Applied Science, Faculty of
Materials Engineering, Department of
DSpace
UBCV
Umashankar, Hariharan
Doctor of Philosophy - PhD
University of British Columbia
Atomistic events govern processes at the microstructural length scale that ultimately affect macroscopic material behavior. For instance, solute segregation to internal interfaces such as grain boundaries (GB) can significantly affect the mechanical properties. Similarly, vacancy interactions with GBs and the diffusion along the GBs are also important phenomena that control the microstructure evolution. Experimental investigations of such processes are laborious, time-consuming, and present technical challenges. This study deals with a computational approach to model solute segregation and vacancy interactions at the atomistic scale using two different simulation methods. The first method uses Density Functional Theory (DFT) calculations to quantify the solute-GB interactions in a BCC Ti- 25 at.% Mo alloy. Electrons are considered explicitly in DFT, serving as a state-of-the-art method for segregation phenomena. A major challenge in studying alloys is the multitude of atomic arrangements of the host species. Results from solutes - Y, Zr and Nb occupying all sites in the alloy yield a spectrum of solution energies. Taking the example of the strongest segregating solute, Yttrium, segregation and solution energies at GBs and in the bulk are calculated across three compositions of Mo - 25, 50 and 75 at.% allowing trends to be mapped for the full composition space. The second approach uses a surrogate model, such as the Atomic Cluster Expansion. Initially, a workflow is established using the training data from Embedded Atom Method (EAM) calculations. Here, properties of bulk, fifteen different <100> symmetric tilt GBs in pure Pt and Au, bulk mixing enthalpies of Pt-Au alloys and vacancy formation energies in bulk Pt-Au alloys are studied. The DFT data for some structures that are chosen based on the EAM surrogate modeling are generated to build a regression model with “ab-initio” accuracy. A property of interest, such as vacancy formation energy in ~ 50 at. % Au for bulk and GB sites is studied. Using a committee-of-models approach, new structures are picked iteratively to improve the model.
Atomistic simulations of crystalline defect interactionsin metallic alloysbyHariharan UmashankarB.Tech., Indian Institute of Technology (IIT) Madras, 2018M.Tech., Indian Institute of Technology (IIT) Madras, 2018A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Materials Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2025© Hariharan Umashankar, 2025The following individuals certify that they have read, and recommend to the Faculty of Grad-uate and Postdoctoral Studies for acceptance, the dissertation entitled:Atomistic simulations of crystalline defect interactions in metallic alloyssubmitted byHariharan Umashankar in partial fulfillment of the requirements for the degreeof Doctor of Philosophy in Materials EngineeringExamining Committee:Matthias Militzer, Professor, Materials Engineering, UBCSupervisorChadwick Sinclair, Professor, Materials Engineering, UBCSupervisory Committee MemberJoerg Rottler, Professor, Physics and Astronomy, UBCSupervisory Committee MemberRoman Krems, Professor, Chemistry, UBCUniversity ExaminerMauricio Ponga de la Torre, Associate Professor, Mechanical Engineering, UBCUniversity ExaminerChuang Deng, Professor, Mechanical Engineering, University of ManitobaExternal ExamineriiAbstractAtomistic events govern processes at the microstructural length scale that ultimately affectmacroscopic material behavior. For instance, solute segregation to internal interfaces such asgrain boundaries (GBs) can significantly affect the mechanical properties. Similarly, vacancyinteractions with GBs and the diffusion along the GBs are also important phenomena thatcontrol the microstructure evolution. Experimental investigations of such processes are labori-ous, time-consuming, and present technical challenges. This study deals with a computationalapproach to model solute segregation and vacancy interactions at the atomistic scale using twodifferent simulation methods.The first method uses Density Functional Theory (DFT) calculations to quantify the solute-GB interactions in a BCC Ti- 25 at.% Mo alloy. Electrons are considered explicitly in DFT,serving as a state-of-the-art method for segregation phenomena. A major challenge in studyingalloys is the multitude of atomic arrangements of the host species. Results from solutes - Y, Zrand Nb occupying all different sites in the alloy yield a spectrum of solution energies. Takingthe example of the strongest segregating solute, Yttrium, segregation and solution energies atGBs and in the bulk are calculated across three compositions of Mo - 25, 50 and 75 at.%,allowing trends to be mapped for the full composition space.The second approach uses a surrogate model, such as the Atomic Cluster Expansion. Ini-tially, a workflow is established using the training data from Embedded Atom Method (EAM)calculations. Here, properties of bulk, fifteen different <100> symmetric tilt GBs in pure Ptand Au, bulk mixing enthalpies of Pt-Au alloys and vacancy formation energies in bulk Pt-Aualloys are studied.The DFT data for some structures that are chosen based on the EAM surrogate modelingare generated to build a regression model with “ab-initio” accuracy. A property of interest,such as the vacancy formation energy in ~ 50 at. % Au for bulk and GB sites is studied. Usinga committee-of-models approach, new structures are picked iteratively to improve the model.iiiLay SummaryMetallic alloys containing many chemical elements are essential in everyday household, trans-portation, and energy applications. Achieving desired properties requires careful microstructuredesign at the micrometer scale (one-tenth of the thickness of a human hair) through thermo-mechanical treatments. Here, atomic-scale defects in the crystalline structure of alloys arecrucial because their interaction with alloying and trace elements influences microstructural evo-lution and properties, including fracture behavior and corrosion resistance. These interactionsoccur on the atomistic length scale (nanometers) and can be characterized with atomic-scalesimulations.In the present work, atomistic simulations are used to study the interactions between solutes,vacancies and grain boundaries in two binary alloy systems - Ti-Mo and Pt-Au alloys. An ab-initio modeling approach is employed for the Ti-Mo alloy. In contrast, the Pt-Au system isinvestigated with a machine learning interatomic potential trained on semi-empirical and ab-initio data.ivPrefaceThe research presented in this thesis was conducted by the author, Hariharan Umashankar,under the supervision of Prof. Matthias Militzer. Most of the simulations and data analysiswere performed by the author in the Department of Materials Engineering at the University ofBritish Columbia (UBC) except for the simulations listed below.The computational resources used for performing simulations in Chapter 4 were done on theMARCONI supercomputer, part of the CINECA high-performance computing in Italy. Theauthor set up the calculations and performed the data analysis himself. Dr. Vsevolod I. Razu-movskiy and Dr. Daniel Scheiber were involved in the discussion and assisted in interpretingthe results presented in Chapter 4. The data for Chapter 5 was performed and collected onthe Niagara supercomputer as part of the Digital Research Alliance of Canada. The authorconducted the data analysis, and the discussions were facilitated together with Prof. MatthiasMilitzer. Results for Chapter 6 and Chapter 7 were performed by the author, and discussionswere held jointly with Prof. Christoph Ortner from the Department of Mathematics at UBC.Some of the choices for generating the machine learning interatomic potential were informed tothe author in discussion with Prof. Ortner and group.Figure 2.1 – Figure 2.24 were taken with permission from the publisher of the cited sources.The results of Chapter 4 have been published under the title, “Modeling solute-grain bound-ary interactions in a BCC Ti-Mo alloy using density functional theory” in Computational Mate-rials Science 229 (2023) 112393, authored by H. Umashankar, D. Scheiber, V.I. Razumovskiyand M. Militzer. Selected results of these studies were presented in the following conferences:Integrated Computational Grain Boundary Engineering (ICGBE) Symposium: THERMEC(Online) 2021, EUROMAT, Online (2021); Canadian Materials Science Conference (CMSC),Toronto, Canada (2022)vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxxv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Ab-initio modeling of bulk and defects in alloys . . . . . . . . . . . . . . . . . 42.1.1 Binary and multi-component bulk simulations . . . . . . . . . . . . . . 42.1.2 Vacancies in binary and multi-component bulk alloys . . . . . . . . . . 72.1.3 Grain boundaries and interfaces in alloys . . . . . . . . . . . . . . . . 102.1.4 Virtual crystal approximation for alloys . . . . . . . . . . . . . . . . . . 112.2 Simulations using semi-empirical models . . . . . . . . . . . . . . . . . . . . . 142.2.1 Grain boundaries in pure metals . . . . . . . . . . . . . . . . . . . . . . 142.2.2 Solutes and vacancies at grain boundaries in metals . . . . . . . . . . 182.2.3 Grain boundaries in alloys . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Machine learning simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.1 Overview of machine learning potentials . . . . . . . . . . . . . . . . . 25vi2.3.2 Types of machine learning interatomic models . . . . . . . . . . . . . . 252.3.3 Computational cost and performance of regression models . . . . . . . 272.3.4 Grain boundaries in pure metals . . . . . . . . . . . . . . . . . . . . . . 292.3.5 ML predictions of GB properties using atomic features . . . . . . . . . 292.3.6 Training set generation for MLIPs . . . . . . . . . . . . . . . . . . . . . 322.4 Summary of literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 Scope and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Solute-grain boundary interactions in a BCC Ti-Mo alloy . . . . . . . . . 374.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Structural models for bulk and GBs . . . . . . . . . . . . . . . . . . . . . . . . 384.3 DFT calculations settings and details . . . . . . . . . . . . . . . . . . . . . . . 384.4 Solute substitution in bulk alloy . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 Grain boundary energy of different GB configurations . . . . . . . . . . . . . . 424.6 GB Segregation in Ti-Mo alloy . . . . . . . . . . . . . . . . . . . . . . . . . . 434.6.1 Segregation energies of Ti/Mo to the GB plane . . . . . . . . . . . . . 434.6.2 Solute segregation energies to the GB plane for GB-I and GB-II . . . . 444.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7.1 Bulk properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7.2 GB properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 Interaction of Y with grain boundary in Ti-Mo alloys . . . . . . . . . . . . 555.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Structural models for alloy bulk and alloy grain boundaries . . . . . . . . . . . 555.3 Bulk and solute in bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.4 GB energies for different alloy configurations . . . . . . . . . . . . . . . . . . . 595.5 Segregation at GBs in Ti-Mo alloys for ten GBs . . . . . . . . . . . . . . . . . 615.5.1 Segregation energies of Ti/Mo to the GB plane . . . . . . . . . . . . . 615.5.2 Yttrium solute segregation energies . . . . . . . . . . . . . . . . . . . . 635.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.6.1 Bulk properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.6.2 Segregation of Y to GBs . . . . . . . . . . . . . . . . . . . . . . . . . . 675.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.8 Limitations of current study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 Surrogate modeling of defects in Pt-Au with EAM accuracy . . . . . . . . 756.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75vii6.2 Structural models for grain boundaries and bulk alloys . . . . . . . . . . . . . 766.2.1 GB structures for elemental Pt and Au . . . . . . . . . . . . . . . . . 766.2.2 Bulk alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.3 Computational workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.3.1 Steps I-III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3.2 Step IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3.3 Step V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3.4 Step VI-VII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.4.1 Bulk properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.4.2 GB properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.4.3 Bulk alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.4.4 Vacancies in bulk alloys . . . . . . . . . . . . . . . . . . . . . . . . . . 986.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117 Vacancy energetics in Pt-Au with DFT accurate surrogate models . . . . 1127.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.2 Structural models for alloy bulk and alloy GB . . . . . . . . . . . . . . . . . . 1137.2.1 Alloy bulk: 51.2 at. % Au alloy . . . . . . . . . . . . . . . . . . . . . . 1137.2.2 Alloy GB: 50 at. % Au alloy . . . . . . . . . . . . . . . . . . . . . . . . 1137.3 Computational workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.3.1 DFT details for training and validation data . . . . . . . . . . . . . . 1147.3.2 ACE model details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.4.1 Vacancy formation in the Pt- 51.2 at. % Au bulk alloy . . . . . . . . . 1187.4.2 Vacancy formation at Σ5 (210) GB in a Pt-50 at. % Au alloy . . . . . . 1257.4.3 Comparison of the computational cost of ACE models with DFT . . . . 1357.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . . 139Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Appendix A Electronic convergence and GB structure in Ti- 25 at.% Mo DFTstudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163A.1 Convergence tests with respect to k-point sampling and plane wave energy cutoffwith Y in bulk Ti - 25 at.% Mo alloy . . . . . . . . . . . . . . . . . . . . . . . 163viiiA.2 Procedure to create ten different configurations of alloy GB structures in Ti - 25at.% Mo alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164A.3 Atomic structures and GB properties of five stable alloy GB configurations in Ti-25 at.% Mo alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Appendix B Structure of GBs for Ti - 50 and 75 at.% Mo alloys . . . . . . . 167B.1 Information about GB and its structure for Ti - 50 at.% Mo alloys . . . . . . . 168B.2 Information about GB and its structure for Ti - 75 at.% Mo alloys . . . . . . . 169Appendix C Simulation settings for constructing training set in VASP . . . 171Appendix D Julia programming scripts for ACE models . . . . . . . . . . . . 172D.1 A Julia script using ACEpotentials.jl for building regression models . . . . . . 172D.2 A Julia script for geometry minimization in JuLIP.jl using a ACE model . . . 174ixList of Tables4.1 GB energy (𝛾𝐺𝐵) listed with concentration of Mo in GB region, GB plane foreach of five stable GB configurations . . . . . . . . . . . . . . . . . . . . . . . 434.2 Comparison of data for volume and bulk modulus from Bakker and DFT-computedvalues from the Materials Project repository for predicting solute segregation en-ergy using Eshelby-type model . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1 The chemical potential [in eV] of Mo (𝜇Mo) and Ti (𝜇Ti) atomic species in Ti -50 and Ti - 75 at. % Mo bulk alloy . . . . . . . . . . . . . . . . . . . . . . . . 586.1 List of all STGBs with misorientation angle (𝜃𝑚𝑖𝑠), Σ value, GB plane and numberof atoms in the computational domain (N𝐺𝐵). . . . . . . . . . . . . . . . . . . 766.2 All the data below correspond to ACE models for reproducing bulk properties ofAu and Pt. ACE model specifications such as correlation order (N𝑐𝑜𝑟𝑟), maximumdegree of polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), number of parameters (𝑁𝑝𝑎𝑟𝑎𝑚),number of rattled and deformed structures (𝑁 𝑠𝑡𝑟𝑠), total number of observationsi.e., energies, forces and virials (N𝑜𝑏𝑠), RMSE errors for observations such asenergies (RMSE𝐸) normalized by number of atoms in the simulation domain,forces (RMSE𝐹) for each atom and virials (RMSE𝑉) normalized by number ofatoms in the simulation domain and weights of energies (𝑤𝐸), forces (𝑤𝐹), virials(𝑤𝑉) used in training of ACE models . . . . . . . . . . . . . . . . . . . . . . . 846.3 Lattice constant (𝑎0), elastic constants(𝐶11, 𝐶12, 𝐶44), vacancy formation (𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 )and migration energy (𝐸𝑚𝑖𝑔𝑣𝑎𝑐 ) predictions for Pt and Au using Pt-EAM, Pt-BK,and Au-EAM, Au-BK models, respectively. . . . . . . . . . . . . . . . . . . . . 84x6.4 All the data shown below correspond to ACE models for reproducing vacancyin bulk properties of Au and Pt. ACE model specifications such as correlationorder (N𝑐𝑜𝑟𝑟), maximum degree of polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), numberof parameters (𝑁𝑝𝑎𝑟𝑎𝑚), number of rattled and deformed structures (𝑁 𝑠𝑡𝑟𝑠), totalnumber of observations i.e., energies, forces and virials (N𝑜𝑏𝑠), RMSE errorsfor observations such as energies (RMSE𝐸) normalized by number of atoms inthe simulation domain, forces (RMSE𝐹) for each atom and virials (RMSE𝑉)normalized by number of atoms in the simulation domain and weights of energies(𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used in training of ACE models. . . . . . . . . . 856.5 Lattice constant (𝑎0), elastic constants(𝐶11, 𝐶12, 𝐶44), vacancy formation (𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 )and migration energy (𝐸𝑚𝑖𝑔𝑣𝑎𝑐 ) predictions for Pt and Au using Pt-EAM, Pt-VACand Au-EAM and Au-VAC models, respectively. . . . . . . . . . . . . . . . . . 866.6 All the data shown below correspond to ACE models for reproducing EAM GBproperties in FCC Au and Pt. ACE model specifications such as correlation order(N𝑏𝑜𝑑𝑦), maximum degree of polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), number ofparameters (𝑁𝑝𝑎𝑟𝑎𝑚), number of rattled and deformed structures (𝑁 𝑠𝑡𝑟𝑠), totalnumber of observations i.e., energies, forces and virials (N𝑜𝑏𝑠), RMSE errorsfor observations such as energies (RMSE𝐸) normalized by number of atoms inthe simulation domain, forces (RMSE𝐹) for each atom and virials ((RMSE𝑉)normalized by number of atoms in the simulation domain and weights of energies(𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used in training of ACE models. . . . . . . . . . 896.7 Ranked (in decreasing order) predicted error of energies of 15 Au GB struc-tures, 𝜎(𝐸𝐺𝐵) normalized by total number of atoms in each structure, 𝑁𝑡𝑜𝑡𝑎𝑙 forAu-GB1, Au-GB2 and Au-GB3 ACE models. The GB structure and predictederror/uncertainty for each model are also listed. . . . . . . . . . . . . . . . . . 916.8 Ranks of decreasing predicted error of energies of 15 Pt GB structures, 𝜎(𝐸𝐺𝐵)normalized by total number of atoms in each structure, 𝑁𝑡𝑜𝑡𝑎𝑙 for Pt-GB1, Pt-GB2 and Pt-GB3 ACE models. The name of the GB structure is also listedalong with the predicted error/uncertainty for each model. . . . . . . . . . . . 92xi6.9 Model specifications for three ACE models, PtAu-BK1, PtAu-BK2 and PtAu-BK3 to simulate Pt-Au bulk alloys. Following are model hyperparameters: cor-relation order (N𝑐𝑜𝑟𝑟), maximum degree of polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡),number of parameters (𝑁𝑝𝑎𝑟𝑎𝑚) and details on training set: number of rattledand deformed structures (𝑁 𝑠𝑡𝑟𝑠), total number of observations i.e., energies, forcesand virials (N𝑜𝑏𝑠), RMSE errors for observations such as energies (RMSE𝐸) nor-malized by number of atoms in the simulation domain, forces (RMSE𝐹) for eachatom and virials (RMSE𝑉) normalized by number of atoms in the simulationdomain and weights of energies (𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used in trainingof ACE models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.10 Predicted error, 𝜎(𝐸𝑎𝑙𝑙𝑜𝑦) normalized by total number of atoms, 𝑁𝑡𝑜𝑡𝑎𝑙 for all 95alloy structures ranked based on decreasing order of predicted error predictedusing PtAu-BK1, PtAu-BK2 and PtAu-BK3 ACE models. . . . . . . . . . . . 976.11 Model hyperparameters, details on training data for ACE model, ACE modelregression and errors after regression of PtAu-VC1, Pt-Au-VC2 and PtAu-VC3models. Correlation order (N𝑐𝑜𝑟𝑟), the maximum degree of the polynomial (𝑛𝑚𝑎𝑥),radial cutoff (𝑟𝑐𝑢𝑡), number of parameters (𝑁𝑝𝑎𝑟𝑎𝑚), number of rattled and de-formed structures (𝑁 𝑠𝑡𝑟𝑠), the total number of observations i.e., energies, forcesand virials (N𝑜𝑏𝑠), RMSE errors for observations such as energies (RMSE𝐸) nor-malized by number of atoms in the simulation domain, forces (RMSE𝐹) for eachatom and virials (RMSE𝑉) normalized by number of atoms in the simulationdomain and weights of energies (𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used in trainingof ACE models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.12 Increasing order of predicted error of all 256 vacancy in Pt or Au site in bulkalloys, 𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖), normalized by total number of atoms in the simulationdomain, 𝑁𝑡𝑜𝑡𝑎𝑙 .. Each column represents data for the three models, i.e., PtAu-VC1, PtAu-VC2 and PtAu-VC3 ACE models. The data inside parenthesis areordered in terms of increasing at. % Au, i.e., 25, 50 and 75 at. % Au. . . . . 1097.1 ACE model parameters such as correlation order (N𝑏𝑜𝑑𝑦), maximum degree ofpolynomial (n𝑚𝑎𝑥), radial cutoff (r𝑐𝑢𝑡) and number of parameters for the potential(N𝑝𝑎𝑟𝑎𝑚) for “ITER0”, “ITER1”, “ITER02” and “ITER3” ACE models. Trainingdata details, i.e., number of structures (𝑁𝑠𝑡𝑟𝑠), total number of atoms (N𝑎𝑡𝑜𝑚𝑠),total number of observables for ACE fit (N𝑜𝑏𝑠). Weights are attached to eachobservation such as weights for energies (𝑤𝐸), forces (𝑤𝐹) and virials (𝑤𝑉) inorder to perform the ACE regression. Root mean squared error (RMSE) forobservations, i.e., energy (RMSE𝐸), force (RMSE𝐹) and virial (RMSE𝑉). . . . 120xii7.2 Decreasing order of uncertainty predictions or predicted error from energy ofvacancy in 125-atom Pt- 51.2 at. % Au bulk alloy, 𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢) normalizedby total number of atoms, 𝑁𝑡𝑜𝑡𝑎𝑙 for ITER0, ITER1, ITER2 and ITER3 ACEmodels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3 Model details and error table for ITER4, ITER5, ITER6 and ITER7 ACE mod-els. ACE model hyperparameters are correlation order (N𝑐𝑜𝑟𝑟), maximum degreeof polynomial (n𝑚𝑎𝑥), radial cutoff distance (r𝑐𝑢𝑡). Details about the trainingdata with number of parameters (N𝑝𝑎𝑟𝑎𝑚), number of structures (N𝑠𝑡𝑟𝑠), numberof atoms (N𝑎𝑡𝑜𝑚𝑠) and number of observations (𝑁𝑜𝑏𝑠). ACE model errors afterfitting: RMSE for energies (RMSE𝐸), forces (RMSE𝐹) and virials (RMSE𝑉).Weights of training data used for fitting of energy (𝑤𝐸), force (𝑤𝐹) and virial(𝑤𝑉). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.4 Uncertainty 𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 ) normalized by 𝑁𝑡𝑜𝑡𝑎𝑙 from a committee size of 100 forall 60 vacancy at Pt/Au GB sites from ITER4, ITER5, ITER6 and ITER7 ACEmodels. Uncertainties are ranked in decreasing order of magnitude. . . . . . . 131A.1 GB energy (𝛾𝐺𝐵), GB excess volume (𝛿𝑉), Mo content at GB region and GBplane for each of the five stable Ti - 25 at.% Mo alloy GB configurations . . . 165B.1 Number of Mo (N𝑀𝑜) and Ti sites (N𝑇𝑖) at GB plane for each of the ten, denotedby GB#, in Ti-50 at.% Mo alloy GBs . . . . . . . . . . . . . . . . . . . . . . . 168B.2 Number of Mo (N𝑀𝑜) and Ti sites (N𝑇𝑖) at GB plane for each of the ten, denotedby GB#, in Ti-75 at.% Mo alloy GBs . . . . . . . . . . . . . . . . . . . . . . . 170xiiiList of Figures2.1 (a) Lattice constants of Mo-Nb bcc alloys calculated using SQS shown in solidsquare (P) adapted from Jiang et al. Experimental data corresponding to 1 isreported from Goldschmidtl et al. and the data corresponding to # is from Cat-terall and Barker (1965) (b) Formation enthalpy of Mo-Nb bcc alloys calculatedusing SQS shown in solid square (P) adapted from Jiang et al. Experimentaldata corresponding to # is taken from Singhal et al. The CPA label correspondsto the Coherent Potential approximation (CPA) method used to calculate theproperties of alloys, data shown is from Sigli et al. . . . . . . . . . . . . . . . . 52.2 Shear moduli calculated for (a) Ti-Mo-Nb , (b)Ti-Mo-Sn, (c) Ti-Mo-Ta, (d) Ti-Mo-Zr and (e) Ti-Nb-Sn ternary systems. For all the systems shown above exceptTi-Nb-Sn, shear modulus increases from the Ti-rich side towards the Mo-rich side. 62.3 Vacancy formation enthalpy computed with DFT as a function of number of Cuin the first nearest neighbor shell (n𝐶𝑢). Different symbols indicate different alloycompositions, squares: 25 at. % Cu, crosses and diamonds: 50 at. % Cu withtwo different SQS structures and open circles: 75 at. % Cu . . . . . . . . . . . 72.4 Vacancy formation energy distributions in Fe-Cr-Ni alloy from (a) SQS1 struc-ture, (b) SQS2 structure and (c) SQS1+SQS2 combined. Red lines correspondto Gaussian distribution fits to the data. . . . . . . . . . . . . . . . . . . . . . 82.5 Vacancy formation energy distributions in Fe-Cr-Ni-Mn alloy shown as a full dis-tribution in (a), full distribution shown as Cr, Fe, Mn and Ni atom replacementsin (b) and the distribution in terms of different local environment, Op0, Op1,Op2, Op3 and Op4. Blue dotted lines correspond to Gaussian distribution fit tothe data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Distribution of vacancy formation energy using DFT in (a) pure metal compo-nents - Al, Sc, Zr, Ti and Hf , (b) binary Hf-Zr alloy, (c) HfTiZr ternary, (d)HfScTiZr, HEA alloy with (e) 5 and (f) 15 at. % Al. Vertical solid and dashedlines refer to mean and composition-weighted linear interpolation prediction fromVegard’s law, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.7 SFEs calculated for Ni and its alloys by DFT (shown in red squares) with thesupercell method and the error bars represent the standard deviation calculatedfrom nine different planes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10xiv2.8 Segregation energy profiles for all GBs computed using the VCA approach (shownin red circles) and the blue joined circles are predicted segregation energies frombond order analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.9 Calculated segregation energies for the W-Re system correlated with (a) Voronoivolume and bond-order parameters (b) Q4 , (c) Q5 and (d) Q6. The red dashedline in (c) represents a linear fit using only the values from two Σ3 tilt GBs,whereas the solid blue line represents a fit using all points. . . . . . . . . . . . 132.10 Schematic representations of how chemical and strain-induced components ofsegregation energy are calculated. The upper and lower panels show the calcu-lations required for GB and bulk geometry, respectively. . . . . . . . . . . . . . 142.11 (a) EAM computed GB energy as a function of disorientation angle for FCC Ni.(b) Scatter plot of 𝛾𝐺𝐵 for the same macroscopic degrees of freedom of each GBin Ni and Al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.12 (a) Scatter plot of GB energy versus disorientation angle for all ~ 43 millionstructures shown with the histogram of GB energy. (b) Box plot of GB energyfor [1 1 1] twist (top), [ 1 1 1] symmetric tilt GBs (middle and bottom), the boxeshighlight the data between the first and third quartiles of the distribution. Thered line shows the median, and the black squares are used to denote the mean. 162.13 The profile for 𝛾-surface for (a) Σ3(111) (b) Σ3(112) and (c) Σ11(113) GB forBCC Fe, left and right panels show the 3-D surface and color contour of GBenergy, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.14 (a) Simulation domain for a FCC Cu Σ5 (310) GB at 800 K with one open surfaceboundary condition on the right to allow for a change in GB density (microscopicdegrees of freedom). The two phases occurring in the phase transformation are(b) normal-kite and (c) split-kite structure separated by a “step” feature in (d).(e) The GB structure is fully transformed with two open surfaces instead of one. 182.15 (a1) - (d1) Metastable GB structures for Σ113 [001](15 1 0), green and orangeatoms represents FCC and GB atoms according to Common Neighbor Analysisin OVITO, sites within black region are used for Mg segregation energy analysis.In (a2) - (d2) geometric feature of Δ𝑉/𝑉0 for all the sites within the black outlinedregion in (a1) - (d1) are represented, where blue and red indicates large and smallatomic volumes. ΔE(𝑖) in (a3) - (d3) represents Mg segregation energy with redand blue indicating unfavorable and favorable segregation, respectively. . . . . 19xv2.16 (a) Histogram plot for Mg segregation energy at all 38 GBs studied by Huber andco-authors. Solid and dashed lines indicate smoothed distribution of the originaldata and data from machine learning prediction model, respectively. Red linesshow the White-Coghlan site occupation for three temperature values and dashedvertical lines show half-occupancy with a bulk concentration of 0.2 at. % Mg.(b) Density of states for all six solutes for symmetric (310) boundary and (c) fora disordered “s” boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.17 (a) Vacancy instability in Al Σ5 (210) GB, the vacancy initially sits at site 2,during the relaxation atom from site 1 moves to 2 leaving the vacancy at site 1.(b) Vacancy moving from site 2 to 1, in Σ13(527) GB in Al. . . . . . . . . . . 212.18 (a) Σ5 (013) [100] GB energy calculated as a function of the bulk concentrationof nickel and different colored lines indicate different temperatures. Some linesstop at intermediate compositions because of a miscibility gap at those temper-atures. At 700 and 800 K, the full composition can be studied since no phaseseparation occurs. (b) GB energy of Σ3(111) with composition of Ni at differenttemperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.19 EAM calculated vacancy formation energy at three GBs in (a-c) pure BCC V,(d-f) random equiatomic TiTaV alloy, and (g-i) equilibrated alloy with Ti segre-gation to GB region. The colored dots in each of the figures in (d) - (i) representdifferent chemical species. Grey squares, red circles and blue triangles indicateTi, Ta and V substitution, respectively. . . . . . . . . . . . . . . . . . . . . . 242.20 RMSE energy, meV/atom (left column) and force, eV/Å errors (right column) forNi, Cu, Li, Mo, Si and Ge. Five MLIPs, including GAP, MTP, SNAP, qSNAPand NNP, are shown along with three semi-empirical potentials – EAM, MEAMand Tersoff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.21 Test error in energies (a proxy for accuracy) as a function of computational cost(estimate for performance) for (a) FCC Cu and (b) diamond Si for ACE, GAP,MTP, qSNAP, SNAP and NN. . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.22 EAM calculated GB energies are shown as a function of the misorientation angle.Representative structures are shown for a few GBs in the inset. The SUs thatare present in the full misorientation are presented at the bottom of the figure.The color bar shows the degree of similarity to FCC bulk. . . . . . . . . . . . 302.23 A correlation color map of the output, segregation energy (Target) with differentinput features, which include physical properties like electronegativity, atomicfeatures for solutes and geometric features. . . . . . . . . . . . . . . . . . . . . 31xvi2.24 A schematic for creating a training set using RandSPG. The top row in each boxrepresents the operation on the structures and the middle row number and thelast row represent a unique set of structures and the total number of structures,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.1 Schematic of all DFT supercells used in the study, gray and red atoms refer to Tiand Mo sites, respectively. (a) 4x4x4 bulk supercell of 128 atom alloy structure(b) GB-I and (c) GB-II alloy structure. Note: The orthogonal GB supercells arerotated slightly to visualize all atomic positions. . . . . . . . . . . . . . . . . . 394.2 DFT calculated bulk solution energy distributions (out of 128 data points) of (a)Y, (b) Zr and (c) Nb in a Ti- 25 at. % Mo alloy. Solution energy distributionsof each solute is split into Ti and Mo site distributions indicated by blue andorange bars, respectively. The vertical dashed lines represent the mean value ofthe individual distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Schematic showing the arrangements of Ti and Mo atoms and structural units(GB region) of the (130) [001] for the GB-I and the GB-II structure are shownin (a) and (b), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 DFT computed segregation energies of Y, Zr and Nb to Ti and Mo host sitesin GB-I (diamond markers) and GB-II (square markers). Individual points arecolored based on whether Ti (blue) or Mo (orange) is substituted at the GBplane. Segregation energies below the dashed line suggest favorable segregation,while above the dashed line suggest unfavorable segregation. . . . . . . . . . . 454.5 Voronoi volume of Ti and Mo sites in the bulk alloy as a function of number ofMo atoms in the 1NN shell (out of 8 atoms). . . . . . . . . . . . . . . . . . . . 464.6 (a)Computed bulk solution energies at the Mo site plotted as a function of theVoronoi volume of the Mo sites before solute substitution. (b) Computed bulksolution energies at the Ti sites plotted as a function of the Voronoi volume ofthe Ti site before solute substitution (c) Computed bulk solution energies for Y,Zr and Nb at both Ti and Mo sites plotted together as a function of Voronoivolume after solute substitution. (d) Computed bulk solution energy for Y, Zrand Nb at both Ti and Mo sites plotted together as a function of Voronoi volumedifference between before and after solute substitution. . . . . . . . . . . . . . 48xvii4.7 (a) Computed bulk solution energies at Mo sites plotted as a function of numberof molybdenum atoms in the first nearest neighbor (1NN) shell around eachsolute-substituted site. (b) Computed bulk solution energies at Ti sites plotted asa function of number of titanium atoms in the first nearest neighbor (1NN) shellaround each solute-substituted site. The trend lines are obtained by averagingthe values at number of sites having the same # of Ti/Mo atoms in the 1NNshell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.8 (a) DFT computed segregation energies of Y, Zr and Nb to Ti (circles) and Mo(squares) host sites in GB-I and GB-II as a function of Voronoi volume of solutesubstituted sites (VX@Ti/Mo) (b)DFT computed segregation energies of Y, Zr andNb to Ti (circles) and Mo (squares) host sites in GB-I and GB-II as a function ofvolume difference between Voronoi volume after and before solute substitutionsat GB plane sites(∆VX−>Ti/Mo) . . . . . . . . . . . . . . . . . . . . . . . . . . 504.9 Comparison of average of segregation energies for Y, Zr and Nb to Ti and Mosites calculated using DFT and Eshelby-type model. . . . . . . . . . . . . . . . 504.10 (a) Eshelby-type model predictions for segregation energies of (3d-, 4d-, 5d-)transition metal solutes to the \Sigma5 GB plane in a Ti-Mo alloy system andaverage segregation energies of Y, Zr and Nb from DFT (b) Atomic volume ofsolute elements as taken from Bakker . . . . . . . . . . . . . . . . . . . . . . . 535.1 The atomic structure of Ti- (a) 50 and (b) 75 at. % Mo bulk alloys used tocompute the interaction of Y in bulk. A representative structure (out of 10 GBs)for GB in (c) 50 and (d) 75 at. % Mo alloys. Substitution of Y is done at theGB plane highlighted by the blue dotted line. Red and grey colored atoms referto Mo and Ti atoms, respectively . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Computed lattice constant from DFT for pure BCC Ti, Ti-25 at. % Mo alloy,Ti-50 at. % Mo alloy, Ti-75 at. % Mo alloy and pure BCC Mo. The dotted blueline connects the lattice constant of pure elements, i.e. BCC Ti and Mo. . . . 575.3 A stacked histogram showing the computed solution energies of Y in a Ti -(a) 50at. % and (b) 75 at. % Mo alloy. The vertical dashed line represents the mean ofthe computed distribution. Orange and blue color bar represents Y substitutionat a Ti and Mo site, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 595.4 Schematic of representative GB structures delineating GB and bulk region for(a) 50 at. % Mo and (b) 75 at. % Mo alloys. GB region is calculated for eachstructure using Common Neighbor Analysis (CNA) in OVITO . . . . . . . . . 60xviii5.5 Computed GB energies plotted as a function of the concentration of Mo in theGB region. Purple circles are the data from the previous chapter on 25 at. %Mo alloy. The dotted line is a linear rule of mixture line connecting the lowestGB energy to the pure BCC Mo Σ5(013) GB energy. A solid line connects theaverage GB energy for each alloy composition. . . . . . . . . . . . . . . . . . 615.6 (a) Scatter plot of Mo and Ti segregation energies at ten different GBs (GB_0....,GB_9) in Ti - 50 at.% Mo alloy with Ti and Mo substitutions shown in blue andorange open circles, respectively. (b) Histogram of segregation energies of Mo andTi shown in blue and orange bars, respectively. Fraction is separately calculatedfor each of the Mo and Ti substitutions. Vertical lines represent average valuesof segregation energies of Mo and Ti. . . . . . . . . . . . . . . . . . . . . . . . 625.7 (a) Scatter plot of Mo and Ti segregation energies at ten different GBs (GB_0....,GB_9) in Ti - 75 at.% Mo alloy with Ti and Mo substitutions shown in blue andorange open circles, respectively. (b) Histogram of segregation energies of Mo andTi shown in blue and orange bars, respectively. Fraction is separately calculatedfor each of the Mo and Ti substitutions. Vertical lines represent average valuesof segregation energies of Mo and Ti. . . . . . . . . . . . . . . . . . . . . . . . 635.8 (a) Scatter plot of segregation energies of Y in 50 at. % Mo alloy at ten differentGBs (GB_0,...., GB_9) with Mo and Ti substitutions shown in orange and blueopen circles, respectively. (b) Histogram of Y segregation energies for both Moand Ti sites shown in blue and orange bars, respectively. Fraction is separatelycalculated for each of the Mo and Ti substitutions. Vertical dotted lines representthe mean of Y segregation energies. . . . . . . . . . . . . . . . . . . . . . . . . 