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

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 classical 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 conditions, 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 discussions 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 elastic 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 different 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 refractory 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 calculations 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 potential for Nb [1]. The “Structure” column lists the crystal structure of each configuration. The “primitive” and “conventional” labels indicate if the supercell is based on the one-atom primitive or the two-atom conventional cell. The liquid configuration starts as a bcc lattice and then melts during the ab initio MD simulation. The “Natoms” column lists the number of atoms in each configuration. The “T ” column lists the temperature of the ab initio MD simulations used to generate the configurations. The “V/V0” column lists the ratio of the supercell volume to the zero-temperature, equilibrium DFT supercell volume. For configuration 13, V0 is the equilibrium volume of the bcc supercell. The “Shear Strain” column indicates the shear strain applied to the supercells, where M and O denote the volume-conserving monoclinic and orthorhombic strains defined in Subsection 3.3.2, respectively. . . 105

4.2 Structural and elastic properties of Nb. The EAM values for the cohesive energy Ecoh, lattice parameter a, bulk modulus B, and elastic constants C11, C12, and C44 of bcc Nb are compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2]. The experimental lattice parameter from Ref. [3] and the experimental elastic constants from Ref. [4] were measured at 4.2 K. The EAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relativetotheenergyofthebccstructure...... 109

4.3 Nb point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumb- h i h i bell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedral in- h i h i terstitials in eV. The values produced by the Nb EAM potential developed in this study are compared to DFT calculations and results from other published EAM, Finnis-Sinclair (F-S), and MEAM potentials. The experimental vacancy formation and migration energy values are from Ref. [5], and the experimental vacancy activation energy values arefromRefs.[5–9]...... 113

xi 4.4 Energies and relaxations of low-index surfaces of bcc Nb. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The numbers in parentheses next to our relaxation values give the number of atomic layers in the supercells. The values produced by the EAM potential developed in this work are compared to DFT calculations and results from other published EAM, F-S, long-range empirical potential (LREP), MEAM, and modiﬁed analytic EAM (MAEAM) potentials. The potential developed in this work closely matches the DFT values for both energies and relaxations. 115

4.5 Structural and elastic properties of Mo. The MEAM values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Mo are compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [10], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure.124

4.6 Mo point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 h i h i dumbbell, 111 dumbbell, 111 crowdion, octahedral, and tetrahe- h i h i dral interstitials in eV. The values produced by the Mo MEAM potential are compared to experimental data, DFT calculations, and results from other published DFT, Finnis-Sinclair (F-S), and model generalized pseudopotential theory (MGPT) potential calculations...... 127

4.7 Energies and relaxations of low-index surfaces of bcc Mo. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the MEAM potential developed in this work are compared to DFT calculations and results from another published MEAM potential...... 127

xii 4.8 Structural and elastic properties of Nb. The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Nb compared to DFT, experiment, and the low-pressure EAM results. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [3], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT and low-pressure EAM results. The energies are relative to the energy of the bcc structure. The MEAM and EAM potentials produce accurate results for the bcc structure, but only the MEAM values for the other crystal structure properties agree well with DFT...... 139

4.9 Nb point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumb- h i h i bell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedral in- h i h i terstitials in eV. The values produced by the high-pressure Nb MEAM potential are compared to the low-pressure Nb EAM potential results and DFT calculations. The MEAM vacancy results agree better with DFT than the EAM values, but the EAM interstitial results agree betterwithDFTthantheMEAMvalues...... 140

4.10 Energies and relaxations of low-index surfaces of bcc Nb. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT and low-pressure EAM calculations. The EAM potential agrees better with DFT than the MEAM potential, but both potentials produce reason- ablevalues...... 142

4.11 Structural and elastic properties of Mo. The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Mo compared to the low-pressure MEAM, DFT and experiment. The experimental value for the cohesive energy is from Ref. [10], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure...... 148

xiii 4.12 Mo point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 h i h i dumbbell, 111 dumbbell, 111 crowdion, octahedral, and tetrahe- h i h i dral interstitials in eV. The values produced by the high-pressure Mo MEAM potential are compared to DFT calculations, experimental data,andthelow-pressureMEAMvalues...... 150

4.13 Energies and relaxations of low-index surfaces of bcc Mo. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations and results from the low-pressure MEAM potential...... 150

4.14 The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc W compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [12], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure...... 155

4.15 W point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumb- h i h i bell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedral in- h i h i terstitials in eV. The values produced by the high-pressure W MEAM potential are compared to DFT calculations and experimental data. The MEAM potential stabilizes the 100 dumbbell and tetrahedral h i interstitials, which relax to either the 111 dumbbell or 111 crow- h i h i dionintheDFTcalculations...... 158

4.16 Energies and relaxations of low-index surfaces of bcc W. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations...... 158

xiv 4.17 The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc V compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure...... 163

4.18 V vacancy properties. The vacancy formation, migration, and activation energies in eV. The values produced by the high-pressure V MEAM potential are compared to DFT calculations and experimental data...... 165

4.19 Energies and relaxations of low-index surfaces of bcc V. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations...... 166

4.20 The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Ta compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure...... 171

4.21 Ta point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumb- h i h i bell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedral in- h i h i terstitials in eV. The values produced by the high-pressure Ta MEAM potential are compared to DFT calculations and experimental data. The MEAM vacancy energies agree reasonably well with DFT and experiment, but the potential overestimates the DFT interstitial energies. 173

xv 4.22 Energies and relaxations of low-index surfaces of bcc Ta. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations...... 173

B.1 The cubic spline knots and boundary conditions for the Nb EAM potential [1]. The first part of the table lists the knots, and the second part of the table lists the boundary conditions on the splines. Spline knots 1-17 for φ and ρ, and spline knots 1-8 for F are optimized by potfit. The adjusted values of knot 0 for φ, ρ, and F , and knot 9 for F are also listed (see text). The coefficients of the cubic polynomials that interpolate between the knots are determined by requiring continuity of the functions and their first and second derivatives, along with theboundaryconditions...... 198

B.2 The cubic spline knots and boundary conditions for the Mo MEAM potential [13]. The coefficients of the cubic polynomials that interpolate between the knots are determined by requiring continuity of the functions and their first and second derivatives, along with the boundary conditions. The first part of the table lists the number of knots N for each spline and the range of the spline variables xmin and xmax. The middle part of the table gives the values at equally spaced spline knots defined by x = x + i(x x )/N, where N is the number i min max − min of spline knots. Finally, the boundary conditions specifying the derivatives of the splines at their end points are listed in the last part of the table...... 199

xvi List of Figures

Figure Page

2.1 Schematic representation of the four categories of classical potentials. The arrows point in the direction of increasing accuracy and complexity...... 18

2.2 The Lennard-Jones potential consists of a long-range attraction and a short-range repulsion. The shape of the curve is typical of many pair potentials...... 20

2.3 The embedded-atom method views the potential energy of a system EAM Vtot as a sum of energy contributions VI associated with embedding atom I in a host system made of the other atoms. VI is the approximation to r given in Eqn. 2.28 in the text...... 24 FZ, I 3.1 Comparison of cubic spline and polynomial interpolation of the function h(x) = 1/(1 + 25x2). The spline c(x) is a piece-wise function com- 2 3 posed of ten cubic polynomials ci(x)= αi + βix + γix + δix . It equals h(x) at the eleven tabulation values xi, and accurately interpolates the function between them. Natural boundary conditions are imposed on the spline. The tenth-order ﬁtted polynomial p(x) exactly reproduces the tabulated values it was ﬁt to, but provides a poor interpolation of h(x) between most of the tabulation points...... 32

3.2 Cubic spline representation of an arbitrary pair potential φ(r). The set of tabulated spline knots (r ,φ(r )) are free parameters that are { i i } adjusted during function optimization to a fitting database. Usually the ri are chosen as equally spaced. In this case, only the φ(ri) are model parameters. The red arrows indicate that these values are free to change during optimization. Radial functions must go to zero at the cutoff distance, so φ(r11) is fixed in this example. It would be difficult to determine an analytic functional form for a potential with this shape, but cubic splines offer a relatively simple parameterization. 34

xvii 3.3 General state of stress for an element within a solid. The ﬁrst index i on the stress components σij indicates the direction of the force, and the second index j refers to the normal of the plane on which the force acts. Normal stresses correspond to i = j, and shear stresses correspond to i = j. Mechanical equilibrium requires σ = σ . . . . 53 6 ij ji 3.4 Vacancy migration in the bcc lattice. a) The unit cell of the ideal bcc lattice. The dashed line is one of the body-diagonals of the cubic cell. b) and d) Two equivalent equilibrium vacancy sites. Vacancy migration can be conceived as the motion of the atom at the body-center location in b) along the body-diagonal to the corner site in d). The atom must overcome an energy barrier to move between the two equilibrium sites. The maximum in the energy barrier occurs in c), where the migrating atom is located midway between the two equilibrium sites along the body-diagonal. Atomic relaxations are not shown, and the surrounding unit cells are removed for clarity...... 69

3.5 Self-interstitials in the bcc lattice. a) The 100 dumbbell, b) 110 h i h i dumbbell, c) 111 dumbbell, d) crowdion, e) octahedral, and f) tetra- h i hedral interstitials. The interstitial atoms are shown in gray. Atomic relaxations are not shown, and the surrounding unit cells are removed forclarity...... 72

3.6 Supercell geometry for surface calculations. The solid region is constructed by replicating the unit cell of the crystal structure. A vacuum region is included to introduce the presence of a surface. The vectors deﬁning the supercell are Ai. DFT and MD calculations for solids are usually carried out under periodic boundary conditions, so surface properties must be tested for convergence with respect to the size of both the solid region and the vacuum region. This ensures that spurious interactions between periodic surface images are suﬃciently reduced...... 73

3.7 Surface relaxation. The spacing between adjacent atomic layers α and ideal β is equal to the spacing in the ideal crystal dα−β before the atomic positions are relaxed. The spacings near a surface in a metal usually surf contract to smaller separations dα−β under relaxation. The change in interplanar distance is greatest for α = 1 and β =2...... 75

xviii 3.8 Edge dislocation in a simple cubic lattice. (a) A defect-free crystal is divided by the plane p-p′. (b) An edge dislocation is formed by shearing dgho to the left by a lattice translation. The dislocation is the row of atoms in red, perpendicular to the plane of the page (the symbol is ⊥ used to label edge dislocations). The distortion of the lattice is largest close to the dislocation line. This region is called the dislocation core. 77

3.9 Perspective view of (a) an edge dislocation, (b) a screw dislocation, and (c) a mixed dislocation. In all three cases, the dislocation is shown in red and the sheared area is shown in gray. The dislocations separate the sheared regions of the crystal from the unsheared regions. . . . . 78

3.10 Construction of the Burgers vector for an edge dislocation. (a) The direction t of the dislocation is chosen to point out of the page. A closed path SABCF is formed around the dislocation. The path is followed counter-clockwise, as determined by the right-hand rule. (b) A similar path is then constructed in an ideal lattice. Here the path fails to close and the Burgers vector b, drawn from the ﬁnish F to the start S of the path, completes the loop. The Burgers vector of an edge dislocation is perpendicular to the dislocation line...... 79

3.11 Construction of the Burgers vector for a screw dislocation. (a) The direction of the dislocation is t. A closed path SABCDEF is formed around the dislocation. The path is followed in the direction given by the right-hand rule. (b) A similar path is then constructed in an ideal lattice. Here the path fails to close and the Burgers vector b, drawn from the ﬁnish F to the start S of the path, completes the loop. The Burgers vector of a screw dislocation is parallel to the dislocation line. 80

3.12 The motion of an edge dislocation under stress. (a)-(f) The motion of the dislocation requires only a relatively small rearrangement of atoms localized near the dislocation core. When the dislocation passes all the way through, the crystal is sheared by a lattice translation b (i.e., the magnitude of the Burgers vector). The same ﬁnal conﬁguration results when an ideal crystal is sheared by a lattice translation, but this required a much larger stress. Edge dislocations move parallel to thedirectionoftheappliedstress...... 81

xix 3.13 The motion of a screw dislocation under stress. (a)-(c) When the dislocation passes all the way through, the crystal is sheared by a lattice translation b (i.e., the magnitude of the Burgers vector). Screw dislocations move perpendicular to the direction of the applied stress. 82

3.14 Schematic of the energy barrier Eb to dislocation motion. (a) For zero stress, the barrier is periodic and the dislocation must overcome the Peierls barrier Ep. (b) If an applied shear stress is less than the Peierls stress, the barrier is still present but lower in value. (c) When the stress equals the Peierls stress, the barrier vanishes and the dislocation glidesthroughthecrystal...... 83

3.15 Diﬀerential displacement maps of the core structures of 1 111 screw 2 h i dislocations in the bcc lattice. (a) Some potentials produce the degenerate, or polarized, core structure. The structures on the left and right have the same energy. (b) Other potentials and DFT produce the nondegenerate, or symmetric, core. (c) In all cases, the dislocation core spreads into three 110 planes of the [111] zone...... 86 { } 3.16 Initial conﬁguration for a bcc screw dislocation with Burgers vector 1 b = 2 [111]. (a) The view along [111]. The dashed line indicates the shear plane. (b) A perspective view shows the planes of atoms spiraling aroundthedislocationline...... 87

3.17 Fixed boundary conditions for the simulation of a single dislocation. The atoms in Region I are allowed to move, while the atoms in Region II are ﬁxed at the positions dictated by elasticity theory. The region outsideRegionIIisvacuum...... 89

3.18 Schematic of mechanical twinning. (a) An ideal crystal subjected to shear stress. (b) A deformed crystal, in which a portion of the crystal called a twin is rotated with respect to the parent crystal lattice. (c) An atomic view of twinning. The atom positions in the parent lattice are gray, and the atom positions in the twin are black. The twin is a mirror image of the parent across the twinning plane A-B...... 91

3.19 Twinning in the bcc lattice. The twinning plane is (112) and the twinning direction is [1¯11].¯ The twin is formed by shifting successive 1 ¯¯ √ (112) planes by 6 [111], corresponding to a shear of 1/ 2. Black atoms are in the plane of the page, and gray atoms lie above and below the page...... 92

xx 3.20 The bcc lattice generated by stacking 100 lattice planes. (a) The { } ideal unit cell, showing the (110) lattice plane as shaded. The atoms in alternating (110) planes are labeled A and B, respectively. (b) View of the bcc lattice above the (110) plane. The atoms in B layers are shown smaller than atoms in A layers for clarity, and lie at a distance a/√2 above and below the A atoms. The bcc crystal is generated by stacking 110 planes in the sequence ABABAB...... 95 { } 3.21 The bcc lattice generated by stacking 112 lattice planes. (a) The { } ideal unit cell, showing the (112)¯ lattice plane as shaded. The atoms in alternating (112)¯ planes are labeled A, B, C, D, E, and F, respectively. (b) View of the bcc lattice above the (110) plane, showing edges of the (112)¯ planes. The atoms labeled B, D, and F lie at a distance a/√2 above and below the atoms labeled A, C, and E. The bcc crystal is generated by stacking 112 planes in the sequence ABCDEFABCDEF... 96 { } 3.22 Geometry of computational cells used for stacking fault calculations. (a) The ideal crystal, showing the fault plane and slip direction. (b) If the supercell lattice vector x3 initially normal to the fault plane is ﬁxed, the distance between faults is d under periodic boundary conditions. The crystal in the primary simulation cell is shown with solid lines, and the periodic image above it is shown with dashed lines. (c) If the supercell lattice vector x3 is tilted as the crystal halves are shifted, the distance between faults is 2d...... 98

4.1 The three cubic splines of the EAM potential for Nb [1]. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. (a) The pair potential φ and (b) the density ρ are functions of the distance r between pairs of atoms. Both of these functions have 17 optimized spline knots and a cutoﬀ radius of 4.750A,˚ which includes ﬁrst-, second-, and third-nearest neighbor interactions in the ideal bcc lattice. (c) The embedding function F depends on the local density n. Eight optimized spline knots parameterize F ...... 106

xxi 4.2 (a) Nb thermal expansion curve. The thermal expansion of the EAM potential developed in this work [1] agrees closely with experiment [14] exp from 0 K to the experimental melting temperature Tmelt = 2742 K. The EAM potential developed in this work slightly overestimates the thermal expansion, while the EAM potential of Guellil and Adams [15] underestimates it. (b) Nb pressure versus volume curve. The experimental data are from shock experiments [16]. The EAM potential in this work agrees well with experiment and DFT calculations from 0 to 75GPa...... 111

4.3 Nb phonon dispersion curves along high-symmetry directions in the Brillouin zone. The EAM potential developed in this study [1] agrees well with experiment [17] for small wave-vectors, but is unable to fully capture the vibrational spectrum. However, the potential improves upon the EAM potentials of Guellil and Adams [15] and Hu et al. [18]. The DFT phonon results closely match experiment over much of the Brillouin zone. The transverse modes in the [ξ00] direction show a plateau around ξ = 0.25, which is an artifact of the interpolation scheme used by the phon codetogeneratethecurves...... 112

4.4 Two-phase melting simulations determine the melting temperature of the Nb EAM potential developed in this work [1]. Initially, half the simulation cell contains liquid Nb and the other half contains bcc Nb. The liquid region of the simulation cell solidiﬁes below the melting temperature, and the solid region melts above the melting temperature. The ﬁgure shows the equilibrium volume for each simulation temperature. There is a sharp increase in volume upon melting. The potential EAM predicts a melting temperature Tmelt = 2686 K, which is within 2% of exp the experimental value Tmelt =2742K...... 116

4.5 γ-surface sections of Nb in the 111 direction [1]. The absence of h i minima indicate that there are no stable stacking faults in the 112 { } and 110 planes along this direction. The DFT curves are smoother { } than the EAM curves, but the overall agreement is good...... 118

xxii 1 4.6 Diﬀerential-displacement maps for the 2 [111] screw dislocation core structure produced by the Nb EAM potential developed in this study [1]. (a) The degenerate core structure results when the atoms are relaxed. (b) Shear stress applied along [111] produces a net displacement of the dislocation in the (1¯12)¯ plane. The ﬁgure shows the core structure when the stress is just above the critical resolved shear stress for dislocationmotion...... 119

1 4.7 The critical resolved shear stress (CRSS) for 2 [111] screw dislocation motion in Nb as a function of the orientation of the maximum resolved shear stress plane (MRSSP), computed with the EAM potential developed in this study [1]. The ﬁgure shows a deviation from Schmid’s law when the angle between the MRSSP and the (101)¯ plane is greater than about +15◦...... 120

4.8 The five cubic splines of the MEAM potential for Mo [13]. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ has 13 knots and a cutoff radius of 5.900 A,˚ the density ρ and three-body term f have 12 knots and a cutoff radius of 5.100 A,˚ the angular term g has 8 knots, and the embedding function F has3knots...... 122

4.9 (a) Mo pressure versus volume curve. The experimental data are from shock experiments [16]. The MEAM potential in this work agrees well with experiment and DFT calculations from 0 to 35 GPa. (b) Mo thermal expansion curve. The thermal expansion of the MEAM potential agrees closely with experiment [14] from 0 to 2000 K. . . . 125

4.10 Mo phonon dispersion curves along high-symmetry directions in the Brillouin zone. The MEAM potential agrees well with experiment [17] andDFT...... 126

4.11 γ-surface sections of Mo in the 111 direction. The absence of minima h i indicate that there are no stable stacking faults in the 112 and 110 { } { } planes along this direction. The MEAM and DFT results agree well, even though no data from conﬁgurations with stacking faults is used toconstructthepotential...... 128

4.12 The ideal shear strength of Mo. The MEAM results compare well with DFT calculations. (a) The energy barrier for twinning W (x). (b) The shear stress τ(x) is the derivative of W (x)...... 129

xxiii 1 4.13 (a) Diﬀerential-displacement maps for the 2 [111] screw dislocation core structure produced by the Mo MEAM potential. The non-degenerate core structure results when the atoms are relaxed. (b) Critical resolved shear stress for dislocation motion in the (101)¯ plane...... 131

4.14 Defect structures formed in Nb under high-pressure shock loading using five different published potentials. (a) The F.EAM potential developed in this work [1] and the A.MFS potential produce deformation twins under shock loading. Atoms with bcc coordination are colored blue, and atoms within the twin boundaries are colored red. The atoms colored yellow correspond to steps on the twin boundaries associated with b = 1 111 twinning dislocations. (b) The J.EAM, D.EAM, and 6 h i D.EFS potentials produce regions that undergo a bcc to face-centered tetragonal (fct) structural phase transformation under shock loading (see the text for the naming convention used for the potentials). Atoms with fct coordination are colored green. The atoms colored yellow correspond to the boundary region between the bcc and fct structures. Twinning is experimentally observed in Nb samples subjected to shock waves, while the bcc-fct transition in unphysical. (c) and (d) show magnifications of the defect structures in (a) and (b), respectively. The figure is from Reference [19]...... 133

4.15 112 γ-surface section in the 111 direction at P = 0 GPa computed { } h i with five different published potentials and DFT. The Nb EAM potential developed in this work is labeled F.EAM. The figure is from Reference[19]...... 134

4.16 112 γ-surface section in the 111 direction at P = 50 GPa com- { } h i puted with five different published potentials and DFT. The Nb EAM potential developed in this work is labeled F.EAM. The figure is from Reference[19]...... 135

4.17 The ﬁve cubic splines of the high-pressure MEAM potential for Nb. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.915 A,˚ the angular term g has 35 knots, and the embedding function F has7knots...... 138

xxiv 4.18 The Nb pressure versus volume and thermal expansion curves of (a) the high-pressure MEAM potential, and (b) the low-pressure EAM potential. The pressure-volume curve of the high-pressure Nb MEAM potential is in excellent agreement with experiment and DFT to over 400 GPa, while the low-pressure EAM potentials deviates drastically above 75 GPa. The thermal expansion of both the high-pressure MEAM potential and low-pressure EAM potential agree well with experiment. Note that the experimental data is shown as points in the pressure- volume ﬁgures, but as a solid line in the thermal expansion ﬁgures. . 141

4.19 The Nb phonon dispersion computed with (a) the low-pressure EAM potential, and (b) the high-pressure MEAM potential. The MEAM potential shows better overall agreement with DFT and experiment than the EAM potential. The MEAM potential produces zone-boundary frequencies at H and N that closely match the DFT and experimental values, in contrast to the much lower EAM values at these points. . 142

4.20 The pressure variation of the 112 and 110 γ-surface sections in { } { } the 111 direction. (a) The low-pressure Nb EAM potential [1] agrees h i reasonable well with DFT for P = 0 GPa, but is unable to produce the correct behavior at elevated pressures. (b) The high-pressure Nb MEAM potential closely matches the DFT calculations over a broad rangeofpressures...... 143

4.21 The pressure variation of the ideal shear strength. (a) The low-pressure Nb EAM potential does not agree with DFT calculations for any pressure. (b) The high-pressure Nb MEAM potential closely matches the DFTresultsoverabroadrangeofpressures...... 144

4.22 The ﬁve cubic splines of the high-pressure MEAM potential for Mo. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.623 A,˚ the angular term g has 35 knots, and the embedding function F has7knots...... 146

xxv 4.23 The pressure-volume and thermal expansion curves of the low-pressure Mo MEAM potential [13] in (a) are compared to results from the high-pressure Mo MEAM potential in (b). The high-pressure potential shows excellent agreement with experiment and DFT to 800 GPa, while the low-pressure potential agrees well to about 35 GPa (see Fig. 4.9). The thermal expansion of the low-pressure model agrees well to 2000 K. The thermal expansion of the high-pressure model is reasonable, but overestimates the experimental curve. Note that the experimental data is shown as points in the pressure-volume ﬁgures, but as a solid lineinthethermalexpansionﬁgures...... 149

4.24 The phonon dispersion of the low-pressure Mo MEAM potential [13] in (a) is compared to results from the high-pressure Mo MEAM potential in (b). Both of the potentials agree well with DFT and experiment. 150

4.25 The pressure variation of the 112 and 110 γ-surface sections in the { } { } 111 direction from the low-pressure Mo MEAM potential [13] in (a) h i are compared to results from the high-pressure Mo MEAM potential in (b). The high-pressure potential agrees closely with DFT, and the low- pressure potential shows large deviations from the DFT calculations. 151

4.26 The pressure variation of the ideal shear strength of the low-pressure Mo MEAM potential [13] in (a) is compared to results from the high- pressure Mo MEAM potential in (b). The high-pressure potential agrees closely with DFT, and the low-pressure potential shows large deviations from the DFT calculations...... 152

4.27 The ﬁve cubic splines of the high-pressure MEAM potential for W. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.665 A,˚ the angular term g has 35 knots, and the embedding function F has7knots...... 154

4.28 (a) The pressure-volume curve of the high-pressure W MEAM potential. The MEAM potential shows excellent agreement with experiment and DFT to 650 GPa. (b) The thermal expansion of the high-pressure MEAM potential is reasonable, but overestimates the experimental curve. Note that the experimental data is shown as points in the pressure-volume ﬁgure, but as a solid line in the thermal expansion ﬁgure...... 156

xxvi 4.29 The phonon dispersion of the high-pressure W MEAM potential agrees well with DFT and experimental data...... 157

4.30 The pressure variation of the 112 and 110 γ-surface sections in the { } { } 111 direction. The high-pressure W MEAM potential agrees closely h i withDFT...... 159

4.31 The pressure variation of the ideal shear strength. The high-pressure W MEAM potential agrees closely with DFT...... 160

4.32 The ﬁve cubic splines of the high-pressure MEAM potential for V. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.271 A,˚ the angular term g has 35 knots, and the embedding function F has7knots...... 162

4.33 (a) The pressure-volume curve of the high-pressure V MEAM potential. The MEAM potential shows excellent agreement with experiment and DFT to 450 GPa. (b) The thermal expansion of the high-pressure MEAM potential is reasonable, but underestimates the experimental curve. Note that the experimental data is shown as points in the pressure-volume ﬁgure, but as a solid line in the thermal expansion ﬁgure...... 164

4.34 The phonon dispersion of the high-pressure V MEAM potential agrees well with DFT and experiment in the [ξ00] direction, but the potential agrees only near the Brillouin zone boundaries in the [ξξξ] and [ξξ0] directions...... 165

4.35 The pressure variation of the 112 and 110 γ-surface sections in the { } { } 111 direction. The high-pressure V MEAM potential agrees closely h i withDFT...... 167

4.36 The pressure variation of the ideal shear strength. The high-pressure V MEAM potential agrees closely with DFT...... 168

xxvii 4.37 The ﬁve cubic splines of the high-pressure MEAM potential for Ta. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.941 A,˚ the angular term g has 35 knots, and the embedding function F has7knots...... 169

4.38 (a) The pressure-volume curve of the high-pressure Ta MEAM potential. The MEAM potential shows excellent agreement with experiment and DFT to 500 GPa. (b) The thermal expansion of the high-pressure MEAM potential agrees well with experiment to 2,500 K, but underestimates the experimental curve at higher temperature. Note that the experimental data is shown as points in the pressure-volume ﬁgure, but as a solid line in the thermal expansion ﬁgure...... 172

4.39 The phonon dispersion of the high-pressure Ta MEAM potential agrees reasonably well with DFT and experiment, except near the [ξξ0] zone boundaryN...... 173

4.40 The pressure variation of the 112 and 110 γ-surface sections in the { } { } 111 direction. The high-pressure Ta MEAM potential agrees closely h i withDFT...... 174

4.41 The pressure variation of the ideal shear strength. The high-pressure Ta MEAM potential agrees closely with DFT...... 175

A.1 Total energy Etotal versus plane-wave cutoﬀ Ecut for V. The calculation is converged when Ecut =550eV...... 182

A.2 Total energy Etotal versus plane-wave cutoﬀ Ecut for Nb. The calculation is converged when Ecut =550eV...... 182

A.3 Total energy Etotal versus plane-wave cutoﬀ Ecut for Ta. The calculation is converged when Ecut =600eV...... 183

A.4 Total energy Etotal versus plane-wave cutoﬀ Ecut for Mo. The calculation is converged when Ecut =600eV...... 183

A.5 Total energy Etotal versus plane-wave cutoﬀ Ecut for W. The calculation is converged when Ecut =600eV...... 184

xxviii A.6 Total energy Etotal and electronic density of states versus Nk for V. The calculation is converged when N = 35, i.e. a 35 35 35 k-point k × × grid...... 185

A.7 Total energy Etotal and electronic density of states versus Nk for Nb. The calculation is converged when N = 31, i.e. a 31 31 31 k-point k × × grid...... 186

A.8 Total energy Etotal and electronic density of states versus Nk for Ta. The calculation is converged when N = 31, i.e. a 31 31 31 k-point k × × grid...... 187

A.9 Total energy Etotal and electronic density of states versus Nk for Mo. The calculation is converged when N = 31, i.e. a 31 31 31 k-point k × × grid...... 188

A.10 Total energy Etotal and electronic density of states versus Nk for W. The calculation is converged when N = 35, i.e. a 35 35 35 k-point k × × grid...... 189

A.11 Total energy Etotal and electronic density of states versus σ for V. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.10eV...... 191

A.12 Total energy Etotal and electronic density of states versus σ for Nb. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.10eV...... 192

A.13 Total energy Etotal and electronic density of states versus σ for Ta. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.20eV...... 193

A.14 Total energy Etotal and electronic density of states versus σ for Mo. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.35eV...... 194

A.15 Total energy Etotal and electronic density of states versus σ for W. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.35eV...... 195

xxix Chapter 1: Introduction

This work develops classical interatomic potentials for the body-centered cubic

(bcc) refractory metals V, Nb, Ta, Mo, and W. Refractory metals are characterized by high strength and melting temperatures above 2,000 K, making them useful for a variety of technological applications. Interatomic potentials model the interactions between atoms in classical molecular dynamics (MD) and Monte Carlo (MC) simulations. The parameters in the potentials in this study are ﬁt to quantum mechanical

ﬁrst-principles calculations based on density-functional theory (DFT). The potentials accurately model the mechanical and thermal properties of the bcc refractory metals over a wide range of temperatures and pressures, providing a sound basis for reliable large-scale MD and MC simulations. The general topic of interatomic potentials is introduced ﬁrst, with emphasis on potentials for simulating metallic systems. Then the bcc refractory metals and alloys are discussed. Finally, the procedure for constructing the potentials in this study is outlined.

1.1 Interatomic Potentials for Metals

The power of computational materials simulations has grown to allow quantum

mechanical DFT calculations of simple material properties on desktop workstations.

Parallel supercomputers greatly extend the capabilities of these methods to complex

1 simulations of thousands of atoms, and accurate predictions of mechanical proper-

ties, defects, vibrational spectra, transition barriers, magnetic properties, and phase

diagrams are possible. Despite the power and availability of advanced computing

resources, many problems still lie beyond the reach of DFT methods due to length-

or time-scale limitations. Reliable simulations of phenomena including solid-state dif-

fusion, complex deformation processes, and phase transformation dynamics require

large numbers atoms and long simulation times inaccessible to quantum-based tech-

niques.

Classical MD simulations based on realistic interatomic potentials oﬀer the compu-

tational eﬃciency needed for large-scale simulation studies. Embedded-atom method

(EAM) [20] and modiﬁed-EAM (MEAM) [21] potentials revolutionized large-scale

MD simulations of metallic systems by removing fundamental limitations of simple

pair-wise interaction models. Pair potentials fail to capture much of the physics of

metals due to their lack of many-body interactions. EAM potentials provide a better

description of metals by implicitly modeling many-body energy contributions. MEAM

potentials extend EAM potentials by including explicit three-body terms, providing

greater ﬂexibility for describing bond-bending interactions.

