LZr:C,Va being 0 γ 1 γ the vacancy formation energy component of the excess energy, − LZr:C,Va − 3 LZr:C,Va 1 γ being the vacancy-vacancy interaction term, and 2 LZr:C,Va being the higher order components. In the excess energy, these are augemented by being linear, quadratic, or cubic in yVa.

165 Figure 6.5 shows the relative magnitude of these diﬀerent contributions to the excess energy as the coeﬃcients of yVa. It is seen that the linear and cubic terms are strongly negative, and the quadratic term is strongly positive and of a greater magnitude. This suggests that the quadratic term, the vacancy-vacancy interactions, will act in opposition to the vacancy formation energy.

0 γ 1 γ Figure 6.5: LZr:C,Va + LZr:C,Va is the coefficient of the linear yVa term in the excess energy, 0 γ 1 γ − LZr:C,Va − 3 LZr:C,Va is the coefficient of the quadratic yVa term in the excess energy, and 1 γ 2 LZr:C,Va is the coefficient of the cubic yVa term in the excess energy. The magnitude of each of these coefficients from the CALPHAD assessment by Fernandez Guillermet [66] is shown as a function of temperature from room temperature to 3700 K (the congruent melting temperature of zirconium carbide).

Figure 6.6 shows the actual energy contribution to the excess energy as a function 0 γ 1 γ 0 γ 1 γ 2 of temperature, that is, LZr:C,Va + LZr:C,Va yVa, − LZr:C,Va − 3 LZr:C,Va yVa, and 1 γ 3 2 LZr:C,Va yVa, for several values of yVa between yVa = 0.02 and yVa = 0.10. This covers a signiﬁcant part of the range of stoichiometry for zirconium carbide.

The red lines show the excess energy contribution that is linear with vacancy con-

166 centration, that is, the term that is related to the vacancy formation energy. At each vacancy concentration, this term decreases in magnitude with temperature. As the vacancy formation energy increases, the magnitude of this term increases, suggesting that it becomes harder to remove carbon atoms from the carbon lattice as more and more atoms are removed.

The green lines show the energy contribution to the excess energy that represents the vacancy-vacancy interaction energy. This is term also decreases in magnitude with temperature for a given vacancy concentration, and increases with an increasing vacancy concentration, as you would intuitively expect. While the vacancy formation energy term is negative, this term is positive, making the energy of the material less negative. However, it is of a smaller magnitude than the term that is linear in vacancy concentration, suggesting that while the formation of vacancies is slowed by vacancy-vacancy interactions, the formation of vacancies is dominant.

The blue lines represent the energy contributions from higher order interactions. These are negative, and increase in magnitude with vacancy concentration. This term is seem to be much smaller then the other lower order terms, although it becomes more comparable to the other terms at high temperatures.

167 Figure 6.6: The three contributions to the excess energy from equation 6.16 are shown as a function of temperature for diﬀerent vacancy concentrations between yVa = 0.02 and yVa = 0.10 from room temperature to 5000 K. The red curves represent the vacancy formation energy term, the green curves represent the vacancy-vacancy interaction energy terms, and the blue curves represent the higher order vacancy interaction energy terms.

Figure 6.7 shows the diﬀerent components as a function of vacancy concentrations for a variety of temperatures. From this plot it is clearer than from Figure 6.6 that 3 the magnitude of the higher order (ytextrmV a) energy term becomes comparable to the vacancy formation and vacancy-vacancy interation energy terms at high temperatures.

168 Figure 6.7: The three contributions to the excess energy from equation 6.16 are shown as a function of vacancy composition for diﬀerent temperatures between 300 K and 4000 K for vacancy concentrations between yVa = 0 and yVa = 0.5. The red curves represent the vacancy formation energy term, the green curves represent the vacancy-vacancy interaction energy terms, and the blue curves represent the higher order vacancy interaction energy terms.

169 6.5 Including the vacancy formation energy explicitly in a CALPHAD assessment

The expression for the reference vacancy formation is found by combining the parameters in the assessment as following:

f 1 ◦ γ ◦ γ ◦ graphite 0 γ 1 γ EVa = − GZr:C + GZr:Va + GC + LZr:C,Va + LZr:C,Va (6.20) NA

◦ γ ◦ γ The end-member parameters GZr:C and GZr:Va are presserved from the previous thermodynamic assessment to maintain consistency with the original assessment [66] and SGTE data [52]. Keeping these original functions, and substituting the calculated function for the vacancy formation energy, it is possible to obtain an expression which 0 γ 1 γ constrains the excess parameters, LZr:C,Va and LZr:C,Va as follows.

1 γ f ◦ γ ◦ γ ◦ graphite 0 γ LZr:C,Va = NAEVa + GZr:C − GZr:Va − GC − LZr:C,Va (6.21)

In this section, the effects of constraining this reference vacancy formation energy to the function fitted to the DFT calculations in Section 6.4 are explored. This constraint was applied in several different ways in order to find the way that best preserved the thermodynamic assessment.

1 γ The ﬁrst approach involved placing a constraint on the parameter LZr:C,Va and reoptimising this parameter under this constraint, and the other parameters are taken straight from the previous assessment. This is described in Section 6.5.2.

0 γ 1 γ The second approach allowed the LZr:C,Va and LZr:C,Va parameters to vary to optimise the parameters to the experimental data, while still fulﬁlling the constraint of reproducing the chosen vacancy formation energy. The excess parameters from the liquid phase were also allowed to vary in order to achieve an optimisation of the phase diagram that is consistent with the experimental data used in the original assessment to a similar degree of error. This is described in Section 6.5.3.

170 It was found that it was not possible to optimise the phase diagram as well as the previous assessment while only changing the excess parameters for the γ (zirconium carbide) phase as in the ﬁrst approach. This can be known intuitively as to recreate the same phase diagram, and changes to one parameter would have to be compensated for in the other parameters. In order to exactly reproduce the same assessment, it would be necessary to also reoptimise parameters in the α (HCP zirconium)and β (BCC zirconium) phases as well as the γ and liquid Gibbs energy parameters. However, changing these parameters slightly was found to have only a small eﬀect on the phase boundary with the γ phase. In the interest of preserving as much of the original assessment as possible, in particular end-member quantities, it was decided only to reoptimise the excess parameters in the γ and liquid phases.

6.5.1 Existing CALPHAD assessment from the literature

The existing thermodynamic assessment of the carbon-zirconium system that we are using here is the widely used assessment that was conducted by Fernandez Guiller- met in 1995 [66]. This CALPHAD assessment used experimental data and some calculated quantities in order to model all of the phases in the carbon-zirconium system. This CALPHAD assessment is reviewed more thoroughly in Chapter 4.

Comparison with experimental data

The phase diagram as assessed by Fernandez Guillermet [66] is shown in Figure 6.8 with the experimental phase diagram data that was used in the thermodynamic assessment superimposed. This data is from Sara, 1965 [226], Rudy et al., 1965, [219], Rudy et al., 1969 [216], Storms and Griﬃn, 1973 [242], Zotov and Kotelnikov, 1975 [282], and Aldesberg et al., 1966 [6]. This data and its associated errors is considered in more depth in Chapter 4.

Due to the very high-temperature nature of these experiments, there is very limited experimental data, and as can be seen, the existing data has signiﬁcant scatter.

171 Figure 6.8: The phase diagram of the carbon-zirconium system from the CALPHAD assessment by Fernandez Guillermet [66], shown with the experimental phase diagram data used in this assessment superimposed [226, 219, 216, 242, 282, 6].

However, it can be seen that the assessed phase diagram reproduces the experimental data well within errors.

Figure 6.9 shows the heat content as a function of temperature for x(C) = 0.497487, as measured by Turchanin and Fesenko [254]. Figure 6.10 shows the heat content as a function of temperature for x(C) = 0.431818, as measured by Turchanin and Fesenko [254]. Figure 6.11 shows the heat content as a function of temperature for x(C) = 0.408284, as measured by Turchanin and Fesenko [254]. Figure 6.12 shows the heat content as a function of temperature for x(C) = 0.494949, as measured by Bolgar et al. [33].

It can be seen that the assessed Gibbs energy functions reproduce the experimental

172 data well within errors.

Figure 6.9: Enthalpy of zirconium car- Figure 6.10: Enthalpy of zirconium carbide as a function of temperature for bide as a function of temperature for x(C) = 0.497487 from 298 K to 4000 K x(C) = 0.431818 from 298 K to 4000 K from the CALPHAD assessment by Fernan- from the CALPHAD assessment by Fernan- dez Guillermet [66], with experimental data dez Guillermet [66], with experimental data measured by Turchanin and Fesenko [254]. measured by Turchanin and Fesenko [254].

Figure 6.11: Enthalpy of zirconium car- Figure 6.12: Enthalpy of zirconium carbide as a function of temperature for bide as a function of temperature for x(C) = 0.408284 from 298 K to 4000 K x(C) = 0.494949 from 298 K to 4000 K from the CALPHAD assessment by Fernan- from the CALPHAD assessment by Fernan- dez Guillermet [66], with experimental data dez Guillermet [66], with experimental data measured by Turchanin and Fesenko [254]. measured by Bolgar et al. [33].

173 However, as we have seen previously, this assessment does not provide a physical vacancy formation energy function, and so we can conclude that this description is limited to describing only properties considered in the optimisation. As such, if we want to ensure that all the types of data that we can calculate or measure are represented as a physically correct quantity in the thermodynamic assessment, we need to include it explicitly in the optimisation. As the quantity that we want to include here, the reference vacancy formation energy, is not represented directly in the existing parameterisation, we enforce a physical vacancy formation energy by constraining the Gibbs energy parameters to fulﬁl the desired relationship.

6.5.2 Adding constraint without optimisation

In order to constrain the Gibbs energy of the γ zirconium carbide phase to reproduce a physical reference vacancy formation energy, without reoptimising any parameters 1 γ from the existing assessment, the LZr:C,Va parameter was ﬁxed according to equation 7.16. The process of doing this with an existing TDB ﬁle in Thermo-Calc is shown in Appendix B.1. Using these parameters, the phase diagram and thermodynamic properties are plotted and shown in Figures 6.13-6.17.

Comparison with experimental data

The phase diagram produced from these parameters under this constraint is shown in Figure 6.13 with the experimental phase diagram data that was used in the thermodynamic assessment superimposed [226, 219, 216, 242, 282, 6].

We can see that the congruent melting temperature has increased by around 500 K, and the phase diagram is no longer reproducing the experimental phase diagram data in any meaningful way. The homogeneity region of the phase diagram has become narrower, and above 1000 K near-stoichiometric zirconium carbide is no longer stable, with the width of stoichiometry decreasing with temperature to 0.469 at. %C at the eutectic temperature, compared to 0.495 at. %C in the original phase diagram.

174 Figure 6.13: The phase diagram of the carbon-zirconium system created from parameters from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy by modifying the ﬁrst order excess energy parameter in the γ phase, without changing any other parameters. The experimental phase diagram data used in the assessment of the original phase diagram is superimposed [226, 219, 216, 242, 282, 6].

Figure 6.14 shows the heat content as a function of temperature for x(C) = 0.497487, as measured by Turchanin and Fesenko [254]. Figure 6.15 shows the heat content as a function of temperature for x(C) = 0.431818, as measured by Turchanin and Fesenko [254]. Figure 6.16 shows the heat content as a function of temperature for x(C) = 0.408284, as measured by Turchanin and Fesenko [254]. Figure 6.17 shows the heat content as a function of temperature for x(C) = 0.494949, as measured by Bolgar et al. [33].

175 Figure 6.14: Enthalpy of zirconium car- Figure 6.15: Enthalpy of zirconium carbide for x(C) = 0.497487 from 298 K to bide for x(C) = 0.431818 from 298 K to 4000 K from the CALPHAD assessment by 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to re- Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy forma- produce a physical reference vacancy formation energy, with experimental data mea- tion energy, with experimental data measured by Turchanin and Fesenko [254]. sured by Turchanin and Fesenko [254].

Figure 6.16: Enthalpy of zirconium car- Figure 6.17: Enthalpy of zirconium carbide for x(C) = 0.408284 from 298 K to bide for x(C) = 0.494949 from 298 K to 4000 K from the CALPHAD assessment by 4000 K from the CALPHAD assessment by Fernandez Guillermet [66] constrained to re- Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy forma- produce a physical reference vacancy formation energy, with experimental data mea- tion energy, with experimental data measured by Turchanin and Fesenko [254]. sured by Bolgar et al. [33].

176 It can be seen in Figures 6.14-6.17 that the heat content data is reproduced to a lesser extent than in the original assessment (shown in Section 6.5.1), but still agrees with the experimental data points within errors. This is a consequence of 1 γ preserving the end-member Gibbs energy functions, and only allowing the LZr:C,Va excess parameter to vary, which prevents significant changes to the thermodynamic functions for this phase.However, it can be seen that there is the most significant deviation from the experimental data and previous assessment at high temperatures, and at near-stoichiometric compositions. This deviation corresponds to the regions on the phase diagram in Figure 6.13 where there are most significant changes visible.

The excess energy parameters

The excess energy is given by

E γ 0 γ 1 γ Gm = yCyVa LZr:C,Va + LZr:C,Va (yC − yVa) (6.22)

1 γ Figure 6.18 shows the ﬁrst order excess energy parameter, LZr:C,Va, from the original CALPHAD assessement [66], and the same parameter under the constraint of reproducing the reference vacancy formation energy as calculated in Section 6.4.2.

1 γ In the original asssessment, LZr:C,Va increases with temperature, which would increase the Gibbs energy of the phase at a given composition as a function of tem- 1 γ perature. However, once the constraint has been applied, the LZr:C,Va parameter decreases signiﬁcantly as a function of temperature, meaning that as the temperature increases, the compound will become increasingly stable. This is seen in the phase diagram in Figure 6.13 where the zirconium carbide phase remains present at higher temperatures than in the original assessment.

From the form of the calculated reference vacancy formation energy, that is decreasing with temperature and becoming negative at high temperature, we would expect that in an assessment that is consistent with with vacancy formation energy, we would have more formation of vacancies than in the original assessment, and

177 1 γ Figure 6.18: The ﬁrst order γ excess parameter, LZr:C,Va from the CALPHAD assessment by Fernandez Guillermet [66] is in red, and the same parameter after it has been constrained such that the excess energy of the γ phase reproduces the calculated reference vacancy formation energy is shown in green, between 298 K and 3700 K (the congruent melting temperature of zirconium carbide). thus a wide range of stoichiometry up to the stoichiometric end-member compound. However, this is not what we see in the phase diagram, where the range of stable stoichiometries is reduced.