645.9 (a) Scatter plot of segregation energies of Y in 75 at. % Mo alloy at ten differentGBs (GB_0,...., GB_9) with Mo and Ti substitutions shown in orange and blueopen circles, respectively. (b) Histogram of Y segregation energies for both Moand Ti sites shown in blue and orange bars, respectively. Fraction is separatelycalculated for each of the Mo and Ti substitutions. The vertical dotted linesrepresent mean of Y segregation energies. . . . . . . . . . . . . . . . . . . . . . 645.10 (a) Voronoi volume of Mo and Ti sites shown in red and grey color on the y-axis,the number of Mo atoms in the first and second nearest neighbor (1NN + 2NN)shell are plotted on the x-axis. The data is from 50 at. % Mo alloy. (b) Voronoivolume of sites in the 75 at. % Mo alloy versus number of Mo atoms in 1NN +2NN shell. The solid lines indicate trend lines connecting average of volume foreach number of Mo atoms in the local neighborhood. . . . . . . . . . . . . . . 66xix5.11 (a) Scatter plot of bulk solution energy of Y at Mo sites in both Ti- 50 and 75at. % Mo alloy shown in red and squares, respectively along with 25 at.% Moalloy data in blue triangles. (b) Scatter plot of bulk solution energy of Y at Tisites in Ti- 50 at. % and Ti- 75 at. % Mo alloy shown by red and green triangles,respectively along with 25 at.% Mo alloy data in blue squares. . . . . . . . . 665.12 Scatter plot of solution energy of Y in bulk Mo and Ti sites are plotted againstthe respective Mo and Ti site volumes for the two alloys studied. The data for25 at.% , 50 at. % Mo and 75 at. % Mo is shown in blue, red and green circles,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.13 Scatter plot of segregation energy of Y to Ti sites as function of (a) Voronoivolume of Ti GB sites, (c) Voronoi volume of Y substituted at Ti sites and (e)volume difference when Y is substituted at Ti sites. Segregation energy of Y atMo sites as function of (b) Voronoi volume of Mo GB sites, (d) Voronoi volumeof Y substituted at Mo sites and (f) volume difference between Y at Mo site andthe original Mo site at GB plane. The data for 25, 50 and 75 at.% Mo alloy isshown in squares, triangles and circles, respectively. . . . . . . . . . . . . . . . 695.14 Elastic properties such as (a) Bulk modulus of Ti-Mo alloys as function of Mocontent in alloys and (b) Shear modulus as a function of concentration of Mo inTi-Mo alloys. Data is obtained from Table 4 as reported in study from Markerand colleagues. Solid lines represent linear rule of mixture law or Vegard’s lawprediction for alloy composition between 25 and 100 at. % Mo. . . . . . . . . . 705.15 (a) Parity plot of average DFT (< 𝐸 𝑠𝑒𝑔,𝐷𝐹𝑇𝑌@𝑇𝑖/𝑀𝑜 >) and Eshelby model (< 𝐸𝑠𝑒𝑔,𝑒𝑙𝑌@𝑇𝑖/𝑀𝑜>)computed segregation energy of Y in both 50 at. % and 75 at. % Mo alloys.(b) Scaling constants (𝐶𝑇𝑖/𝑀𝑜) used for the Eshelby-type model fits for volumeof the Mo (orange square) and Ti (blue circles) sites. . . . . . . . . . . . . . . 715.16 Comparison of segregation energy computed with DFT (orange squares) andEshelby-type model (blue squares) for segregation of Y at (a) Mo and (b) Ti in50 at. % Mo alloy and segregation of Y at (c) Mo and (d) Ti in 75 at. % Moalloy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.17 Parity plot of average DFT (< 𝐸 𝑠𝑒𝑔,𝐷𝐹𝑇𝑌@𝑇𝑖/𝑀𝑜 >) and Eshelby model (< 𝐸𝑠𝑒𝑔,𝑒𝑙𝑌@𝑇𝑖/𝑀𝑜>)computed segregation energy of Y to Mo sites (orange solid circles) and Ti sites(blue solid circles) in both 50 at. % and 75 at. % Mo alloys. The Eshelby modelis developed for an averaged scaling constants (𝐶𝑇𝑖/𝑀𝑜) from 25, 50 and 75 at.%Mo alloy and the elastic moduli predicted by Vegard’s law. . . . . . . . . . . . 73xx6.1 Visualization and sequence of individual steps used in the workflow. Each bubbleblock refers to either preparing training data, making predictions using an inter-atomic potential for relaxation, or building an ACE model. The arrows indicatethe direction in which the workflow is implemented. . . . . . . . . . . . . . . 786.2 Representative structures from training set of Pt-BK and Au-BK ACE modelsconsisting of rattled (a) bulk Au, (b) bulk Pt and cell deformed (c) bulk Au and(d) bulk Pt. Yellow and grey colored atoms are Au and Pt species, respectively.Atom sizes are reduced to visualize the structures better and help see the out-of-lattice atom positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.3 Training data consisting of observations i.e. energies, forces, and virials, for Pt-BK (top panel) and Au-BK (bottom panel), respectively. Frequency (log-scale)of observations plotted on the y-axis of energies (subfigures a, d) of structures,forces (subfigures b, e) of atoms and virials of structures (subfigures c, f) in thetraining set plotted in the x-axis. . . . . . . . . . . . . . . . . . . . . . . . . . 826.4 Parity plot for (a) energies, (b) forces and (c) virials computed using the ACEmodels (y-axis) and EAM potential (x-axis) for Au. The bottom panel corre-sponds to the parity plot for Pt in the case of (d) energies, (e) forces, and (f)virials predicted from ACE and EAM models. . . . . . . . . . . . . . . . . . . 836.5 The vacancy migration path shown as the relative energy of images (y-axis)plotted as a function of a number of images used in the Nudged Elastic Band(NEB) (x-axis) method to calculate the migration barrier energy. Figure (a)shows the three model results (Au-EAM, Au-BK and Au-VAC) for Au, andfigure (b) panel shows the results from Pt-EAM, Pt-BK and Pt-VAC models. . 866.6 EAM relaxed ground-state structures for three representative GBs – (a)Σ5b (31 0) 36.87 ◦, (b) Σ13a (0 1 5) 22.62 ◦and (c) Σ145a (0 9 8), 83.17 ◦ from thecurrent study. GB structures are visualized using centrosymmetry parameterscomputed using OVITO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.7 EAM calculated GB energies, 𝛾𝐺𝐵 (y-axis) reported in mJ m−2 plotted as afunction of misorientation angle (x-axis) for <100> STGBs in Pt (solid bluecircles) and Au (solid orange circles). . . . . . . . . . . . . . . . . . . . . . . . 886.8 Predictions from the ACE model (shown in the y-axis) for Au (left panel) andPt (right panel) are plotted as a function of misorientation angle plotted on thex-axis. Solid x markers represent data from EAM potential and hollow circlesmarkers represent the ACE model predictions from Au-GB1, Au-GB2 and Au-GB3 and Pt-GB1, Pt-GB2 and Pt-GB3 for Au and Pt, respectively. . . . . . . 90xxi6.9 Parity plot for predicted error (or) uncertainty for energy (y-axis), 𝜎(𝐸𝐺𝐵) nor-malized by number of atoms in the GB structure (𝑁𝑡𝑜𝑡𝑎𝑙) versus normalized trueerror, Δ𝐸𝐺𝐵/𝑁𝑡𝑜𝑡𝑎𝑙 (x-axis), of 15 GB structures in Au (top panel) and Pt (bot-tom panel). The left, middle and right panels show the results from the GB1,GB2 and GB3 models, respectively. Dotted black line represents perfect parityline, i.e., Y=X as a visual guide to the eye. . . . . . . . . . . . . . . . . . . . . 936.10 EAM computed mixing enthalpy (Δ𝐻𝑚𝑖𝑥) plotted as a function of the compositionof Au, 𝑐𝐴𝑢 in Pt-Au bulk alloys. Blue semi-transparent blue circles (o) represententhalpies for five structures in each composition and blue solid cross (x) linemarkers represent the average of mixing enthalpy in each composition. . . . . . 946.11 Mixing enthalpy predictions from (a) PtAu-BK1, (c) PtAu-BK2 and (e) PtAu-BK3 compared with EAM potential predictions. Parity plot for mixing enthalpy(Δ𝐻𝑚𝑖𝑥) predictions using (b) PtAu-BK1, (d) PtAu-BK2 and (f) PtAu-BK3 ACEmodels compared with EAM potential predictions (plotted on the x-axis) shownin blue circles. Perfect parity, i.e., Y=X, is represented by a red dotted line. . 966.12 Predicted error, 𝜎(𝐸𝑎𝑙𝑙𝑜𝑦) (shown in y-axis) and true error, Δ𝐸𝑎𝑙𝑙𝑜𝑦 (shown in x-axis) for all 95 Pt-Au alloy structures normalized by total number of total atoms,𝑁𝑡𝑜𝑡𝑎𝑙 using (a) PtAu-BK1, (b) PtAu-BK2 and (c) PtAu-BK3 ACE models. Thedotted black line is used to represent perfect parity, i.e., Y=X. . . . . . . . . . 986.13 (a) Atomic structure of 256-atom Pt-25 at. % Au bulk alloy visualized withOVITO. A Pt site is highlighted with an arrow where a vacancy will be situated.(b) Atomic structure of a 255-atom vacancy in alloy structure with vacancy’slocation highlighted with an arrow. (c) The Widom structure of Au atom sub-stituted at a previous Pt site pointed out using an arrow. Yellow and grey atomsrefer to Au and Pt species, respectively. . . . . . . . . . . . . . . . . . . . . . . 996.14 Histogram of chemical potential (𝜇) distribution of Pt and Au predictions fromEAM potential shown in orange and blue bars, respectively. Predictions for Pt-25 at. % Au, Pt- 50 at. % Au and Pt- 75 at. % Au alloys are shown in (a), (b)and (c), respectively. Solid vertical lines represent the mean of each distribution. 1006.15 Stacked histogram of vacancy formation energy distributions at both Pt and Ausites for (a) Pt-25 at. % Au, (b) Pt- 50 at. % Au and (c) Pt-75 at. % Au alloy,orange and blue bars represent data from Pt and Au sites, respectively. Thefraction of sites from all 256 bulk sites is represented in y-axis. . . . . . . . . . 1016.16 Average vacancy formation energy (shown in blue solid circles) along with stan-dard deviation of distribution (blue error bars) plotted as a function of alloycomposition of Au (shown in x-axis). Black solid lines represent the trend of av-erage vacancy formation energy, and the orange dotted line represents the lineartrend connecting pure Pt and Au. . . . . . . . . . . . . . . . . . . . . . . . . . 102xxii6.17 Chemical potential of Au (𝜇𝐴𝑢) and Pt (𝜇𝑃𝑡) shown in left and right panel,respectively. Predictions from PtAu-VC1 ACE model (plotted in y-axis) andEAM potential (shown in x-axis) are represented with blue circles. Perfect parityline, i.e., Y=X is represented in red dotted line and a band of +/- 0.1 eV areused as guide to human eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.18 Parity plot for vacancy formation energy at 256 sites in three bulk alloys pre-dictions from PtAu-VC1 ACE model (in y-axis) and EAM potential (in x-axis).Red dotted line represents perfect parity between predictions, Y=X and greyregion represents +/- 0.1 eV band around the perfect parity line as a guide tothe human eye. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.19 Parity plot for vacancy formation energy at both Au and Pt sites for (a) Pt-25 at. % Au, (b) Pt-50 at. % Au and (c) Pt-75 at. % Au. Predictions fromPtAu-VC2 ACE model (y-axis) is plotted along with predictions from EAMinteratomic potential (x-axis) shown in blue circles (o). Perfect parity, i.e., Y=Xis represented with red dotted line and a grey colored band of +/- 0.1 eV is shownas a guide to the eye. The black squares represent the structures included in thetraining set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.20 EAM calculated vacancy formation energy as a function of number of Au atomsin the 1NN local neighborhood (out of 12 atoms) at Au sites (top panel) and Ptsites (bottom panel) for Pt-25 at. % Au (left panel), Pt-50 at. % Au (middlepanel) and Pt-75 at. % Au (right panel) alloy. Black box markers represent therepresentative sites included for PtAu-VC3 ACE model from each number of Auatoms in the 1NN neighbor shell for both sites for three alloy compositions. . . 1066.21 Parity plot for predictions of all 256 vacancy formation energies (in blue hollowcircles) from PtAu-VC3 ACE model (y-axis) and PtAu-EAM potential (x-axis)in (a) 25 at. % Au, (b) 50 at. % Au and (c) 75 at. % Au bulk alloys. Reddotted line represents the Y=X perfect parity line, and the grey region shows a+/- 0.1 eV band around the Y=X parity plot. The red line and the band areused to guide the eye. The black squares represent the structures included in thetraining set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.22 Normalized uncertainty (y-axis) of vacancy in Pt/Au site in bulk alloy, 𝜎𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢/𝑁𝑡𝑜𝑡𝑎𝑙from ACE model plotted with normalized true error (x-axis) between ACE andEAM model, Δ𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢/𝑁𝑡𝑜𝑡𝑎𝑙for PtAu-VC1 (top panel), PtAu-VC2 (middlepanel) and PtAu-VC3 (bottom panel). The left panel is for 25 at. % Au alloy,middle panel for 50 at. % Au alloy and right panel for 75 at. % Au alloy for allthree models, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110xxiii7.1 The structure of 125-atom bulk alloy structure used for calculating vacancy for-mation energy containing 51.2 at. % Au visualized by OVITO. Yellow and greyatoms represent Au and Pt species, respectively. . . . . . . . . . . . . . . . . . 1137.2 Simulation domain of 50 at. % Au Σ5(210) Pt-Au alloy GB visualized withOVITO. Yellow and grey-colored atoms represent Au and Pt species, respectively. 1147.3 Individual structure types for the training set used for the initial “ITER0” ACEmodel, left and right panels indicate bulk and defect structures ,i.e., vacancyin bulk and GB. The top and bottom panels show the pure components (FCCPt and Au) and Pt-Au alloy structures. The top left panel shows some of therattled and deformed structures of 125-atom FCC Pt and Au structures. Topright depicts some of the rattled and deformed 255-atom vacancy in FCC Ptand Au, 114-atom Σ5(310) GB structure, 120-atom Σ5(210) GB structure. Thebottom left panel shows one of the rattled and deformed 125-atom alloy Pt-Aubulk structures out of a total of 7 compositions, and the bottom right panelshows one of the rattled and deformed structures of one composition120-atomΣ5(210) GB structure out of a total of 3 compositions. . . . . . . . . . . . . . 1167.4 Workflow diagram for iterative ACE model fitting involving different steps: (I)collecting all initial structures relevant for property of interest, (II) creating anACE model using Bayesian Linear Regression (BLR), (III) this steps involvesmultiple sub-steps – (A) Calculate vacancy formation energies, (B) evaluate un-certainty using committee-of-models approach, (C) Pick ten vacancy andWidom-type structure that show top ten uncertainty (D) add them into training set to beused for next iteration ACE model regression and (E) retrain ACE model usingupdated training set using BLR and (IV) stop the iterations when uncertain-ties are below an user-defined threshold value (or) ACE model predictions showgood agreement with reference model predictions (or) after a certain number ofiterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.5 Histogram of DFT-PBEsol computed vacancy formation energy (𝐸 𝑓 𝑜𝑟𝑚,𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) forall 125 sites in Pt-51.2 at. % Au alloy, each colored bar represents a fraction ofsites with vacancy formation energy within the bin size. Yellow and grey barsdepict the vacancy formation energy at Au and Pt sites, respectively. . . . . . 1197.6 Parity plot for (a) energies of Pt, Au and Pt-Au alloy structures (b) forces [eV/Å]of Pt and Au atoms and (c) virial of Pt, Au and Pt-Au alloy structures computedwith “ITER-0” ACE model (plotted on the y-axis) and DFT-PBEsol (plottedon the x-axis) using hollow blue circles. The total number of data points forenergies are 1006 (total number of structures), and the total number of atoms is131967. The dotted red line illustrates perfect parity, i.e., the Y=X line betweenACE and DFT data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120xxiv7.7 Parity plot of vacancy formation energy in Pt- 51.2 at. % Au bulk alloy using(a) ITER0, (b) ITER1, (c) ITER2 and (d) ITER3 ACE model with ACE andDFT predictions plotted on y and x axis, respectively. The perfect parity, i.e.,Y=X is shown by a red dashed line and a grey band is used as a visual guideto represent data within +/- 0.1 eV of the perfect parity line. The black coloredmarkers show the structures added to the training set. The red star markersshow top ten uncertainties in all of the 125 structures for the current iteration. 1227.8 Scatter plot of normalized predicted error, i.e., uncertainty predictions 𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢)/𝑁𝑡𝑜𝑡𝑎𝑙(plotted on the y-axis) and normalized true error, Δ𝐸𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢/𝑁𝑡𝑜𝑡𝑎𝑙 (plotted onthe x-axis) of all four ACE models – ITER0, ITER1, ITER2 and ITER3 modelshown in green, black, blue and red triangles, respectively. Perfect parity, i.e.,Y=X, is plotted using the dotted black line. . . . . . . . . . . . . . . . . . . . 1247.9 Atomic structure of 120-atom Pt- 50 at. % Au alloy GB colored according tolocal environment, i.e., FCC bulk and GB sites. FCC bulk and GB atomic sitesare colored by green and grey colors, respectively. The GB plane is highlightedwith a grey semi-transparent line and the structural units (SU) are highlightedand connected by blue dotted lines. . . . . . . . . . . . . . . . . . . . . . . . . 1257.10 Histogram of DFT computed vacancy formation (𝐸 𝑓 𝑜𝑟𝑚,𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) at 60 GB sitesshown with colored bars for fraction of sites within a specified bin size. Yellowand grey colored bars represent data from Au and Pt sites, respectively. . . . . 1287.11 Histogram of DFT computed chemical potential of (a) Au and Pt sites (60 intotal) in Pt - 50 at. % Au alloy Σ5(210) and (b) Au and Pt sites (125 in total)in bulk FCC Pt- 51.2 at. % Au alloy is shown in (a) and (b), respectively.Thesolid blue line for each histogram represents the mean of each chemical potentialdistribution and the dotted line along with the arrow represents the shift in meanof Au and Pt chemical potential between GB and bulk. Yellow and grey coloredbars represent chemical potential for Au and Pt, respectively. . . . . . . . . . . 1297.12 Parity plot for vacancy formation energy at Pt/Au sites in Σ5(210)GB Pt- 50at. % Au alloy. Y-axis for (a), (b), (c) and (d) represents data from ACEmodels ITER4, ITER5, ITER6 and ITER7. X-axis data is the reference modelpredictions from DFT-PBEsol. The red dotted line represents perfect parity,i.e., Y=X and a +/- 0.1 eV region around perfect parity is represented by agrey band. The black colored markers show the structures added to the trainingset. The red star markers show top ten uncertainties in all of the 120 vacancystructures for the current iteration. . . . . . . . . . . . . . . . . . . . . . . . . 130xxv7.13 Scatter plot of normalized predicted, 𝜎(𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢)/𝑁𝑡𝑜𝑡𝑎𝑙 (shown in y-axis) andnormalized true error, |Δ𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 |/𝑁𝑡𝑜𝑡𝑎𝑙 (shown in x-axis) shown in green,black, blue and red circles for the ITER4, ITER5, ITER6 and ITER7 models,respectively. Perfect parity line, Y=X is shown by a black dotted line. . . . . . 1327.14 Three examples of structures undergoing structural changes when performinggeometry relaxation with vacancy at GB sites. Displacement vectors (repre-sented with black arrows) are shown for atoms with significant movement in thestructure. Atoms show the initial structure before relaxation, and the arrowsdesignate the final positions of atoms. The grey semi-transparent line indicatesthe GB plane. The left panel of structures represent the DFT relaxed structures,while the right panel indicate ACE relaxed structures. Yellow and grey atomsrepresent gold and platinum species, respectively. . . . . . . . . . . . . . . . . 1337.15 Schematic of potential energy surface (PES) shown as a function of configura-tional coordinates predicted by DFT (green semi-transparent line) and Surrogatemodel (orange semi-transparent line). The two minima highlight the energies oftwo different configurations (blue-colored atoms shown with bonds) and a possi-ble relaxation path between the two configurations. Red dots represent samplingstructures to build the surrogate model. The final structures for surrogate modeland DFT are shown in orange and red solid circles. The schematic is taken withpermission from the open access article of Mosquera-Lois and co-authors. . . . 135A.1 The detailed supercell setup along with the results for convergence with respect tok-point sampling and plane wave energy cutoff is shown in (a) and (b), respectively. 163A.2 Numbering of planes in the y-direction of the bulk slab of the Σ5 GB, 10 differentplanes lead to 10 different GBs. Grey and red atoms refer to Ti and Mo sites,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164A.3 Atomic structures of all five stable GB structures of configuration 1, 2, 3, 4 and5 (from Table A.1) shown in (a), (b), (c), (d) and (e), respectively. Grey and redatoms refer to Ti and Mo sites, respectively. . . . . . . . . . . . . . . . . . . . 166B.1 Alloy structure containing 60 atoms for (a) 50 at.% Mo and (b) 75 at.% Mo alloy.Each y-plane is highlighted using a blue arrow and a number that will used todistinguish different GB that are constructed and to calculate Y segregationenergies. Red colored and grey colored atoms correspond to Mo and Ti species,respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167xxviB.2 Visualization of atomic structure of 50 at.% Mo alloy GBs that are labelled inthe text as (a) GB_0, (b) GB_1, (c) GB_2, (d) GB_3, (e) GB_4, (f) GB_5,(g) GB_6, (h) GB_7, (i) GB_8 and (j) GB_9. Red colored and grey coloredatoms correspond to Mo and Ti species, respectively. . . . . . . . . . . . . . . 169B.3 Visualization of atomic structure of 75 at.% Mo alloy GBs that are labelled inthe text as (a) GB_0, (b) GB_1, (c) GB_2, (d) GB_3, (e) GB_4, (f) GB_5,(g) GB_6, (h) GB_7, (i) GB_8 and (j) GB_9. Red colored and grey coloredatoms correspond to Mo and Ti species, respectively. . . . . . . . . . . . . . . 170xxviiList of Symbols𝐴𝐺𝐵 Area of Grain boundary simulation domain, Å2𝑎0 Lattice constant, Å𝐶11,𝐶12,𝐶44 Elastic constants, eV/Å3ETi,alloy,EMo,alloy Per-atom energy of Ti/Mo in Ti-Mo alloy, eV𝐸 𝑠𝑜𝑙X@Ti,𝐸𝑠𝑜𝑙X@Mo Solution energy of solute X in Ti-Mo alloy at Ti / Mo site, eV𝐸𝑠𝑒𝑔𝑀𝑜@𝑇𝑖,𝐸𝑠𝑒𝑔𝑇𝑖@𝑀𝑜 Segregation energy of Mo/Ti to Ti/Mo host site, eV𝐸𝑠𝑒𝑔X@Ti,𝐸𝑠𝑒𝑔X@Mo Segregation energy of solute X at Ti/Mo site, eV𝐸𝑠𝑒𝑔Y@Ti,𝐸𝑠𝑒𝑔Y@Mo Segregation energy of solute Y at Ti/Mo site, eV𝐸 𝑠𝑜𝑙Y@Ti,𝐸𝑠𝑜𝑙Y@Mo Solution energy of solute Y at Ti/Mo site, eV𝐸𝑟𝑒𝑝 Energy of representative site in bulk, eV𝐸128,𝐸125 Energy of 128-atom, 125-atom supercell, eV𝐸𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 , 𝐸𝑚𝑖𝑔𝑣𝑎𝑐 Formation and migration energy of vacancy in bulk, eV𝐸𝑒𝑎𝑚𝐺𝐵 , 𝐸𝐴𝐶𝐸𝐺𝐵 Energy of GB supercell calculated by EAM and ACE model, eV𝐸𝑒𝑎𝑚𝑎𝑙𝑙𝑜𝑦, 𝐸𝐴𝐶𝐸𝑎𝑙𝑙𝑜𝑦 Energy of Pt-Au bulk alloy structure calculated byEAM and ACE model, eVxxviii𝐸𝑒𝑎𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢, 𝐸𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 Energy of vacancy at Pt/Au in site “i” , eV𝐸𝑓 𝑜𝑟𝑚,𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 Vacancy formation energy of Pt/Au site in bulk, eV𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢,𝑖 Energy of supercell with vacancy at Pt/Au site “i” in bulk alloy, eV𝐸𝐵𝐾𝑎𝑙𝑙𝑜𝑦 Energy of alloy bulk supercell, eV𝐸𝑓 𝑜𝑟𝑚,𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 Vacancy formation energy of Pt/Au site in GB region, eV𝐸𝑠𝑒𝑔,𝑒𝑙𝑋@𝑇𝑖/𝑀𝑜Eshelby-model predicted segregation energy of solute X toTi/Mo site, eV𝐸𝐵𝐾,𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢, 𝐸𝐵𝐾,𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 Energy of supercell with vacancy at Pt/Au site in bulk alloycalculated by DFT and ACE, eVE𝐺𝐵VC@Pt/Au,i Energy of supercell with vacancy in Pt/Au site in GB region, eVE𝐵𝐾,𝐼VC@Pt,i/Au,iEnergy of supercell with vacancy in Pt/Au site in bulk alloystructure calculated with model 𝐼 in the committee-of-models, eV<E𝐵𝐾,𝐼VC@Pt,i/Au,i>Mean of energy of supercell with vacancy in Pt/Au site in bulk alloycalculated with all the models in the committee-of-models, eVE𝐺𝐵,𝐼VC@Pt,i/Au,iEnergy of supercell with vacancy in Pt/Au site in GB regioncalculated with model 𝐼 in the committee-of-models, eV<E𝐺𝐵,𝐼VC@Pt,i/Au,i>Mean of energy of supercell with vacancy in Pt/Au site in GB regioncalculated with all the models in the committee-of-models, eV𝐸𝐺𝐵,𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢, 𝐸𝐺𝐵,𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢Energy of supercell with vacancy at Pt/Au site in GB regioncalculated by DFT and ACE, eV𝐸𝐵𝐾𝑎𝑙𝑙𝑜𝑦 Energy of alloy bulk supercell, eV𝐸𝐺𝐵𝑎𝑙𝑙𝑜𝑦 Energy of alloy grain boundary supercell, eV𝐸𝐺𝐵 Energy of GB supercell of pure system, eVxxixE𝐵𝐾Pt,i–>Au,i, E𝐵𝐾Au,i–>Pt,i Energy of Widom-type structure in bulk, eVE𝐺𝐵Pt,i–>Au,i, E𝐺𝐵Au,i–>Pt,i Energy of Widom-type structure in GB region, eV𝐺𝐴 Shear modulus of host A, eV/Å3𝐾𝐴, 𝐾𝐵 Bulk modulus of host A and solute B, eV/Å3𝑁𝐴𝑢, 𝑁𝑃𝑡 Number of Au/Pt atoms, –𝑁𝐺𝐵 Number of atoms in grain boundary supercell, –𝑁𝑡𝑜𝑡𝑎𝑙 Total number of atoms in the supercell, –𝑁𝑐𝑜𝑟𝑟 Correlation order of body-order expansion in ACE, –𝑁𝑐𝑚𝑡𝑒 Number of models in the committee-of-models, –𝑁𝑜𝑏𝑠 Number of observations in training set, –𝑁𝑝𝑎𝑟𝑎𝑚 Number of ACE model parameters, –𝑁𝑠𝑡𝑟𝑠 Number of structures in training set for ACE model, –𝑛𝑚𝑎𝑥 Maximum degree of polynomial used in ACE model, –𝑄4, 𝑄5, 𝑄6 Steinhardt’s bond orientational order parameters, –𝑅𝑀𝑆𝐸𝐸 ,𝑅𝑀𝑆𝐸𝐹 ,𝑅𝑀𝑆𝐸𝑉 Root Mean Square Error for energy, eV, force, eV/Å, and virial, eV𝑟𝑐𝑢𝑡 Radial cutoff hyperparameter in ACE model, Å𝑉𝑠𝑖𝑡𝑒𝑠 Voronoi volume of Mo/Ti sites, Å3𝑉𝑋,𝑖 Voronoi volume of solute X at site “i”, Å3𝑉𝑇𝑖/𝑀𝑜,𝑖 Voronoi volume of Ti/Mo site “i”, Å3xxx𝑤𝐸 , 𝑤𝐹 , 𝑤𝑉 Weights of energy, force and virial observables in ACE regression, –𝛾GB Grain Boundary energy, mJ m−2Δ(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) True error for energy of vacancy at Pt/Au site supercell, eVΔ(𝐸𝐺𝐵) True error for energy of GB supercell, eVΔ(𝐸𝑎𝑙𝑙𝑜𝑦) True error for energy of alloy bulk supercell, eVΔ(𝐸𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) True error for energy of vacancy at Pt/Au in alloy bulk supercell, eVΔ(𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) True error for energy of vacancy at Pt/Au in alloy GB supercell, eVΔ𝐻𝑚𝑖𝑥 Enthalpy of mixing, meV/at.∆VX−>Ti/Mo Voronoi volume difference when solute “X” substituted at Ti/Mo, Å3𝛿𝑉Grain boundary excess volume, Å𝜃𝑚𝑖𝑠 Misorientation angle, ◦𝜇𝐵𝐾𝑃𝑡/𝐴𝑢,𝑖 Chemical potential of Pt/Au site “i” in bulk, eV𝜇Pt@GB,i, 𝜇Au@GB,i Chemical potential of Pt/Au site “i” in GB, eVΣCoincidence Site Lattice value, –𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) Uncertainty of energy of vacancy in Pt/Au site , eV𝜎(𝐸𝑎𝑙𝑙𝑜𝑦) Uncertainty of energy of Pt-Au alloy bulk structure, eV𝜎(𝐸𝐺𝐵) Uncertainty of energy of grain boundary structure, eV𝜎(𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) Uncertainty of energy of vacancy in Pt/Au site at GB supercell, eV𝜎(𝐸𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) Uncertainty of energy of vacancy in Pt/Au site at bulk supercell, eVxxxiList of Abbreviations1NN First Nearest Neighbor2NN Second Nearest NeighborACE Atomic Cluster ExpansionAPB Anti-Phase BoundaryAPT Atom Probe TomographyATAT Alloy Theoretic Automated ToolkitBCC Body Centered CubicCALPHAD CALculation of PHAse DiagramsCNA Common Neighbor AnalysisDFT Density Functional TheoryEAM Embedded Atom MethodFCC Face Centered CubicFIRE Fast Inertial Relaxation EngineGAP Gaussian Approximation PotentialGB Grain BoundaryHCP Hexagonal Close PackedMAE Mean Absolute ErrorMC/MD Monte Carlo / Molecular DynamicsMCSQS Monte Carlo Special Quasirandom StructuresxxxiiMLIP Machine Learning Interatomic PotentialMM Molecular MechanicsMP Materials ProjectMS Molecular StaticsOVITO Open Visualization ToolPAW Projector Augmented WavePBE Perdew-Burke-ErnzerhofRMSE Root Mean Square ErrorSNAP Spectral Neighbor Analysis PotentialSQS Special Quasirandom StructuresSRO Short Range OrderSTGBs Symmetric Tilt Grain BoundariesVASP Vienna Ab-initio Simulation PackageVCA Virtual Crystal ApproximationVFE Vacancy Formation EnergyXC Exchange-CorrelationxxxiiiAcknowledgementsThis graduate program has taught me many invaluable life lessons. First and foremost, I wouldlike to thank my research supervisor, Prof. Matthias Militzer, for his support and supervision.Thank you to my supervisory committee members - Prof. Joerg Rottler and Prof. ChadwickSinclair, for their valuable feedback and insights.Visiting and collaborating with the Materials Center Leoben (MCL) in Austria was a mean-ingful experience that provided a solid foundation for my thesis work. I would especially like tothank Dr. Vsevolod Razumovskiy and Dr. Daniel Scheiber for hosting me at the ComputationalMaterials Design group and for their patience in answering my DFT questions and discussingsolute segregation science.Collaborating with Prof. Christoph Ortner in the Department of Mathematics gave me theskills to leverage the ACE models developed by his group, which were used in my research. Heand his group introduced me to the Julia programming language and ACE methodology, whichhelped shape my thesis. Some of the programming and technical discussions with Dr. LiamHuber during my early PhD years were invaluable.I also want to acknowledge the time I spent with the Microstructure research group at UBC-MTRL and the countless social sessions we used to discuss science, among other topics. Along list of current students and alumni who have been supportive and helpful throughout thisjourney: Maddie, Mariana, Nicolas, Arthur, Ayush, Siavash, Ali K., Zhuo, Shixin, Minghui,Shuheng, Freddy, Yuri, Nobu, Ali D., Naveen, Sabyasachi, Vinoth and Guru. I have alwaysappreciated the companionship of the IITM groups, particularly the Meta wing, where weenjoyed exploring various cities in North America. I am also grateful to the ICME group forthe WhatsApp calls and the memorable trip to Chicago.Ottawa and Toronto have always felt like a second home to me. I truly enjoyed the timespent there with my close family and their extended family, as well as the delicious dosas andmutton rolls I savored. Finally, I want to thank Appa, Amma, and my sister. They wereconstant pillars of emotional support throughout this long journey.xxxivDedicationTo mywho nurtured and cared for me during my childhood.I dedicate this thesis to your lives.xxxvChapter 1IntroductionMany materials that encompass everyday life are metallic alloys. These alloys contain amultitude of chemical elements that create desirable properties to fulfill the requirements forthe applications of these alloys. For instance, stainless steel is an iron-based alloy with at leastfive elements such as Fe, Mn, Cr, C, and Al, to give it corrosion resistance and high ductility [1].Other alloying elements are added to improve mechanical properties and resistance to oxidationand creep, including nickel, molybdenum, copper, titanium, aluminum, silicon, niobium, nitro-gen and carbon. Designing these technologically relevant alloys remains a big scientific andtechnological challenge. In 2011, the Materials Genome initiative was established to developa roadmap to advance materials design and discover new materials. A successful example ofthis approach was achieved by QuesTek® Innovations LLC with the Computational MaterialsDesign approach to develop a next-generation gear steel Ferrium C64 [2]. The strengtheningmechanisms of these alloys are associated with defects in their crystalline structure, i.e. grainboundaries, dislocations, solutes and precipitates. For example, the interaction of solutes withgrain boundaries can be used to control the grain size resulting from thermo-mechanical pro-cessing. The latter mechanism is currently being pursued to create tough and strong Pt-Aunanocrystalline alloys for improving high-cycle fatigue resistance [3].Due to the structurally complex nature of grain boundaries, alloying elements/solutes oftensegregate to them, facilitating reduction in grain growth at operating conditions that are wellabove room temperature [4]. In order to quantify the solute segregation behavior at grainboundaries, various experimental and simulation techniques have been developed. One recentlydeveloped tool is Atom Probe Tomography (APT), which enables atomistic scale resolutions [5].Many published studies [6–8] have been conducted to quantify solute segregation to individualinterfaces in multi-component alloys such as nickel-based superalloys, steels and titanium alloys.The main limitations of these studies are that unveiling physical mechanisms of solute segre-gation phenomena is often challenging, and characterizing multiple interfaces inside a material1is time-consuming. An alternative method of studying this phenomenon is to use computersimulations [9].Various approaches exist on atomic scale modeling of these interactions, one is the “first-principles” approach where the electrons inside the material are treated explicitly, and anotheris semi-empirical, where the electronic interactions are captured via surrogate models. The firstapproach was applied to study grain boundary (GB) segregation and cohesion of various solutesin, for example, Ni, Ti, Fe, and Mg host systems [10–15]. However, these models are computa-tionally expensive and often involve millions of CPU hours on large superclusters. For instance,a report [16] published for the United Kingdom (UK) supercomputer, ARCHER, states thatabout 5.4 million node hours are spent on a commercial electronic structure theory code suchas the Vienna Ab-initio Simulation Package (VASP) among other codes. Such consumption ofCPU hours also contributes to global carbon dioxide emissions, which makes them undesirablefor use in the long term in the current fight against climate change.This motivates the other types of models, traditionally known as semi-empirical models, toprovide predictions at a lower computational cost but at the expense of reduced accuracy. Inparticular, the embedded atom method (EAM) and its variations where a physically informedinteratomic potential is used to parameterize the complex potential energy surface of metalsand binary alloys [17–19]. The parameterization was often done to reproduce simple materialproperties such as the lattice constant, bulk modulus, vacancy formation energy and surfaceenergy. [20,21] Such a model provided a good starting point to capture solute segregation effectsin metals and alloys and unveil its complex distributional nature [22–24]. However, these inter-actions are limited to the predictive abilities of the interatomic potential used to simulate them.Thus, these interactions are not accurately captured quantitatively. A recent advancement inbuilding interatomic potentials with the help of training structures data, i.e., providing struc-tural and atomic data from electronic structure calculations, allowed revisiting this challenge.This opportunity opens possibilities for exploring computational complexity versus accuracyfor simulating materials for a specific purpose, such as solute/vacancy segregation in alloys.This thesis work investigates the interactions of solute and vacancies in the bulk and grainboundaries in binary alloys using atomistic simulations. The thesis is organized as follows:Chapter 2 reviews the recent studies on modeling chemically complex alloys using DFT, fol-lowed by a discussion on structural descriptors of grain boundaries, and finally some discussionon surrogate models available in the literature to model these complex defect interactions. Abackground on modeling different types of defects in metallic alloys, along with some back-ground on DFT and Atomic cluster expansion (ACE) models, is also provided. Chapter 3provides the objectives and scope of the thesis research. Chapter 4 illustrates the use of DFTcalculations to compute the interactions of solutes in the bulk and grain boundaries in a Ti-Mo alloy. Chapter 5 provides an extension of these simulations to Ti-Mo alloys with different2compositions using the example of one solute, Y, segregating to GBs for the full compositionspace. Chapter 6 provides an alternate approach to DFT simulations by using the recentlydeveloped Atomic Cluster Expansion (ACE) surrogate models to simulate the bulk and GBsin Pt and Au and binary bulk Pt-Au alloys. In Chapter 7, ACE models trained on DFT dataare presented to determine vacancy interactions at bulk and GBs in Pt-Au alloys. Chapter 8provides a summary of the thesis along with recommendations for future work.3Chapter 2Literature review2.1 Ab-initio modeling of bulk and defects in alloys2.1.1 Binary and multi-component bulk simulationsThis section covers DFT simulations for binary and multi-component alloy systems relevantto this thesis. A successful method applied to study solid solution behavior in random metallicalloys is the Special Quasi-random Structures (SQS) method developed by Zunger and co-workers [25]. Even with a relatively small number of atoms, in Figure 2.1(a), lattice constantsof Mo-Nb binary alloys as a function of the composition of Nb calculated from SQS are comparedwith experiments, and there is good agreement between them. A key observation is that thelattice constant of the Mo-Nb alloys increases linearly from pure Mo to the pure Nb side.Since the development of this method, computational power has increased greatly, allowing formodeling larger alloy supercell sizes and accurate thermodynamics of alloys. Structural modelshave gone from 2, 4, 8 and 16-atom supercells [26] to 96-atom and 128-atom supercells formodeling HCP high entropy alloys (HEAs) [27] and BCC HEAs [28], respectively. With suchan increase in system size, the errors in computed properties of the alloy due to having a smallsupercell have been reduced.Another study by Raabe et al. [29] calculated the lattice properties of Ti-Mo and Ti-Nballoys, including lattice constants, solution enthalpies and Young’s modulus. They reportedthat Mo is a stronger stabilizer of BCC Ti than Nb. Using density functional theory, Jianget al. [26] studied the Mo-Nb system and computed the formation enthalpies across the entirecomposition space. They observed that Nb dissolves in Mo, as shown in figure 2.1(b). Theformation energy across the full composition space shows a symmetric profile. For the 50 at. %Nb alloy, the composition where Mo and Nb show the most favorable interaction, the mixingenthalpy is calculated to be ∼ -10 kJ/mol. Zhou et al. [30] studied the Ti-Zr system and found4that Zr is a neutral element, as it has minimal impact on the phase stability of BCC Ti. All ofthe mentioned systems were modeled for bulk alloys (relevant to the current study), achievingsome success in comparing calculated properties with experimental data.Figure 2.1: (a) Lattice constants of Mo-Nb bcc alloys calculated using SQS shown in solid square(P) adapted from [26]. Experimental data corresponding to1 as reported by Goldschmidtl etal. [31] and the data corresponding to # is from Catterall and Barker (1965) [32] (b)Formationenthalpy of Mo-Nb bcc alloys calculated using SQS shown in solid square (P) adapted from[26]. Experimental data corresponding to # is taken from Singhal et al. [33] The “CPA” labelcorresponds to the Coherent Potential approximation (CPA) method used to calculate theproperties of alloys, data shown is from Sigli et al. [34]Adapted from [26] with