Most published potentials are based on simple analytic functions with parameters fit to a small number of experimentally measured properties. These potentials generally reproduce the properties they are fit to, but often have limited utility for simulating properties outside their fitting range. Ercolessi and Adams [22] proposed a method to incorporate information about forces and energies from a wide variety of local atomic environments by fitting the potential parameters to data from

2 DFT calculations. Potentials constructed using this method have a greater range of applicability than analytic potentials.

This study uses the method of Ercolessi and Adams to generate EAM and MEAM potentials for the bcc metals V, Nb, Ta, Mo, and W. The optimal parameter values accurately mimic forces, energies, and stresses from DFT calculations for a variety of atomic environments. Experimental data is not used to ﬁt the potentials in this work, since diﬀerences between experiment and DFT lead to inconsistencies in the

ﬁtting data. Comparison of physical properties computed using the potentials to DFT calculations and experimental data indicates that the potentials provide accurate models of thermal and mechanical behavior over a wide range of temperatures and pressures.

1.2 Body-Centered Cubic Refractory Metals

The bcc transition metals are key components of many high performance structural materials, and the deformation mechanisms in bcc metals and alloys have been a focus of intense research for over 50 years. References [23–29] review experimental and modeling studies of deformation and plasticity in bcc materials. Line defects in the crystalline lattice called dislocations are central to understanding plastic deformation in bcc metals, and the large lattice resistance to the motion of screw dislocations explains the high strength of the bcc metals. Less well understood is the role of ﬁne details of the dislocation core structure in determining dislocation motion at ﬁnite temperature, and deformation mechanisms under non-equilibrium and high strain- rate loading conditions. Interatomic potentials are an essential tool for studying the deformation of bcc metals over a wide range of temperature and strain conditions

3 since they provide the necessary balance of accuracy, computational eﬃciency, and atomic-level resolution for detailed studies of defect behavior.

The deformation mechanisms in elemental bcc metals are fundamentally interest- ing, and important for understanding defects in industrial alloys with bcc constituents.

The high strength and high melting temperatures of the bcc refractory metals V, Nb,

Ta, Mo, and W make them attractive candidate materials for advanced alloys. A brief list of alloys that contain the bcc refractory metals indicates their technological importance for a wide range of applications: many high-strength steels [30] and Ni– based superalloys contain Mo [31]; Ti–based Gum Metal alloys with unique elastic and plastic deformation properties contain V, Nb, or Ta [32]; non-toxic Ti–Nb-based shape memory alloys are candidate materials for biomedical implants [33]; Nb–, Mo–, and W–based alloys are candidate structural materials in fusion reactors [34]. Atom- istic modeling is a key component in modern materials development, and accurate interatomic potentials are needed for a fundamental understanding of deformation and transformation processes in alloys. The potentials developed in this work for elemental bcc metals provide a solid foundation for describing the interactions between like-species in alloy systems with bcc components.

1.3 Scope

The following chapters present information necessary to understand, generate, and test interatomic potentials for metallic systems. Chapter 2 begins with a discussion of the approximations to quantum mechanics leading to simpliﬁed potential energy functions that depend only on the coordinates of the atomic nuclei. These interatomic

4 potentials are the model of atomic interactions necessary for classical molecular dynamics and Monte Carlo simulations. The hierarchy of interatomic potentials for metallic systems is detailed, with thorough descriptions of two-body pair potentials and many-body EAM and MEAM potentials.

Chapter 3 describes the process of generating accurate first-principles DFT data to which the parameters in the potential functions are fit. This involves choosing the most appropriate exchange-correlation functional and pseudopotential for each metal, along with the criteria governing the numerical convergence of the DFT calculations. This chapter also explains the calculations used to evaluate the quality of the interatomic potentials. The calculations test a wide range of physical properties, some included in the fitting database and others not in the database.

Chapter 4 discusses the V, Nb, Ta, Mo, and W potentials developed in this work, including their range of applicability. EAM and MEAM potentials generated from low-pressure DFT data are good models of many low-pressure properties of the bcc phase, but fail to provide quantitative agreement with DFT at high pressure.

Databases containing low- and high-pressure DFT calculations produce potentials that agree well with DFT and experimental results over a broad range of pressures and temperatures. These high-pressure potentials accurately model a wide variety of properties at ambient conditions, and produce the correct pressure dependence of properties related to mechanical deformation.

Chapter 5 summarizes the results of the thesis, and discusses limitations and possible extensions of the potentials developed in this work. The potentials are accurate models for a wide range of applications, and serve as a viable basis for developing potentials for alloys containing the bcc refractory metals.

5 Chapter 2: Classical Interatomic Potentials

The fields of condensed matter physics, materials science, and chemistry have greatly benefited from quantum mechanical-based calculations. Density-functional theory, configuration interaction methods, and coupled cluster techniques are accurate first-principles methods capable of providing solid theoretical understanding of complex systems ranging from atoms and molecules to technological materials. De- spite the successes of these methods, there are many problems beyond their scope due to size or time-scale limitations. The computational time of ab initio calculations

scales as the number of particles cubed or worse, rendering the methods computa-

tionally demanding (conventional density-functional theory codes can simulate only a

few thousand atoms on massively parallel supercomputers). Simulations of phenom-

ena involving long-range strain ﬁelds or long-wavelength ﬂuctuations, including phase

transformations, plasticity, and deformation under shock loading, require computa-

tional methods capable of reaching length scales beyond the current capabilities of

ab initio calculations. Equally challenging to ﬁrst-principles methods are phenomena

occurring over long times, including annealing of damaged regions in a crystal lattice,

grain growth, and diﬀusion.

Molecular dynamics (MD) is a computational method for studying the dynamics

of systems of interacting particles [35–42]. The particles exert forces on each other,

6 and the system evolves in time according to classical mechanics. Numerically, the simulation time is discretized into small intervals and the particle positions, velocities, and forces are updated sequentially at each time-step. Time-steps on the order of a femtosecond are typically required to resolve atomic vibrations in condensed phases.

The computational time of MD simulations utilizing short-ranged classical potential energy functions, i.e., classical potentials, scales linearly with the number of particles in the system, allowing simulations of millions to billions of particles and timescales beyond nanoseconds. Many problems of practical interest can be addressed with

MD, extending far beyond the reach of ﬁrst-principles methods. Of course there are limitations to classical potential models, perhaps the most severe being the accuracy of the interparticle forces. Since the physics of a given material is encoded in the potential model, the results of an MD simulation are only as good as the underlying potential.

The idea of an eﬀective interatomic potential to describe the interaction between atoms is an old idea, and a large amount of work has been done on potential development. The aim of this chapter is to provide a brief introduction to the potentials most relevant for simulations of metallic systems, and little will be said about potentials for semiconductors, molecules, polymers, biological systems, or a multitude of other potentials for specialized applications. The literature on interatomic potentials for metals alone is vast, and an exhaustive review is beyond the scope of this thesis.

Therefore, only a brief summary of simple potentials will be given before the discussion of the eﬀective many-body potentials relevant to the research summarized in this thesis. More information about interatomic potentials is found in Refs. [43–50].

7 2.1 Justiﬁcation for Classical Potentials

This section outlines the theoretical basis for classical interatomic potentials. The

first subsection discusses the Born-Oppenheimer approximation, which decouples the full Schrödinger equation into separate equations for the electrons and nuclei. It will be seen that the nuclear equation includes a nuclear kinetic energy term and an effective potential energy that includes both nucleus-nucleus interactions as well as kinetic and potential energy contributions from the electrons. The existence of this effective potential is the basis for approximate classical interatomic potentials.

In the second subsection it is shown that under most circumstances of interest, the nuclei behave as classical particles. The Schr¨odinger equation for the nuclei can then be replaced by classical equations of motion, with forces determined by an eﬀective classical potential depending only on nuclear coordinates.

2.1.1 Born-Oppenheimer Approximation

The systems of interest in condensed matter physics, materials science, and chemistry are described fundamentally as a system of interacting nuclei and electrons.

Such systems are governed by the quantum mechanical Schr¨odinger equation:

∂ Hˆ Ψ( r , r ,t)= i~ Ψ( r , r ,t) , (2.1) { i} { I } ∂t { i} { I } where Hˆ is the Hamiltonian operator and Ψ ( r , r ,t) is the many-body wave { i} { I } function. The set of electronic coordinates is denoted r and the set of nuclear { i} coordinates is r . The Hamiltonian consists of a kinetic energy operator for the { I } electrons Tˆe, a kinetic energy operator for the nuclei TˆN , a potential energy operator for electron-electron interactions Vˆee, a potential energy operator for electron-nucleus

8 interactions VˆeN , and a potential energy operator for nucleus-nucleus interactions

VˆNN :

Hˆ = Tê + TˆN + Vêe + VêN + VˆNN . (2.2)

The kinetic energy operators in the coordinate representation are

Nel ~2 2 Tˆ = r , (2.3) e − 2m∇ i i=1 X

Nnuc ~2 ˆ 2 TN = rI , (2.4) − 2MI ∇ I=1 X where ~ is Planck’s constant divided by 2π, m is the mass of an electron, MI is the mass of nucleus I, ri is the position variable of electron i, rI is the position variable of nucleus I, Nel is the number of electrons, and Nnuc is the number of nuclei. The

interactions between the particles are Coulomb interactions, and the potential energy

operators in the coordinate representation are

N el e2 Vˆ = , (2.5) ee 4πǫ r r j>i=1 0 i j X | − |

Nel Nnuc 2 e ZI VˆeN = , (2.6) − 4πǫ0 ri rI i=1 I=i X X | − |

Nnuc 2 e ZI ZJ VˆNN = , (2.7) 4πǫ0 rI rJ J>I=1 X | − | where e is the magnitude of the electron’s charge, ǫ0 is the permittivity of free space, and ZI is the atomic number of nucleus I.

9 The analytic solution of Schr¨odinger’s equation for a complex, multi-atom system

is not feasible, and exact numerical solutions exist only for some atoms and small

molecules. Therefore, a great deal of eﬀort has been applied to developing approx-

imate methods of solution. An important simpliﬁcation of the Hamiltonian stems

from the large diﬀerence in mass between the electrons and nuclei. Even in the most

unfavorable case of a nucleus consisting of a single proton, the ratio of electron mass

to nuclear mass is 1:1836. This large diﬀerence means that the electronic motion is

much faster than the motion of the nuclei, and the electrons can adjust very rapidly

to changes in the nuclear coordinates. Therefore, it is reasonable to suppose that the

electronic motion can be uncoupled from the nuclear motion and as the nuclei move,

the electronic wave function adjusts instantaneously such that the electrons always

remain in the same state.

The formal method for implementing this simpliﬁcation is the Born-Oppenheimer

approximation [51–55]. Since the electronic motion is rapid, the ﬁrst step is to regard

the nuclei as ﬁxed and solve the electronic problem for the given nuclear positions.

This is the clamped nuclei approximation. The kinetic energy of the nuclei is then

zero, and the Hamiltonian Hˆe governing the electrons is

Hê = Tê + Vêe + VêN + VˆNN , (2.8)

where the terms have the same meaning as before. The electronic wave function φn

satisﬁes a time-independent Schr¨odinger equation for this Hamiltonian:

Hˆ φ ( r , r )= E ( r )φ ( r , r ), (2.9) e n { i} { I } e,n { I } n { i} { I }

10 where the electronic eigenfunctions φ ( r , r ) and eigenvalues E ( r ) contain n { i} { I } e,n { I } the r as parameters. This dependence on r is expected since the electronic { I } { I } wave functions and eigenvalues will vary as the nuclear coordinates change.

The full wave function of the system Ψ( r , r ,t) is then expanded in the set { i} { I } of electronic wave functions:

Ψ( r , r ,t)= θ ( r ,t)φ ( r , r ), (2.10) { i} { I } n { I } n { i} { I } n X where θ ( r ,t) is the wave function of the nuclei for electronic state n. If we assume n { I } that the motion of the nuclei does not excite the electrons, the expansion in Eqn. 2.10 reduces to

Ψ( r , r ,t)= θ ( r ,t)φ ( r , r ). (2.11) { i} { I } n { I } n { i} { I } For the purpose of constructing classical potential models, we assume that this assumption is valid and adopt this simple product form for Ψ( r , r ,t). Further- { i} { I } more, we assume that the electrons are always in their ground state n = 0 since the potentials developed in this study are constructed from data obtained using conventional ground-state density-functional theory calculations. Thus, the starting point is

Ψ( r , r ,t)= θ ( r ,t)φ ( r , r ). (2.12) { i} { I } 0 { I } 0 { i} { I } Inserting this wave function into the full Schr¨odinger equation yields

11 ∂ Nnuc ~2 r r ~ 2 r r φ0( i , I ) i rI + Ee,0( I ) θ0( I ,t)= { } { } − ∂t − 2MI ∇ { } { } I=1 ! X Nnuc ~2 2 r r r r r r r r 2 I θ0( I ,t) I φ0( i , I )+ θ0( I ,t) rI φ0( I , i ) . 2MI ∇ { } · ∇ { } { } { } ∇ { } { } I=1 X (2.13) Next, consider the contributions to the energy of the system by the terms on the

right-hand-side. The ﬁrst term is proportional to

∗ 1 ∗ dr φ ( r , r ) r φ ( r , r )= r dr φ ( r , r )φ ( r , r ), { i} 0 { i} { I } ∇ I 0 { i} { I } 2∇ I { i} 0 { i} { I } 0 { i} { I } Z Z (2.14)

which vanishes due to the normalization of the electronic wave function. This follows

since the electronic eigenfunctions can be chosen as real [52]. The contribution to

the energy from the second term on the right-hand-side is small and can often be

neglected. This is demonstrated by considering the worst case of electrons tightly

bound to the nucleus, as these are expected to be most aﬀected by the nuclear mo-

tion [54]. In this case, φ ( r , r )= φ ( r r ) and the energy contributions are 0 { i} { I } 0 { i − I } proportional to

~2 r ∗ r r 2 r r d i φ0( i , I ) rI φ0( i , I ) − { } { } { } 2MI ∇ { } { } Z ~2 ∗ 2 = dri φ0( ri , rI ) ri φ0( ri , rI ) − { } { } { } 2MI ∇ { } { } Z ~2 m ∗ 2 = dr φ ( r , r ) r φ ( r , r ), (2.15) −M { i} 0 { i} { I } 2m∇ i 0 { i} { I } I Z which is m/MI times the kinetic energy of the electrons. When this term is retained the approximation is called adiabatic. When it is not included the approximation is

12 called the Born-Oppenheimer approximation. This mass ratio is of the order 10−6 to

10−5 for the bcc refractory metals, so it will be neglected.

The ﬁnal result is

Nnuc ~2 ∂ 2 r r ~ r rI + Ee,0( I ) θ0( I ,t)= i θ0( I ,t). (2.16) − 2MI ∇ { } { } ∂t { } I=1 ! X This is a Schrödinger equation for the nuclear wave function, in which the nuclei are subjected to a potential energy E ( r ). This is the Born-Oppenheimer potential e,0 { I } energy surface containing contributions from the direct nucleus-nucleus interactions, as well as kinetic and potential energy contributions from the electrons. Therefore, within the Born-Oppenheimer approximation each nucleus moves under the influence of an effective potential energy that depends only on the coordinates of the nuclei.

Note that the Born-Oppenheimer approximation results in a separation of the coupled electron-nuclei system into an independent electronic problem and a nuclear problem in which the electronic eﬀects are subsumed into an eﬀective potential.

Before moving on to the next step in defining classical potentials, more should be said about the validity of the Born-Oppenheimer approximation. A full analysis and discussion of the applicability of the Born-Oppenheimer approximation is beyond the scope of this thesis, but it should be kept in mind that there are situations in which the assumptions leading to Eqn. 2.16 break down. When transitions between electronic states become important, so-called “non-Born-Oppenheimer” terms must be included in the nuclear Hamiltonian. References [56, 57] discuss such corrections for atoms and molecules. At first glance it appears that the Born-Oppenheimer approximation in not applicable to metals, since their is no energy band gap and an infinitesimal energy is needed to excite electrons. However, excitations are usually

13 conﬁned to a narrow region about the Fermi surface at temperatures of interest,

and the neglect of non-Born-Oppenheimer terms has little eﬀect on most properties

accessible to classical simulations [55]. Therefore, we proceed under the assumption

that the Born-Oppenheimer approximation is valid.

2.1.2 Classical Nuclei Approximation

Within the Born-Oppenheimer approximation, the nuclear wave function θ ( r ,t) 0 { I } obeys a Schr¨odinger equation containing an eﬀective potential that depends only on the nuclear coordinates, Eqn. 2.16. However, under appropriate conditions the nuclei are massive enough to be treated as classical particles. A well known condition for treating an ensemble of particles classically is when the mean interparticle spacing is much larger than the thermal de Broglie wavelength Λ [58,59]:

2π~2 Λ= , (2.17) sMI kBT where kB is Boltzmann’s constant and T is temperature. Note that the larger the mass and the higher the temperature, the smaller the wavelength and the more classical the behavior. As an example consider the lightest bcc refractory metal V. The mass of V is 50.942 u and the equilibrium nearest neighbor distance in the bcc solid is d = 2.62 A.˚ Λ becomes comparable to d around T = 1 K, demonstrating that the

nuclei of bcc refractory metals can safely be treated as classical particles unless the

temperature is very low.

When the classical approximation is valid, the Schr¨odinger equation for the nuclei

Eqn. 2.16 can be replaced by classical equations of motion, e.g., Newton’s second law:

14 M ¨r = r E ( r ), (2.18) I I −∇ I e,0 { I } where the gradient of the eﬀective potential E ( r ) gives the forces on the nu- e,0 { I } clei. Ref. [41] provides more details on justifying the replacement of the Schr¨odinger equation with classical equations.

There are two main routes that can now be chosen for simulating the classical dynamics of the interacting nuclei. The first is ab initio molecular dynamics (MD) [41], in which the effective potential energy is determined by direct electronic structure calculations each time the nuclei move. This is an accurate method for studying the dynamics of interacting many-body systems, but it is very computationally demanding. The second route is classical MD, in which the potential energy E ( r ) e,0 { I } is approximated by a fixed functional form V ( r ) called a classical interatomic { I } potential:

E ( r ) V ( r ). (2.19) e,0 { I } ≈ { I } Classical MD is not as accurate as ab initio MD, but it is much less computationally demanding and allows access to simulations of phenomena occurring over large length-scales and long time-scales. The purpose of this study is developing accurate and reliable classical interatomic potentials V ( r ) for simulating the bcc refractory { I } metals V, Nb, Ta, Mo, and W.

15 2.2 Classiﬁcations of Interatomic Potentials

The total potential energy Vtot of a given system can be expanded in classical potentials involving pairs of atoms V2(rI , rJ ), triplets of atoms V3(rI , rJ , rK ), etc. [35,

46]:

Vtot = V2(rI , rJ )+ V3(rI , rJ , rK )+ ... (2.20) I

computationally eﬃcient potentials for condensed systems. Very high-order terms

would have to be included in the series to accurately describe the energy, making the

computations too demanding [46]. Instead, low-order truncations of the series with

the direct N-body terms replaced by eﬀective N-body interactions that include aver-

aged higher-order contributions can provide both the accuracy and speed necessary

for meaningful large-scale simulations. Interatomic potentials for metals and semi-

conductors can be classiﬁed into four broad categories based on the number of terms

retained in the expansion, and how higher-order terms are approximated [46]: (1) pair

potentials; (2) cluster potentials; (3) pair functionals; and (4) cluster functionals.

(1) Pair potentials model the potential energy of the system as a sum of eﬀective

two-body interactions:

eﬀ Vtot = V2 (rI , rJ ) (2.21) I

ent than the pair potential for a condensed phase of these atoms. Pair potentials

16 can successfully model some types of systems, but have limited applicability for met-

als. Nevertheless, pair interactions comprise the most familiar interatomic potentials

and they set the stage for the discussion of the more realistic many-body potentials

developed in this study. Section 2.2.1 contains more information on pair potentials.

(2) Cluster potentials contain eﬀective interactions beyond two-body terms, and can accordingly account for more physics than pair potentials:

eﬀ eﬀ Vtot = V2 (rI , rJ )+ V3 (rI , rJ , rK )+ ... (2.22) I

tional burden associated with calculating four-body and higher-order terms. Three-

body contributions depend on the bonding angles, leading to a better description of

systems with covalent character.

eﬀ (3) Pair functionals supplement the two-body pair interaction V2 (rI , rJ ) with a

term that depends on the sum of two-body functions h2(rI , rJ ):

eﬀ Vtot = V2 (rI , rJ )+ F h2(rI , rJ ) , (2.23) I

boring atoms J. Note that if F is a linear function of its argument, Eqn. 2.23

reduces to a pair potential. The non-linearity of F therefore implicitly accounts for

some many-body eﬀects through a sum of pair-wise terms. Pair functionals greatly

improve over pair potentials in the description of metallic systems due the approxi-

mation of these important many-body interactions. The embedded-atom method po-

tentials [20,60] discussed in Section 2.3.1 are pair functionals.

17 Figure 2.1: Schematic representation of the four categories of classical potentials. The arrows point in the direction of increasing accuracy and complexity.

(4) Cluster functionals attempt to combine the virtues of pair functionals and cluster potentials by supplementing the argument of F in Eqn. 2.23 with higher order

terms h3(rI , rJ , rK ), ... :

eﬀ Vtot = V2 (rI , rJ )+ F h2(rI , rJ ), h3(rI , rJ , rK ), ... . (2.24) I

sion of terms beyond h2 better represents many-body eﬀects and bond directionality.

The modiﬁed embedded-atom method potentials [21,61] presented in Section 2.3.2 are

cluster functionals. Figure 2.1, adapted from Ref. [46], schematically illustrates the

relative accuracy and complexity of the four classes of potentials.

18 2.2.1 Pair Potentials

Pair potentials are the simplest form of interatomic potential. The pair interaction between two atoms I and J is

V eff (r , r )= V eff ( r r ) V eff (r ), (2.25) 2 I J 2 | I − J | ≡ 2 IJ where rIJ is the scalar distance between the atoms. The total potential energy of a system described by pair potentials is Eqn. 2.21. One of the earliest and most commonly used pair interactions is the Lennard-Jones potential [62], which consists of a short-range repulsive term and a long-range attractive interaction:

σ 12 σ 6 V eﬀ (r) = 4ǫ , (2.26) 2 r − r where ǫ and σ can be regarded as empirical parameters determined by ﬁtting to

experimental or ab initio data. The Lennard-Jones potential gives a reasonably accu-

rate description of the closed-shell noble gas elements, in which the attractive forces

arises from ﬂuctuating dipole-dipole interactions. The 1/r6 dependence of the attrac-

tive term can be derived using second-order perturbation theory, while the functional

form of the repulsive term is harder to derive from ﬁrst-principles. Therefore, the

1/r12 form is chosen for simplicity. Figure 2.2 plots the Lennard-Jones potential

versus interatomic distance.

Many other pair potentials exist, represented either as analytic functions like the

Lennard-Jones example, as numerical tabulations, or as interpolating functions such

as cubic splines. Regardless of form, pair potentials cannot account for much of the

physics of metallic and semiconducting systems due to the absence of many-body

19 Figure 2.2: The Lennard-Jones potential consists of a long-range attraction and a short-range repulsion. The shape of the curve is typical of many pair potentials.

eﬀects that cannot be modeled using two-body terms. Some of the deﬁciencies of pair potentials include [46, 63]: (1) pair potentials predict a linear dependence of bond energy on coordination number, while for metals the relationship is non-linear; (2)

f the ratio of the vacancy formation energy Evac to the cohesive energy Ecoh is about one for pair potentials, while for metals it is about 0.2 to 0.3; (3) surfaces usually relax outward in pair models, in contrast to metals whose surfaces usually relax inward; (4) cubic metals can show drastic deviations from the Cauchy elastic constant relation

C12/C44 = 1, which rigorously holds for pair interactions.

2.2.2 Higher-Order Potentials

The clear deﬁciencies of pair potentials motivated the development of more so- phisticated potential models that approximate the many-body contributions to the

20 energy. Early attempts included supplementing the pair interactions with a volume- dependent term [64–66]. This remedies the Cauchy relation problem, but cannot properly describe defects since the volume is ill-defined near regions where the crystal lattice is distorted. The first major breakthroughs came in the 1980’s with the proposal of embedded-atom method [20], Finnis-Sinclair [67], effective-medium theory [68], and glue model [63] potentials. These models describe the local environment of an atom using an intermediate quantity that is well defined at all points in space, thereby removing the difficulties associated with volume-dependent potentials. The arguments used to derive these potentials differ, but the final functional forms are similar. The embedded-atom method is discussed in the next section, since it is based on density-functional theory which determines the fitting data used to construct the potentials in this study.

2.3 Many-Body Potentials for Real Metals

Embedded-atom method (EAM) potentials extend the two-body interactions contained in pair potentials by including a term that implicitly models many-body interactions, thus providing a much improved description of metallic systems. EAM potentials are classiﬁed as pair functionals, which were presented in Section 2.2. Mod- iﬁed embedded-atom method (MEAM) potentials supplement the implicit many-body contributions contained in EAM with explicit three-body terms. This provides a more

ﬂexible and accurate representation of crystals with directional bonding. MEAM potentials are classiﬁed as cluster functionals, which were also presented in Section 2.2.

The discussion focuses on the theoretical underpinnings of the models, with the practical issues of determining functional forms deferred to Chapter 3.

21 2.3.1 Embedded-Atom Method Potentials

Density-functional theory (DFT) [69,70] asserts that the ground-state energy of a

many-body system is a functional of the electronic density nel. Extensive discussions

of DFT and its applications are found in Refs. [41, 55, 71–81]. A corollary to DFT

states that the energy Eimp required to embed an impurity atom Z in a system at

position r is a functional r of the electronic density n of the host system I FZ, I host before the impurity is introduced [82,83]:

E = r [n ]. (2.27) imp FZ, I host The embedded-atom method (EAM) [20, 50, 60, 84–86] views each atom in a system as an “impurity” in a host made up of all the other atoms. The embedding energy functional r is approximated as a sum of pair interactions φ , plus an FZ, I SI SJ embedding function FSI that depends on the local density nI at the position of the

“impurity” atom:

r [n ] φ (r )+ F (n ), (2.28) FZ, I host ≈ SI SJ IJ SI I XJ6=I where rIJ is the scalar distance between atoms I and J. Non-linearity of the em-

bedding function is required to account for many-body interactions, as discussed in

Section 2.2. The local density nI is the sum of spherically averaged densities ρSJ contributed by the surrounding atoms J:

nI = ρSJ (rIJ ). (2.29) XJ6=I

22 The SI subscripts on the functions φSI SJ , FSI , and ρSI indicate that they depend

on the species of the atoms. In the original formulation of EAM the functions ρSI represented the actual electronic density of atom I, but in modern potential develop-

ment ρSI need not be strictly interpreted as an electronic density. Flexible functional

representations can allow ρSI to become negative, in contrast to the electronic density which is positive deﬁnite.

EAM The total potential energy Vtot is obtained by summing the energy contributions from all the atoms in the system, taking care not to double-count pair interactions:

EAM Vtot = φSI SJ (rIJ )+ FSI ρSJ (rIJ ) . (2.30) J>I I ! X X XJ6=I

An N-component system requires N(N + 1)/2 pair potentials φSI SJ , N density func-

tions ρSI , and N embedding functions FSI . The special case of a binary system A-B requires seven functions φAA, φAB, φBB, ρA, ρB, FA, and FB, while a monoatomic system requires only three functions φ, ρ, and F . Figure 2.3 illustrates the EAM view of a collection of atoms, where each atom’s energy results from embedding it within the remaining atoms.

It is tempting to interpret the pair term in the EAM energy expression Eqn. 2.30 as an electrostatic ion-ion interaction and the embedding term as an energy contribution due to electronic interactions, but this simple division is not a unique description. The

EAM energy is invariant under the transformations

φ (r ) φ (r ) α ρ (r ) α ρ (r ), SI SJ IJ → SI SJ IJ − SI SJ IJ − SJ SI IJ F (n ) F (n )+ α n , (2.31) SI I → SI I SI I

23 EAM Figure 2.3: The embedded-atom method views the potential energy of a system Vtot as a sum of energy contributions VI associated with embedding atom I in a host system made of the other atoms. V is the approximation to r given in Eqn. 2.28 in the I FZ, I text.

where the αSI are arbitrary constants. This indicates that contributions to the pair

energy can be transferred to the embedding energy without aﬀecting the total energy

of the system. The pair and embedding terms must therefore be taken together for

the potential to have physical meaning. EAM potentials are also invariant under the

transformations

ρ (r ) βρ (r ), SI IJ → SI IJ F (n ) F (n /β), (2.32) SI I → SI I

where β is an arbitrary constant. This means that the density units are arbitrary.

These transformations lead to a large number of potentials with diﬀerent functional

forms but identical physics. This degeneracy can be lifted by imposing constraints

on the densities and embedding functions [87]. A common choice is to scale the

densities functions ρSI such that the local density nI is conﬁned to a given range

24 such as ( 1, 1), and to set the derivatives of the embedding functions F to zero at − SI the average density for each type of atom. EAM potentials have been applied to a

large number of metallic systems over the last three decades, with examples found in

Refs. [1,15,84,88–90].

2.3.2 Modiﬁed Embedded-Atom Method Potentials

Modiﬁed embedded-atom method (MEAM) potentials [21, 91] extend the EAM

formalism by including three-body bond bending terms in the local density nI . The explicit inclusion of three-body interactions provides a more general description of the density, resulting in a better physical representation of bonds with directional character. The angular dependence of the density in the original formulation of

MEAM contains a series of four terms with s-, p-, d-, and f-orbital character. These

ﬁxed functional forms were replaced by a more general angular dependence in a later formulation of MEAM [61].

MEAM The total potential energy Vtot for a system described by this later version of

MEAM is

MEAM Vtot = φ(rIJ )+ F (nI ), (2.33) J>I I X X where the density nI is now modiﬁed to

nI = ρ(rIJ )+ f(rIJ )f(rIK )g(cos θJIK ). (2.34) XJ6=I K>JX6=I The functions f(rIJ ) and g(cos θJIK ) model the three-body interactions, where θJIK is the bond angle centered on atom I within the triplet of atoms I, J, and K. φ, F , ρ, f, and g are represented as cubic splines in this formulation, which provides greater

25 ﬂexibility than the ﬁxed analytic functional forms used in the original development.

This spline-based form of MEAM has not yet been applied to alloys, and there are

several possible ways of generalizing the functions for multicomponent systems. For

example, the function f(rIJ ) can be modiﬁed to fSJ (rIJ ) or fSI SJ (rIJ ), and g(cos θJIK )

can be modiﬁed to gSI (cos θJIK ), gSJ SK (cos θJIK ), or gSJ SI SK (cos θJIK ). The best way

to index the functions is still unknown.

The MEAM potential reduces to an EAM potential if f = 0 or g = 0, and

it reduces to the Stillinger-Weber model for covalent materials [92] if ρ = 0 and

F (n ) n . The MEAM energy is invariant under the following transformations: I ∝ I

ρ(r ) αρ(r ), IJ → IJ f(r ) f(r )/β, IJ → IJ g(cos θ ) αβ2g(cos θ ), JIK → JIK F (n ) F (n /α), (2.35) I → I where α and β are arbitrary constants.

Applications of the original form of MEAM include semiconductors [21,91,93,94], bcc and fcc metals [93,95,96], hcp metals [97], and binary alloy systems [91,93]. The spline-based version of MEAM discussed in this section has been applied to Si [61],

Ti [98], and Mo [13]. This work develops spline-based EAM and MEAM potentials for the bcc refractory metals V, Nb, Ta, Mo, and W. The next chapter discusses the cubic spline representation of the potential functions, and ﬁtting procedures to determine the spline parameters for a given material.