As the vacancy formation energy is lower following the application of the constraint, it is more likely that vacancies would form, and hence the stoichiometric (or near- stoichiometric) compound is no longer stable on the phase diagram. Having this lower (and decreasing) vacancy formation energy destabilises the near-stoichiometric compound meaning that more vacancies are present in the stable state.

178 6.5.3 Optimising γ, liquid, and β-Zr excess parameters

As the set of Gibbs energy parameters under the constraint of the physical reference vacancy formation energy no longer reproduced the experimental thermodynamic and phase diagram data, it was necessary to adjust some of the other parameters within the assessment in order to account for the changes caused by constraining the excess energy. Appendix B.2 shows how it is possible to apply this constraint to the setup ﬁles for an optimisation using Thermo-Calc.

Choice of parameters to optimise

The parameters were optimised using this data in order to reproduce the original assessment [66] as closely as possible. To be consistent with the previous assessment, the end member parameters were kept the same, and only the excess parameters that do not directly encode physical data were allowed to vary in the optimisation.

When reoptimising the parameters, the signs and approximate magnitude of each parameter was preserved from the previous optimisation to keep the assessment physically realistic. In order to reproduce the original phase diagram, as well as varying parameters describing the γ phase, it was found to be necessary to allow the parameters of other phases to also vary in order to compensate for the changes to the the parameters describing the γ phase.

It was found that it was not possible to produce a phase diagram that reproduced the experimental data and the original assessment to a satisfactory extent, under the constraint of including the calculated reference vacancy formation energy, without optimising the parameters describing the Gibbs energy of the γ, liquid, and β phases. In order to fully account for the changes made to the γ phase when applying the constraint, it is also necessary to modify the α phase, but in practice for this system it was found that it was not necessary to do so in order to produce the desired description, and so the α phase parameters from the original assessment were preserved to keep the description as similar to the original as possible.

179 Parameterisation of the γ phase

The Redlich-Kister expansion of the excess parameters for the γ phase in the assessment by Fernandez Guillermet [66] has coefficients for T ln(T ) and T 2 terms as in equation 6.5 which were determined by fitting to experimental heat capacity data at various non-stoichiometric compositions [66]. In order to reproduce this data in the same way as the original assessment, these coefficients were kept the same, and only the coefficients of the non-T dependent and linear terms were allowed to vary in the optimisation. Thus within the zeroth order γ phase excess parameter, only terms a7 and a8 from equation 6.5 were varied, to maintain consistency with the fitting to the experimental data used in the previous assessment. The first order excess parameter for this phase was constrained by the proposed function for the vacancy formation energy, and so the coefficients of the terms in this parameter were not adjusted explicitly.

Parameterisation of the liquid phase

The excess Gibbs energy of the liquid phase in the currently accepted assessment of this system [66] is parameterised as following:

E liquid h0 liquid 1 liquid 2 liquid 2i Gm = xCxZr LC,Zr + LC,Zr (xC − xZr) + LC,Zr (xC − xZr) (6.23) where 0 liquid LC,Zr = a15 + a16T (6.24)

1 liquid LC,Zr = a17 (6.25)

2 liquid LC,Zr = a18 (6.26)

0 liquid 1 liquid 2 liquid LC,Zr , LC,Zr , and LC,Zr were allowed to vary in the optimisation, although they were not allowed to change sign or order of magnitude.

180 Parameterisation of the β phase

The excess Gibbs energy of the β phase in the currently accepted assessment of this system [66] is parameterised as following:

E β 0 β Gm = xCxZr LC,Zr (6.27) where 0 β LC,Zr = a19 (6.28)

0 β LC,Zr was allowed to vary in the optimisation, but was not allowed to change sign or order of magnitude.

Parameters varied in the optimisation

0 γ 0 liquid 1 liquid 2 liquid 0 β In total, LZr:C,Va, LC,Zr , LC,Zr , LC,Zr , and LC,Zr were allowed to vary in the optimisation. This gives a total of five excess parameters which have seven coefficients 1 γ that were allowed to vary. As LZr:C,Va was under the applied constraint, the total number of excess parameters that could vary was reduced by one, meaning that there were four fewer coefficients that could be changed in the optimisation. Including new physical data by placing a constraint on the Gibbs energy parameters reduces the number of degrees of freedom in the optimisation, which is desirable in order to get the most physically meaningful fit.

Data used in the optimisation

When optimising parameters, the set of experimental data used in the previous assessment [66] was reproduced, as far as possible. However, as a very small amount of data was available, the optimisation was challenging, particularly as the optimisation algorithms in PARROT are not designed to deal with constrained parameters. For this reason, as is often done in other studies, the opimisation process was done

181 largely by varying the parameters by hand until an approximate ﬁt was found. In the early stages of the optimisation, invented data points along the phase boundaries of the existing assessment were also used to guide the optimisation to produce a phase diagram as close to the previous assessment as possible using the PARROT module in Thermo-Calc [13]. Once an approximate phase diagram was produced using these invented data points, they were removed and the parameters were optimised using the real experimental data points used in the previous assessment until a converged set of parameters was found.

The previous CALPHAD assessment of the carbon-zirconium system was conducted with a relatively small amount of experimental phase diagram and thermodynamic data. For consistency, the same set of data was used in the reoptimisation of this system. The data used in the assessment is summarised here, and examined more thoroughly in Chapter 4.

Experimental phase diagram data from three studies was used in the assessment. Measurements of the liquidus from Adelsberg et al. [6] were used. Sara [226] reported measurements of the solidus, and liquid single phase and γ+liquid two phase regions. Rudy et al. [219, 216] measured the γ+liquid two phase region and liquid single phase region, as well as the solidus and liquidus.

The thermodynamic data consisted of data from six studies. Baker et al. [16] reports the enthalpy of formation of diﬀerent compositions of ZrCx. Kantor and Fomichev [119] report the enthalpy as a function of temperature for various stoichiometries of

ZrCx. Bolgar et al. [33] give the enthalpy as a function of temperature for stoichiometric zirconium carbide. Turchanin and Fesenko [254] report measurements of the enthalpy of several compositions of zirconium carbide over a range of temperatures. Andrievskii et al. [14] report measurements of the activity of zirconium in zirconium carbide as a function of composition at 2500 K. Storms and Griﬃn [242] report measurements of the activity of zirconium in zirconum carbide as a function of composition at 2103K, and as a function of temperature for ZrC0.96.

182 Process of reoptimisation

It was possible to adjust the parameters for one phase at a time, before allowing them all to vary together, in order to ensure that each phase reproduced the data relating to that phase well.

As the graphite phase is modelled as having no solubility of other elements within it, there are no excess parameters relating to this phase. As we preserved all end member Gibbs energy parameters from the previous assessment, in adjusting the γ/γ+graphite phase boundary, only the excess parameters for the γ phase could be changed. By using the thermodynamic data to guide the form of the Gibbs energy for the γ phase, these parameters were varied until the γ/γ+graphite phase boundary was satisfactory, and the thermodynamic data was ﬁtted well.

To adjust the other side of the homogeneity range of the γ phase, the α-Zr and β-Zr excess parameters could be varied, keeping the gamma parameter ﬁxed. The α/α+γ phase boundary did not need to be modiﬁed, and thus the original parameters for the α phase were kept the same as in the previous assessment [66]. The Gibbs energy parameters for the β phase were then adjusted slightly.

With the γ, α, and β parameters ﬁxed, the liquid excess parameters were then varied such that the solidus, liquidus, and invariant eutectic temperatures were in agreement with the experimental data and previous assessment.

Finally, all of the parameters were allowed to vary in order to ensure that a true global minimum energy had been found. The resulting optimised parameters can be found in Appendix A.1.

Comparison with experimental data

The phase diagram produced from these parameters under this constraint is shown in Figure 6.19 with the experimental phase diagram data that was used in the thermodynamic assessment superimposed [226, 219, 216, 242, 282, 6].

183 Figure 6.19: The phase diagram of the carbon-zirconium system created from parameters from the CALPHAD assessment by Fernandez Guillermet [66] constrained to reproduce a physical reference vacancy formation energy by modifying the ﬁrst order excess energy parameter in the γ phase, and reoptimising the excess parameters in the γ, liquid, and β phases. The experimental phase diagram data used in the assessment of the original phase diagram is superimposed [226, 219, 216, 242, 282, 6].

It can be seen that the phase diagram in Figure 6.19 reproduces the experimental phase diagram data within the scatter of the experimental data to approximately the same extent as the original assessment. The congruent melting temperature in the reoptimisation is around 40 K higher than in the original assessment, which is within the range of uncertainty of the majority of the data, which is 50-100 K. The invariant eutectic point (γ+liquid+graphite) is maintained at the same composition, and the temperature has decreased by around 30 K. All other invariant points are reproduced exactly.

The zirconium-rich liquidus is steeper than in the previous assessment and appears

184 not to reproduce the experimental points from Adelsberg et al. [6] along it. However, these data have a signiﬁcant error associated with them, and are reproduced within this uncertainty.

The boundaries of the single phase zirconium carbide region have changed significantly from the previous assessment. The carbon composition at the boundary at the γ+liquid+graphite eutectic temperature is now xC = 0.472 as compared to xC = 0.495 originally. However, there is no clear experimental data to indicate where this point should lie that this new value is in disagreement with. As a result of introducing the lower (and decreasing to become negative) reference vacancy formation energy into the assessment, it would be expected that vacancies would be more likely to form, and thus the near-stoichiometric zirconium carbide would be less stable. The other side of the single phase region has similarly moved to create a narrower range. While there is some experimental data measuring this phase boundary, there is signiﬁcant error associated with these measurements, and the reoptimised phase diagram is not in disagreement with them beyond their uncertainty.

In the original phase diagram, the width of the single phase region can be seen to be clearly decreasing with temperature below 1100 K. Although historical experimental phase diagrams often show the edges of the single phase region as being vertical with temperature, when applying the CALPHAD, it is necessary to close all single phase regions at the same composition in physically realistic assessment. By visually extrapolating the phase boundaries in the original assessment, it can be seen that the phase boundaries on either side will meet at some temperature below 298 K (which is the limit of validity of the assessed parameters). The phase diagram produced using the reoptimised parameters does not have this obvious narrowing at low temperatures above 500K. However, it can be seen that below around 600 K, the gradient of the phase boundary has changed and it is in fact creating a narrower phase region as the temperature decreases. This suggests that the shape of the single phase region produced by the set of reoptimised Gibbs energy functions is physically realistic.

It would be possible to continue this optimisation to the point that it perfect reproduces all of the invariant temperatures in the previous assessment, but as it was

185 optimised to the stage that it reproduced the phase diagram and experimental data satisfactorily and to the same extent as the original assessment, no further optimisation was done beyond this point. More details on how this goodness of ﬁt was quantiﬁed are in Section 6.5.3.

Figure 6.20 shows the heat content as a function of temperature for x(C) = 0.497487, as measured by Turchanin and Fesenko [254]. Figure 6.21 shows the heat content as a function of temperature for x(C) = 0.431818, as measured by Turchanin and Fesenko [254]. Figure 6.22 shows the heat content as a function of temperature for x(C) = 0.408284, as measured by Turchanin and Fesenko [254]. Figure 6.23 shows the heat content as a function of temperature for x(C) = 0.494949, as measured by Bolgar et al. [33].

Figure 6.20: Enthalpy of zirconium car- Figure 6.21: Enthalpy of zirconium carbide as a function of temperature for bide as a function of temperature for x(C) = 0.497487 from 298 K to 4000 K x(C) = 0.431818 from 298 K to 4000 K from the CALPHAD assessment by Fernan- from the CALPHAD assessment by Fernan- dez Guillermet [66] constrained to reproduce dez Guillermet [66] constrained to reproduce a physical reference vacancy formation en- a physical reference vacancy formation energy, and reoptimising the excess parameters ergy, and reoptimising the excess parameters in the γ, liquid, and β phases, with experi- in the γ, liquid, and β phases, with experimental data measured by Turchanin and Fe- mental data measured by Turchanin and Fe- senko [254]. senko [254].

186 Figure 6.22: Enthalpy of zirconium car- Figure 6.23: Enthalpy of zirconium carbide as a function of temperature for bide as a function of temperature for x(C) = 0.408284 from 298 K to 4000 K x(C) = 0.494949 from 298 K to 4000 K from the CALPHAD assessment by Fernan- from the CALPHAD assessment by Fernan- dez Guillermet [66] constrained to reproduce dez Guillermet [66] constrained to reproduce a physical reference vacancy formation en- a physical reference vacancy formation energy, and reoptimising the excess parameters ergy, and reoptimising the excess parameters in the γ, liquid, and β phases, with experi- in the γ, liquid, and β phases, with experimental data measured by Turchanin and Fe- mental data measured by Bolgar et al. [33]. senko [254].

The functions at the different compositions show that the quality of the fit is best for compositions further from stoichiometry. In Figure 6.20 (x(C) = 0.497487) it can be seen that the fit of the function to the experimental data shown is worse than in Figures 6.21-6.23, but it is still reproducing the data within experimental uncertainty.

It was found that while using the parameterisation from the assessment by Fernandez Guillermet [66], it was not possible to reproduce the same phase boundary between the single-phase γ region and the two-phase γ and graphite region up to 3200 K. The graphite Gibbs energy is described with the SGTE carbon (graphite) Gibbs energy function, with no solubility of zirconium in graphite modelled. As such this phase could not be modiﬁed in the current approach. Keeping this phase the same, the competing γ phase is modiﬁed to produce this phase boundary. Two parameters in the γ phase Gibbs energy were adjusted, the temperature independent and linear

187 0 γ coefficients in LZr:C,Va. While maintaining certain physical constraints such as preventing artifical kinks forming in the phase boundary, it was not found to be possible to adjust the phase boundary to be as close to perfect stoichiometry as in the previous assessment. However, from the experimental data, there is some disagreement of the positioning of the γ/γ+graphite phase boundary, with many experimental studies finding the peak stoichiometry to be lower than described in the CALPHAD assessment by Fernandez Guillermet. As discussed in Section 6.5.2, the presence of more vacancies in the stable phase than in the previous assessment can be seen as a consequence of a lower vacancy formation energy, meaning that more vacancies will naturally form in the compound.

Quantifying the quality of the reoptimisation

In order to compare the ﬁt of the Gibbs energy functions from the assessment to the experimental data, a sum-of-squares analysis was conducted to compare the R2 values for the experimental data used in the ﬁt and the original and the reoptimised assessments.

2 R is the sum of the squared vertical distance from the data points, yi, and the function f(xi, {a}). 2 X 2 R = [yi − f(xi, a1, a2, ..., an)] (6.29)

This was done using the in-built sum of squares of errors calculator in PARROT [13], where all data used in the thermodynamic optimisation was considered and given an equal weighting.