permission fromElsevier.Another set of alloys relevant to the current study, Pt-Au and Pt-Ir alloys studied usingDFT, was conducted by Saksena and co-authors [35]. Across the full composition space, bothalloys exhibit a positive enthalpy of formation, indicating that A-B host atoms do not mix.From experiments, both alloys do show a miscibility gap at low temperatures. Using DFTsimulations, Pt-Ir alloys show a slightly higher enthalpy of formation, ∼ 5.5 kJ/mol, comparedto the Pt-Au system with ∼ 4.5 kJ/mol. The composition for the highest formation enthalpyis at ∼ 37 at. % Au and ∼ 50 at. % Ir for Pt-Au and Pt-Ir alloys, respectively.Studying elastic properties also provides insight into materials design for various alloys, andDFT provides trend estimates for elastic constants. Investigating a ternary alloy system, i.e.increasing in chemical complexity, a few studies modeling such an alloy system relevant tothe current study. Marker et al. [36] investigated the elastic properties (bulk modulus, shearmodulus and Young’s modulus) of Ti-X-Y systems (X≠Y=Mo, Nb, Ta) using a CALPHAD-type approach. This approach allows for predicting elastic constants with only a few selectedDFT calculations. The elastic stiffness coefficients of a multi-component system modeled usingthe CALPHAD approach:5𝐸 (𝑐𝑎𝑙𝑙𝑜𝑦) =∑︁c𝑋 · 𝐸𝑋 +∑︁c𝑋c𝑌 (𝐿0 + 𝐿1(𝑐𝑋 − 𝑐𝑌 )) +∑︁c𝑋c𝑌c𝑍𝐿2 (2.1)In Equation (2.1), the first term is for all the elemental pure systems, 𝑐𝑋 is the atom fractionof the component X in the alloy, and 𝐸𝑋 is referred to as the elastic stiffness coefficient of thepure elements. The second and third terms are calculated over all binary and ternary systems.The interaction coefficients 𝐿𝑜, 𝐿1 for the binary systems and 𝐿2 for the ternary systems arefound by fitting Equation (2.1) to a few DFT data in their previously published work [37] forpure elements and selected binary systems. One of the main findings is that the same trendexists for stiffness coefficients in Ti-X-Y due to the stabilizing effects of Mo, Nb and Ta for Ti,as shown in Figure 2.2. For instance, the shear modulus increases from the Ti-rich side towardsthe Mo-rich side for all the above systems except Ti-Nb-Sn.Figure 2.2: Shear moduli calculated for (a) Ti-Mo-Nb , (b)Ti-Mo-Sn, (c) Ti-Mo-Ta, (d) Ti-Mo-Zr and (e) Ti-Nb-Sn ternary systems using a CALPHAD-type modeling approach. Adaptedfrom [38] with permission from Elsevier.62.1.2 Vacancies in binary and multi-component bulk alloysVacancies are point defects present in metallic alloys. Diffusion of atoms from one regionto another occurs with the help of vacancies and proves vital for phenomena such as diffu-sional creep at high temperatures [39]. When modeling vacancies in metallic alloys using first-principles and molecular statics calculations, one can calculate vacancy formation and vacancymigration barrier energies at all of the bulk sites.One of the earliest studies to model vacancies in binary alloys was by Zhang and Sluiter [40].Their work investigated vacancy formation energies in Cu-Ni alloys using a 16-atom supercellwith DFT calculations. Figure 2.3 shows the vacancy formation energies at a few selected Niand Cu sites with different local neighborhoods for alloys containing 25% at. %, 50 at. % and75 at. % Cu. There exists a clear trend where the vacancy formation energy decreases withincreasing the number of Cu atoms in the nearest neighbor shell.Figure 2.3: Vacancy formation enthalpy computed with DFT as a function of number of Cu inthe first nearest neighbor shell (n𝐶𝑢). Different symbols indicate different alloy compositions,squares: 25 at. % Cu, crosses and diamonds: 50 at. % Cu with two different SQS structuresand open circles: 75 at. % Cu. Adapted from [40] with permission from the American PhysicalSociety.In random ternary alloys such as BCC Fe-20 at. % Cr- 10 at. % Ni alloy, vacancy formationenergetics was studied by Zhang and co-workers [41]. They calculated vacancy formation energyat 256 sites in a 256-atom SQS supercell (SQS1) and found that the mean formation energywas 2.04 eV with a standard deviation of 0.11 eV. In order to confirm the repeatability ofthis data, another SQS structure (SQS2) was used. The mean for this structure was 2.01 eV,and the standard deviation of a fitted Gaussian distribution was 0.13 eV. The authors thencombined SQS1 and SQS2 results, which led to a distribution with a mean of 2.03 eV and a7standard deviation of 0.16 eV. This suggests that sufficient sampling of different sites is crucialfor estimating the broadening of the vacancy formation energy distribution. The results areshown in Figure 2.4. Thus, obtaining a statistically converged full distribution for vacancyformation energy is crucial.Figure 2.4: Vacancy formation energy distributions in Fe-Cr-Ni alloy from (a) SQS1 structure,(b) SQS2 structure and (c) SQS1+SQS2 combined. Red lines correspond to Gaussian distribu-tion fits to the data. Adapted from [41] with permission from Elsevier.In another study for a quaternary alloy system, Li et al. [42] studied vacancy energeticsand magnetic effects in an equimolar NiFeMnCr high entropy alloy. The resulting vacancyformation energy distribution has a mean of 1.96 eV and a spread of 0.7 eV. The distributionis shown in Figure 2.5(a), the distribution is also denoted by the host atom that was replacedto create a vacancy in Figure 2.5(b) and finally, as a function of the local environment aroundthe vacancy in Figure 2.5(c). “Op0” environment implies a uniform local environment of threeatoms of each species denoted by (3,3,3,3). “Op1” refers to one operation of deviation from“Op0”, for example, (3,4,2,3) is three atoms of Ni, four atoms of Fe, two atoms of Mn and threeatoms of Cr. Op3 and Op4 are two and three operations of deviation from “Op0”, respectively.Figure 2.5: Vacancy formation energy distributions in Fe-Cr-Ni-Mn alloy shown as a full dis-tribution in (a), full distribution shown as Cr, Fe, Mn and Ni atom replacements in (b) andthe distribution in terms of different local environment, Op0, Op1, Op2, Op3 and Op4. Bluedotted lines correspond to the Gaussian distribution fit to the data. Adapted from [42] withpermission from Elsevier.8A recent study by Zhang et al. [27] used a SQS approach to model various binary and multi-component alloys of 192-atom HCP Al-Hf-Sc-Ti-Zr high entropy alloys (HEAs). A majorfinding is illustrated in Figure 2.6, which shows vacancy formation energy as a continuousdistribution. Broadening of the vacancy formation energies is seen when going from the HfZrbinary alloy system to the quinary HEAs. This indicates that the increase in the number ofchemical elements in the alloy leads to a broadening in the calculated vacancy formation energy.Adding Ti into the alloy, i.e. from HfZr to TiHfZr alloys, a decrease of 0.11 eV is noticed in themean vacancy formation energy. In contrast, adding Sc to the ternary alloy increases the meanvacancy formation energy by 0.23 eV. Finally, adding Al into the quaternary alloy decreasesthe mean formation energy by 0.11 eV and even further by 0.11 eV when adding 15 at. % ofAl.Figure 2.6: Distribution of vacancy formation energy using DFT in (a) pure metal components- Al, Sc, Zr, Ti and Hf , (b) binary Hf-Zr alloy, (c) HfTiZr ternary, (d) HfScTiZr, HEA alloywith (e) 5 and (f) 15 at. % Al. Vertical solid and dashed lines refer to mean and composition-weighted linear interpolation prediction from Vegard’s law, respectively. Adapted from [27]with permission from Elsevier.92.1.3 Grain boundaries and interfaces in alloysExtended planar defects, including grain boundaries and heterophase interfaces or interphaseboundaries, are challenging to model using atomistic simulations. Often, a large simulationdomain size is required. Additionally, the relaxed ground-state structures are computationallyexpensive to calculate compared to the bulk.Zhao et al. [43] studied stacking fault energies (SFEs) in FCC concentrated solid solutionalloys (CSA) of several equiatomic Ni-based alloy systems using a 108-atom supercell withnine different [111] planes and a vacuum of 10 Å. Collinear spin-polarized DFT calculationswere carried out to account for magnetic effects in all alloys. In NiCo, NiCoCr, NiCoFeCr, andNiCoFeCrMn, SFEs are calculated to be negative and unstable. Adding Co into the binaries hasthe largest effect on reducing SFEs compared to NiFe and NiPd systems. Adding Cr decreasesSFE significantly more than Fe for ternaries, while Mn is better at reducing SFE than Pdin quinaries. This effect can be attributed to the role of 𝑑−electrons in determining SFE intransition metals and their alloys [44].Figure 2.7: SFEs calculated for Ni and its alloys by DFT (shown in red squares) with thesupercell method and the error bars represent the standard deviation calculated from ninedifferent planes. Taken from [43] with permission from Elsevier.An important class of alloys, Ni-based superalloys for high-temperature mechanical prop-erties, contain some antiphase (APB) boundaries in the Ni3Al phase. In a recent study byChen et al. [45], the effect of solutes on the (111) APB energies was examined using DFT.Solutes such as Ti and Rh show opposite trends, increasing and decreasing the APB energy,respectively. Meanwhile, solutes such as Co do not affect the APB energy and is comparableto the intrinsic APB energy without solutes.10Shen et al. [46] studied two GBs - Σ3(111) [110] and Σ5(013) [100] in U-Mo and U-Nb alloysystems. Due to the mechanically unstable nature of BCC U, these two alloying elements areadded to stabilize the GB structures. It is found that the GB for BCC U - 10, 20 at. % Mophase is unstable and has a negative GB energy. After a critical amount of Mo and Nb, thecalculated GB energy becomes positive, thus stabilizing the GB structure.2.1.4 Virtual crystal approximation for alloysIn order to circumvent the issue of sampling different alloy structures or to use a largedomain to capture all the randomness of an alloy, a mean-field approach is very attractive.Virtual crystal approximation (VCA) is a popular approach to describe the chemical disorderin solid solutions [47]. In the framework of VCA, each lattice site is occupied by a virtual alloyatom, and its pseudopotential is a weighted linear combination of the pseudopotential of eachof its constituents. It has to be noted that the basic alloy properties, such as lattice constant,elastic moduli and solution energies, are accurately modeled using this method. In addition, itwas found that VCA proved to be a valuable tool for modeling alloys composed of atoms withsimilar atomic radius and electronic structure [48].Using the VCA approach, Scheiber and co-workers [49] studied segregation of Re soluteand cohesion at different GBs in a BCC W- 25 at. % Re alloy including nine symmetric tiltGBs, one twist and five mixed twist-tilt GBs. Due to the mean-field approach of VCA, it waspossible to study many different geometries for a highly alloyed system. Segregation energiesof Re to different sites were calculated, and the strongest segregation energy was found to be-0.75 eV. Further, segregation was predicted to be weaker for low-energy Σ3 < 110 > {111}GB, pure twist and mixed GBs. Furthermore, the work of separation was computed, and thelargest effect was seen for Σ3 < 110 > {111} GB with a 13% increase due to the segregation ofRe. The authors performed bond-order analysis [50] to supplement the computed segregationenergy data. The results of the segregation profiles are shown in Figure 2.8.The authors analyzed trends of calculated segregation energies by quantifying how differenteach site is from a bulk crystal structure. Voronoi Volume and bond-order parameters Q4,Q5 and Q6 were used to study the local atomic environment at each GB site. All the trendsare reported in Figure 2.9. Voronoi volume failed to explain all the variations in segregationenergies, suggesting that size alone was not playing a decisive role. The bond-order parameterQ5 correlated very well with segregation energies. The bond order for each atom around all11Figure 2.8: Segregation energy profiles for all GBs computed using the VCA approach (shownin red circles) and the blue joined circles are predicted segregation energies from bond orderanalysis. Taken from [49] with permission from Elsevier.neighboring sites in a sphere of cutoff radius, 𝑟𝑐𝑢𝑡 , is defined by the spherical harmonics for theset of vectors r𝑖 pointing to the neighbors:𝑄𝑙 =√√4𝜋2𝑙 + 1+𝑙∑︁𝑚=−𝑙|𝑞𝑙𝑚 |2 (2.2)𝑞𝑙𝑚 =1𝑁∑︁𝑟𝑖<𝑟𝐶𝑢𝑡𝑌𝑚𝑙 (r𝑖) (2.3)Here, 𝑌𝑚𝑙 (r) are the spherical harmonics, and N is the number of neighboring sites within thespecified cutoff. A cutoff of 3.7 Å was used, including the BCC alloy system’s first and secondnearest neighbor. The bond order parameters, Q4 and Q6 are generally used to identify FCC,BCC and HCP structures [50], although it was found that the segregation energy correlatedwith the 𝑄5 parameter.12Figure 2.9: Calculated segregation energies for the W-Re system correlated with (a) Voronoivolume and bond-order parameters (b) Q4, (c) Q5 and (d) Q6. The red dashed line in (c)represents a linear fit using only the values from two Σ3 tilt GBs, whereas the solid blue linerepresents a fit using all points. Taken from [49] with permission from Elsevier.Yan et al. [51] investigated the segregation of Y and B in a Σ3 [110] (111) grain boundary inBCC Ti-Mo and Ti-V alloys using VCA. In this case, segregation energy is a single represen-tative value since only one type of atom exists. Segregation of Y and B to the GB plane wasstudied for Ti- 40 at. % Mo alloy and Ti-60 at. % V alloy. In the case of Ti-Mo alloy, it wasfound to be -0.51 eV, a negative value implying segregation. The contributions from structural(strain) and chemical are found to be -0.16 and -0.36 eV, respectively. Meanwhile, B was foundto be anti-segregating, with 0.15 eV as segregation energy, and, primarily, chemical contribu-tions of 0.14 eV to the segregation energy. The contributions from chemical and structuralcomponents are calculated as shown in Figure 2.10. The chemical contribution is the energydifference when substituting a solute at GB versus at bulk. For the structural contributionto segregation energy, the further relaxation when a solute is added needs to be taken intoaccount.13Figure 2.10: Schematic representations of how chemical and strain-induced components ofsegregation energy are calculated. The upper and lower panels show the calculations requiredfor GB and bulk geometry, respectively. Taken from [51] with permission from the AmericanPhysical Society.2.2 Simulations using semi-empirical models2.2.1 Grain boundaries in pure metalsIn a typical microstructure of metallic alloys, a multitude of GBs are present. The propertiesof such GBs and their structures are crucial to understanding polycrystalline material phenom-ena. GBs have five macroscopic degrees of freedom, including the misorientation angle, the axisof rotation, and the GB plane. In order to successfully model the variability of properties withGBs, molecular statics and dynamics calculations are attractive. GB energy is a property thatcan be calculated with such simulations. A seminal work on calculating GB energies was doneby Olmsted and co-workers [52]. Their survey of computed GB energies for 388 distinct GBs inNi and Al was the first and most comprehensive work in the literature. The potentials used forNi and Al were developed by Foiles and Hoyt [53] and Ercolessi and Adams [54], respectively.One of the results from their study is shown in Figure 2.11(a). The GB energy approaches zeroat low disorientation angles since there is a minimal degree of lattice mismatch between thetwo grains.14Figure 2.11: (a) EAM computed GB energy as a function of disorientation angle for FCC Ni.(b) Scatter plot of 𝛾𝐺𝐵 for the same macroscopic degrees of freedom of each GB in Ni and Al.Taken from [52] with permission from Elsevier.An important trend emerged from the study of Olmsted and co-authors: the GB energy forthe same GB for Al and Ni show a linear trend as seen in Figure 2.11(b). Thus, the ratio of GBenergies for Al and Ni is related to the basic properties of these metals. The product of shearmoduli and lattice constant (𝜇𝑣𝑜𝑖𝑔ℎ𝑡 · 𝑎0) and the product of C44 and lattice constant betweenAl and Ni were close to the scaling factor for such GB energy comparison. However, theyconcluded that this hypothesis needs to be tested across other metals to ensure transferabilityto other systems.Expanding on the work by Olmsted and co-workers, a comprehensive work on a plethora ofGBs spanning the 5-D crystallographic space in FCC Al was conducted by Homer et al. [55].GBs with different rotation axes and planes consisting of a total of 7304 were chosen to calculatetheir structure and energy using the EAM developed by Mishin and co-authors [56]. Thecalculated GB energy and its distribution for all 43,009,235 metastable GB structures for the7304 GBs is shown in Figure 2.12. The metastable structures arise because of the multiplicity ofmicroscopic degrees of freedom for every unique macroscopic description of a GB. The mean andstandard deviation of the entire distribution are found to be 546 and 55 mJ m−2, respectively.Twist GBs are calculated to have the lowest and the highest GB energies, and many of thesestates are close to the minimum energy, indicating that different starting unique configurationsstill found the final minimum energy configuration. Even with such large datasets, the generaltrend still follows the model of Read-Shockley [57], where the GB energy peaks at around30 - 40◦ and then slightly lowered for higher disorientation angles. Having a repository ofdifferent metastable states also allows for predicting states through which a GB could pass asit undergoes migration or deformation [58].15Figure 2.12: (a) Scatter plot of GB energy versus disorientation angle for all ~ 43 millionstructures shown with the histogram of GB energy. (b) Box plot of GB energy for [1 1 1] twist(top), [ 1 1 1] symmetric tilt GBs (middle and bottom), the boxes highlight the data betweenthe first and third quartiles of the distribution. The red line shows the median, and the blacksquares are used to denote the mean. Taken from [55] with permission from Elsevier.To systematically study the microscopic degrees of freedom, 𝛾-surfaces of GBs are used.Translations for the in-plane directions of a GB are used to compute a 𝛾-surface. Terentyevet al. [59] studied <110> tilt GBs in BCC Fe with different available interatomic potentials topredict the energetically favorable configurations for the sliding pathway of a few GBs. Outof the four potentials, the potential developed by Mendelev and co-authors [60] is used tocompute the 𝛾−surfaces since it was fit to DFT data for self-interstitials, generalized stackingfault energy, liquid structures and melting point temperature. In Figures 2.13(a) - (c), GBenergies for various configurations translated in-plane are plotted for three GBs. The datasuggests that there are GB configurations exhibiting minima and saddle points for three GBsstudied here, i.e. Σ3(111), Σ3(112) and Σ11(113). In the context of the GB sliding phenomena,the configurations at minima and saddle points prove vital.Similar to bulk phases of metals, phase transformations occurring at GBs are now studiedin the literature using simulation methods. The seminal work by Frolov et al. [61] showedthat a reversible first-order transformation between two GB phases exists in a high-angle GBof FCC Cu. At 800 K, the GB structure transforms from the normal-kite phase to the split-kite phase, as shown in Figures 2.14(b) and (c), respectively. Since this report, there have beenmany more investigations on other GBs in metallic systems to analyze the phase transformationhappening at the GBs at various temperatures and stresses. Meiners and co-authors [62] studiedthe transformations at symmetric and asymmetric [111] tilt GBs in copper. They showed theexistence of two phases, i.e. domino and pearl-shaped phases at Σ19 (178¯) GB. Recently, Brink16Figure 2.13: The profile for 𝛾-surface for (a) Σ3(111) (b) Σ3(112) and (c) Σ11(113) GB forBCC Fe, left and right panels show the 3-D surface and color contour of GB energy, respectively.Taken from [59] with permission from Elsevier.et al. [63] used the available EAM potentials to test the universality of GB phases in FCC metalssuch as Ni, Cu, Pd, Ag, Au and Al. Their investigation found that a range of high-angle [111]symmetric tilt GBs, namely Σ7, Σ13b, Σ49, and Σ19b, exhibited phase transformation in thetemperature range between 700 K to 1100 K. They also noted that available EAM potentialshave numerical smoothness issues when evaluating the second derivatives of the potential energy.This issue calls for developing interatomic potentials that allow for predicting the kinetics ofdefect transformations without exhibiting any artifacts during the simulation.17Figure 2.14: (a) Simulation domain for a FCC Cu Σ5 (310) GB at 800 K with one open surfaceboundary condition on the right to allow for a change in GB density (microscopic degreesof freedom). The two phases occurring in the phase transformation are (b) normal-kite and(c) split-kite structure separated by a “step” feature in (d). (e) The GB structure is fullytransformed with two open surfaces instead of one. Taken from [61] with permission fromSpringer Nature.2.2.2 Solutes and vacancies at grain boundaries in metalsThe phenomenon of solute segregation to GBs impacts mechanical response and fractureresistance, even in small quantities. With metastability in GB structures, it is crucial to quantifyits impact on solute segregation. Mahmood et al. [23] used atomistic simulations using EAMpotential to study Mg segregation to metastable GB structures from the family of [001] and[110] symmetric tilt GBs in FCC Al. One such GB, Σ113 [001](15 1 0), and its metastable statesare shown in Figures 2.15(a1) - (d1), it is found that segregation energies strongly correlate withthe relative change in volume at GB sites and volume of Al bulk lattice site. This trend is alsoseen in other metastable GBs (MS1, MS2, MS3 and MS4) with different microscopic degreesof freedom (DOFs) with the same macroscopic DOFs seen in Figures 2.15 (a2) - (d2) and (a3)- (d3). Segregation to 22 total GBs used in this study strongly correlated with the structuraldescriptor ΔV/V0. Thus, the authors constructed a Gaussian process regression model to makepredictions on other GBs in Al using the available data from their study. This formalismalso provides variability in GB segregation energy. It moves away from using a single-valuedquantity, which can then be used to predict solute concentration at GBs at finite temperatures.18Figure 2.15: (a1) - (d1) Metastable GB structures for Σ113 [001](15 1 0), green and orangeatoms represent FCC and GB atoms according to Common Neighbor Analysis in OVITO, siteswithin black region are used for Mg segregation energy analysis. In (a2) - (d2) the geometricfeature of Δ𝑉/𝑉0 for all the sites within the black outlined region in (a1) - (d1) are represented,where blue and red indicate large and small atomic volumes. ΔE(𝑖) in (a3) - (d3) representsMg segregation energy, with red and blue indicating unfavorable and favorable segregation,respectively. Taken from [64] with permission from Springer Nature under Creative CommonsCC BY license.Another seminal work that studied the segregation of different solutes to Al was done byHuber and co-authors [24]. In this study, an extensive dataset of over a million segregationenergies for six solute species (Pb, Co, Mg, Fe, Ni and Ti) at 38 GBs was created usingavailable Al-based EAM potentials [65–69]. They studied GB occupation with temperatureusing Langmuir-McLean (LM) and White-Coghlan (WC) models that take segregation energiesas input. In Figure 2.16(a), segregation energies of Mg across all 38 GBs. Using a singleaveraged segregation energy, it was found that Langmuir-McLean models were inadequate tocapture the full distribution of segregation and fit using the WC model, which is more suitable.Solute concentration at the GB in the WC model using segregation energy at the GB site 𝑖,𝐸𝑋@𝑖𝑠𝑒𝑔 for all GB sites, 1 to 𝑁𝐺𝐵 is given by:𝑐𝐺𝐵 =1𝑁𝐺𝐵𝑁𝐺𝐵∑︁𝑖=111 + 1−𝑐𝑏𝑢𝑙𝑘𝑐𝑏𝑢𝑙𝑘 · exp(𝐸𝑋@𝑖𝑠𝑒𝑔𝑘𝐵𝑇)As temperature increases from 0 K to 600 K, the solute segregation level decreases from0 to -0.2 to -0.35 eV. There exists a distribution or spectrum for solute segregation energiesto a more general GB (denoted by GB “s”) as shown in Figures 2.16(b) and (c), respectively.Overall, it was shown that the following trend holds good for segregation in FCC Al: Pb >Co > Mg > Fe > Ni > Ti. A simple physics-based model was developed based on the datagenerated that required only information from undecorated GB, thus reducing the number of19calculations from the total number of GB sites to just one. A simple linear model with ΔV andΔN (change in number of bonds at a GB site) proved useful in interpreting the results of thesegregation energies but showed a prediction error of 0.06 to 0.17 eV for all the solutes. Usinga machine learning model with decision trees and geometric features provided reduced errorsat the cost of interpretability.Figure 2.16: (a) Histogram plot for Mg segregation energy at all 38 GBs studied by Huberand co-authors [70]. Solid and dashed lines indicate a smoothed distribution of the originaldata and data from the machine learning prediction model, respectively. Red lines show theWhite-Coghlan site occupation for three temperature values and dashed vertical lines showhalf-occupancy with a bulk concentration of 0.2 at. % Mg. (b) The density of states for all sixsolutes for a symmetric (310) boundary and (c) for a disordered “s” boundary. Taken from [70]with permission from Springer Nature under Creative Commons CC BY license.Wagih and Schuh [71] developed a workflow using EAM potentials to evaluate solute-soluteinteractions and GB segregation energy spectrum in non-dilute Al-Mg alloys. Using MC/MDsimulations, solute concentrations at GB were calculated in a polycrystalline sample, and thesolute-solute interaction parameter, 𝜔, was used to fit the atomistic data. The interactionparameter is introduced by the following equation:𝑋𝐺𝐵1 − 𝑋𝐺𝐵 =𝑋𝑏𝑢𝑙𝑘1 − 𝑋𝑏𝑢𝑙𝑘 𝑒𝑥𝑝(−𝐸𝑠𝑒𝑔 + 𝜔 · 𝑋𝐺𝐵𝑘𝐵𝑇)When using X𝑡𝑜𝑡= 0.1% to 10% at 450 K, three parameters are required to characterize theskew-normal distribution of the GB segregation spectrum and one parameter, 𝜔, to evaluatethe solute-solute interactions. The Fowler-Guggenheim (FG) approach was adopted instead ofLM or WC models to evaluate the solute-solute interactions.Suzuki and Mishin [72] studied vacancy formation at GBs in FCC metals such as Al and Cuusing EAM potentials. They found that vacancies in some GBs tend to delocalize and causesevere atomic relaxations that drastically modify the local GB structure. These delocalized20vacancies tend to have low formation energies. Examples of such instabilities are shown inFigures 2.17(a) and (b), and the cause of such instability was due to tension and compressionstresses at each of the sites in the GB. Their study was limited to symmetric tilt GBs. Futurestudies are required to corroborate these findings in other GB types such as twist, asymmetricGBs, etc., and GBs in BCC metals need to be investigated. This work laid the groundwork forrevealing the atomic mechanisms of vacancy-mediated diffusion along GBs.Figure 2.17: (a) Vacancy instability in Al Σ5 (210) GB, the vacancy initially sits at site 2,during the relaxation atom from site 1 moves to 2 leaving the vacancy at site 1. (b) Vacancymoving from site 2 to 1, in Σ13(527) GB in Al. Taken from [72] with permission from SpringerNature.Tschopp and co-authors [73] studied vacancy formation energies at 170 GBs in BCC Fe within15 Å of the GB plane. A strong correlation was found between GB energy and the mean vacancyformation energy for each GB, with increasing GB energy, the mean vacancy formation energy(VFE) decreases. However, their study did not present the effect of vacancy reconstructionswhen analyzing these results. Singh and Parashar [74] studied vacancy formation energies at [00 0 1] and [0 1 1 0] tilt axis GBs in HCP Zr using the interatomic potential from Mendelev andAckland [75]. They found the same conclusion as Tschopp and co-authors: boundaries withhigher GB energy are likely to have low VFEs. VFEs calculated within 2 Å showed significantdeviation from the bulk value of ∼ 2 eV, and in GB with misorientation of 30 ◦ with ∼ 263 000atoms, the VFE approached zero eV at the center of the GB. VFEs are found to be lower forasymmetric tilt GBs than symmetric tilt GBs.2.2.3 Grain boundaries in alloysUsing Molecular statics simulations, Tikhonchev and co-authors [76] calculated the GB en-ergies of four different GBs in BCC Fe-Cr alloys using an EAM potential. This potential [77]was developed to accurately predict the mixing enthalpy of random ferromagnetic FeCr alloysat low and high Cr concentrations to mimic the experimentally observed paramagnetic state ofthe alloy. Using an EAM potential instead of DFT allowed for considering many GB configu-rations for different GB types for the full composition space. In all the GBs studied, the GB21energy lies below the linear rule of mixture lines, except for a few data points in Σ5(210) at 5at. % Cr and Σ5 (310) GB at 15 and 30 at. % Cr. Using the data, the authors could developthe enthalpy of mixing at GBs and then fit them to Redlich-Kister polynomials [78], and GB(111) predicted the strongest reduction in mixing enthalpy, since it deviated the most from thelinear rule.Fischer and co-authors [79] developed a new EAM potential for Cu-Ni alloys by fitting theexperimental data on mixing enthalpy and taking available data from the literature. Usingthis potential, GB energy for two GBs, Σ5 and Σ3 twin GB, was calculated as a function oftemperature from 400 K to 800 K. The results of GB energy are shown in Figure 2.18 where thereis a less pronounced increase on the copper-rich side for both GBs, and increases strongly whenapproaching the pure nickel side. This effect was primarily attributed to the strong segregationof copper on the nickel-rich side. Additionally, Cu and Ni have a lattice mismatch of ∼ 2.7%,and this elastic effect also contributes to GB energy. For Σ3 GB, the elastic contribution is 70%at 400 K, while the remaining is assigned to segregation. It is important to note that the lowestGB energy in the entire composition space was found to be on the pure Cu side. Meanwhile,some systems exist [80] where the minimum GB energy occurs at intermediate compositionswith both constituents segregating to the GB.Figure 2.18: (a) Σ5 (013) [100] GB energy calculated as a function of the bulk concentration ofnickel and different colored lines indicate different temperatures. Some lines stop at intermediatecompositions because of a miscibility gap at those temperatures. At 700 and 800 K, the fullcomposition can be studied since no phase separation occurs. (b) GB energy of Σ3(111) withcomposition of Ni at different temperatures. Taken from [77] with permission from Elsevier.Investigating ternary alloys such as Fe-30 at. % Ni - 22 at. % Cr, Dai et al. [81] studiedcoherent (CTB) and incoherent twin GB structure (ITB) and the formation of Cr clusters usingMC/MD simulations during tensile and compression loading. Using the available potential forFe-Cr-Ni alloy [82], they found no significant segregation of Ni and Cr existed in coherent GBs.On the other hand, significant segregation tendency was found at incoherent GBs at highertemperatures of 1000 K. A snapshot of the formation of Shockley partial dislocations showed22that higher (65 at. %) Cr concentrations clusters existed near these defects and segregationof Ni and Cr depletion happened at the CTB providing an increase in yield strength of thosesamples.Recently, vacancy interactions at multi-component alloy GBs have also been studied withEAM potentials. One such system is the equiatomic ternary alloy system TiTaV, where vacancyformation energies (VFEs) were calculated at three GBs, Σ3(111), Σ3(112) and Σ19(331) [83].Apart from generating a random GB sample, they also equilibrated their GB structure usingMonte-Carlo/molecular dynamics to show Ti atoms preferentially segregate to those GBs. Re-sults for VFEs are shown in Figure 2.19, in the case of pure BCC V, vacancies are readilyformed at some sites in Σ3 (111) and Σ19 (331), this trend is also seen in alloys, with lowestenergies reaching ∼ -2 eV. Random equiatomic alloy GBs of the Σ3(111) and Σ19 show favorableVFEs compared to Σ3 (112) like the case of pure BCC V. In the case of equilibrated structures,Figures 2.19(g)-(i), VFEs are high for Ta, followed by V and Ti vacancies. Another importantnote is that the VFEs in the equilibrated sample are significantly weakened compared to pureBCC V due to segregation of Ti at GBs. Further analysis shows that Ti-rich clusters are co-hesively weak and provide favorable segregation to vacancies to the GBs. At the same time,Ta-rich are relatively stronger and prefer to remain in the bulk region.23Figure 2.19: EAM calculated vacancy formation energy at three GBs in (a-c) pure BCC V,(d-f) random equiatomic TiTaV alloy, and (g-i) equilibrated alloy with Ti segregation to GBregion. The colored dots in each of the figures in (d) - (i) represent different chemical species.Grey squares, red circles and blue triangles indicate Ti, Ta and V substitution, respectively.Taken from an open-access MDPI article [83].First-order phase transformations at GBs, like in metals, also occur in complex concen-trated alloy systems. Cao et al. [84] studied phase transition at Σ5(210) symmetric tilt GB inequiatomic random FCC CrCoNi alloy. At low temperatures, the GB structure consists of aseries of “normal kite” units, and after 600 K, the GB structures start to transform into “filledkite” units. Finally, at 1300 K and above, the GB region becomes large (>10Å), and premeltingbegins. It is also noted that the split kite structure found in phase transformations at GBs inpure FCC Cu [61,85] was not found in this study. Using MC/MD simulations, they also foundthat phase transition at GB can be delayed in the presence of Ni-clustering and Co-Cr orderingconfigurations at the GB region.242.3 Machine learning simulations2.3.1 Overview of machine learning potentialsMachine learning interatomic potentials (MLIPs) are regression models that predict energiesand forces with the input of atomic structures in 3-D space. These models are referred to assurrogate models since they are used to replace the underlying potential energy surfaces (PES),which are otherwise obtained from high-cost DFT simulations, for instance. On the other hand,these models are more expensive than the semi-empirical potentials discussed in the last section.In principle, these machine learning models use local atomic “descriptors” as inputs and basisfunctions or neural networks as basis functions to train on. A crucial success of MLIPs is thatthe total energy of the system can now be decomposed into a sum of atomic site energies:𝐸 = Σ𝑎𝑡𝑜𝑚𝑠𝑖 𝜀({𝑟𝑖 𝑗 }),where 𝑟𝑖 𝑗 = 𝑟 𝑗 − 𝑟𝑖 is the relative position between atoms 𝑖 and 𝑗 , where an atom 𝑖 is thecentral atom and index 𝑗 runs over the atoms in a fixed neighborhood of atom 𝑖 within a cutoffradius of 4-6 Å, i.e., |𝑟𝑖 𝑗 | < 𝑟𝑐𝑢𝑡 . The locality of such interactions, especially in metals, allowsone to construct such models confidently.Below, the types of models available in the literature, along with their computational cost andperformance are reviewed. Next, GBs simulated using MLIPs, and atomic features representingGB structures are presented. Finally, some information on datasets required for training andfoundation models is also reviewed from the literature.2.3.2 Types of machine learning interatomic modelsBehler and Parrinello [86] started using generalized neural-network potentials (NNPs) totrain regression models in 2007, where the input layer consisted of nodes encoding the cartesiancoordinates of each atom (using symmetry functions), and the output layer consisted of nodesfor the energy of each atom. This study aimed to simulate the dynamic properties of bulkSi, and the NNP model reasonably reproduced radial distribution function (RDF) to DFTaccuracy. They also noted that their NNP was five orders of magnitude faster than the DFTcalculations, and NNPs scale linearly with system size, unlike DFT.Another effort to model the PES without electrons was the Gaussian Approximation Poten-tials (GAP) pioneered by Barto´k et al. [87] in 2010. Gaussian process (GP) regression [88, 89]25and Gaussian basis functions are used to find the parameters of such a potential. Surface en-ergies and elastic constants of diamond were calculated, and good matching was found withDFT. Phonon properties and thermal expansion were also studied and validated with DFTusing quasiharmonic approximation at low temperatures. GAP models enable a kernel-basedmethod without using NN to link with structural descriptors for fitting.Spectral neighbor analysis potentials (SNAP) introduced by Thompson and co-authors [90]use a linear relationship between site energies and bispectrum components. Building a modelwith least-squares linear regression to simulate solid and liquid transition metal Ta gave goodresults for a melting point of 2790 K compared with 3085 +/- 130 K from DFT [91]. Itwas found that SNAP potentials gave the correct shape and size of Peierls barrier for screwdislocation motion in BCC Ta when compared to other available empirical potentials [92, 93].An extension to SNAP potentials was constructed, referred to as quadratic SNAP (qSNAP),that has lower root mean squared errors (RMSEs) but at the expense of a larger number ofparameters [94]. An embedding energy term (like in EAMs) is added to SNAPs to result inqSNAP transforming a four-body SNAP to seven-body potential, thus capturing data from adiverse training structure set. Quadratic SNAP was shown to be a special case of NN with twohidden layers, while linear SNAP consists of only one hidden layer.Moment Tensor potentials (MTPs) developed by Shapeev [95] was another class of system-atically improvable interatomic potential. Polynomial basis sets were used along with momenttensors of structural descriptors. Using invariant polynomials, MTPs are optimal in terms ofcost and performance. For example, with the same number of parameters, 2.8 % RMSE and∼ 3 ms/atom cost and 4.2 % RMSE and ∼ 135 ms/atom cost for MTP and GAP, respectively.Additionally, Podryabinkin and Shapeev [96] provided an on-the-fly active learning strategy toimprove existing MTPs for a particular scenario outside the training domain. Following this,Drautz [97] used an atomic cluster expansion (ACE) approach to parameterize the interactionmodel for atoms. This approach was developed as a complete descriptor of the local atomicenvironment, a major issue in earlier models discussed by Pozdnyakov and co-authors [98].Further, it was found that ACE models provided the optimum in terms of cost-performancetrade-offs compared to MTP, SNAP, GAP and NNs [99].On the other hand, other research groups focus on developing physically-informed MLIPsby extending current MEAM potentials. These potentials are “ultra-fast” (UF) compared tocurrent generations of MLIPs [100,101]. Vita and Trinkle used highly non-linear functions in thedensity embedding term and called them, s-MEAM, a low-complexity MLIP. In the s-MEAMformalism, the energy of each atom, 𝑖, is written as:26𝐸 =∑︁𝑖𝐸𝑖 =∑︁𝑖[𝐹𝑐𝑖 (𝜌𝑖) +∑︁𝑖< 𝑗𝜙𝑐 𝑗 (𝑟𝑖 𝑗 )](2.4)𝜌𝑖 =∑︁𝑗≠𝑖𝜌𝑖 +∑︁𝑗<𝑘; 𝑗 ,𝑘≠𝑖𝑓𝑐 𝑗 (𝑟𝑖 𝑗 ) 𝑓𝑐𝑘 (𝑟𝑖𝑘 )𝑔𝑐 𝑗𝑐𝑘 [cos(𝜃𝑖 𝑗 𝑘 )] (2.5)In Equation (2.4), 𝐸𝑖 is composed of repulsive pair interaction term (𝜙) and an embeddingfunction, 𝐹, for each atomic species, 𝑐𝑖. The density term in Equation (2.5), 𝜌𝑖, is the sum of2-body and 3-body (products of 𝑓 and 𝑔) contributions, and cos(𝜃𝑖 𝑗 𝑘 ) is the cosine of the bondangle formed by atoms, 𝑖, 𝑗 and 𝑘 with 𝑖 at the center. For A-A, A-B, and B-B species bonds,𝑔𝐴𝐴, 𝑔𝐴𝐵, and 𝑔𝐵𝐵 need to be parameterized, respectively. The five functions 𝜙, 𝜌, 𝑓 , 𝑔, 𝐹are parameterized by the cubic Hermite splines [102]. This could be compared to a single-layer NN or a low-order MTP. However, the functions used are readily interpretable, i.e. 