26 Chapter 3: Generating and Testing Classical Potentials

The embedded-atom method (EAM) and modiﬁed embedded-atom method

(MEAM) potentials introduced in Chapter 2 provide an accurate description of metallic bonding, while providing the computational eﬃciency necessary for large-scale molecular dynamics and Monte Carlo simulations. The MEAM formalism includes explicit three-body energy contributions in addition to the implicit many-body effects modeled by EAM potentials. The total potential energy of monoatomic systems described by EAM and MEAM potentials is

(M)EAM Vtot = φ(rIJ )+ F (nI ). (3.1) J>I I X X The local density nI describes the bonding environment of atom I due to density

contributions from surrounding atoms. The EAM density at the location of atom I

nI = ρ(rIJ ), (3.2) XJ6=I where each atom J surrounding atom I contributes a density ρ(rIJ ). The MEAM density is

27 nI = ρ(rIJ )+ f(rIJ )f(rIK )g(cos θJIK ), (3.3) XJ6=I K>JX6=I where the EAM two-body contributions ρ(rIJ ) are supplemented by contributions

from triplets of atoms I, J, and K. The angle θJIK is the bonding angle of the triplet, centered on atom I.

Equations 3.1-3.3 give the general form of EAM and MEAM potentials, where the functions φ, ρ, F , f, and g depend on the system under consideration. A large number of functional forms have been reported in the literature, ranging from forms based on quantum mechanical derivations (see, e.g., Refs. [68, 99]), to simple but rather arbitrary choices of analytic functions with a small number of ﬁtting parameter (see, e.g., Refs. [100,101]). These models are often convenient for their simplicity and ease of ﬁtting, but are usually of limited accuracy for a broad range of simulations.

Another representation, which is used for the potentials developed in this work, is to determine the function values at a finite number of points by fitting to a diverse collection of physical data, and interpolate between the points using smooth functions such as cubic polynomials (see, e.g., Refs. [1, 13, 22, 61, 90, 98]). Typical fitting data include forces, energies, and stress tensor components. This method of generating potentials is called the force-matching method, pioneered by Ercolessi and Adams [22].

These functional forms are usually more diﬃcult to optimize due to the complexity of the parameter space, but the interpolating functions can provide much more ﬂexibility than simple analytic functions for describing complicated functional dependencies.

The parameters in the interpolating functions are optimized to a suitable set of

ﬁtting data. This produces a large number of candidate potentials which must be screened to determine if they are adequate for materials simulations. The screening

28 process consists of testing how well the potentials reproduce the ﬁtting data, and how

well they interpolate between and extrapolate beyond the ﬁtting data. This involves

using the potentials to compute a large number of physical properties, and compar-

ing the results to experimental data and ab initio calculations. Quality potentials are able to accurately reproduce the fitting data used in their construction, and predict properties not explicitly included in the fitting database. Many revisions of the parameter choices, fitting data, and testing procedures are required to produce highly accurate potentials capable of modeling a wide range of behavior for a large variety of simulations.

The first section in this chapter discusses the cubic spline representation of EAM and MEAM functions. This is followed by a section describing the density-functional theory (DFT) calculations utilized to construct the data for fitting the functions. The next section contains information about numerical optimizers, necessary for determining the “best” spline parameters for a given fitting database. The final section discusses testing procedures for determining if a given candidate potential is acceptable.

This involves using the potential to compute a large number of physical properties, and comparing the results to experimental data and ab initio calculations. Quality

potentials are able to accurately reproduce the ﬁtting data used in their construction,

and predict properties not explicitly included in the ﬁtting database.

3.1 Cubic Spline-Based Interatomic Potentials

If a function h(x) is known at a ﬁnite number of points, various interpolating

schemes can be used to determine the values at intermediate points. A polynomial

of order m 1 will exactly pass through the m known points, but can oscillate wildly −

29 if m is large. This oﬀers a poor description of h(x) except for points at which the function is already known. Often a better alternative is a set of piece-wise functions that interpolate smoothly between the known values, with constraints imposed to guarantee continuity of the functions and some of their derivatives. If the functions are cubic polynomials, the resulting interpolation is called a cubic spline.

2 3 In a cubic spline representation, the polynomial ci(x) = αi + βix + γix + δix interpolates between the tabulated values of h(x) at xi and xi+1. The ci’s form a piece-wise function when taken together, where m 1 cubic functions are needed to − interpolate between m tabulated values of h(x). Thus, a total of 4(m 1) coefficients − must be determined. The following conditions provide 4(m 1) 2 equations for the − − coefficients, and ensure that the cubic polynomials pass through the tabulated values and that they are continuous along with their first and second derivatives:

c (x )= h(x ), 1 i

cm−1(xm)= h(xm),

c (x )= c (x ), 1 i m 2, i i+1 i+1 i+1 ≤ ≤ − c′ (x )= c′ (x ), 1 i m 2, i i+1 i+1 i+1 ≤ ≤ − c′′(x )= c′′ (x ), 1 i m 2, (3.4) i i+1 i+1 i+1 ≤ ≤ − where primes indicate derivatives. Boundary conditions imposed on the first cubic function c1(x) and the last cubic function cm−1(x) determine the final two conditions needed to solve for all the coefficients. A commonly used choice is called natural boundary conditions, which require the second derivative of the first cubic function

30 ′′ c1(x) to vanish at the ﬁrst tabulated point x1, and the second derivative of the last

′′ cubic function cm−1(x) to vanish at the last tabulated point xm:

′′ c1(x1) = 0,

′′ cm−1(xm) = 0. (3.5)

Physical considerations may require alternative choices for the boundary conditions.

Forces between atoms are determined by diﬀerentiation of the interatomic potential,

and the forces must vanish beyond the interaction range of the potential. When the

potential functions are parameterized as cubic splines, this is enforced by requiring

the ﬁrst derivatives of radial functions to vanish at the cutoﬀ distance of the potential.

Figure 3.1 illustrates these ideas by comparing cubic spline interpolation with polynomial interpolation for the function h(x) = 1/(1 + 25x2). The thick black line is h(x), and the black points are eleven equally spaced values of h(x). The tenth-

10 i order ﬁtted polynomial p(x) = i=0 aix (shown in red) exactly passes through the tabulated points, but poorly predictsP values of h(x) over most of the domain. A lower- order polynomial may be more well-behaved, but it will not reproduce the tabulated values exactly. A much better approximation to h(x) is provided by cubic spline interpolation (shown in gray). The smooth spline composed of piece-wise functions ci(x) passes through the tabulated values, and accurately approximates h(x) over the whole domain.

Cubic splines can be used to represent interatomic potential functions, but the functions are not known a priori. Instead, the tabulation points are free parameters determined by ﬁtting to experimental data, ab initio calculations, or a combination

31 Figure 3.1: Comparison of cubic spline and polynomial interpolation of the function h(x) = 1/(1 + 25x2). The spline c(x) is a piece-wise function composed of ten cubic 2 3 polynomials ci(x) = αi + βix + γix + δix . It equals h(x) at the eleven tabulation values xi, and accurately interpolates the function between them. Natural boundary conditions are imposed on the spline. The tenth-order ﬁtted polynomial p(x) exactly reproduces the tabulated values it was ﬁt to, but provides a poor interpolation of h(x) between most of the tabulation points.

32 of the two. The cubic spline interpolates between these points, providing intermediate values. Consider the example of the pair interaction φ(r) in EAM and MEAM potentials, tabulated as the set of points (r ,φ(r )), (r ,φ(r )), ..., (r ,φ(r )), { 1 1 2 2 m−1 m−1 (r ,φ(r )) . The tabulated values are called spline knots. Usually, r and r are m m } 1 m

fixed based on physical considerations. The rest of the ri and all of the φ(ri) can be treated as fitting parameters, but typically the ri are chosen as evenly spaced. In this case the free parameters in the model are the values of the potential φ(ri), which are optimized to a fitting database. Figure 3.2 shows the cubic spline representation of a generic potential φ(r), and illustrates that the degrees of freedom in the model are the knot values φ(ri).

Cubic splines can accurately mimic complicated functional forms, providing a very flexible means of describing interatomic potentials. This eliminates the very difficult task of deriving (or guessing) explicit functional dependencies suitable for describing complex interactions, and removes any bias by the person developing the potential. Rather, the fitting data itself determines not only the parameter values but also the shapes of the functions themselves. Cubic spline-based potentials have been applied to a variety of systems, including semiconductors [61], bcc metals [1,13,88], fcc metals [22,90], hcp metals [98], and quasicrystal alloys [102]. The next sections discuss constructing databases of fitting data, and numerical optimizers for determining the spline parameters for a given material.

3.2 Accurate First-Principles Fitting Data

The force-matching method [22] utilized to construct the potentials in this work relies on a database of accurate DFT calculations of forces, energies, and stresses for

33 Figure 3.2: Cubic spline representation of an arbitrary pair potential φ(r). The set of tabulated spline knots (r ,φ(r )) are free parameters that are adjusted during { i i } function optimization to a fitting database. Usually the ri are chosen as equally spaced. In this case, only the φ(ri) are model parameters. The red arrows indicate that these values are free to change during optimization. Radial functions must go to zero at the cutoff distance, so φ(r11) is fixed in this example. It would be difficult to determine an analytic functional form for a potential with this shape, but cubic splines offer a relatively simple parameterization.

34 various conﬁgurations of atoms (see, e.g., Refs. [41,55,69–81] for thorough discussions of DFT). The parameters in the cubic-spline potentials are optimized to reproduce this data as well as possible, within the inherent limitations of the classical potential models. The optimization process proceeds by minimizing a least-squares error function ( k ) that depends on the set of spline knots k : Z { } { }

N data 2 ( k )= W 2 Q(M)EAM ( k ) QDFT , (3.6) Z { } i i { } − i i=1 X DFT where the Wi are weighting factors, Qi is a force component, energy, or stress tensor component in the DFT fitting database, Q(M)EAM ( k ) is the corresponding i { } value predicted by a (M)EAM potential defined by spline knots k , and N is the { } data total number of fitting data.

The numerical optimization of the classical potential functions is based on the freely available force-matching code potfit [87, 102]. The parameter space of cubic spline-based (M)EAM potentials is complicated and contains many local minima. Therefore, global numerical optimizers must be utilized to search for acceptable minimum solutions of the error function ( k ). The current stable version Z { } of potfit optimizes spline-based EAM functions using either the simulated annealing [103–105] or diﬀerential evolution [106] global optimizers, in combination with the

Powell method [105,107,108] for local optimization. The diﬀerential evolution scheme implemented in potfit does not work well for optimizing spline-based potentials.

Our research group has modiﬁed the potfit code to optimize spline-based MEAM potentials. Simulated annealing does not handle well the extra complexities introduced by the angular terms in MEAM potentials, so we have also implemented a genetic algorithm global optimizer [109–111]. The genetic algorithm ﬁnds low-lying

35 local minima in the MEAM parameter space more eﬃciently than simulated anneal-

ing.

The Vienna Ab initio Simulation Package (vasp) [79, 112] generates the DFT

ﬁtting data. vasp is a widely used commercial DFT code based on plane-wave ex-

pansions of the Kohn-Sham orbitals, using pseudopotentials to represent the atomic

nuclei and core electrons. In addition to the choice of the type of pseudopotential

and the exchange-correlation functional, the main factors controlling the accuracy

and numerical convergence of the DFT calculations are the number of reciprocal-

space k-points used to sample the Brillouin zone in numerical sums, the number of plane-waves included in the expansion of the Kohn-Sham orbitals, and the width of the Fermi-surface smearing used to speed numerical convergence for metallic systems.

The next two subsections discuss these issues, and present the convergence parameters determined for the bcc refractory metals.

3.2.1 Exchange-Correlation Functionals and Pseudopotentials

Exchange-Correlation Functionals

The Hohenberg-Kohn theorem [69] of DFT states that the energy E (and all other physical properties) of an interacting many-electron system is a functional of

the electronic density n. The argument of a functional is enclosed in square brackets

to distinguish it from a function, e.g., the energy functional is E = E[n]. The energy is composed of the electronic kinetic energy T [n], the electronic potential energy V [n],

and the interaction energy of the electrons with an external potential provided by the

nuclei Vext:

36 E[n]= T [n]+ V [n]+ Vext(r)n(r)dr. (3.7) Z The kinetic energy and electronic potential energy functionals are universal in the

sense that they are the same for all N-electron systems. Hence, the external potential

distinguishes between diﬀerent physical systems. The energy functional is equal to

the ground-state energy E0 if the electronic density is equal to the ground-state

density n0. For all other densities, the energy is greater than the ground-state energy.

The nuclear-nuclear interaction is an additive constant within the Born-Oppenheimer

approximation and will not be included in the discussion.

The Hohenberg-Kohn theorem guarantees the existence of the energy functional,

but does not provide a scheme for computing it. A scheme for practical DFT cal-

culations is provided by the Kohn-Sham formalism [70]. The Kohn-Sham method

considers an auxiliary system of N noninteracting electrons in an eﬀective potential

Vs(r). The energy functional for the noninteracting system is

Es[n]= Ts[n]+ Vs(r)n(r)dr, (3.8) Z where the kinetic energy functional Ts[n] is diﬀerent than the kinetic energy functional

T [n] of the interacting electrons. The noninteracting electrons occupy the single-

particle Kohn-Sham orbitals ϕn(r), which satisfy the Kohn-Sham eigenvalue equations

~2 2 + V (r) ϕ (r)= ǫ ϕ (r). (3.9) −2m∇ s n n n These equations are single-particle Schr¨odinger equations with the potential Vs(r).

The density of the auxiliary system is computed by summing over the densities arising

from the N occupied Kohn-Sham orbitals:

37 N n(r)= ϕ (r) 2. (3.10) | n | i=1 X The potential Vs(r) is chosen such that the ground-state density of the auxiliary system is the same as the ground-state density of the true interacting system. Once the density is obtained from the noninteracting system, it can be used in the energy functional for the interacting system. It is advantageous to use the auxiliary system, since it is much simpler to perform calculations on noninteracting electrons.

The form of Vs(r) that produces the correct density is determined by rewriting the energy functional of the true system in terms of the noninteracting kinetic energy:

e2 n(r)n(r′) E[n]= T [n]+ drdr′ + V (r)n(r)dr s 2 r r′ ext ZZ | − | Z e2 n(r)n(r′) + T [n] T [n]+ V [n] drdr′ . − s − 2 r r′ ZZ | − | e2 n(r)n(r′) T [n]+ drdr′ + V (r)n(r)dr + E [n]. (3.11) ≡ s 2 r r′ ext xc ZZ | − | Z The energy is now written as a sum of the noninteracting kinetic energy, the direct

Hartree contribution to electronic potential energy, the external potential energy, and the term in braces called the exchange-correlation energy Exc[n]. Expressions for the

ﬁrst three terms are known, and all the unknown information about the electronic interactions is contained in the universal functional Exc[n]. The exact functional form of the exchange-correlation energy is unknown, but approximations are available that yield accurate results for many systems.

The ground-state density, which minimizes the energy functional, is obtained by setting the functional derivative of the energy to zero subject to the constraint that the density must integrate to the total number of electrons Nelec:

38 δ E[n] µ n(r)dr N = 0, (3.12) δn − − elec Z where µ is a Lagrange multiplier that enforces the constraint. Carrying out the

diﬀerentiation for the interacting system gives

δT [n] n(r′) s + e2 dr′ + V (r)+ v [n]= µ, (3.13) δn r r′ ext xc Z | − | where the exchange-correlation potential vxc is deﬁned as

δE [n] v = xc . (3.14) xc δn

Setting the constrained derivative of the noninteracting energy functional to zero yields

δT [n] s + V (r)= µ. (3.15) δn s

These two equations will be equivalent, and hence produce equivalent ground-state densities, if

n(r′) V (r)= V (r)+ e2 dr′ + v (r). (3.16) s ext r r′ xc Z | − |

Accurate approximations to vxc(r) implemented in vasp include the local density approximation (LDA) [70] and diﬀerent generalized gradient approximations. The

′ true vxc(r) depends on the density at all points in space r , i.e., it is a non-local functional. The LDA assumes that vxc(r) can be approximated as a local functional that depends on the density only at position r, and that the density varies slowly

39 enough that the exchange-correlation energy locally equals that of a homogeneous

electron gas. With these assumptions, the exchange-correlation energy is

LDA Exc [n]= ǫxc[n]n(r)dr, (3.17) Z

where ǫxc[n] is the exchange-correlation energy per electron of a homogeneous electron gas.

Generalized gradient approximations (GGAs) of the exchange-correlation energy attempt to improve upon the LDA by including information about the gradient of the density. The GGA exchange-correlation energy can be written as

EGGA[n]= f(n(r), n(r))dr. (3.18) xc ∇ Z There are several GGAs with diﬀerent functional forms for f(n(r), n(r)). The ∇ Perdew-Wang (PW91) [113] and Perdew-Burke-Ernzerhof (PBE) [114] GGAs are

implemented in vasp. GGA functionals generally perform better than LDA, with

improvements in properties including lattice constants and bond angles, cohesive en-

ergies, activation barriers, and elastic behavior.

Pseudopotentials

DFT calculations can be divided into two broad classes, depending on the num-

ber of electrons explicitly included in the calculations. All-electron DFT calculations

include both the core and valence electrons, while pseudopotential DFT calculations

replace the nuclei and core electrons with an eﬀective potential that interacts with

the valence electrons. Pseudopotentials provide a good description of materials since

40 the contributions of the tightly-bound core electrons to bonding are usually not sig-

niﬁcant. The semicore states in the highest ﬁlled shells are often important in tran-

sition metal systems, and must be treated explicitly. Pseudopotential calculations

have a lower computational burden than all-electron calculations due to fewer elec-

tronic states, and smoother wave-functions requiring fewer expansion basis functions.

vasp is a pseudopotential code that implements either ultra-soft (US) pseudopoten-

tials [115,116] or projector augmented-wave (PAW) pseudopotentials [117,118].

A full discussion of diﬀerent pseudopotentials and their construction is beyond

the scope of this thesis. Rather, results are presented comparing several exchange-

correlation functional and pseudopotential combinations available in vasp for Nb

properties. The method giving the best overall agreement with experiment was chosen

for all subsequent DFT calculations for the bcc refractory metals.

Table 3.1 shows this comparison for the lattice constant a, bulk modulus B, and

elastic constants C′ =(C C )/2 and C of bcc Nb. The combination of the PBE 11 − 12 44 exchange-correlation functional with the PAW pseudopotential that treats the 4s- and 4p-semicore states as valence states provides the best overall physical description of the tabulated properties. Explicitly treating the semicore states as valence states is more computationally intensive, but provides a better description of properties.

This is especially true under conditions where atoms closely approach one another, such as high-pressure simulations and in the vicinity of certain lattice defects. The

PBE-PAW combination is also the most modern, widely used method oﬀered by vasp for metallic systems. vasp provides PAW pseudopotentials with the highest s- and p-semicore states treated as valence states for V and Nb, and PAW pseudopotentials with the highest p-semicore states treated as valence states for Ta, Mo, and W. These

41 ′ XC Functional / PP a (A)˚ B (GPa) C (GPa) C44 (GPa) LDA / US 3.228 191 70 -5 LDA / US-pv 3.265 190 61 9 LDA / PAW-pv 3.264 195 63 14 LDA / PAW-sv 3.248 192 65 13 PW91 / US 3.292 170 61 1 PW91 / US-pv 3.326 168 55 11 PW91 / PAW-pv 3.323 173 56 15 PW91 / PAW-sv 3.310 170 58 15 PBE / PAW-pv 3.322 172 58 21 PBE / PAW-sv 3.309 172 59 22 Experiment 3.303 173 60 31

Table 3.1: Comparison of calculated lattice constant a, bulk modulus B, and elastic ′ constants C =(C11 C12)/2 and C44 of Nb to experimental values. 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 diﬀerent 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 with experiment.

pseudopotentials and the PBE functional are used to construct the ﬁtting data for the potentials developed in this study.

3.2.2 Convergence Tests

The numerical convergence of DFT calculations is crucial for accurate and reliable property calculations. The three main convergence parameters governing the quality of a vasp calculation are the number of plane-wave basis functions used to expand the Kohn-Sham orbitals ϕn(r), the number of k-points in reciprocal space used to

42 numerically sample integrals over the Brillouin zone, and the smearing width used to smooth the sharp transition between occupied and unoccupied states in metals.

Plane-Wave Expansion of the Kohn-Sham Orbitals

The solutions of Schr¨odinger’s equation in a periodic system must satisfy Bloch’s theorem, which states that the wave-functions are products of a plane-wave and a function with the periodicity of the system unk(r):

ik·r ψnk(r)= e unk(r), (3.19)

where unk(r)= unk(r + R) and R is a translation period. The index n is called the band index, which speciﬁes the diﬀerent allowed solutions for a given wave-vector k.

The Kohn-Sham orbitals ϕn(r) are solutions of a single-particle Schr¨odinger equation for an eﬀective potential Vs(r). A simulation supercell is a simulation cell constructed from smaller cells, usually the primitive or unit cell of the crystal structure. vasp employs periodic boundary conditions to mimic a bulk sample with a small supercell, so the ϕn(r) must satisfy Bloch’s theorem:

ik·r ϕnk(r)= e unk(r), (3.20)

where the function unk(r) is periodic in the supercell lattice vectors R. The periodic function unk(r) is expanded in plane-waves with wave-vectors commensurate with the supercell dimensions, i.e., the reciprocal lattice vectors G corresponding to the supercell lattice vectors R:

43 ik·r iG·r i(k+G)·r ϕnk(r)= e cn,Ge = cn,k+Ge . (3.21) G G X X An exact description of ϕnk(r) requires an inﬁnite number of plane-waves for the expansion in Eqn. 3.21. It is reasonable to assume that low-energy plane-waves are more important than high-energy ones, however, and that the sum can be truncated at plane-waves with magnitudes less than some upper cutoﬀ value Gcut:

i(k+G)·r ϕnk(r)= cn,k+Ge . (3.22) k G | + X|

Ecut:

~2G2 E = cut . (3.23) cut 2m

The number of plane-waves included in the wave-function expansions in vasp is speciﬁed by the value of Ecut. Convergence with respect to the size of the basis set is therefore determined by increasing the value of Ecut until the properties of interest do not change within a speciﬁed tolerance. One of the main advantages of the plane- wave basis set is this simple way to systematically improve convergence by increasing a single parameter. Another advantage that makes plane-wave especially useful for solids is their natural periodicity.

Sampling the Brillouin Zone

Many quantities Q in DFT calculations, such as total energy, are expressed as integrals over the Brillouin zone in reciprocal-space. These integrals are approximated

44 in numerical codes as ﬁnite sums over a discrete number of k-points in the Brillouin

zone:

8π3 Q = f(k)q(k)dk f(k)q(k), (3.24) BZ ≈ V k Z X where V is the supercell volume, and f(k) speciﬁes the occupation of state k. The

accuracy of the calculations increase as the density of the sampling grid increases,

but the computational cost also increases.

There are several methods for choosing eﬃcient k-point grids. The Monkhorst-

Pack method [119] implemented in vasp generates a grid of k-points evenly spaced throughout the Brillouin zone. Similar to the plane-wave convergence test, the number of k-points (speciﬁed as N N N divisions the reciprocal-space lattice vectors) 1 × 2 × 3 is increased until properties of interest do not change within a speciﬁed tolerance. All vasp calculations in this study use Monkhorst-Pack grids centered on the Γ-point, i.e., k = 0.

Fermi Surface Smearing

The final convergence criterion is specific to metallic systems. The occupied states are separated from the unoccupied states in metals by the Fermi surface. At zero temperature, this is a sharp boundary that is difficult to represent in numerical calculations. In terms of Brilloiun zone integrations, the occupation function f(k) in

Eqn. 3.24 is a step function at zero temperature.

Tetrahedron methods [120–123] provide an accurate method for sampling the ﬁne details of the Fermi surface in metals. The value of the integrand is known at the points of the sampling grid, and k-space is ﬁlled with tetrahedra whose vertices lie

45 on the grid points. The value of the integrand is found at intermediate k-points by

interpolating along the tetrahedra between the known values, leading to accurate es-

timates of the Brillouin zone integrals. Tetrahedron methods provide accurate energy

values, but forces can be wrong by up to 10%.

This diﬃculty in producing accurate forces is removed by replacing the sharp func-

tion f(k) by a smooth function that preserves the accuracy of the sum. The method of Methfessel and Paxton [124] implemented in vasp uses Hermite polynomial expansions to provide a smooth function for Brillouin zone integration. The extent of the smoothing is controlled by a single parameter, the smearing width σ. The optimal value of σ depends on the density of k-points. If σ is too large the energy will deviate signiﬁcantly from the accurate tetrahedron method value, and if σ is too small there will be spurious oscillations in the electronic density of states.

Procedure for Setting Convergence Parameters for Metals

The proper values of E , N N N , and σ for the bcc metals are determined cut 1 × 2 × 3 by the following procedure:

(1) Determine Ecut: The value of Ecut is independent of the k-point grid, and a very dense grid (47 47 47) is used at ﬁrst to ensure converged results with respect to × × the number of k-points. The energy of the ideal bcc primitive cell is computed for increasing values of Ecut using the tetrahedron method of Bl¨ochl [123]. The converged

DFT value for the lattice constant is not yet known, so the experimental value is used. The total energy Etotal versus Ecut is plotted, and the smallest value of Ecut is chosen for which oscillations in the energy stabilize and an increase in Ecut produces

46 a change of Etotal less than 1 meV. This level of convergence produces accurate values

for both energies and forces.

(2) Determine the k-space grid N N N : Since the bcc cell has cubic symmetry, 1 × 2 × 3 N = N = N N . The value of E determined in step (1) is used to compute 1 2 3 ≡ k cut

Etotal and electronic density of states (DOS) for increasing values of Nk using the tetrahedron method of Bl¨ochl. Etotal versus Nk is plotted, and the DOS for the different values of Nk are plotted. The smallest value of Nk is chosen that produces a total energy within 1 meV of the value from the Nk = 47 calculation, while accurately representing the DOS near the Fermi level from the Nk = 47 calculation.

(3) Determine the smearing parameter σ: The value of Ecut from step (1) and the value of Nk from step (2) are used to compute Etotal and the DOS for increasing values

of σ using the ﬁrst-order Methfessel-Paxton method [124] to smooth the Fermi sur-

face. Etotal versus σ is plotted, and the DOS for the diﬀerent values of σ are plotted.

Large values of σ smooth out sharp features in the DOS. The value of σ is chosen that produces a total energy within 1 meV of the value from the Nk = 47 tetrahedron method calculation, while retaining most of the sharp features present in the DOS near the Fermi level from the Nk = 47 calculation.

Appendix A presents the various DFT calculations used to determine the convergence parameters used in this work for the bcc refractory metals. The results are summarized in Table 3.2. All DFT calculations utilize the PBE exchange-correlation functional and PAW pseudopotentials.

47 Element E (eV) N N N σ (eV) cut 1 × 2 × 3 V 550 35 35 35 0.10 × × Nb 550 31 31 31 0.10 × × Ta 600 31 31 31 0.20 × × Mo 600 31 31 31 0.35 × × W 600 35 35 35 0.35 × ×

Table 3.2: Convergence parameters used for DFT calculations for the bcc refractory 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 calculations 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.

3.3 Tests for Interatomic Potentials

The potentials generated by optimizing cubic spline parameters to DFT calculations must be tested to ensure they produce physically acceptable behavior. Spline- based potentials require a relatively large number of parameters to model a diverse

fitting database. This results in a complicated multi-dimensional parameter space with a large number of local minima, and numerical optimizers often get trapped when searching for the global minimum. This leads to a large number of potentials with different physical behavior, even though they have the same number of parameters and are fit to the same data. Therefore, each potential must be thoroughly tested against properties included in the fit as well as properties outside of the fitting database. This section discusses the calculation of properties for verifying the quality of interatomic potentials. The discussion is general, with specific details of

48 convergence, simulation details, and comparisons to DFT and experiment deferred to

Chapter 4 where speciﬁc potentials for the ﬁve bcc refractory metals are presented.

3.3.1 Equations of State

One of the most basic tests for an interatomic potential is how well it can reproduce the energy E as a function of crystal volume V for the lowest energy crystal structure.

The E versus V relation is an equation of state that determines the equilibrium lattice constants, the equilibrium energy E0 with respect to a chosen reference energy, and the bulk modulus B. The lowest energy crystal structure of V, Nb, Ta, Mo, and W is bcc. This structure is cubic and therefore has only one equilibrium lattice constant a0. The bulk modulus B is deﬁned as

1 ∂P 1 ∂2E B = = , (3.25) −V ∂V V ∂V 2 T T where P is pressure and T is temperature. The bulk modulus measures the stiﬀ- ness of a solid against compression by hydrostatic pressure, and must be positive for mechanical stability in cubic crystals [125,126].

The energy-volume equation of state is determined by calculating the energy E of the ideal bcc crystal at a number of diﬀerent volumes V . All standard DFT and

MD codes compute the energy of a given configuration of atoms. In practice, 20 evenly spaced values of V within the range 0.90V V 1.10V produce an accurate 0 ≤ ≤ 0 representation of the equation of state for determining a0, E0, and B. A low-order polynomial can then be fit to the data, and differentiated to find a0, E0, and B [49].

It is somewhat more convenient to ﬁt the third-order Birch-Murnaghan equation of state [126,127] to the E-V data:

49 2 3 9 V 2/3 9 V 2/3 E (V )= E + BV 0 1 + BV (B′ 4) 0 1 , (3.26) BM 0 8 0 V − 16 0 − V − " # " # where the equilibrium volume V0, energy E0, bulk modulus B, and pressure derivative

′ ′ of the bulk modulus B are ﬁtting parameters. Thus, E0, B, B , and a0 (obtained from

V0) are obtained from the ﬁt, eliminating the further analysis required by a polynomial

ﬁt. The resulting values can be compared with DFT values and experimental data.

Another important test for an interatomic potential is its ability to determine the volume as a function of pressure. All standard DFT and MD codes compute the pressure for a given simulation cell geometry, so the pressure-volume equation of state is computed in a manner similar to the energy-volume equation of state. The P -V relation is usually reported only for compression, where it can be compared to DFT calculations as well as data from shock experiments.

A potential should also predict the proper ground state structure of a material to prevent unphysical phase transformations in dynamical simulations. In principle, the stability of the ground state structure can be verified by comparing the E-V equation of state for the expected lowest energy structure to the equations of state of other crystal structures. There are an infinite number of possible structures, however, so it cannot be known with certainty if a potential has the proper ground state. A common practice is to compare the E-V relations of many known structures, and also perform finite temperature MD simulations to test if the expected ground state structure is stable against perturbations. The relative stability of the structures and the shapes of the E-V curves can be compared to DFT calculations, offering a stringent test of a potential’s ability to model changes in coordination number and bonding environments.

50 3.3.2 Elastic Constants

Materials deform when external forces are applied. If the distortions remain when

the forces are removed, the material has undergone plastic deformation. Plasticity

is a complex phenomenon involving the motion of extended defects. If the material

returns to its original state after the forces are removed, the deformation is elastic.

An interatomic potential should properly describe deformation under various loading conditions. The elastic response of a material is determined by various elastic constants, which relate the applied forces to the resultant deformations. Elasticity theory is discussed in many books and articles, e.g., Refs. [49,125,126,128,129].