For the original set of parameters without the constraint of our VFE calculation imposed, R2 = 4.62337446×108. The reoptimised phase diagram has R2 = 4.60521314× 108. It can be seen that the R2 values are similar for both assessments, and hence the phase diagrams both ﬁt the phase boundary data to the same extent. In fact, the R2 value after the reoptimisation can be seen to be slightly lower, indicating that the reoptimisation reproduces the experimental data slightly better than the origi-

188 nal assessment. Although the reoptimised parameters do not exactly reproduce the original phase diagram, with certain invariant points and phase boundaries having moved, we can see that overall it is still in agreement with the experimental data to the same extent as the original assessment.

By considering these R2 values, the reoptimisation of the system under the constraint of having a physical vacancy formation energy can be seen to reproduce the phase diagram and thermodynamic experimental data to a similar degree of uncertainty as the previous thermodynamic assessment [66], as well having a vacancy formation energy that is consistent with expectations of physical behaviour.

Gibbs energies of each phase

The Gibbs energy of each phase as a function of composition from the original and reoptimised assessments is shown in Figures 6.24-6.29. Figures 6.24, 6.26, and 6.28 show the Gibbs energies of the previous assessment by Fernandez Guillermet [66] at 2000 K, 2500 K, and 3000 K respectively. Figures 6.25, 6.27, and 6.29 show the Gibbs energies of the reoptimised assessment from this work at 2000 K, 2500 K, and 3000 K respectively.

The black curves in Figures 6.24-6.29 is the Gibbs energy of the liquid phase, the green curve is the Gibbs energy of the γ zirconium carbide phase, the red curve is the Gibbs energy of the β-Zr phase, and the turquoise curve is the Gibbs energy of the α-Zr phase. The point seen on at the C-rich edge is the graphite Gibbs energy at this temperature (which goes not exist at other compositions beyond at x(C) = 1. By examining these plots, it is possible to visually approximate the phase boundaries by estimating the points of common tangent.

The γ (green), liquid (black), and β (red) phases have had some parameters reoptimised, so we shall compare these.

At each temperature, we can see that the phase boundaries remain at about the same place, as we see from the reoptimised phase diagram in Figure 6.19.

189 Figure 6.24: Gibbs energy of each phase Figure 6.25: Gibbs energy of each phase in in the carbon-zirconium system as a func- the carbon-zirconium system as a function of tion of carbon concentration at 2000 K from carbon concentration at 2000 K from the re- the CALPHAD assessment by Fernandez optimised assessment incorporating the ref- Guillermet [66]. erence vacancy formation energy function.

Figure 6.26: Gibbs energy of each phase Figure 6.27: Gibbs energy of each phase in in the carbon-zirconium system as a func- the carbon-zirconium system as a function of tion of carbon concentration at 2500 K from carbon concentration at 2500 K from the re- the CALPHAD assessment by Fernandez optimised assessment incorporating the ref- Guillermet [66]. erence vacancy formation energy function.

190 Figure 6.28: Gibbs energy of each phase Figure 6.29: Gibbs energy of each phase in in the carbon-zirconium system as a func- the carbon-zirconium system as a function of tion of carbon concentration at 3000 K from carbon concentration at 3000 K from the re- the CALPHAD assessment by Fernandez optimised assessment incorporating the ref- Guillermet [66]. erence vacancy formation energy function.

The Gibbs energy of the liquid phase (black) appears to be very similar in the original and reoptimised assessments at all of the temperatures shown, although it becomes slightly more negative in the zirconium-rich composition range for the reoptimised parameters. As the signs and orders of magnitude of the coeﬃcients of the parameters are not allowed to change, the Gibbs energy function still varies smoothly and continuously without any unphysical behaviour.

The Gibbs energy for the β phase (red) has not changed signiﬁcantly from the original assessment, as only one parameter was changed in the reassessment, which was a large positive constant excess parameter. Parameters of this type serve largely to stabilise a phase, and often are an arbitrarily large amount when there is a limited amount of data available to ﬁt them exactly [161]. The β phase is slightly less negative between the end members in the reoptimised assessment, as a consequence of the excess parameter, which is originally large and negative, being reduced slightly in magnitude.

The Gibbs energy for the γ phase (green) has changed most signiﬁcantly in the reoptimisation, because the constraint applied to the excess parameters means that

191 they have greater changes made to them then the adjustments made in the liquid and β phases. The end member Gibbs energy is fixed to the original value, as only the excess parameters were changed. However, at low carbon content, instead of a constantly decreasing Gibbs energy of the γ phase as in the original assessment, there is a miscibility gap in the Gibbs free energy that becomes more prominent as the temperature increases. Although this behaviour appears in the range where the γ phase is not stable, and thus does not affect the phase diagram or thermodynamic properties of the stable phases at any temperature and composition, ideally we would not have a significantly different shape of the reoptimised function as compared to the original Gibbs energy function.

The lattice stability of FCC-Zr

This miscibility gap suggests that the lattice stability of FCC-Zr (the end member energy) should be higher, in order to prevent the Gibbs energy from having to curve back down as xC → 0. We have retained the SGTE Gibbs energy function for FCC- Zr [52] in the reoptimisation, but examination of the parameters in the SGTE Gibbs energy function for this phase shows that there would be room to modify these in order to remove this miscibility gap from the reoptimised Gibbs energy function.

FCC-zirconium does not precipitate as a stable phase without the presence of other elements as interstitial defects to stabilise it [62]. The SGTE Gibbs energy function [52] is given as an energy addition to the SGTE SER Gibbs energy function (HCP- Zr): ◦ γ SER γ−α γ−α ◦ α GZr:Va − HZr = ∆H − ∆S · T + GZr (6.30) where ∆Hγ−α = 7600 J/mol and ∆Sγ−α = 0.9 J/mol/K. These values for ∆Hγ−α and ∆Sγ−α are from Saunders et al. [230] where they are extrapolated from binary phases where FCC-Zr is an end member, rather than having been calculated for the metastable phase itself.

In the literature, there are several values for ∆Hγ−α reported as well as the values

192 from Saunders et al. above. Kaufman was the ﬁrst to estimate ∆Hγ−α to be around 3400 J/mol and also estimated ∆Sγ−α=0 J/mol/K. Using one-electron theory, Skriver [238] obtained a value of ∆Hγ−α = 3400 J/mol. DFT calculations from Wang et al. [263] give ∆Hγ−α = 3690 J/mol. With these calculations of the lattice stability of FCC-Zr being consistently similar and diﬀerent to the value obtained by Saunders et al. there is evidence to suggest that the lattice stability of FCC-Zr is not physically ◦ γ accurate, and thus there is freedom to reoptimise the parameters in GZr:Va without loss of any physical information.

Although in this study, for consistency with other systems, including higher order systems, we did not want to change the end member states, in order to explore the eﬀects of adjusting the lattice stability of FCC-Zr, ∆Hγ−α and ∆Sγ−α were allowed to vary in the optimisation. It was found that this gave a more physically realistic set of Gibbs energy parameters without having to make signiﬁcant changes to the parameters, and gave more freedom to adjust the shape of the single phase zirconium carbide region of the phase diagram.

This gives some evidence that the current SGTE Gibbs energy function for FCC-Zr is not representing the reality of the physics of this phase, and suggests that we should be open to modifying it using insight given by modern ﬁrst principles techniques. This also shows that although it was found that it was possible to create a phase diagram which reproduces the experimental data to the same extent as the original by reoptimising certain excess parameters, in a full reassessment where parameters do not need to be so carefully preserved it is possible to adjust the shape of the Gibbs energy functions further so that they are as we would physically expect even in their metastable states.

Comparing the excess energy parameters

Figure 6.31 shows the relative magnitude of these diﬀerent contributions to the excess energy as the coeﬃcients of yVa following the reoptimisation of the parameters under the constraint of reproducing the calculated reference vacancy formation energy.

193 Figure 6.30 shows the same curves for the original parameters on the same scale for reference.

0 γ 1 γ Figure 6.30: ( LZr:C,Va + LZr:C,Va) (red) Figure 6.31: The coefficients of the linear is the coefficient of the linear yVa term in (red), quadratic (green), and cubic (blue) in 0 γ 1 γ the excess energy, (− LZr:C,Va − 3 LZr:C,Va) yVa terms of the excess energy of the γ phase (green) is the coefficient of the quadratic yVa following the reoptimisation of the excess γ, 1 γ term in the excess energy, and (2 LZr:C,Va) liquid, and β Gibbs energy parameters from (blue) is the coefficient of the cubic yVa the CALPHAD assessment from Fernandez term in the excess energy. The magnitude Guillermet [66] under the constraint of re- of each of these coefficients from the CAL- producing the calculated reference vacancy PHAD assessment by Fernandez Guillermet formation energy are shown as a function [66] is shown as a function of temperature of temperature from room temperature to from room temperature to 3700 K. 3700 K.

We can see that for both the original and reoptimised sets of parameters, the linear and cubic terms are negative, and the quadratic term is strongly positive and of a greater magnitude. As discussed in Section 6.4.2, this suggests that the vacancy- vacancy interaction energies, represented by the quadratic term, act to oppose the vacancy formation energy and higher order terms.

It can be seen that following the reoptimisation, the gradient of each term has changed from increasing to decreasing, and vice versa. Additionally, the magnitude of each term is around twice its previous magnitude following the reoptimisation. The linear term, representing the vacancy formation energy (the red line) can be seen to be decreasing with temperature, as we would expect following the incorporation of a vacancy formation energy of that form. In order to counter this behaviour and

194 stabilise the vacancies, the quadratic term then has to increase with temperature.

Figures 6.32 and 6.33 show the actual energy contribution to the excess energy as a function of temperature for the original assessment and the reoptimised assess- 0 γ 1 γ 0 γ 1 γ 2 ment respectively. That is, LZr:C,Va + LZr:C,Va yVa, − LZr:C,Va − 3 LZr:C,Va yVa, 1 γ 3 and 2 LZr:C,Va yVa, for several values of yVa between yVa = 0.02 and yVa = 0.10.

Figure 6.32: The three con- Figure 6.33: The three contributions to the excess energy tributions to the excess en- from equation 6.16 are shown as a function of temperature ergy from equation 6.16 are for diﬀerent vacancy concentrations between yVa = 0.02 shown as a function of tem- and yVa = 0.10 from room temperature to 5000 K, from perature for diﬀerent vacancy the reoptimised assessment. The red curves represent the concentrations between yVa = vacancy formation energy term, the green curves represent 0.02 and yVa = 0.10 from the vacancy-vacancy interaction energy terms, and the blue room temperature to 5000 K, curves represent the higher order vacancy interaction en- from the CALPHAD assess- ergy terms. These can be compared with the curves from ment by Fernandez Guiller- the original CALPHAD optimisation in Figure 6.32. met [66].

The red lines show the excess energy contribution that is linear with vacancy concentration, that is, the term that is related to the vacancy formation energy. Opposite to in the original assessment, at each vacancy concentration, this term from the

195 reoptimisation increases in magnitude with temperature, becoming more negative. This term now decreases with temperature following the reoptimisaion. Intuitively we would expect the vacancy formation energy to decrease with temperature, as we see from the DFT calculations in Section 6.4.

As the vacancy concentration increases, the vacancy formation energy at each temperature becomes more negative. This suggests that at higher vacancy concentration, vacancies are more likely to form in the carbon lattice.

The green lines show the energy contribution to the excess energy that represents the vacancy-vacancy interaction energy. Like the term representing the vacancy formation energy, this term also increases in magnitude with temperature for a given vacancy concentration after the reoptimisation, and increases with an increasing vacancy concentration, which is expected as there is a likely to be a higher probability of having more vacancy-vacancy interactions with a higher vacancy concentration. It is also physically intuitive that at a higher temperature there would be more vacancy-vacancy interactions.

While the vacancy formation energy term is negative, the vacancy-vacancy interaction term is positive, which makes the energy of the material less negative overall. However, as in the original optimisation, it is of a smaller magnitude than the term that is linear in vacancy concentration, suggesting that while the formation of vacancies is slowed by vacancy-vacancy interactions, the formation of vacancies is dominant. The diﬀerence between the vacancy formation energy term and the vacancy-vacancy interaction term becomes larger at high temperatures, indicating that as the temperature increases, formation of vacancies beomes very prominent in the material. This is supported by the narrowing of the range of phase stability in the phase diagram.

The blue lines represent the energy contributions from higher order interactions. These are negative, and increase in magnitude with vacancy concentration. This term is seem to be much smaller then the other lower order terms, and becomes less signiﬁcant compared to the other terms at high temperatures and higher vacancy

196 Figure 6.34: The three Figure 6.35: The three contributions to the excess en- contributions to the excess ergy from equation 6.16 are shown as a function of vacancy energy from equation 6.16 concentration at diﬀerent temperatures between300 K and are shown as a function 4000 K, between yVa = 0 and yVa = 0.5 from room tem- of vacancy concentration perature to 5000 K, from the reoptimised assessment. The at diﬀerent temperatures red curves represent the vacancy formation energy term, the between300 K and 4000 K, green curves represent the vacancy-vacancy interaction en- between yVa = 0 and ergy terms, and the blue curves represent the higher order yVa = 0.5 from room tem- vacancy interaction energy terms. These can be compared perature to 5000 K, from the with the curves from the original CALPHAD optimisation CALPHAD assessment by in Figure 6.34. Fernandez Guillermet [66]. concentrations following the reoptimisation.

Figures 6.34 and 6.35 show the diﬀerent components as a function of vacancy concentrations for a variety of temperatures.

If the excess energy is overall negative, the non-stoichiometric compound is more stable than the end member compounds. In Figure 6.35 we can visually sum the three energy contributions to get the total excess energy. It can be seen that at low vacancy concentration, the negative linear term is dominant, meaning that the formation of vacancies will be dominant. However, the quadratic term becomes

197 relatively more signiﬁcant at intermediate vacancy concentrations, indicating that the interactions between vacancies will act to prevent vacancies from forming. The quadratic term increases faster than the linear term with vacancy concentration, preventing vacancies forming beyond a certain point. This explains why, despite there being a negative vacancy formation energy, the compound is stable at intermediate stoichiometries.

6.6 Summary and conclusions

In thermodynamic modelling, it is often challenging to accurately represent the properties and behaviour relating to the defects in compounds, whether they are ordered or disordered defects. There are many models to represent the ordering of compounds that can be used in the CALPHAD method [161], but the formation and interaction energies relating to defects are not explicitly considered in these assessments, although these properties are implicitly included in the data.