𝜙,𝜌, 𝑓 and embedding function 𝐹, forming different interatomic potential components. Twohyperparameters are required – number of spline knots in each spline and the x-positions ofeach of those knots.2.3.3 Computational cost and performance of regression modelsSince MLIPs parameterize a high-dimensional space, it is thus important to illustrate itscapabilities on a cost-performance basis. In 2020, a work by Zuo and co-authors tried to dothis [103]. Pareto fronts for computational accuracy, i.e., how well an interatomic potentialreproduces energies, are studied by changing the parameter space of these potentials. In Fig-ure 2.20, errors for semi-empirical potentials are at least one to two orders of magnitude higherthan MLIPs. It is also clear that MEAM potential’s error for BCC Mo is half of the EAM,showing the appropriateness of MEAM for BCC crystal structure. Furthermore, qSNAP alsoshowed lower errors than its linear version, SNAP in BCC Mo. Out of all the MLIPs, GAPand MTP show similar error estimates and the lowest among the five MLIPs studied here.27Figure 2.20: RMSE energy, meV/atom (left column) and force, eV/Å errors (right column) forNi, Cu, Li, Mo, Si and Ge. Five MLIPs, including GAP, MTP, SNAP, qSNAP and NNP, areshown along with three semi-empirical potentials – EAM, MEAM and Tersoff. Taken from [103]with permission from the American Chemical Society.As described in [100], simple and interpretable interatomic potentials such as s-MEAMmodels , the errors are similar to NNs and SNAP and two to three times larger than forGAP and MTP. These errors are typically ten to hundred times smaller than those of EAMand MEAM, as reported by Zuo and co-authors [103]. For example, in FCC Cu, s-MEAMshows 1.83 meV/atom as a training set error for energies, while the energy error reported forEAM was 7.29 meV/atom. With the introduction of ACE potentials, Lysogorskiy and co-authors [99] provided a quantitative study to show that ACE potentials are the latest MLIPsthat show optimal front on cost versus performance. This performance already shows that ACEpotentials are suitable for building regression models to simulate metallic systems, and this iswhat will be followed for the rest of this thesis. The implementation details are provided inthe documentation inside ACEpotentials.jl and can be accessed here [104] and in the softwarepaper by Witt and co-authors [105].28Figure 2.21: Test error in energies (a proxy for accuracy) as a function of computational cost(estimate for performance) for (a) FCC Cu and (b) diamond Si for ACE, GAP, MTP, qSNAP,SNAP and NN. Taken from [99] with permission from Springer Nature.2.3.4 Grain boundaries in pure metalsDragoni and co-authors [106] used GAP potentials to study the 𝛾-surface for Σ3 (110) and(112) GBs in BCC 𝛼-Fe. They found good quantitive agreement with DFT data. This agree-ment is not surprising since GAP descriptors consider rotational and translational symmetriesand have a diverse set of training data consisting of 150,000 local atomic environments.Another study by Nishiyama et al. [107] created MLIPs to predict grain boundaries inFCC metals. Their training set structures contain no grain boundary structures and use bulkstructures of Ag, Al, Pd, Cu, Au and Pt to perform rattle and shear deformations on them. TheMLIP predictions for all metals showed good agreement with DFT for selected special GBs. Anartificial neural network was developed by Yokoi et al. [108] to predict the vibrational propertiesof GBs in FCC Al. It was found that Σ9 asymmetric tilt GBs showed larger vibrational entropiesthan the symmetric Σ9 GBs, and thus the ATGBs are more stable at elevated temperatures.Twin, Σ19, Σ7 and Σ13 GBs in HCP Mg were studied using MTP by Poul and co-authors [109]and showed good agreement with DFT by systematically changing the level of MTPs. Finally,it is noted that there are no current studies on GBs in alloys simulated using MLIPs.2.3.5 ML predictions of GB properties using atomic featuresApart from building interatomic potentials / PES regressors to study, researchers have usedstructural descriptors to classify GB structures and relate them to GB properties such as GBenergy and GB segregation energy in metals and alloys. This kind of analysis is incredibly usefulin analyzing the physical origins of such properties. Priedeman et al. studied the structuralunits (SU) using the Smooth Overlap of Atomic Positions descriptor at <100 > GBs in FCCNi [110]. In Figure 2.22, five SUs were identified and classified to cover the GB structures in29the entire misorientation range of <1 0 0 > GB structures. Out of them, “A” and “B” areFCC bulk, “C” SU is found in Σ5 (012) 53.13◦, “I” is found in GBs between Σ5 (013) 36.9 ◦andΣ5 (012) 53.13 ◦ and SK and EK units are found in low misorientation angle regions. Theyargue that while these classifications existed in the literature [73, 111], this study shows theseinvestigations using local GB structure classification using SOAP descriptor in great detail.Figure 2.22: EAM calculated GB energies are shown as a function of the misorientation angle.Representative structures are shown for a few GBs in the inset. The SUs that are present in thefull misorientation are presented at the bottom of the figure. The color bar shows the degreeof similarity to FCC bulk. Taken from [110] with permission from Elsevier.From the data repository of Huber for FCC Al [70], the authors showed that building gradientboosted regression random forest using some of the atomic features that were seen as important.In the supplementary document, nineteen features were analyzed in all the GB and GBs-solutestructures. For larger solutes such as Pb, Δ𝑉 and Δ𝑉3 , Voronoi volume difference is clearlya strong predictor for Pb solute segregation energies, where size plays a crucial role. For theother five solutes, the area-to-volume ratios, 𝐴/𝑉 , of the Voronoi polyhedra (used to calculatethe Voronoi volume) for the undecorated GB site is a strong predictor. This predictor alsotakes into account the nearest-neighbor information encoded in it.Dösinger et al. [112] studied the efficacy of structural descriptors and active learning strate-gies for prediction of Re segregation to W- 25 at. %Re GBs. The dataset consisted of 219segregation energies for 15 different tilt and mixed twist-tilt GBs, and 180 cartesian coordi-nates (x, y, z) of 60 atoms around each segregation site were taken as input. It has beenpreviously noted that the Q5 Steinhardt parameter gave a promising result of R2> 0.8. Usinga Gaussian process regression model, the lowest RMSE of 0.05 eV was achieved using the R15descriptor (the cartesian coordinates of 15 nearest neighbors) , among other descriptors usedin the study.30Messina et al. [113]used machine learning approaches to predict Al segregation to 30 HCPMg symmetric tilt <0 0 0 1> GBs. Various local atomic environment quantifiers, such asgeometric descriptors (Voronoi volume, Voronoi faces, coordination number, common-neighboranalysis etc.) and per-atom quantities such as potential energy (pe), forces and stresses wereused to build a predictive model for GB segregation energies with RMSE of 6.21 meV. Thisframework enabled to predict segregation energies on a large GB with 19,624 atoms with GBplane {26, 1¯7,9¯,0}/{9¯, 1¯7, 26, 0} with just several seconds, while MS could have been 4-6 ordersof magnitude higher. This cost will be substantial for larger GB structures and can be used tofasten solute atom swap states in kinetic Monte Carlo simulations.Zhang et al. investigated five GB structures in Al and 44 solutes for a total of 924 segregationenergies were calculated. An important feature that was inspected was the Voronoi volume ofsolute atoms in calculating segregation energies as seen in Figure 2.23. Solutes such as K, Ba,etc., showed the strongest segregation tendency compared to Mg and Cu solutes, which aresimilar sized to Al, showing negligible segregation. Different ML algorithms, including linearregression models, decision tree models, support vector regression models and an artificialneural network (ANN) are used to study the atomic features in Figure 2.23. The highest modelaccuracy was found for the ANN model, 𝑅2 = 0.958, which is expected since ANN presentsthe highest model complexity among the ML models. GB property-feature surrogate modelrelationships are incredibly valuable in fast and accurate estimates of segregation energies inmetals. However, it should be noted that prediction is not made on a very different GB structurethan the ones in the training set.Figure 2.23: A correlation color map of the output, segregation energy (Target) with differentinput features, which include physical properties like electronegativity, atomic features for so-lutes and geometric features. Taken from [114] with permission from Elsevier.31Figure 2.24: A schematic for creating a training set using RandSPG. The top row in eachbox represents the operation on the structures and the middle row number and the last rowrepresent a unique set of structures and the total number of structures, respectively. Takenfrom [109] with permission from the American Physical Society.2.3.6 Training set generation for MLIPsA crucial part of training MLIPs is the generation of training set structures. A main bot-tleneck is which structures are added to the training set and which structures these models aretested on. A recent effort by Poul et al. [109] used an approach to model defects in Mg andthermomechanical effects in Mg. The software used in their work is RandSPG [115], whichgenerates a structure consisting of a small number of atoms using all the 230 space groupsavailable in crystals [116]. This is done by setting the limit for the total number of atoms inthe supercell and a minimum separation distance between atoms. The training set generationscheme is shown in Figure 2.24. A main advantage of this approach is that the cost of creatingthe training data is relatively cheap since the number of atoms is less than 10. This allows asystematic sampling of all the different atomic environments in Mg. On the other hand, the ∼80,000 structures in the training set are relatively large set and get even larger for alloy systems.To develop GAP MLIPs for predicting GB structures and energies in graphene, Guo andco-authors [117] used a more conservative approach in developing their training set. Theyidentified only three GBs (Σ 43, Σ 91 and Σ 103) to be included in the training set , out ofthe total 21 GBs that were of interest. This analysis was performed using the Structural UnitModel [118] of symmetric tilt GBs in graphene and identified those GBs. The RMSE error onenergies and forces for training and testing set were 3 meV/at. and 65 meV/Å, and, 3 meV/at.and 91 meV/Å, respectively. The GAP model accurately predicted GB structures, energies,forces and phonon properties compared to DFT.32Subramanyam and Perez [119] developed a material-agnostic approach to developing trainingsets for MLIPs using an information-entropy driven approach. In simple terms, the objectivefunction should penalize redundant information and promote information diversity in the train-ing set. ACE potentials were developed for unary systems such as W, Al, Be, Re, Os, and C.Five GBs were tested, and the highest relative error of 25% was found for Σ3[11¯0](112) andΣ3[11¯0](110) and the lowest for Σ5[100](013) with a 1% error.In the case of alloys, Hodapp and Shapeev [120] designed MTPs to predict stacking faultenergies (SFEs) for MoNbTa medium-entropy alloys and their corresponding binary systems.A key aspect of building a training set was how to effectively sample the composition space tomake accurate predictions for the desired composition space. They found that starting from auniformly sampled initial training set in the amount of Mo and Nb in MoNbTa ternary alloy gaveaccurate predictions of SFEs. A more recent study by Poul and co-authors [121] extended theRandSPG approach to model MLIPs for the ternary alloy system, Mg-Al-Ca. The automatedsmall symmetric structure training approach was used to construct a training set automaticallywith only a few parameters, such as the total number of atoms in each training configurationalong with its composition, how many structures are sampled along relaxation trajectory inDFT, and the amount of random perturbed structures added. They reproduced point andextended defects for binary and ternary systems to a mean average error of 135.7 meV and 0.92meV/Å2, respectively.A final note on all of the above cited studies is that they provide a publicly available reposi-tory of the training data and a working version of MLIPs developed for interested users to testthem. This is a major aspect of the MLIPs training process since the data used is the mostcrucial factor in the prediction accuracy and is part of best practices for machine learning inmaterials science [122].332.4 Summary of literature reviewBased on the above literature review, it is evident that ab-initio simulations are only availablefor a few GBs because of the associated computational cost of simulating a large computationaldomain. Additionally, there exist a few studies on vacancies and solute interactions in pure hostsystems, but only a few exist for alloys due to the various local host-vacancy configurations.Calculations using semi-empirical applications are more common and can simulate large systemsizes. However, they do suffer from reproducing properties with ab-initio accuracy. To tacklethis, MLIPs present an interesting opportunity to build interatomic potentials to predict prop-erties with first-principles accuracy. However, no studies exist to evaluate these interatomicpotentials for solute or vacancy interactions in alloys. Thus, the scope and objectives of thisthesis are outlined in detail in the next chapter.34Chapter 3Scope and objectivesThis thesis is focused on investigating defect-solute and defect-vacancy interactions in binarymetallic alloys, particularly emphasizing grain boundaries (GBs) as the defect of interest. Mod-eling a chemically complex system, the number of permutations/combinations of solute-hostpairs grows significantly compared to the case of pure systems. Not much has been done tostudy GB solute segregation in alloys with DFT. A rigorous approach to calculating propertiesat each site has never been explored in the literature. A Ti-Mo alloy was used as a modelsystem since Mo solute was added as a BCC Ti phase stabilizing element. Going further, solutesegregation calculations for various alloy compositions are far fewer in the literature and willgive insights into these interactions in the full composition space.With the latest development of MLIPs, ACE potentials serve as a good alternative to modelinteractions in alloys. Instead of a solute, vacancy serves as a point defect that interacts withGBs in Pt-Au alloys. With Pt-Au alloys, the main motivation is that they are candidate alloysfor solute-stabilized nanocrystalline alloys. ACE models are proven to be computationallycheaper while retaining DFT accuracy.These models require structures in the training set, and the quantity of these structuresrequired for accurate prediction still remains unclear. Especially in the case of a multitudeof host-vacancy pairs in alloys, recommendations are required for what set of structures areincluded in the training for a global alloy composition.35Given the above set of aims to meet, the following specific objectives are identified:1. Quantify the solution and segregation energies of three solutes, i.e. Y, Zr and Nb, at GBsin a binary Ti-25 at.% Mo alloys using DFT calculations.2. Determine the solute segregation energy of Y in Ti-Mo alloys with different Mo alloyingcontent using DFT calculations.3. Develop a workflow for ACE models with EAM data as input and illustrate the use ofuncertainty-based ACE model training for Pt-Au alloys.4. Quantify vacancy formation energies in Pt-Au bulk and GBs using ACE models trainedon DFT data.Each chapter documents methodology including how atomic structures were generated, sim-ulations settings and equations for thermodynamic quantities of interest. For Chapter 4 andChapter 5, details about DFT calculations, relaxation and GB construction procedures areprovided. In Chapter 6 and Chapter 7, details on reference interatomic potential/model, ACEmodel training data and ACE model regression settings are mentioned.36Chapter 4Modeling solute-grain boundaryinteractions in a BCC Ti-Mo alloyusing DFT 14.1 OverviewIn this chapter studies are presented on the segregation of solute species in a binary alloy, Ti-25 at. % Mo. Three solutes, Y, Zr and Nb are chosen in our study as they are common alloyingelements in 𝛽-Ti alloys. They are, for example, added to produce fine-grained microstructures[123] and low-modulus bone implants for biomedical applications [124, 125]. Further, thesethree solutes are neighbors in the periodic table with different atomic sizes. Using densityfunctional theory (DFT) calculations, the binary Ti-25 at. % Mo alloy is represented withinthe SQS approach and all possible configurations with the solute are included when evaluatingthe energetics of the bulk and GB structure. Representative values are computed and analyzedfor bulk solution energies and segregation energies of Y, Zr and Nb to the Σ5(130)[001] GB.Further, segregation of the host elements (Ti and Mo) to the GB plane are calculated and finallythe computed energies are analyzed in terms of the local volume and chemistry. The calculatedsegregation energies are then used to develop an Eshelby-type model to predict the segregationenergies of other solutes which have not been calculated using DFT. In this analysis, the latticeconstant of the studied alloy, the bulk volume of solutes and the elastic properties (bulk andshear moduli) of solutes and host elements are used to estimate solute segregation energies.1The results presented in this chapter have been published. See Umashankar, H., Scheiber, D., Razumovskiy,V. I., & Militzer, M. (2023). Modeling solute-grain boundary interactions in a bcc Ti-Mo alloy using densityfunctional theory. Computational Materials Science 229, 112393.374.2 Structural models for bulk and GBsTi-25 at. % Mo is chosen as the alloy composition for our calculations. For creating the alloystructures, the Warren-Cowley short range order (SROs) parameter [126, 127] is computed forsix consecutive nearest neighbor (NN) shells. A structure with SRO=0 for first NN and SRO< 0.01 for the other five shells is considered as a representative Ti-Mo alloy structure. For thebulk structure, a 128 atom supercell is used with dimensions of 12.85×12.85×12.85 Å as shownin Figure 4.1. In order to obtain energies of a Ti and Mo atom in a Ti- 25at. % Mo structure, a125 atom supercell is also created (which contains 93 Ti with 31Mo + 1Mo), i.e., with a slightlyhigher (25.6 at. % ) Mo content. The SROs of the 125 atom supercell is set to be similar tothat of the 128-atom supercell. For creating the GB structure, first a 60-atom bulk structureis generated with similar SROs as the 128-atom supercell with x, y, z axes aligned as [130],[3¯10] and [001] crystallographic directions respectively. A Σ5(130)[001] GB bcc Ti-25 at. %Mo structure with dimensions of 10.16×20.32×9.64 Å, containing 120 atoms is constructed bychoosing a y-plane to mirror in the 60 atom supercell. Ten different alloy GB structures arepossible to create with this approach.4.3 DFT calculations settings and detailsDFT calculations are conducted using the Vienna ab initio simulation package (VASP) withthe (PAW) pseudo potentials [128, 129], and the (PBE) generalized gradient approximation(GGA) [130] for the exchange-correlation functional to arrive at the ground-state structuresand energies. To truncate the plane-wave expansion, 360 eV energy cutoff is adopted. A k-mesh density of 0.02 Å−1is employed to sample the Brillouin zone. The outermost electronshell (and number of electrons) in the pseudo potentials used for Ti, Mo, Y, Zr and Nb are 4s(12), 5s (14), 5s (11), 5s (12), 5s (13), respectively. A force cutoff of 0.01 eVÅ−1is adopted forthe positions only relaxation step and an energy cutoff of 10−8 eV is used for the volume onlyrelaxation step. Periodic boundary conditions without vacuum are adopted since otherwisethe free surfaces cause instability in the simulations. The Birch-Murnaghan equation of state(EoS) method [131] is employed to relax the GB and GB-solute structures. It is often foundthat varying the supercell length perpendicular to the GB plane changes the excess volume ofthe GB, 𝛿𝑉 , and thus the GB energy [132]. In our study, the length of the supercell normalto the GB plane is varied in steps of 0.2 Å to change the volume. Four such calculations areperformed in order to obtain the minimum energy for each GB and this GB structure is thenused for any further calculations. Finally, solute substitution is only done at the GB habit planein order to preserve symmetry. For the post-processing step, modules from pymatgen [133] andpyiron [134] codebases are used to perform the EoS fits and obtain the relevant informationfor Voronoi volume and local chemistry at each site, i.e., number of Mo and Ti atoms around38each atom in the first nearest neighbor (1NN) shell. Bulk HCP Y, Zr and BCC Nb simulationsare setup to determine the energy-per-atom of these solutes that is required to calculate thesolution and segregation energies of these solutes.Figure 4.1: Schematic of all DFT supercells used in the study, gray and red atoms refer to Tiand Mo sites, respectively. (a) 4x4x4 bulk supercell of 128 atom alloy structure (b) GB-I and(c) GB-II alloy structure. Note: The orthogonal GB supercells are rotated slightly to visualizeall atomic positions.4.4 Solute substitution in bulk alloyAll 128 sites in the alloy bulk structure are substituted by a third species, i.e., the solute.There are two host species (Ti and Mo) substitutions for the solute. Since each of the sites hasa different local environment, there is a distribution in the solute solution energy. In order tocompute bulk solution energies, we first calculate the energy of one Mo atom (EMo,alloy) andone Ti atom (ETi,alloy) in the Ti-25 at. % Mo alloy to obtain the chemical potential of the twospecies using:39EMo,alloy = [E125(93, 32, 0) − 3132 · E128(96, 32, 0)] (4.1)ETi,alloy =13 · [E128(96, 32, 0) − E125(93, 32, 0)] (4.2)In the above notation, the number of atoms of each species is given inside the parenthesis.The first number refers to the majority species, Ti, the next two numbers refers to the countof Mo atoms and the solute species, respectively. Equations (4.1) and (4.2) represent thechange in alloy composition by one Mo and Ti atom, respectively, in the framework of theadopted supercells. Hence equation (4.1) provides the energy of one Mo atom in the alloy andequation (4.2) the energy of one Ti atom.The per-atom energy or chemical potential of Mo and Ti in the Ti- 25 at%. Mo alloy iscalculated to be -11.22 and -7.76 eV, respectively. In contrast, the per-atom energy of Mo andTi in their respective BCC structure is calculated to be -10.93 and -7.71 eV, respectively. Bulksolution energy of solute X at a Ti site is defined as follows:EsolX@Ti = E128(95, 32, 1) − E128(96, 32, 0) − EX,ref + ETi,alloy (4.3)Similarly, for substitution of solute, X at a Mo site one has:EsolX@Mo = E128(96, 31, 1) − E128(96, 32, 0) − EX,ref + EMo,alloy (4.4), where EX,ref refers to the energy of a single solute atom in their ground state structure. ForY and Zr, the HCP structure and for Nb, the BCC crystal structure is relaxed to obtain therespective per-atom solute reference energies.40Figure 4.2: DFT calculated bulk solution energy distributions (out of 128 data points) of (a)Y, (b) Zr and (c) Nb in a Ti- 25 at. % Mo alloy. Solution energy distributions of each soluteis split into Ti and Mo site distributions indicated by blue and orange bars, respectively. Thevertical dashed lines represent the mean value of the individual distributions.The two distributions (EsolX@Ti and EsolX@Mo) can be reconciled into a single distribution EsolXfor each solute as shown in Figure 4.2 using the approach given by Equation (4.1) and Equa-tion (4.2). These distributions can be described with normal distributions characterized bya mean value and standard deviation. Note, in the case of pure systems, these distributionscollapse into a single value or a Dirac delta distribution. The atomic sites which are at themean value are considered as “representative sites” for the solutes in the bulk. From Figure 4.2,we find that Y (mean: 1.29 eV, std: 0.16 eV) has a strong tendency to not dissolve in the bulkfollowed by Zr (mean: 0.3 eV, std: 0.1 eV). This can be rationalized based on the size of soluteatoms, Y being a larger solute atom has the least tendency to dissolve in the bulk. The rangefor Y solution energies is 1.0 - 1.7 eV and for Zr, the range of solution energies is 0.1 - 0.5 eV.On the other hand, Nb, has comparatively minor solution energies with a mean of 0.03 eV and astandard deviation of 0.08 eV. Favorable dissolution, i.e. EXsol < 0 occurs for Nb when replacingMo atoms in the original bulk alloy as illustrated in Figure 4.2(c). Although there exists asingle distribution, minor differences between the individual distributions for Ti and Mo hostsubstitutions are found. For Y, Zr and Nb, the difference between < EsolX@Ti > and < EsolX@Mo >are 45, 48 and 56 meV, respectively using the mean values of the distributions shown above.Representative bulk sites are identified when the solute substitutes at a Ti site and Mo site,respectively. The solution energies for these representative Ti and Mo sites (Erep(95, 32, 1) andErep(96, 31, 1)) are 1.33, 0.29, 0.04 eV and 1.28, 0.31, 0.03 eV for Y, Zr and Nb, respectively.414.5 Grain boundary energy of different GB configura-tionsWhen creating the Ti-25 at. % Mo alloy GB structure, there are many possibilities onmirroring across the y-plane in the bulk supercell. This leads to different arrangements ofTi and Mo sites in the GB structure and as a result there are varying concentrations of Moat the GB. There are ten possible GBs in the present case but only five of them have stableconfigurations. The other five configurations contain less than 20 at. % Mo in either the bulkor the GB region of the supercell and lead to significant reconstruction.In order to characterize the concentration of Mo at the GB, the structural unit model ofa Σ5 alloy GB is used, as highlighted in Figure 4.3. From the schematic, it is clear that thestructural repeat units (5 atoms for each unit in Σ5) can have different arrangements of Moatoms in them. To quantify the concentration of Mo in the GB region, the ratio of number ofMo (small-sized) atoms to the number of GB (shown in dark violet) atoms is used.Figure 4.3: Schematic showing the arrangements of Ti and Mo atoms and structural units(GB region) of the (130) [001] for the GB-I and the GB-II structure are shown in (a) and (b),respectively. To make the distinction between Ti and Mo, Ti and Mo species are shown by smalland large spheres, respectively. The GB region atoms are shown in dark violet and are identifiedby Common Neighbor Analysis (CNA) modifier in OVITO [135]. Note: The orthogonal GBsupercells are rotated slightly to visualize all atomic positions.The GB energy (𝛾GB) is computed as follows:𝛾GB = [EGB(90, 30, 0) − 90 · ETi,alloy − 30 · EMo,alloy]/(2 · AGB) (4.5)Here, AGB refers to the area of the GB, and a factor of two is introduced to account for twoGBs since periodic boundary conditions are used. EGB(90, 30, 0) refers to the energy of the Ti-25at. % Mo GB supercell, which varies according to individual arrangements of Ti and Moatoms in the GB structure. Table 4.1 summarizes the GB energy together with the chemicalcompositions of the GB region as well as in the GB planes.42Table 4.1: GB energy (𝛾𝐺𝐵) listed with concentration of Mo in GB region, GB plane for eachof five stable GB configurationsConfigurationMo content in the GBregion (at. % Mo, outof 60 atoms)Mo content at GBplane (at. % Mo, outof 12 atoms)GB energy, 𝛾𝐺𝐵 (mJm−2)1 20 16.6 8952 (GB-I) 23.3 33.3 11003 (GB-II) 23.3 50 9944 26.7 33.3 10835 30 33.3 1011As shown in Table 4.1, the GB energies are found to be approximately 1 J m−2, with a 10%variation from 0.9 J m−2 to 1.1 J m−2. Within this range of variation, there is no clear trendwith the Mo content in the GB plane or GB region. Nevertheless, the lowest GB energy isfor the structure with the lowest Mo concentration at the GB region and GB plane. Similarly,Shen et al. [46] predicted a non-linear variation of GB energy in bcc U as function of Mo or Nbcontent. Further, the role of domain size has been evaluated by conducting a simulation witha 240-atom cell for the GB-II configuration. The cell size has been increased in the direction ofGB normal, i.e., to minimize the interactions across the grain. In this case, the calculated GBenergy is 966 mJ 𝑚−2, i.e. within 30 mJ 𝑚−2 of the GB energy of the 120 atom cell such that itis concluded that this smaller domain size is sufficient for convergence of the GB simulations.This difference is well within the reported value for change in GB energy with system size asseen in work by Jin et al. [136]. For further details on all five stable grain boundaries and theirrespective grain boundary properties, please refer to Appendix A.4.6 GB Segregation in Ti-Mo alloy4.6.1 Segregation energies of Ti/Mo to the GB planeThe two GBs with 23.3 at. % Mo in the GB region which is close to the alloy composition areselected for solute substitution in the GB habit plane. Based on the local GB site chemistries,there are six different segregation sites for each GB, i.e. 3 sites for Ti and Mo each at GB-Iand 4 and 2 sites for Ti and Mo sites at GB-II, respectively. The segregation energies of Moand Ti in the binary Ti-Mo system to the GB plane are calculated as follows:43EsegMo@Ti = [EGB(89, 31, 0) − EGB(90, 30, 0)] − [Erep(95, 33, 0) − E128(96, 32, 0)] (4.6)EsegTi@Mo = [EGB(91, 29, 0) − EGB(90, 30, 0)] − [Erep(97, 31, 0) − E128(96, 32, 0)] (4.7)In order to calculate the two quantities above, we need a representative site for a Mo atomat a Ti site in the bulk alloy and similarly for a Ti atom at a Mo site. We take the average of 96data points for Mo substitution to be Erep(95, 33, 0; bulk) and the average of 32 data points forTi substitution to be Erep(97, 31, 0; bulk). For the GBs, all sites at the GB plane are substitutedfor Ti and Mo at Mo and Ti sites, respectively and the results are included in Figure 4.4. Interms of statistics, we have 7 sites from GB-I (3 sites) and GB-II (4 sites) available for Mosubstitution. Similarly, we have 5 sites from GB-I (3 sites) and GB-II (2 sites) available for Tisubstitution. Segregation energies of Ti have a mean of -0.33 eV and range from -0.44 to -0.23eV. On the other hand, Mo segregation has a mean of 0.44 eV and a rather large range from0.1 to 0.69 eV. These results indicate that Ti has a tendency to segregate to the GBs whereasMo tends to de-segregate.4.6.2 Solute segregation energies to the GB plane for GB-I and GB-IIThe segregation energy of a solute, X to a Ti site is calculated as follows:EsegX@Ti = [EGB(89, 30, 1) − EGB(90, 30, 0)] − [Erep(95, 32, 1) − E128(96, 32, 0)] (4.8)Similarly, for a Mo site, the segregation energy is computed as:EsegX@Mo = [EGB(90, 29, 1) − EGB(90, 30, 0)] − [Erep(96, 31, 1) − E128(96, 32, 0)] (4.9)Figure 4.4 shows the segregation energies of the three solutes in GB-I and GB-II. For Y,there is a case when substituting Ti in GB-II leads to a structural change of the GB resultingin a large (-2.02 eV) segregation energy which is not shown in Figure 4.4. Overall, there is aclear trend for the segregation energies of the three solutes. Y has the strongest attraction tothe GBs followed by Zr whereas Nb shows only comparatively weak attraction when replacingMo but not when replacing Ti. This is consistent with their bulk misfit as shown above based44on solution energies (see Figure 4.2), i.e. order of sizes of solutes are as follows: Y > Zr > Nband the larger the differences in atomic sizes between the solute and the host, the larger thesegregation energy. Further, for all solutes replacing Mo in the GB is preferred as compared toTi substitution which is consistent with the segregation energies of Mo and Ti at the GB sitesin the binary Ti-Mo system, i.e. there is a preference of Ti segregation and de-segregation ofMo.Figure 4.4: DFT computed segregation energies of Y, Zr and Nb to Ti and Mo host sites inGB-I (diamond markers) and GB-II (square markers). Individual points are colored based onwhether Ti (blue) or Mo (orange) is substituted at the GB plane. Segregation energies belowthe dashed line suggest favorable segregation, while above the dashed line suggest unfavorablesegregation.4.7 Discussion4.7.1 Bulk propertiesIn the bulk alloy, there 128 sites of Ti and Mo in total. The Voronoi volume of these sitesis shown in Figure 4.5 as a function of the number of Mo atoms in the first nearest neighbor(1NN) shell of each atomic site. With an increasing number of Mo atoms in the 1NN shell,the local Voronoi volume increases for Mo and decreases for Ti sites, respectively. Further, thelocal volume of Ti sites is in general larger than that of Mo sites.In order to decouple the local chemistry and local volume effects on the bulk solution en-ergies, the trends between solution energies and Voronoi volume before solute substitution areinvestigated first. Note that a common approach to that is the “ghost impurity model” [137],but for the complex case of an SQS structure, this requires substantial computational efforts.As shown in Figures 4.6(a) and (b), the trends for Y, Zr and Nb’s bulk solution energy are thesame except for Nb on Ti sites where the solution energies are negligible. In general, there is a45decrease of solution energies with increasing site volume. This trend is particularly pronouncedfor Y when replacing Ti. Secondly, solution energies at both Ti and Mo sites as a functionof Voronoi volume after solute substitution are shown in Figure 4.6(c) indicating that solutesolution energies at both host sites are correlated with the Voronoi volume of the substitutedsolute.Figure 4.5: Voronoi volume of Ti and Mo sites in the bulk alloy as a function of number of Moatoms in the 1NN shell (out of 8 atoms).The negative slope in Figure 4.6(c) can be rationalized by size effects, Y being a larger soluteatom prefers to dissolve in sites which are larger in size in the distribution of available sites.Similarly, Zr has a moderate linear dependence with the Voronoi volume indicating the sizeeffects persists although to a smaller extent than for Y. In the case of Nb, the dependencewith the volume is negligible and this can be rationalized by the fact that Nb is a smallersolute compared to the other two solutes. Finally, in Figure 4.6(d), the solution energies areplotted as a function of Voronoi volume difference before and after solute substitution. Thevolume differences are larger when the solutes substitute Mo sites than Ti sites. Additionally, itshould be noted that for each solute, there is no correlation of solution energies with the volumedifferences. As both the volume before and after relaxation with solute substitution show acorrelation with the Voronoi volume, but not the change in Voronoi volume, this indicates thatthe relaxation volume has only a minor influence on the segregation tendency.In addition to site volume, the local neighborhood can affect the bulk solution energy dis-tribution. To analyze the potential chemistry effect the number of Mo/Ti atoms in the 1NNshell can be considered as a measure. The results of this analysis are shown in figure 4.7. Asnoted in ref. [138], the slopes of the trend lines are essentially the interaction parameters be-tween the solute and the host atom that the solute is being substituted for. Slopes of the lines46of Nb (-0.1 eV/at) and Zr (-0.09 eV/at) substituting a Mo site indicate towards to favorabledissolution when increasing the number of Mo atoms in the local chemical neighborhood. Thiscould be because of chemical interaction (i.e., electronegativity) between Nb and Mo as foundin ref. [139] when studying high-entropy alloys containing Mo and Nb. Another key observationis the dissolution of Y in a Ti site (slope of -0.08 eV/at); it becomes favorable to dissolve aY atom in a Ti site with local neighborhoods having a higher number of Ti atoms. This canbe rationalized by the strain-relief argument where a Y atom likes to dissolve at the Ti siteswith a larger Voronoi volume as compared to the Mo sites as seen in Figures 4.6(a) and (b),respectively. This variation indicates that there might be synergistic effects of local volume andchemistry for the case of Y substituting at Ti. Finally, there is almost no correlation betweenNb and Zr substitution energies at Ti sites and the number of Ti in the first neighbor shell. Thisis true for Y substitution at Mo sites as well. Thus, the dominant effect on the bulk solutionenergies is the local Voronoi volume for these cases and it should be noted that the local volumeis also dependent on the local arrangement of Mo and Ti atoms around the substituted site(see Figure 4.5).47Figure 4.6: (a)Computed bulk solution energies at the Mo site plotted as a function of theVoronoi volume of the Mo sites before solute substitution. (b) Computed bulk solution energiesat the Ti sites plotted as a function of the Voronoi volume of the Ti site before solute substitution(c) Computed bulk solution energies for Y, Zr and Nb at both Ti and Mo sites plotted togetheras a function of Voronoi volume after solute substitution. (d) Computed bulk solution energyfor Y, Zr and Nb at both Ti and Mo sites plotted together as a function of Voronoi volumedifference between before and after solute substitution.4.7.2 GB propertiesThe differences in solute segregation to Ti and Mo sites are analyzed in further detail.First, a discussion on Voronoi volume of the undecorated and decorated GB site is presented.A trend analysis of segregation energies with Voronoi volume of clean GB sites shows thatthe segregation energies of various solute-host pairs are essentially independent of the Voronoivolume of clean GB sites. Next, the trends of solute segregation energies as a function ofVoronoi volume of solute substituted sites is presented as shown in Figure 4.8(a) where at leastfor Y the magnitude of the segregation energy increases with site volume. The trends for Zr48Figure 4.7: (a) Computed bulk solution energies at Mo sites plotted as a function of numberof molybdenum atoms in the first nearest neighbor (1NN) shell around each solute-substitutedsite. (b) Computed bulk solution energies at Ti sites plotted as a function of number of titaniumatoms in the first nearest neighbor (1NN) shell around each solute-substituted site. The trendlines are obtained by averaging the values at number of sites having the same # of Ti/Moatoms in the 1NN shell.and Nb remain negligible. Although the individual data for Ti and Mo substitution seem tohave similar volume ranges, the segregation energies vary for both cases.However, the volume difference before and after solute substitution provides a clearer trendfor the resulting segregation energies. This is shown in Figure 4.8(b), where the x-axis representsthe change in site volume between solute segregated site and that of the clean GB site, i.e.