A familiar example is a linear spring obeying Hooke’s law. If a such spring is stretched from its initial length l0 to a ﬁnal length l1, it will exert a force F that tends to restore it to its original length. Hooke’s law states that this force is proportional to the change in length:

F = k(l l ) k∆l, (3.27) − 1 − 0 ≡ − where k is a proportionality constant called the spring constant. The negative sign ensures that the restoring force is opposite to the direction of the displacement. A stretched spring stores elastic potential energy Eelastic proportional to the square of the displacement:

1 E = k (∆l)2 . (3.28) elastic 2

Hooke’s law is not a fundamental law of nature, but rather an approximate relation

that holds for many materials under suﬃciently small displacements. If a spring is

51 stretched beyond its elastic limit, the relation between force and displacement is no longer linear.

When a three-dimensional bulk material deforms under an imposed load, it is more convenient to normalize the forces by the areas over which they are applied.

Stress σ is deﬁned as force per unit area A:

F σ = . (3.29) A

Normal stress occurs if the force is perpendicular to the area, and shear stress occurs if the force is parallel to the area. The resulting deformations are called strains ǫ, deﬁned as the deformation ∆l divided by the undeformed dimension l0:

∆l ǫ = . (3.30) l0 Figure 3.3 shows the general state of stress for an inﬁnitesimal element within a

solid. Elasticity is a continuum concept, so the “inﬁnitesimal” element is macroscop-

ically small but still contains many atoms. Forces can point in diﬀerent directions

on a given plane of the inﬁnitesimal element. A general stress is denoted σij, where

the ﬁrst index i gives the direction of the force and the second index j gives the

normal direction of the plane on which the force acts. Normal stresses correspond to

i = j, and shear stresses correspond to i = j. The σ form a nine-component stress 6 ij

tensor σ¯. Mechanical equilibrium requires zero net torque on the body, so σij = σji.

Therefore,σ ¯ is symmetric:

σ11 σ12 σ13 σ¯ = σ σ σ . (3.31)  12 22 23  σ13 σ23 σ33   52 Figure 3.3: General state of stress for an element within a solid. The ﬁrst index i on the stress components σij indicates the direction of the force, and the second index j refers to the normal of the plane on which the force acts. Normal stresses correspond to i = j, and shear stresses correspond to i = j. Mechanical equilibrium requires 6 σij = σji.

53 When the material deforms, the point located at x =(x1, x2, x3) moves to a new

′ ′ ′ ′ position x =(x1, x2, x3). A general strain can be deﬁned in terms of the displacement

of the point u = x′ x as −

∂ui ǫij = , (3.32) ∂xj which corresponds to Eqn. 3.30 as a change in length normalized by an undeformed length. This can be rewritten as

1 ∂u ∂u 1 ∂u ∂u ǫ = i + j + i j ǫS + ǫA, (3.33) ij 2 ∂x ∂x 2 ∂x − ∂x ≡ ij ij j i j i S A where ǫij is a symmetric contribution and ǫij is an antisymmetric contribution. The

antisymmetric part corresponds to rigid rotations, so it is removed and the strain is

redeﬁned as the symmetric part:

1 ∂u ∂u ǫ = i + j . (3.34) ij 2 ∂x ∂x j i

The ǫij form a symmetric strain tensor ǫ¯:

ǫ11 ǫ12 ǫ13 ǫ¯ = ǫ ǫ ǫ . (3.35)  12 22 23  ǫ13 ǫ23 ǫ33 Hooke’s law generalizes to a tensor equation relating stress and strain:

σij = Cijklǫkl, (3.36) Xkl where the elastic constants Cijkl generalize the spring constant k in Hooke’s law. The

generalized elastic energy stored in a deformed body is

54 1 E = C ǫ ǫ . (3.37) elastic 2 ijkl ij kl Xijkl The Cijkl form an 81-component elastic constant tensor C¯, but not all of the compo-

nents are independent. The symmetry of the stress tensor σij = σji requires

Cijklǫkl = Cjiklǫkl, (3.38) Xkl Xkl which implies Cijkl = Cjikl. The symmetry of the strain tensor ǫkl = ǫlk requires

Cijklǫkl = Cijlkǫkl, (3.39) Xkl Xkl which implies Cijkl = Cijlk. The elastic energy must also be invariant under inter- change of indices:

1 1 C ǫ ǫ = C ǫ ǫ , (3.40) 2 ijkl ij kl 2 klij kl ij Xijkl Xijkl which implies Cijkl = Cklij. Thus, the following relations hold between the elastic

tensor components:

Cijkl = Cjikl = Cijlk = Cklij. (3.41)

This reduces the number of independent components from 81 to 21. The indices ij can take six possible values, and Table 3.3 lists the possible values of kl for each of these choices.

The index pairs ij and kl can take only six possible pairs of values each, so Voigt notation replaces the double indices ij by a single index α that ranges from one to six. Table 3.4 lists the correspondence between the double index values and the single

55 ij kl Number of allowed kl values 11 11, 12, 13, 22, 23, 33 6 12 12, 13, 22, 23, 33 5 13 13, 22, 23, 33 4 22 22, 23, 33 3 23 23, 33 2 33 33 1

Table 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.

index values in Voigt notation. The Cijkl are written Cαβ, and the 81-component tensor composed of the Cijkl becomes a symmetric 36-component tensor composed of the Cαβ. The stress tensor becomes

σ11 σ12 σ13 σ1 σ6 σ5 σ¯ = σ σ σ σ σ σ , (3.42)  12 22 23  ≡  6 2 4  σ13 σ23 σ33 σ5 σ4 σ3 and the strain tensor is    

ǫ11 ǫ12 ǫ13 ǫ1 ǫ6/2 ǫ5/2 ǫ¯ = ǫ ǫ ǫ ǫ /2 ǫ ǫ /2 . (3.43)  12 22 23  ≡  6 2 4  ǫ13 ǫ23 ǫ33 ǫ5/2 ǫ4/2 ǫ3 The stress-strain relation in Voigt notation  is 

σα = Cαβǫβ, (3.44) Xβ=1 and the elastic energy is

56 Original notation: Voigt notation: ij or kl α or β 11 1 22 2 33 3 23 4 13 5 12 6

Table 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 i = j. ijkl ≡ αβ ij ≡ α ii ≡ α ij ≡ α 6

1 6 E = C ǫ ǫ . (3.45) elastic 2 αβ α β α,βX=1 Crystal symmetries further reduce the number of independent components of Cαβ.

For cubic crystals, there are only three independent elastic constants C11, C12, and

C44. The elastic constant tensor for cubic systems is [126]

C11 C12 C12 0 0 0 C12 C11 C12 0 0 0  C C C 0 0 0  C¯ = 12 12 11 . (3.46)  0 0 0 C 0 0   44   0 0 0 0 C 0   44   0 0 0 0 0 C   44    The bulk modulus B must be positive for mechanical stability, and is related to C11

and C12 in cubic crystals by [126]

C + 2C B = 11 12 > 0 (3.47) 3

Mechanical stability in cubic systems also requires [126]

57 C C C′ 11 − 12 > 0, (3.48) ≡ 2 and

C44 > 0. (3.49)

The stress-strain relations for cubic materials are

σ1 = C11ǫ1 + C12ǫ2 + C12ǫ3,

σ2 = C12ǫ1 + C11ǫ2 + C12ǫ3,

σ3 = C12ǫ1 + C12ǫ2 + C11ǫ3,

σ4 = C44ǫ4,

σ5 = C44ǫ5,

σ6 = C44ǫ6. (3.50)

The cubic stress-strain relations Eqn. 3.50 oﬀer a method for computing the elastic constants C11, C12, and C44. A strain tensorǫ ¯ is devised to pick out speciﬁc elastic

constants, or linear combinations of elastic constants. Then the primitive vectors

ai deﬁning the simulation cell in a DFT or MD calculation are deformed at several

′ values of this strain to new vectors a i according to

a′ =(1¯ +ǫ ¯) a , (3.51) i · i where 1¯ is the 3 3 identity tensor. The atoms must be relaxed after straining the × cell to ensure they have zero net force. The resulting stress tensor is computed, and

58 for small strains the stress versus strain will be linear. The slope of the plotted data

gives the particular elastic constant combination corresponding to the strain tensor.

The simplest example for calculating C11 and C12 uses the strain

x 0 0 ǫ¯ = 0 0 0 . (3.52)   0 0 0 The corresponding stress components are 

σ1 = C11x,

σ2 = C12x,

σ3 = C12x,

σ4 = 0,

σ5 = 0,

σ6 = 0. (3.53)

In practice, strains of 1 2% are well within the elastic limit. If x is varied between, − say, 0.01 and 0.01 in steps of 0.001, the slope of σ versus x is C and the slope of − 1 11

σ2 or σ3 versus x is C12.

This simple strain changes the volume of the cell, and it is preferable to use volume- conserving strains to eliminate pressure-volume contributions. The orthorhombic strain [126]

x 0 0 ǫ¯ = 0 x 0 . (3.54)  − x2  0 0 1−x2 conserves volume and produces the stress components

59 x2 σ = C x C x + C , 1 11 − 12 12 1 x2 −x2 σ = C x + C x + C , 2 − 11 12 12 1 x2 x2 − σ = C , 3 11 1 x2 − σ4 = 0,

σ5 = 0,

σ6 = 0. (3.55)

The slope of (σ σ )/4 versus x is C′ = (C C )/2. The bulk modulus B = 1 − 2 11 − 12 ′ (C11 + 2C12)/3 and C are solved for C11 and C12.

The monoclinic strain [126]

x 0 2 0 x ǫ¯ = 2 0 0 . (3.56)  x2  0 0 4−x2 conserves volume and produces the stress components

x2 σ = C , 1 12 4 x2 −x2 σ = C , 2 12 4 x2 −x2 σ = C , 3 11 4 x2 − σ4 = 0,

σ5 = 0,

σ6 = C44x. (3.57)

The slope of σ6 versus x is C44.

60 3.3.3 Vibrational Frequencies

The atoms in a crystal are not ﬁxed at the equilibrium sites of the ideal lattice.

External inﬂuences and thermal eﬀects cause the atoms to vibrate about the ideal

sites, and even at zero temperature there is still the quantum mechanical zero-point

motion associated with oscillating masses. The motion of the atoms will generally

be very complicated, but any vibrational state can be described in terms of a super-

position of normal modes. A normal mode is a simpler vibration with a well-deﬁned

frequency. The analysis of normal mode frequencies begins by assuming that the

forces between the atoms will be approximately harmonic if the atomic displace-

ments are suﬃciently small. Harmonic means that the potential energy is quadratic

in atomic displacements.

The ith Cartesian component of the equilibrium position of atom α in lattice cell

n is Rnαi, and the components of the displacement of this atom from equilibrium are unαi. The potential energy of the crystal is expanded in a Taylor series about the

equilibrium positions:

1 ∂2V V (u + R )= V (R )+ u u + ..., (3.58) nαi nαi nαi 2 ∂R ∂R nαi mβj nαi nαi mβj Xmβj where the partial derivatives are evaluated at the equilibrium positions, and the term linear in unαi vanishes since the energy is a minimum at the equilibrium positions.

The harmonic approximation truncates this expansion after the quadratic term. This is usually justiﬁed at low temperatures since it is expected that the displacements from equilibrium are small compared to the lattice spacing. The equilibrium energy

61 V (Rnαi) is an unimportant constant for analyzing lattice vibrations, so it will be set

to zero in the subsequent discussion.

The starting point for analyzing lattice vibrations is then

1 ∂2V 1 V (u + R )= u u V mβju u , (3.59) nαi nαi 2 ∂R ∂R nαi mβj ≡ 2 nαi nαi mβj nαi nαi mβj nαi Xmβj Xmβj

mβj where the elements of the force-constant matrix Vnαi are deﬁned as

2 mβj ∂ V Vnαi = . (3.60) ∂Rnαi∂Rmβj The ith component of the force on atom α in cell n is then

F = V mβju . (3.61) nαi − nαi mβj Xmβj mβj An important symmetry of the matrix elements Vnαi follows from the translational invariance of the lattice. If the whole lattice is translated by a Bravais lattice vector

R′, the energy of the crystal is unaﬀected. This means that the force-constants depend on the relative positions of lattice cells R R , not the absolute positions R and n − m n

Rm:

mβj (m−n)βj Vnαi = V0αi . (3.62)

The potential energy and force components are then

1 V (u + R )= V (m−n)βju u , (3.63) nαi nαi 2 0αi nαi mβj nαi Xmβj and

62 F = V (m−n)βju . (3.64) nαi − 0αi mβj Xmβj The equations of motion for the atoms are given by Newton’s second law:

M u¨ = V (m−n)βju (3.65) α nαi − 0αi mβj Xmβj A solution for this set of coupled diﬀerential equations is a plane-wave with frequency

ω and wave-vector k:

1 i(k·Rn−ωt) unαi = uαi(k)e (3.66) √Mα Substitution of Eqn. 3.66 into Eqn. 3.65 yields

1 k R R ω2u (k)+ V (m−n)βjei ·( m− n)u (k) − αi 0αi βj ≡ m MαMβ Xβj X ω2u (k)+ Dβj(pk)u (k) = 0, (3.67) − αi αi βj Xβj βj where the elements Dαi (k) of the dynamical matrix are deﬁned as

βj 1 (m−n)βj ik·(Rm−Rn) Dαi (k)= V0αi e . (3.68) m MαMβ X βj Note that the Dαi (k) are independentp of n due to the symmetry introduced in Eqn. 3.62.

Equation 3.67 can be expressed in matrix notation as

Dˆ (k) u = ω(k)2u. (3.69) ·

63 The solutions of this eigenvalue equation give the vibrational frequencies ω(k). The

frequencies depend on k since the dynamical matrix depends on k. The frequencies

are also periodic in reciprocal lattice vectors G (ω(k) = ω(k + G)), due to the periodicity of the dynamical matrix (Dˆ (k)= Dˆ (k + G)). The dynamical matrix of a three-dimensional crystal with s atoms per unit cell has dimension 3s. Thus, for each value of k there are 3s values of ω. The dependence of ω on k is called the phonon dispersion relation.

The eigenfrequencies ω(k) of a solid can be computed with an interatomic potential or DFT via the phonon code phon [130], which utilizes the small-displacement method. Recall that, within the harmonic approximation, the force components are given by Eqn. 3.64:

F = V (m−n)βju , nαi − 0αi mβj Xmβj where umβj is the displacement of atom β in cell m along direction j, and Fnαi is

th the i force component on atom α in cell n. For small umβj, the elements of the

(m−n)βj force-constant matrix V0αi are approximately

(m−n)βj Fnαi V0αi . (3.70) ≈ −umβj Thus, the elements of the force-constant matrix (and therefore the dynamical matrix) are computed by displacing the atoms in the unit cell and calculating the resulting forces induced on the other atoms. Since the force calculations are done in periodic supercells, the size of the cell must be large enough to ensure converged results. phon further uses the point group symmetry of the lattice to reduce the number of necessary calculations. A detailed description of phon is found in Ref. [130]. The

64 phonon dispersion relations of a classical potential can be compared with DFT calcu-

lations and experimental measurements. More information about lattice vibrations

is found in, e.g., Refs. [52,53,128,131–133].

3.3.4 Thermal Expansion

Thermal expansion calculations evaluate the response of a classical potential to

changes in temperature, and test the anharmonicity of the interatomic interactions.

Thermal expansion is determined by computing the change in lattice constant with

temperature using NP T -ensemble MD simulations. The pressure P is typically set to atmospheric pressure, and simulations are performed at several increasing temperature values T from room temperature to the melting point. Classical MD is accurate for evaluating thermal expansion above the Debye temperature, but fails at low temperatures where quantum effects are important [15]. The simulation time for each temperature must be long enough for the system to thermally equilibrate, and the crystal lattice to expand to its equilibrium dimensions for the given pressure and temperature. Thermal equilibration can be determined by several techniques such as correlation function methods, but it is often sufficient to plot the value of lattice constant versus time and visually examine when the fluctuations stabilize around an average value.

The equilibrium value of the lattice constant for a given temperature T is de-

termined by averaging the lattice constant values at each simulation time-step after

equilibration is reached. The thermal expansion curve α(T ) is obtained by computing the percent change in lattice constant a(T ) with respect to the room temperature lattice constant a0, and plotting the values versus temperature:

65 a(T ) a α(T )= − 0 100%. (3.71) a0 × The thermal expansion curve must also be tested for convergence with respect to the number of particles N in the simulation cell. DFT calculations of thermal expansion are computationally intensive, so the results of an interatomic potential are tested against experimental data.

3.3.5 Melting Temperature

Solid-state properties are the main focus of this study, but melting temperatures are also investigated since they provide a stringent test for interatomic potentials.

Many EAM potentials reported in the literature produce melting temperatures more than 30% lower than experimental values [134]. Simulations near the melting point sample a wide range of local atomic environments and test a potential’s ability to accurately predict a phase transition.

Accurate melting temperatures are computed from MD simulations using the method of coexisting phases [134–137]. An intial configuration of atoms is generated, consisting of a solid region and a liquid region in contact with each other. The solid-liquid interface provides nucleation sites for melting or solidification, removing overheating or undercooling issues associated with single-phase simulations. NP T - ensemble MD simulations are performed on the initial configuration. If the simulation temperature is below the melting point, the solid region grows as the liquid solidifies.

If the temperature is above the melting point, the liquid region grows as the solid melts. An accurate melting temperature is obtained by narrowing down this temperature “window” around the melting point. Melting is identiﬁed by a sharp increase

66 in the equilibrium volume as the system passes from solid to liquid. The simulations

are performed under periodic boundary conditions so the number of atoms in the

primary simulation cell must be large enough to eliminate ﬁnite-size eﬀects.

3.3.6 Point Defects

All solids contain defects that disrupt the periodicity of the crystal lattice. The most important point defect in metals is the vacancy [138, 139], which is an atom missing from the lattice. The equilibrium concentration of vacancies cvac increases with temperature T as

f c e−Evac/kBT , (3.72) vac ∝

f where Evac is the vacancy formation energy. As vacancies diﬀuse through a solid due to the thermal motion of the atoms, they must overcome the vacancy migration

m energy barrier Evac to move from one equilibrium site to the next. The rate of vacancy- assisted atomic diﬀusion rvac increases with temperature as

f m r e−(Evac+Evac)/kBT e−Qvac/kBT , (3.73) vac ∝ ≡ where Qvac is the vacancy activation energy. Vacancies play a dominant role in solid- state diﬀusion, and inﬂuence many material processes such as creep and dislocation motion.

The vacancy formation energy is computed in DFT and MD calculations by removing an atom from the lattice of an N-atom supercell. If the unit cell lattice vectors are denoted ai, the supercell lattice vectors are Ai = miai with integer mi.

The system now contains N 1 atoms, and the void left by vacancy results in forces −

67 on the atoms. The atomic positions must be relaxed until the forces are zero. The

supercell lattice vectors are kept ﬁxed at the ideal supercell values to mimic the bulk

solid away from the vacancy. In a real material, the atom removed from the vacancy

location moves to another point in the solid such as the surface [49]. The calculated

N energy of the ideal N-atom supercell Eideal must therefore be corrected by the scale factor (N 1)/N. The vacancy formation energy Ef is calculated by subtracting − vac the energy of the relaxed (N 1)-atom vacancy supercell EN−1 from the corrected − vac energy of the ideal N-atom supercell:

N 1 Ef = EN−1 − EN . (3.74) vac vac − N ideal The periodic boundary conditions employed by DFT and MD codes result in a periodic array of vacancies. The vacancy in the primary simulation cell interacts with its periodic images, so the size of the supercell must be increased to reduce these spurious interactions. The supercell is suﬃciently large when the formation energy

f Evac converges within a required tolerance.

The vacancies in a solid can diﬀuse due to thermal motion of the atoms. An atom

vibrating about its equilibrium lattice site can move into an adjacent vacancy, and the

vacancy will then be located at the previously occupied lattice site. The atom must

m overcome the vacancy migration energy barrier Evac to move between the two sites. In the bcc lattice, two equivalent equilibrium vacancy sites are the body-center location and the corner of the unit cell. The maximum in the energy barrier occurs when the diﬀusing atom is located half-way between these two sites along the body-diagonal of the cubic unit cell. Figure 3.4 depicts vacancy migration in a bcc solid.

68 Figure 3.4: Vacancy migration in the bcc lattice. a) The unit cell of the ideal bcc lattice. The dashed line is one of the body-diagonals of the cubic cell. b) and d) Two equivalent equilibrium vacancy sites. Vacancy migration can be conceived as the motion of the atom at the body-center location in b) along the body-diagonal to the corner site in d). The atom must overcome an energy barrier to move between the two equilibrium sites. The maximum in the energy barrier occurs in c), where the migrating atom is located midway between the two equilibrium sites along the body-diagonal. Atomic relaxations are not shown, and the surrounding unit cells are removed for clarity.

69 The migration barrier can be computed by removing an atom from the lattice,

and moving an adjacent atom to the position corresponding to the maximum in the

energy barrier. The positions of the surrounding atoms must then be relaxed until

they have zero net force. The resulting energy of the (N 1)-atom supercell is EN−1. − mig The energy barrier is measured with respect to the energy of an (N 1)-atom supercell − N−1 with the vacancy at its equilibrium position Evac , so the vacancy migration energy

m barrier Evac is

Em = EN−1 EN−1. (3.75) vac mig − vac

m f The size of the supercell must be increased until Evac converges, as in the case of Evac.

The nudged elastic-band method [140–143] can be used to compute the energy barriers of more complicated processes, where the maximum in the energy barriers cannot easily be determined by symmetry. The vacancy formation and migration energies calculated with an interatomic potential can be compared with DFT calculations and experimental measurements.

A second class of point defects of interest are interstitials. An interstitial is an extra atom inserted into the crystal lattice. When the extra atom is the same as the other atoms in a monoatomic solid, it is called a self-interstitial. Figure 3.5 shows the six important self-interstitials for bcc lattices: a) the 100 dumbbell, b) the 110 h i h i dumbbell, c) the 111 dumbbell, d) the 111 crowdion, e) the octahedral, and f) the h i h i tetrahedral interstitials. Interstitials distort the crystal lattice more than vacancies, resulting in much larger formations energies. The interstitial migration energies are lower than the vacancy migration energy, however. Thus, the concentration of interstitials is low compared to the vacancy concentration and interstitials usually have

70 a much smaller eﬀect on material properties. Interstitials are important in irradi-

ated materials, where high energy collisions can knock large numbers of atoms away

from their equilibrium sites. Interstitial formation energies are computed similar to

the vacancy formation energy, but larger supercells are needed for convergence. Ex-

perimental values for interstitial energies are not available for bcc metals, so DFT

calculations test the accuracy of interstitial properties from interatomic potentials.

3.3.7 Surface Properties

Surface calculations test the ability of a potential to account for large changes in

coordination number from the bulk solid. The formation energies and relaxed geome-

tries of the low-index 100 , 110 , and 111 surfaces are most relevant for cubic { } { } { } crystals. DFT and MD codes used for calculations of solids usually employ periodic

boundary conditions to eliminate surface eﬀects for mimicking a bulk material. Sur-

faces can still be studied within this computational framework by employing a large

supercell with a solid region and a vacuum region outside the surface of interest.

The dimensions of the solid and vacuum must be large enough to remove unwanted

interactions between periodic surface images. Therefore, surface properties must be

tested for convergence by increasing the size of both the solid and the vacuum regions

until the property values don’t change within a required tolerance. Figure 3.6 shows

a schematic supercell geometry for surface calculations.

Surface formation energies are computed by constructing an N-atom periodic supercell by repeating the unit cells of the ideal lattice and adding a vacuum region above the surface of interest. Periodic boundary conditions produce two surfaces in the supercell. The atoms near the surfaces experience forces due to the absence of

71 Figure 3.5: Self-interstitials in the bcc lattice. a) The 100 dumbbell, b) 110 dumbbell, c) 111 dumbbell, d) crowdion, e) octahedral,h andi f) tetrahedralh interstitials.i The interstitialh i atoms are shown in gray. Atomic relaxations are not shown, and the surrounding unit cells are removed for clarity.

72 Figure 3.6: Supercell geometry for surface calculations. The solid region is constructed by replicating the unit cell of the crystal structure. A vacuum region is included to introduce the presence of a surface. The vectors deﬁning the supercell are Ai. DFT and MD calculations for solids are usually carried out under periodic boundary conditions, so surface properties must be tested for convergence with respect to the size of both the solid region and the vacuum region. This ensures that spurious interactions between periodic surface images are suﬃciently reduced.

73 atoms in the vacuum, so their positions must be relaxed until they have zero net force.

The presence of the surfaces increases the energy of the relaxed N-atom supercell with vacuum above the energy of the N-atom ideal supercell. The formation energy of the

(ijk) (ijk) surface Esurf is computed as

EN EN E(ijk) = surf − ideal (3.76) surf 2A(ijk)

N where Esurf is the energy of the relaxed N-atom supercell with the vacuum region,

N (ijk) Eideal is the energy of the N-atom ideal structure, and A is the area of one of the surfaces in the supercell with vacuum. The factor of 2 accounts for the fact that the vacuum region produces two surfaces under periodic boundary conditions. Note that the unit of surface energy is energy/area.

Surface relaxations are also an important test for an interatomic potential. Metal surfaces usually relax inwards, while pair potentials often predict outward expansions.

This is remedied by the many-body EAM and MEAM potentials. Surface relaxations are expressed as the percent change in spacing between atomic planes parallel to the surface, with respect to the corresponding interplanar spacing in the ideal crystal.

(ijk) The relaxation ∆α−β of the adjacent atomic layers α and β parallel to the (ijk) surface is

dsurf dideal (ijk) α−β − α−β ∆α−β = ideal 100%, (3.77) dαβ ×

surf where dα−β is the spacing between the α and β atomic layers in the relaxed supercell

ideal with vacuum, and dα−β is the corresponding spacing in the ideal crystal. The largest

(ijk) change in spacing is usually ∆1−2 for the layers closest to the vacuum, α = 1 and

β = 2. Figure 3.7 shows a schematic representation of surface relaxation. Surface

74 Figure 3.7: Surface relaxation. The spacing between adjacent atomic layers α and β is ideal equal to the spacing in the ideal crystal dα−β before the atomic positions are relaxed. surf The spacings near a surface in a metal usually contract to smaller separations dα−β under relaxation. The change in interplanar distance is greatest for α = 1 and β = 2.

energies are usually measured on polycrystalline samples, providing a value averaged over many surface geometries. Surface relaxations have not been measured for most bcc metal surfaces. Therefore, DFT calculations provide the best direct comparisons for evaluating the accuracy of surface properties of an interatomic potential.

75 3.3.8 Dislocations

Crystals deform under stress, and the resulting behavior generally falls into one of two categories. (1) The crystal will return to its initial state upon unloading if the applied stress is suﬃciently small. This is the elastic behavior discussed in connection

with the elastic constants presented in Subsection 3.3.2. (2) The stress causes the

material to strain past its elastic limit, and when the stress is removed the crystal is

permanently deformed. This is called plastic deformation, and most metals plastically

deform for strains of order 0.01%. Plastic deformation is a complex process involving

line defects in the crystal lattice called dislocations. Dislocations play a dominant

role in plasticity, and inﬂuence a wide variety of materials properties including creep,

ductility, hardness, friction, and crystal growth. This study focuses on the atomic

structure of ideal dislocations, and the stresses required for dislocation motion at zero

temperature. Other aspects of plasticity and dislocations are thoroughly discussed in

many books and review articles, e.g., Refs. [138,144–149].

Early attempts to explain plasticity assumed that the deformation occurs by ho-

mogeneous slipping of atomic planes past each other in ideal crystals [150]. This

requires simultaneously breaking all the atomic bonds across the plane, and leads to

shear stresses 103 to 105 times greater than experimentally observed values. This

large discrepancy led to the proposal that local disturbances in the crystal lattice

called dislocations move under much lower stress, causing the lattice to shear as they

pass through the crystal [151–153].

76 Figure 3.8: Edge dislocation in a simple cubic lattice. (a) A defect-free crystal is divided by the plane p-p′. (b) An edge dislocation is formed by shearing dgho to the left by a lattice translation. The dislocation is the row of atoms in red, perpendicular to the plane of the page (the symbol is used to label edge dislocations). The distortion of the lattice is largest close⊥ to the dislocation line. This region is called the dislocation core.

Dislocation Geometry

The two basic types of dislocations are edge dislocations and screw dislocations.

Figure 3.8 shows a schematic of an edge dislocation in a simple cubic lattice. In

Figure 3.8(a), the ideal crystal is divided by the horizontal plane p-p′. The dislocation is formed by shearing the region dgho to the left by a lattice translation. Figure 3.8(b) shows the lattice after the atoms relax. The dislocation is the edge of the extra half- plane of atoms d- above the plane p-p′. The dislocation is the line of red atoms ⊥ perpendicular to the plane of the page, denoted by . Note that the edge dislocation ⊥ is perpendicular to the shear direction. The dislocation core is the highly distorted region near the dislocation line. Figure 3.9(a) shows a perspective view of the edge dislocation.

77 Figure 3.9: Perspective view of (a) an edge dislocation, (b) a screw dislocation, and (c) a mixed dislocation. In all three cases, the dislocation is shown in red and the sheared area is shown in gray. The dislocations separate the sheared regions of the crystal from the unsheared regions.

A screw dislocation is generated similar to the edge case, except region dgho in

Figure 3.8 is sheared perpendicular to the plane of the page. Figure 3.9(b) shows a perspective view of a screw dislocation in a simple cubic lattice. In contrast to the edge case, the screw dislocation is parallel to the shear direction. A general curved dislocation shown in Figure 3.9(c) has mixed edge-screw character. In all cases, dislocations form the boundary separating a sheared region of the lattice from an unsheared region.

The Burgers Vector

The dislocations shown in Figures 3.8 and 3.9 are schematic and focus on the simple cubic lattice. Dislocations in the bcc lattice are more complicated, and often diﬃcult to visualize. Therefore, it is useful to have a simple, general way to quantify the properties of a dislocation. The Burgers vector b provides such a description. The magnitude of b is the amount by which the crystal is sheared by the dislocation, equal

78 Figure 3.10: Construction of the Burgers vector for an edge dislocation. (a) The direction t of the dislocation is chosen to point out of the page. A closed path SABCF is formed around the dislocation. The path is followed counter-clockwise, as determined by the right-hand rule. (b) A similar path is then constructed in an ideal lattice. Here the path fails to close and the Burgers vector b, drawn from the ﬁnish F to the start S of the path, completes the loop. The Burgers vector of an edge dislocation is perpendicular to the dislocation line.

to a lattice translation. The direction of b gives the shear direction. A dislocation is

fully speciﬁed by its Burgers vector b and line direction.

The Burgers vector is constructed by ﬁrst specifying a unit vector t that is tan-

gent to the dislocation line. There are two possible directions for t and the choice is

arbitrary. Once a direction is chosen, a closed path consisting of lattice translations

is formed around the dislocation. This is the Burgers circuit. The path is traversed

in the direction given by a right-hand rule: the thumb of the right hand points along

t and the ﬁngers curl in the direction of the path. Figure 3.10(a) shows the Burgers

circuit SABCF for a crystal containing an edge dislocation. The same path is then

formed in an ideal lattice. In this case the path fails to close, and the Burgers vector

b completes the circuit. The Burgers vector is chosen to point from the ﬁnish of the

79 Figure 3.11: Construction of the Burgers vector for a screw dislocation. (a) The direction of the dislocation is t. A closed path SABCDEF is formed around the dislocation. The path is followed in the direction given by the right-hand rule. (b) A similar path is then constructed in an ideal lattice. Here the path fails to close and the Burgers vector b, drawn from the ﬁnish F to the start S of the path, completes the loop. The Burgers vector of a screw dislocation is parallel to the dislocation line.

path F to the starting point S. This choice, along with the right-hand rule method for choosing the path direction, is called the ﬁnish-start/right-hand (FS/RH) convention [144, 148]. Figure 3.11 shows the Burgers vector for a screw dislocation. The

Burgers vector is perpendicular to t for edge dislocations and parallel to t for screw dislocations. In the general case of curved dislocations, b is at an angle to t.