A quantity that is not usually explictly considered in CALPHAD assessments, the reference vacancy formation energy, was defined as the energy difference between stoichiometric zirconium carbide and defective zirconium carbide, where a single carbon atom has been removed to the graphite phase. This quantity was extracted from an existing CALPHAD assessement [66] and compared with calculated DFT data up to high temperatures. It was found that the existing CALPHAD assessment did not describe a reference formation energy that was similar to the one calculated using first principles. As such, certain parameters within the existing CALPHAD assessment were reoptimised under the constraint of fitting this calculated reference vacancy formation energy, thus incorporating the reference vacancy formation energy into the thermodynamic assessment as an addition piece of physical data.

By reoptimising certain parameters within an existing thermodynamic description for the carbon-zirconium system [66], it was shown that it is possible to create a thermodynamic description that encodes a physically-based description of the vacancy

198 formation energy without losing compatibility with other data used in the assessment. The experimental data used in the assessment has shown to be reproduced by the reoptimised assessment to the same extent as in the original thermodynamic assessment, using a sum-of-squares analysis.

This was done by reoptimising excess parameters within the zirconium carbide (γ), liquid, and β-Zr phases. A method has been demonstrated whereby a temperature- dependent function for the vacancy formation energy may be used to constrain the parameters of the Gibbs energy functions during an assessment, which can be applied either to existing assessments or during an original optimisation.

In this study, the temperature dependence of the reference vacancy formation energy in zirconium carbide was calculated using DFT and ﬁtted with a quadratic function. This work may be extended to include defect formation energies of other forms, for instance by considering also the formation energies of other defects, defects on multiple lattices, and defect-defect interaction energies.

6.6.1 Further work

In applying the reference vacancy formation as a constraint on the excess energy, the 1 γ number of degrees of freedom in the system was reduced as the LZr:C,Va parameter was no longer allowed to vary in the optimisation. As such, this constraint could be compensated for by adding an additional degree of freedom in the form of an additional excess parameter. Having an additional excess parameter in the γ phase would give more freedom to adjust the boundaries of the single phase region and the solidus and liquidus lines, and would help prevent unwanted miscibility gaps appearing in the Gibbs energy function.

In this work, only one way of constraining the excess energy parameters of the γ phase 1 γ was explored, which was to directly deﬁne the LZr:C,Va in terms of other parameters 0 γ and quantities, including the LZr:C,Va term, which was allowed to vary in the optimisation. While it was possible to reoptimise the parameters such that the Gibbs

199 energy of the phase was similar to before the assessment, the behaviour of different components of the excess energy (when separated into different orders of yVa dependence) individually changed significantly, as we saw in Section 6.5.3. Futhermore, the components that are of different orders of yVa are related and dependent on each other, although it is not physically meaningful to have a direct relation between the vacancy formation energy, vacancy-vacancy interaction energy, and higher order interaction terms in this way. To improve on the reoptimised description, constraining the excess energy in other ways should be considered.

The way that the excess energy is parameterised in the CALPHAD approach has been suﬃcient in the past to describe the thermodynamics of systems such that they reproduce the experimental and calculated phase diagram and thermodynamic data. However, as improvements to the methodology of calculations allow more complex quantities such as defect-defect interaction energies to be calculated, the currently used Redlich-Kister polynomials no longer allow inclusion of all the available physics correctly. Although I have shown that it is possible to include quantities such as the defect formation energy and defect interaction energies in a CALPHAD assessment without any deterioration to the quality of the assessment, including such quantities by means of constraining parameters in the excess energy is not ideal, and in fact introduces unphysical relations between parameters as each quantity separately constrains the same parameters.

As more calculated information becomes available for thermodynamic assessments, it becomes necessary to reformulate the excess energy parameterisation to reﬂect this. By adopting a defect-centric approach to the excess energy, it is possible to include various types of data that relate to the defect properties of materials, without losing any of the information already encoded in the excess energy.

6.6.2 The future of the CALPHAD approach

Part of the ethos of the CALPHAD approach is to critically analyse all pieces of data and only use trustworthy and reliabale data in an assessment. This critical

200 analysis should also be applied to the quality of the unary descriptions, even if they come from SGTE [52]. The SGTE unary Gibbs energy descriptions are widely used and trusted, and most are the result of large amounts of experimental measurement and high quality fitting. However, they cannot be used as end member descriptions without consideration of their quality, simply for the sake of maintaining consistency with other studies. Some descriptions, particularly of metastable phases such as FCC-Zr, were fitted with approximations and very sparse data many years ago, and the development of first principles techniques has allowed us to understand more about these phases since then. However, updating SGTE descriptions to incorporate this new knowledge is rarely done, because the need to thus update all systems using the unary description as an end member is weighed against the improvement in quality. As such, many in the CALPHAD community are using descriptions for the unary phases that have the potential to be improved.

There are published assessments of unary phases, such as the graphite description from Hallstedt et al. used in the DFT reference vacancy formation energy calculation in this chapter [93] which have been developed as a result of the need to have a description of the Gibbs energy at low temperatures (as SGTE descriptions are usually limited to have validity only above 298 K) and incorporate new experimental and ﬁrst principles insight. While such descriptions may be used in some circumstances, they are rarely used in CALPHAD assessments as use of non-standard end members limits the use of the assessment in larger thermodynamic databases.

As more and more first principles calculations show inaccuracies and inconsistencies in the SGTE unary Gibbs energy functions, there is more evidence to improve the set of functions as a whole. If there are only minor changes made to an end member parameter (preserving the sublattice modelling and the stable phases), it is not significantly challenging to update binary systems that are reliant on it as long as the experimental data used is available and well documented. This is usually not possible, because while the experimental data used is reported, the exact weightings and method used to obtain the converged set of parameters are not, and obtaining and formatting the experimental data from the literature to the input files is extremely

201 time consuming. Despite this, it is highly unusual for an assessor to share such input ﬁles widely.

There are some projects, such as the Materials Genome Project [116] that aim to consistently map materials properties, and as such collect all of the experimental information and metadata related to phase diagram assessments, so that it is not necessary for each person doing an assessment to spend large amounts of time collect- ing and managing data when it has been done in the past [42]. In order to maximise the eﬀectiveness of the CALPHAD approach in understanding the thermodynamics of multicomponent systems, projects such as the Materials Genome Project are of vital importance in ensuring that it is as easy as possible to enure that future thermodynamic assessments are both consistent and of a high quality.

202 Chapter 7

Thermodynamic assessment of the carbon-zirconium phase diagram using the CALPHAD approach

7.1 Introduction

In the two decades since the assessment of the carbon-zirconium system by Fernandez Guillermet [66], there have been some studies on the carbon-zirconium system, both experimental and theoretical, which have given us more insight into the thermodynamic properties than was previously available. These new data were highlighted and examined in detail in Chapter 4. These data considered regions of the phase diagram where very little information was available at the time of the previous assessment, as well as using modern experimental measurement techniques and fabrication methods to accurate determine key phase boundaries. These studies suggest that the existing assessment by Fernandez Guillermet does not accurately describe all of the phase boundaries, particularly in regions where the assessment was extrapolated due to a lack of experimental data.

Using all available experimental data, a full new thermodynamic assessment of the carbon-zirconium system is presented here. Some parameters from the assessment

203 Phase Structure Formulae Sublattice model (Pearson symbol, prototype, Strukturbereicht) GRAPHITE A9 hP4, C(graphite), A9 graphite-C (C)1 HCP A3 hP2, Mg, A3 α-Zr (Zr)2(C,Va)1 BCC A2 cI2, W, A2 β-Zr (Zr)1(C,Va)3 FCC A1 cF8, NaCl, A1 γ-ZrC (Zr)1(C,Va)1 LIQUID (Zr)1(C)1

Table 7.1: Crystal structure data of all condensed phases in the C-Zr system. by Fernandez Guillermet have been retained from the previous assessment, where they were ﬁtted directly to experimental data that is not contradictory to any new experimental or ﬁrst principles data.

7.2 Thermodynamic modelling

The thermodynamic modelling framework used by Fernandez Guillermet [66] was maintained in this assessment, with some parameters modiﬁed to reproduce the new experimental data in the assessment.

7.2.1 Phases in the carbon-zirconium system

The ﬁrst experimental phase diagram for the carbon-zirconium system was produced by Sara et al. in 1963 [229]. Since then, there have been many other experimental assessments of the phase diagram [226, 219, 6, 216, 241] that all report ﬁve stable condensed phases in the carbon-zirconium system, which are listed in Table 7.1.

The solid phase, hexagonal graphite, is reported to have very limited (<1 %) solubility of zirconium, and so this was not allowed in the previous modelling of the graphite phase. The unary Gibbs energy function used is from the Scientiﬁc Group Thermo- data Europe (SGTE) database [52] for consistency with other published assessments, but is as published by Gusafson [90].

204 Both the α (HCP) and β (BCC) phases of zirconium have some solubility of carbon, which is described by allowing some carbon atoms to occupy the otherwise unfilled octahedral interstitial sites. The nonstoichiometric γ ZrCx phase can be described as a lattice of metastable FCC zirconium, where octahedral interstitial sites are mostly filled with carbon atoms. These phases are referred to as interstitial phases [75, 85, 249, 66] and are described using a two sublattice version of the CEF [102, 12], where there is a zirconium sublattice of HCP, BCC, or FCC structure, and the octahedral interstitial sites form their own sublattice filled with varying numbers of carbon atoms and vacancies. The sublattice models for each phase are given in Table 7.1.

7.2.2 Gibbs energy parameterisation

The molar Gibbs energies of each interstitial phase φ with sublattice model (Zr)1(C,Va)c (where c, the number of interstitial sites per metallic atom takes the value c = 0.5 for the α phase, c = 3 for the β phase, and c = 1 for the γ phase) can be written as

φ ◦ φ ◦ φ E φ Gm = yC GZr:C + yVa GZr:Va + cRT (yC ln yC + yVa ln yVa) + Gm (7.1)

where yi is the site fraction of component i, meaning the fraction of available sites occupied by i, as described by Sundman and Agren˚ [243], as outlined in Sections 2.3.6 ◦ φ and 2.3.7. GZr:C refers to the Gibbs energy of the phase where all interstitial sites ◦ φ are ﬁlled with carbon atoms, and GZr:Va refers to the energy of the pure zirconium state, which was taken to be the Gibbs energy function of that phase from the SGTE database [52] for consistency with other thermodynamic assessments.

The Gibbs energy of each phase is referred to the enthalpy of a standard (stable element reference (SER)) state recommended by SGTE [52] which is deﬁned as the most stable state of the pure elements at 298.15 K and atmospheric pressure.

E φ The excess Gibbs energy of the φ phase, Gm, represents the diﬀerence between the measured quantites and those predicted in the ideal solution model. It is described

205 by a Redlich-Kister polynomial [209] written as

E φ h0 φ 1 φ i Gm = yCyVa LZr:C,Va + LZr:C,Va (yC − yVa) (7.2)

φ from equation 2.29, where both of the excess Gibbs energy parameters, LZr:C,Va, have temperature dependence with the similar forms

0 φ 0 φ φ φ φ 2 LZr:C,Va = AZr:C,Va + BZr:C,Va · T + CZr:C,Va · T · ln T + DZr:C,Va · T (7.3) and 1 φ 1 φ φ φ φ 2 LZr:C,Va = AZr:C,Va + BZr:C,Va · T + CZr:C,Va · T · ln T + DZr:C,Va · T (7.4)

It can be seen that we have retained the same Gibbs energy models described in Section 6.1.2.

Exponents of CALPHAD sometimes regard these phenomenological parameters as accounting for the interactions between carbon atoms and vacancies on the carbon sublattice, but it is more physical to think of them as parameterising the eﬀect on the Gibbs energy, including both entropy and potential energy, of vacancy-vacancy interactions. It should not be forgotten that the exact entropy implicitly includes an important excess part, beyond the ideal solution part that is represented explicitly in equation 7.1.

1 φ φ φ For the α and β phases, the LZr:C,Va, CZr:C,Va, and DZr:C,Va parameters are assigned a zero value, meaning that the α and β phases are described with a regular solution approximation [66]. All other parameters were fitted to the available experimental data, where the final choice of parameters are chosen such that they reproduce the experimental data as well as possible without using so many parameters as to overfit the data.

◦ φ Each of the stoichiometric Gibbs energies, GZr:C, was taken with reference to the SER enthalpies as ◦ φ SER SER ◦ φ GZr:C − HZr − cHC = ∆ GZr:C (7.5)

206 and ﬁtted as a function of temperature with parameters

◦ φ 2 3 −1 −3 GZr:C = a + bT + cT ln T + dT + eT + fT + gT (7.6)∆ where the coeﬃcients were determined by ﬁtting to available experimental data. The Gibbs end member energies of the α and β phases were taken from studies of the metastable carbides ZrC0.5 and ZrC3 respectively, where the modelling was based on studies of other transition metal carbides and nitrides [63, 185, 86].

◦ φ The other end member Gibbs energies, GZr:Va, are the pure element Gibbs energy functions as given in the SGTE database [52], originally from the assessment of zirconium in various structures by Fernandez Guillermet [63].

The liquid phase is described using a substitutional solution model

liquid ◦ liquid ◦ liquid E liquid Gm = xC GC + xZr GZr + RT (xC ln xC + xZr ln xZr) + Gm (7.7) where xi is the atomic fraction of component i in the liquid phase. The Gibbs energy of pure carbon and zirconium in the liquid phase were taken from Gustafson [90] and Fernandez Guillermet [63] respectively. The excess Gibbs energy term was described using a Redlich-Kister polynomial [209] as

E liquid h0 liquid 1 liquid 2 liquid 2i Gm = xCxZr LC,Zr + LC,Zr (xC − xZr) + LC,Zr (xC − xZr) (7.8) where the parameters are phenomenologically determined by ﬁtting to experimental data using 0 liquid 0 liquid 0 liquid LC,Zr = AC,Zr + BC,Zr · T (7.9)

1 liquid 1 liquid LC,Zr = AC,Zr (7.10)

2 liquid 2 liquid LC,Zr = AC,Zr (7.11) where only the zeroth order liquid excess Gibbs energy parameter has a temperature dependence.

207 7.3 Experimental information available

The experimental information in the literature for the carbon zirconium system is reviewed in Chapter 4. For each set of data, a decision was made whether to use it in the optimisation or to use it as a point of comparison with the ﬁnal assessment. Only the experimental data that was used in the optimisation is described in this section, and other data that was compared with the resulting assessment is shown in Section 7.5.2.

7.3.1 Unary Gibbs energy functions

Since the previous assessment of the carbon-zirconium system by Fernandez Guiller- met [66], the recommended Gibbs energy functions for the pure elements from SGTE [52] that make up the descriptions of the end members have been updated to allow for an improved descripton of some zirconium phases. The latest version of the SGTE unary parameters are thefore used in this CALPHAD assessment.