∆VX−>Ti/Mo = VX,i − VTi/Mo,i. Here, VX,i is the Voronoi volume of solute at GB site i andVTi/Mo,i is the Voronoi volume of the Ti/Mo atom at the same GB site i. A general observationis that a volume increase favors segregation whereas a decrease in volume leads to a positive,i.e. unfavorable, segregation energy as is the case for Nb on Ti sites. Nb segregating to a Tisite occupies a smaller volume than its undecorated counterpart site which is associated withpositive (unfavorable) segregation energies. Further, a general trend is that the volume changesfor Mo sites are larger than for Ti sites and hence this further indicates why the magnitudeof favorable segregation energies to a Mo site is larger. This volume difference scales linearlywith the solute segregation energy, where a larger volume difference implies a stronger solutesegregation.49Figure 4.8: (a) DFT computed segregation energies of Y, Zr and Nb to Ti (circles) and Mo(squares) host sites in GB-I and GB-II as a function of Voronoi volume of solute substitutedsites (VX@Ti/Mo) (b)DFT computed segregation energies of Y, Zr and Nb to Ti (circles) andMo (squares) host sites in GB-I and GB-II as a function of volume difference between Voronoivolume after and before solute substitutions at GB plane sites(∆VX−>Ti/Mo)A consistent shift of segregation energies when replacing Mo versus Ti is also predicted forY, Zr and Nb, i.e. the attraction for these solutes to the GB is larger when replacing Mo.Excluding the outlier (-0.25 eV) for Y at Ti host substitution, differences between< EsegX@Ti >and < EsegX@Mo > for Y, Zr and Nb are 0.41, 0.41 and 0.50 eV, respectively. However, thesedifferences do not exactly match with the difference of 0.77 eV between Ti and Mo segregationenergies in the Ti-Mo system (see Figure 4.4). Thus, there is also a solute specific role inchanging the GB site volume.Figure 4.9: Comparison of average of segregation energies for Y, Zr and Nb to Ti and Mo sitescalculated using DFT and Eshelby-type model.50The above analysis indicates that elastic interactions are significant for the magnitude ofthe segregation energies. In order to obtain average segregation energies, the evaluation of anEshelby-type approach may be considered using information of the undecorated GB structuresas proposed in previous studies [140,141]. From Figure 4.8(b), it is seen that there is primarilyelastic contribution to the segregation energies of solutes.Eshelby [142, 143] first proposed a quantification of the elastic energy contribution for in-troducing an inclusion or precipitate in a soft matrix material. In the case of GB segregation,solute atoms are considered as having different volume than the host atom and the elasticenergy can be computed by bringing a solute atom from the bulk to the GB site. This wasproposed by Scheiber et al. [144] with an expression which captures these interactions based onan Eshelby-type model:Eseg,elX@Ti/Mo =−(VA − VB)2( 32 · VAGA + 2 ·VBKB )+ (VGBA − VB)2( 32 ·VGBAGA + 2 ·VBKB )+ −(VGBA − VA)2( 32 ·VGBAGA + 2 ·VAKA )(4.10)On the right hand side of the above equation, the first term accounts for energy releaseby removing the solute atom B from the bulk host site, A. The second term refers to energycontributing to the addition of solute atom B into the host site A at the GB associated with adifferent volume, 𝑉𝐺𝐵𝐴 . Finally, the third term is the energy associated with the removal of hostA atom from the GB site. Here, VA, VB and VGBA refers to bulk volume of host and solute aswell as volume of the GB sites, respectively. The elastic properties, GA, KA and KB are shearmodulus of host, bulk modulus of host and solute, respectively.These properties for the host atom are taken to be that of Ti-Mo alloys: 100 GPa for Young’smodulus from [51] and a Poisson’s ratio of 0.33 was assumed which allowed to calculate 𝐺𝐴 and𝐾𝐴 to be 98 GPa and 37.6 GPa, respectively. Here,𝑉𝐴 is determined based on the calculatedlattice constant for the Ti-Mo alloy of 3.213 Å. The reference [51] used for obtaining the Young’smodulus is based on DFT calculations. In the same reference, there is a reasonable agreementobserved between DFT calculations and experiments. Hence, the framework for calculating thebulk and shear modulus of the alloy is consistent with other data used in the Eshelby model.In the case of Y, Zr and Nb as solutes (B), the properties are taken from Bakker [145]. Asproposed by Scheiber et al. [144], a scaling constant 𝑐 can be introduced for 𝑉𝐺𝐵𝐴 to improvefits to the calculated segregation energies by changing the clean GB site volume by multiplyingit with 𝑐. Since, there exists a difference in solute segregation energies when substituting Ti orMo sites (see Figure 4.4 ), two scaling factors (𝑐𝑇𝑖 and 𝑐𝑀𝑜) are quantified, one for each hostspecies. The scaling factors 𝑐𝑇𝑖 and 𝑐𝑀𝑜, are found to be 1.2 and 1.5, respectively reflecting thelarger volume change of Mo sites after solute substitution.51Table 4.2: Comparison of data for volume and bulk modulus from Bakker [145] and DFT-computed values from the Materials Project repository for predicting solute segregation energyusing Eshelby-type modelSpecies,crystalstructureBulkmodulus (K)fromBakker [145],GPaBulkmodulus (K)fromMaterialsProject, GPaBulk volume(𝑉𝐵) fromBakker [145],Å3/at.Bulk volume(𝑉𝐵) fromMaterialsProject,Å3/at.Nb, BCC 170.3 174 17.94 18.31Zr, BCC 83.35 89 23.25 23Y, HCP 36.62 41 33.04 32.85Whilst, these models do not capture the true complexity of each site they still give overalltrends of solute segregation energy predictions for the considered alloy system. The predictionsfrom the model based on elastic interactions are summarized in Figure 4.9 indicating that theY and Zr average segregation energies can be explained well within this framework, while Nbpredictions are a bit off from the DFT calculated values. Thus, other factors as e.g. chemicalinteractions may affect the segregation energies of smaller-sized solutes such as Nb. Still, thedeviations are within 0.25 eV and general trends among the solutes are predicted correctly,nonetheless. If reasonable predictions could be made in this way one would not have theneed for performing various solute-GB calculations. Huber et al. [140] have shown that solutesegregation energies of Pb (over-sized) and Mg (smaller) in Al scale linearly with Voronoi volumeof clean sites in symmetric and general tilt boundaries. Using this as a motivation, the Eshelbymodel with the Voronoi volume of the clean GB sites can be used to estimate the average GBsolute segregation energies for other transition metal (TM) solutes in Ti- 25 at%. Mo alloy.In Figure 4.10(a), the predicted energies are presented using the scaling constants obtainedabove for Ti and Mo sites. Table 4.2 presents the data for bulk modulus and volume of solutesobtained from Bakker [145], comparing them with the values reported through DFT calculationsfrom the Materials Project repository [146]. This comparison supports the notion that the datafrom Bakker [145] is comparable to the findings reported in the DFT calculations indicatingthat these values can be used without the need for performing additional DFT calculations.The observed trends correlate strongly with the solute volume shown in Figure 4.10(b). Thelarger solutes show similar behavior as Y and Zr, i.e. they have a tendency to segregate to theGB with a preference for the Mo sites. On the other hand, smaller solutes, i.e. those in the52middle of the transition metals series, have a tendency to de-segregate and the repulsion fromthe Mo sites is larger than that from the Ti sites. Overall, the simplified analysis provides a firstguidance regarding the solute-GB interaction for the given alloy provided elastic contributionsare dominant. This is the case for large absolute values of the segregation energy. For absolutesegregation energies below 0.15 eV (see Figure 4.10), chemical corrections may be needed. Thesimplified analysis identifies those transition metal solutes that can significantly contribute toGB enrichment and include Sc, Y, Zr, Cd, La, Hf, and Hg with a negative segregation energy,i.e. EsegX@Ti/Mo < -0.5 eV.Figure 4.10: (a) Eshelby-type model predictions for segregation energies of (3d-, 4d-, 5d-)transition metal solutes to the Σ5 GB plane in a Ti-Mo alloy system and average segregationenergies of Y, Zr and Nb from DFT (b) Atomic volume of solute elements as taken fromBakker [145]4.8 SummaryThis chapter lays out a first-principles approach for investigating solute segregation in highlyalloyed systems i.e., here Ti-25 at. % Mo based on the SQS approach. The main findings ofthis work are as follows:531. A rigorous approach has been applied to determine solution energies in the bulk by con-sidering all possible permutations in the simulation domain. The resulting solution energydistributions provide the basis to identify representative bulk energies for each consideredsolute. In the present case, i.e. for Y, Zr and Nb, elastic interactions dominate such thatthe volume of the bulk site rather than its local chemistry determine the mean solutionenergy which increase with solute size, i.e. Nb < Zr < Y. In these elastically dominatedcases, the identification of the averaged bulk volume site in the alloy without solutes canbe used to calculate the representative solution energy.2. The solute segregation energies to GBs is also dominated by elastic interactions. Thesituation at the GB is, however, more complicated as the elastic interaction may beaffected by the local chemistry. In the investigated cases, the change in GB volumesites before and after solute substitution has been identified as the critical aspect thatdetermines the solute segregation energy. This volume change has been found to dependon which of the host species is replaced, i.e. it is larger for Mo sites than for Ti sites.Thus, the magnitude of segregation energies are larger when Mo sites are replaced andoverall the segregation energies follow the solute size with Y having the largest attractionto the GBs followed by Zr and Nb.3. Using an Eshelby-type approach with a host species dependent scaling factor for thegrain boundary site volume provides a satisfactory description of the mean segregationenergies computed from the DFT simulations for strongly segregating solutes (here Y, Zr),i.e. solutes that are significantly larger than the host atoms. The Eshelby-type analysishas been extended to three different series of transition metals showing a clear trend ofsegregation energies with solute size, i.e. larger solutes are more attracted to smaller hostsites (Mo) and smaller solutes are less repelled from larger host sites (Ti).54Chapter 5Interaction of Y with grain boundaryin Ti-Mo alloys5.1 OverviewThis chapter describes the segregation studies of Yttrium in Ti-Mo alloys for three compo-sitions of 25, 50 and 75 at. % Mo. Yttrium was chosen since it was found to be the strongestsegregating solute element among the three solutes studied in Chapter 4. Using density func-tional theory (DFT) calculations, the binary Ti- 50 and 75 at. % Mo alloys are representedwithin the Special Quasi-random (SQS) approach, and all possible configurations for the Yt-trium solute are included when evaluating the energetics of the bulk and GB structure. Repre-sentative values are computed and analyzed for bulk solution energies and segregation energiesof Y to the Σ5(130)[001] GB in Ti- 50 and 75 at. % Mo alloys. Further, the segregation of thehost elements (Ti and Mo) to the GB plane for ten different GB configurations are calculated.Trends across the three compositions - 25, 50 and 75 at. % Mo are analyzed, and some of theanalysis from Chapter 4 are used here, too.5.2 Structural models for alloy bulk and alloy grain bound-ariesThe alloy theoretic automatic toolkit (ATAT) [147] was used to create a 128-atom (4x4x4)Ti-Mo bulk structure for 50 at. % and 75 at. % Mo alloy. The structures are visualizedwith OVITO [135] in Figures 5.1(a) and (b). In order to create the GB structure for the twocompositions, an approach listed in the previous chapter is followed. First, a 60-atom supercellis created using ATAT, the main difference in this chapter is that all the 10 atomic planes areused to create a Σ5(013) [100] alloy GB. These structures are now used to calculate the GB55energy and segregation energy of Y to GB sites. All the Ti and Mo atoms at the GB plane areused to substitute for Y. as shown in Figures 5.1(c) and (d). Visualization of all the ten GBsused to calculate segregation energies of Y for the 50 at. % Mo alloy and 75 at. % Mo alloy isshown in Appendix B. Periodic GB calculations are performed using the same simulation setupand input settings as in Chapter 4. The calculation for pure BCC Mo Σ5 (013) GB is also setup in this Chapter to calculate its GB energy. This calculation is done with the same setup ofchanging the GB supercell volume by varying the length of the supercell perpendicular to theGB plane direction.c) d)b)a)TiMoGB plane GB planeFigure 5.1: The atomic structure of Ti- (a) 50 and (b) 75 at. % Mo bulk alloys used to computethe interaction of Y in bulk. A representative structure (out of 10 GBs) for GB in (c) 50 and(d) 75 at. % Mo alloys. Substitution of Y is done at the GB plane highlighted by the bluedotted line. Red and grey colored atoms refer to Mo and Ti atoms, respectively.5.3 Bulk and solute in bulkThe lattice constants of Ti-Mo alloys are shown in Figure 5.2. The lattice constants displaya negative deviation from the Vegard’s rule of mixture. For the Ti - 50 at. % Mo alloy, thedeviation from Vegard’s law is the largest. Bulk BCC Ti can be simulated here to get thelattice constant, only when a defect is introduced the structure becomes dynamically unstable.56Figure 5.2: Computed lattice constant from DFT for pure BCC Ti, Ti-25 at. % Mo alloy, Ti-50at. % Mo alloy, Ti-75 at. % Mo alloy and pure BCC Mo. The dotted blue line connects thelattice constant of pure elements, i.e. BCC Ti and Mo.Starting with the Ti - 50 at. % Mo alloy, each site in the bulk alloy is substituted for Y,and then the solution energy is calculated according to Equation (5.1) and Equation (5.2) atMo and Ti substitution, respectively. In Equation (5.1) to Equation (5.4), the first numberin the parenthesis represents the number of Mo atoms, the second number represents numberof Ti atoms in the structure. The number of Mo and Ti atoms in the 128-bulk alloy for thiscomposition are 64 (𝑁128𝑀𝑜 ) and 64 (𝑁128𝑇𝑖 ) atoms, respectively.𝐸solY@Mo=𝐸Y@Ti(𝑁128𝑀𝑜 -1,𝑁128𝑇𝑖 ,1) -𝐸alloy(𝑁128𝑀𝑜 ,𝑁128𝑇𝑖 ,0) +𝜇Mo(1,0,0) -𝐸Y,hcp(0, 0, 1) (5.1)𝐸solY@Ti=𝐸Y@Mo(𝑁128𝑀𝑜 ,𝑁128𝑇𝑖 -1,1) -𝐸alloy(𝑁128𝑀𝑜 ,𝑁128𝑇𝑖 ,0) +𝜇Ti(0,1,0) -𝐸Y,hcp(0, 0, 1) (5.2)The chemical potential of the alloy is calculated by increasing the overall Mo content ofthe bulk. This is done using a different 125-atom structure with a composition close to 50at. % Mo, i.e., a structure with 64 Mo atoms (𝑁125𝑀𝑜 ) and 61 Ti atoms (𝑁125𝑇𝑖 ), leading to acomposition of 51.2 at. % Mo. The chemical potential of Mo and Ti in the alloy is calculatedby Equation (5.3) and Equation (5.4). The chemical potential is reported in Table 5.1.𝜇Ti=1(𝑁128𝑇𝑖 − 𝑁125𝑇𝑖 )[𝐸128(𝑁128𝑀𝑜 , 𝑁128𝑇𝑖 )−𝐸125(𝑁125𝑀𝑜 ,𝑁125𝑇𝑖 ) ] (5.3)𝜇Mo=1(𝑁128𝑇𝑖 − 𝑁125𝑇𝑖 )[𝐸125(𝑁125𝑀𝑜 ,𝑁125𝑇𝑖 ) -𝑁125𝑇𝑖𝑁125𝑀𝑜𝐸128(𝑁128𝑀𝑜 ,𝑁128𝑇𝑖 )] (5.4)The calculated solution energies show a continuous distribution with a mean of 1.64 eV anda standard deviation of 0.13 eV, as shown in Figure 5.3 (a). A stronger tendency to not dissolve57Table 5.1: The chemical potential [in eV] of Mo (𝜇Mo) and Ti (𝜇Ti) atomic species in Ti - 50and Ti - 75 at. % Mo bulk alloyMo content 𝜇Mo[eV] 𝜇Ti[eV]50 at. % -11.090 -7.81575 at. % -11.035 -7.921in the bulk is observed compared to the Ti - 25 at. % Mo alloy. Representative sites used tocalculate segregation energy later are picked when close to the mean value of the distribution.For this alloy composition, the representative solution energy for Mo and Ti site is 1.65 and1.63 eV, respectively. A negligible difference of 0.02 eV between the two sites is observed.For the Ti - 75 at. % Mo alloy, the chemical potential is calculated by using a different125-atom structure with a composition close to 75 at. % Mo, a structure with 93 Mo atoms(𝑁125𝑀𝑜 ) and 32 Ti atoms (𝑁125𝑇𝑖 ), leading to 74.4 at. % Mo. The chemical potentials of Mo andTi atom are computed using Equation (5.5) and Equation (5.6), respectively. The number ofMo and Ti atoms in the 128-bulk alloy are 96 (𝑁128𝑀𝑜 ) and 32 (𝑁128𝑇𝑖 ) atoms for this composition,respectively.𝜇Mo=𝑁128𝑇𝑖𝑁128𝑀𝑜[𝐸128(𝑁128𝑇𝑖 , 𝑁128𝑀𝑜 )−𝐸125(𝑁125𝑇𝑖 ,𝑁125𝑀𝑜 ) ] (5.5)𝜇Ti=[𝐸125(𝑁125𝑇𝑖 ,𝑁125𝑀𝑜 ) -𝑁125𝑀𝑜𝑁128𝑀𝑜𝐸128(𝑁128𝑇𝑖 ,𝑁128𝑀𝑜 )] (5.6)Solution energies of Y for Mo and Ti sites are calculated using Equation (5.8) and Equa-tion (5.7), respectively. Calculated solution energies of Y for the 75 at. % Mo alloy are shownin Figure 5.3 (b).58Figure 5.3: A stacked histogram showing the computed solution energies of Y in a Ti -(a)50 at. % and (b) 75 at. % Mo alloy. The vertical dashed line represents the mean of thecomputed distribution. Orange and blue color bar represents Y substitution at a Ti and Mosite, respectively.𝐸solY@Ti=𝐸Y@Ti(𝑁128𝑀𝑜 ,𝑁128𝑇𝑖 -1,1) -𝐸alloy(𝑁128𝑀𝑜 ,𝑁128𝑇𝑖 ,0) +𝜇Ti(0,1,0) -𝐸Y,hcp(0, 0, 1) (5.7)𝐸solY@Mo=𝐸Y@Mo(𝑁128𝑀𝑜 -1,𝑁128𝑇𝑖 ,1) -𝐸alloy(𝑁128𝑀𝑜 ,𝑁128𝑇𝑖 ,0) +𝜇Mo(1,0,0) -𝐸Y,hcp(0, 0, 1) (5.8)The distribution has a mean of 2.07 eV with a standard deviation of 0.13 eV. The tendencyto dissolve in bulk is even lower for this alloy compared to alloys containing 25 and 50 at. %Mo. In this case, representative site for Mo and Ti substitution is 2.06 and 2.09 eV, respectivelywith a negligible difference of -0.03 eV is observed.5.4 GB energies for different alloy configurationsDifferent GBs are constructed to study the variation of GB energies with Mo content in theGB region. The chemical potentials from Table 5.1 are used to calculate the GB energies.It is important to distinguish between alloy composition and Mo content at the GB. In59Figure 5.4, this distinction is made, the GB region refers to the atoms that are not of BCClocal structural arrangement. In each GB structure, 60 out of the 120 atoms belong to theGB region, and the rest are in the bulk region. The amount of Mo is calculated for the GBregion in each structure and is used as a metric to compare the GB energy for different alloycompositions..Figure 5.4: Schematic of representative GB structures delineating GB and bulk region for (a)50 at. % Mo and (b) 75 at. % Mo alloys. GB region is calculated for each structure usingCommon Neighbor Analysis (CNA) in OVITO [135].The following equation is used to compute GB energies for a Ti-Mo alloy:𝛾GB = [EGB(NTi,NMo) − NTi · 𝜇Ti − NMo · 𝜇Mo]/(2 · AGB) (5.9)Here, 𝑁𝑇𝑖 and 𝑁𝑀𝑜 are the number of Ti and Mo atoms in the GB supercell, respectively.For 50 at.% Mo alloy, there are 60 atoms for Ti and Mo and for 75 at.% Mo alloy there are90 Mo and 30 Ti atoms. The chemical potentials for the 50 and 75 at.% Mo alloy are takenfrom Table 5.1. The results using Equation (5.9) for the two compositions along with 25 at.%Mo alloy are presented in Figure 5.5. The average of GB energy for each alloy composition liesabove the linear rule of mixture for GB energy connecting the lowest GB energy to the pure Moside. To analyze the pure Mo Σ5 GB, data from [132] is used as a reference and good matchingwith the present study’s data is achieved. This provides more confidence in the trends emergingfrom GB energy as a function of increasing Mo content in the GB region.60Figure 5.5: Computed GB energies plotted as a function of the concentration of Mo in theGB region. Purple circles are the data from the previous chapter on 25 at. % Mo alloy. Thedotted line is a linear rule of mixture line connecting the lowest GB energy to the pure BCC MoΣ5(013) GB energy. A solid line connects the average GB energy for each alloy composition.“Scheiber2016” refers to the BCC Mo Σ5(013) GB from [132]5.5 Segregation at GBs in Ti-Mo alloys for ten GBs5.5.1 Segregation energies of Ti/Mo to the GB planeFirst, segregation of host elements to the GB is presented here. Segregation energy iscomputed for all 60 sites, six sites for each GB for ten GBs. In the 50 at. % Mo, out of the60 sites, 30 of them are Mo sites and the rest are Ti. This is not obvious since the GB regionhas different amount of Mo than the overall Mo content. The solution energy of the Mo and Tirepresentative site is -0.18 eV and -0.14 eV, respectively. Using this representative Ti and Mosite, Ti and Mo segregation energies are calculated using Equation (5.10) and Equation (5.11),respectively.𝐸segTi@Mo=[𝐸Ti@Mo,GB -𝐸GB] -[𝐸𝑟𝑒𝑝,𝑇𝑖@𝑀𝑜,𝑏𝑘 - 𝐸𝑏𝑘 ] (5.10)𝐸segMo@Ti=[𝐸Mo@Ti,GB -𝐸GB] -[𝐸𝑟𝑒𝑝,𝑀𝑜@𝑇𝑖,𝑏𝑘 - 𝐸𝑏𝑘 ] (5.11)In the above equations, 𝐸𝑏𝑘 and 𝐸𝐺𝐵 refer to the energy of alloy bulk and GB, respectively.The two terms required for segregation of Mo to Ti site are: 𝐸𝑀𝑜@𝑇𝑖,𝐺𝐵 and 𝐸𝑟𝑒𝑝,𝑀𝑜@𝑇𝑖,𝑏𝑘 whichrefer to energy of Mo at Ti site GB supercell and energy of representative Mo at Ti site inbulk structure. For Ti segregation at the Mo site, 𝐸𝑇𝑖@𝑀𝑜,𝐺𝐵 and 𝐸𝑟𝑒𝑝,𝑇𝑖@𝑀𝑜,𝑏𝑘 are the energyof GB supercell with Ti substituted at the Mo GB site and energy of representative Ti atthe Mo site alloy bulk structure. For the Ti-50 at. % Mo alloy, all the individual Mo and61Ti segregation energies to each site are shown in Figure 5.6(a) and the histogram of all suchenergies in Figure 5.6(b). Overall, Ti prefers to segregate to Mo sites, and Mo does not liketo segregate to Ti sites. The average segregation energy of Mo and Ti are 0.63 and -0.28 eV,respectively. The standard deviation for Mo and Ti segregation energy distributions are 0.19and 0.2 eV, respectively.Figure 5.6: (a) Scatter plot of Mo and Ti segregation energies at ten different GBs (GB_0....,GB_9) in Ti - 50 at.% Mo alloy with Ti and Mo substitutions shown in blue and orangeopen circles, respectively. (b) Histogram of segregation energies of Mo and Ti shown in blueand orange bars, respectively. Fraction is separately calculated for each of the Mo and Tisubstitutions. Vertical lines represent average values of segregation energies of Mo and Ti.In the case of Ti-75 at. % Mo alloy, there are 45 and 15 sites for Mo and Ti substitutions,respectively. Different representative sites are required for this alloy composition. The repre-sentative solution energy of Mo and Ti at Ti and Mo sites is 0.01 and -0.1 eV, respectively.All the individual Mo and Ti segregation energies to each site are shown in Figure 5.7(a) andthe histogram of all such energies is presented in Figure 5.7(b). This alloy composition showsthe same preference for segregation as the 50 at. % Mo alloy. However, the magnitude ofde-segregation of Mo to Ti sites is reduced compared to the Ti- 50 at. % Mo alloy. The segre-gation of Ti to Mo sites remains relatively the same. The mean segregation energy of Mo andTi are 0.38 and -0.31 eV, respectively. The standard deviation is 0.14 eV for both Mo and Tisegregation energy distributions.62Figure 5.7: (a) Scatter plot of Mo and Ti segregation energies at ten different GBs (GB_0....,GB_9) in Ti - 75 at.% Mo alloy with Ti and Mo substitutions shown in blue and orangeopen circles, respectively. (b) Histogram of segregation energies of Mo and Ti shown in blueand orange bars, respectively. Fraction is separately calculated for each of the Mo and Tisubstitutions. Vertical lines represent average values of segregation energies of Mo and Ti.5.5.2 Yttrium solute segregation energiesWith Y substituting at all sites in the GB planes of the ten GBs, more data is generatedto account for the chemical complexity of Ti-Mo alloys. Each site at the GB now representsa different local neighborhood and shows a wider variability in the segregation energies of Y.The segregation energies at Mo and Ti sites are calculated according to Equation (5.12) andEquation (5.13).𝐸segY@Mo=[𝐸Y@Mo,GB -𝐸GB] -[𝐸𝑟𝑒𝑝,𝑌@𝑀𝑜,𝑏𝑘 - 𝐸𝑏𝑘 ] (5.12)𝐸segY@Ti=[𝐸Y@Ti,GB -𝐸GB] -[𝐸𝑟𝑒𝑝,𝑌@𝑇𝑖,𝑏𝑘 - 𝐸𝑏𝑘 ] (5.13)In the Ti-50 at. % Mo alloy, for all the ten GBs, the data sorted by each GB is shownin Figure 5.8(a) and the distribution for both segregation energies is shown as a histogramin Figure 5.8(b). The mean Y segregation energies at Mo and Ti sites are -1.14 and -0.75eV, respectively. The standard deviation for the Mo and Ti substitution is 0.23 and 0.20 eV,respectively. The tendency of segregation of Y to both Mo and Ti sites is higher compared tothe Ti- 25 at. % Mo alloy.63Figure 5.8: (a) Scatter plot of segregation energies of Y in 50 at. % Mo alloy at ten differentGBs (GB_0,...., GB_9) with Mo and Ti substitutions shown in orange and blue open circles,respectively. (b) Histogram of Y segregation energies for both Mo and Ti sites shown in blueand orange bars, respectively. Fraction is separately calculated for each of the Mo and Tisubstitutions. Vertical dotted lines represent the mean of Y segregation energies.In the Ti-75 at. % Mo alloy, more Mo (smaller) sites are available at the GB plane. SinceMo does not prefer to be at the GB thus the segregation energy of Y is higher for this alloycomposition. Segregation of Y to all ten GBs is presented in Figure 5.9(a), and the histogram ofthe resulting segregation energy distributions is shown in Figure 5.9(b). The mean Y segregationenergy distribution to Mo sites is -1.45 eV, indicating a 27% increase in segregation tendencyfrom Ti-50 at. % Mo alloy. For Ti sites, the mean Y segregation energy is -1.1 eV, there is a47% increase in segregation tendency compared to the 50 at. % Mo alloy. Since there are moreMo sites, the segregation energy of Y shows a larger standard deviation of 0.17 eV comparedto a standard deviation of 0.08 eV for Y segregation at Ti sites.Figure 5.9: (a) Scatter plot of segregation energies of Y in 75 at. % Mo alloy at ten differentGBs (GB_0,...., GB_9) with Mo and Ti substitutions shown in orange and blue open circles,respectively. (b) Histogram of Y segregation energies for both Mo and Ti sites shown in blueand orange bars, respectively. Fraction is separately calculated for each of the Mo and Tisubstitutions. The vertical dotted lines represent mean of Y segregation energies.645.6 Discussion5.6.1 Bulk propertiesAll 128 sites containing Mo and Ti sites in the alloy are analyzed in detail to understandthe characteristics of both 50 at. % and 75 at. % Mo alloys. In Figures 5.10(a) and (b), theVoronoi volume of each site is shown as a function of number of Mo atoms in the first and secondnearest neighbor shell surrounding each site for the Ti-50 at. % and Ti-75 at. % Mo alloys,respectively. Since 1NN (0.87 · 𝑎𝐵𝐶𝐶) and 2NN (𝑎𝐵𝐶𝐶) distances are close in the BCC crystallattice, hence, both shells are considered for characterizing local chemical neighborhood. Thereis a total of 14 atoms in 1NN and 2NN (8 + 6) shells in the BCC crystal lattice, as evidentfrom the figure, most of the data is centered around the alloy composition, i.e., 7-8 Mo atomsin Ti-50 at. % Mo alloy. In the 50 at. % Mo alloy, the volume ranges from 15.25 to 16.1 Å3and 15.9 to 16.8 Å3 for Mo and Ti sites, respectively. On average, Ti sites with more Mo atomsaround it are smaller than those with fewer Mo atoms. In contrast, for Mo sites, there is adegree of variability in the data, however, on average there is no significant change in volumewith local chemical neighborhood.In the Ti-75 at. % Mo alloy, both Mo and Ti sites show a decreasing trend with increasingMo atoms in the surrounding local environment. Volume ranges are narrower in comparisonto the Ti-50 at. % Mo alloy, with the Mo range being 15.4 - 15.92 Å3 and Ti sites range from15.7 to 16.1 Å3. Among the Ti sites, the smallest volume is recorded for all 14 1NN+2NNneighboring sites that contain Mo, which aligns with the atomic sizes of Mo and Ti in the alloy.Most of the data for Mo sites in this alloy composition is for 9-11 Mo atoms in the surroundingneighborhood.65Figure 5.10: (a) Voronoi volume of Mo and Ti sites shown in red and grey color on the y-axis,the number of Mo atoms in the first and second nearest neighbor (1NN + 2NN) shell are plottedon the x-axis. The data is from 50 at. % Mo alloy. (b) Voronoi volume of sites in the 75 at. %Mo alloy versus number of Mo atoms in 1NN + 2NN shell. The solid lines indicate trend linesconnecting average of volume for each number of Mo atoms in the local neighborhood.Since Y is a large solute atom, Voronoi volumes can be used as a way to evaluate Y solutionenergies. First, it is worthwhile to study the solution energy of Y in bulk trends with originalMo and Ti sites where Y substitution is made. In Figures 5.11(a) and (b), solution energy ofY at Mo and Ti sites in both alloy compositions is shown as a function of Mo and Ti bulkvolumes, respectively. From the trend of solution energy versus volume of Ti sites, there is aclear indication of solution energy decreasing with increasing volume of Ti sites. In the caseof analyzing solution energy with volume of Mo sites, there is no specific trend that emergesand only clustering of data is seen. Other effects such as chemical neighborhood around eachMo site may contribute to this behavior. For the Ti-75 at. % Mo alloy, the data suggests thatthere is not much appreciable change in solution energy with increasing volume of Ti sites.Figure 5.11: (a) Scatter plot of bulk solution energy of Y at Mo sites in both Ti- 50 and 75 at.% Mo alloy shown in red and squares, respectively along with 25 at.% Mo alloy data in bluetriangles. (b) Scatter plot of bulk solution energy of Y at Ti sites in Ti- 50 at. % and Ti- 75 at.% Mo alloy shown by red and green triangles, respectively along with 25 at.% Mo alloy datain blue squares.66Finally, the volume of Y after substitution in Ti and Mo sites in both compositions areanalyzed in Figure 5.12. The strain-relief argument is followed where Yttrium’s solution energydecreases with increasing Y site volume. There is also continuity in the linear trend betweensolution energies and volume of Y at sites for 50 and 75 at.% Mo alloy, which seems to connectat ~ 1.9 eV of solution energy and 17.25 Å3 for site volume of Y solute. Adding the Ti -25at.% Mo alloy, the solution energy trend is extended to larger volumes of Y solute at Ti andMo sites.Figure 5.12: Scatter plot of solution energy of Y in bulk Mo and Ti sites are plotted againstthe respective Mo and Ti site volumes for the two alloys studied. The data for 25 at.% , 50 at.% Mo and 75 at. % Mo is shown in blue, red and green circles, respectively.5.6.2 Segregation of Y to GBsIn order to analyze Y GB segregation energy trends, three different volume metrics areconsidered. First, the variation of GB segregation energy with initial (before substitution of Y)volume of Ti and Mo sites are shown in Figures 5.13(a) and (b), respectively. In the case of Tisites at GB, segregation energy of Y at Ti sites in 50 at. % Mo alloy are mostly clustered around-0.6 eV and 17.35 Å3 and no particular trend is found. The range of Ti site volumes is from 16.5to 18.35 Å3 for both alloy compositions. For the higher Mo content alloys, segregation energyof Y at Ti sites shows a slight linear trend of increasing segregation tendency with increasingvolume. This is consistent with the strain-relief argument at the GB sites. For Mo sites at GB,67same trend emerges where 50 at. % Mo alloy shows almost variability with initial Mo GB sites’volume until 17.2 Å3 and then a weak linear trend emerges for larger volumes. For the 75 at.% Mo alloys, the data is centered around -1.5 eV segregation energy and 17.62 Å3. The sitevolume range of Mo is from 16.5 to 18.35 Å3 for both alloy compositions. In the 75 at. % Moalloy, the range of volume is small, from 17.25 to 17.85 Å3.In Figures 5.13(c) and (d) segregation energy of Y is presented in terms of volume of the Ysolute substituted at Ti and Mo sites, respectively. A linear trend is observed for Ti sites in the50 at. % Mo alloy while the trend is less evident for the 75 at. % Mo alloy. The volume rangefor Y at Ti sites for the lower Mo content alloy is from 19.5 to 22.7 Å3, on the other hand thehigher Mo content alloy has a narrow range from 19.2 to 20.4 Å3. In the case of Y at Mo sites,there is clustering of segregation energies for the 75 at. % Mo alloy around -1.4 eV and 19.75Å3. In the case of the Ti- 50 at. % Mo alloy, there is a weak linear dependence of segregationenergy with volume of Y at Mo GB sites until 21Å3 and then segregation energies plateau at-1.2 eV when further increasing the volume of Y at Mo sites.Finally, segregation energy of Y is analyzed with volume differences at Mo and Ti sites whensubstituting Y solute. In Figure 5.13(e), there is a linear trend of segregation energy withincreasing volume difference of Y at Ti sites in 50 at. % Mo alloy, the volume difference rangesfrom 2.0 to 5.5 Å3. For the 75 at. % Mo alloy, the data is more clustered around 2Å3 of volumedifference and has a narrow range from 1.8 to 2.2 Å3. Volume difference at Mo GB sites forboth alloys are shown in Figure 5.13(f), the strain-relief argument does not hold for 75 at. %Mo alloy. In this case, clustering around 2.2 Å3 and range from 1.8 to 2.6 Å3 and even showsthe segregation tendency decreases when increasing volume difference, which is opposite to thestrain-relief argument. In the 50 at. % Mo alloy, there is a weak trend of increasing segregationtendency with increasing volume difference, volume difference ranges from 2.5 to 5 Å3. Afterthe volume analysis, it would be worthwhile to perform the Eshelby analysis for 50 and 75 at.% Mo alloys for Y segregation at both Mo and Ti sites.68Figure 5.13: Scatter plot of segregation energy of Y to Ti sites as function of (a) Voronoi volumeof Ti GB sites, (c) Voronoi volume of Y substituted at Ti sites and (e) volume difference whenY is substituted at Ti sites. Segregation energy of Y at Mo sites as function of (b) Voronoivolume of Mo GB sites, (d) Voronoi volume of Y substituted at Mo sites and (f) volumedifference between Y at Mo site and the original Mo site at GB plane. The data for 25, 50 and75 at.% Mo alloy is shown in squares, triangles and circles, respectively.In order to use the Eshelby equation (Equation (5.14)) for estimating segregation energies,some properties of the alloy such as bulk (𝐾𝐴) and shear modulus (𝐺𝐴) are required. It is alsoestablished that DFT calculated values serve as a good starting point for these properties. Froma high-throughput DFT study of Ti-Mo alloys by Marker and co-authors [37], these propertiesare used to perform the Eshelby analysis. These elastic properties are summarized in Figure 5.14where a positive deviation from Vegard’s law is observed in Figure 5.14(a) for bulk modulus ofthe alloy (KA). For shear modulus of the alloys, GA, a weak negative deviation from Vegard’slaw is seen in Figure 5.14(b) . Other input properties are volume of host lattice (VA), i.e.,69estimated from the lattice constants of Ti-Mo alloys as presented in Figure 5.2. The data forvolume of solute elements (VB) are taken from Bakker [145] for Y. Finally, the Voronoi volumeof GB sites (VGBTi/Mo) before substitution are used from Figures 5.13(a) and (b) for Ti and Mosites, respectively. Similar to the analysis in the previous chapter, a constant scaling factor forMo and Ti sites is used such that the volume is scaled as follows: VGBA = CTi/Mo · VGBTi/Mo forall sites of each species.Eseg,elX@Ti/Mo =−(VA − VB)2( 32 · VAGA + 2 ·VBKB )+ (VGBA − VB)2( 32 ·VGBAGA + 2 ·VBKB )+ −(VGBA − VA)2( 32 ·VGBAGA + 2 ·VAKA )(5.14)Figure 5.14: Elastic properties such as (a) Bulk modulus of Ti-Mo alloys as function of Mocontent in alloys and (b) Shear modulus as a function of concentration of Mo in Ti-Mo alloys.Data is obtained from Table 4 as reported in study from Marker and colleagues [37]. Solidlines represent the linear rule of mixture law or Vegard’s law prediction for alloy compositionbetween 25 and 100 at. % Mo.In generating the Eshelby fits, all data points of average segregation energies computedfrom DFT and predicted from the Eshelby-type model for two host elements and two alloycompositions are shown in Figure 5.15(a). Y at the Mo site is greater than Y at the Ti siteand the corresponding segregation energies for 75 at. % Mo alloy is higher compared to 50 at.