Dislocation Motion Under Stress

The slip plane of a dislocation is the plane containing the dislocation’s Burgers

vector and the dislocation line. The motion of a dislocation in the slip plane due to

shear stress is called glide. The local rearrangement of a small number of atomic bonds

80 Figure 3.12: The motion of an edge dislocation under stress. (a)-(f) The motion of the dislocation requires only a relatively small rearrangement of atoms localized near the dislocation core. When the dislocation passes all the way through, the crystal is sheared by a lattice translation b (i.e., the magnitude of the Burgers vector). The same ﬁnal conﬁguration results when an ideal crystal is sheared by a lattice translation, but this required a much larger stress. Edge dislocations move parallel to the direction of the applied stress.

near the dislocation core is required for a dislocation to glide a distance b through

the lattice, opposed to the large number of bonds that would have to be broken for

homogeneous shear of the ideal crystal. Figures 3.12 and 3.13 show the motion of

edge and screw dislocations under shear stress, respectively. The lattice is sheared by

a lattice translation when the dislocation passes through the crystal, as in the ideal

shear case, but the required stress is much lower for crystals with dislocations.

Dislocations usually glide in close packed directions, since this requires minimal

shear and lattice distortion. The slip planes on which the dislocations move are usu-

ally planes of highest atomic density, since these planes have the largest interplanar

81 Figure 3.13: The motion of a screw dislocation under stress. (a)-(c) When the dislocation passes all the way through, the crystal is sheared by a lattice translation b (i.e., the magnitude of the Burgers vector). Screw dislocations move perpendicular to the direction of the applied stress.

spacing and weakest bonding. The close-packed directions in bcc crystals are 111 h i directions, and along these directions the interatomic separations correspond to Burg-

ers vectors b = 1 111 . The primary slip planes in bcc crystals are 110 and 112 2 h i { } { } planes. Edge dislocations have higher mobility than screw dislocations in bcc metals, leading to low concentrations of edge dislocations. Therefore, the plasticity of bcc metals is dictated primarily by b = 1 111 screw dislocations. 2 h i At zero temperature, a dislocation must overcome an energy barrier called the

Peierls barrier Ep to translate through the lattice. This energy barrier is periodic in lattice translations due to the periodicity of the crystal. The Peierls barrier is symmetric when no stress is applied to the crystal, but under shear stress the barrier becomes larger in the direction opposite to the shear and smaller in the direction parallel to the shear. When the shear stress reaches a critical value called the Peierls stress τp, the barrier in the direction parallel to the shear vanishes and the dislocation translates through the lattice. Figure 3.14 illustrates the behavior of the Peierls

82 Figure 3.14: Schematic of the energy barrier Eb to dislocation motion. (a) For zero stress, the barrier is periodic and the dislocation must overcome the Peierls barrier Ep. (b) If an applied shear stress is less than the Peierls stress, the barrier is still present but lower in value. (c) When the stress equals the Peierls stress, the barrier vanishes and the dislocation glides through the crystal.

barrier under stress. At ﬁnite temperatures, thermal ﬂuctuations can assist in moving dislocations for τ <τp.

Schmid’s law [154, 155] states that slip begins when the resolved shear stress on the slip plane in the slip direction reaches a critical value (i.e. the Peierls stress τP), independent of both the orientation of the crystal and non-shear components of the applied stress. If a bcc crystal with slip plane (101)¯ is loaded in pure shear σ on a plane at angle χ to the slip plane, Schmid’s law takes the form

τ σ = P . (3.78) cos χ

83 The plane on which the shear stress is applied is called the maximum resolved shear

stress plane (MRSSP), and the critical value of σ is called the critical resolved shear

stress (CRSS). If the slip plane is (1¯12),¯ χ is still measured with respect to (101)¯ and

Schmid’s law takes the form

τ σ = P , (3.79) cos(χ + 30◦) since the angle between (101)¯ and (1¯12)¯ is 30◦.

Schmid’s law holds well for some metals, e.g. Mg, but breaks down for bcc metals [23–27, 29]. The deviations from Schmid’s law can be traced to the twinning- antiwinning asymmetry of the bcc lattice and the dependence of the CRSS on non- shear components of the applied stress [28, 146]. Twinning is discussed in Subsec- tion 3.3.9 on ideal shear strength.

Elastic Properties of Dislocations

The atomic displacements far from a dislocation core can be modelled using anisotropic elasticity theory. The discussion is limited to screw dislocations since they govern the plastic response of bcc metals. A point r in an ideal lattice moves

to r′ under strain. The displacement of the point is u = r′ r. Classical anisotropic − elasticity predicts that a screw dislocation in a cubic material with Burgers vector

b = bzˆ and line direction zˆ produces atomic displacements

ux = 0,

uy = 0, b y u = tan−1 √A , (3.80) z 2π x 84 where the anisotropy ratio A = 2C /(C C ) [144]. Elasticity theory is not valid 44 11 − 12 near the dislocation core where the atomic displacements are large, so atomic-scale calculations must be used to determine the properties of the core region. However, the elastic solution provides a useful starting conﬁguration for introducing dislocations into a computational simulation cell.

Screw Dislocations in BCC Metals

The 1 111 screw dislocations control the plastic response on bcc metals. Elas- 2 h i ticity theory predicts the long-range displacement ﬁeld due to the dislocation, but

the structure of the highly-distorted core region must be obtained from atomistic

simulations. Classical potentials for bcc metals produce two diﬀerent types of screw

dislocation core structures upon relaxation of the atomic positions, depending on the

detailed nature of the potential functions [148]. Figure 3.15 shows diﬀerential dis-

placement maps [156] of the two types of core structures. Diﬀerential displacement

maps represent the relative atomic displacements in the direction perpendicular to

the plane of the page as arrows pointing between atoms. The atoms are projected

onto the (111) plane, and the lengths of the arrows are scaled such that an arrow

connects two atoms if the diﬀerence in their displacements is b/3. The shadings of

the atoms indicate that there are three repeating layers of atoms in the [111] direction

in the ideal crystal.

The two structures in Figure 3.15(a) have the same energy and are are called

degenerate, or polarized. The structure in Figure 3.15(b) is called non-degenerate,

or symmetric. Some interatomic potentials produce the degenerate core, while other

potentials and DFT produce the symmetric core [148]. In all cases, the core spreads

85 (a)

(b) (c) (110) (011)

[111] (101)

1 Figure 3.15: Diﬀerential displacement maps of the core structures of 2 111 screw dislocations in the bcc lattice. (a) Some potentials produce the degenerateh , ori polarized, core structure. The structures on the left and right have the same energy. (b) Other potentials and DFT produce the nondegenerate, or symmetric, core. (c) In all cases, the dislocation core spreads into three 110 planes of the [111] zone. { } onto three intersecting 110 planes. When symmetric cores are subjected to shear { } stress, they polarize and behave similar to degenerate cores [148].

Computational Modeling of Dislocations

Dislocation calculations test the plastic deformation properties of an interatomic potential. A 1 111 screw dislocation is generated in a computational cell by displac- 2 h i ing the atoms according to Eqn. 3.80, where the [111] cubic lattice direction must be oriented along the zˆ-direction of the simulation cell. Figure 3.16 shows the initial

1 ¯ conditions for a single 2 [111] screw dislocation with (110) slip plane.

86 Figure 3.16: Initial conﬁguration for a bcc screw dislocation with Burgers vector 1 b = 2 [111]. (a) The view along [111]. The dashed line indicates the shear plane. (b) A perspective view shows the planes of atoms spiraling around the dislocation line.

87 Proper boundary conditions are crucial for reliable dislocation simulations. Pe-

riodic boundary conditions applied in the direction parallel to the dislocation line

to model an inﬁnitely long dislocation. The boundary conditions perpendicular to

the dislocation line must be carefully chosen to model an isolated dislocation. The

simulation cell is divided into an outer region in which the atomic positions are ﬁxed

according to the elastic solution Eqn. 3.80, and an inner region in which the atomic

positions are free to move. The atomic positions will smoothly transition from the

highly distorted core region to the continuum elastic region if the inner region is

large enough, eﬀectively mimicking an isolated dislocation in an inﬁnite lattice. Fig-

ure 3.17 shows a schematic of this ﬁxed boundary condition. The stable core structure

is obtained by relaxing the positions of the atoms in the inner region using, e.g., the

conjugate-gradient method [157,158].

The critical resolved shear stress (CRSS) for dislocation motion is determined

for diﬀerent values of the angle χ between the maximum resolved shear stress plane

(MRSSP) and the slip plane by straining the simulation cell containing the dislocation.

The strain is increased in small increments, and the atoms in the inner region are allowed to relax after each increase in strain. The dislocation moves when the resulting stress reaches the CRSS for the given value of χ. Classical potentials can easily

handle the large number of atoms necessary when using ﬁxed boundary conditions,

while DFT calculations are limited to much smaller numbers of atoms and require

specialized boundary conditions.

88 Figure 3.17: Fixed boundary conditions for the simulation of a single dislocation. The atoms in Region I are allowed to move, while the atoms in Region II are ﬁxed at the positions dictated by elasticity theory. The region outside Region II is vacuum.

89 3.3.9 Ideal Shear Strength

The maximum shear stress a defect-free crystal can sustain without plastically deforming is the ideal shear strength. Real metals usually deform at much lower stresses due to the presence of defects such as dislocations, but the ideal shear strength is an important quantity in plasticity and fracture theories [159,160] and serves as a valuable test of deformation properties of classical potentials.

A commonly used measure of the ideal shear strength [160] involves the formation of an ideal mechanical twin. A twin is a region of the crystal lattice that is rotated with respect to the parent lattice, but retains the crystal structure of the parent.

In the simplest cases, the parent and twin are mirror images of each other across a boundary called the twinning plane. Figure 3.18 shows a schematic twin formed by shearing the lattice.

Twinning is an important plastic deformation mode for metals, and observed twins in bcc crystals result from shearing on 112 planes in 111 directions. Figure 3.19 { } h i shows a (112)[1¯11]¯ twin in a bcc lattice formed by shifting successive (112) planes by

1 ¯¯ √ 6 [111], corresponding to a shear of stw = 1/ 2. Shearing in the opposite direction is called antitwinning, which requires greater stress than twinning [160].

The ideal bcc crystal is generated from the primitive lattice vectors

a a = ( xˆ + yˆ + zˆ), 1 2 − a a = (xˆ yˆ + zˆ), 2 2 − a a = (xˆ + yˆ zˆ). (3.81) 3 2 −

90 Figure 3.18: Schematic of mechanical twinning. (a) An ideal crystal subjected to shear stress. (b) A deformed crystal, in which a portion of the crystal called a twin is rotated with respect to the parent crystal lattice. (c) An atomic view of twinning. The atom positions in the parent lattice are gray, and the atom positions in the twin are black. The twin is a mirror image of the parent across the twinning plane A-B.

91 Figure 3.19: Twinning in the bcc lattice. The twinning plane is (112) and the twinning ¯¯ 1 ¯¯ direction is [111]. The twin is formed by shifting successive (112) planes by 6 [111], corresponding to a shear of 1/√2. Black atoms are in the plane of the page, and gray atoms lie above and below the page.

92 The shear deformation corresponding to twinning is eﬀected by straining the primitive

vectors as [160]

1 x a1 a1 + ( xˆ yˆ + zˆ), → 6 stw − − 1 x a2 a2 + ( xˆ yˆ + zˆ), → 6 stw − − a a . (3.82) 3 → 3

As the strain parameter x varies from 0 to stw and the crystal changes into a twin, the energy E(x) increases from the ideal bcc value to a maximum at x = stw/2, and

then decreases back to the ideal value at x = stw. The corresponding shear stress

τ(x) is

1 dE(x) τ(x)= , (3.83) V0 dx

where V0 is the atomic volume. The ideal shear strength is deﬁned as the maxi-

mum shear stress along the twinning deformation path of the crystal. The energy is

computed with DFT or a classical potential at a number of values of x, and the result-

ing energy versus strain values are numerically diﬀerentiated to determine the shear

stress. A large number of x-values, of order 1000, is needed to obtain smooth τ(x)

curves. The ideal shear strength produced by a classical potential is tested against

DFT calculations.

3.3.10 Generalized Stacking Faults

Ideal crystal structures consist of atoms arranged on periodic lattice sites. An

equivalent picture of the crystal structure is atoms arranged in planes that are stacked

93 in a well-deﬁned sequence. Figure 3.20(a) shows a unit cell of the bcc structure, high-

lighting the (110) lattice plane. Figure 3.20(b) shows that a bcc crystal is generated by

stacking 110 planes in the sequence ABABAB... Alternatively, Fig. 3.21 shows that { } the lattice is generated by stacking 112 planes in the sequence ABCDEFABCDEF... { } Common plastic-deformation modes in the bcc refractory metals include the slipping

of 110 or 112 lattice planes past each other via the motion of 1 111 screw dislo- { } { } 2 h i cations, and the formation of mechanical twins under shear on 112 planes in 111 { } h i directions.

As planes of atoms slide past each other during plastic deformation, the normal

stacking sequence of the planes is disrupted creating a stacking fault. As the planes of atoms slip past each other, the energy of the crystal changes periodically due to the periodicity of the lattice. A plot of the excess energy with respect to the ideal structure per unit fault area versus displacement is a two-dimensional surface called a

γ-surface [161], or generalized stacking fault. Since bcc metals slip in 111 directions h i on 110 or 112 planes, the 111 sections through the 110 and 112 γ-surfaces { } { } h i { } { } are most relevant for plastic deformation. Stacking fault energies inﬂuence the motion

and structure of dislocations and twins (see, e.g., Refs. [19,144,148,149]), and oﬀer a

measure of the ideal shear strength of a material (see, e.g., Ref. [162]).

Generalized stacking fault energies are computed by sliding two halves of an ideal

crystal with respect to each other in a given direction on a given plane. The fault

energies are calculated for incremental displacements over one period of the fault

plane. DFT and MD codes utilize periodic boundary conditions for simulating bulk

solids, so the fault energies are computed in a periodic supercell geometry. If the

supercell lattice vectors are ﬁxed and one axis is normal to the fault plane, Fig. 3.22

94 Figure 3.20: The bcc lattice generated by stacking 100 lattice planes. (a) The ideal unit cell, showing the (110) lattice plane as shaded.{ The} atoms in alternating (110) planes are labeled A and B, respectively. (b) View of the bcc lattice above the (110) plane. The atoms in B layers are shown smaller than atoms in A layers for clarity, and lie at a distance a/√2 above and below the A atoms. The bcc crystal is generated by stacking 110 planes in the sequence ABABAB... { }

95 Figure 3.21: The bcc lattice generated by stacking 112 lattice planes. (a) The ideal unit cell, showing the (112)¯ lattice plane as shaded.{ The} atoms in alternating (112)¯ planes are labeled A, B, C, D, E, and F, respectively. (b) View of the bcc lattice above the (110) plane, showing edges of the (112)¯ planes. The atoms labeled B, D, and F lie at a distance a/√2 above and below the atoms labeled A, C, and E. The bcc crystal is generated by stacking 112 planes in the sequence ABCDEFABCDEF... { }

96 shows that the simulation cell will contain two faults separated by half the length

of the simulation cell. An alternative is tilting the supercell lattice vector that is

initially normal to the fault plane in the undeformed crystal by the slip distance.

This eﬀectively doubles the distance between the periodic images of the stacking

fault, reducing the number of atoms required to achieve numerical convergence of

the fault energies. The positions of the atoms in the direction normal to the fault

plane can be relaxed, but full relaxation will cause atoms to relax to their ideal

positions since the faults are unstable in bcc crystals. Stable stacking faults have

not been experimentally observed in bcc metals, and calculations based on DFT and

classical potentials conﬁrm the absence of local minima on the 110 and 112 γ- { } { } surfaces [148]. The bcc stacking fault energies predicted by classical potentials are therefore tested against DFT calculations.

97 Figure 3.22: Geometry of computational cells used for stacking fault calculations. (a) The ideal crystal, showing the fault plane and slip direction. (b) If the supercell lattice vector x3 initially normal to the fault plane is ﬁxed, the distance between faults is d under periodic boundary conditions. The crystal in the primary simulation cell is shown with solid lines, and the periodic image above it is shown with dashed lines. (c) If the supercell lattice vector x3 is tilted as the crystal halves are shifted, the distance between faults is 2d.

98 Chapter 4: Results and Applications of Classical Potentials

The force-matching method [22] discussed in the last chapter is applied to developing cubic spline-based EAM and MEAM potentials for the bcc refractory metals V, Nb, Ta, Mo, and W. The ﬁtting data used to construct the potentials consists of forces, energies, and stresses for various conﬁgurations of atoms, computed via the DFT plane-wave code vasp. The PBE generalized gradient approximation

(GGA) functional accounts for electronic exchange and correlation energy, and projec-

tor augmented-wave (PAW) pseudopotentials represent the interaction of the nuclei

and core electrons with the valence electrons. The convergence criteria discussed in

Subsection 3.2.2 and Appendix A ensure numerically converged results. The ﬁtted po-

tentials compute the wide-ranging physical properties introduced in Subsections 3.3.1-

3.3.10, and comparison of the results to DFT calculations and experimental data tests

the accuracy of the potentials.

The adjustable parameters in cubic spline-based interatomic potentials are the

number of spline knots in each function, the values of the spline knots, the ﬁtting

weights Wi in the error function Eqn. 3.6, the configurations included in the fitting database, and the outer cutoff radii of the radial functions. The smallest interatomic separation in the fitting database fixes the inner cutoff radii of the radial functions.

The optimal spline knot values for a given set of the other adjustable parameters

99 minimize the error function Eqn. 3.6. Searching the parameter space for accurate po-

tentials requires varying the adjustable parameters, and minimizing the error function

for each set of parameters. The initial values of the knots must be speciﬁed before

optimization begins. The initial choice for the embedding function F is a quadratic

dependence on n, and the initial choice for the remaining functions is a linear dependence on their respective arguments. This initial condition imposes minimal bias on the ﬁnal optimized values.

Parameter Space for EAM and MEAM Potentials

The number of knots in each potential function can be varied along with the ﬁtting database, set of weights, and cutoﬀ radii. The following example shows that a full systematic exploration of the EAM and MEAM parameter spaces is not possible with current methods. If the number of knots in each of the three function in an EAM potential is varied from, say, 5 to 20, this results in 163 = 4, 096 candidates for the optimal number of knots. The global optimizers that minimize the error function

Eqn. 3.6 often get stuck in local minima corresponding to unphysical potentials, so

10-100 independent trials should be performed with different seeds for the random number generator that dictates the initial search direction in parameter space. This corresponds to 40,960-409,600 potentials that must be thoroughly tested for each choice of fitting data, weights, and cutoffs. The situation is much worse for MEAM potentials, since there are five functions and 165 = 1, 048, 576 candidate choices for the optimal number of knots. Additionally, MEAM potentials require roughly 100 times more computational effort to optimize than EAM potentials. A larger upper bound for the maximum number of knots leads to even more possibilities.

100 The large number of candidate potentials coupled to the inﬁnite freedom in the

choice of fitting data, cutoff radii, and fitting weights leads to an optimization problem

of overwhelming complexity. Progress can only be made by restricting the search for

optimal potentials to a small subspace in the full parameter space. A simple method to

determine a reasonable number of spline knots involves computing the error function

in Eqn. 3.6 for the ﬁtting database, and for an independent testing database of DFT

calculations. If the error for the testing database is much larger than the error for

the ﬁtting database, there are likely too many knots in the potential [87,163,164].

Comparisons of physical properties computed with the potentials to DFT and

experiment further test for the optimal number of knots. Potentials with too few

knots show large deviations from DFT and experiment, and too many knots creates

a large number of local minima in the parameter space corresponding to unphysical

potentials. Tests of thousands of potentials with diﬀerent choices of the adjustable

parameters show that potentials ﬁt to low-pressure data with 3-10 knots for the em-

bedding function and 12-20 knots for the other functions produce similar errors for

the ﬁtting and testing databases, and yield good results for physical properties. Po-

tentials ﬁt to low- and high-pressure data require more knots due to the increased

complexity of the ﬁtting data. High-pressure potentials with 5-10 knots for the em-

bedding function and 25-35 knots for the other functions typically yield good results.

Increased weighting of the energies and stresses relative to the forces improves physical properties. Energy and stress weights of 100 relative to force weights of

1 yield physically reasonable potentials with force errors less than 25%. Potentials generated using signiﬁcantly smaller energy and stress weights produce properties

101 that do not agree well with DFT and experiment, and larger weights do not improve the ﬁts.

Potentials are also constructed for different values of the radial cutoffs. The cutoff radii are increased incrementally, with each increase leading to the inclusion of an extra neighbor shell of atoms in the ideal bcc structure. The bcc MEAM and EAM potentials investigated in this study behave differently with increasing radial cutoff.

Larger values of the cutoﬀ radii lead to improved results for MEAM potentials, at the expense of higher computational cost. The quality of EAM potentials increases only up to the inclusion of third nearest-neighbor interactions. Longer-ranged EAM potentials usually improve the energy values of higher-energy crystal phases at the expense of the properties of the bcc phase. MEAM potentials do not have this limitation since they have explicit three-body terms that account for the variety of bonding angles present in these crystal structures.

The fitting database is initially confined to data directly relevant for a small number of properties. The lattice constant, bulk modulus, and pressure-volume curve are determined by supercells constructed from the bcc primitive cell for a range of volumes around the equilibrium volume. A bcc supercell with an atom removed con- strains the vacancy formation energy, and strained supercells constrain the elastic constants. Finite temperature MD snapshots for the bcc and liquid phases sample a variety of local atomic environments, leading to accurate forces for classical MD simulations and thermal properties. The potentials resulting from this fitting data are tested on other properties such as interstitial formation energies, stacking fault

102 curves, and surface properties. DFT data is added to the ﬁtting database, if necessary, to improve the properties not well described by the initial ﬁtting data.

Potentials for the BCC Metals

The ﬁrst section in this chapter presents results for an EAM potential for Nb [1].

The potential successfully predicts a wide range of thermal and mechanical properties at low pressure, including surface properties, thermal expansion, melting temperature, and stacking fault energies. The next section presents a MEAM potential for Mo [13], which is also successful at predicting Mo properties at low-pressure. The Nb EAM potential was used by a research group at Los Alamos National Laboratory to simulate the behavior of Nb under high-pressure shock loading conditions [19]. This study reveals that the behavior of the potential is qualitatively correct to 50 GPa, but the stacking fault energies under pressure are too low compared to DFT.

Analysis of thousands of EAM and MEAM potentials generated from low-pressure

DFT data reveals that high-pressure DFT data is required in the fitting database to produce potentials capable of modeling the bcc metals at both low and high pressure. The analysis also shows that high-pressure bcc properties and the energies and structures of higher-energy crystal phases from MEAM calculations agree much better with DFT than the corresponding EAM calculations. These properties are not only important for high-pressure simulations, but are also relevant for alloys that can exist in different crystal phases depending on the thermodynamic conditions. The last section of this chapter presents results for MEAM potentials fit to low- and high- pressure DFT data that show the MEAM formalism is capable of accurately modeling the bcc metals over a wide range of pressures.

103 The force-matching method has been used to develop a Ta potential [88], but

the work presented in this thesis is the ﬁrst application of the method to Nb, Mo,

W, and V. This thesis also presents the ﬁrst work using the force-matching method

to develop potentials for high-pressure applications. These single-element potentials

lay the groundwork for developing accurate potentials for alloys containing the bcc

refractory metals.

4.1 Force-Matched EAM Potential for Nb

The EAM potential for Nb [1] is constructed by ﬁtting the cubic spline knots to

the forces, energies, and stresses for the atomic conﬁgurations listed in Table 4.1. The

optimization is performed by the simulated annealing algorithm in the potfit code.

The number of knots in each function, the outer cutoff radii of φ(r) and ρ(r), the weights W in the error function ( k ) (Eqn. 3.6), and the configurations in the i Z { } fitting database are varied until reasonable potentials are obtained.

The configurations are generated from ab initio MD simulations using the vasp code. The fitting database is constructed in two steps. First, the MD simulations are used to generate realistic atomic trajectories for the simulation conditions listed in the table. These calculations use a relatively low convergence criteria to reduce the computational burden. A single k-point is used, and Ecut is set to the default value of 219.927 eV from the vasp pseudopotential file. The MD simulations run for 400 steps with a 3 fs time-step.

The second step in constructing the fitting data uses the fully converged k-point grid and energy cutoff to determine accurate forces, energies, and stresses for the atomic configurations from the final step of the MD simulations. The fitting data

104 Conﬁguration Structure Natoms T (K) V/V0 Shear Strain 1 bcc, primitive 125 300 0.90 None 2 bcc, primitive 125 300 1.00 None 3 bcc, primitive 125 300 1.10 None 4 bcc, primitive, vacancy 124 300 1.00 None 5 bcc, conventional 128 300 1.00 2%, M − 6 bcc, conventional 128 300 1.00 1%, M − 7 bcc, conventional 128 300 1.00 1%, M 8 bcc, conventional 128 300 1.00 2%, M 9 bcc, conventional 128 300 1.00 2%, O − 10 bcc, conventional 128 300 1.00 2%, O 11 bcc, primitive 125 1200 1.00 None 12 bcc, primitive 125 2200 1.00 None 13 liquid, primitive 125 5000 1.00 None 14 fcc, primitive 125 300 1.00 None 15 hcp, conventional 128 300 1.00 None

Table 4.1: Configurations in the DFT database used to construct the EAM potential for Nb [1]. The “Structure” column lists the crystal structure of each configuration. The “primitive” and “conventional” labels indicate if the supercell is based on the one-atom primitive or the two-atom conventional cell. The liquid configuration starts as a bcc lattice and then melts during the ab initio MD simulation. The “Natoms” column lists the number of atoms in each configuration. The “T ” column lists the temperature of the ab initio MD simulations used to generate the configurations. The “V/V0” column lists the ratio of the supercell volume to the zero-temperature, equilibrium DFT supercell volume. For configuration 13, V0 is the equilibrium volume of the bcc supercell. The “Shear Strain” column indicates the shear strain applied to the supercells, where M and O denote the volume-conserving monoclinic and orthorhombic strains defined in Subsection 3.3.2, respectively.

105 (a) (b)0.5 (c) -3.0 2.0 0.4 1.5 Pair potential "Electronic density" -3.5 Embedding function 0.3 1.0 -4.0 (eV) ρ

0.2 (eV) 0.5 φ

Spline knots F -4.5 0.0 0.1 Cubic polynomials 0.0 -0.5 -5.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.0 0.2 0.4 0.6 0.8 1.0 r (A) r (A) n

Figure 4.1: The three cubic splines of the EAM potential for Nb [1]. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. (a) The pair potential φ and (b) the density ρ are functions of the distance r between pairs of atoms. Both of these functions have 17 optimized spline knots and a cutoﬀ radius of 4.750A,˚ which includes ﬁrst-, second-, and third-nearest neighbor interactions in the ideal bcc lattice. (c) The embedding function F depends on the local density n. Eight optimized spline knots parameterize F .

consists of 1,895 forces (5,685 Cartesian force components), 15 energy values, and 90

stress tensor components for the conﬁgurations in Table 4.1. Figure 4.1 shows the

optimized splines for the three functions comprising the EAM potential. The pair

potential φ and the density ρ contain 17 optimized spline knots and have an outer cutoﬀ radius of 4.750 A.˚ The embedding function F contains 8 optimized knots, and

ρ is scaled such that n lies in the range 0 n 1. Appendix B lists the values of ≤ ≤ the spline knots.

Fitting Errors

The ﬁtting errors produced by the potential for the quantities in the ﬁtting database indicate the quality of the potential. The errors associated with numerically small DFT data are typically greater than the errors from larger data. Since

106 very small values are inherently inaccurate, weighting the terms in the error calcula-

tions by the magnitudes of the DFT values produces a more relevant measure of the

quality of a potential. The weighted relative RMS deviation of the energies, stresses,

or force magnitudes is

N Q QEAM QDFT 2 ∆Q = ω i − i 100%, (4.1) RMS v i QDFT × u i=1 i uX where Qi is an energy, a stresst tensor component, or a force magnitude, and NQ is the respective number of such quantities in the ﬁtting database. The scaled magnitudes of the DFT data ωi weight the terms in the sum:

DFT Qi ωi = | | . (4.2) NQ QDFT j=1 | j | The weighted RMS error of the energiesP is only 0.1%. The diagonal components of the stress tensors are also accurately reproduced with a 6% weighted RMS error.

In contrast, the weighted RMS error for the oﬀ-diagonal components of the stress

tensors has the much larger value of 307%. The oﬀ-diagonal values are very small

however, even for the strained supercells. Despite this large error, the elastic con-

stants produced by the potential are reasonable. The weighted RMS deviation of the

force magnitudes is 25%. The weighted average angular deviation of the EAM force

directions from the DFT force directions is 15.1◦, which is determined from

atoms

θavg = ωiθi, (4.3) i=1 X where θi is the angle between the EAM force on atom i and the DFT force on atom

i, and ωi is the corresponding scaled DFT force magnitude.

107 Physical Properties for Testing the Potential

In addition to reasonable ﬁtting errors, the validity of an interatomic potential

for modeling a material is determined by how well it describes physical properties. A

basic requirement for a potential is the ability to stabilize the correct ground-state

structure, and accurately reproduce the lattice constant a, bulk modulus B, and

elastic constants C11, C12, and C44 of this structure. Table 4.2 shows that the Nb

EAM values of a, B, C11, C12, and C44 for the bcc structure compare well with DFT

calculations and experimental data. The table also lists the energies and structural

parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti crystal structures. The β-W and

β-Ta structures are metastable phases of W and Ta, respectively, and ω-Ti is a high- pressure phase of Ti. Comparison of the EAM values to DFT results shows that the

EAM potential does not predict highly accurate values for these structures, but it does predict that they all have higher energies than the bcc lattice. These results, coupled with ﬁnite temperature MD simulations of the bcc structure that produce no phase transition, strongly suggest that the bcc structure is the ground-state of the

EAM potential.

Figure 4.2(a) shows the thermal expansion of the EAM potential from 0 K to

exp the experimental melting temperature Tmelt = 2742 K. The curve is computed using NP T -ensemble MD simulations of 8,192 atoms at P = 1 atm, for 138 temperatures in the range 0

108 Property EAM [1] GGA-PBE [1] Experiment

Ecoh (eV/atom) 7.09 7.10 7.57 a (A)˚ 3.308 3.309 3.303 B (GPa) 172 172 173

C11 (GPa) 244 251 253

C12 (GPa) 136 133 133

C44 (GPa) 32 22 31

∆Efcc−bcc (meV/atom) 187 324 ...

afcc (A)˚ 4.157 4.217 ...

∆Ehcp−bcc (meV/atom) 187 297 ...

ahcp (A)˚ 2.940 2.867 ...

chcp (A)˚ 4.800 5.238 ...

∆EβW−bcc (meV/atom) 77 104 ...

aβW (A)˚ 5.280 5.296 ...

∆EβTa−bcc (meV/atom) 105 83 ...

aβTa (A)˚ 10.200 10.184 ...

cβTa (A)˚ 5.313 5.371 ...

∆EωTi−bcc (meV/atom) 167 201 ...

aωTi (A)˚ 4.845 4.887 ...

cωTi (A)˚ 2.735 2.678 ...