7.3.2 Phase diagram data

The γ+liquid+graphite eutectic temperature and composition

In the last decades, techniques to measure the ultra-high temperature phase transitions have been developed and used in various studies to precisely determine invariant temperatures such as eutectic temperatures in systems such as carbon-zirconium for use as a calibration point [165, 97, 269]. This has meant that the γ+liquid+graphite eutectic temperature has been measured with very high accuracy, notably in studies by Woolliams et al. [269] and Hartmann [96], with further studies by Jackson et al. [112] and Manara et al. [164]. Hartmann reports a eutectic temperature at 3155±1 K, in agreement with early studies by Woolliams et al., which is lower than the eutectic temperature chosen in the previous CALPHAD assessment of the

208 carbon-zirconium by Fernandez Guillermet of 3200 K as a compromise between the scatter of experimental data [9, 46, 202, 59, 262, 226, 219, 216, 6, 282] with associated uncertainties of around 50 K on each measurement [66]. The temperature measured by Hartmann [96] was taken as the invariant temperaure in the optimisation, with the liquid composition being taken from studies by Sara [226], Rudy et al. [218] and Storms [241].

The γ + liquid + β eutectic temperature and composition

The β + liquid + γ invariant point has been studied extensively in the literature[99, 20, 59, 226, 219, 218, 241, 182, 216, 237, 242, 148, 24, 186] with agreement on the invariant temperature but some scatter in the measurement of the compositions of each phase. Temperature measurements from Bhatt et al. [24] were included in the optimisation with the compositions allowed to vary within the experimentally measured range.

The α + β + γ eutectic temperature and composition

The α + β + γ three phase equilibrium point has been determined in the literature by several studies [99, 20, 226, 219, 241, 237, 148, 186]. In these studies, there is agreement on the invariant temperature, but some scatter in the compositions of each phase. The measurements by Sara [226] were included in the optimisation.

The γ and graphite liquidus

Techniques such as the pulsed laser melting techniques developed by Manara et al. [165] have also been applied to other phase transitions in the carbon-zirconium system in studies by Jackson et al. [113, 114, 164], Sheindlin et al. [234] and Kondratyev et al. [135] where improvements to modern fabrication techniques [112, 115, 79, 95] were

209 also used to ensure the best possible accuracy in both temperature and composition of specimens.

Jackson et al. [112, 114, 164] measured the liquidus and the solidus of the γ phase over a range of carbon compositions from 0.32 at.% C and 0.72 at.% C. There were no existing measurements on the liquidus in the carbon-rich region in the literature, and very few consistent and accurate measurements of the liquidus temperature in the zirconium-rich region, where the experimental diﬃculty lead to a great deal of scatter and large uncertainties in the reported data [74, 8, 80, 38, 59, 189, 218, 282, 229, 226, 219, 216]

Measurements by Jackson et al. provide a consistent study across a range of composition, and demonstrate that the shape of the liquidus might be ﬂatter than in the previous thermodynamic assessment, and the gamma+liquid+graphite eutectic composition may be at a higher carbon composition than estimated previously. Measurements using a similar technique by Kondratyev et al. also suggest that the eutectic composition should be at a higher carbon content, and provide the only measurement of the liquidus of graphite+(C,Zr)-liquid available in the literature.

Experimental data points from Jackson et al. and Kondratyev et al. were included in the optimisation, along with data from Adelsberg et al. [6], Sara [226], and Rudy et al. [219, 216].

The solidus

Jackson et al. [112, 114, 164] also provide measurements of the solidus temperature at several compositions near the congruent melting temperature. Although these fall within the range of the scatter of the existing measurements [229, 226, 59, 219, 216], it is helpful to have a consistent set of data over a large composition range. The experimental data from Jackson et al. suggests that the range of stoichiometry for the single phase zirconium carbide region should be slightly wider than in the previous assessment by Fernandez Guillermet [66].

210 Experimental data from Jackson et al. was included in the optimisation along with experimental data from Sara [226], and Rudy et al. [219, 216].

7.3.3 Thermodynamic data

Heat capacity of zirconium carbides

Fernandez Guillermet considered the experimentally measured heat capacity of ZrCx from several studies [265, 257, 170, 172, 199, 134], but due to the signiﬁcant scatter amongst the experimental data (that can be seen in Figure 7.1), Fernandez Guiller- ◦ γ met determined several of the parameters in GZr:C from a single set of experimental heat capacity data.

The heat capacity is deﬁned as

∂2G C = −T (7.12) P ∂T 2 P,Ni

◦ γ hence for the parameterisation of GZr:C the heat capacity of stoichiometric ZrC in the γ phase can be written as

2 −2 −4 CP = c − 2dT − 6eT − 2fT − 12gT (7.13)

For stoichiometric and near-stoichiometric zirconium carbide, there is a great deal of scatter in the experimentally measured heat capacity, and there are no studies examining a large range of temperatures. Fernandez Guillermet used a single set of experimental data from Westrum and Feick [265] measured from 5 K to 350 K to the parameters in equation 7.13, which could then be extrapolated to give the heat capacity at higher temperatures where no reliable data was available.

There are several calculations of thermodynamic properties of zirconium carbide available in the literature, such as those from Iikubo et al. [111] and Abdollahi [2].

211 These studies report calculations of temperature dependent thermodynamic properties using the quasiharmonic approximation of DFT. Duﬀ et al. report ﬁrst principles calculations of the thermodynamic properties of zirconium carbide, including the heat capacity, from 0 K to the melting point, taking into account the quasiharmonic, electronic, and anharmonic contributions to the free energy [54]. These results were used as a benchmark for comparison with the CALPHAD assessment. These results show remarkable consistency with the heat capacity and Gibbs energy from the thermodynamic assessment by Fernandez Guillermet.

Figure 7.1: Heat capacity of stoichiometric zirconium carbide as a function of temperature from the CALPHAD assessment by Fernandez Guillermet [66], DFT calculations by Duﬀ et al. [54], and experimentally measured values [265, 257, 170, 172, 199, 134, 275].

Zapata Solvas reports experimentally measured values for the heat capacity of stoichiometric zirconium carbide up to 1100 K [275] which can also be seen to be consistent with the existing heat capacity parameterisaton from Fernandez Guillermet.

212 As the new experimental and ﬁrst principles thermodynamic data from Zapata Solvas [275] and Duﬀ et al. [54] are consistent with the heat capacity of stoichiometric zirconium carbide from the existing optimisation, these parameters were preserved when the reassessment of the system was considered.

Heat content

Measurements of the heat content of zirconium carbides have been reported by several authors [181, 173, 155, 33, 120, 255, 252, 253, 254, 72, 233]. There have been no reported experiments of this quantity in the open literature since the previous CALPHAD assessment of this system by Fernandez Guillermet [66] although several first principles studies such as those by Iikubo et al. [111], Abdollahi [2], and Duff et al. [54] calculate thermodynamic properties of the stoichiometric carbide that include the enthalpy. The Gibbs energy of the stoichiometric γ phase was not modified from the assessment by Fernandez Guillermet, but the Gibbs energy of the non-stoichiometric phase was determined by including heat content measurements at different stoichiometries from Bolgar et al. [33], Turchanin and Fesenko [254], and Kantor and Fomichev [120] in the optimisation of the parameters.

Activity

The activity has been measured experimentally by Storms and Griﬃn [242] and Andrievskii et al. [14] as a function of carbon content (at 2103 K and 2500 K respectively) and as a function of temperature for ZrCx in equilibrium with graphite by Storms and Griﬃn [242]. All activity measurements were included in the optimisation of the Gibbs energy parameters.

213 Enthalpy of formation at 298.15 K

The room temperature enthalpy of formation of the γ phase has been reported in the experimental literature by numerous authors [163, 201, 47, 3, 103, 16, 109, 138, 10, 167, 148, 70] using many diﬀerent methods. The experimental values from Baker et al. [16] were the only enthalpy of formation values used in the thermodynamic optimisation.

There are several ﬁrst principles studies in the literature examining the 0 K enthalpy of formation such as Zhang et al. [279], Razumovskiy et al. [208] and Yu et al. [273], which were considered as a point of comparison, but were not used in the thermodynamic assessment.

7.4 Determination of parameters

7.4.1 Choice of parameters to vary

The Gibbs energy parameterisation is summarised below, with the parameters that were varied in the optimisation indicated, and the source of the parameter where they were obtained from elsewhere.

The unary Gibbs energy functions were taken as the recommended Gibbs energy functions from SGTE [52].

The parameters that were ﬁtted directly to thermodynamic data by Fernandez Guillermet were kept the same in this assessment, as there is no new thermodynamic data, experimental or calculated, that is in disagreement with the data that these functions were ﬁtted to.

The graphite phase

◦ graphite SER GC − HC taken exactly from SGTE recommended values [52]

214 The α phase

◦ α SER GZr:Va − HZr taken exactly from SGTE recommended values [52]

◦ α SER SER GZr:C−0.5HC −HZr taken exactly from thermodynamic assessment by Fernandez Guillermet [66]

The description of the end member metastable Zr2C compound was taken from Fer- nandez Guillermet, as it was ﬁtted to estimates of the thermodynamic quantities of the metastable carbides following the approach by Fernandez Guillermet and Grim- vall [69, 67, 68, 65, 71].

0 α LZr:C,Va = a1

The excess parameter of the α phase was allowed to vary in the assessment by Fernandez Guillermet to ﬁt the phase diagram data, and so these were also allowed to vary in this assessment.

The β phase

◦ β SER GZr:Va − HZr taken exactly from SGTE recommended values [52]

◦ β SER SER GZr:C − 3HC − HZr taken exactly from thermodynamic assessment by Fernandez Guillermet [66]

The description of the end member metastable ZrC3 compound was taken from Fer- nandez Guillermet, as it was ﬁtted to estimates of the thermodynamic quantities of the metastable carbides following the approach by Fernandez Guillermet and Grim- vall [69, 67, 68, 65, 71].

215 0 β LZr:C,Va = a2

The excess parameter of the β phase was allowed to vary in the assessment by Fernandez Guillermet to ﬁt the phase diagram data, and so these were also allowed to vary in this assessment.

The γ phase

◦ γ SER GZr:Va − HZr taken exactly from SGTE recommended values [52]

◦ γ SER SER GZr:C − HC − HZr taken exactly from thermodynamic assessment by Fernandez Guillermet [66]

The descriptions of the end member ZrC compound was taken from Fernandez Guillermet as it was ﬁtted directly to measurements of the heat capacity and heat content and estimates of the enthalpy and entropy following the approach by Fer- nandez Guillermet and Grimvall [69, 67, 68, 65, 71].

0 α 2 LZr:C,Va = a3 + a4T + a5T ln T + a6T

1 α 2 LZr:C,Va = a7 + a4T + a5T ln T + a6T

2 The coefficients of the T ln T and T terms (a5 and a6) in the γ excess parameters are fitted directly to measured heat capacities at different stoichiometries of ZrCx by

Fernandez Guillermet, and so these were from the existing assessment. The a3, a4, and a5 parameters were allowed to vary in the optimisation as they are not directly fitted to data and instead optimised to find the values that best fit all of the available data.

216 The liquid phase

◦ liq SER GC − HC taken exactly from SGTE recommended values [52]

◦ liq SER GZr − HZr taken exactly from SGTE recommended values [52]

0 liq LZr,C = a8 + a9T

1 liq LZr,C = a10

2 liq LZr,C = a11

The excess Gibbs energy parameters for the liquid phase description were ﬁtted to the phase diagram data in the assessment by Fernandez Guillermet. In this assessment they were allowed to vary in the optimisation of the parameters to the experimental information alongside the other varied parameters.

7.4.2 Optimisation strategy

The optimisation, by which the parameters were varied iteratively until the best ﬁt to all experimental data used in the assessment is found, was done using the PARROT module in the software Thermo-Calc [13].

The phase boundary measured by Kondratyev et al. [135] is the graphite liquidus. As the graphite Gibbs energy function is taken directly from the SGTE database [52], and has no solubility modelled, there are no parameters that were varied in the graphite phase. Therefore the excess liquid parameters were varied to modify this phase boundary to be consistent with the data from Kondratyev et al., instead of the estimation by Fernandez Guillermet [66]. Once these points were ﬁtted, the excess parameters in the γ phase were varied to reproduce the eutectic temperature from

217 Hartmann [96], and the other liquidus and solidus data, particularly the carbon-rich γ liquidus measured by Jackson et al. [112, 114], as this represents a phase boundary that was not experimentally measured at the time of the previous assessment. The thermodynamic data relating to the γ phase was also considered in the optimisation.

It was found that the excess Gibbs energy parameters for the liquid and γ phase had to be varied together to find a compromise between fitting the data from Kon- dratyev et al. and the other phase boundaries and thermodynamic data. The phase diagram found reproduces the recent experiments from Hartmann, Jackson et al. and Kondratyev et al. significantly better than the original assessment by Fernandez Guillermet.

Finally, the excess Gibbs energy parameters in the α and β phases were considered, in order to ensure that the phase boundaries, solubilities, and invariant temperatures were consistent with experiment. The β+γ+liquid invariant point was ﬁxed by varying only the γ and liquid excess parameters, and so the Gibbs energy parameters in the β phase remained entirely the same as in the assessment from Fernandez Guillermet. The α+β+γ invariant point was ﬁxed by varying the excess Gibbs energy parameter in the α phase slightly to restore the desired invariant temperature.

Finally, all the parameters varied in the optimisation were allowed to vary together to ensure that a global equilibrium had been found.

7.5 Results and discussion/comparison with experiment/previous assessment

The optimised phase diagram from Fernandez Guillermet [66] is shown in Figure 7.2, and the optimised phase diagram from this assessment is shown in Figure 7.3. It can be seen that the phase boundaries are broadly the same, with the main changes being made to the carbon-rich eutectic temperature, and the shape of the γ phase homogeneity region. The optimised Gibbs energy parameters can be found in Ap-

218 pendix A.2. In this section, the new assessment of the carbon-zirconium system is compared with experimental and theoretical data, as well as with the thermodynamic assessment by Fernandez Guillermet [66].

Figure 7.2: Carbon-zirconium phase di- Figure 7.3: Carbon-zirconium phase diagram from the assessment by Fernandez agram from the assessment shown in this Guillermet [66]. work.

7.5.1 Invariant temperatures

The invariant points in this assessment of the carbon-zirconium system are presented in Table 7.2. The invariant points from the assessment from Fernandez Guillermet [66] are also shown for comparison.

Davey, 2017 Fernandez Guillermet, 1995 Invariant Composition (at. % C) Temperature (K) Composition (at. % C) Temperature (K) α + β + γ 39.8 1159 37.9 1159 β + γ + liquid 35.7 2127 35.9 2127 γ + liquid + graphite 49.8 3155 49.5 3200 Congruent melt of γ 46.8 3710 45.8 3700

Table 7.2: Invariant points in the carbon-zirconium system in the optimised thermodynamic assessment.