% Mo alloy. It is also noted that the segregation energy of Y to Mo in 50 at. % Mo alloy isslightly greater (-1.14 eV compared to -1.1 eV) than segregation energy of Y at Ti in 75 at. %Mo alloy. The scaling constants for Voronoi volume of Mo and Ti sites for the Eshelby-modelsfor the four cases are analyzed as a function of alloy composition in Figure 5.15(b). A slightdecrease is noticed for volume scaling constant for 50 at. % Mo alloy compared to 25 at. % Moalloy. This behavior can be attributed to smaller volumes for both Mo and Ti sites found in 50at. % compared to 25 at. %. For the 75 at. % Mo alloy, the volumes are even smaller than 50at. % Mo alloy but the segregation energies are increased. This increases in the scaling factorfor volumes of the GB sites in the 75 at. % Mo alloy compared to 50 at. % Mo alloy.70Figure 5.15: (a) Parity plot of average DFT (< 𝐸 𝑠𝑒𝑔,𝐷𝐹𝑇𝑌@𝑇𝑖/𝑀𝑜 >) and Eshelby model (< 𝐸𝑠𝑒𝑔,𝑒𝑙𝑌@𝑇𝑖/𝑀𝑜>)computed segregation energy of Y in both 50 at. % and 75 at. % Mo alloys. (b) Scalingconstants (𝐶𝑇𝑖/𝑀𝑜) used for the Eshelby-type model fits for volume of the Mo (orange square)and Ti (blue circles) sites.The results from the Eshelby analysis for Mo and Ti sites in the Ti - 50 at.% Mo alloyare presented in Figures 5.16(a) and (b), respectively. The segregation energies at the Ti GBsites follow the Eshelby predictions closely except for a few exceptions, e.g. at ~ 19.0 Å3.The predictions for the Mo GB sites do not seem to do well like the case of Ti sites, as seenin Figure 5.16(b), where the volume scaling does not hold for this host site. For the higherMo content alloy, the Eshelby predictions are compared with DFT calculations for Mo and Tisites in Figures 5.16(c) and (d), respectively. The Eshelby-type model predictions for Ti arereasonable compared to that for Mo sites as seen in the Ti-50 at. % Mo alloy. The highestdiscrepancy in predictions for Ti sites is ~ 10 % where DFT predicts ~ -0.9 eV and Eshelbypredicts ~ -1.0 eV at a volume of ~17.5 Å3. Meanwhile, the Eshelby predictions for Mo sitesunder-predict and over-predict DFT computed values for the full volume range.71Figure 5.16: Comparison of segregation energy computed with DFT (orange squares) andEshelby-type model (blue squares) for segregation of Y at (a) Mo and (b) Ti in 50 at. % Moalloy and segregation of Y at (c) Mo and (d) Ti in 75 at. % Mo alloy.The two crucial pieces of information for using the Eshelby model in Ti - 50 at. % and 75at. % Mo alloys are: (i) elastic properties of the alloy and (ii) scaling constants for the volumeof Ti and Mo GB sites, 𝐶𝑇𝑖 and 𝐶𝑀𝑜, respectively. With simple assumptions for the two inputs,the Eshelby analysis is repeated assuming a Vegard’s law prediction with just Ti - 25 at. %alloy and pure BCC Mo and averaged scaling constants of 1.18 and 1.38 for 𝐶𝑇𝑖 and 𝐶𝑀𝑜 forthe three alloy compositions (see dashed line in Figure 5.15). The mean segregation energypredicted by this Eshelby model, along with DFT computed mean Y segregation energy, isshown in Figure 5.17. The Eshelby model results for all the host sites and alloy compositionsshow small deviation from the parity, i.e. Y=X. The two sets of data that deviate from theparity line are Y segregation to Mo sites in 25 at.% Mo alloy and Y segregation to Ti sites in75 at.% Mo alloy.72Figure 5.17: Parity plot of average DFT (< 𝐸 𝑠𝑒𝑔,𝐷𝐹𝑇𝑌@𝑇𝑖/𝑀𝑜 >) and Eshelby model (< 𝐸𝑠𝑒𝑔,𝑒𝑙𝑌@𝑇𝑖/𝑀𝑜>)computed segregation energy of Y to Mo sites (orange solid circles) and Ti sites (blue solidcircles) in both 50 at. % and 75 at. % Mo alloys. The Eshelby model is developed for anaveraged scaling constants (𝐶𝑇𝑖/𝑀𝑜) from 25, 50 and 75 at.% Mo alloy and the elastic modulipredicted by Vegard’s law.5.7 Summary1. An extension of framework to calculate solution and segregation energies in Ti-Mo alloysis developed for higher Mo alloy compositions. Trends of lattice properties of bulk alloys,solution energy of Y in bulk, solution energy of Mo and Ti in bulk, segregation of Mo andTi to GBs and segregation of Y to GBs for 25 at.%, 50 at. % and 75 at. % Mo alloys areinvestigated. These properties provide a computer-aided alloy design approach for Ti-Moalloys.2. The solution tendency of Y decreases with increasing Mo content in the alloy, and solutionenergy scales linearly with the volume of Y substituted at Mo and Ti sites in the bulk.Similarly, the segregation of Mo and Ti is stronger with higher Mo content, i.e. 75 at.%Mo than 50 at.% Mo alloy than 25 at.% Mo alloy. With increasing Mo alloy content,segregation tendency of Y solute to both Mo and Ti sites is increased.3. The volume and volume differences of Ti sites are shown to scale linearly with segregationenergies of Y. Eshelby analysis proves to be useful for predicting average Y segregationenergies and comparing them to DFT-calculated ones. This is especially true for Ti sitesat GB but not for Mo sites. This can be attributed to other effects, such as chemicaleffects, when substituting Mo sites at GB which does not prefer to be at the GB.735.8 Limitations of current studyThe current DFT studies of Ti-Mo alloys have two key limitations:1. The GB sites at the GB plane are only considered. This is primarily because there wassignificant GB reconstruction when solutes were substituted at off-GB plane sites for theTi-25 at.% Mo alloy. Hence, this site type was used for our calculations. It should beemphasized that other sites at GBs also contribute to the segregation energy distribution.Although Jin et al. [136] have shown that for larger solutes like Nb in BCC Fe, the GBplane site shows the strongest segregation energy.2. The present DFT calculations are limited to one particular type of grain boundary, i.e. aΣ5 GB. In a polycrystalline material, various other types of GBs are present. Furthermore,Ito and Sawada [148] had studied 10 different GBs in FCC Fe and found that trends foraverage segregation energies between Σ5 GB and those for all GBs did not change. Suhaneet al. [149] studied the effect of solute segregation at different GBs on retarding graingrowth in the presence of various alloying elements in Fe. They found that segregationtrends for the Σ5 GB can be considered as representative among other GBs. Thesefindings suggest that the trends for solutes obtained for this GB can be transferred toother general GBs. Furthermore, in studies of Jin and co-authors [136] indicate that thesite volumes of this GB are representative of the site volumes found in a general GB. Thisfurther motivates that the calculated segregation energies at this GB are representativefor other GBs in a polycrystal.Even so the current computational workflow can be further extended to study other GBs,including solute-solute interactions and this would require an enormous amount of computa-tional resources since it requires many more DFT calculations. An approach to combat thisdisadvantage is to build a surrogate model to compute these interactions in an inexpensive way.74Chapter 6Surrogate modeling of defects in Pt-Auwith EAM accuracy6.1 IntroductionA computational workflow needs to be established to create a surrogate model, and trainingdata based on EAM data is a good start for that purpose. This is because creating EAM datais inexpensive compared to ab-initio data. At the time of writing this thesis, there are no EAMpotentials in the NIST repository for Ti-Mo alloys. Hence, the focus is shifted to another binaryalloy system, i.e. Pt-Au alloys for which an EAM potential is available [150].This alloy system is known for high strength and toughness due to nanocrystalline-sizedgrains stabilized with the help of Au segregation to GBs [150,151]. Also, Au solute segregationto triple junctions predicted by atomistic simulations of Pt-Au alloys by Barnett and co-authors[152] are shown to also contribute to the thermal stability of nanometer-sized grains.This alloy system can also exhibit high hardness [153] and high wear resistance [154–156].Using molecular dynamics, Pradhan et al. [154] investigated the abrasive wear mechanism withincreasing Au segregation to GBs. An increased resistance to wear for Pt- 0.07 at.% Au alloyat 1020 K is observed compared to pure Pt due to Au segregation to GBs. These simulationsare consistent with experimental studies [157] on Pt-Au alloys.Given these practical implications for the Pt-Au alloy system, crystalline defects in this alloysystem are modeled using an ACE surrogate model with EAM data to establish a computationalworkflow for building MLIPs.756.2 Structural models for grain boundaries and bulk al-loys6.2.1 GB structures for elemental Pt and AuThere are not many publicly and open-sourced codes available for creating GB structures,and GBs are known to have different macroscopic degrees of freedom that need to be defined. Inthis thesis, the GBcode software developed by Hadian and colleagues [158] is used to constructvarious GB structures. This open-sourced code provides a systematic procedure to create GBsin cubic crystal lattices. In this chapter, symmetric tilt GBs (STGBs) are chosen such that theyare equally spaced in the misorientation space as listed in Table 6.1. The table provides de-tails on the misorientation angle, Σ value, GB plane and the number of atoms for all the STGBs.Table 6.1: List of all STGBs with misorientation angle (𝜃𝑚𝑖𝑠), Σ value, GB plane and numberof atoms in the computational domain (N𝐺𝐵).𝜃𝑚𝑖𝑠 (◦) Σ GBplane(h k l)N𝐺𝐵7.63 113a (0 1 15) 217012.68 41a (0 1 9) 123218.93 37a (0 1 6) 113622.62 13a (0 1 5) 58828.07 17a (0 1 4) 76836.87 5a (0 1 3) 34843.603 29a (0 2 5) 88053.13 5b (0 1 2) 68458.11 53 (0 9 5) 60061.93 17b (0 5 3) 38467.38 13b (0 3 2) 60071.08 37b (0 7 5) 55273.740 25a (0 4 3) 355277.32 41b (0 5 4) 126483.27 145a (0 9 8) 454476Further, to define microscopic degrees of freedom for each GB, the program allows for aminimum cutoff distance between the two grains. This feature is incredibly useful to ensureatoms do not get too close when subject to geometry relaxation. A cutoff of 0.4-0.6 𝑎0 (latticeconstant) is chosen to provide a good starting point for creating initial GB structures. Thesestructures represent some of low-angle and high-angle GBs in the <100> misorientation space.This choice of low and high-angle GBs allows to systematically study the use of different trainingstructures to predict the GB energies. The Σ value is typically associated with high-angle GBswith misorientation angles of at least 15◦. Low-angle GBs are dislocation arrays but the Σnotation has been extended to these GBs as well, see e.g. Table 3.1 in [159].6.2.2 Bulk alloysIn order to create a bulk alloy with varying compositions of Au, a method based on counts ofatoms is used instead of the conventional method of using probabilities with Monte Carlo SpecialQuasi-random Structures (MCSQS) code in the Alloy Theoretic Automated Toolkit (ATAT)software package [160]. First, a Pt bulk with 256 atoms is taken, a random seed is picked,and a request of random numbers is placed within the numpy package [161] in the Pythonprogramming language with the number of Au atoms in the bulk alloy as input. Then, thoserandom numbers are taken as the index of Au atoms in the alloy structure to be substituted.This method is cheaper to use than querying MCSQS for alloy structures. This is especially truesince a relatively large number of alloy structures are used in this chapter, i.e. 95 structures.The compositions of these structures range from 5 at. % Au to 95 at. % Au, increasing in stepsof 5 at. % Au. Five random alloy structures for each composition are generated.For the vacancy formation in bulk alloys, only three alloy structures consisting of Pt-25, 50and 75 at. % Au alloys are used. Since only a few structures are required, MCSQS in ATATsoftware is used to create the bulk alloy structures. The correlation functions are calculateduntil the sixth nearest neighbor shell, and the objective function is computed only until paircorrelations. Once the correlations are below a threshold value such as 0.01, the final alloystructure that maximizes the objective function is taken. The expressions for correlation andobjective functions are provided in van de Walle and co-authors [162].6.3 Computational workflowAn emphasis is placed on using a computational workflow to (i) create training set struc-tures for developing an ACE interatomic potential, (ii) making predictions using the generatedACE potential and (iii) then analyzing the results with the results from the molecular statics(MS) simulations with the EAM potential [150]. A schematic of this workflow is presented inFigure 6.1.77Figure 6.1: Visualization and sequence of individual steps used in the workflow. Each blockrefers to either preparing training data, making predictions using an reference potential, i.e.EAM, for relaxation, or building an ACE model. The arrows indicate the direction in whichthe workflow is implemented. The workflow is repeated to produce different models for a fewof the properties studied.6.3.1 Steps I-IIIFirst, the EAM potential is used to relax the structures to its ground state. This process isstep (I) in the workflow, and this constitutes reference data. Along with this, one can computethe relevant thermodynamic property, e.g. the mixing enthalpy of the alloy. For the secondstep, a few of the structures are chosen that are assumed to adequately describe the propertyof interest. For example, this could be selecting 3-4 ground-state structures of the 50 at. % Aualloy to create an ACE model.The third step involves creating different samples of the selected structures from Step (II)as shown in Figure 6.1. This step is done in two separate operations, first operation is referredto as “rattled” samples, where each spatial coordinate of atoms in the simulation domain ischanged from its relaxed configuration. In practice, rattled structures are created by movingeach atom in three directions by a small amount for all atoms, i.e., a random decimal sampledfrom a normal distribution with mean at zero and standard deviation, 𝑐𝑟𝑡𝑙 , provided by theuser. There could be N𝑖 samples for each chosen parameter 𝑐𝑖𝑟𝑡𝑙 . The choice of the standarddeviation value, 𝑐𝑟𝑡𝑙 , depends on a fraction (i.e. 5-10%) of the first nearest neighbor distance,i.e. 𝑎0/√2 in FCC crystals.The next operation is done to create the “deformed” structures of the simulation supercell. Adeformation matrix, deform, of the following type is used: deform = I+ 𝑐𝑑𝑓 𝑚 · [rand(3, 3) −0.5],where I is the identity matrix, rand(3,3) is a matrix consisting of random numbers between0 and 1 for each of the nine matrix elements. The matrix from the simulation cell, i.e. lattice78vectors for each direction, is multiplied by the deform matrix. The strength of the deformationis controlled by the 𝑐𝑑𝑓 𝑚 parameter and with M𝑖 samples for each parameter 𝑐𝑖𝑑𝑓 𝑚. Here, M andN are user choices for number of structures to be included in the training set. It is importantto note that each operation is done separately. The choice of the deformation parameter, 𝑐𝑑𝑓 𝑚is dependent on which structures are useful in the simulation of target properties. Here, someof the choices are made based on inputs and feedback from the Ortner group in the departmentof Mathematics at UBC.Figure 6.2: Representative structures from training set of Pt-BK and Au-BK ACE modelsconsisting of rattled (a) bulk Au, (b) bulk Pt and cell deformed (c) bulk Au and (d) bulk Pt.Yellow and grey colored atoms are Au and Pt species, respectively. Atom sizes are reduced tovisualize the structures better and help see the out-of-lattice atom positions.Rattled structures of 256-atom supercell are shown in Figures 6.2(a) and (c) for bulk Pt andAu, respectively. The atoms are displaced from the perfect FCC positions in these structures.Bulk FCC Pt and Au are used to create deformed structures seen in Figures 6.2(b) and (d),respectively where the simulation box is not cubic anymore.6.3.2 Step IVStep (IV) involves compiling all the sampled structures into one file using XYZ file format. Thisfile consists of energy, force of atoms in structures, and virials of structures computed with EAM.The virials are defined by the stress components (eV/Å3) of the simulation cell multiplied by79its volume (Å3). The evaluation of energies, forces and virials for these structures is based onone step simulation run with JuLIP.jl code using the EAM potential. This is equivalent to “run0” command [163] inside the LAMMPS code to evaluate and output the above observationswithout moving the atoms or taking a timestep. This is essentially the same step done in thenext Chapter but using a DFT calculator such as VASP in order to generate DFT trainingdata.6.3.3 Step VHyperparameters for ACE modelIn order to perform step (V), one needs to choose the hyperparameters of the ACE potential,three choices need to be made for the correlation order of potential (N𝑐𝑜𝑟𝑟), the maximumdegree of polynomials used for generating the potential (𝑛𝑚𝑎𝑥) and a radial cutoff, 𝑟𝑐𝑢𝑡 , fortruncating the cluster expansion for the interaction of the many-body system. Even thoughthese are independent hyperparameters, it is important to choose a sufficiently large radialcutoff distance that encompasses the interactions behind the choice of the correlation order.To provide some rationale behind the choice of radial cutoff, the following simulation can beperformed. A central atom in the simulation domain is selected, and atoms surrounding thisatom within the radial cutoff distance are fixed in space. An atom outside this cutoff distanceis moved slightly, e.g. 0.2 Å, and the change in force at the central atom is measured and willbe minimal, i.e. 0.01 eV/Å. By using this setup, a large enough radial cutoff distance can beestablished.Furthermore, weights of the types of observations, i.e. energies, forces, and virials, arerequired to perform the regression of the ACE model. Typically, energies are weighted thehighest among the three since they are less frequent data and are important for predictingenergy-related target properties. Forces are usually assigned more emphasis than virials sincethey are responsible for finding the minimum-energy configurations. Once these choices aremade, an ACE potential consisting of model parameters, 𝑁𝑝𝑎𝑟𝑎𝑚, on the order of 1000, can begenerated.Bayesian Linear RegressionThe ACE regression model predicts energy, force, and virial data by minimizing the differencebetween predicted and observed values for the training data. In the Bayesian approach, eachmodel parameter is treated as a Gaussian distribution.The objective of Bayesian linear regression is to parameterize this distribution, i.e. findthe mean and standard deviation based on the training data. This is done by maximizing the80logarithm of the likelihood, which identifies the most likely model parameters that fit the data.Finally, a finite number of values are randomly sampled from the distribution for each modelparameter to create the committee-of-models. More details can be found in the work by Wittand co-authors [105].6.3.4 Step VI-VIIFor step (VI), the base ACE potential is used to evaluate energy predictions on the complete listof reference (ground-state) structures that were created in step (I). It should be noted that thisis different from the training set structures since those structures included rattled and deformedversions of a few pre-selected structures. Results from this step provide a reliable way to assesshow the generated ACE potential performs against predictions from the EAM potential.In the next step, i.e. step (VII), the committee of potential is used to generate an uncertaintymeasure that is the standard deviation of the 𝑁𝑐𝑚𝑡𝑒 predictions. This uncertainty providesinsight into how well the structure being tested is captured within the training set. Uncertaintypredictions of energies are made for all structures within reference data, i.e. the database ofground-state structures relevant for property prediction.Step (VIII) compares true error, i.e. the energy difference between EAM prediction andACE model predictions, with uncertainties, which is the standard deviation of energy from thecommittee-of-models.It should also be noted that this workflow is a preliminary one compared to what is presentedin the next Chapter using DFT data as input. In the next Chapter, the iterative process isrefined to include fewer structures in each iteration and improve the ACE model predictions.6.4 Results and discussion6.4.1 Bulk propertiesPure bulkStarting with rattled bulk Pt and Au structures and deformed bulk Pt and Au, two ACE models,namely, Pt-BK and Au-BK, are developed for Pt and Au, respectively. Model hyperparametersused are correlation order of 3 (body order of 4), 5.5 Å is used for radial cutoff consistent withthe cutoff of the underlying EAM potential and the maximum degree of polynomials is 12.Typically, for metallic systems, a 4-body interaction (or 3-correlation order) is sufficient for81modeling many-body interaction. For this section, five different values – 0.04, 0.08, 0.12, 0.16and 0.2 Å are used for the standard deviation, 𝑐𝑟𝑡𝑙 , and ten structures are sampled for eachvalue. A total of 50 structures each having 256 atoms are created, and this gives 50 energies,38400 (3*256*50) forces and 300 (6*50) virials, yielding a total of 38750 observations.Further, cell deformed structures where the strength of the deformation is controlled by the𝑐𝑑𝑓 𝑚 parameter, and a set of values namely, 0.01, 0.02, 0.03, 0.04 and 0.05 is used. For eachparameter value, ten structures are sampled. This operation also gives 38750 observations, andboth datasets are combined to get a total of 77500 observations for development of the ACEmodel. The details of training data and ACE model fits are presented in Table 6.2.Before making predictions, it is essential to present some of the details in the training data.The observations, namely, energies, forces and virials for the training dataset for the Pt-BKand Au-BK models, are shown in Figures 6.3(a), (b), (c) and (d), (e), (f), respectively. Inthe case of Au structures, the range for energies is between ∼ -775 eV to -675 eV, indicatingstructures close to ground-state and higher energy states are sampled, forces range from ∼ -3eV/Å to 2.5 eV/Å also sampling equilibrium and highly displaced atoms. Deformed structuresfrom equilibrium FCC Au cell parameters lead to a virial range from ∼ -450 eV to 1200 eV,indicating samples from both tensile and compressed structures. For Pt structures, energiesvary from ∼-800 to -1400 eV, forces range from ∼-8 to 8.2 eV/Å, and virials range from ∼ -1000eV to 4000 eV, also including equilibrium, non-equilibrium, tensile and compressed structures.Figure 6.3: Training data consisting of observations i.e. energies, forces, and virials, for Pt-BK(top panel) and Au-BK (bottom panel), respectively. Frequency (log-scale) of observationsplotted on the y-axis of energies (subfigures a, d) of structures, forces (subfigures b, e) of atomsand virials of structures (subfigures c, f) in the training set plotted in the x-axis.82The observations from the ACE model on the training data compared with EAM modelpredictions are presented in Figures 6.4(a), (b), (c) and (d), (e), (f) for Au-BK and Pt-BKmodel, respectively. The root mean squared error (RMSE) for each observation is shown inTable 6.2. These results are comparable to most of the regression models published in theliterature [164–167].Figure 6.4: Parity plot for (a) energies, (b) forces and (c) virials computed using the ACEmodels (y-axis) and EAM potential (x-axis) for Au. The bottom panel corresponds to theparity plot for Pt in the case of (d) energies, (e) forces, and (f) virials predicted from ACE andEAM models.The lattice constant (𝑎0) and elastic constants (𝐶11, 𝐶12, 𝐶44) of Au and Pt are calculatedand compared with those obtained using EAM as shown in Table 6.3. Elastic constants are cal-culated with ACE potentials as implemented in the matscipy code base [168]. Good matchingof between ACE and EAM model predictions and the largest relative error found is for 𝐶44 (2.8%), and including more shear-deformed structures in the training set will help with reducingthis relative error.83Table 6.2: All the data below correspond to ACE models for reproducing bulk properties of Auand Pt.ACE model specifications such as correlation order (N𝑐𝑜𝑟𝑟), maximum degree of polynomial(𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), number of parameters (𝑁𝑝𝑎𝑟𝑎𝑚), number of rattled and deformedstructures (𝑁 𝑠𝑡𝑟𝑠), total number of observations i.e., energies, forces and virials (N𝑜𝑏𝑠), RMSEerrors for observations such as energies (RMSE𝐸) normalized by number of atoms in the sim-ulation domain, forces (RMSE𝐹) for each atom and virials (RMSE𝑉) normalized by number ofatoms in the simulation domain and weights of energies (𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used intraining of ACE models.ModelnameN𝑐𝑜𝑟𝑟 n𝑚𝑎𝑥 r𝑐𝑢𝑡[Å]N𝑝𝑎𝑟𝑎𝑚N𝑠𝑡𝑟𝑠 N𝑜𝑏𝑠 RMSE𝐸[meV/at.]RMSE𝐹[meV/Å]RMSE𝑉[meV/at.]w𝐸 w𝐹 w𝑉Au-BK3 12 5.5 2396 100 77500 0.34 4.85 12.08 60.0 5.0 1.0Pt-BK 3 12 5.5 2396 100 77500 0.51 5.13 25.42 60.0 5.0 1.0Table 6.3: Lattice constant (𝑎0), elastic constants(𝐶11, 𝐶12, 𝐶44), vacancy formation (𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 )and migration energy (𝐸𝑚𝑖𝑔𝑣𝑎𝑐 ) predictions for Pt and Au using Pt-EAM, Pt-BK, and Au-EAM,Au-BK models, respectively.Au [Au-EAM, Au-BK] Pt [Pt-EAM,Pt-BK]𝑎0 [Å] 4.153, 4.152 3.975, 3.976𝐶11[GPa] 168.4, 166.13 264.3, 257.5𝐶12[GPa] 134.8, 134.2 221.9, 219.2𝐶44[GPa] 44.3, 43.2 68.8, 66.5𝐸𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 [eV] 0.72, 0.80 1.89, 1.84𝐸𝑚𝑖𝑔𝑣𝑎𝑐 [eV] 0.64, 0.53 0.84, 1.00Vacancy in bulkIn order to calculate vacancy formation and migration energies, the current dataset needs tobe extended. Another model, i.e. Au-VAC and Pt-VAC with 100 structures of rattled andcell deformed versions each of an EAM relaxed mono-vacancy ground-state structure, is addedinto the database for simulating Au and Pt, respectively. For rattling operation, 𝑐𝑟𝑡𝑙 standarddeviations of normal distributions where displacements are sampled are 0.04, 0.08, 0.1, 0.12 and0.14 Å. In deformed structures, the 𝑐𝑑𝑓 𝑚 values used are 0.01, 0.02, 0.025, 0.03 and 0.035. ACEmodel parameters and the number of training set observations are shown in Table 6.4. Vacancy84Table 6.4: All the data shown below correspond to ACE models for reproducing vacancy inbulk properties of Au and Pt. ACE model specifications such as correlation order (N𝑐𝑜𝑟𝑟),maximum degree of polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), number of parameters (𝑁𝑝𝑎𝑟𝑎𝑚),number of rattled and deformed structures (𝑁 𝑠𝑡𝑟𝑠), total number of observations i.e., energies,forces and virials (N𝑜𝑏𝑠), RMSE errors for observations such as energies (RMSE𝐸) normalized bynumber of atoms in the simulation domain, forces (RMSE𝐹) for each atom and virials (RMSE𝑉)normalized by number of atoms in the simulation domain and weights of energies (𝑤𝐸), forces(𝑤𝐹), virials (𝑤𝑉) used in training of ACE models.ModelnameN𝑐𝑜𝑟𝑟 n𝑚𝑎𝑥 r𝑐𝑢𝑡[Å]N𝑝𝑎𝑟𝑎𝑚N𝑠𝑡𝑟𝑠 N𝑜𝑏𝑠 RMSE𝐸[meV/at.]RMSE𝐹[meV/Å]RMSE𝑉[meV/at.]w𝐸 w𝐹 w𝑉Au-VAC3 16 5.5 8604 200 154700 0.29 6.39 21.89 60.0 5.0 1.0Pt-VAC3 16 5.5 8604 200 154700 0.29 3.45 13.95 60.0 5.0 1.0formation energy, 𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 , is calculated using Equation (6.1). In Equation (6.1), the energyof 256-atom bulk and 255-atom vacancy in the bulk structure after geometry optimizationcorresponds to 𝐸𝑏𝑢𝑙𝑘 and 𝐸𝑣𝑎𝑐, respectively.𝐸𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 = 𝐸𝑣𝑎𝑐 − 𝑁𝑏𝑢𝑙𝑘 − 1𝑁𝑏𝑢𝑙𝑘 · 𝐸𝑏𝑢𝑙𝑘 (6.1)Vacancy migration energy, 𝐸𝑚𝑖𝑔𝑣𝑎𝑐 , is calculated as energy difference between initial and saddleconfigurations, i.e. E(image#3) - E(image#0), as shown in Figure 6.5. The results for themigration path of both species are shown in Figure 6.5.After training the ACE models with the extended dataset, the lattice constant, elastic con-stants, vacancy formation, and migration energy predictions are calculated using VAC models,as listed in Table 6.5. Perfect matching for formation energy in Au is achieved, but an errorof 30-40 meV is seen for the formation energy in Pt and migration energy in both Au and Pt.This small error already shows that reasonable predictions can be obtained for both properties.85Table 6.5: Lattice constant (𝑎0), elastic constants(𝐶11, 𝐶12, 𝐶44), vacancy formation (𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 )and migration energy (𝐸𝑚𝑖𝑔𝑣𝑎𝑐 ) predictions for Pt and Au using Pt-EAM, Pt-VAC and Au-EAMand Au-VAC models, respectively.Au [Au-EAM, Au-VAC] Pt [Pt-EAM, Pt-VAC]𝑎0 [Å] 4.153, 4.153 3.975, 3.976𝐶11[GPa] 168.4, 167.8 264.3, 258.0𝐶12[GPa] 134.8, 134.9 221.9, 216.1𝐶44[GPa] 44.3, 42.6 68.8, 66.88𝐸𝑓 𝑜𝑟𝑚𝑣𝑎𝑐 [eV] 0.72, 0.72 1.89, 1.87𝐸𝑚𝑖𝑔𝑣𝑎𝑐 [eV] 0.64, 0.67 0.84, 0.88Upon inspecting the mono-vacancy migration path in Au and Pt as plotted in Figures 6.5(a)and (b), respectively, it is clear that the saddle point configuration for the vacancy migrationpath is incorrectly predicted using Au-BK and Pt-BK models. Using the VAC models to predictvacancy migration energy, the difference for Au is 30 meV and for Pt is 40 meV, i.e. in bothcases there is a small quantitative difference. Including a few rattled and deformed structuresaround saddle point structure will yield better results than the current VAC models.Figure 6.5: The vacancy migration path shown as the relative energy of images (y-axis) plottedas a function of a number of images used in the Nudged Elastic Band (NEB) (x-axis) methodto calculate the migration barrier energy. Figure (a) shows the three model results (Au-EAM,Au-BK and Au-VAC) for Au, and figure (b) panel shows the results from Pt-EAM, Pt-BK andPt-VAC models.866.4.2 GB propertiesThe list of fifteen [001] symmetric tilt GBs (STGBs) used to calculate GB energy with EAMpotential is shown in Table 6.1. The GB energy, 𝛾𝐺𝐵, is calculated using Equation (6.2), with𝐸𝐺𝐵 as the energy of the ground-state GB structure and a factor of 2 is used since there aretwo GBs in the simulation cell due to periodic boundary conditions (PBCs). Three of thefifteen GB EAM-predicted ground-state structures are shown in Figure 6.6. In Figure 6.6(a),the commonly-studied GB structure in FCC systems, Σ5 (310) is shown along with structuralunits (SUs) repeated along the GB. The GB structure for Σ13a (150) is shown in Figure 6.6(b),where this GB exhibits a different SU compared to Σ5(310). A low-angle GB such as Σ145a(980) is represented in Figure 6.6(c), where an array of two dislocations is identified for thisGB. These GBs provide a diverse dataset to test the ACE model prediction on different GBswith varying training datasets.Figure 6.6: EAM relaxed ground-state structures for three representative GBs – (a)Σ5b (2 10) 53.13 ◦, (b) Σ13a (0 1 5) 22.62 ◦and (c) Σ145a (0 9 8), 83.17 ◦ from the current study. GBstructures are visualized using centrosymmetry parameters computed using OVITO [135].The GB energies, 𝛾𝐺𝐵, predicted using the EAM model are plotted in Figure 6.7 as a functionof misorientation angle (𝜃𝑚𝑖𝑠). It is evident that GB energies scale with the lattice constantand shear modulus of the two metals – Pt and Au, as also reported in [52]. The GBs in Aushows lower GB energies than Pt, and the relative trend is the same for both metals.𝛾𝐺𝐵 =(𝐸𝐺𝐵 − 𝑁𝐺𝐵 · 𝐸𝑏𝑢𝑙𝑘/𝑁𝑏𝑢𝑙𝑘 )2 · 𝐴𝐺𝐵 (6.2)87Figure 6.7: EAM calculated GB energies, 𝛾𝐺𝐵 (y-axis) reported in mJ m−2 plotted as a functionof misorientation angle (x-axis) for <100> STGBs in Pt (solid blue circles) and Au (solid orangecircles).For the first set of models, a dataset that includes 75 bulk rattled configurations sampledfrom a distribution of rattled strength, 0.1, 0.2, 0.3, 0.35 and 0.40 Å and another bulk Pt/Audeformed structures with deformation strength of 0.05, 0.1, 0.15, 0.175 and 0.2 for Pt and Aueach. There are 150 structures with 116250 observations, and ACE model details along withtraining errors are reported in Table 6.6 for both Au and Pt.In the next model, Au-GB2 and Pt-GB2, the Σ5 (012)[100] and Σ5 (013)[100] GB are addedinto the training set for three reasons. First, these two GBs are commonly studied in theliterature, and second, they have a structural unit that is present in most of <100> tilt GBs asmentioned by Priedeman and others [110], and third, their domain size is reasonably small tobe computed by most models like EAM and DFT. The computational domain size of both Σ5GBs are listed in Table 6.1, rattling strength is chosen to be 0.15, 0.25, 0.28 Å for six rattledstructures for a total of 18 structures. For deformed structures, 𝑐𝑑𝑓 𝑚 values of 0.075, 0.125 and0.14 are taken with six structures for a total of 18 structures. The number of observations alongwith ACE model details are shown in Table 6.6.For the third model for simulating Pt and Au GBs, GB3, training sets from both models,GB1 and GB2, are combined to compare GB energy and uncertainty predictions with EAMGB energy true error energy predictions. As seen in Table 6.6, the RMSE error of the Au-GB3model for energy, forces and virials are similar to the Au-GB1 model, whereas the Pt-GB3model shows a sum of errors of the Pt-GB1 and Pt-GB2 models.88Table 6.6: All the data shown below correspond to ACE models for reproducing EAM GBproperties in FCC Au and Pt. ACE model specifications such as correlation order (N𝑏𝑜𝑑𝑦),maximum degree of polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), number of parameters (𝑁𝑝𝑎𝑟𝑎𝑚),number of rattled and deformed structures (𝑁 𝑠𝑡𝑟𝑠), total number of observations i.e., energies,forces and virials (N𝑜𝑏𝑠), RMSE errors for observations such as energies (RMSE𝐸) normalizedby number of atoms in the simulation domain, forces (RMSE𝐹) for each atom and virials((RMSE𝑉) normalized by number of atoms in the simulation domain and weights of energies(𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used in training of ACE models..ModelnameN𝑐𝑜𝑟𝑟 n𝑚𝑎𝑥 r𝑐𝑢𝑡[Å]N𝑝𝑎𝑟𝑎𝑚N𝑠𝑡𝑟𝑠 N𝑜𝑏𝑠 RMSE𝐸[meV/at.]RMSE𝐹[meV/Å]RMSE𝑉[meV/at.]w𝐸 w𝐹 w𝑉Au-GB13 14 5.5 4676 150 116250 0.31 6.76 25.43 60.0 5.0 1.0Au-GB23 14 5.5 4676 72 111960 0.23 10.33 9.7 60.0 5.0 1.0Au-GB33 14 5.5 4676 222 228210 0.32 8.79 22.66 60.0 5.0 1.0Pt-GB13 14 5.5 4676 150 116250 0.40 3.82 16.57 60.0 5.0 1.0Pt-GB23 14 5.5 4676 72 111960 0.32 6.84 13.59 60.0 5.0 1.0Pt-GB33 14 5.5 4676 222 228210 0.63 7.04 23.55 60.0 5.0 1.0The ACE model predictions are compared with EAM predictions of GB energy, 𝛾𝐺𝐵 (calcu-lated using Equation (6.2)) and plotted for all six models in Figure 6.8. For GB1 models, bothAu and Pt lead to reasonable predictions for low-angle GBs, and this is true since most of theseGB structures have bulk regions in between and only consist of periodic array of dislocations.Without any GB structures, the high-angle GBs’ predictions are over-predicted by the Au-GB1ACE model while the Pt-GB1 ACE model under-predicts GB energies. The next set of models,Au-GB2 and Pt-GB2 models show better matching of predictions with EAM model predic-tions. With the Pt-GB2 model, high-angle GBs between 𝜃𝑚𝑖𝑠 between 28 and 53◦ showed aslight mismatch of relative error of 2% between the ACE and EAM models (see Figure 6.8(d)).On the other hand, the GB energies of all fifteen GB structures are predicted well by the Au-GB2 model. In the final model, both Au-GB3 and Pt-GB3 perfectly match EAM-predicted GBenergies, as shown in Figures 6.8(e) and (f), respectively. As a next step, comparing uncertainty,i.e. standard deviation of energies from ACE models, with the true error, i.e. the absolute valueof the difference of model energy and EAM energy prediction, would be helpful.89Figure 6.8: Predictions from the ACE model (shown in the y-axis) for Au (left panel) andPt (right panel) are plotted as a function of misorientation angle plotted on the x-axis. Solidx markers represent data from EAM potential and hollow circles markers represent the ACEmodel predictions from Au-GB1, Au-GB2 and Au-GB3 and Pt-GB1, Pt-GB2 and Pt-GB3 forAu and Pt, respectively.Some comments on the accuracy of the ACE model predictions are required here. Typically,a GB simulation is conducted to compare against experimental results. In experiments, thedetermination of GB energies is presented with a number of sources of errors and uncertainties,as shown by Rohrer and co-authors [169,170]. They show that there is a 10% error bar for themost frequently observed GB energies measured in FCC Ni. Most of the ACE model resultsexcept for a few GBs in the GB1 model, are well within the +/- 10% of reference EAM GBenergies.Uncertainties are computed for the energy of GB supercell structures with the help of the𝑁𝑐𝑚𝑡𝑒 ACE committee-of-models and are calculated as follows:𝜎(𝐸𝐺𝐵) =√√1𝑁𝑐𝑚𝑡𝑒𝑁𝑐𝑚𝑡𝑒∑︁𝑖=1(𝐸 𝑖𝐺𝐵− < 𝐸 𝑖𝐺𝐵 >)2(6.3)90Here, 𝐸 𝑖𝐺𝐵 are the predictions of energy of GB structures (see Equation (6.2)), and < 𝐸𝑖𝐺𝐵 > isthe mean value of the distribution of energy of GB structures computed by ACE committee-of-models. This standard deviation measure only depends on the ACE model’s energy predictionsand not the EAM model. This measure can be calculated for all fifteen GBs and is shown inTable 6.7. Since each GB has a different number of atoms, standard deviation predictions arenormalized by the number of atoms in each GB structure. Additionally, a scaling factor of1200 (average of number of atoms in all fifteen GBs) is used to scale the small magnitude ofuncertainties. This scaling is done to present meaningful numbers in the normalized uncertaintyvalues (see Table 6.7). Here, the energy of the supercell with the GB (𝐸𝐺𝐵) is used rather thanGB energy because the main source of uncertainty for GB energy comes from the term 𝐸𝐺𝐵(see Equation (6.2)) since 𝐸𝑏𝑢𝑙𝑘 is captured well within these models.Table 6.7: Ranked (in decreasing order) predicted error of energies of 15 Au GB structures,𝜎(𝐸𝐺𝐵) normalized by total number of atoms in each structure, 𝑁𝑡𝑜𝑡𝑎𝑙 for Au-GB1, Au-GB2and Au-GB3 ACE models. The GB structure and predicted error/uncertainty for each modelare also listed.Rank # Au-GB1:1200*𝜎(𝐸𝐺𝐵)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],GB structureAu-GB2:1200*𝜎(𝐸𝐺𝐵)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],GB structureAu-GB3:1200*𝜎(𝐸𝐺𝐵)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],GB structure1 1053.05, Σ17a 82.68, Σ25a 38.66, Σ17b2 926.21, Σ17b 82.08, Σ145a 37.35, Σ17a3 849.13, Σ41a 77.39, Σ113a 36.84, Σ13b.... .... .... ....15 96.41, Σ145a 9.66, Σ5(210) 9.06, Σ25aFrom Table 6.7, a clear trend emerges: since Au-GB1 only contains bulk Au in training data,the highest three uncertainties correspond to GB structures that are high-angle GBs, implyingthese structures are not well captured in the training data. The GB with the lowest uncertaintyis a small-angle GB, Σ145a, consistent with the training data. For the Au-GB2 model, in thetop three uncertainties, two GBs are low-angle GBs, which are not in the training set, althoughthe low-angle, Σ25a GB, is also seen as the highest uncertainty. The lowest uncertainty is nowattributed to one of the training set GBs, i.e., Σ5(210) GB. In the Au-GB3 model, all three topuncertainties are for high-angle GBs, and the lowest is for Σ25a GB.91Table 6.8: Ranks of decreasing predicted error of energies of 15 Pt GB structures, 𝜎(𝐸𝐺𝐵)normalized by total number of atoms in each structure, 𝑁𝑡𝑜𝑡𝑎𝑙 for Pt-GB1, Pt-GB2 and Pt-GB3 ACE models. The name of the GB structure is also listed along with the predictederror/uncertainty for each model.Rank # Pt-GB1:1200*𝜎(𝐸𝐺𝐵)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],GB structurePt-GB2:1200*𝜎(𝐸𝐺𝐵)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],GB structurePt-GB3:1200*𝜎(𝐸𝐺𝐵)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],GB structure1 457.40, Σ17a 67.55, Σ17b 114.23, Σ17b2 412.17, Σ5(310) 57.72, Σ145a 80.22, Σ13b3 410.92, Σ5(210) 56.99, Σ25a 51.73, Σ41a.... .... .... ....15 34.84, Σ145a 6.96, Σ5(210) 11.96, Σ5(210)The uncertainty for Pt GB structures is presented in Table 6.8. The absolute values of theseerrors from Pt ACE models are lower than GBs in Au for three Au models. In the case ofthe Pt-GB1 model, the top three correspond to high-angle GBs, which are not included in thetraining set. The structure with the lowest uncertainty is a low-angle GB, which is well-capturedwith bulk Pt rattled and deformed structures. For the Pt-GB2 model the structures with thehighest uncertainties are high-angle GBs, i.e., Σ25𝑎 and Σ17b and a low-angle GB, Σ145𝑎, isnot well-captured with just Σ5GBs training data. The lowest uncertainty structure is the sameGB as in the training set, i.e., Σ5(210) GB. Finally, for the Pt-GB3 ACE model with both bulkand Σ5 GBs structures in the training set – the model predicts Σ5 GB as the structure with thelowest uncertainty. Out of the top three uncertainties, Σ41a is a low-angle GB while the otherare high-angle GBs. To get lower uncertainties on them, they can be included in the trainingset for the next set of models.The true error can be computed based on the ACE model and the EAM model predictions.Since the interest is GB energy i.e. 