Table 4.2: Structural and elastic properties of Nb. The EAM values for the cohesive energy Ecoh, lattice parameter a, bulk modulus B, and elastic constants C11, C12, and C44 of bcc Nb are compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2]. The experimental lattice parameter from Ref. [3] and the experimental elastic constants from Ref. [4] were measured at 4.2 K. The EAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure.

109 potential developed by Guellil and Adams [15] which underestimates the thermal expansion.

Figure 4.2(b) shows the pressure variation of the EAM potential as a function of the relative volume V/V0, where V0 is the zero-pressure volume. The curve is computed using NP T -ensemble MD simulations of 8,192 atoms at T = 293 K, for 50 pressures in the range 0

The spectrum of phonon frequencies as a function of wave-vector is determined using the phon code. Figure 4.3 compares the EAM results to DFT calculations and experimental data [17] in the [ξ00], [ξξξ], and [ξξ0] directions in reciprocal space. The

DFT values are computed using an 8 8 8 supercell based on the bcc primitive cell. × × The DFT results closely match experiment over most of the Brillouin zone, but the transverse modes in the [ξ00] direction show a plateau around ξ = 0.25 not present in the experimental data. This discrepancy is not physical, but rather an artifact of the interpolation scheme used by phon in generating the curves. The phonon frequencies are computed exactly at only a small number of points in the Brillouin zone, and phon interpolates between these exact points to generate smooth curves. More exact points

(i.e., supercells larger than 8 8 8) are required to remove this discrepancy. This × × is easily accomplished with the EAM potential, but is prohibitively expensive with

DFT. The EAM potential accurately matches the experimental phonon frequencies for

110 okare elwt xeietadDTcluain rm0 from calculations EAM DFT The versus and experiment pressure [16]. with Nb experiments well shock (b) agrees from work it. are underestimates data whi experimental [15] The expansion, Adams thermal and the Guellil overestimates of slightly work this in oeta eeoe nti ok[]are lsl ihexpe expansi with thermal closely The agrees temperature melting [1] curve. experimental work the expansion this in thermal developed Nb potential (a) 4.2: Figure (a) (b)

Pressure (GPa) 100 ( L − L 0 ) / L 0 (%) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 20 40 60 80 0 . . . 1.0 0.9 0.8 0.7 0 0010 0020 3000 2500 2000 1500 1000 500 0 Relative volume Experimental datafit EAM, GuellilandAdams EAM, thisworkat1atm Temperature (K) T melt exp 111 72K h A oeta developed potential EAM The K. 2742 = Experimental data GGA−PBE EAM, 293K V / 0 iet[4 rm0Kto K 0 from [14] riment eteEMpotential EAM the le o7 GPa. 75 to oeta nthis in potential no h EAM the of on ouecurve. volume tm o nesiil.Teaoi oiin r eae using v relaxed are for positions atoms atomic The 249 interstitials. use for calculations atoms 31, DFT and The properties, vacancy properties. for terstitial atoms 8,191 with supercells use Hu develop and potentials [15] EAM Adams the than bu closely N, more and H experiment boundaries match zone the at featur agreement the poor of in some results This reproduce to unable is but vectors, wave small phon lta around plateau pcrm oee,teptnilipoe pnteEMpotenti agree ca EAM Hu fully [1] the and to study upon [15] unable improves this Adams potential is in the but However, wave-vectors, developed spectrum. small potential for EAM [17] dir The periment high-symmetry along zone. curves dispersion louin phonon Nb 4.3: Figure vrmc fteBilunzn.Tetases oe nte[ the in modes transverse The zone. Brillouin the of much over Frequency (THz) al . it on eeteege o b h A calculatio EAM The Nb. for energies defect point lists 4.3 Table . 0.5 0.0 2 4 6 8 Γ oet eeaetecurves. the generate to code ξ 0 = [ξ 00] tal. et tal. et . 5 hc sa riato h neplto ceeue ythe by used scheme interpolation the of artifact an is which 25, [18]. 1] h F hnnrslscoeymthexperiment match closely results phonon DFT The [18]. 1.0 Η

112 EAM, thiswork GGA−PBE, thiswork EAM, Huetal. EAM, GuellilandAdams Experimental data [ξ ξξ] 0.5 h conjugate-gradient the tr h vibrational the pture ξ

0 ieto hwa show direction 00] 0.0 si h spectrum. the in es Γ cin nteBril- the in ections 5 tm o in- for atoms 251 cnis n 251 and acancies, h A results EAM the t db uli and Guellil by ed l fGelland Guellil of als elwt ex- with well s si hswork this in ns [ξ ξ0] 0.5 Ν 2 4 6 8 f m f f f f f f Method Evac Evac Qvac E100 E110 E111 Ecrd Eoct Etet EAM [1] 3.10 0.77 3.87 4.50 3.83 4.09 4.02 4.36 4.37 GGA-PBE [1] 2.72 0.55 3.27 4.76 4.31 3.95 3.99 4.89 4.56 Experiment 2.60-3.10 0.60-1.60 3.20-4.70 ...... EAM [15] 2.88 0.97 3.85 ...... EAM [18] 2.76 0.64 3.40 4.44 4.39 4.74 4.93 4.43 4.73 F-S [165] ...... 4.13 3.99 ... 4.10 4.23 4.26 F-S [166] ...... 4.821 4.485 4.795 4.857 ...... F-S [167] 2.48 0.91 3.39 4.85 4.54 4.88 4.95 4.91 4.95 MEAM [168] 2.75 0.54 3.29 ...... MEAM [169] 2.75 0.57 3.32 ... 2.56 ......

Table 4.3: Nb point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumbbell, 111 dumbbell, 111 crowdion, octahedral, andh tetrahedrali interstitialsh i in eV. Theh val-i ues producedh byi the Nb EAM potential developed in this study are compared to DFT calculations and results from other published EAM, Finnis-Sinclair (F-S), and MEAM potentials. The experimental vacancy formation and migration energy values are from Ref. [5], and the experimental vacancy activation energy values are from Refs. [5–9].

method [157,158] in all cases. The table compares the results for our EAM and DFT

calculations to other published EAM [15, 18], Finnis-Sinclair (F-S) [165–167], and

MEAM [168, 169] potentials. The experimental measurements produce a wide range

of values, and most of the results are in reasonable agreement with the experimental

data and the DFT calculations.

Surface properties are presented in Table 4.4. The EAM calculations use slab-

geometry supercells with 600 atomic layers under periodic boundary conditions. A

large vacuum region is introduced to generated two free surfaces. The DFT calcula-

tions are performed using 24-, 36-, and 48-layer slabs for the 100 surface, and 12-, { } 18-, and 24-layer slabs for the 110 and 111 surfaces. A vacuum region 10 A˚ thick { } { }

113 separates the periodic surface images. The atoms in the EAM and DFT calculations

are relaxed using the conjugate-gradient method. The diﬀerent numbers of layers

in the DFT calculations reveal the convergence of the DFT values with respect to

system size. The energy values for the diﬀerent numbers of layers vary by 1 meV/A˚2

or less, and change by less than 0.2 meV/A˚2 when the vacuum thickness is increased to 15 A.˚ The DFT surface relaxation values show a variation of 3.1% or less as the number of layers change. The EAM and DFT surface calculations performed in this study agree very well for both energy values and relaxations. Most of the results from previously published EAM [15, 18], F-S [170], long-range empirical potential

(LREP) [171], MEAM [93,169], and modiﬁed analytic EAM (MAEAM) [172] studies are in reasonable agreement with DFT.

The method of coexisting phases determines the melting temperature for the

EAM potential. The NP T -ensemble MD simulation supercells contain at least 16,500 atoms, and each simulation runs for 5,000,000 steps with a 1 fs time-step. The equilibrium volume for each temperature value is determined by averaging over the last

5,000 simulation steps. Simulation supercells with 130,000 atoms produce the same melting temperature as the 16,500-atom simulations, indicating that the results are converged with respect to system size. Figure 4.4 shows the equilibrium volume as a function of temperature for P = 1 atm. The volume of the system changes abruptly upon melting at 2686 5 K. This melting temperature is in excellent agreement with ± the experimental melting temperature of 2742 K, with an error of only 2%.

Figure 4.5 shows sections through the 112 and 110 γ-surfaces in the 111 { } { } h i direction. The EAM calculations use supercells with 60,000 atoms to determine the unrelaxed and relaxed γ-surface energies. The supercell for the 112 γ-surface { }

114 {110} {110} {100} {100} {111} {111} Method Esurf ∆12 Esurf ∆12 Esurf ∆12 EAM [1] 127 -5.0(600) 147 -13.9(600) 154 -27.0(600) GGA-PBE [1] 131 -3.9(12) 146 -12.4(24) 149 -30.7(12) -3.9(18) -13.0(36) -28.4(18) -4.5(24) -12.3(48) -27.6(24) EAM [15] 113 -1.6 123 +0.52 ...... EAM [18] 108 ... 120 ...... F-S [170] 104 -5.1 122 -16.0 ...... LREP [171] 112 ... 131 ... 146 ... MEAM [93] 117 ... 174 ... 126 ... MEAM [169] 155 -7.3 169 -12.5 182 -35.5 MAEAM [172] 110 ... 125 ... 143 ...

Table 4.4: Energies and relaxations of low-index surfaces of bcc Nb. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The numbers in parentheses next to our relaxation values give the number of atomic layers in the supercells. The values produced by the EAM potential developed in this work are compared to DFT calculations and results from other published EAM, F-S, long- range empirical potential (LREP), MEAM, and modiﬁed analytic EAM (MAEAM) potentials. The potential developed in this work closely matches the DFT values for both energies and relaxations.

115 20.6 = 2686 K at 1 atm 20.4 Tmelt

20.2 3 20.0

19.8

Volume (A / atom) 19.6

19.4 2600 2700 2800 2900 Temperature (K)

Figure 4.4: Two-phase melting simulations determine the melting temperature of the Nb EAM potential developed in this work [1]. Initially, half the simulation cell contains liquid Nb and the other half contains bcc Nb. The liquid region of the simulation cell solidiﬁes below the melting temperature, and the solid region melts above the melting temperature. The ﬁgure shows the equilibrium volume for each simulation temperature. There is a sharp increase in volume upon melting. The EAM potential predicts a melting temperature Tmelt = 2686 K, which is within 2% of the exp experimental value Tmelt = 2742 K.

116 has 3,000 atomic layers, and the supercell for the 110 γ-surface has 2,000 atomic { } layers. In the relaxed EAM calculations, the atoms are allowed to move only in

the direction perpendicular to the fault plane, since the stacking faults are unstable.

The DFT calculations use supercells with 24 atomic layers for the 112 γ-surface { } calculations, and 12 atomic layers for the 110 calculations. The supercell axis { } initially perpendicular to the fault plane is tilted by the slip distance to eﬀectively

reduce the number of necessary atomic layers by half. The EAM results agree well

with the DFT calculations, but the DFT curves are smoother than the EAM curves.

However, relaxation smooths some of the bumpiness present in the unrelaxed EAM

curves.

The EAM potential produces the degenerate core structure for 1 111 screw dis- 2 h i locations, as Figure 4.6(a) shows. The core structure is determined by conjugate-

gradient relaxation of a 900,000-atom supercell. The atoms are arranged in a cylin-

drical slab oriented such that the x-axis is along the [121]¯ direction, the y-axis is along

[101],¯ and the z-axis is along [111]. The radius of the cylinder is 60 nm, and has 15

(111) planes in the z-direction. The atoms are initially displaced according to the anisotropic elastic strain ﬁeld produced by a screw dislocation. Periodic boundary conditions are applied in the [111] direction. Atoms greater than 58 nm from the center of the cylinder are ﬁxed and atoms less than 58 nm from the center are allowed to relax.

Shear stress applied to the crystal in the maximum resolved shear stress plane

(MRSSP) at an angle χ to the (101)¯ plane causes the dislocation to move when the

stress reaches the critical resolved shear stress (CRSS) for dislocation motion. The

dislocation moves in the (1¯12)¯ plane for all values of χ between 30◦ and +30◦, −

117 (a) 0.08 1.28 EAM relaxed 0.07 {112} EAM unrelaxed 1.12 ) 2 0.06 GGA−PBE unrelaxed 0.96

0.05 0.80 ( J/m (eV/A 0.04 0.64 2 0.03 0.48 ) 0.02 0.32

Fault energy 0.01 0.16 0.00 0.00 0.0 0.2 0.4 0.6 0.8 1.0 Normalized displacement along 111 (b) 0.08 1.28 0.07 {110} 1.12 ) 2 0.06 0.96

0.05 0.80 ( J/m (eV/A 0.04 0.64 2 0.03 0.48 ) 0.02 0.32

Fault energy 0.01 0.16 0.00 0.00 0.0 0.2 0.4 0.6 0.8 1.0 Normalized displacement along 111

Figure 4.5: γ-surface sections of Nb in the 111 direction [1]. The absence of minima indicate that there are no stable stacking faultsh i in the 112 and 110 planes along this direction. The DFT curves are smoother than the{ EAM} curves,{ but} the overall agreement is good.

118 (a) (110)

(211)

(101)

(112)

(011) (121) (b) (011) (112)

(101)

(211)

(110)

1 Figure 4.6: Diﬀerential-displacement maps for the 2 [111] screw dislocation core structure produced by the Nb EAM potential developed in this study [1]. (a) The degenerate core structure results when the atoms are relaxed. (b) Shear stress applied along [111] produces a net displacement of the dislocation in the (1¯12)¯ plane. The ﬁgure shows the core structure when the stress is just above the critical resolved shear stress for dislocation motion.

as Figure 4.6(b) shows. Slip on 112 and 110 planes has been experimentally { } { } observed in Nb single crystals [173–175]. The variation of the CRSS with χ is shown in Figure 4.7, which reveals the breakdown of the Schmid law for χ greater than about

15◦.

The Nb EAM potential developed in this work accurately models a wide range of physical properties at low to moderate pressures. It has been applied by other research groups to study diverse phenomena in Nb including intrinsic localized phonon

119 (112) (101) (211) 4.0

3.5

3.0

2.5

2.0

1.5 Schmid law

1.0 Critical resolved shear stress (GPa) -30 -20 -10 0 10 20 30 The angle between MRSSP and (101) (degrees)

1 Figure 4.7: The critical resolved shear stress (CRSS) for 2 [111] screw dislocation motion in Nb as a function of the orientation of the maximum resolved shear stress plane (MRSSP), computed with the EAM potential developed in this study [1]. The ﬁgure shows a deviation from Schmid’s law when the angle between the MRSSP and the (101)¯ plane is greater than about +15◦.

120 modes [176,177], melting [178], and shock-induced plastic deformation [19]. The shock

simulation study reveals that the potential produces qualitatively correct behavior at

high pressure, but it may not be adequate for quantitative high-pressure calculations.

The shock results are presented in Section 4.3, and potentials developed for high-

pressure applications are presented in Section 4.4.

4.2 Force-Matched MEAM Potential for Mo

The MEAM potential for Mo developed in this work [13] is constructed by ﬁtting the cubic spline knots to a database of DFT forces, energies, and elastic constants.

The EAM potential for Nb was ﬁt to data from relatively large supercells (124–

128 atoms) at finite temperatures. Accordingly, the Nb database contains a large number of force values (5,685 Cartesian force components). Analysis shows that accurate potentials can also be obtained by fitting to databases that replace some of the large finite-temperature supercells with smaller zero-temperature supercells.

This is advantageous due to the reduced computational burden associated with both generating the DFT database and optimizing the potentials. The ﬁtting database for the Mo MEAM potential contains the zero-temperature relaxed 53-atom vacancy supercell, zero-temperature 1-atom primitive bcc cells with lattice parameters in the range 2.95 A˚ a 3.35 A,˚ and the zero temperature bcc elastic constants. Addi- ≤ ≤ tionally, the database contains three 125-atom supercells at 1270 K (bcc structure),

2320 K (bcc structure), and 5270 K (liquid), respectively. The weighted RMS force magnitude error for the MEAM potential is 9%, and the weighted average angular deviation of the force directions in 5.5◦. Figure 4.8 shows the optimized splines for

121 Figure 4.8: The five cubic splines of the MEAM potential for Mo [13]. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ has 13 knots and a cutoff radius of 5.900 A,˚ the density ρ and three-body term f have 12 knots and a cutoff radius of 5.100 A,˚ the angular term g has 8 knots, and the embedding function F has 3 knots.

the ﬁve MEAM functions, and Appendix B lists the values of the spline knots.

Physical Properties for Testing the Potential

As with the Nb EAM potential, the Mo MEAM potential is tested by thorough calculations of a wide range of physical properties. The size of the supercells, time- steps and simulation durations for MD runs, atomic relaxation method, etc. used to compute physical properties are similar to the EAM calculations discussed in the

122 previous section. Table 4.5 lists the equilibrium lattice parameter, bulk modulus, and

elastic constants of the bcc phase, along with the energy values of other candidate

crystal structures. The MEAM values for the bcc phase agree well with experiment

and DFT. The MEAM potential does not predict highly accurate values for the other

crystal structures, but it does predict that they all have higher energies than the

bcc phase. These results, coupled with ﬁnite temperature MD simulations of the

bcc structure that produce no phase transformation, strongly suggest that the bcc

structure is the stable ground-state of the MEAM potential. The ﬁtting databases

used to construct the high-pressure MEAM potentials presented in Section 4.4 include

data for these high-energy structures, and the potentials accurately reproduce the

DFT results. EAM potentials ﬁt to data for multiple crystal structures are not able

to mimic the DFT results, indicating that the angular terms in MEAM are important

for modeling multiple phases.

Figure 4.9(a) shows the pressure-volume curve of the MEAM potential. The

MEAM result closely matches DFT and data from shock experiments [16] from 0 to 35 GPa. Beyond this pressure, the MEAM curve develops a small kink around

55 GPa. Figure 4.9(b) compares the thermal expansion of the MEAM potential to experiment [14]. The potential agrees well with experiment up to 2000 K. Two-phase melting simulations produce a melting temperature of T MEAM = 3220 10 K, which melt ± exp deviates about 11% from the experimental value Tmelt = 2900 K. The phonon spectrum produced by the MEAM potential is displayed in Fig-

ure 4.10. The phonon frequencies are plotted as a function of wave-vector along

high-symmetry directions in the Brillouin zone. The MEAM results closely match

123 Property MEAM [13] GGA-PBE [13] Experiment

Ecoh (eV/atom) 6.82 6.25 6.82 a (A)˚ 3.167 3.169 3.147 B (GPa) 253 263 270

C11 (GPa) 441 462 479

C12 (GPa) 158 163 165

C44 (GPa) 96 102 108

∆Efcc−bcc (meV/atom) 391 418 ...

afcc (A)˚ 3.931 4.013 ...

∆Ehcp−bcc (meV/atom) 415 433 ...

ahcp (A)˚ 2.743 2.765 ...

chcp (A)˚ 4.692 4.905 ...

∆EβW−bcc (meV/atom) 266 96 ...

aβW (A)˚ 5.026 5.058 ...

∆EβTa−bcc (meV/atom) 280 168 ...

aβTa (A)˚ 9.719 9.752 ...

cβTa (A)˚ 5.048 5.113 ...

∆EωTi−bcc (meV/atom) 332 404 ...

aωTi (A)˚ 4.616 4.681 ...

cωTi (A)˚ 2.595 2.572 ...

Table 4.5: Structural and elastic properties of Mo. The MEAM values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Mo are compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [10], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure.

124 ( a ) 1 0 0 E x p e r i m e n t D F T - P B E 8 0 )

a M E A M P

G 6 0 (

e r

u 4 0 s s e r

P 2 0

0 0 . 8 0 . 9 1 . 0 V / V 0 ( b ) 1 . 5 E x p e r i m e n t a l d a t a f i t M E A M

) 1 . 0 % (

L 0 . 5

∆

0 . 0 500 1000 1500 2000 T e m p e r a t u r e ( K )

Figure 4.9: (a) Mo pressure versus volume curve. The experimental data are from shock experiments [16]. The MEAM potential in this work agrees well with experiment and DFT calculations from 0 to 35 GPa. (b) Mo thermal expansion curve. The thermal expansion of the MEAM potential agrees closely with experiment [14] from 0 to 2000 K.

125 8 8 )

z 6 6 H T (

c 4 4 n e q u e r 2 E x p e r i m e n t 2 F D F T - P B E M E A M Γ [ξ00] ΗΝ[ξξξ] Γ [ξξ0]

Figure 4.10: Mo phonon dispersion curves along high-symmetry directions in the Brillouin zone. The MEAM potential agrees well with experiment [17] and DFT.

DFT and experimental measurements [17], reﬂecting the accuracy of the force match-

ing and elastic constants.

Tables 4.6 and 4.7 present point defect and surface properties, respectively. The

point defect values are compared to experimental data and published DFT [13, 179,

180], F-S [181], and model generalized pseudopotential theory (MGPT) potential [182]

calculations. Most of the results agree reasonably well with DFT and experiment, with

the exception of the MGPT interstitial energies. The MEAM surface calculations are

compared to DFT calculations and other published MEAM results [169]. The MEAM

results from this work agree reasonably well with DFT, but the energetic ordering

of the surfaces is not the same as DFT. The energy ordering of the other MEAM

potential also disagrees with DFT.

The 112 and 110 γ-surface sections in the 111 direction are plotted in Fig- { } { } h i ure 4.11. The MEAM results agree well with the DFT calculations. The MEAM

126 f m f f f f f f Method Evac Evac Qvac E100 E110 E111 Ecrd Eoct Etet MEAM [13] 2.96 1.64 4.60 7.82 7.68 7.66 7.64 8.07 8.20 GGA-PBE [13] 2.79 1.22 4.01 8.90 7.66 7.52 7.52 9.05 8.47 Experiment [5] 3.00-3.24 1.35-1.62 4.35-4.86 ...... GGA-PBE [179] 2.96 1.28 4.24 9.00 7.58 7.42 7.42 9.07 8.40 GGA-PBE [180] ...... 8.77 7.51 7.34 7.34 8.86 8.20 F-S [181] 2.5 1.3 3.8 7.2 7.0 7.3 7.2 7.6 7.6 MGPT [182] 2.9 1.6 4.5 16.3 10.9 14.2 13.9 17.5 14.9

Table 4.6: Mo point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumbbell, 111 h i h i h i dumbbell, 111 crowdion, octahedral, and tetrahedral interstitials in eV. The values produced byh thei Mo MEAM potential are compared to experimental data, DFT calculations, and results from other published DFT, Finnis-Sinclair (F-S), and model generalized pseudopotential theory (MGPT) potential calculations.

{110} {110} {100} {100} {111} {111} Method Esurf ∆12 Esurf ∆12 Esurf ∆12 MEAM [13] 164 -4.5 180 -11.0 201 -23.6 GGA-PBE [13] 174 -4.4 200 -12.3 186 -20.8 MEAM [169] 180 -3.3 195 -3.3 211 -14.0

Table 4.7: Energies and relaxations of low-index surfaces of bcc Mo. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the MEAM potential developed in this work are compared to DFT calculations and results from another published MEAM potential.

127 1 2 0 ( a ) ( 1 1 2 ) 1 0 0 8 0

6 0 ) 2 Å / 4 0 M E A M r e l a x e d V

e M E A M u n r e l a x e d

m 2 0 D F T - GGA u n r e l a x e d (

gy r

e 1 2 0

n ( b ) ( 0 1 1 ) e

lt 1 0 0 u a F

8 0 6 0 4 0 2 0

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 R e l a t i v e d i s p l a c e m e n t a l o n g [111]

Figure 4.11: γ-surface sections of Mo in the 111 direction. The absence of minima indicate that there are no stable stacking faultsh ini the 112 and 110 planes along this direction. The MEAM and DFT results agree well,{ even} thoug{ h} no data from conﬁgurations with stacking faults is used to construct the potential.

energies slightly underestimate the DFT values and the MEAM 112 curve has less { } asymmetry than the DFT curve, but the overall the results match well. Figure 4.12 shows the reasonable agreement of the ideal shear strength predicted by the MEAM model with the corresponding DFT result. Part (a) of the ﬁgure displays the energy barrier for twinning deformation, and (b) shows the corresponding shear stress computed from the derivative of the energy barrier.

The MEAM potential produces the symmetric structure for the core of the 1 111 2 h i screw dislocation, as Figure 4.13(a) shows. Similar core structures are produced in

128 0 . 4

) ( a ) V e (

0 . 3 W

r c e i r r a

b 0 . 2

g y r e

n 0 . 1 D F T - GGA

E M E A M

ؔ 1 0 c ( b ) ) a P 5 G (

s s

e 0 r x t

s c

r - 5 e a h S - 1 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 R e l a t i v e s h e a r x

Figure 4.12: The ideal shear strength of Mo. The MEAM results compare well with DFT calculations. (a) The energy barrier for twinning W (x). (b) The shear stress τ(x) is the derivative of W (x).

129 DFT [183,184] and tight-binding [185] calculations. Figure 4.13(b) shows the variation

of the critical resolved shear stress (CRSS) for dislocation motion as a function of

the orientation of the maximum resolved shear stress plane (MRSSP). The screw

dislocation moves in the (101)¯ plane when the shear stress reaches the CRSS for all

orientations of the MRSSP between 30◦ and +30◦. The ﬁgure shows that the CRSS − deviates from the prediction of the Schmid law. For a MRSSP angle of 0◦, the MEAM value for the CRSS of 1.79 GPa is in excellent agreement with the DFT value of 1.8

GPa [184].

The MEAM potential accurately models a wide range of Mo properties at low to moderate pressures. High-pressure tests of the potential presented in Section 4.3 reveal that it may not be adequate for reliable high-pressure simulations. Potentials developed for high-pressure applications are presented in Section 4.4

4.3 Application of the Nb EAM Potential to Shock Simula- tions

A research group at Los Alamos National Laboratory simulated shock-wave induced deformation in Nb using ﬁve diﬀerent published interatomic potentials [19].

They tested the EAM potential developed in this work [1] (labeled F.EAM in Fig- ures 4.14-4.16), an EAM potential by Johnson and Oh [101] (labeled J.EAM), an EAM potential by Demkowicz and Hoagland [186] (labeled D.EAM), an extended Finnis-

Sinclair potential by Dai et al. [187] (labeled D.EFS), and a modiﬁed Finnis-Sinclair potential by Ackland and Thetford [166] (labeled A.MFS). All of these potentials are constructed from low-pressure data, and all of them are based on analytic functions except the F.EAM force-matched cubic-spline potential developed in this work.

130 (a) (110) (121) (011)

(211) (112)

(101) (101)

(112) (211)

(011) (121) (110) (b) (112) (101) (211) 2 . 6

2 . 4

2 . 2

2 . 0

1 . 8 Schmid law CRSS (GPa) 1 . 6 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 Angle between MRSSP and (101) (degree)

1 Figure 4.13: (a) Diﬀerential-displacement maps for the 2 [111] screw dislocation core structure produced by the Mo MEAM potential. The non-degenerate core structure results when the atoms are relaxed. (b) Critical resolved shear stress for dislocation motion in the (101)¯ plane.

131 Shock waves are generated by slamming a piston with velocity up against simulation supercells containing 2.5 million Nb atoms. The value of up for each potential

is chosen just large enough to induce structural changes in the samples. The F.EAM

and A.MFS potentials produce regions that undergo mechanical twinning, while the

J.EAM, D.EAM, and D.EFS potentials produce regions that transform from the bcc

structure to the face-centered tetragonal (fct) structure. The twinning deformation is

experimentally observed in shocked Nb samples, while the bcc to fct transition is an

unphysical artifact of the J.EAM, D.EAM, and D.EFS potentials [19]. Figure 4.14

shows deformed supercell corresponding to these two processes.

Recall from Subsection 3.3.9 that mechanical twins in bcc metals are formed by

shearing the lattice in the 112 plane by 1 111 . The Los Alamos researchers exam- { } 6 h i ined the 112 generalized stacking faults produced by the ﬁve potentials, and traced { } the two classes of defects formed by shock waves to diﬀerences in the fault energies

near 1 111 . Figure 4.15 shows the 112 stacking fault sections in the 111 direc- 6 h i { } h i tion produced by the ﬁve potentials at P = 0 GPa. The F.EAM potential developed

in this work closely matches DFT over the full displacement range, and the A.MFS

potential agrees reasonably well with DFT. The J.EAM, D.EAM, and D.EFS poten-

tials produce local minima in their stacking fault energy curves which disagrees with

DFT. The potentials behave diﬀerently in the middle of the curves, but all produce

similar stacking fault energies at a displacement corresponding to the twinning shear

1 111 . 6 h i Figure 4.16 shows the stacking fault energy curves at P = 50 GPa. The F.EAM

and A.MFS energies increase with pressure and develop no local minima in qualitative

agreement with DFT, but the increase in energy is too low compared to DFT. The

132 Figure 4.14: Defect structures formed in Nb under high-pressure shock loading using five different published potentials. (a) The F.EAM potential developed in this work [1] and the A.MFS potential produce deformation twins under shock loading. Atoms with bcc coordination are colored blue, and atoms within the twin boundaries are colored red. The atoms colored yellow correspond to steps on the twin boundaries 1 associated with b = 6 111 twinning dislocations. (b) The J.EAM, D.EAM, and D.EFS potentials produceh i regions that undergo a bcc to face-centered tetragonal (fct) structural phase transformation under shock loading (see the text for the naming convention used for the potentials). Atoms with fct coordination are colored green. The atoms colored yellow correspond to the boundary region between the bcc and fct structures. Twinning is experimentally observed in Nb samples subjected to shock waves, while the bcc-fct transition in unphysical. (c) and (d) show magnifications of the defect structures in (a) and (b), respectively. The figure is from Reference [19].

133 Figure 4.15: 112 γ-surface section in the 111 direction at P = 0 GPa computed with five different{ } published potentials andh DFT.i The Nb EAM potential developed in this work is labeled F.EAM. The figure is from Reference [19].

J.EAM, D.EAM, and D.EFS potentials develop local maxima in their stacking fault energies at the twinning displacement. This stabilizes the fct structure, leading to the bcc to fct transformation produced by these potentials in shock simulations.

The Nb EAM potential developed in this work produces the same qualitative behavior as DFT for the pressure dependence of the 112 stacking fault energy to 50 { } GPa, which leads to the expected twinning deformation under shock loading. The

potential does not agree quantitatively with the DFT results, so the next section

presents potentials that accurately model the pressure dependence of properties rel-

evant to mechanical deformation.

134 Figure 4.16: 112 γ-surface section in the 111 direction at P = 50 GPa computed { } h i with five different published potentials and DFT. The Nb EAM potential developed in this work is labeled F.EAM. The figure is from Reference [19].

4.4 Potentials for High-Pressure Applications

This section develops force-matched MEAM potentials for Nb, Mo, W, V, and

Ta from ﬁtting databases containing DFT forces, energies, and stresses from atomic conﬁgurations at low- and high-pressures. The cubic spline knots are optimized using a genetic algorithm. The potentials compute many properties that agree well with

DFT and experiment over a wide range of pressures and temperatures, providing accurate models of the bcc metals for diverse applications. The potentials introduced in this section are called high-pressure potentials to distinguish them from the low- pressure potentials for Nb and Mo presented in Sections 4.1 and 4.2, respectively.