219 7.5.2 Comparison with experimental data

Phase diagram

The assessed phase diagram is shown in Figure 7.4 with all the experimental phase diagram data used in this assessment superimposed [112, 114, 226, 219, 216, 6, 135, 96]. Figures 7.5 and 7.6 show the phase diagrams from Fernandez Guillermet [66] and this assessment respectively, with the same data superimposed, which can be used for comparison.

Figure 7.4: Phase diagram of the carbon-zirconium system as assessed in this study, with experimental data superimposed [114, 135, 96, 219, 216, 226, 6].

220 Figure 7.5: C-Zr phase diagram from Figure 7.6: C-Zr phase diagram from this the thermodynamic assessment by Fernan- thermodynamic assessment with experimen- dez Guillermet [66] with experimental data tal data superimposed. superimposed.

The γ + liquid + graphite three phase equilibrium

The γ + liquid + graphite invariant point has been measured in several studies in the literature which have a great deal of scatter in the measured values [9, 46, 205, 59, 262, 128, 226, 219, 6, 218, 241, 182, 216, 237, 282, 148, 186, 66]. It can be seen that the new optimisation has a reduced γ + liquid + graphite eutectic temperature at 3155 K, as compared to 3200 K in the previous assessment. This is consistent with the measurements of this temperature by Hartmann et al. [96, 97] and Woolliams et al. [269] using pulsed laser melting techniques to determine the invariant temperature with very little uncertainty (<1 K). The invariant composition at this three phase equilibrium is at a higher carbon composition than in the assessment by Fernandez Guillermet, being at 70.1 at.% C in the new assessment, compared to 67.6 at.% C previously. This is consistent with the measurements of the carbon-rich ZrCx liquidus by Jackson et al. [114, 112] that provided the ﬁrst measurements of this region.

221 The graphite liquidus

A measurement of the liquidus of the graphite phase by Kondratyev et al. [135] was not fully reproduced by the assessment. The elemental analysis of the sample used in this study demonstrated that there were signiﬁcant sources of error in the measurement, and so this data point can be used to give guidance where there was previously no known data, rather than being a precise measurement. It can be seen that the CALPHAD assessment in this study is more consistent with this data point than the previous assessment by Fernandez Guilleremet.

The ZrCx solidus and liquidus

The ZrCx liquidus has been modified slightly from the previous assessment to be flatter and wider near the congruent melting point. This is consistent with the experimental data from Jackson et al. [112, 114] that provides consistent measurements of the liquidus and solidus temperatures across a wider range of compositions than any other existing experimental studies in the open literature. The solidus and liquidus measurements from Sara [226] and Rudy et al. [219, 216] have a significant degree of scatter, and many samples were reported as being melted or contaminated during testing. The CALPHAD assessment reproduces this data within the range of associated error for data points that are considered reliable. The liquidus measurements by Adelsberg et al. are reproduced within the range of experimental uncertainty.

The congruent melting temperature of the γ phase

The measurements of the congruent melting temperature in the experimental literature have a great deal of scatter and associated experimental uncertainty, with measurements from 3400-4016 K reported in the literature [74, 8, 80, 38, 59, 189, 267, 226, 219, 241, 218, 216, 237, 282, 148, 186, 66]. Measurements of the solidus and liquidus by Jackson et al. [112, 114] using the pulsed laser method suggest that

222 the congruent melting point of the samples used is around 3680 K, with an associated error of around 40 K. However, elemental analysis of the samples used shows the presence of oxygen and nitrogen as a contaminant. Jackson reports that the presence of these elements lowers the melting temperature of zirconium carbide and so the true congruent melting temperature will be higher than this. The congruent melting temperature of zirconium carbide from the thermodynamic assessment is 3710 K, which is consistent with the experimental studies and their respective errors. The composition of the congruent melting point was determined by the thermodymic calculations and is at a slightly higher carbon content than the previous assessment by Fernandez Guillermet, but is consistent with the range of experimentally measured values in the literature.

The boundaries of the ZrCx single phase region

The boundaries of the single phase ZrCx region have been measured in the literature in several studies [229, 226, 59, 219, 216, 241, 282, 260, 114] with a great deal of contradiction between studies. As these measurements were contradictory, only the measurements by Sara [226] and Rudy et al. [219, 216] were used to guide the assessment far from the congruent melting temperature, and additional measurements by Jackson et al. [112, 114] were used at high temperatures. The phase boundaries found in the thermodynamic assessment are consistent with the α + β + γ three phase equilibrium point, and consistent with the experimentally measured phase boundaries within the experimental error. Below 2000 K the width of the single phase region is seen to be narrower in the new thermodynamic assessment than in the assessment by Fernandez Guillermet [66]. It is consistent with experimentally drawn phase diagrams in the literature [226, 216, 6, 186] and is as consistent with the experimental data as the previous assessment.

223 The β + liquid + γ three phase equilibrium

The β + liquid + γ invariant point has been reported to be between 2078 K and 2136 K in the literature [99, 20, 59, 226, 219, 218, 241, 182, 216, 237, 242, 148, 24, 186, 66]. In this study the temperature measured by Bhatt et al. [24] of 2127 K was chosen. Bhatt et al. do not report an invariant composition, so the invariant point was determined in the thermodymic assessment to be 35.7 at.% C which is consistent with the range of values measured experimentally.

The α + β + γ three phase equilibrium

The α + β + γ three phase equilibrium point has been determined in the literature by several studies with agreement on the temperature and some scatter in the composition [99, 20, 226, 219, 241, 237, 148, 186, 66]. The invariant temperature from Sara [226] of 1159 K was used as the invariant temperature in this assessment, and the compositon is within the range of the measurements, at 39.8 at.% C.

Solubility of carbon in zirconium and zirconium in carbon

The solubility of carbon in zirconium has been experimentally measured by Sara [226], Kubaschewski-von-Goldbeck [148], and Storms [241], all of whom report the solubility of carbon in β-Zr being very low, and the solubility of carbon in α-Zr being negligible. The solubility of zirconium in graphite is reported as negligible in the literature, and the sublattice modelling in the assessment by Fernandez Guillermet does not allow for any solubility [66]. The carbon and zirconium solubility in the description provided here is consistent both with the experimentally measured values and with the assessment by Fernandez Guillermet.

224 Thermodynamic data

The experimental data available for the carbon-zirconium system consists of measurements of the enthalpy of formation, heat capacity, heat content, and activity at various compositions and temperatures. These data are compared with the optimised Gibbs energy functions and the previous thermodynamic assessment by Fernandez Guillermet [66] in this section.

Heat content

◦ γ ◦ γ Measurements of the heat content, Hm(T ) − Hm(T0) where T0 is a reference temperature, are reported in the literature for diﬀerent stoichiometries by Kantor and Fomichev [119], Bolgar et al. [33], Turchanin and Feskenko [254], and others [181, 155, 173, 242, 233]. The experimental data used in the thermodynamic assessment are shown superimposed on the calculated heat content from the assessment by Fer- nandez Guillermet [66] and from this assessment in Figures 7.7, 7.9, 7.11, 7.13, and Figures 7.8, 7.10, 7.12, 7.14 respectively.

Kantor and Fomichev report measurements of the heat content as a function of temperature for stoichiometric zirconium carbide. This is preserved from the assessment by Fernandez Guillermet and shown in Figure 7.15.

It can be seen that although the thermodynamic functions from this assessment are slightly diﬀerent to the assessment by Fernandez Guillermet, they reproduce the experimental data very well.

225 Figure 7.7: Heat content from Fernandez Figure 7.8: Heat content from the assess- Guillermet [66] with experimental data from ment in this study with experimental data Bolgar et al. [33]. from Bolgar et al. [33].

Figure 7.9: Heat content from Fernandez Figure 7.10: Heat content from the assess- Guillermet [66] with experimental data from ment in this study with experimental data Turchanin and Fesenko [254]. from Turchanin and Fesenko [254].

226 Figure 7.11: Heat content from Fernandez Figure 7.12: Heat content from the assess- Guillermet [66] with experimental data from ment in this study with experimental data Turchanin and Fesenko [254]. from Turchanin and Fesenko [254].

Figure 7.13: Heat content from Fernandez Figure 7.14: Heat content from the assess- Guillermet [66] with experimental data from ment in this study with experimental data Turchanin and Fesenko [254]. from Turchanin and Fesenko [254].

227 Figure 7.15: Heat content of stoichiometric zirconium carbide from Fernandez Guillermet [66] with experimental data from Kantor and Fomichev [119].

Activity

Figures 7.16 and 7.17 show the activity of zirconium as a function of composition at 2103 K from the thermodynamic assessments from Fernandez Guillermet [66] and this study respectively, with experimental data from Storms and Griﬃn [242] superimposed. It can be seen that neither assessment reproduce the experimental data exactly, but both account for the experimental data within the range of experimental uncertainty as it is reported.

Figures 7.18 and 7.19 show the activity of zirconium as a function of composition at 2500 K from the thermodynamic assessments from Fernandez Guillermet [66] and this study respectively, with experimental data from Andrievskii et al. [14] superimposed. It can be seen that the thermodynamic function from this assessment reproduces the experimental data better than the assessment from Fernandez Guillermet. However, both reproduce the experimental data within the range of experimental uncertainty as reported in the literature.

228 Figure 7.16: Activity of zirconium as a Figure 7.17: Activity of zirconium as a function of composition at 2103 K from the function of composition at 2103 K from the assessment by Fernandez Guillermet [66], assessment in this study, with experimental with experimental data from Storms and data from Storms and Griﬃn [242]. Griﬃn [242].

Figure 7.18: Activity of zirconium as a function of composition at 2500 K from the Figure 7.19: Activity of zirconium as a assessment by Fernandez Guillermet [66], function of composition at 2103 K from the with experimental data from Andrievskii et assessment in this study, with experimental al. [14]. data from Andrievskii et al. [14].

The thermodynamic optimisation involves ﬁnding parameters that are a compromise

229 between ﬁtting each data point. While the experimentally measured activity data by Storms and Griﬃn [242] in Figures 7.16 and 7.17 are reproduced better in the assessment by Fernandez Guillermet than in the assessment presented here, the data from Andrievskii et al. [14] shown in Figures 7.18 and 7.19 are reproduced better in the new assessment.

Heat capacity

The Gibbs energy function for the stoichiometric ZrC has not been changed in the thermodynamic assessment, so remains the same as shown in Figure 7.1 where it can be seen that the low temerature experimental data from Westrum and Feick [265] is reproduced very well, and the scatter of high temperature heat capacity data considered in the assessment by Fernandez Guillermet [257, 170, 172, 199, 134] is approximately reproduced. The experimental data from Zapata Solvas [275] is reproduced well, as is the theoretical data from DFT calculations by Duﬀ et al. [54] which include quasiharmonic, electronic, and anharmonic contributions to the free energy.

Westrum and Feick [265] report measurements of the heat capacity of zirconium carbide with formula ZrC0.96 from 5 K to 350 K, which are shown alongside the heat capacity for this stoichiometry from the thermodynamic assessment. The parameters that are ﬁtted directly to the heat capacity are retained from the assessment by Fernandez Guillermet [66] with only the linear and temperature independent parameters varied in the optimisation. As such the thermodynamic assessment still reproduces the experimental data very well, as can be seen in Figure 7.20.

Enthalpy of formation

Figure 7.21 shows the enthalpy of formation at 298.15 K of zirconium carbide at diﬀerent carbon concentrations from 0 at. % C (FCC-Zr) to 50 at. % C (stoichiometric zirconium carbide). The enthalpy of formation for various stoichiometries has been

230 Figure 7.20: Heat capacity of ZrC0.96 as measured by Westrum and Feick [265] and from the thermodynamic assessment in this study. reported as experimental values by various authors [163, 201, 47, 3, 103, 16, 109, 138, 10, 167, 148]. The experimentally measured values from Baker et al. [16], Mah [3], Mah and Boyle [163], and Kornilov et al. [138] that were considered in this assessment are shown in Figure 7.21 alongside the enthalpies of formation from the CALPHAD assessments by Fernandez Guillermet [66] (red) and this study (black).

It can be seen that the assessment from Fernandez Guillermet reproduces the majority of the data (particularly the substoichiometric measurements by Baker et al. and Kornilov et al.) better than the new assessment, but a function consistent with the substoichiometric data from Mah is given instead. There is signiﬁcant error on the measurements of the enthalpy of formation reported in the literature, and the enthalpy of formation from this assessment is still in agreement with the experimental data from Baker et al. and Kornilov et al. within the range of experimental uncertainty.

231 Figure 7.21: Molar enthalpy of formation for zirconium carbide as a function of carbon content at 298.15 K from thermodynamic assessments by Fernandez Guillermet [66] and this study, compared with experimentally measured values [16, 3, 163, 138].

7.6 Vacancy formation energy

The CALPHAD assessment described so far in this chapter has been done without considering the calculations of the vacancy formation energy and their inclusion in CALPHAD assessments as described in Chapter 6. This has been done to enable the optimised parameters to be most compatible with existing thermodynamic databases. In this section, the approach described in Chapter 6 is used to incorporate the calculated reference vacancy formation energy into the thermodynamic assessment.

Figure 7.22 shows the reference vacancy formation energy from the CALPHAD assessment by Fernandez Guillermet [66], the thermodynamic assessment decribed above, and from the DFT calculations [244] described in Chapter 6. It can be seen that the reference vacancy formation energy from the new assessment shows a decreasing trend, but does not agree with the DFT calculations of the reference vacancy formation energy at low temperature as the same quantity from the assessment by Fernandez Guillermet does.

232 Figure 7.22: Reference vacancy formation energies (as deﬁned in Chapter 6) as calculated using DFT (red), and extracted from the CALPHAD assessment from Fernandez Guillermet [66] (blue) and from the assessment shown above (green).

In order to obtain a set of Gibbs energy functions that were consistent with the calculated vacancy formation energy, the excess Gibbs energy parameters for the γ phase were constrained to produce the calculated vacancy formation energy and certain parameters were reoptimised, as in Chapter 6, where only the excess Gibbs energy parameters that are not directly ﬁtted to experimental data are changed.