𝛾𝐺𝐵, the focus is on the 𝐸𝐺𝐵 term in Equation (6.2). TheEAM potential provides a prediction namely, 𝐸𝑒𝑎𝑚𝐺𝐵 for the 15 GBs, while 𝐸𝐴𝐶𝐸𝐺𝐵 gives the ACEmodel prediction; the following expression is used to calculate the true error:|Δ(𝐸𝐺𝐵) | =| 𝐸𝑒𝑎𝑚𝐺𝐵 − 𝐸 𝐴𝐶𝐸𝐺𝐵 | (6.4)This true error, |Δ(𝐸𝐺𝐵) |, needs to be normalized by a number of atoms in each GB simulationdomain, 𝑁𝑡𝑜𝑡𝑎𝑙 and compared with the uncertainty for all fifteen GBs and all six models created.92This comparison is made in Figure 6.9, where Au and Pt models are presented in (a), (b), (c) and(d), (e), (f), respectively. Ideally, it would be expected that a high uncertainty also correspondsto a high true error and should match in magnitude, i.e., Y=X. In fact, this is partially true forAu-GB1 and Pt-GB1 (as shown in (a) and (d) subfigures), which exhibit a slope of 1 for thecomparison. While the errors do not fall on the Y=X line, the relative trends of the errors canbe used as a measure of what GBs can be added in the next set of models that can be created.Also, the uncertainties are lower when compared to true errors for both cases. In Figures 6.9(b)and (e), the uncertainties and true errors for Au-GB2 and Pt-GB2 models are shown. For a fewGBs in the Au-GB2 model the uncertainties are larger than the true errors. While other GBsin the Au-GB2 and Pt-GB2 models, the uncertainties are lower in comparison to the true error.The predicted error and true error for Au-GB3 and Pt-GB3 models as shown in Figures 6.9(c)and (f), respectively. Overall, there is a much improved correlation between uncertainties andtrue errors as compared to the GB1 models.Figure 6.9: Parity plot for predicted error (or) uncertainty for energy (y-axis), 𝜎(𝐸𝐺𝐵) normal-ized by number of atoms in the GB structure (𝑁𝑡𝑜𝑡𝑎𝑙) versus normalized true error, Δ𝐸𝐺𝐵/𝑁𝑡𝑜𝑡𝑎𝑙(x-axis), of 15 GB structures in Au (top panel) and Pt (bottom panel). The left, middle andright panels show the results from the GB1, GB2 and GB3 models, respectively. The dottedblack line represents the perfect parity line, i.e., Y=X, as a visual guide to the eye.936.4.3 Bulk alloysThe Pt-Au bulk alloys with varying compositions are studied using ACE and EAM modelpredictions, focusing on the mixing enthalpy of the alloys, Δ𝐻𝑚𝑖𝑥 expressed as:Δ𝐻𝑚𝑖𝑥 = (𝐸𝑎𝑙𝑙𝑜𝑦 − 𝑁𝑃𝑡𝐸𝑃𝑡 − 𝑁𝐴𝑢𝐸𝐴𝑢)/𝑁𝑎𝑙𝑙𝑜𝑦 (6.5)In Equation (6.5), 𝐸𝑎𝑙𝑙𝑜𝑦 refers to the energy of Pt-Au alloy structure computed with EAMor ACE models with 𝑁𝑃𝑡 and 𝑁𝐴𝑢 as the number of atoms in the alloy and N𝑡𝑜𝑡𝑎𝑙 being the totalnumber of atoms in the alloy (here, it is 256). The energy of single Pt and Au in the FCC bulkstructure is denoted by 𝐸𝑃𝑡 and 𝐸𝐴𝑢, respectively. Mixing enthalpies for all 95 of Pt- 5- 95 at.% Au alloy structures are presented in Figure 6.11, and it depicts that all mixing enthalpiesare positive in magnitude, representing repulsive interaction between Pt and Au. The highestmixing enthalpy occurs at ~ 35 at. % Au with an enthalpy of ~ 60 meV/at. This data point isclose to the miscibility gap predicted at 39 at. % Au by Okamoto and Massalski [171].Figure 6.10: EAM computed mixing enthalpy (Δ𝐻𝑚𝑖𝑥) plotted as a function of the compositionof Au, 𝑐𝐴𝑢 in Pt-Au bulk alloys. Blue semi-transparent blue circles (o) represent enthalpies forfive structures in each composition and blue solid cross (x) line markers represent the averageof mixing enthalpy in each composition.Three ACE models are created with training datasets of varying bulk alloy compositionsdenoted by PtAu-BK1, PtAu-BK2 and PtAu-BK3. The PtAu-BK1 model includes rattledconfigurations with displacements sampled from a normal distribution of 0.15, 0.2, 0.22, 0.25,and 0.28 Å for the standard deviation of the distribution with 10 structures for each rattlingstrength parameter. Deformed structures are added with 𝑐𝑑𝑓 𝑚 parameters consisting of 0.075,0.1, 0.11, 0.125 and 0.14 values, as well as 10 structures for each value. In PtAu-BK1, onlythe Pt - 50 at. % Au alloy of 5 structures (that have different arrangements of Au atoms)94is added to the training data. Bulk FCC Pt and Au with the same values for rattling anddeformation strengths have also been added. This choice leads to a total of 700 structuresin the dataset and 542500 observations. In the second model, PtAu-BK2, values of 0.15, 0.2,0.22 and 0.25 Å and 0.075, 0.1, 0.11 and 0.125 are used for rattling strength and deformationstrength, respectively. Five structures are sampled for each of the above strength values. Thestructures added here are bulk Pt, Au and three structures of Pt- 25, 50 and 75 at. % Au alloysfor a total of 650 structures. This selection ensures the same number of configurations as thePtAu-BK1 model are taken for the PtAu-BK2 model. Finally, in the PtAu-BK3 model, sevencompositions, namely, Pt- 5, 10, 25, 50, 75, 90 and 95 at. % Au alloys with two structures andbulk Pt and Au are used as the training set. The same values for rattling and deform parameteris used, with five structures sampled for each value. This gives 640 structures along with 496000observations for this model. The ACE model training and error tables after training are shownin Table 6.9.Table 6.9: Model specifications for three ACE models, PtAu-BK1, PtAu-BK2 and PtAu-BK3,to simulate Pt-Au bulk alloys. Following are model hyperparameters: correlation order (N𝑐𝑜𝑟𝑟),maximum degree of polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), number of parameters (𝑁𝑝𝑎𝑟𝑎𝑚)and details on training set: number of rattled and deformed structures (𝑁 𝑠𝑡𝑟𝑠), total numberof observations i.e., energies, forces and virials (N𝑜𝑏𝑠), RMSE errors for observations such asenergies (RMSE𝐸) normalized by number of atoms in the simulation domain, forces (RMSE𝐹)for each atom and virials (RMSE𝑉) normalized by number of atoms in the simulation domainand weights of energies (𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used in training of ACE models.ModelnameN𝑐𝑜𝑟𝑟 n𝑚𝑎𝑥 r𝑐𝑢𝑡[Å]N𝑝𝑎𝑟𝑎𝑚N𝑠𝑡𝑟𝑠 N𝑜𝑏𝑠 RMSE𝐸[meV/at.]RMSE𝐹[meV/Å]RMSE𝑉[meV/at.]w𝐸 w𝐹 w𝑉PtAu-BK13 12 5.5 2396 700 542500 0.30 15 12.26 60.0 5.0 1.0PtAu-BK23 12 5.5 2396 650 503750 0.38 17.66 14.54 60.0 5.0 1.0PtAu-BK33 12 5.5 2396 640 496000 0.39 18 14.033 60.0 5.0 1.0The ACE prediction model for PtAu-BK1, PtAu-BK2 and Pt-BK3 are compared with EAMpredictions in Figures 6.11(a), (c) and (e), respectively. The PtAu-BK1 ACE model qualita-tively captures the shape of the mixing enthalpy curve as seen in Figure 6.11(a), for a quantita-tive comparison, a parity plot between ACE and EAM predictions is shown in Figure 6.11(b).95This model over-predicts the enthalpy when compared to EAM for all compositions. For parityof predictions from PtAu-BK2 and PtAu-BK3 models with EAM predictions, the comparisonis shown in Figures 6.11(d) and (f), respectively. A reasonably good match between EAM andACE model predictions is found for these two models.Figure 6.11: Mixing enthalpy predictions from (a) PtAu-BK1, (c) PtAu-BK2 and (e) PtAu-BK3compared with EAM potential predictions. Parity plot for mixing enthalpy (Δ𝐻𝑚𝑖𝑥) predictionsusing (b) PtAu-BK1, (d) PtAu-BK2 and (f) PtAu-BK3 ACE models compared with EAMpotential predictions (plotted on the x-axis) shown in blue circles. Perfect parity, i.e., Y=X, isrepresented by a red dotted line.Uncertainties from the committee-of-models for PtAu-BK1, PtAu-BK2 and PtAu-BK3 arecomputed for Pt-Au alloy structures using the following expression:96𝜎(𝐸𝑎𝑙𝑙𝑜𝑦) =√√1𝑁𝑐𝑚𝑡𝑒𝑁𝑐𝑚𝑡𝑒∑︁𝑖=1(𝐸 𝑖𝑎𝑙𝑙𝑜𝑦− < 𝐸 𝑖𝑎𝑙𝑙𝑜𝑦 >)2(6.6)In the above equation (Equation (6.6)), the uncertainty of Pt-Au alloy structure is given by𝜎(𝐸𝑎𝑙𝑙𝑜𝑦) predicted from committee size of 100, 𝐸 𝑖𝑎𝑙𝑙𝑜𝑦 gives the individual energy predictionsand the mean of those predictions is denoted by < 𝐸 𝑖𝑎𝑙𝑙𝑜𝑦 >. For all three models and all 95 alloystructures, uncertainties are tabulated in Table 6.10. In the case of PtAu-BK1, the top threeuncertainties are for alloy compositions of 20 and 25 at. % Au alloy structures while the lowestuncertainty is for Pt-50 at. % Au alloy, which is included in the training set. For the PtAu-BK2model, the top three uncertainty are 35, 40 and 45 at. % Au, close to the composition of themiscibility gap. The lowest uncertainty is for a composition included in the training set, i.e.,75 at. % Au. For the third model, PtAu-BK3, the top three predicted errors are 40, 50 and 55at. % Pt structure while 95 at. % Au alloy structures show the smallest uncertainty.Table 6.10: Predicted error, 𝜎(𝐸𝑎𝑙𝑙𝑜𝑦) normalized by total number of atoms, 𝑁𝑡𝑜𝑡𝑎𝑙 for all 95alloy structures ranked based on decreasing order of predicted error predicted using PtAu-BK1,PtAu-BK2 and PtAu-BK3 ACE models.Rank # PtAu-BK1:256*𝜎(𝐸𝑎𝑙𝑙𝑜𝑦)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],composition [at. %Au]PtAu-BK2:256*𝜎(𝐸𝑎𝑙𝑙𝑜𝑦)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],compositionPtAu-BK3:256*𝜎(𝐸𝑎𝑙𝑙𝑜𝑦)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.],composition1 383.63, 20 11.00, 35 16.60, 402 371.34, 25 10.75, 40 13.89, 503 364.32, 20 10.53, 45 13.85, 55.... .... .... ....95 2.23, 50 3.55, 75 2.65, 95As a next step, uncertainties are compared with the true errors for the Pt-Au alloy structure,true error, |Δ(𝐸𝑎𝑙𝑙𝑜𝑦)| that is computed by the following expression:|Δ(𝐸𝑎𝑙𝑙𝑜𝑦) | =| 𝐸𝑒𝑎𝑚𝑎𝑙𝑙𝑜𝑦 − 𝐸 𝐴𝐶𝐸𝑎𝑙𝑙𝑜𝑦 | (6.7)Predictions from EAM and the ACE model are denoted by 𝐸𝑒𝑎𝑚𝑎𝑙𝑙𝑜𝑦 and 𝐸𝐴𝐶𝐸𝑎𝑙𝑙𝑜𝑦, respectively.The true error, |Δ(𝐸𝑎𝑙𝑙𝑜𝑦) |, is the absolute value of the difference between the two predictions.The uncertainties are compared against the true error for the three ACE models and all 9597alloy structures. In Figure 6.12(a), the predicted and true errors for the PtAu-BK1 model areprovided with the lowest data points corresponding to the five structures of 50 at. % Au alloys.This behavior is consistent with the dataset because those were included in this model. Mostof the uncertainty data are cluster around 10e−2 - 10e−3 eV/at. While true error extends from10−5 to ~ 10−2 eV/at. This trend is typical for models that were trained with Bayesian LinearRegression with data that do not have large noise. Data from atomistic simulations are typicallylow-noise data since they are mostly deterministic except for the use of convergence settingsfor arriving at the final energies [172]. This flatness of uncertainties with a wide variation oftrue error for both PtAu-BK2 and PtAu-BK3 models is also seen in Figures 6.12(b) and (c),respectively.Figure 6.12: Predicted error, 𝜎(𝐸𝑎𝑙𝑙𝑜𝑦) (shown in y-axis) and true error, Δ𝐸𝑎𝑙𝑙𝑜𝑦 (shown inx-axis) for all 95 Pt-Au alloy structures normalized by total number of total atoms, 𝑁𝑡𝑜𝑡𝑎𝑙 using(a) PtAu-BK1, (b) PtAu-BK2 and (c) PtAu-BK3 ACE models. The dotted black line is usedto represent perfect parity, i.e., Y=X.6.4.4 Vacancies in bulk alloysEAM model predictionsThe next property of interest to calculate is the vacancy formation energy in Pt-Au bulk alloys.Here, three compositions – Pt - 25, 50, 75 at. % Au alloy of 256-atom size supercell are usedand a vacancy is substituted at all of the 256 sites. All the essential structures required tocalculate vacancy formation energy in alloys are illustrated in Figure 6.13.First, the bulk alloy is relaxed to ground-state using EAM and 𝐸𝑎𝑙𝑙𝑜𝑦 denotes its ground-stateenergy before a vacancy is created. A site is identified where the vacancy will be situated, as98shown in Figure 6.13(a), and this site is identified here as Pt. Second, the vacancy is created atthis site, as shown in Figure 6.13(b). This structure is relaxed to ground-state to get 𝐸𝑣𝑎𝑐@𝑃𝑡,𝑖, i.e., the relaxed energy of the vacancy at site, i. Finally, another structure is also required tocalculate the chemical potential, 𝜇𝑃𝑡,𝑖 of i𝑡ℎ site as shown in Figure 6.13(c). This structure isreferred to as the Widom-type substitution structure [173, 174], where the original Pt speciesis now substituted with the other species in the alloy, i.e. Au. This swap leads to a slightchange in the alloy composition to 25.4 at. % Au instead of 25 at. % Au. The energy of theground-state structure is known as 𝐸𝑊𝐼𝐷𝑂𝑀 , or, in this case, 𝐸𝐵𝐾𝐴𝑢𝑖−>𝑃𝑡𝑖 , the other case is whenPt replaces an existing Au atom to calculate 𝜇𝐴𝑢,𝑖 and E𝐵𝐾Pt,i –>Au,i.Figure 6.13: (a) Atomic structure of 256-atom Pt-25 at. % Au bulk alloy visualized withOVITO [135]. A Pt site is highlighted with an arrow where a vacancy will be situated. (b)Atomic structure of a 255-atom vacancy in alloy structure with vacancy’s location highlightedwith an arrow. (c) Widom structure of Au atom substituted at a previously Pt site pointedout using an arrow. Yellow and grey atoms refer to Au and Pt species, respectively.The vacancy formation energy (𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢𝑖) for all bulk sites is calculated using the followingexpression, Equation (6.8):𝐸𝑓 𝑜𝑟𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖 = 𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖 − 𝐸𝑎𝑙𝑙𝑜𝑦 + 𝜇𝑃𝑡/𝐴𝑢,𝑖 (6.8)In the above equation, 𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖 represents the energy of the ground-state structure of thevacancy at a Pt/Au site in the bulk alloy. The ground-state energy of the alloy bulk structureis denoted by 𝐸𝑎𝑙𝑙𝑜𝑦. In order to calculate the chemical potential of Pt or Au site, 𝜇𝑃𝑡/𝐴𝑢,𝑖, thefollowing equations are used:NPt𝜇Pt,𝑖 + NAu𝜇Au,i = Ealloy (6.9)The total energy of bulk alloy is sum of the number of Pt atoms, 𝑁𝑃𝑡 , multiplied by the Ptchemical potential, 𝜇𝑃𝑡,𝑖, and the number of Au atoms, 𝑁𝐴𝑢 multiplied by Au chemical potential,99𝜇𝐴𝑢,𝑖 as expressed in Equation (6.9). For calculating the chemical potential, there are two cases:(i) in the case of a Au atom at a bulk site being replaced by a Pt atom (Pt𝑖 –> Au𝑖):𝜇Pt@BK,i − 𝜇Au@BK,i = E𝐵𝐾Pt,i–>Au,i(𝐸𝑊𝐼𝐷𝑂𝑀) − Ealloy (6.10)(OR)(ii) In the case of a Pt atom at a bulk site being replaced by a Au atom (Au𝑖 –> Pt𝑖):𝜇Au@BK,i − 𝜇Pt@BK,i = E𝐵𝐾Au,i–>Pt,i(𝐸𝑊𝐼𝐷𝑂𝑀) − Ealloy (6.11)The left side of both the above equations, Equation (6.10) and Equation (6.11), refers to theenergy difference after and before the substitution of Au or Pt in Pt or Au original bulk site,respectively. The right side of the equation is the energy difference between Widom and alloystructure.The chemical potentials for Au and Pt species in the three alloys, namely, Pt- 25 at. %, 50 at.% and 75 at. % Au are shown in Figures 6.14(a), (b) and (c), respectively. Before mentioningthe chemical potential in the alloy, it is worthwhile to refer to the energy/chemical potentialof a single Au and Pt atom in pure FCC structure as -3.033 and -5.506 eV, respectively. Themean value of the Au and Pt chemical potentials in Pt- 25, 50 and 75 at. % Au alloys are-2.89 and -5.47 eV, -3.04 and -5.39, -3.05 and -5.38 eV, respectively. For the 50 at. % Au alloy,the chemical potential of both Au and Pt is of similar spread, with a standard deviation of0.036 eV. Meanwhile, for the leaner Au composition (25 at. % Au), the spread for Au chemicalpotential is larger than the Pt’s, i.e., 0.057 versus 0.019 eV. This trend is consistent becausemost Pt atoms are surrounded by a few Au atoms and are still in the FCC Pt-like environment.Figure 6.14: Histogram of chemical potential (𝜇) distribution of Pt and Au predictions fromEAM potential shown in orange and blue bars, respectively. Predictions for Pt- 25 at. % Au,Pt- 50 at. % Au and Pt- 75 at. % Au alloys are shown in (a), (b) and (c), respectively. Solidvertical lines represent the mean of each distribution.100On the other hand, the case for Au atoms have a high amount of Pt atoms around them,and hence, the chemical potential of Au is different from the energy of FCC pure Au. In Pt-75at. % Au alloy, the standard deviation of 𝜇𝑃𝑡 shows a larger value, 0.043 eV, while the chemicalpotential of Au shows a smaller standard deviation, 0.014 eV.Figure 6.15: Stacked histogram of vacancy formation energy distributions at both Pt and Ausites for (a) Pt-25 at. % Au, (b) Pt- 50 at. % Au and (c) Pt-75 at. % Au alloy, orange andblue bars represent data from Pt and Au sites, respectively. The fraction of sites from all 256bulk sites is represented in y-axis.Results for vacancy formation energy distribution from the EAM potential are presented inFigures 6.15(a), (b) and (c) for Pt - 25 at. %, 50 at. % and 75 at. % Au alloys, respectively.The mean of the vacancy formation energy distribution (of 256 data points) changes from 1.44to 1.04 to 0.8 eV for Pt- 25, 50, 75 at. % Au alloys, respectively. The spread of each distributionis measured by standard deviation, which changes from 0.22 to 0.23 to 0.16 eV for the Pt- 25,50, 75 at. % Au alloys, respectively. A statistical summary of the vacancy formation energydistributions is shown in Figure 6.16 by presenting the mean and standard deviation of thedistributions. The mean vacancy formation energy follows a negative deviation from Vegard’slaw prediction [175,176]. Including the standard deviation with the mean, some of the sites fallabove Vegard’s law prediction for 25 at. % Au alloy. It is also important to note that the 75at. % Au alloy has a smaller standard deviation than the other two alloys. This trend indicatesthat the vacancy formation energy tends to fall close to each other for this rich Au composition.101Figure 6.16: Average vacancy formation energy (shown in blue solid circles) along with standarddeviation of distribution (blue error bars) plotted as a function of alloy composition of Au(shown in x-axis). Black solid lines represent the trend of average vacancy formation energy,and the orange dotted line represents the linear trend connecting pure Pt and Au.ACE model training and predictionsThree ACE models are developed to make predictions to be compared with the EAM data forthe three alloys, Pt- 25, 50 and 75 at. % Au. For the first PtAu-VC1 model, the same trainingdataset as the PtAu-BK2 model is used, as it contain the three alloys that are investigated here.Values of 0.15, 0.2, 0.22 and 0.25 Å and 0.075, 0.1, 0.11 and 0.125 are used for rattling strengthand deformation strength, respectively. Five structures for each strength value are sampled,the total number of structures and observations are reported in Table 6.11. It is important tonote no vacancy or Widom-type structures are added into the training set for this model. Themodel errors and other regression details are provided in Table 6.11.102Table 6.11: Model hyperparameters, details on training data for ACE model, ACE modelregression and errors after regression of PtAu-VC1, Pt-Au-VC2 and PtAu-VC3 models. Corre-lation order (N𝑐𝑜𝑟𝑟), the maximum degree of the polynomial (𝑛𝑚𝑎𝑥), radial cutoff (𝑟𝑐𝑢𝑡), numberof parameters (𝑁𝑝𝑎𝑟𝑎𝑚), number of rattled and deformed structures (𝑁 𝑠𝑡𝑟𝑠), the total numberof observations i.e., energies, forces and virials (N𝑜𝑏𝑠), RMSE errors for observations such asenergies (RMSE𝐸) normalized by number of atoms in the simulation domain, forces (RMSE𝐹)for each atom and virials (RMSE𝑉) normalized by number of atoms in the simulation domainand weights of energies (𝑤𝐸), forces (𝑤𝐹), virials (𝑤𝑉) used in training of ACE models.ModelnameN𝑐𝑜𝑟𝑟 n𝑚𝑎𝑥 r𝑐𝑢𝑡[Å]N𝑝𝑎𝑟𝑎𝑚N𝑠𝑡𝑟𝑠 N𝑜𝑏𝑠 RMSE𝐸[meV/at.]RMSE𝐹[meV/Å]RMSE𝑉[meV/at.]w𝐸 w𝐹 w𝑉Pt-AuVC13 14 5.5 4676 650 503750 0.29 17 12.81 60.0 5.0 1.0Pt-AuVC23 14 5.5 4676 584 452024 0.36 16 12.84 60.0 5.0 1.0PtAu-VC33 14 5.5 4676 584 452024 0.35 15.9 12.93 60.0 5.0 1.0Before discussing PtAu-VC2 model, another thermodynamic quantity, i.e. chemical potentialof Pt/Au sites, are compared between PtAu-VC1 model and EAM model predictions. Thechemical potential of Au and Pt species at all 256 sites for Pt- 25 at. % Au alloy computed byPtAu-VC1 and ACE model are shown in Figures 6.17(a) and (b), respectively. In the case ofchemical potential of Au (𝜇𝐴𝑢) predicted by PtAu-VC1, most of the sites are captured within+/- 0.03 eV except for a few sites and the both extremes of the data. For chemical potentialof Pt, 𝜇𝑃𝑡 , all of the data predicted by the ACE model is well inside the +/- 0.03 eV window,even better than 𝜇𝐴𝑢 ’s parity.Figure 6.17: Chemical potential of Au (𝜇𝐴𝑢) and Pt (𝜇𝑃𝑡) shown in left and right panel, respec-tively. Predictions from PtAu-VC1 ACE model (plotted in y-axis) and EAM potential (shownin x-axis) are represented with blue circles. Perfect parity line, i.e., Y=X is represented in reddotted line and a band of +/- 0.1 eV are used as guide to human eye.103The results for the vacancy formation energy of all 256 sites from PtAu-VC1 model arenow compared with the EAM predictions in Figures 6.18(a), (b) and (c) for Pt- 25, 50 and 75at.% Au alloys, respectively. It is seen for all compositions that the ACE model predictionsunder estimate the EAM model predictions. Another key observation is that the relative trendof vacancy formation energies is maintained with the ACE model when compared to EAMpredictions. In other words, a relatively lower vacancy formation energy predicted from EAMis also predicted as a lower formation energy in ACE, in relative terms. To get quantitivematching with EAM predictions, it is necessary to add alloy Pt-Au structures with vacanciesinto the training set as a next step.Figure 6.18: Parity plot for vacancy formation energy at 256 sites in three bulk alloys predic-tions from PtAu-VC1 ACE model (in y-axis) and EAM potential (in x-axis). Red dotted linerepresents perfect parity between predictions, Y=X and grey region represents +0.1 eV bandaround the perfect parity line as a guide to the human eye.The reasoning behind the choice of the 0.1 eV band is that, ideally, these surrogate modelswill be developed in the next Chapter to replace DFT simulations. Typically, an estimate of0.025-0.03 eV (𝑘𝐵𝑇 , with T~300K) for DFT simulations is used, and the 0.1 eV error band is amuch larger window to get the ACE model predictions within the reference model predictions.Furthermore, an energy window of 0.1 eV is suitable for comparing the model predictionsagainst experimental errors in measuring vacancy formation energy in metals. This discrepancy104is illustrated in works by Gillan [177] and Jacucci et al [178] when comparing predictions fromsimulations with experimental measurements.For the next model, PtAu-VC2, vacancies at bulk Au and Pt sites are selected at random forall three compositions to be included in training set. A total of 8 vacancy structures are includedalong with the 8 Widom-type structures each for all three compositions. The vacancy formationenergy for these pre-selected sites are shown as black square markers in Figures 6.19(a), (b)and (c). A total of 48 structures are rattled and deformed with 2 samples each using 0.15 and0.22 Å for rattling strength as well as 0.075 and 0.11 for deformation strength, respectively. Forvacancy in bulk and alloy bulk type structures, there are 384 rattled and deformed structuresand 100 structures each of rattled and deformed of Pt and Au from Pt-BK and Au-BK ACEmodels are added. The model details along with regression errors and training data set size forthis model (PtAu-VC2) are listed in Table 6.11.Figure 6.19: Parity plot for vacancy formation energy at both Au and Pt sites for (a) Pt-25 at.% Au, (b) Pt-50 at. % Au and (c) Pt-75 at. % Au. Predictions from PtAu-VC2 ACE model(y-axis) is plotted along with predictions from EAM interatomic potential (x-axis) shown inblue circles (o). Perfect parity, i.e., Y=X is represented with red dotted line and a grey coloredband of +/- 0.1 eV is shown as a guide to the eye. The black squares represent the structuresincluded in the training set.ACE model predictions for vacancy formation energy in all 256 sites for Pt-25, 50 and 75at. % Au alloys along with EAM model predictions are shown in Figures 6.19(a), (b) and(c), respectively. The ACE model predictions for Pt-25 at. % Au alloy show good matchingwith EAM model predictions for lower vacancy formation energies (𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢), i.e. 0.8-1.2eV, above this value, ACE model under predicts the 𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 value with regards to EAMmodel predictions. This could be because those randomly selected sites are not high in vacancy105formation energy for this composition. Hence, they do not capture this vacancy formationenergy regime accurately within 0.1 eV of EAM predictions.In the Pt-50 at. % Au alloy, the range of 0.4-1.0 eV matches well between ACE and EAMmodel predictions, while the values above that range are over predicted by the ACE modelcompared to the EAM model. For Pt-75 at. % Au alloy, all the individual vacancy formationenergies computed by the ACE model are well captured within +/- 0.1 eV of the EAM modelpredictions. A systematic approach is required to assess the trend of vacancy formation energywith the local atomic first nearest neighborhood (1NN), and this classification process will helpwith identifying the essential structures that need to be added to the training set. Vacancyformation energies computed with EAM at Au and Pt sites are shown with the number of Auatoms in the 1NN neighborhood for Pt-25 at. % Au , 50 at. % and 75 at. % Au alloys inFigures 6.20(a), (b), (c) and (d), (e), (f), respectively.Figure 6.20: EAM calculated vacancy formation energy as a function of number of Au atomsin the 1NN local neighborhood (out of 12 atoms) at Au sites (top panel) and Pt sites (bottompanel) for Pt-25 at. % Au (left panel), Pt-50 at. % Au (middle panel) and Pt-75 at. % Au(right panel) alloy. Black box markers represent the representative sites from each number ofAu atoms in the 1NN neighbor shell for both host sites for three alloy compositions.Representative sites close to the mean value of formation energy at each number of Au (N𝐴𝑢)in 1NN shell are identified. These sites for Pt-25 at. % Au alloys have 0 to 7 Au atoms in their1NN shell for a vacancy at both Au and Pt sites in the bulk alloy. The number of Au atomsfor Pt-50 at. % Au are 3 to 10 Au atoms and 1 to 9 Au atoms in 1NN shell at Au and Pt sitesin the alloy, for the case of Pt-75 at. % Au alloy, the range for number of Au atoms is from 6to 12 Au atoms for both Au and Pt sites in the alloy.106The training set of the next model, PtAu-VC3, is derived from this analysis, where repre-sentative sites with a number of Au atoms from 2 to 5, 5 to 8 and 7 to 10 atoms are chosenfor Pt - 25, 50 and 75 at. % bulk alloy compositions for each Au and Pt site. Along with thevacancy at those sites (255-atom structure), Widom structures (256-atom structure) are alsoincluded in the training set. So, a total of four structures for three compositions (25, 50 and 75at.% Au) and two species (Pt and Au sites), gives 24 structures to give a total of 48 structures.The vacancy formation energy for these pre-selected sites are shown as black square markers inFigures 6.21(a), (b) and (c). The same deformation and rattling parameters as for the modelPtAu-VC2 are used along with two samples for each parameter to give 384 structures in thetraining set. Finally, the bulk Pt and Au structures of 100 each are derived from Pt-BK andAu-BK models to give a total of 584 structures in the training set for this model. Other details,such as model regression parameters and choice of hyperparameters for this ACE model, areprovided in Table 6.11.Figure 6.21: Parity plot for predictions of all 256 vacancy formation energies (in blue hollowcircles) from PtAu-VC3 ACE model (y-axis) and PtAu-EAM potential (x-axis) in (a) 25 at.% Au, (b) 50 at. % Au and (c) 75 at. % Au bulk alloys. Red dotted line represents theY=X perfect parity line, and the grey region shows a +/- 0.1 eV band around the Y=X parityplot. The red line and grey band are used to guide the eye. The black squares represent thestructures included in the training set.With the ACE model, PtAu-VC3, predictions are provided along with the EAM modelpredictions in Figures 6.21(a), (b) and (c) for Pt -25, 50 and 75 at. % Au alloys, respectively.Prediction comparison for Pt-25 at. % Au alloy shows the same trend as the previous model(but quantitative deviations are larger), PtAu-VC2, with range of a 0.8-1.2 eV predicted wellby the ACE model while larger values are under-predicted by the ACE model in comparison107with EAM predictions. The PtAu-VC3 model well captures the predictions for Pt- 50 at. %Au alloy compared to the previous model, although some of the values in the range of 1.0-1.2eV of 𝐸 𝑓 𝑜𝑟𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 are over predicted by ACE compared to EAM. The higher Au composition, 75at. %, shows good matching of ACE predictions along with EAM model predictions.Out of all the energy terms in Equation (6.8), the main contribution to the predicted error isthe energy term of vacancy at a Pt/Au site in the bulk alloy (𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢). Using the committeeof ACE potentials with size of 100, predicted error or uncertainty for the vacancy at Pt and Ausite in bulk alloy, 𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) is given by:𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) =√√1𝑁𝑐𝑚𝑡𝑒𝑁𝑐𝑚𝑡𝑒∑︁𝑖=1(𝐸 𝑖𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢− < 𝐸 𝑖𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 >)2(6.12)In the above equation (Equation (6.12)), predictions of energy of the supercell with a va-cancy at Pt or Au site in the alloy is depicted by 𝐸 𝑖𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 for a particular model 𝑖 in theensemble of models, the mean of those values are represented by < 𝐸 𝑖𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 >. The uncer-tainty for the energy of supercell with a vacancy in Pt or Au site in the bulk alloy for the threealloys is summarized in Table 6.12. It is seen that the uncertainties for all 256 sites in the firstmodel, PtAu-VC1, are higher than in the two other models. This behavior is expected sincethe training set contains no structures with vacancies. In the case of the PtAu-VC2 model, theuncertainties performs slightly better than the PtAu-VC3 model. It is also observed that theuncertainty is larger for the top three sites in Pt-25 at. % Au alloy compared to the other twocompositions in the PtAu-VC3 model.108Table 6.12: Increasing order of predicted error of all 256 vacancy in Pt or Au site in bulk alloys,𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖), normalized by total number of atoms in the simulation domain, 𝑁𝑡𝑜𝑡𝑎𝑙 . Eachcolumn represents data for the three models, i.e., PtAu-VC1, PtAu-VC2 and PtAu-VC3 ACEmodels. The data inside parenthesis are ordered in terms of increasing at. % Au, i.e., 25, 50and 75 at. % Au.Rank # PtAu-VC1:256*𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.]for sites in (25 at. % ,50 at. % , 75 at. %Au)PtAu-VC2:256*𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.]for sites in (25 at. % ,50 at. % , 75 at. %Au)PtAu-VC3:256*𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢,𝑖)/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.]for (25 at. % , 50 at.% , 75 at. % Au)1 (29.71, 34.28, 35.42) (11.47, 8.33, 8.95) (15.02,11.49, 10.33)2 (29.66, 33.87, 31.30) (7.69, 8.29, 8.85) (11.30, 10.20, 9.80)3 (29.43, 32.71, 31.03) (7.68, 7.83, 8.27) (10.96, 8.90, 9.21).... .... .... ....256 (18.56, 18.26, 18.59) (3.65, 4.01, 3.61) (4.48, 3.87, 3.74)As a next step, uncertainties are compared with the true error of the vacancy of the Pt/Ausite in the bulk alloy. The true error for the energies of the supercells with a vacancy at thePt/Au site in the bulk alloys is calculated using the following expression:|Δ(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) | =| 𝐸𝑒𝑎𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 − 𝐸 𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 | (6.13)In the above equation, the true error is calculated by taking the absolute value of thedifference between the energy of the supercell with a vacancy in all 256 Pt/Au sites computedby EAM, 𝐸𝑒𝑎𝑚𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢, and the same energy computed by the ACE model, 𝐸𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢. Thismeasure is readily computable using a less expensive model such as EAM, the uncertaintiesare compared with the true errors to see how well they correlate. This comparison is made forall three models and three compositions in Figure 6.22. In the first model, PtAu-VC1, all theuncertainties and true errors are clustered at ~ 10−4 and 10−2 eV/at., respectively. This trendis seen for all three compositions and is explained by the fact that training data contains novacancies. Hence, the model trained on such data designates all the structures with a vacancyin them as similar. For the cases of PtAu-VC2 and PtAu-VC3 models, there are a few datapoints that over-predict the uncertainty compared to the true error, while others under-predictthe uncertainty compared to true error. The variation of uncertainty with true error seemsrelatively flat, i.e., uncertainty does not vary with changing the true error. There are two109hypotheses as to why this might happen. First, the EAM model is a low-complexity modelwith only a few parameters, so the ACE model does not show proper scaling of uncertaintieswith true error. Second, the underlying data from the EAM model are of low-noise quality.The use of Bayesian Linear Regression (BLR) may not be suitable, and some improvementsin the regression methodology are needed to give better-scaled uncertainties. Some efforts inthe literature, such as Conformal predictions [179, 180] and pointwise optimal parameter sets(POPS) [172] regression, can help to remedy this issue. In the next chapter, the training datais switched to DFT instead of EAM and the issue of low-dimensional training data causingproblems with uncertainty scaling is tested.Figure 6.22: Normalized uncertainty (y-axis) of vacancy in Pt/Au site in bulk alloy,𝜎(𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢)/𝑁𝑡𝑜𝑡𝑎𝑙 from ACE model plotted with normalized true error (x-axis) betweenACE and EAM model, Δ𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢/𝑁𝑡𝑜𝑡𝑎𝑙for PtAu-VC1 (top panel), PtAu-VC2 (middle panel)and PtAu-VC3 (bottom panel). The left panel is for 25 at. % Au alloy, middle panel for 50 at.% Au alloy and right panel for 75 at. % Au alloy for all three models, respectively.1106.5 Summary1. In this chapter, a workflow for creating ACE models with EAM reference data and testingthem for various thermodynamic properties was presented. This provides an opportunityto create systematically customizable ACE models to model properties in Pt and Au, bulkPt-Au alloys and vacancies in bulk alloys. Training data were selected and target struc-ture uncertainties were quantified. A committee-of-models approach provides uncertaintyquantification (UQ) for atomistic simulation for metallic systems. Also, since these arelinear models, uncertainty quantification can be evaluated without additional cost [181].2. Specifically, bulk Pt and Au, vacancy in bulk Pt and Au, bulk alloys Pt-Au are well-predicted with ACE models with different choices of training data. It is seen that Σ5(210)and (310) GBs are a good representative for <100> tilt GBs that are chosen in this study.For bulk alloys, sampling structures with different Au content such as 25, 50 and 75 at.% Au provides a good agreement with EAM data.3. On the other hand, vacancy formation energy in bulk alloys for Pt- 25, 50 and 75 at.% Au alloys are not well captured with a hand-crafted training set. Other approachessuch as active learning and hyperactive learning exist to systematically include trainingstructures to predict better matching with reference data and to explore the potentialenergy surfaces (PES) better [182–185].111Chapter 7Vacancy energetics in Pt-Au with DFTaccurate surrogate models7.1 IntroductionUniversal pre-trained machine learning potentials (UPMLIPs) trained on DFT data such asMACE [186], Orb [187], CHGNet [188], SevenNet [189] and M3GNet [190] have shown promisein fast screening of candidate structures containing multiple chemical species [191]. In thecontext of Pt-Au alloys, Zhu and co-authors [192] found that the above-mentioned UPMLIPsyielded different formation energy curves for Pt-Au alloys as a function of Au composition inthe alloy. Only the Orb MLIP provides a reasonable agreement with DFT calculated formationenergies. The average relative error in formation energy for all Au compositions was found tobe 1.9% when compared to DFT. Casillas-Trujillo et al. [193] evaluated the mixing energiesof 21 disordered binary alloys using different MLIPs and two different approximations for theXC functional in DFT. Their work found that all of the MLIPs showed qualitative differenceswhen compared to DFT results. Since UPMLIPs do not quantitatively represent the energeticsof Pt-Au alloys, it is recommended to build an alternative Pt-Au alloy MLIP to reproduce theDFT-computed property of interest.Recently, the thermodynamic stability of Pt-Au and other Pt-based alloy spherical nanopar-ticles (NPs) up to 6 nm in diameter was studied using moment tensor potentials (MTPs) alongwith active learning techniques [194]. This MLIP resulted in reasonable agreement betweenground-state configurations from MTPs and experiments. With the help of MTPs, differentmorphologies of Pt-Au nanoparticles at large scales and various compositions of Au in thenanoparticles were investigated. However, no other current studies exist for extended defectssuch as GBs in the Pt-Au system using DFT and ACE surrogate models.The following work will be a continuation from the previous chapter, ACE models will be112developed to predict vacancy formation energies in alloy bulk and GB for an alloy compositionof ~ 50 at. % Au in the context of DFT simulations. Insights from the previous chapter willbe taken to create an initial training data set for a preliminary ACE model to simulate theabove quantities of interest. A workflow describes on how to iteratively add a training set intothis initial set to better match properties from the reference model, i.e. DFT simulations. Thisworkflow is followed for both cases, alloy bulk and GB, and comments are made on why themodel works or does not work in these cases.7.2 Structural models for alloy bulk and alloy GB7.2.1 Alloy bulk: 51.2 at. % Au alloyThe atomic structure for the alloy composition of 51.2 at.