First Principles-Based Interatomic Potentials for Modeling the Body-Centered Cubic Metals V, Nb, Ta, Mo, and W Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Michael Richard Fellinger, B.S., M.S. Graduate Program in Physics The Ohio State University 2013 Dissertation Committee: John W. Wilkins, Advisor Jon Pelz Nandini Trivedi Michael Mills c Copyright by Michael Richard Fellinger 2013 Abstract Accurate large-scale materials simulations depend crucially on high-quality clas- sical interatomic potentials. This study constructs embedded-atom method (EAM) and modified embedded-atom method (MEAM) interatomic potentials for the body- centered cubic (bcc) metals V, Nb, Ta, Mo, and W from data generated by first- principles density-functional theory (DFT) calculations. Comparisons of a wide range of computed materials properties to DFT calculations and experimental data test the quality of the potentials. The analysis reveals that EAM and MEAM potentials generated from low-pressure DFT data accurately model many properties of the bcc metals at low to moderate pressure, but MEAM potentials generated from low- and high-pressure data are needed for quantitative high-pressure simulations. These high- pressure potentials capture much of the physics of the bcc metals at ambient con- ditions, produce the correct energies and geometries of multiple crystal phases, and correctly model the pressure dependence of mechanical properties. The potentials provide a reliable method for studying the deformation of bcc metals over a broad range of temperature and strain conditions, and also offer a viable starting point for constructing accurate potentials for technologically important alloys containing the bcc metals. ii Acknowledgments I thank John Wilkins for his guidance and support. This work would not have been possible without him. I thank Hyoungki Park and Jeremy Nicklas for perform- ing calculations and writing code, and for thoughtful discussions about interatomic potentials. All aspects of this project were improved by their efforts. Richard Hennig and Dallas Trinkle guided me during the introductory phases of this study, and dis- cussions with them over the years provided useful insights that improved the quality of this work. Many other people have helped me either professionally or personally and de- serve recognition. I thank my family, Jonathan Pelz, Nandini Trivedi, Michael Mills, Sheikh Akbar, Trisch Longbrake, Alisha Bruderly, Amita Wadhera, James Davis, Cheasequah Blevins, Robert Guidry, Sheldon Bailey, Julia Young, Geoffrey Smith, Leslie Smith, Alex Yuffa, Bruce Hinds, Marissa Hinds, Kevin Driver, William Parker, Lena Osborne, Adam Hauser, Justin Link, Bill Schneider, Russell Peck, Charles Place, Rakesh Tiwari, Mehul Dixit, John Draskovic, Grayson Williams, James Stapleton, Veronica Stapleton, Greg Viera, Catherine Sundt, Patrick Smith, Nicholas Harmon, David Massey, Reni Ayachitula, Mike Hinton, Michael Boss, Greg Sollenberger, Kevin Knobbe, Ty Nelson, Kevin Troudt, Kirk Troudt, Jeff Stevens, Kerry Highbarger, Re- becca Daskalova, Alex Daskalova, and Jake Livesay. iii I was supported for two years at Ohio State as a Fowler fellow, and by the United States Department of Energy the rest of the time. Computing resources were provided by the Ohio Supercomputing Center and the National Energy Research Scientific Computing Center. iv Vita June 20, 1980 ...............................Born–Brighton, Colorado, USA 1998–2002 ..................................Undergraduate Student, Colorado School of Mines, Golden, Colorado 2002–2004 ..................................Master’s Student, Colorado School of Mines, Golden, Colorado 2004–2006 ..................................Fowler Fellow, Department of Physics, The Ohio State University, Columbus, Ohio 2006–present ...............................Graduate Research Associate, Depart- ment of Physics, The Ohio State Uni- versity, Columbus, Ohio Publications Research Publications M. T. Lusk, M. R. Fellinger, and P. D. Beale, Grain-Boundary Free Energy Via Thermodynamic Integration, J. Chem. Phys., 124:064707, 2006. M. R. Fellinger, H. Park, and J. W. Wilkins, Force-Matched Embedded-Atom Method Potential for Niobium, Phys. Rev. B, 81:144119, 2010. H. Park, M. R. Fellinger, T. J. Lenosky, W. W. Tipton, D. R. Trinkle, S. P. Rudin, C. Woodward, J. W. Wilkins, and R. G. Hennig, Ab Initio Based Empirical Potential Used to Study the Mechanical Properties of Molybdenum, Phys. Rev. B, 85:214121, 2012. v Fields of Study Major Field: Physics vi Table of Contents Page Abstract....................................... ii Acknowledgments.................................. iii Vita......................................... v ListofTables.................................... x ListofFigures ................................... xvii 1. Introduction.................................. 