In the reoptimisation, the excess Gibbs energy of the γ phase is parameterised as

E γ 0 γ 1 γ Gm = yCyVa LZr:C,Va + LZr:C,Va (yC − yVa) (7.14)

233 where 0 γ 0 γ 0 γ 0 γ 0 γ 2 LZr:C,Va = AC,Zr + BC,Zr · T + CC,Zr · T · ln T + DC,Zr · T (7.15)

1 γ and the LZr:C,Va parameter is constrained by the vacancy formation energy from the DFT calculations, which is ﬁtted with a quadratic form and incorporated as

1 γ f ◦ γ ◦ γ ◦ graphite 0 γ LZr:C,Va = NAEVa + GZr:C − GZr:Va − GC − LZr:C,Va (7.16)

0 γ 0 γ From the γ phase, AC,Zr and BC,Zr were allowed to vary in the reoptimisation. The excess Gibbs energy of the liquid phase was then parameterised as

E liquid h0 liquid 1 liquid 2 liquid 2i Gm = xCxZr LC,Zr + LC,Zr (xC − xZr) + LC,Zr (xC − xZr) (7.17) where 0 liquid 0 liquid 0 liquid LC,Zr = AC,Zr + BC,Zr · T (7.18)

1 liquid 1 liquid 1 liquid LC,Zr = AC,Zr + BC,Zr · T (7.19)

2 liquid 2 liquid LC,Zr = AC,Zr (7.20)

An additional temperature dependence was added in the ﬁrst order excess liquid Gibbs energy parameter in order to better describe the system. All liquid excess Gibbs energy parameters were allowed to vary in the optimisation.

It was found that it was not necessary to adjust any excess Gibbs energy parameters in the α or β phases and so they were not varied in the optimisation. The optimised parameters can be found in Appendix A.3.

The resulting phase diagram can be seen in Figure 7.24 compared with experimental data, with the phase diagram calculated earlier in the chapter shown in Figure 7.23 for reference. It can be seen that the γ single phase region has become narrower as we saw in Chapter 6, where the near-stoichiometric compound is no longer stable as the vacancy formation energy becomes lower or negative. It can be seen that there is good agreement with the experimentally measured phase boundaries within the

234 range of the scatter of the data, but the γ + graphite + liquid invariant point can be seen to be at a lower carbon content in the reoptimised assessment, contrary to the experimentally measured graphite liquidus from Kondratyev et al. [135].

It can be seen that the homogeneity range of zironium carbide is narrower after the constraint on the vacancy formation energy has been applied. This can be explained in the same way as in Chapter 6 where the lowering of the vacancy formation energy means that the stoichiometric phase is no longer likely to be stable at high temperatures. By considering the contributions to the excess energy of diﬀerent order of vacancy concentration as in Section 6.5.3 we can see that the vacancy-vacancy interactions and higher order terms then destabilise the compound beyond a certain vacancy concentration.

Figure 7.23: Carbon-zirconium phase di- Figure 7.24: Carbon-zirconium phase dia- agram from the assessment including recent gram from the assessment including recent experimental data, with experimental data experimental data under the constraint of superimposed [114, 135, 96, 219, 216, 226, 6]. reproducing the calculated vacancy formation energy, with experimental data superimposed [114, 135, 96, 219, 216, 226, 6].

Figures 7.26, 7.28, 7.30, and 7.32 show the heat content of the γ phase at several diﬀerent stoichiometries as a function of temperature, compared with experimentally measured values from Bolgar et al. [33] and Turchanin and Fesenko [254]. The same quantities are shown from the optimisation without the inclusion of the vacancy for-

235 Figure 7.25: Heat content from the as- Figure 7.26: Heat content from the assess- sessment including recent experimental data ment including recent experimental data un- with experimental data from Bolgar et al. der the constraint of reproducing the calcu- [33]. lated vacancy formation energy, with experimental data from Bolgar et al. [33] mation energy in Figures 7.25, 7.27, 7.29, and 7.31 for reference. It can be seen that the reoptimised parameters reproduce the thermodynamic quantities well within the range of experimental uncertainty, with the worst agreement being at near stoichiometric compositions. This suggests that the phase boundary could be improved with further optimisations, which may be improved by varying other excess parameters in the optimisation.

The Gibbs energy functions resulting from reoptimising certain parameters under the constraint of fulﬁlling a calculated reference vacancy formation energy can be seen to describe the phase diagram and thermodynamic functions well, producing a description that is consistent with all recent experimental and theoretical data relating to the carbon-zirconium system.

236 Figure 7.27: Heat content from the as- Figure 7.28: Heat content from the assessment including recent experimental data sessment including recent experimental data with experimental data from Turchanin and under the constraint of reproducing the cal- Fesenko [254]. culated vacancy formation energy, with experimental data from Turchanin and Fesenko [254].

Figure 7.29: Heat content from the as- Figure 7.30: Heat content from the assessment including recent experimental data sessment including recent experimental data with experimental data from Turchanin and under the constraint of reproducing the cal- Fesenko [254]. culated vacancy formation energy, with experimental data from Turchanin and Fesenko [254].

237 Figure 7.31: Heat content from the as- Figure 7.32: Heat content from the assessment including recent experimental data sessment including recent experimental data with experimental data from Turchanin and under the constraint of reproducing the cal- Fesenko [254]. culated vacancy formation energy, with experimental data from Turchanin and Fesenko [254].

7.7 Summary and conclusions

In this chapter, a new CALPHAD assessment of the carbon-zirconium system has been presented, that takes into account all relevant experimental and theoretical data. This assessment is consistent with the sublattice modelling and methods used by Fernandez Guillermet [66] in a previous assesment, with changes made to update the unary Gibbs energy functions to those now recommended by SGTE [52], and to incorporate new experimental and theoretical insights that have been made since the previous assessment in 1995. The new assessment was kept as similar as possible to the original assessment where there was no new information that were inconsistent with the assessment by Fernandez Guillermet. This was done in order to be as consistent as possible with systems that may use the carbon-zirconium assessment by Fernandez Guillermet as a sub-system, which allows the use of this system with only minor changes to the higher order parameters.

The experimental information available in the literature on this system are very scat-

238 tered and have signiﬁcant experimental error associated with them. This assessment uses the optimisation procedure used in the CALPHAD method [161], implemented by the PARROT module in Thermo-Calc [13] to produce a thermodynamic description of the system that considers all reliable experimental data and its associated uncertainty. By ﬁnding optimal values for describing the Gibbs energy of each phase in this way, a consistent description is obtained for both the phase diagram and the thermodynamic properties as a function of composition and temperature.

The obtained model has been shown to reproduce the experimental data within the range of scatter or experimental uncertainty, and can be seen to be in better agreement with experimental phase boundary measurements than the assessment by Fernandez Guillermet. This is particularly seen in regions where there was no data available at the time of the previous assessment, or where signiﬁcant eﬀorts have been made to precisely determine invariant temperatures in the system.

In the ﬁrst CALPHAD assessment shown, the calculated reference vacancy formation energy shown in Chapter 6 was not included in the assessment, and only conventional types of data were included in the optimisation. This was done in order to make the Gibbs energy parameters as compatible as possible with existing thermodynamic databases of higher order systems.

As shown in Chapter 6, it is also be possible to include calculated values for defect formation energies related to this system, in order to be consistent with a type of first principles data that is not normally included explicitly in a CALPHAD assessment. Following the method outlined in Chapter 6, some parameters from the new assessment were reoptimised under the constraint that the vacancy formation energy is the form of a function fitted to first principles calculations of the vacancy formation energy from Mellan [244], and another optimised set of Gibbs energy parameters describing the carbon-zirconium system were produced. It was shown that when the calculated vacancy formation energy is included, the zirconium carbide γ phase is no longer stable at carbon contents as high as in the original assessment, as the calculated vacancy formation energy suggests that the formation of vacancies is favourable, destabilising the stoichiometric compound.

239 240 Chapter 8

Conclusions and future direction

Zirconum and hafnium carbides and borides are widely-used ultra-high temperature ceramics (UHTCs) with applications in the aerospace and nuclear industries, amongst others. For such applications, precise and consistent descriptions of the phase stability and properties of these materials is essential, and CALPHAD modelling can be used to combine all available thermodynamic and phase diagram data and associated uncertainties into a single description.

The aim of this work was to evaluate the state of the thermodynamic descriptions of the zirconium and hafnium carbides and borides considering both experimental and calculated data, and the thermodynamic (CALPHAD) modelling used in such descriptions. In examining the currently available descriptions and data, particular attention was paid to the uncertainties associated with each measurement. As these are UHTCs with melting temperatures in excess of 3000 K, there are significant challenges in obtaining accurate measurements or calculations near the melting point. These materials also oxidise readily in air, meaning that the presence of any oxygen during these experiments can have a significant effect on the outcome. Furthermore, the transition metals zirconium and hafnium are often mutually contaminated as a result of their production processes, which may affect the measurements.

A large part of CALPHAD assessments is ﬁnding, organising, and considering every

241 piece of experimental and calculated phase diagram and thermochemical data and its associated metadata. As well as making a decision as to which data to include directly in the thermodynamic optimisation, the associated uncertainty and scatter in context of the other available data informs the weighting that each piece of data is given. As many of the experimental investigations of single element or binary compounds are decades old and published in obscure sources or different languages, it can be difficult and time consuming to obtain all of the available data that is needed to perform a thermodynamic assessment. This work is often repeated multiple times by different authors working on the same system, as it is highly unusual in the CALPHAD community to share formatted experimental data or the way that the data was weighted in an optimisation.

This reluctance to openly share such work is being challenged by projects such as the Materials Genome Initiative [116] that aim to gather and distribute all available experimental and theoretical data such that thermodynamic assessments can be carried out more easily and consistently without the need to spend significant amounts of time searching for data in the literature. The Materials Project was introduced as a way to sort and organise the wealth of first principles data being produced that can provide valuable insight to a great deal of material science studies, leading to a resurgence in CALPHAD assessments of low order systems incorporating first principles results. Eventually, the Materials Project database concept was expanded to also record experimental data in a variety of formats that can be helpful in a CAL- PHAD assessment through the Materials Genome Initiative. In order to maximise the effectiveness of the CALPHAD method, consistency and complete consideration of all data is vital, and contributing to projects such as the Materials Project is a key step in making this generation of CALPHAD assessments as high quality as possible.

Although the use of standard descriptions of end member Gibbs energies as recommended by SGTE [52] has been very valuable in ensuring that the thermodynamic assessments that were conducted over the last 26 years were largely compatible, allowing the development of databases of many elements, the widespread use of these descriptions has frozen the accuracy of CALPHAD assessments at the level of un-

242 derstanding that we had of these elements in 1991. Particularly the descriptions of metastable compounds given in SGTE contain large numbers of approximations. Thanks to improvements to and the widespread availability of ﬁrst principles calculations, we now have a greater understanding of these phases, giving the possibility of improving the descriptions. However, as the use of SGTE functions is so established, there is reluctance in many thermodynamic assessments to use other functions, despite the availability of improved descriptions, and updating the SGTE descriptons in wide use has the consequence of requiring reassessments of all systems that use them as an end member.

There has been a push for the use of improved descriptions, with possibility to include tables of data directly in the database ﬁles to avoid the errors associated with ﬁtting polynomial functions to the data being introduced by Thermo-Calc [251]. In combi- nation with work done by the Materials Project, reassessments of low order systems will be accelerated, meaning that updates to the recommended unary descriptions from SGTE can be incorporated with greater ease in the future.

First principles calculations have revived the CALPHAD community and can provide a lot of insight into the way that phases should be modelled with the CALPHAD approach, as well as providing data where there is a lack of, or conflicting information in the experimental literature. Part of being proficient in the CALPHAD approach has always been to be able to take insights from a variety of experimental sources, being aware of the advantages and pitfalls of each experimental method. These days, there are vast quantities of first principles calculations available in the literature, and as such an understanding of whether a calculation is reliable or valuable is also necessary.

As well as being able to provide calculations of thermodynamic properties such as enthalpies of formation or heat capacities that can be used in place of experimental data in a thermodynamic assessment, ﬁrst principles calculations can give us insight in ways that have not previously been considered as part of thermodynamic assessments as a matter of course, such as defect formation or defect-defect interaction energies. In order to obtain a consistent high quality description of a system, it is

243 necessary to ensure that it is compatible with as much physical understanding as is available. Developments in methodology mean that more types of information are becoming available, and as such eﬀorts to incorporate such data that has not previously been explicitly considered become increasingly important.

Describing the nature and effects of point defects is a challenge that is the subject of many studies in the first principles literature. These calculations are improving our understanding of the structures within non-stoichiometric compounds that are challenging to assess experimentally, thermal effects, and interstitially and substitu- tionally defective structures.

By considering the eﬀects of substitutional defects on the structure and thermodynamic properties of the hafnium and zirconium carbides and borides, we obtained information that could be directly included in a ternary CALPHAD assessment, and provided insight into the solution models that should be used to describe the interactions between the transition metal boride or carbide phases. By considering the stability of metal substitutions in the compounds from experimental literature and ﬁrst principles calculations, it was found that the ideality of the modelling in the B- Hf-Zr an C-Hf-Zr ternary systems as they exist in the open literature is inadequate to describe the behaviour of the solid solutions between the diboride and carbide phases.

Defect formation energies are not usually explicitly included in CALPHAD modelling, although there are several examples of calculations of such properties in the literature for compounds such as zirconium carbide that are vacancy-stabilised. Such properties are implicitly included in the Gibbs energies parameterised in the CAL- PHAD approach, but rarely compared with experimentally measured or theoretically calculated values. Comparing the calculated values with a value extracted from the CALPHAD assessment of the carbon-zirconium system revealed that despite the existing assessment describing the experimental data well, it was not providing a physical representation of the vacancy formation energy.

The excess Gibbs energy of each phase in the currently accepted CALPHAD parame-

244 terisation is fitted to available data to describe the difference between measurements of properties and the value from an ideal solution model between end members. It was found that by constraining the excess Gibbs energy to a different form and reoptimising the excess parameters accordingly, it was possible to force a CALPHAD assessment to fulfil a chosen function for the vacancy formation energy without any loss of information in terms of reproducing the experimental data used in the thermodynamic optimisation. This was demonstrated for the assessment of the carbon- zirconium system by Fernandez Guillermet [66] where it was seen that the sum of the squares of the differences between the experimental data and the thermodynamic description was not increased by including this additional piece of information. Fur- thermore, introducing this piece of information as a constraint on the Gibbs energy parameters reduces the number of degrees of freedom in the system, meaning that it is more physically meaningful.

Using experimental information that was not available at the time of the previous CALPHAD assessment that elucidates certain ambiguities in the phase diagram, a reassessment of the carbon-zirconium system was conducted. This was done using the traditional CALPHAD approach where ﬁrst principles calculations of defect related properties were not considered, and also by incorporating the vacancy formation energy using the method presented.