The ﬁtting database for each high-pressure potential contains: (1) 17 bcc primitive cells with volumes in the range 0.40V V 1.20V ; (2) two 125-atom 0,bcc ≤ ≤ 0,bcc

135 bcc supercells at ﬁnite temperatures and one 125-atom liquid supercell at ﬁnite tem-

perature; (3) the 53-atom unrelaxed vacancy supercell and the 53-atom unrelaxed

vacancy migration supercell; (4) the 48-atom supercells with unrelaxed 100 , 110 , { } { } and 111 surfaces; (5) three 48-atom supercells corresponding to different points { } along the 110 γ-surface section in the 111 direction at P = 0 GPa, and three { } h i similar configurations at P = 100 GPa; (6) five 36-atom supercells corresponding to different points along the 112 γ-surface section in the 111 direction at P = 0 GPa, { } h i and five similar configurations at P = 100 GPa; (7) three 30-atom β-Ta unit cells with volumes in the range 0.95V V 1.05V , three 8-atom β-W unit 0,β−Ta ≤ ≤ 0,β−Ta cells with volumes in the range 0.95V V 1.05V , three 3-atom ω-Ta 0,β−W ≤ ≤ 0,β−W unit cells with volumes in the range 0.95V V 1.05V , seven 2-atom hcp 0,ω−Ti ≤ ≤ 0,ω−Ti unit cells with volumes in the range 0.85V V 1.15V , three fcc primitive 0,hcp ≤ ≤ 0,hcp cells with volumes in the range 0.95V V 1.05V ; (8) 20 2-atom bcc unit 0,fcc ≤ ≤ 0,fcc cells with monoclinic strain for pressures in the range 0 GPa P 100 GPa, and 20 ≤ ≤ 2-atom bcc unit cells with orthorhombic strain for pressures in this pressure range;

(9) eight bcc primitive cell corresponding to diﬀerent points along the ideal shear twinning path for each of the ﬁve pressures 0 GPa, 25 GPa, 50 GPa, 75 GPa, and

100 GPa.

The following subsections compare the properties of the high-pressure Mo, W, V,

Nb, and Ta MEAM potentials to the low-pressure potentials, DFT calculations, and experimental data. The high-pressure potentials provide a better overall description of low-pressure properties than the low-pressure potentials. The high-pressure potentials also properly describe the pressure dependence of the generalized stacking faults and ideal shear strength. The low-pressure potentials produce reasonable values for these

136 properties at P = 0 GPa, but they do not agree with DFT at high pressure. The high-pressure potentials are labeled MEAM-HP in the following tables and ﬁgures.

4.4.1 High-Pressure MEAM Potential for Nb

Figure 4.17 shows the optimized splines of the high-pressure Nb MEAM potential.

The radial functions φ, ρ, and f each have 30 spline knots and an outer cutoﬀ radius of 6.915 A.˚ The inner cutoﬀ for the radial functions is 2.073 A.˚ The embedding function F has 7 knots and the angular term g has 35 knots. The high-pressure

MEAM potentials require more knots than the low-pressure MEAM potential since the ﬁtting databases for the high-pressure potentials contain more data than the low-pressure database.

Table 4.8 compares bcc properties computed with the high-pressure Nb MEAM potential to DFT calculations, experimental data, and the low-pressure Nb EAM results discussed in Section 4.1. The table also compares the energies and lattice parameters of higher-energy structures computed with the high-pressure potential to low-pressure EAM and DFT calculations. The bcc properties computed with the

EAM and MEAM potentials agree well with DFT and experiment. The high-pressure

MEAM values for the higher-energy crystal structures agree much better with DFT than the low-pressure EAM potential results.

Figure 4.18 compares the pressure versus volume curves and thermal expansion curves of the high-pressure Nb MEAM potential to the low-pressure EAM potential.

The high-pressure model closely matches the pressure-volume results from experiment to 175 GPA and DFT to over 400 GPa, while the EAM potential deviates greatly

137 Figure 4.17: The ﬁve cubic splines of the high-pressure MEAM potential for Nb. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.915 A,˚ the angular term g has 35 knots, and the embedding function F has 7 knots.

138 Property MEAM-HP GGA-PBE [1] Experiment EAM [1]

Ecoh (eV/atom) 7.10 7.10 7.57 7.09 a (A)˚ 3.311 3.309 3.303 3.308 B (GPa) 173 172 173 172

C11 (GPa) 246 251 253 244

C12 (GPa) 137 133 133 136

C44 (GPa) 23 22 31 32

∆Efcc−bcc (meV/atom) 323 324 ... 187

afcc (A)˚ 4.218 4.217 ... 4.157

∆Ehcp−bcc (meV/atom) 298 297 ... 187

ahcp (A)˚ 2.853 2.867 ... 2.940

chcp (A)˚ 5.305 5.238 ... 4.800

∆EβW−bcc (meV/atom) 105 104 ... 77

aβW (A)˚ 5.298 5.296 ... 5.280

∆EβTa−bcc (meV/atom) 83 83 ... 105

aβTa (A)˚ 10.196 10.184 ... 10.200

cβTa (A)˚ 5.368 5.371 ... 5.313

∆EωTi−bcc (meV/atom) 201 201 ... 167

aωTi (A)˚ 4.897 4.887 ... 4.845

cωTi (A)˚ 2.669 2.678 ... 2.735

Table 4.8: Structural and elastic properties of Nb. The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Nb compared to DFT, experiment, and the low-pressure EAM results. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [3], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT and low-pressure EAM results. The energies are relative to the energy of the bcc structure. The MEAM and EAM potentials produce accurate results for the bcc structure, but only the MEAM values for the other crystal structure properties agree well with DFT.

139 f m f f f f f f Method Evac Evac Qvac E100 E110 E111 Ecrd Eoct Etet MEAM-HP 2.91 0.39 3.30 5.27 5.10 4.16 4.17 5.27 5.38 GGA-PBE [1] 2.72 0.55 3.27 4.76 4.31 3.95 3.99 4.89 4.56 Experiment [5] 2.60-3.10 0.60-1.60 3.20-4.70 ...... EAM [1] 3.10 0.77 3.87 4.50 3.83 4.09 4.02 4.36 4.37

Table 4.9: Nb point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumbbell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedralh i interstitialsh i in eV. Theh valuesi producedh byi the high-pressure Nb MEAM potential are compared to the low-pressure Nb EAM potential results and DFT calculations. The MEAM vacancy results agree better with DFT than the EAM values, but the EAM interstitial results agree better with DFT than the MEAM values.

above 75 GPa. Both potentials produce thermal expansion curves in good agreement with the experimental data.

Figure 4.19 compares the phonon spectra of the low-pressure Nb EAM potential to the high-pressure MEAM results. The MEAM potential agrees with experiment and

DFT better than the EAM potential, especially near the Brillouin zone boundaries H and N. The improved phonon spectrum of the MEAM potential reﬂects the ability of

MEAM to better reproduce the forces in the ﬁtting database than the EAM potential does.

Tables 4.9 and 4.10 compare the Nb point defect and surface properties of the low-pressure EAM and high-pressure MEAM potentials to DFT, respectively. The

MEAM vacancy energy values are closer to the DFT results than the EAM values, but both potentials produce reasonable vacancy properties. The EAM interstitial and surface properties agree better with DFT than MEAM, but again both potentials produce reasonable values.

140 Figure 4.18: The Nb pressure versus volume and thermal expansion curves of (a) the high-pressure MEAM potential, and (b) the low-pressure EAM potential. The pressure-volume curve of the high-pressure Nb MEAM potential is in excellent agreement with experiment and DFT to over 400 GPa, while the low-pressure EAM potentials deviates drastically above 75 GPa. The thermal expansion of both the high- pressure MEAM potential and low-pressure EAM potential agree well with experiment. Note that the experimental data is shown as points in the pressure-volume ﬁgures, but as a solid line in the thermal expansion ﬁgures.

141 Figure 4.19: The Nb phonon dispersion computed with (a) the low-pressure EAM potential, and (b) the high-pressure MEAM potential. The MEAM potential shows better overall agreement with DFT and experiment than the EAM potential. The MEAM potential produces zone-boundary frequencies at H and N that closely match the DFT and experimental values, in contrast to the much lower EAM values at these points.

{110} {110} {100} {100} {111} {111} Method Esurf ∆12 Esurf ∆12 Esurf ∆12 MEAM-HP 135 -2.7 155 -3.7 163 -10.8 GGA-PBE [1] 131 -3.9 146 -12.4 149 -30.7 EAM [1] 127 -5.0 147 -13.9 154 -27.0

Table 4.10: Energies and relaxations of low-index surfaces of bcc Nb. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT and low-pressure EAM calculations. The EAM potential agrees better with DFT than the MEAM potential, but both potentials produce reasonable values.

142 Figure 4.20: The pressure variation of the 112 and 110 γ-surface sections in the 111 direction. (a) The low-pressure Nb{ EAM} potential{ } [1] agrees reasonable wellh withi DFT for P = 0 GPa, but is unable to produce the correct behavior at elevated pressures. (b) The high-pressure Nb MEAM potential closely matches the DFT calculations over a broad range of pressures.

Figure 4.20 compares the pressure variation of the 112 and 110 γ-surfaces { } { } from 0 GPa to 100 GPa produced by the low-pressure EAM and high-pressure MEAM potentials to DFT calculations. Figure 4.21 compares the pressure variation of the ideal shear strength of the two potentials. The high-pressure potential shows excellent agreement with the DFT calculations, while the low-pressure potential in not capable of modeling the pressure dependence of these properties.

143 Figure 4.21: The pressure variation of the ideal shear strength. (a) The low-pressure Nb EAM potential does not agree with DFT calculations for any pressure. (b) The high-pressure Nb MEAM potential closely matches the DFT results over a broad range of pressures.

144 This subsection shows that the low-pressure EAM potential produces accurate results for many low-pressure properties of the bcc phase, but fails to produce the correct energies of higher-energy crystal structures. The EAM potential also fails at describing the pressure variation of the stacking fault energy curves and the ideal shear strength. The fitting database for the EAM potential does not contain any configurations with the β-Ta, β-W, or ω-Ti structures. EAM potentials fit to bcc data and data from these structures produce better results for these structures, but at the expense of bcc properties. Likewise, EAM potentials fit to low-pressure data and high- pressure stacking fault and ideal shear strength data produce better high-pressure results at the expense of the low-pressure properties. The MEAM potential agrees well with DFT and experiment for low-pressure properties, high-pressure properties, and higher-energy crystal structures properties. MEAM potentials contain explicit three- body interactions that accurately model a broader range of structures and material behavior than simpler EAM potentials.

4.4.2 High-Pressure MEAM Potential for Mo

Figure 4.22 shows the optimized splines of the high-pressure Mo MEAM potential.

The radial functions φ, ρ, and f each have 30 spline knots and an outer cutoﬀ radius of

6.623 A.˚ The inner cutoﬀ for the radial functions is 2.011 A.˚ The embedding function

F has 7 knots and the angular term g has 35 knots.

Table 4.11 compares bcc properties computed with the high-pressure Mo MEAM potential to DFT calculations, experimental data, and the low-pressure Mo MEAM potential discussed in Section 4.2. The table also compares the energies and lattice parameters of higher-energy structures computed with the high-pressure potential to

145 Figure 4.22: The ﬁve cubic splines of the high-pressure MEAM potential for Mo. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.623 A,˚ the angular term g has 35 knots, and the embedding function F has 7 knots.

146 low-pressure MEAM and DFT calculations. All the properties listed in the table

produced by the high-pressure potential agree well with DFT and experiment, and

the properties of the higher-energy crystal structures is greatly improved by the high-

pressure MEAM potential.

Figure 4.23 compares the pressure versus volume curves and thermal expansion

curves of the high-pressure MEAM potential to the low-pressure MEAM potential.

The high-pressure model accurately matches the pressure-volume results from exper-

iment to 200 GPa and DFT to 800 GPa, which is a great improvement over the

low-pressure potential. The thermal expansion of the low-pressure model agrees well

with experiment to 2000 K. The high-pressure potential shows reasonable agreement

with experiment, but overestimates the experimental data. Figure 4.24 compares the

phonon spectra of the low- and high-pressure Mo MEAM potentials. Both potentials

produce results that closely match DFT and experiment over much of the Brillouin

zone.

Tables 4.12 and 4.13 compare the point defect and surface properties of the low-

and high-pressure Mo MEAM potentials to DFT, respectively. The vacancy and

surface results for the high-pressure MEAM potential are in excellent agreement with

DFT. The energetic ordering of the surfaces is correct for the high-pressure potential,

while the low-pressure potential has the incorrect ordering for the 100 and 111 { } { } surfaces. Both potentials produce reasonable values for the interstitial energies.

Figure 4.25 compares the pressure variation of the 112 and 110 γ-surfaces from { } { } 0 GPa to 100 GPa produced by the low- and high-pressure MEAM potentials to DFT calculations. Figure 4.26 compares the pressure variation of the ideal shear strength of the two potentials. The high-pressure potential shows excellent agreement with

147 Property MEAM-HP GGA-PBE [13] Experiment MEAM [13]

Ecoh (eV/atom) 6.25 6.25 6.82 6.82 a (A)˚ 3.169 3.169 3.147 3.167 B (GPa) 263 263 270 253

C11 (GPa) 462 462 479 441

C12 (GPa) 164 163 165 158

C44 (GPa) 106 102 108 96

∆Efcc−bcc (meV/atom) 417 418 ... 391

afcc (A)˚ 4.012 4.013 ... 3.931

∆Ehcp−bcc (meV/atom) 431 433 ... 415

ahcp (A)˚ 2.770 2.765 ... 2.743

chcp (A)˚ 4.888 4.905 ... 4.692

∆EβW−bcc (meV/atom) 94 96 ... 266

aβW (A)˚ 5.057 5.058 ... 5.026

∆EβTa−bcc (meV/atom) 164 168 ... 280

aβTa (A)˚ 9.751 9.752 ... 9.719

cβTa (A)˚ 5.113 5.113 ... 5.048

∆EωTi−bcc (meV/atom) 403 404 ... 332

aωTi (A)˚ 4.689 4.681 ... 4.616

cωTi (A)˚ 2.563 2.572 ... 2.595

Table 4.11: Structural and elastic properties of Mo. The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Mo compared to the low-pressure MEAM, DFT and experiment. The experimental value for the cohesive energy is from Ref. [10], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure.

148 Figure 4.23: The pressure-volume and thermal expansion curves of the low-pressure Mo MEAM potential [13] in (a) are compared to results from the high-pressure Mo MEAM potential in (b). The high-pressure potential shows excellent agreement with experiment and DFT to 800 GPa, while the low-pressure potential agrees well to about 35 GPa (see Fig. 4.9). The thermal expansion of the low-pressure model agrees well to 2000 K. The thermal expansion of the high-pressure model is reasonable, but overestimates the experimental curve. Note that the experimental data is shown as points in the pressure-volume ﬁgures, but as a solid line in the thermal expansion ﬁgures.

149 Figure 4.24: The phonon dispersion of the low-pressure Mo MEAM potential [13] in (a) is compared to results from the high-pressure Mo MEAM potential in (b). Both of the potentials agree well with DFT and experiment.

f m f f f f f f Method Evac Evac Qvac E100 E110 E111 Ecrd Eoct Etet MEAM-HP 2.73 1.22 3.99 7.53 7.73 7.54 7.52 7.99 7.71 GGA-PBE [13] 2.79 1.22 4.01 8.90 7.66 7.52 7.52 9.05 8.47 Experiment [5] 3.00-3.24 1.35-1.62 4.35-4.86 ...... MEAM [13] 2.96 1.64 4.60 7.82 7.68 7.66 7.64 8.07 8.20

Table 4.12: Mo point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumbbell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedralh i interstitialsh i in eV. The valuesh i produced byh thei high-pressure Mo MEAM potential are compared to DFT calculations, experimental data, and the low-pressure MEAM values.

{110} {110} {100} {100} {111} {111} Method Esurf ∆12 Esurf ∆12 Esurf ∆12 MEAM-HP 173 -3.7 200 -10.1 189 -25.2 GGA-PBE [13] 174 -4.4 200 -12.3 186 -20.8 MEAM [13] 180 -3.3 195 -3.3 211 -14.0

Table 4.13: Energies and relaxations of low-index surfaces of bcc Mo. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations and results from the low-pressure MEAM potential.

150 Figure 4.25: The pressure variation of the 112 and 110 γ-surface sections in the 111 direction from the low-pressure Mo MEAM{ } potential{ } [13] in (a) are compared h i to results from the high-pressure Mo MEAM potential in (b). The high-pressure potential agrees closely with DFT, and the low-pressure potential shows large deviations from the DFT calculations.

the DFT calculations, while the low-pressure potential in not capable of accurately modeling the pressure dependence of these properties.

This subsection shows that the low-pressure MEAM potential produces accurate results for many low-pressure properties of the bcc phase, but fails to produce the correct energies of higher-energy crystal structures. The low-pressure MEAM potential also fails at describing the pressure variation of the stacking fault energy curves

151 Figure 4.26: The pressure variation of the ideal shear strength of the low-pressure Mo MEAM potential [13] in (a) is compared to results from the high-pressure Mo MEAM potential in (b). The high-pressure potential agrees closely with DFT, and the low-pressure potential shows large deviations from the DFT calculations.

152 and the ideal shear strength. The ﬁtting database for the low-pressure MEAM po-

tential does not contain any configurations with the fcc, hcp, β-Ta, β-W, or ω-Ti structures. The low-pressure fitting database must be supplemented with data from the higher-energy crystal structures and high-pressure bcc configurations to produce a MEAM potential that agrees well with DFT and experiment for low-pressure properties, high-pressure properties, and higher-energy crystal structures properties.

4.4.3 High-Pressure MEAM Potential for W

Figure 4.27 shows the optimized splines of the high-pressure W MEAM potential.

The radial functions φ, ρ, and f each have 30 spline knots and an outer cutoﬀ radius of

6.665 A.˚ The inner cutoﬀ for the radial functions is 2.035 A.˚ The embedding function

F has 7 knots and the angular term g has 35 knots.

Table 4.14 compares bcc properties computed with the high-pressure W MEAM potential to DFT calculations and experimental data. The table also compares the energies and lattice parameters of higher-energy structures computed with the high- pressure potential to DFT calculations. All the properties computed by the high- pressure potential agree well with DFT and experiment.

Figure 4.28 compares the pressure versus volume curves and thermal expansion curves of the high-pressure MEAM potential to experiment and DFT. The high- pressure potential accurately matches the pressure-volume results from experiment to

275 GPa and DFT to 650 GPa. The thermal expansion of the high-pressure potential shows reasonable agreement with experiment, but overestimates the experimental data. Figure 4.29 shows that the phonon spectra of the high-pressure W MEAM potential compares well with DFT calculations and experimental data.

153 Figure 4.27: The ﬁve cubic splines of the high-pressure MEAM potential for W. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.665 A,˚ the angular term g has 35 knots, and the embedding function F has 7 knots.

154 Property MEAM-HP GGA-PBE Experiment

Ecoh (eV/atom) 8.31 8.31 8.90 a (A)˚ 3.189 3.189 3.163 B (GPa) 304 304 314

C11 (GPa) 532 513 533

C12 (GPa) 190 199 205

C44 (GPa) 151 142 163

∆Efcc−bcc (meV/atom) 478 479 ...

afcc (A)˚ 4.041 4.044 ...

∆Ehcp−bcc (meV/atom) 496 496 ...

ahcp (A)˚ 2.798 2.781 ...

chcp (A)˚ 4.891 4.957 ...

∆EβW−bcc (meV/atom) 91 91 ...

aβW (A)˚ 5.087 5.089 ...

∆EβTa−bcc (meV/atom) 171 173 ...

aβTa (A)˚ 9.805 9.811 ...

cβTa (A)˚ 5.151 5.149 ...

∆EωTi−bcc (meV/atom) 542 541 ...

aωTi (A)˚ 4.699 4.699 ...

cωTi (A)˚ 2.613 2.612 ...

Table 4.14: The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc W compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [12], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure.

155 Figure 4.28: (a) The pressure-volume curve of the high-pressure W MEAM potential. The MEAM potential shows excellent agreement with experiment and DFT to 650 GPa. (b) The thermal expansion of the high-pressure MEAM potential is reasonable, but overestimates the experimental curve. Note that the experimental data is shown as points in the pressure-volume ﬁgure, but as a solid line in the thermal expansion ﬁgure.

156 Figure 4.29: The phonon dispersion of the high-pressure W MEAM potential agrees well with DFT and experimental data.

Tables 4.15 and 4.16 compare the point defect and surface properties of the high- pressure W MEAM potential to DFT and experiment, respectively. The vacancy and surface results for the high-pressure MEAM potential are in excellent agreement with DFT. The MEAM interstitial energies generally agree well with the DFT values. However, the MEAM potential stabilizes the 100 dumbbell and tetrahedral h i interstitials, which relax to either the 111 dumbbell or 111 crowdion in the DFT h i h i calculations.

Figures 4.30 and 4.31 compare the pressure variation of the 112 and 110 { } { } γ-surfaces and the ideal shear strength produced by the high-pressure W MEAM potential to DFT calculations. The high-pressure potential shows excellent agreement with the DFT calculations from 0 to 100 GPa.

The high-pressure W potential has similar accuracy to the high-pressure Nb and

Mo potentials. It produces results for low- and high-pressure bcc properties, and the

157 f m f f f f f f Method Evac Evac Qvac E100 E110 E111 Ecrd Eoct Etet MEAM-HP 3.14 1.83 4.97 10.50 10.58 10.39 10.39 11.04 10.50 GGA-PBE 3.17 1.70 4.87 ... 10.64 10.31 10.31 12.42 ... Experiment [5] 3.10-4.10 1.70-1.78 4.80-5.88 ......

Table 4.15: W point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumbbell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedralh i interstitialsh i in eV. The valuesh i produced byh thei high-pressure W MEAM potential are compared to DFT calculations and experimental data. The MEAM potential stabilizes the 100 dumbbell and tetrahedral interstitials, which relax to either the 111 dumbbellh ori 111 crowdion in the DFT calculations. h i h i

{110} {110} {100} {100} {111} {111} Method Esurf ∆12 Esurf ∆12 Esurf ∆12 MEAM-HP 195 -5.5 235 -13.5 228 -32.2 GGA-PBE 200 -3.8 245 -11.5 216 -21.6

Table 4.16: Energies and relaxations of low-index surfaces of bcc W. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations.

158 Figure 4.30: The pressure variation of the 112 and 110 γ-surface sections in the 111 direction. The high-pressure W MEAM{ potential} { agrees} closely with DFT. h i

159 Figure 4.31: The pressure variation of the ideal shear strength. The high-pressure W MEAM potential agrees closely with DFT.

160 properties of higher-energy crystal structures that agree well with DFT and experi-

ment.

4.4.4 High-Pressure MEAM Potential for V

Figure 4.32 shows the optimized splines of the high-pressure V MEAM potential.

The radial functions φ, ρ, and f each have 30 spline knots and an outer cutoﬀ radius of

6.271 A.˚ The inner cutoﬀ for the radial functions is 1.830 A.˚ The embedding function

F has 7 knots and the angular term g has 35 knots.

Table 4.17 compares bcc properties computed with the high-pressure V MEAM potential to DFT calculations and experimental data. The table also compares the energies and lattice parameters of higher-energy structures computed with the high- pressure potential to DFT calculations. The properties computed by the high-pressure potential agree well with DFT and experiment, except the low value of the elastic constant C44.

Figure 4.33 compares the pressure versus volume curves and thermal expansion curves of the high-pressure V MEAM potential to experiment and DFT. The high- pressure potential accurately matches the pressure-volume results from experiment to

125 GPa and DFT to 450 GPa. The thermal expansion of the high-pressure potential shows reasonable agreement with experiment to 1,500 K. At higher temperatures the agreement is not as good. Figure 4.34 shows the phonon spectra of the high-pressure

V MEAM potential. The MEAM results compare well with DFT calculations and experimental data in the [ξ00] direction, but only at the zone boundaries in the [ξξξ] and [ξξ0] directions.

161 Figure 4.32: The ﬁve cubic splines of the high-pressure MEAM potential for V. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.271 A,˚ the angular term g has 35 knots, and the embedding function F has 7 knots.

162 Property MEAM-HP GGA-PBE Experiment

Ecoh (eV/atom) 5.45 5.44 5.31 a (A)˚ 3.002 3.001 3.024 B (GPa) 181 182 157

C11 (GPa) 266 267 232

C12 (GPa) 139 139 119

C44 (GPa) 16 24 46

∆Efcc−bcc (meV/atom) 245 243 ...

afcc (A)˚ 3.821 3.823 ...

∆Ehcp−bcc (meV/atom) 256 254 ...

ahcp (A)˚ 2.615 2.615 ...

chcp (A)˚ 4.689 4.691 ...

∆EβW−bcc (meV/atom) 49 47 ...

aβW (A)˚ 4.796 4.795 ...

∆EβTa−bcc (meV/atom) 39 38 ...

aβTa (A)˚ 9.314 9.287 ...

cβTa (A)˚ 4.796 4.791 ...

∆EωTi−bcc (meV/atom) 133 129 ...

aωTi (A)˚ 4.453 4.457 ...

cωTi (A)˚ 2.396 2.389 ...

Table 4.17: The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc V compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure.

163 Figure 4.33: (a) The pressure-volume curve of the high-pressure V MEAM potential. The MEAM potential shows excellent agreement with experiment and DFT to 450 GPa. (b) The thermal expansion of the high-pressure MEAM potential is reasonable, but underestimates the experimental curve. Note that the experimental data is shown as points in the pressure-volume ﬁgure, but as a solid line in the thermal expansion ﬁgure.

164 Figure 4.34: The phonon dispersion of the high-pressure V MEAM potential agrees well with DFT and experiment in the [ξ00] direction, but the potential agrees only near the Brillouin zone boundaries in the [ξξξ] and [ξξ0] directions.

f m Method Evac Evac Qvac MEAM-HP 2.82 0.32 3.14 GGA-PBE 2.42 0.45 2.87 Experiment [5] 2.10-2.20 0.50-1.20 2.60-3.40

Table 4.18: V vacancy properties. The vacancy formation, migration, and activation energies in eV. The values produced by the high-pressure V MEAM potential are compared to DFT calculations and experimental data.

Tables 4.18 and 4.19 compare the vacancy and surface properties of the high-

pressure V MEAM potential to DFT and experiment, respectively. The vacancy

and surface results for the high-pressure MEAM potential agree well with DFT or

experiment.

Figures 4.35 and 4.36 compare the pressure variation of the 112 and 110 { } { } γ-surfaces and the ideal shear strength produced by the high-pressure V MEAM

165 {110} {110} {100} {100} {111} {111} Method Esurf ∆12 Esurf ∆12 Esurf ∆12 MEAM-HP 140 -9.1 155 -7.4 175 -19.5 GGA-PBE 151 -5.2 156 -11.9 170 -19.1

Table 4.19: Energies and relaxations of low-index surfaces of bcc V. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations.

potential to DFT calculations. The high-pressure potential shows good agreement with the DFT calculations from 0 to 100 GPa.

The high-pressure V MEAM potential underestimates C44 and the thermal expan-

sion curve, and produces phonon dispersion curves in the [ξξξ] and [ξξ0] directions

that do not agree well with DFT and experiment. The potential accurately mod-

els many other bcc properties at low- and high-pressures, and reproduces the DFT

energies and structures of higher-energy crystal phases.

4.4.5 High-Pressure MEAM Potential for Ta

Figure 4.37 shows the optimized splines of the high-pressure Ta MEAM potential.

The radial functions φ, ρ, and f each have 30 spline knots and an outer cutoﬀ radius of

6.941 A.˚ The inner cutoﬀ for the radial functions is 2.119 A.˚ The embedding function

F has 7 knots and the angular term g has 35 knots.

Table 4.20 compares bcc properties computed with the high-pressure Ta MEAM

potential to DFT calculations and experimental data. The table also compares the

166 Figure 4.35: The pressure variation of the 112 and 110 γ-surface sections in the 111 direction. The high-pressure V MEAM{ potential} { agrees} closely with DFT. h i

167 Figure 4.36: The pressure variation of the ideal shear strength. The high-pressure V MEAM potential agrees closely with DFT.

168 Figure 4.37: The ﬁve cubic splines of the high-pressure MEAM potential for Ta. The points are the optimized spline knots and the solid lines are cubic polynomials that interpolate between the knots. The pair potential φ, density ρ, and three-body term f have 30 knots and a cutoﬀ radius of 6.941 A,˚ the angular term g has 35 knots, and the embedding function F has 7 knots.

169 energies and lattice parameters of higher-energy structures computed with the high-

pressure potential to DFT calculations.The properties computed by the high-pressure

potential agree well with DFT and experiment.

Figure 4.38 compares the pressure versus volume curves and thermal expansion

curves of the high-pressure Ta MEAM potential to experiment and DFT. The high-

pressure potential accurately matches the pressure-volume results from experiment

to 225 GPa and DFT to 500 GPa. The thermal expansion of the high-pressure

potential shows good agreement with experiment to 2,500 K. At higher temperatures

the potential underestimates the thermal expansion. Figure 4.39 shows the phonon

spectra of the high-pressure Ta MEAM potential. The MEAM results agree fairly

well with DFT calculations and experimental data in the [ξ00] and [ξξξ] directions, but the agreement is not as good in the [ξξ0] direction.

Tables 4.21 and 4.22 compare the point defect and surface properties of the high- pressure V MEAM potential to DFT and experiment, respectively. The vacancy and surface results for the high-pressure MEAM potential agree reasonably well with DFT or experiment. The MEAM potential overestimates most of the interstitial energies by more than 1 eV though.

Figures 4.40 and 4.41 compare the pressure variation of the 112 and 110 { } { } γ-surfaces and the ideal shear strength produced by the high-pressure Ta MEAM potential to DFT calculations. The high-pressure potential shows good agreement with the DFT calculations from 0 to 100 GPa.

The high-pressure Ta MEAM potential overestimates the interstitial formation energies. The potential accurately models many other bcc properties at low- and

170 Property MEAM-HP GGA-PBE Experiment

Ecoh (eV/atom) 8.47 8.46 8.10 a (A)˚ 3.322 3.322 3.303 B (GPa) 200 195 194

C11 (GPa) 285 265 266

C12 (GPa) 158 159 158

C44 (GPa) 72 75 87

∆Efcc−bcc (meV/atom) 246 242 ...

afcc (A)˚ 4.222 4.227 ...

∆Ehcp−bcc (meV/atom) 282 280 ...

ahcp (A)˚ 2.872 2.903 ...

chcp (A)˚ 5.245 5.147 ...

∆EβW−bcc (meV/atom) 35 32 ...

aβW (A)˚ 5.318 5.316 ...

∆EβTa−bcc (meV/atom) 4 2 ...

aβTa (A)˚ 10.252 10.253 ...

cβTa (A)˚ 5.350 5.342 ...

∆EωTi−bcc (meV/atom) 229 226 ...

aωTi (A)˚ 4.867 4.868 ...

cωTi (A)˚ 2.726 2.729 ...

Table 4.20: The high-pressure MEAM (MEAM-HP) values for the cohesive energy, lattice parameter, bulk modulus, and elastic constants of bcc Ta compared to DFT and experiment. The experimental value for the cohesive energy is from Ref. [2], the experimental lattice parameter is from Ref. [11], and the experimental elastic constants are from Ref. [4]. The MEAM values for the energies and lattice parameters of the fcc, hcp, β-W, β-Ta, and ω-Ti structures are compared to DFT results. The energies are relative to the energy of the bcc structure.

171 Figure 4.38: (a) The pressure-volume curve of the high-pressure Ta MEAM potential. The MEAM potential shows excellent agreement with experiment and DFT to 500 GPa. (b) The thermal expansion of the high-pressure MEAM potential agrees well with experiment to 2,500 K, but underestimates the experimental curve at higher temperature. Note that the experimental data is shown as points in the pressure- volume ﬁgure, but as a solid line in the thermal expansion ﬁgure.