1 1.1 Interatomic Potentials for Metals . 1 1.2 Body-Centered Cubic Refractory Metals . 3 1.3 Scope .................................. 4 2. Classical Interatomic Potentials . 6 2.1 Justification for Classical Potentials . 8 2.1.1 Born-Oppenheimer Approximation . 8 2.1.2 Classical Nuclei Approximation . 14 2.2 Classifications of Interatomic Potentials . 16 2.2.1 Pair Potentials . 19 2.2.2 Higher-Order Potentials . 20 2.3 Many-Body Potentials for Real Metals . 21 2.3.1 Embedded-Atom Method Potentials . 22 2.3.2 Modified Embedded-Atom Method Potentials . 25 vii 3. Generating and Testing Classical Potentials . 27 3.1 Cubic Spline-Based Interatomic Potentials . 29 3.2 AccurateFirst-PrinciplesFittingData . 33 3.2.1 Exchange-Correlation Functionals and Pseudopotentials . 36 3.2.2 Convergence Tests . 42 3.3 Tests for Interatomic Potentials . 48 3.3.1 EquationsofState ....................... 49 3.3.2 ElasticConstants........................ 51 3.3.3 Vibrational Frequencies . 61 3.3.4 ThermalExpansion ...................... 65 3.3.5 MeltingTemperature. 66 3.3.6 Point Defects . 67 3.3.7 SurfaceProperties . 71 3.3.8 Dislocations........................... 76 3.3.9 IdealShearStrength . 90 3.3.10 Generalized Stacking Faults . 93 4. Results and Applications of Classical Potentials . 99 4.1 Force-Matched EAM Potential for Nb . 104 4.2 Force-Matched MEAM Potential for Mo . 121 4.3 Application of the Nb EAM Potential to Shock Simulations . 130 4.4 Potentials for High-Pressure Applications . 135 4.4.1 High-Pressure MEAM Potential for Nb . 137 4.4.2 High-Pressure MEAM Potential for Mo . 145 4.4.3 High-Pressure MEAM Potential for W . 153 4.4.4 High-Pressure MEAM Potential for V . 161 4.4.5 High-Pressure MEAM Potential for Ta . 166 5. Conclusion................................... 177 5.1 SummaryofResults .......................... 177 5.2 Limitations and Possible Extensions of This Work . 179 Appendices 181 A. Convergence Parameters for DFT Calculations . 181 A.1 Plane-Wave Cutoff Energy Ecut Convergence . 181 A.2 k-PointConvergence . 184 viii A.3 FermiSurfaceSmearing . 190 B. Optimized Spline Knots for EAM and MEAM Potentials . 196 B.1 NbEAMPotential ........................... 196 B.2 MoMEAMPotential.......................... 197 Bibliography .................................... 200 ix List of Tables Table Page 3.1 Comparison of calculated lattice constant a, bulk modulus B, and elas- tic constants C′ = (C C )/2 and C of Nb to experimental val- 11 − 12 44 ues. The calculations are performed using vasp with different choices of exchange-correlation (XC) functionals and pseudopotentials (PP). LDA is the local density approximation, PW91 and PBE are two dif- ferent versions of generalized gradient approximations (GGAs), US is an ultra-soft pseudopotential, and PAW is a projector-augmented wave pseudopotential. Pseudopotentials labeled pv treat the 4p-semicore states as valence states, and pseudopotentials labeled sv treat the 4p- and 4s-semicore states as valence states. The PBE functional combined with the PAW-sv pseudopotential produces the best overall agreement withexperiment. ............................. 42 3.2 Convergence parameters used for DFT calculations for the bcc refrac- tory metals. The first column specifies the metal, the second column is the plane-wave energy cutoff used to determine the number of plane- wave basis functions, the third column gives the grid of k-points, and the fourth column is the Fermi surface smearing parameter. All the cal- culations use the PBE GGA exchange-correlation functional and PAW pseudopotentials. The pseudopotentials for V and Nb treat the highest s- and p-semicore states as valence states, and the pseudopotentials for Ta, Mo, and W treat the highest p-semicore states as valence states. 48 3.3 Possible values for the indices of Cijkl. The elastic constants obey the relation Cijkl = Cjikl = Cijlk = Cklij. This limits the values of ij to six possible choices. The table list the possible values of kl for each choice of ij. The total number of possibilities is 6 + 5 + 4 + 3 + 2 + 1 = 21. Thus, there are at most 21 independent values of Cijkl. ........ 56 x 3.4 Voigt notation for stress-strain variables. The six possible pairs of values taken by ij or kl are replaced by six single values taken by α or β. In Voigt notation C C , σ σ , ǫ ǫ , and ǫ ǫ /2 for ijkl ≡ αβ ij ≡ α ii ≡ α ij ≡ α i = j..................................... 57 6 4.1 Configurations in the DFT database used to construct the EAM po- tential for Nb [1]. The “Structure” column lists the crystal structure of each configuration. The “primitive” and “conventional”