Further to demonstrating that it is possible to include the vacancy formation energy in a thermodynamic assessment, it was shown how other quantities such as defect- defect interaction energies could also be incorporated in a similar way. However, by adapting the current Redlich-Kister polynomial parameterisation of the excess Gibbs energy to represent these quantities through applying constraints on the excess energy parameters, it is not possible to avoid a non-physical relationship between terms. Certain parts of the CALPHAD parameterisation of the Gibbs energy are very ﬂexible for adaptation to describe defect-related properties, such as the use of sublattices to describe ordered vacancies. However, in order to fully describe the defect properties that it is now possible to calculate, it is necessary to build a new representation of the Gibbs excess energy that is compatible with these physical

245 quantities.

Finally, it is important to note that while first principles calculations may have reached an accuracy comparable to experimental measurements for some quantities [179], new experimental data is still needed to fill gaps where data is lacking and remove ambiguity where data is conflicting. By combining developments in experimental techniques with willingness to adapt the CALPHAD methodology where it is required, the CALPHAD approach can continue to grow as a powerful technique in material research. Work such as was shown in this thesis that is heavily based on both first principles and experimental data can help to localise areas where the understanding of these systems is poor, and can shine light on where experiments are particularly needed for use in their desired applications.

246 Appendix A

Assessed Gibbs energy functions

A.1 Assessed Gibbs energy functions incorporating a calculated vacancy formation energy into the CALPHAD assessment of the carbon-zirconium system by Fernandez Guillermet [66]

The graphite phase

◦ graphite SER −4 2 −1 GC −HC = −17368.441+170.73T −24.3T ln T −4.723×10 T +2562600T

◦ graphite SER 8 −2 10 −3 GC − HC =−2.643 × 10 T + 1.2 × 10 T [52]

The α phase: (Zr)1(C,Va)0.5

◦ α SER 2 GZr:Va − HZr = −7827.595 + 125.64905T − 24.1618T ln T − 0.00437791T

◦ α SER −1 GZr:Va − HZr =+34971T : 298.15 K < T < 2128 K [52]

◦ α SER GZr:Va − HZr = −26085.921 + 262.724183T − 42.144T ln T

◦ α SER 31 −9 GZr:Va − HZr =−1.342896 × 10 T : 2128 K < T < 6000 K [52]

247 ◦ α SER SER 2 GZr:C −0.5HC −HZr = −115822.7+212.2971T −36.10565T ln T −0.001375489T

◦ α SER −7 3 −1 8 −3 GZr:C − 0.5HC =−1.361587 × 10 T + 217131T − 1.9505689 × 10 T [66]

0 α 4 LZr:C,Va = 1.140114 × 10

The β phase: (Zr)1(C,Va)3

◦ β SER −4 2 GZr:Va − HZr = −525.539 + 124.9457T − 25.607406T ln T − 3.40084 × 10 T

◦ β −9 3 −1 −11 4 GZr:Va =−9.729 × 10 T + 25233T − 7.6143 × 10 T : 298.15 K < T < 2128 K [52]

◦ β SER GZr:Va − HZr = −30705.955 + 264.284163T − 42.144T ln T

◦ β SER 32 −9 GZr:Va − HZr =+1.276058 × 10 T : 2128 K < T < 6000 K [52]

◦ β SER SER 2 GZr:C − 3HC − HZr = −142838.2 + 631.7121T − 96.28173T ln T − 0.001856037T

◦ β SER −8 3 −1 9 −3 GZr:C − 3HC =−9.2968513 × 10 T + 2261356T − 7.933899 × 10 T [66]

0 β 5 LZr:C,Va = −1.900 × 10

The γ phase: (Zr)1(C,Va)1

◦ γ SER ◦ α GZr:Va − HZr = GZr:Va + 7600 − 0.9T [52]

◦ γ SER SER 2 GZr:C − HC − HZr = −224784.9 + 297.0288T − 48.14055T ln T − 0.001372273T

◦ γ SER SER −7 3 −1 8 −3 GZr:C − HC − HZr =−1.015994 × 10 T + 517213T − 8.30054316 × 10 T [66]

0 γ 4 2 LZr:C,Va = −4.89368989 × 10 − 32.9254621T + 6.042424T ln T − .001326472T

248 1 γ f ◦ γ ◦ γ ◦ graphite 0 γ LZr:C,Va = NAEVa + GZr:C − GZr:Va − GC − LZr:C,Va

f 4 −3 2 where EVa = 7.7828512 × 10 − 37.962382T − 3.4400088 × 10 T

The liquid phase: (Zr)1(C)1

◦ liq SER ◦ graphite GC − HC = GC + 117369 − 24.63T [52]

◦ liq SER ◦ α GZr − HZr = GZr:Va + 18147.69 − 9.080812T

◦ liq SER 22 7 GZr − HZr =+1.6275 × 10 T : 298.15 K < T < 2128 K [52]

◦ liq SER GZr − HZr = −8281.26 + 253.812609T − 42.144T ln T : 2128 K < T < 6000 K [52]

0 liq 5 LZr,C = −2.92198496 × 10 + 3.59279633T

1 liq 4 LZr,C = −1.91086545 × 10

2 liq 4 LZr,C = 6.04801797 × 10

249 A.2 Assessed Gibbs energy functions from the CAL- PHAD assessment of the carbon-zirconium system presented in Chapter 7

The graphite phase

◦ graphite SER −4 2 −1 GC −HC = −17368.441+170.73T −24.3T ln T −4.723×10 T +2562600T

◦ graphite SER 8 −2 10 −3 GC − HC =−2.643 × 10 T + 1.2 × 10 T [52]

The α phase: (Zr)1(C,Va)0.5

◦ α SER 2 GZr:Va − HZr = −7827.595 + 125.64905T − 24.1618T ln T − 0.00437791T

◦ α SER −1 GZr:Va − HZr =+34971T : 298.15 K < T < 2128 K [52]

◦ α SER GZr:Va − HZr = −26085.921 + 262.724183T − 42.144T ln T

◦ α SER 31 −9 GZr:Va − HZr =−1.342896 × 10 T : 2128 K < T < 6000 K [52]

◦ α SER SER 2 GZr:C −0.5HC −HZr = −115822.7+212.2971T −36.10565T ln T −0.001375489T

◦ α SER −7 3 −1 8 −3 GZr:C − 0.5HC =−1.361587 × 10 T + 217131T − 1.9505689 × 10 T [66]

0 α 4 LZr:C,Va = 1.140114 × 10

The β phase: (Zr)1(C,Va)3

◦ β SER −4 2 GZr:Va − HZr = −525.539 + 124.9457T − 25.607406T ln T − 3.40084 × 10 T

◦ β −9 3 −1 −11 4 GZr:Va =−9.729 × 10 T + 25233T − 7.6143 × 10 T : 298.15 K < T < 2128 K [52]

◦ β SER GZr:Va − HZr = −30705.955 + 264.284163T − 42.144T ln T

250 ◦ β SER 32 −9 GZr:Va − HZr =+1.276058 × 10 T : 2128 K < T < 6000 K [52]

◦ β SER SER 2 GZr:C − 3HC − HZr = −142838.2 + 631.7121T − 96.28173T ln T − 0.001856037T

◦ β SER −8 3 −1 9 −3 GZr:C − 3HC =−9.2968513 × 10 T + 2261356T − 7.933899 × 10 T [66]

0 β LZr:C,Va = −223221.3 [66]

The γ phase: (Zr)1(C,Va)1

◦ γ SER ◦ α GZr:Va − HZr = GZr:Va + 7600 − 0.9T [52]

◦ γ SER SER 2 GZr:C − HC − HZr = −224784.9 + 297.0288T − 48.14055T ln T − 0.001372273T

◦ γ SER SER −7 3 −1 8 −3 GZr:C − HC − HZr =−1.015994 × 10 T + 517213T − 8.30054316 × 10 T [66]

0 γ 2 LZr:C,Va = −10 − 44.0132T + 6.042424T ln T − .001326472T

1 γ 4 2 LZr:C,Va = −3.10 × 10 − 44.0132T + 6.042424T ln T − .001326472T

The liquid phase: (Zr)1(C)1

◦ liq SER ◦ graphite GC − HC = GC + 117369 − 24.63T [52]

◦ liq SER ◦ α 22 7 GZr − HZr = GZr:Va + 18147.69 − 9.080812T + 1.6275 × 10 T : 298.15 K < T < 2128 K [52]

◦ liq SER GZr − HZr = −8281.26 + 253.812609T − 42.144T ln T : 2128 K < T < 6000 K [52]

0 liq 5 LZr,C = −2.870574 × 10 + 12.9T

1 liq 4 LZr,C = −6.191238 × 10

251 2 liq 4 LZr,C = 1.40 × 10

252 A.3 Assessed Gibbs energy functions incorporating a calculated vacancy formation energy into the CALPHAD assessment of the carbon-zirconium system presented in Chapter 7

The graphite phase

◦ graphite SER −4 2 −1 GC −HC = −17368.441+170.73T −24.3T ln T −4.723×10 T +2562600T

◦ graphite SER 8 −2 10 −3 GC − HC =−2.643 × 10 T + 1.2 × 10 T [52]

The α phase: (Zr)1(C,Va)0.5

◦ α SER 2 GZr:Va − HZr = −7827.595 + 125.64905T − 24.1618T ln T − 0.00437791T

◦ α SER −1 GZr:Va − HZr =+34971T : 298.15 K < T < 2128 K [52]

◦ α SER GZr:Va − HZr = −26085.921 + 262.724183T − 42.144T ln T

◦ α SER 31 −9 GZr:Va − HZr =−1.342896 × 10 T : 2128 K < T < 6000 K [52]

◦ α SER SER 2 GZr:C −0.5HC −HZr = −115822.7+212.2971T −36.10565T ln T −0.001375489T

◦ α SER −7 3 −1 8 −3 GZr:C − 0.5HC =−1.361587 × 10 T + 217131T − 1.9505689 × 10 T [66]

0 α 4 LZr:C,Va = 1.140114 × 10

The β phase: (Zr)1(C,Va)3

◦ β SER −4 2 GZr:Va − HZr = −525.539 + 124.9457T − 25.607406T ln T − 3.40084 × 10 T

◦ β −9 3 −1 −11 4 GZr:Va =−9.729 × 10 T + 25233T − 7.6143 × 10 T : 298.15 K < T < 2128 K [52]

253 ◦ β SER GZr:Va − HZr = −30705.955 + 264.284163T − 42.144T ln T

◦ β SER 32 −9 GZr:Va − HZr =+1.276058 × 10 T : 2128 K < T < 6000 K [52]

◦ β SER SER 2 GZr:C − 3HC − HZr = −142838.2 + 631.7121T − 96.28173T ln T − 0.001856037T

◦ β SER −8 3 −1 9 −3 GZr:C − 3HC =−9.2968513 × 10 T + 2261356T − 7.933899 × 10 T [66]

0 β LZr:C,Va = −223221.3 [66]

The γ phase: (Zr)1(C,Va)1

◦ γ SER ◦ α GZr:Va − HZr = GZr:Va + 7600 − 0.9T [52]

◦ γ SER SER 2 GZr:C − HC − HZr = −224784.9 + 297.0288T − 48.14055T ln T − 0.001372273T

◦ γ SER SER −7 3 −1 8 −3 GZr:C − HC − HZr =−1.015994 × 10 T + 517213T − 8.30054316 × 10 T [66]

0 γ 4 2 LZr:C,Va = −3.0 × 10 − 44.0132T + 6.042424T ln T − .001326472T

1 γ f ◦ γ ◦ γ ◦ graphite 0 γ LZr:C,Va = NAEVa + GZr:C − GZr:Va − GC − LZr:C,Va

f 4 −3 2 where EVa = 7.7828512 × 10 − 37.962382T − 3.4400088 × 10 T

The liquid phase: (Zr)1(C)1

◦ liq SER ◦ graphite GC − HC = GC + 117369 − 24.63T [52]

◦ liq SER ◦ α GZr − HZr = GZr:Va + 18147.69 − 9.080812T

◦ liq SER 22 7 GZr − HZr =+1.6275 × 10 T : 298.15 K < T < 2128 K [52]

254 ◦ liq SER GZr − HZr = −8281.26 + 253.812609T − 42.144T ln T : 2128 K < T < 6000 K [52]

0 liq 5 LZr,C = −2.1763050 × 10 − 19.01140T

1 liq 4 LZr,C = 1.5300 × 10 − 12.90774T

2 liq 4 LZr,C = 4.0 × 10

255 256 Appendix B

Constraining the VFE within Thermo-Calc input ﬁles

B.1 Constraining the vacancy formation energy within a TDB ﬁle

For an existing set of Gibbs energy parameters, it is possible to force the assessment to represent a specific vacancy formation energy function, by constraining one of the parameters. For the case of zirconium carbide, the modified TDB file is shown in Figure B.1.

A chosen vacancy formation energy is enforced by constraining the ﬁrst order excess 1 γ γ parameter, LZr:C,Va with equation 7.16, where a chosen function for the vacancy f formation energy can be used as EVa.

1 γ To apply this constraint in the TDB file, all parameters but LZr:C,Va (G(FCC,Zr:C,VA;1) in the TDB shown) remain the same as in the original TDB. Each part of this parameter is defined as a function in the TDB, which then are combined to create a constrained parameter. Defining the first order excess parameter in terms of other parameters can cause the TDB to load incorrectly, and so it is safer to redefine the parameters that the first order excess parameter is dependent on as functions.

257 Figure B.1: The FCC phase part of the TDB file that constrains the first order excess parameter in the γ phase to fulfil a chosen vacancy formation energy. The original parameter is shown and commented out in the TDB.

258 f In this case, EVa is not defined as a separate function, but defined in the parameter, although it could equally be expressed separately as a function in the TDB and called 1 γ in the definition of the LZr:C,Va parameter.

1 γ In doing this, the previous value of the LZr:C,Va parameter is lost, and it is replaced with a parameter that ensures that the vacancy formation energy is consistent with f a chosen function, EVa.

B.2 Constraining the vacancy formation energy within a setup ﬁle for an assessment

In order to conduct an optimisation of a system while also ensuring that the vacancy formation energy is fulfilled by a given function, a constraint can be written into the setup file. In order to apply this constraint by only changing the excess parameters, optimised parameters for the end members must be already known (by assessing without constraint, or in this case, known from a previous assessment) and defined at functions within the setup file. The remaining parameters can be defined as variables to be optimised with the constraint on the vacancy formation energy given in equation 7.16 defined using these functions. An example of this is shown in Figure B.2 where it can be seen that the first order excess parameter, G(FCC˙A1,ZR:C,VA;1), is comprised of variables also defined in the zeroth order excess parameter and previously defined functions that represent fixed quantities.

259 Figure B.2: Extract from the setup ﬁle for carbon-zirconium. The parameters to be optimised can be deﬁned in terms of the variables to be optimised and the end-member parameters as functions.

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