172 Figure 4.39: The phonon dispersion of the high-pressure Ta MEAM potential agrees reasonably well with DFT and experiment, except near the [ξξ0] zone boundary N.

f m f f f f f f Method Evac Evac Qvac E100 E110 E111 Ecrd Eoct Etet MEAM-HP 3.39 0.62 4.00 7.40 6.37 6.05 6.05 7.22 5.98 GGA-PBE 2.91 0.73 3.64 6.04 5.55 4.83 4.82 6.08 5.86 Experiment [5] 2.20-3.10 0.70-1.90 2.90-5.00 ......

Table 4.21: Ta point defects. The vacancy formation, migration, and activation energies, and the formation energies of the 100 dumbbell, 110 dumbbell, 111 dumbbell, 111 crowdion, octahedral, and tetrahedralh i interstitialsh i in eV. The valuesh i produced byh thei high-pressure Ta MEAM potential are compared to DFT calculations and experimental data. The MEAM vacancy energies agree reasonably well with DFT and experiment, but the potential overestimates the DFT interstitial energies.

{110} {110} {100} {100} {111} {111} Method Esurf ∆12 Esurf ∆12 Esurf ∆12 MEAM-HP 149 -4.3 172 -4.3 183 -13.1 GGA-PBE 146 -4.5 155 -12.7 169 -23.7

Table 4.22: Energies and relaxations of low-index surfaces of bcc Ta. The surface {ijk} ˚2 {ijk} energies Esurf are in meV/A . The relaxation values ∆12 are the relative percent change in the interplanar spacing of the ﬁrst two surface layers upon relaxation. The values produced by the high-pressure MEAM potential developed in this work are compared to DFT calculations.

173 Figure 4.40: The pressure variation of the 112 and 110 γ-surface sections in the 111 direction. The high-pressure Ta MEAM{ potential} { agrees} closely with DFT. h i

174 Figure 4.41: The pressure variation of the ideal shear strength. The high-pressure Ta MEAM potential agrees closely with DFT.

175 high-pressures, and reproduces the DFT energies and structures of higher-energy crystal phases.

This section presents results for potentials generated from low-pressure data, and potentials generated from low- and high-pressure data. The low-pressure potentials are good models of the bcc structure at low pressure, but they fail to produce the correct variation of the stacking faults and ideal shear strength with pressure. EAM potentials ﬁt to low- and high-pressure data show improved high-pressure behavior, but at the expense of the low-pressure properties. MEAM potentials ﬁt to low- and high-pressure data accurately model the bcc metals over a wide range of pressures and temperatures.

176 Chapter 5: Conclusion

This work develops highly accurate cubic spline-based EAM and MEAM potentials for the bcc refractory transition metals V, Nb, Ta, Mo, and W. The force- matching method fits the spline knots to forces, energies, and stresses from DFT calculations. EAM and MEAM potentials fit to low-pressure DFT data accurately model wide-ranging properties at low to moderate pressures, but fail for high-pressure calculations. EAM potentials fit to low- and high-pressure DFT data produce improved high-pressure behavior, but the accuracy of the low-pressure properties becomes worse. MEAM potentials fit to low- and high-pressure DFT data successfully model the pressure-dependence of properties related to mechanical deformation, while retaining the low-pressure accuracy necessary for predictive modeling at ambient conditions.

5.1 Summary of Results

EAM and MEAM potentials ﬁt to forces, energies, and stresses from DFT calculations model the bcc transition metals. This work presents the ﬁrst application of the force-matching method to constructing potentials for V, Nb, Mo, and W, and also the

ﬁrst application of the method to generating potentials applicable for high-pressure studies of mechanical properties and deformation.

177 The low-pressure Nb EAM potential developed in this work is ﬁt to a database of

forces, energies, and stresses for conﬁgurations of atoms under diﬀerent temperature

and strain conditions at low to moderate pressures. The potential accurately mod-

els bcc properties at low pressure, including (1) structural and elastic properties, (2)

properties of point defects (vacancies and self-interstitial atoms), linear defects (dislo-

cations), and planar defects (surfaces and stacking faults), and (3) ﬁnite temperature

eﬀects (thermal expansion and melting). The low-pressure Mo MEAM potential ﬁt

to DFT data at low to moderate pressures also accurately models wide-ranging bcc

properties at low-pressures.

The low-pressure potentials fail to describe the correct pressure dependence of mechanical properties relevant for ﬁnite strain-rate plasticity and shock simulations.

They also fail at predicting the correct energies and structural parameters of higher- energy crystal phases not included in the ﬁtting database. EAM potentials ﬁt to

DFT data from multiple crystal structures produce better agreement with DFT for the high-energy structures, but at the expense of the bcc properties. This is due to the lack of explicit three-body bond-bending terms, since MEAM potentials ﬁt to data from multiple crystal structures produce accurate results for all phases included in the ﬁtting database.

The low-pressure Nb EAM and Mo MEAM potentials are ﬁt to low- to moderate- pressure DFT data from either zero-temperature compressed primitive bcc cells, or

ﬁnite temperature MD data of the bcc and liquid phases. Tests of the pressure dependence of the generalized stacking faults and ideal shear stress reveal that this

ﬁtting data is insuﬃcient to produce quantitatively correct mechanical behavior at high-pressure. EAM potentials generated from databases supplemented with data

178 relevant for high-pressure stacking faults and ideal shear strength are unable to produce physically reasonable results at low and high pressure. On the other hand,

MEAM potentials generated from this data accurately model many properties of the bcc phase across a wide range of pressures. They can also account for the energetics of multiple crystal phases if relevant data is included in the ﬁtting database. The success of the MEAM potentials in accurately describing the pressure dependence of mechanical properties while also modeling multiple crystal phases has implications for high-pressure shock simulations, and also for extending the potentials to alloy systems with multiple crystal phases.

5.2 Limitations and Possible Extensions of This Work

The potentials developed in this work are applicable to modeling diverse properties over a wide range of simulation conditions, and open the possibility for accurate studies of bcc metals under extreme non-equilibrium simulation conditions. A large number of calculations test the potentials, but one should always be wary of applying any interatomic potential to problems outside of the range it is designed for. The Nb shock simulations discussed in Section4.3 clearly illustrate this point: five published interatomic potentials generated from low-pressure fitting data produce high-pressure results that are qualitative at best. The high-pressure MEAM potentials developed in this study remedy the deficiencies of the low-pressure models, but care and thought must be exercised when applying the potentials to study new phenomena.

The potentials constructed in this thesis also oﬀer a viable starting point for designing high-quality potentials for alloy systems with bcc components. A common method for generating potentials for multicomponent systems uses existing potentials

179 for the interactions between atoms of the same type. A variety of methods for constructing the interactions between dissimilar species have been proposed, including a simple procedure that averages the pair potentials of the diﬀerent atoms to obtain the interactions between species [188] and a method the ﬁts the cross-interactions to alloy mixing enthalpies [189]. An alternative is again provided by the force-matching method, for which the cross-terms are parameterized by cubic splines determined by

ﬁtting to a DFT database of forces, energies, and stresses for alloy conﬁgurations.

This oﬀers a promising path towards accurate alloy potentials based on the single- element potentials developed in this study.

In summary, the potentials developed in this work offer an improved description of the bcc refractory metals over potentials developed in previous studies. A better understanding of the applicability and limitations of EAM and MEAM models for bcc metals is also obtained. The high-pressure potentials offer a new opportunity for accurate simulations of bcc metals under extreme conditions, and afford a viable starting point for designing potentials for multicomponent alloy systems.

180 Appendix A: Convergence Parameters for DFT Calculations

The convergence parameters for the DFT calculations are determined by the method outlined in Section 3.2.2. This Appendix presents the details of the DFT calculations used to determine the values for the plane-wave cutoﬀ energy Ecut, the

k-point grid N N N , and the Fermi surface smearing parameter σ for the bcc k × k × k refractory metals V, Nb, Ta, Mo, and W. All calculations use the PBE GGA exchange correlation functional and PAW pseudopotentials. The V and Nb pseudopotentials treat the highest s- and p-semicore states as valence states. The Ta, Mo, and W

pseudopotentials treat the highest p-semicore states as valence states.

A.1 Plane-Wave Cutoﬀ Energy Ecut Convergence

Figures A.1-A.5 show the variation of total energy Etotal with plane-wave cut- oﬀ energy Ecut. The calculations are carried out using the tetrahedron method of

Bl¨ochl [123]. The experimental lattice constant is used for each metal. To ensure convergence with respect to k-points, a very dense grid of k-points is used (47 47 47). × ×

The values of Ecut chosen for the metals are Ecut = 550 eV for V and Nb, and

Ecut = 600 eV for Ta, Mo, and W.

181 Total Energy vs. Plane-Wave Energy Cutoff for V

-9.114

L -9.115 eV H total

E -9.116

-9.117

350 400 450 500 550 600 650

Ecut HeVL

Figure A.1: Total energy Etotal versus plane-wave cutoﬀ Ecut for V. The calculation is converged when Ecut = 550 eV.

Total Energy vs. Plane-Wave Energy Cutoff for Nb

-10.218

-10.220 L eV H -10.222 total E

-10.224

-10.226 350 400 450 500 550 600 650

Ecut HeVL

Figure A.2: Total energy Etotal versus plane-wave cutoﬀ Ecut for Nb. The calculation is converged when Ecut = 550 eV.

182 Total Energy vs. Plane-Wave Energy Cutoff for Ta -11.8500

-11.8505 L eV H -11.8510 total E -11.8515

-11.8520 350 400 450 500 550 600 650 700

Ecut HeVL

Figure A.3: Total energy Etotal versus plane-wave cutoﬀ Ecut for Ta. The calculation is converged when Ecut = 600 eV.

Total Energy vs. Plane-Wave Energy Cutoff for Mo -10.8415 -10.8420 -10.8425

L -10.8430 eV H -10.8435 total

E -10.8440 -10.8445 -10.8450 -10.8455 350 400 450 500 550 600 650 700

Ecut HeVL

Figure A.4: Total energy Etotal versus plane-wave cutoﬀ Ecut for Mo. The calculation is converged when Ecut = 600 eV.

183 Total Energy vs. Plane-Wave Energy Cutoff for W

-12.946

L -12.947 eV H total

E -12.948

-12.949

350 400 450 500 550 600 650 700

Ecut HeVL

Figure A.5: Total energy Etotal versus plane-wave cutoﬀ Ecut for W. The calculation is converged when Ecut = 600 eV.

A.2 k-Point Convergence

Figures A.6-A.10 show the variation of total energy Etotal and electronic density of states (DOS) near the Fermi level with the number of k-points in each reciprocal- space direction, N N N . The DOS for each choice of N is compared to the DOS k × k × k k calculation with Nk = 47, and all the DOS curves are shifted such that the Fermi level is at E = 0. The calculations are carried out using the tetrahedron method of

Bl¨ochl [123] and use the Ecut values determined in Section A.1. The experimental lattice constant is used for each metal. The values of Nk chosen for the metals are

Nk = 31 for Nb, Ta, and Mo, and Nk = 35 for V and W.

184 Total Energy vs. k-Points for V Electronic Density of States vs. Energy for V -9.1163 3.0

- L

9.1164 1 - 2.5 = -9.1165 Nk 19 eV H L -9.1166 2.0 Nk=47 eV H - 9.1167 1.5 total - E 9.1168 -9.1169 1.0

-9.1170 Density of0.5 States -9.1171 15 20 25 30 35 40 45 -1.0 -0.5 0.0 0.5 1.0

Nk E HeVL

Electronic Density of States vs. Energy for V Electronic Density of States vs. Energy for V 3.0 3.0 L L 1 1

- 2.5 - 2.5 Nk=23 Nk=27 eV eV H H 2.0 Nk=47 2.0 Nk=47

1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for V Electronic Density of States vs. Energy for V 3.0 3.0 L L 1 1

- 2.5 - 2.5 Nk=31 Nk=35 eV eV H H 2.0 Nk=47 2.0 Nk=47

1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for V Electronic Density of States vs. Energy for V 3.0 3.0 L L 1 1

- 2.5 - 2.5 Nk=39 Nk=43 eV eV H H 2.0 Nk=47 2.0 Nk=47

1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.6: Total energy Etotal and electronic density of states versus Nk for V. The calculation is converged when N = 35, i.e. a 35 35 35 k-point grid. k × ×

185 Total Energy vs. k-Points for Nb Electronic Density of States vs. Energy for Nb -10.2245 2.5 L 1 - Nk=19

eV 2.0 H

L = -10.2250 Nk 47 eV H 1.5 total E -10.2255 1.0

Density of0.5 States -10.2260 15 20 25 30 35 40 45 -1.0 -0.5 0.0 0.5 1.0

Nk E HeVL

Electronic Density of States vs. Energy for Nb Electronic Density of States vs. Energy for Nb 2.5 2.5 L L 1 1 - - Nk=23 Nk=27

eV 2.0 eV 2.0 H H Nk=47 Nk=47 1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Nb Electronic Density of States vs. Energy for Nb 2.5 2.5 L L 1 1 - - Nk=31 Nk=35

eV 2.0 eV 2.0 H H Nk=47 Nk=47 1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Nb Electronic Density of States vs. Energy for Nb 2.5 2.5 L L 1 1 - - Nk=39 Nk=43

eV 2.0 eV 2.0 H H Nk=47 Nk=47 1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.7: Total energy Etotal and electronic density of states versus Nk for Nb. The calculation is converged when N = 31, i.e. a 31 31 31 k-point grid. k × ×

186 Total Energy vs. k-Points for Ta Electronic Density of States vs. Energy for Ta -11.8516 2.5 L

-11.8517 1 - 2.0 Nk=19 - eV 11.8518 H

L N =47 - 1.5 k eV 11.8519 H - total 11.8520 1.0 E -11.8521 0.5 - 11.8522 Density of States -11.8523 0.0 15 20 25 30 35 40 45 -1.0 -0.5 0.0 0.5 1.0

Nk E HeVL

Electronic Density of States vs. Energy for Ta Electronic Density of States vs. Energy for Ta 2.5 2.5 L L 1 1 - 2.0 - 2.0 Nk=23 Nk=27 eV eV H = H = 1.5 Nk 47 1.5 Nk 47

1.0 1.0

0.5 0.5 Density of States Density of States 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Ta Electronic Density of States vs. Energy for Ta 2.5 2.5 L L 1 1 - 2.0 - 2.0 Nk=31 Nk=35 eV eV H = H = 1.5 Nk 47 1.5 Nk 47

1.0 1.0

0.5 0.5 Density of States Density of States 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Ta Electronic Density of States vs. Energy for Ta 2.5 2.5 L L 1 1 - 2.0 - 2.0 Nk=39 Nk=43 eV eV H = H = 1.5 Nk 47 1.5 Nk 47

1.0 1.0

0.5 0.5 Density of States Density of States 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.8: Total energy Etotal and electronic density of states versus Nk for Ta. The calculation is converged when N = 31, i.e. a 31 31 31 k-point grid. k × ×

187 Total Energy vs. k-Points for Mo Electronic Density of States vs. Energy for Mo -10.8438 1.4 L 1

-10.8440 - Nk=19

eV 1.2 - H L 10.8442 Nk=47 eV H -10.8444 1.0

total - E 10.8446 0.8 -10.8448 0.6 -10.8450 Density of States 0.4 15 20 25 30 35 40 45 -1.0 -0.5 0.0 0.5 1.0

Nk E HeVL

Electronic Density of States vs. Energy for Mo Electronic Density of States vs. Energy for Mo 1.4 1.4 L L 1 1 - - Nk=23 Nk=27

eV 1.2 eV 1.2 H H Nk=47 Nk=47 1.0 1.0

0.8 0.8

0.6 0.6 Density of States Density of States

0.4 0.4 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Mo Electronic Density of States vs. Energy for Mo 1.4 1.4 L L 1 1 - - Nk=31 Nk=35

eV 1.2 eV 1.2 H H Nk=47 Nk=47 1.0 1.0

0.8 0.8

0.6 0.6 Density of States Density of States

0.4 0.4 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Mo Electronic Density of States vs. Energy for Mo 1.4 1.4 L L 1 1 - - Nk=39 Nk=43

eV 1.2 eV 1.2 H H Nk=47 Nk=47 1.0 1.0

0.8 0.8

0.6 0.6 Density of States Density of States

0.4 0.4 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.9: Total energy Etotal and electronic density of states versus Nk for Mo. The calculation is converged when N = 31, i.e. a 31 31 31 k-point grid. k × ×

188 Total Energy vs. k-Points for W Electronic Density of States vs. Energy for W -12.9480 0.9 L

- 1

12.9482 - 0.8 Nk=19 eV - H L 12.9484 0.7 Nk=47 eV H -12.9486 0.6 total E -12.9488 0.5

-12.9490 0.4 Density of States -12.9492 0.3 15 20 25 30 35 40 45 -1.0 -0.5 0.0 0.5 1.0

Nk E HeVL

Electronic Density of States vs. Energy for W Electronic Density of States vs. Energy for W 0.9 0.9 L L 1 1

- 0.8 - 0.8 Nk=23 Nk=27 eV eV H 0.7 H 0.7 Nk=47 Nk=47 0.6 0.6

0.5 0.5

0.4 0.4 Density of States Density of States 0.3 0.3 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for W Electronic Density of States vs. Energy for W 0.9 0.9 L L 1 1

- 0.8 - 0.8 Nk=31 Nk=35 eV eV H 0.7 H 0.7 Nk=47 Nk=47 0.6 0.6

0.5 0.5

0.4 0.4 Density of States Density of States 0.3 0.3 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for W Electronic Density of States vs. Energy for W 0.9 0.9 L L 1 1

- 0.8 - 0.8 Nk=39 Nk=43 eV eV H 0.7 H 0.7 Nk=47 Nk=47 0.6 0.6

0.5 0.5

0.4 0.4 Density of States Density of States 0.3 0.3 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.10: Total energy Etotal and electronic density of states versus Nk for W. The calculation is converged when N = 35, i.e. a 35 35 35 k-point grid. k × ×

189 A.3 Fermi Surface Smearing

Figures A.11-A.15 show the variation of total energy Etotal and electronic density

of states (DOS) near the Fermi level with the Fermi surface smearing width σ. The calculations are carried out using ﬁrst-order Methfessel-Paxton smearing [124], and use the Ecut values determined in Section A.1 and the k-points determined in Sec- tion A.2. The DOS for each choice of σ is compared to the tetrahedron method DOS calculation, and all the DOS curves are shifted such that the Fermi level is at E = 0.

The experimental lattice constant is used for each metal. The values of σ chosen for

the metals are σ = 0.10 eV for V and Nb, σ = 0.20 eV for Ta, and σ = 0.35 for Mo

and W.

190 Total Energy vs. Smearig Width for V Electronic Density of States vs. Energy for V -9.1160 3.0 L 1

- - 2.5 9.1165 Σ=0.05 eV eV H

L 2.0 -9.1170 Tetrahedron eV H 1.5 total -9.1175 E 1.0 -9.1180 Density of0.5 States -9.1185 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 -1.0 -0.5 0.0 0.5 1.0 Σ HeVL E HeVL

Electronic Density of States vs. Energy for V Electronic Density of States vs. Energy for V 3.0 3.0 L L 1 1

- 2.5 - 2.5 Σ=0.10 eV Σ=0.15 eV eV eV H H 2.0 Tetrahedron 2.0 Tetrahedron

1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for V Electronic Density of States vs. Energy for V 3.0 3.0 L L 1 1

- 2.5 - 2.5 Σ=0.20 eV Σ=0.25 eV eV eV H H 2.0 Tetrahedron 2.0 Tetrahedron

1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for V Electronic Density of States vs. Energy for V 3.0 3.0 L L 1 1

- 2.5 - 2.5 Σ=0.30 eV Σ=0.35 eV eV eV H H 2.0 Tetrahedron 2.0 Tetrahedron

1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.11: Total energy Etotal and electronic density of states versus σ for V. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.10 eV.

191 Total Energy vs. Smearing Width for Nb Electronic Density of States vs. Energy for Nb - 10.2240 2.5 L

-10.2242 1 - Σ=0.05 eV

-10.2244 eV 2.0 H

L Tetrahedron -10.2246 eV

H 1.5 -10.2248

total - E 10.2250 1.0 -10.2252

-10.2254 Density of0.5 States -10.2256 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 -1.0 -0.5 0.0 0.5 1.0 Σ HeVL E HeVL

Electronic Density of States vs. Energy for Nb Electronic Density of States vs. Energy for Nb 2.5 2.5 L L 1 1 - Σ=0.10 eV - Σ=0.15 eV

eV 2.0 eV 2.0 H Tetrahedron H Tetrahedron 1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Nb Electronic Density of States vs. Energy for Nb 2.5 2.5 L L 1 1 - Σ=0.20 eV - Σ=0.25 eV

eV 2.0 eV 2.0 H Tetrahedron H Tetrahedron 1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Nb Electronic Density of States vs. Energy for Nb 2.5 2.5 L L 1 1 - Σ=0.30 eV - Σ=0.35 eV

eV 2.0 eV 2.0 H Tetrahedron H Tetrahedron 1.5 1.5

1.0 1.0

Density of0.5 States Density of0.5 States

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.12: Total energy Etotal and electronic density of states versus σ for Nb. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.10 eV.

192 Total Energy vs. Smearing Width for Ta Electronic Density of States vs. Energy for Ta -11.8508 2.5 L

-11.8510 1 - 2.0 Σ=0.05 eV

- eV 11.8512 H

L Tetrahedron - 1.5 eV 11.8514 H - total 11.8516 1.0 E -11.8518 0.5 - 11.8520 Density of States -11.8522 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 -1.0 -0.5 0.0 0.5 1.0 Σ HeVL E HeVL

Electronic Density of States vs. Energy for Ta Electronic Density of States vs. Energy for Ta 2.5 2.5 L L 1 1 - 2.0 Σ=0.10 eV - 2.0 Σ=0.15 eV eV eV H H 1.5 Tetrahedron 1.5 Tetrahedron

1.0 1.0

0.5 0.5 Density of States Density of States 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Ta Electronic Density of States vs. Energy for Ta 2.5 2.5 L L 1 1 - 2.0 Σ=0.20 eV - 2.0 Σ=0.25 eV eV eV H H 1.5 Tetrahedron 1.5 Tetrahedron

1.0 1.0

0.5 0.5 Density of States Density of States 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Ta Electronic Density of States vs. Energy for Ta 2.5 2.5 L L 1 1 - 2.0 Σ=0.30 eV - 2.0 Σ=0.35 eV eV eV H H 1.5 Tetrahedron 1.5 Tetrahedron

1.0 1.0

0.5 0.5 Density of States Density of States 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.13: Total energy Etotal and electronic density of states versus σ for Ta. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.20 eV.

193 Total Energy vs. Smearing Width for Mo Electronic Density of States vs. Energy for Mo 1.6

-10.8450 L 1 1.4 - Σ=0.05 eV eV

H 1.2 L Tetrahedron

eV - H 10.8455 1.0

total 0.8 E 0.6 -10.8460

Density of0.4 States 0.2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 -1.0 -0.5 0.0 0.5 1.0 Σ HeVL E HeVL

Electronic Density of States vs. Energy for Mo Electronic Density of States vs. Energy for Mo 1.6 1.6 L L

1 1.4 1 1.4 - Σ=0.10 eV - Σ=0.15 eV eV eV

H 1.2 H 1.2 Tetrahedron Tetrahedron 1.0 1.0 0.8 0.8 0.6 0.6

Density of0.4 States Density of0.4 States 0.2 0.2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Mo Electronic Density of States vs. Energy for Mo 1.6 1.6 L L

1 1.4 1 1.4 - Σ=0.20 eV - Σ=0.25 eV eV eV

H 1.2 H 1.2 Tetrahedron Tetrahedron 1.0 1.0 0.8 0.8 0.6 0.6

Density of0.4 States Density of0.4 States 0.2 0.2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for Mo Electronic Density of States vs. Energy for Mo 1.6 1.6 L L

1 1.4 1 1.4 - Σ=0.30 eV - Σ=0.35 eV eV eV

H 1.2 H 1.2 Tetrahedron Tetrahedron 1.0 1.0 0.8 0.8 0.6 0.6

Density of0.4 States Density of0.4 States 0.2 0.2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.14: Total energy Etotal and electronic density of states versus σ for Mo. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.35 eV.

194 Total Energy vs. Smearig Width for W Electronic Density of States vs. Energy for W -12.9486 1.0 L 1 - - 12.9488 0.8 Σ=0.05 eV eV H

L Tetrahedron

eV -

H 12.9490 0.6 total E -12.9492 0.4

-12.9494 Density of States 0.2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 -1.0 -0.5 0.0 0.5 1.0 Σ HeVL E HeVL

Electronic Density of States vs. Energy for W Electronic Density of States vs. Energy for W 1.0 1.0 L L 1 1 - - 0.8 Σ=0.10 eV 0.8 Σ=0.15 eV eV eV H H Tetrahedron Tetrahedron 0.6 0.6

0.4 0.4 Density of States Density of States 0.2 0.2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for W Electronic Density of States vs. Energy for W 1.0 1.0 L L 1 1 - - 0.8 Σ=0.20 eV 0.8 Σ=0.25 eV eV eV H H Tetrahedron Tetrahedron 0.6 0.6

0.4 0.4 Density of States Density of States 0.2 0.2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Electronic Density of States vs. Energy for W Electronic Density of States vs. Energy for W 1.0 1.0 L L 1 1 - - 0.8 Σ=0.30 eV 0.8 Σ=0.35 eV eV eV H H Tetrahedron Tetrahedron 0.6 0.6

0.4 0.4 Density of States Density of States 0.2 0.2 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 E HeVL E HeVL

Figure A.15: Total energy Etotal and electronic density of states versus σ for W. The horizontal line in the energy plot is the tetrahedron method energy. The optimal value is σ = 0.35 eV.

195 Appendix B: Optimized Spline Knots for EAM and MEAM Potentials

This Appendix lists the spline knots and boundary conditions for the potentials

presented in Chapter 4. The knot values have been rounded to eight decimal places.

The full precision values for the Nb EAM potential are in Ref. [1], and the full precision

values for the Mo MEAM potential are in Ref. [13].

B.1 Nb EAM Potential

Table B.1 lists the optimized spline knots for the Nb EAM potential developed in

this work [1]. In MD simulations, ﬂuctuations can move atoms closer together than

the minimum interatomic distance in the ﬁtting database. The potfit program

accounts for this by extending φ and ρ to r values smaller than the inner cutoﬀ radius. The cubic polynomials in the range 2.073

196 steep cubic function. Setting F equal to 4.828 eV at n = 1.264 determines the ﬁnal coeﬃcient.

Despite these modiﬁcations, the potential is not repulsive enough for high-temperature and high-pressure simulations, where atoms closely approach one another. To overcome this limitation, we modify φ for 1.738

We also modify the extension of F for small n values to properly describe the cohesive energy. The minimum of the EAM energy per atom versus volume curve for bcc Nb equals the cohesive energy, but the embedding energy is not zero for n = 0 when the atoms are far apart. Therefore, we replace the potfit modification for small n by a different cubic polynomial. We choose three coefficients to ensure continuity of the embedding function and its first and second derivatives at n = 0.0775. We determine the final coefficient by setting F (0) = 0.

B.2 Mo MEAM Potential

Table B.2 lists the optimized spline knots for the Mo MEAM developed in this work [13]. The modiﬁcations to the EAM potential discussed in Section B.1 are not necessary for the MEAM potential.

197 i ri (A)˚ φ(ri) (eV) ρ(ri) ni F (ni) (eV) 0 1.7383750 5.64480806 0.68317602 0.00000000 0.00000000 1 2.0730000 1.95203249 0.41866138 0.07749294 3.34728569 − 2 2.2403125 1.09403598 0.24814239 0.20927966 4.54633449 − 3 2.4076250 0.51088585 0.13515113 0.34106638 4.89345623 − 4 2.5749375 0.08234334 0.06780203 0.47285311 4.95023644 − 5 2.7422500 0.17755065 0.03707860 0.60463983 4.94497069 − − 6 2.9095625 0.31133193 0.02383416 0.73642655 4.84569948 − − 7 3.0768750 0.39000438 0.01322667 0.86821328 4.74371759 − − 8 3.2441875 0.40555115 0.00859424 1.00000000 4.55614221 − − 9 3.4115000 0.35188220 0.00902608 1.26357345 4.82834839 − 10 3.5788125 0.25163493 0.01322871 − 11 3.7461250 0.14537802 0.01610260 − 12 3.9134375 0.07811976 0.01119941 − 13 4.0807500 0.04722050 0.00740724 − 14 4.2480625 0.03283083 0.00241663 − − 15 4.4153750 0.02123653 0.00257247 − − 16 4.5826875 0.00649537 0.00051588 − − 17 4.7500000 0.00000000 0.00000000

′′ ′′ ′′ φ (r1) = 0 ρ (r1) = 0 F (n1) = 0 ′ ′ ′′ φ (r17) = 0 ρ (r17) = 0 F (n8) = 0

Table B.1: The cubic spline knots and boundary conditions for the Nb EAM potential [1]. The first part of the table lists the knots, and the second part of the table lists the boundary conditions on the splines. Spline knots 1-17 for φ and ρ, and spline knots 1-8 for F are optimized by potfit. The adjusted values of knot 0 for φ, ρ, and F , and knot 9 for F are also listed (see text). The coefficients of the cubic polynomials that interpolate between the knots are determined by requiring continuity of the functions and their first and second derivatives, along with the boundary conditions.

198 x xmin xmax N φ r (A)˚ 2.0118713 5.9000000 13 ρ r (A)˚ 2.0118713 5.1000000 12 f r (A)˚ 2.0118713 5.1000000 12 U n 95.8550744 32.1224593 3 − − g cos θ 1.0000000 0.9998790 8 − i φ(ri) (eV) ρ(ri) f(ri) F (ni) (eV) g(xi) 0 4.63243873 26.49444997 3.38822768 1.50152668 0.12986931 − − − 1 1.75248915 17.20509099 2.42073620 0.35608978 0.37932159 − − 2 0.43824370 8.514689240 1.38368225 2.32296219 0.00526915 − − 3 0.01579726 4.57648474 0.41817143 0.33754097 − − − 4 0.06830008 3.26571489 0.36611329 0.45406155 − − − − 5 0.05926324 2.48041575 0.55432360 0.20067429 − − − − 6 0.08698198 1.29720468 0.22735802 0.94204584 − − − − 7 0.05888102 0.12383911 0.05025722 6.81741287 − − − − 8 0.03143077 0.00063192 0.02618391 − 9 0.01914110 0.22702167 0.00038479 − − 10 0.00706338 0.08116555 0.00061945 − − − 11 0.00117183 0.00000000 0.00000000 − 12 0.00000000

′ ′ −1 ′ −1 ′ ′ i φ (ri) (eV/A)˚ ρ (ri) (A˚ ) f (ri) (A˚ ) F (ni) (eV) g (xi) 0 -11.52990417 0.00000000 0.00000000 0.02291577 2.61429618 N 0.00000000 0.00000000 0.00000000 0.13078892 24.32895534 −

Table B.2: The cubic spline knots and boundary conditions for the Mo MEAM potential [13]. The coefficients of the cubic polynomials that interpolate between the knots are determined by requiring continuity of the functions and their first and second derivatives, along with the boundary conditions. The first part of the table lists the number of knots N for each spline and the range of the spline variables xmin and xmax. The middle part of the table gives the values at equally spaced spline knots defined by x = x + i(x x )/N, where N is the number of spline knots. Finally, the i min max − min boundary conditions specifying the derivatives of the splines at their end points are listed in the last part of the table.

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