% Au is generated with MCSQS codeinside the ATAT suite of programs [160]. Initially, a structure of 5x5x5 of a primitive cellof FCC Pt is created. All atoms in the SQS program are set to be either Pt or Au species.Correlations for the final alloy structure are less than 0.01, the clusters considered here aretwo-body clusters only, and the number of neighbors is six nearest neighbors (2·a0). In thefinal structure, 64 atoms of Au and 61 atoms of Pt are in the structure as shown in Figure 7.1,yielding a composition of 51.2 at. % Au. This structure will be used to calculate vacancyformation energy at all Pt and Au sites using DFT and ACE models in sec. 7.4.1.Figure 7.1: The structure of 125-atom bulk alloy used for calculating vacancy formation energycontaining 51.2 at. % Au visualized by OVITO [135]. Yellow and grey atoms represent Au andPt species, respectively.7.2.2 Alloy GB: 50 at. % Au alloyIn order to create an alloy GB structure, first, a 120-atom Σ5 (210) Pt GB is created using the“GBcode” developed by Hadian and colleagues [158]. This structure is relaxed in VASP, which113is then used as input for the MCSQS code in ATAT. The inputs for the MCSQS code are asfollows: (i) use only pairs for calculating correlations, (ii) provide the overall composition, i.e.50 at.% Au and (ii) stop the MCSQS code when the correlations are below 0.01. The resultingstructure relaxed using DFT is shown in Figure 7.2. This structure will be used to calculatevacancy formation energy at sites in the GB region in 7.4.2.Figure 7.2: Simulation domain of 50 at. % Au Σ5(210) Pt-Au alloy GB visualized with OVITO[135]. Yellow and grey-colored atoms represent Au and Pt species, respectively.7.3 Computational workflow7.3.1 DFT details for training and validation dataThe major difference between this chapter and the previous one is the type of training dataused for creating ACE models. This chapter uses DFT simulations with PBEsol [195] (revisedPerdew-Burke-Ernzerhof GGA XC for solids) XC functional to create training and validationdata. The PBEsol XC was chosen because the original PBE XC had systemic issues predictingproperties such as lattice constant, bulk moduli [196] and surface energies [197]. In a paper byHaas and co-authors [198], a survey of lattice constants of solids is done with XC functionals,and it was shown that PBEsol was sufficiently able to capture the lattice constant of Pt andAu elemental solids within ~ 0.5% of experimental data. While it was also shown that LocalDensity Approximation (LDA) XC was a preferred choice for noble metals, there was littleinformation on whether this XC would work for the Pt-Au alloy system.In terms of creating training data using DFT calculations, there are some fundamental dif-ferences between how it is done with EAM models and DFT, i.e., the user needs additional114parameters for electronic convergence. This chapter uses the Vienna Ab-initio Simulation Pack-age (VASP) [199] to calculate the observables, i.e., energies, forces and virials. These choicesare explained in detail in Appendix C.The next step is to create validation data with the same simulation settings as the trainingdata to compare energies and relaxed geometries. The DFT simulations for training and testdata have to be done in the same way in terms of setup and input parameters. The finalground-state structures from these DFT simulations are then compared to the relaxed ground-state structures from the relaxation of ACE models and also used to compare predictions ofquantities of interest between ACE and DFT.7.3.2 ACE model detailsFrom the previous chapter, some training structures are added to start to predict the vacancyformation energies in alloy bulk and GB in Pt-Au alloys. First, the structures from the puresystems are sampled, these include the following: Au and Pt bulk FCC 125-atom structures,vacancy in bulk Au and Pt along with Au and Pt GB structures of two representative GBs –Σ5(012) and Σ5(013) GBs with 114 and 120-atom supercell sizes are used. These structuresare rattled and deformed using the same formulation as explained in Chapter 6. For the alloystructures, one bulk alloy for each composition of 6.25, 12, 25, 50, 75, 87.5 and 93.75 at. % Auis taken. For alloy GB, structures of Σ5(012) Pt- 25, 50 and 75 at. % Au content are added.All of these structures are rattled and deformed to obtain a dataset of 1006 structures, thedetails for the number of observations in the dataset are listed in Table 7.1. One representativestructure from each type of structure, i.e., bulk and defect, along with pure element/alloy types,are shown in Figure 7.3. One important thing to note is that no vacancy in alloy bulk and GBstructures has been added to the dataset at this moment.115Figure 7.3: Individual structure types for the training set used for the initial “ITER0” ACEmodel, left and right panels indicate bulk and defect structures ,i.e., vacancy in bulk and GB.The top and bottom panels show the pure components (FCC Pt and Au) and Pt-Au alloystructures. The top left panel shows some of the rattled and deformed structures of 125-atomFCC Pt and Au structures. The top right depicts some of the rattled and deformed 255-atom vacancy in FCC Pt and Au, 114-atom Σ5(310) GB structure, and 120-atom Σ5(210) GBstructure. The bottom left panel shows one of the rattled and deformed 125-atom alloy Pt-Aubulk structures out of a total of 7 compositions, and the bottom right panel shows one of therattled and deformed structures of one composition of 120-atom Pt- 25 at. % Au Σ5(210) GBstructure out of a total of three compositions.In order to predict vacancy formation energies with adequate accuracy of DFT predictions,a workflow is presented, and a schematic of this process is shown in Figure 7.4. Step (I) isexplained in the previous paragraph, for the next step, an ACE model referred to as “ITER0”needs to be created. Since this is a large training dataset with rich information, ACE modelhyperparameters such as the correlation order of 3 with the maximum degree of polynomialto be 16 are chosen along with a larger radial cutoff distance, 6.5 Å compared to the previouschapter on the EAM data fitting. All the models developed in this Chapter are constrained toa maximum value of 𝑛𝑚𝑎𝑥 = 16, as attempts to increase it further resulted in out-of-memoryerrors when executed on a single compute node of the Niagara compute cluster as a part of theDigital Research Alliance of Canada. A typical script written in Julia programming languagefor this step is given in Appendix D.1. These actions correspond to step (II) in the workflow.116Figure 7.4: Workflow diagram for iterative ACE model fitting involving different steps: (I)collecting all initial structures relevant for property of interest, (II) creating an ACE model usingBayesian Linear Regression (BLR), (III) this steps involves multiple sub-steps – (A) Calculatevacancy formation energies, (B) evaluate uncertainty using committee-of-models approach, (C)Pick ten vacancy and Widom-type structure that show top ten uncertainty (D) add them intotraining set to be used for next iteration ACE model regression and (E) retrain ACE modelusing updated training set using BLR and (IV) stop the iterations when uncertainties are belowan user-defined threshold value (or) ACE model predictions show good agreement with referencemodel predictions (or) after a certain number of iterations.In the next step, step (III-A), vacancy formation energies are calculated for 125 sites in thebulk Pt- 51.2 at. % Au alloy using the base ACE model (this is discussed in detail in 7.4.1).For step (III-B), the committee of models (size of 100) is evaluated for all 125 structures ofground-state relaxed vacancy at Pt/Au sites configurations from the base ACE model. Thisaction is repeated for other structures that are needed to calculate the vacancy formation energysuch as Widom-type insertions where a Pt and Au atom is replaced in a site where Au andPt species were originally present, respectively. In step (III-C), all the 125 structures with avacancy and 125 Widom-type structures are organized with decreasing order of uncertainty,i.e. 𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢) and 𝜎(𝐸𝑊𝐼𝐷𝑂𝑀), respectively. The top ten structures for each category ofstructures (vacancy and Widom) are identified that will be added into the training set forthe next iteration of ACE model fitting. This action corresponds to step (III-D). The finalpart of step (III) is to re-train the ACE model with the modified training dataset with thesame hyperparameters. The code version for the ACEpotentials.jl package used in this thesisis 0.6.10. For reproducibility purposes, it is also noted that JuLIP version 0.14.8 is used forgeometry optimization in Julia.1177.4 Results and discussion7.4.1 Vacancy formation in the Pt- 51.2 at. % Au bulk alloyVacancy formation energy at a site in bulk Pt - 51.2 at. % Au alloy is calculated by theexpression given below:Eform,BKvac@Pt/Au,i = E𝐵𝐾VC@Pt/Au,i − E𝐵𝐾alloy + 𝜇𝐵𝐾Pt/Au,i (7.1)In the above equation (Equation (7.1)), energy of ground-state structure of vacancy in Pt orAu site “i” in the bulk alloy is given by E𝐵𝐾VC@Pt/Au,i, the energy of ground-state structure of thebulk alloy is E𝐵𝐾alloy and the chemical potential of that Pt/Au site “𝑖” is denoted by 𝜇𝐵𝐾Pt/Au,i. Inorder to calculate the chemical potential of that site “𝑖”, the following equations (Equation (7.2)and Equation (7.3)/Equation (7.4)) are used.The total energy of the alloy structure, E𝐵𝐾alloy is decomposed into contributions from energiesof all Pt atoms in the bulk, NPt, each with chemical potential of 𝜇Pt@BK,𝑖 and the energiesof all the Au atoms in the bulk, NAu, each with chemical potential of 𝜇Au@BK,i. This energydecomposition is represented mathematically using Equation (7.2) as given below:NPt𝜇Pt@BK,𝑖 + NAu𝜇Au@BK,i = E𝐵𝐾alloy (7.2)In order to determine the chemical potential of the Au or Pt site, Widom-type Pt/Au inser-tions are made in the Au and Pt sites of the alloy. This operation changes the alloy’s overallcomposition, allowing for the determination of the chemical potential.The case of Au atom in the alloy is represented in the Equation (7.3) below. The 𝐸𝐵𝐾𝑊𝐼𝐷𝑂𝑀is the energy of the ground-state structure in which a Pt atom is replaced where the originalAu species was located (site “𝑖”), this is also designated as E𝐵𝐾Pt,i–>Au,i for this case.𝜇Pt@BK,i − 𝜇Au@BK,i = E𝐵𝐾Pt,i–>Au,i(𝐸𝐵𝐾𝑊𝐼𝐷𝑂𝑀) − E𝐵𝐾alloy (7.3)On the left-hand side of Equation (7.3), the energy difference when a Pt atom is replacing anAu atom in site “i” is expressed as a chemical potential difference. On the right side, the energydifference between the Widom structure and the original alloy structure is represented, equatedto the chemical potential difference as expressed on the left side.For the case of the Pt atom in the site “i”, the Au atom will be replacing Pt and then theequation to solve for chemical potential becomes the following:𝜇Au@BK,i − 𝜇Pt@BK,i = E𝐵𝐾Au,i–>Pt,i(𝐸𝐵𝐾𝑊𝐼𝐷𝑂𝑀) − E𝐵𝐾alloy (7.4)118The 𝐸𝐵𝐾𝑊𝐼𝐷𝑂𝑀 term in this case, is the energy of the ground-state structure where an Auatom is replaced at a Pt site (site “i”). This is also designated as E𝐵𝐾Au,i–>Pt,i On the rightside of Equation (7.4), the energy difference between the Widom structure and the originalalloy structure is represented, and this is equal to the chemical potential difference between Ausubstituted atom and the original Pt site as expressed on the left side.Figure 7.5: Histogram of DFT-PBEsol computed vacancy formation energy (𝐸 𝑓 𝑜𝑟𝑚,𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) for all125 sites in Pt-51.2 at. % Au alloy, each colored bar represents a fraction of sites with vacancyformation energy within the bin size. Yellow and grey bars depict the vacancy formation energyat Au and Pt sites, respectively.Chemical potentials of Au and Pt for all 125 sites form a distribution. The mean of thechemical potential of Au and Pt in bulk is given by -3.92 and -6.87 eV, respectively. Thestandard deviation of the chemical potential distribution of Pt and Au is calculated to be 0.032and 0.03 eV, respectively. In Figure 7.5, the distribution of vacancy formation energy at allthe sites calculated using DFT is shown as a histogram for the 125-atom bulk alloy structure.The mean and the standard deviation of the distribution are found to be 0.92 and 0.13 eV,respectively.119Table 7.1: ACE model parameters such as correlation order (N𝑐𝑜𝑟𝑟), maximum degree of poly-nomial (n𝑚𝑎𝑥), radial cutoff (r𝑐𝑢𝑡) and number of parameters for the potential (N𝑝𝑎𝑟𝑎𝑚) for“ITER0”, “ITER1”, “ITER02” and “ITER3” ACE models. Training data details, i.e., numberof structures (𝑁𝑠𝑡𝑟𝑠), total number of atoms (N𝑎𝑡𝑜𝑚𝑠), total number of observables for ACE fit(N𝑜𝑏𝑠). Weights are attached to each observation such as weights for energies (𝑤𝐸), forces (𝑤𝐹)and virials (𝑤𝑉) in order to perform the ACE regression. Root mean squared error (RMSE) forobservations, i.e., energy (RMSE𝐸), force (RMSE𝐹) and virial (RMSE𝑉).ModelnameN𝑐𝑜𝑟𝑟n𝑚𝑎𝑥 r𝑐𝑢𝑡[Å]N𝑝𝑎𝑟𝑎𝑚N𝑠𝑡𝑟𝑠 N𝑎𝑡𝑜𝑚𝑠 N𝑜𝑏𝑠 RMSE𝐸[meV/at.]RMSE𝐹[meV/Å]RMSE𝑉[meV/at.]w𝐸 w𝐹 w𝑉ITER0 3 16 6.5 8604 1006 131967 402943 2.18 14 20.36 20.0 4.0 1.0ITER1 3 16 6.5 8604 1026 134457 410553 2.16 14 20.19 20.0 4.0 1.0ITER2 3 16 6.5 8604 1046 136947 418163 2.15 15 20.01 20.0 4.0 1.0ITER3 3 16 6.5 8604 1066 139437 425773 2.13 15 19.83 20.0 4.0 1.0The iterative fitting process is repeated for the three iterations after creation of the ITER0model to give three more models. i.e., ITER1, ITER2 and ITER3 models. The specifics of thesemodels along with the “metrics” of fit and root mean squared error in observations (𝑅𝑀𝑆𝐸𝐸 ,𝑅𝑀𝑆𝐸𝐹 and 𝑅𝑀𝑆𝐸𝑉), are shown in Table 7.1. To provide the reader with the range of valuesseen in each observation and how the ACE model predictions are for those observations usingthe training dataset, the parity plot of energies, forces and virials are shown in Figures 7.6(a),(b) and (c), respectively.Figure 7.6: Parity plot for (a) energies of Pt, Au and Pt-Au alloy structures (b) forces [eV/Å] ofPt and Au atoms and (c) virial of Pt, Au and Pt-Au alloy structures computed with “ITER-0”ACE model (plotted on the y-axis) and DFT-PBEsol (plotted on the x-axis) using hollow bluecircles. The total number of data points for energies are 1006 (total number of structures), andthe total number of atoms is 131967, i.e., data points for force parity plot and 6036 (1006*6)points for virials. The dotted red line illustrates perfect parity, i.e., the Y=X line between ACEand DFT data.120The individual data points in the energy parity plot (in subfigure (a)) show a cluster ofpoints. This is because there are different types and categories of structures, i.e., bulk/defectand pure/alloy structures and energies of these individual types fall within one cluster. Forforce parity, it is seen that force is evenly distributed above and below 0 eV/Å, sampling bothclose to equilibrium and out-of-equilibrium structures. In the case of virials, the data showsstructures exhibit compressive and tensive stress components.In Figure 7.7 the predictions from the iterative ACE model fitting are compared with DFTpredictions, i.e., reference model predictions. In ITER0, the model under-predicts some of theDFT data values near the range ~ 0.5 - 0.65 eV and also for ranges between 0.85 and 1.05 eV.Furthermore, a few DFT data points close to 1.1 - 1.2 eV are also under-predicted and a over-prediction of one DFT calculated point at ~ 1.07 eV. The parity metrics for this iteration aremean absolute error (MAE) of 0.058 eV and an RMSE of 0.075 eV. In the next iteration, modelITER1 gets the predictions from ~ 0.85 to 1.05 eV into the +/- 0.1 eV band. Furthermore, thepoints in the low energy region (~ 0.5 to 0.65 eV) are inside the +/- 0.1 eV band. The accuracyshown by this model is 0.045 eV for MAE and an RMSE of 0.058 eV.121Figure 7.7: Parity plot of vacancy formation energy in Pt- 51.2 at. % Au bulk alloy using (a)ITER0, (b) ITER1, (c) ITER2 and (d) ITER3 ACE model with ACE and DFT predictionsplotted on y and x axis, respectively. The perfect parity, i.e., Y=X is shown by a red dashedline and a grey band is used as a visual guide to represent data within +/- 0.1 eV of the perfectparity line. The black colored markers show the structures added to the training set. The redstar markers show top ten uncertainties in all of the 125 structures for the current iteration.In the next iteration, model ITER2 is evaluated against DFT predictions in Figure 7.7(c).Not much changes are observed in accuracy improvement, but the uncertainties for valuesoutside the +/- 0.1 eV band are correctly identified for some of the outliers. This observationis true for DFT predictions at ~ 0.95 eV and ~ 1.07 eV, which the ACE model over-predicts.This model shows a MAE of 0.0436 eV and RMSE of 0.0565 eV, and as noted previously, theimprovements are marginal compared to the previous iteration.For the final iteration, ITER3, overall, the vacancy formation energy in all sites is predictedby ACE within DFT predictions except for four data points that are under-predicted. TheMAE for this iteration is 0.041 eV, and RMSE is 0.0525 eV. Those structures which exhibit thetop ten uncertainties could be added to the training set in the next iteration to provide betterpredictions.122Another aspect of using a committee of models in the current ACE models is to evaluatethe uncertainties for each structure of interest. This measure will provide on how reliable theextrapolation of such a model is to structures of interest from the training set. For all fourmodels studied here, those values are included in Table 7.2 in decreasing order of uncertainties.This uncertainty information is also provided in Figure 7.7 as a way to illustrate the iterativeACE fitting process.Uncertainty is quantified only for structure with vacancy in Pt/Au site (𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢) ), sincethis is the largest uncertainty out of three in Equation (7.1). Using the following expression,𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢) is measured as:𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢) =√√1𝑁𝑐𝑚𝑡𝑒𝑁𝑐𝑚𝑡𝑒∑︁𝐼=1(𝐸𝐵𝐾,𝐼𝑉𝐶@𝑃𝑡,𝑖/𝐴𝑢,𝑖− < 𝐸𝐵𝐾,𝐼𝑉𝐶@𝑃𝑡,𝑖/𝐴𝑢,𝑖 >)2(7.5)With iterations, it is clear that the uncertainties for all 125 structures (vacancy in bulk) arereducing in magnitude. For iteration, ITER0, the uncertainties range from ~ 40 meV/at. to 18meV/at, this range changes to ~ 20 to ~ 12 meV/at for ITER1 model. For the next iteration,ITER1 is also seen with the vacancy formation energy predictions in Figure 7.7(b). In ITER2,the range of uncertainties is from 18.2 meV/at. to 9.75 meV/at. For the final iteration, ITER3,the range becomes 15.74 meV/at. to 9.68 meV/at. for the structure with the lowest uncertainty.Table 7.2: Decreasing order of uncertainty predictions or predicted error from energy of vacancyin 125-atom Pt- 51.2 at. % Au bulk alloy, 𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢) normalized by total number of atoms,𝑁𝑡𝑜𝑡𝑎𝑙 for ITER0, ITER1, ITER2 and ITER3 ACE models.Rank#ITER0:125*𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙 [meV/at.]ITER1:125*𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙 [meV/at.]ITER2:125*𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙 [meV/at.]ITER3:125*𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙 [meV/at.]1 39.88 20.12 18.25 15.742 31.77 20.099 16.69 15.593 29.22 19.66 16.51 15.50.... .... .... .... ....125 17.72 12.14 9.76 9.68123A final analysis of predicted error, i.e., the uncertainty of energy of structure with a vacancyin bulk is shown along with the true error, Δ(𝐸𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) in Figure 7.8. The following expressionprovides the true error, Δ(𝐸𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢):Δ(𝐸𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) =| 𝐸𝐵𝐾,𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 − 𝐸𝐵𝐾,𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 | (7.6)The energy prediction of DFT for a vacancy at the Pt or Au site is given by 𝐸𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢. Itis important to note that this energy comes from the validation dataset, i.e., the structures arerelaxed with DFT, and the final ground-state energy is taken to be 𝐸𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢. For the term𝐸 𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 , the base ACE model from each iteration are used to relax the structures with avacancy in them and the final energy is taken to be 𝐸 𝐴𝐶𝐸−𝑚𝑒𝑎𝑛𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 .In the previous chapter, the same structure, i.e. EAM relaxed structure, is used to makeone-step prediction with a surrogate. In this chapter, the surrogate is used to relax an initialstructure while the same structure is relaxed using DFT, and the final energies are comparedalong with uncertainties on those final structures.In Figure 7.8, uncertainty versus true error data is shown for all four ACE models. Noneof the models show a good parity on the Y=X line. Also, the data shows a typical “flatness”issue, as seen in the case of EAM data. However, the predicted error goes down slightly foreach iteration, as is evident from Table 7.2. Furthermore, the density of points where theuncertainties are higher than true errors also goes up with increasing the number of iterations.Figure 7.8: Scatter plot of normalized predicted error, i.e., uncertainty predictions𝜎(𝐸𝐵𝐾𝑉𝐶@𝑃𝑡/𝐴𝑢)/𝑁𝑡𝑜𝑡𝑎𝑙 (plotted on the y-axis) and normalized true error, Δ𝐸𝐵𝐾𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢/𝑁𝑡𝑜𝑡𝑎𝑙 (plot-ted on the x-axis) of all four ACE models – ITER0, ITER1, ITER2 and ITER3 model shownin green, black, blue and red triangles, respectively. Perfect parity, i.e., Y=X, is plotted usingthe dotted black line.1247.4.2 Vacancy formation at Σ5 (210) GB in a Pt-50 at. % Au alloyAs an extension to calculating vacancy formation in bulk, the vacancy formation energy atGB sites are studied using DFT and iteratively fit ACE models. In order to do this, a detailedexplanation of the GB structures needs to be presented. In Figure 7.9, atoms are identified asFCC bulk and GB sites using Common neighbor Analysis (CNA) [50] in OVITO [135] basedon their local neighborhood. Out of the GB sites, first the GB plane is identified with thegray semi-transparent line and the atoms which represent structural units (SU) of the Σ5 (012)GB are shown with dotted blue lines and labeled SU1 and SU2. The site on the GB plane isreferred to as the “𝑎” site. The two sites on the left side of GB plane are 𝑏 and 𝑐 based ontheir distance from the GB plane. Similarly, the GB sites on the right side of the GB planeare designated as 𝑏’ and 𝑐’. These sites are illustrated for one such SU, i.e., SU2 and are thusreferred to SU2- 𝑎 etc., and they can also be applied to SU1 using a prefix – SU1.Figure 7.9: Atomic structure of 120-atom Pt- 50 at. % Au alloy Σ5(210) GB colored accordingto local environment, i.e., FCC bulk and GB sites. FCC bulk and GB atomic sites are colored bygreen and grey colors, respectively. The GB plane is highlighted with a grey semi-transparentline and the structural units (SU) are highlighted and connected by blue dotted lines.The iterative fitting process is repeated for a vacancy at GB sites in Pt- 50 at. % Au alloyand the results are presented in sec. 7.4.2. The final ACE model iteration, ITER3, from thevacancy in bulk alloy case, is taken, and then the structures showing the highest uncertaintiesare added into the training set. In this case, these structures are of two types: (i) vacancy ata GB site and (ii) Widom-type insertion at a Pt/Au GB site. The entire process is repeated125for four iterations, resulting in ITER4, ITER5, ITER6 and ITER7 ACE models. The specificsof the training set for each model creation step, with model accuracy in predicting the energy,forces and virial observations, are reported in Table 7.3.Table 7.3: Model details and error table for ITER4, ITER5, ITER6 and ITER7 ACE models.ACE model hyperparameters are correlation order (N𝑐𝑜𝑟𝑟), maximum degree of polynomial(n𝑚𝑎𝑥), radial cutoff distance (r𝑐𝑢𝑡). Details about the training data with number of parameters(N𝑝𝑎𝑟𝑎𝑚), number of structures (N𝑠𝑡𝑟𝑠), number of atoms (N𝑎𝑡𝑜𝑚𝑠) and number of observations(𝑁𝑜𝑏𝑠). ACE model errors after fitting: RMSE for energies (RMSE𝐸), forces (RMSE𝐹) andvirials (RMSE𝑉). Weights of training data used for fitting of energy (𝑤𝐸), force (𝑤𝐹) and virial(𝑤𝑉).ModelnameN𝑐𝑜𝑟𝑟n𝑚𝑎𝑥 r𝑐𝑢𝑡[Å]N𝑝𝑎𝑟𝑎𝑚N𝑠𝑡𝑟𝑠 N𝑎𝑡𝑜𝑚𝑠 N𝑜𝑏𝑠 RMSE𝐸[meV/at.]RMSE𝐹[meV/Å]RMSE𝑉[meV/at.]w𝐸 w𝐹 w𝑉ITER4 3 16 6.5 8604 1086 141827 433083 2.182 14 19.78 20.0 4.0 1.0ITER5 3 16 6.5 8604 1106 144217 440393 2.1 14 19.66 20.0 4.0 1.0ITER6 3 16 6.5 8604 1126 146607 447703 2.09 15 19.55 20.0 4.0 1.0ITER7 3 16 6.5 8604 1146 148997 455013 2.07 15 19.43 20.0 4.0 1.0The equation used to calculate formation energy of a vacancy at a Pt/Au GB site,Eform,GBvac@Pt/Au,i,is shown in Equation (7.7).Eform,GBvac@Pt/Au,i = E𝐺𝐵VC@Pt/Au,i − E𝐺𝐵alloy + 𝜇𝐺𝐵Pt/Au,i (7.7)The energy of ground-state structure of vacancy in Pt or Au in site “i” at GB is givenby E𝐺𝐵VC@Pt/Au,i and the energy of ground-state alloy GB structure is given by E𝐺𝐵alloy while thechemical potential of Pt/Au in that site “i” is given by 𝜇𝐺𝐵Pt/Au,i.For calculating the chemical potential of Pt or Au sites at GB, information from the chemicalpotential of Au and Pt in the bulk alloy (with slightly different composition of Au) is required.The mean of chemical potential of Au and Pt in bulk is denoted as < 𝜇𝐵𝐾Au > and < 𝜇𝐵𝐾Pt >,respectively. In Equation (7.8), the total energy of alloy GB supercell (denoted on the righthand side of equation) is decomposed into the following components. First, the bulk Pt and Ausites with each having the mean chemical potential from the bulk is used to express the energycontribution from the bulk sites, i.e., N𝐵𝐾Pt < 𝜇𝐵𝐾Pt > +N𝐵𝐾Au < 𝜇𝐵𝐾Au >. The energy of all Pt and126Au GB sites are represented as N𝐺𝐵Pt 𝜇𝐺𝐵Pt,𝑖 and N𝐺𝐵Au 𝜇𝐺𝐵Au,i, respectively, where site “i” is where thevacancy will be inserted. The final expression is shown to be:N𝐵𝐾Pt < 𝜇𝐵𝐾Pt > +N𝐵𝐾Au < 𝜇𝐵𝐾Au > +N𝐺𝐵Pt 𝜇𝐺𝐵Pt,𝑖 + N𝐺𝐵Au 𝜇𝐺𝐵Au,i = E𝐺𝐵alloy (7.8)When there is an Au atom at the GB site, the Widom-type insertion structure is madesuch that a Pt atom is swapped in the place of the Au atom and the following expression,Equation (7.9), is used to calculate the chemical potential of Au at that GB site “i”:𝜇Pt@GB,i − 𝜇Au@GB,i = E𝐺𝐵Pt,i–>Au,i(𝐸𝐺𝐵𝑊𝐼𝐷𝑂𝑀) − E𝐺𝐵alloy (7.9)For the case of Pt atom, the same equation will need to be modified such that when an Auatom is inserted into the original GB site as written in Equation (7.10):𝜇Au@GB,i − 𝜇Pt@GB,i = E𝐺𝐵Au,i–>Pt,i(𝐸𝐺𝐵𝑊𝐼𝐷𝑂𝑀) − E𝐺𝐵alloy (7.10)The distribution of vacancy formation energy at all 60 Pt and Au GB sites is shown inFigure 7.10. The mean and standard deviation of this distribution are 0.68 and 0.53 eV,respectively. This shows that the mean of vacancy formation energies is lower when comparedto bulk sites, and the distribution is much broader when compared to the distribution of vacancyformation energy in bulk.127Figure 7.10: Histogram of DFT computed vacancy formation (𝐸 𝑓 𝑜𝑟𝑚,𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) at 60 sites shownwith colored bars for fraction of sites within a specified bin size. Yellow and grey colored barsrepresent data from Au and Pt sites, respectively.Not many results exist in the literature on using DFT for studying vacancy interaction withGBs. Hence, a few works from the literature based on MS/MD simulations are referred hereto put the current results in perspective. Bai and colleagues [200] studied the interaction ofpoint defects (vacancy and interstitial) with GBs in FCC Cu. With GBs considered strongsinks for vacancies, the formation energies are reduced by 40 % for asymmetric Σ11 and Σ45GB compared to the bulk. While for the Σ3 twin GB, a weak interaction with vacancy wasfound (reduction in vacancy formation energy was less than 0.8 %). In their study, the vacancyformation energies reported for the Σ11 GB showed negative formation energies leading toan interstitial-vacancy annihilation event. All their results confirmed that the interaction ofvacancies with a given GB is very sensitive to its structure and energy. In the work by Hallilet al. [201], vacancy formation energy was studied in FCC Ni GBs. Here, vacancy formationenergies below 0 eV were found to have a large magnitude of displacement (0 – 0.6 Å). Inanother work by Uberuaga and co-authors [202], the interaction of vacancies with general GBsin FCC Cu was inspected. It was shown that there is a correlation between the segregation ofvacancies and the magnitude of atomic volumes, i.e. vacancies were attracted to sites of smalleratomic volume.To illustrate the differences between vacancy formation energy in bulk and GB, the chemicalpotential distribution of Pt and Au in both cases is shown in Figure 7.11. The mean andstandard deviation of 𝜇𝐺𝐵𝑃𝑡/𝐴𝑢,𝑖 for Pt and Au species are -6.66 and -3.77 eV and 0.073 eV forboth cases, respectively. The chemical potential distribution of Pt and Au in GB is twice aswide (0.07 versus 0.03 eV) compared to that of Pt and Au in the bulk. The main reasonbehind the wider chemical potential distribution is because of more complex structures at GBs128as compared to FCC bulk region. Furthermore, the difference of mean of chemical potentialdistribution for Pt and Au between GB and bulk cases are 0.21 and 0.15 eV, respectively. Thisaffects the vacancy formation energy (see Equation (7.7)) and shifts the energy distributionaccording to this difference in the mean value.Figure 7.11: Histogram of DFT computed chemical potential of (a) Au and Pt sites (60 in total)in Pt - 50 at. % Au alloy Σ5(210) and (b) Au and Pt sites (125 in total) in bulk FCC Pt -51.2 at. % Au alloy is shown in (a) and (b), respectively.The solid blue line for each histogramrepresents the mean of each chemical potential distribution and the dotted line along with thearrow represents the shift in mean of Au and Pt chemical potential between GB and bulk.Yellow and grey colored bars represent chemical potential for Au and Pt, respectively.The iteratively fitted ACE model predictions are shown along with DFT data for vacancyformation energies at GB sites in Figure 7.12. For the ITER4 model, the parity plot does notlook good and there are both under and over predictions by the ACE model up to +/- 1.2 eVwithin DFT predictions. However, some of the sites are correctly identified to be out of paritywith the uncertainty measures, such as the one predicted to be 1.5 eV with DFT. This iterationachieves RMSE and MAE of 0.35 and 0.2 eV, respectively.129Figure 7.12: Parity plot for vacancy formation energy at Pt/Au sites in Σ5(210)GB Pt- 50 at.% Au alloy. Y-axis for (a), (b), (c) and (d) represents data from ACE models ITER4, ITER5,ITER6 and ITER7. X-axis data is the reference model predictions from DFT-PBEsol. Thered dotted line represents perfect parity, i.e., Y=X and a +/- 0.1 eV region around perfectparity is represented by a grey band. The black colored markers show the structures addedto the training set. The red star markers show top ten uncertainties in all of the 120 vacancystructures for the current iteration.In the ITER5 model, most of the points that showed under predictions are inside the parityregion of +/- 0.1 eV except for two data points at DFT predictions of 0.8 and 1.0 eV. An RMSEand a MAE of 0.248 and 0.166 eV is realized in this iteration. In ITER6, improvement of onlyone point outside the 0.1 eV band shows under-prediction compared to DFT. Furthermore,there is only a marginal improvement in RMSE at 0.246 eV and MAE at 0.166 eV with noimprovement.In ITER7, all the under-predicted values from previous iterations are within the 0.1 eV band,and only data that show over-prediction by ACE model exists outside. This prediction trendwill be later explained by using comparing the relaxed structures between ACE model andDFT. Also, the error measures such as RMSE and MAE with 0.251 and 0.174 eV go up slightlycompared to previous iterations.130Since there is no vacancy at GB site structure in the training set, this structure shows thelargest contribution to calculating uncertainties for vacancy formation energies. The uncertaintyfor this structure, 𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 ), is calculated as:𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢) =√√1𝑁𝑐𝑚𝑡𝑒𝑁𝑐𝑚𝑡𝑒∑︁𝐼=1(𝐸𝐺𝐵,𝐼𝑉𝐶@𝑃𝑡,𝑖/𝐴𝑢,𝑖− < 𝐸𝐺𝐵,𝐼𝑉𝐶@𝑃𝑡,𝑖/𝐴𝑢,𝑖 >)2(7.11)In Equation (7.11), energy of vacancy in GB site using the ACE model, 𝐼, (out of 100) isdepicted by 𝐸𝐺𝐵,𝐼𝑉𝐶@𝑃𝑡,𝑖/𝐴𝑢,𝑖 and the mean of 100 ACE model predictions for each vacancy in alloyGB site, 𝑖, is denoted by < 𝐸𝐺𝐵,𝐼𝑉𝐶@𝑃𝑡,𝑖/𝐴𝑢,𝑖 >. The predicted errors, i.e. uncertainties, for thisstructure is shown in Table 7.4. In ITER4, the predicted error ranges from ~ 32 meV/at. to~ 15 meV/at. and, in the next iteration, ITER5, the highest uncertainty is reduced by ~ 8meV/at. and the range is reduced to be between ~ 24 meV/at. and ~ 12 meV/at. This is alsoevidenced by an improvement in the parity plot shown in Figure 7.12(b). In ITER6, the highestuncertainty drops by 6.5 meV/at. and range is between 17.3 and 10.6 meV/at. and showing areduction compared to ITER5. In the final iteration, ITER7, the highest uncertainty is onlyreduced by ~ 2.5 meV/at and the lowest uncertainty goes below 10 meV/at.Table 7.4: Uncertainty 𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 ) normalized by 𝑁𝑡𝑜𝑡𝑎𝑙 from a committee size of 100 forall 60 vacancy at Pt/Au GB sites from ITER4, ITER5, ITER6 and ITER7 ACE models.Uncertainties are ranked in decreasing order of magnitude.Rank # ITER4:120*𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.]ITER5:120*𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.]ITER6:120*𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.]ITER7:120*𝜎(𝐸𝐺𝐵𝑉𝐶@𝑃𝑡/𝐴𝑢𝑖 )/𝑁𝑡𝑜𝑡𝑎𝑙[meV/at.]1 31.89 23.78 17.24 14.812 27.49 23.10 16.70 14.693 27.05 17.79 16.41 14.35.... .... .... .... ....60 14.79 12.07 10.63 9.17The uncertainties, i.e. predicted errors are compared with true errors of the supercell energyof GB structures with a vacancy and are shown in Figure 7.13. The true error normalized by131the total number of atoms, |Δ𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 |/𝑁𝑡𝑜𝑡𝑎𝑙 , is calculated using the energy of ground-statestructure of vacancy in the GB site as obtained by DFT (𝐸𝐺𝐵,𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) and ACE (𝐸𝐺𝐵,𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢),respectively, i.e.Δ(𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢) =| 𝐸𝐺𝐵,𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 − 𝐸𝐺𝐵,𝐴𝐶𝐸𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 | (7.12)Figure 7.13: Scatter plot of normalized predicted, 𝜎(𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢)/𝑁𝑡𝑜𝑡𝑎𝑙 (shown in y-axis) andnormalized true error, |Δ𝐸𝐺𝐵𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 |/𝑁𝑡𝑜𝑡𝑎𝑙 (shown in x-axis) shown in green, black, blue andred circles for the ITER4, ITER5, ITER6 and ITER7 models, respectively. Perfect parity line,Y=X is shown by a black dotted line.The true error ranges from ~ 8·10−4 to ~ 10−2 eV/at. in ITER4 model. In this iteration,the uncertainty is under predicted compared to the true error. In the next three iterations,the uncertainty goes down as seen in Table 7.4. There are few sites where uncertainty is overpredicted compared to the true error and some sites start to fall on the parity line. Anotherway to add structures in the training set in the next iteration is that structures with high trueerror can be added. However, it is important to note that 𝐸𝐺𝐵,𝐷𝐹𝑇𝑣𝑎𝑐@𝑃𝑡/𝐴𝑢 is expensive to calculatesince there exists many combinations of vacancy at different host sites.To further inspect the differences in the parity plot in Figure 7.12(d), it is worthwhile tocompare the final relaxed structures using the DFT and the ITER7 ACE model. Three examplesof relaxed vacancy at GB site structures are shown in Figures 7.14(a), (c), (e) and (b), (d), (f),respectively. In Figure 7.12(d), these examples are denoted by site-𝛼, 𝛽,𝛾 and, are chosen sincethey show the largest discrepancy between ACE and DFT predictions. For the first example, a132vacancy (a missing Pt atom) is at the GB plane, as seen in (a) and (b). The vacancy formationenergy is calculated to be 0.12 eV with DFT. The ACE predicted vacancy formation energyat this site is 1.33 eV. The DFT relaxed structure is explained first: The gold atom at SU2-𝑐’site moves to the vacant site (SU2-𝑎) at the GB plane, leaving a vacant site behind. The goldatom from SU2-𝑏’ moves into the newly formed vacant site, SU2-𝑐’, and two gold atoms fromSU2-𝑏 and SU2-𝑐’ slightly move towards the SU2-𝑏’ position. No structural changes in theGB structure are observed when relaxed with the ACE model. The reason could be that thepotential energy surface (PES) minima might be different for both models.Figure 7.14: Three examples of structures undergoing structural changes when performinggeometry relaxation with vacancy at GB sites. Displacement vectors (represented with blackarrows) are shown for atoms with significant movement in the structure. Atoms show the initialstructure before relaxation, and the arrows designate the final positions of atoms. The greysemi-transparent line indicates the GB plane. The left panel of structures represent the DFTrelaxed structures, while the right panel indicate ACE relaxed structures. Yellow and greyatoms represent gold and platinum species, respectively.133For the second instance of structure comparisons between DFT and ACE (see Figures 7.14(c)and (d)), a vacancy is created at the Au site in SU1-𝑐. In the DFT relaxed structure, a goldatom from the SU1-𝑏 site moves into the vacant site. This movement is followed by another goldatom from the SU1-𝑏 site into the newly created SU1-𝑏 vacant site. In the case of relaxationwith the ACE model, a gold atom at the SU1-𝑏 site slightly moves towards the SU1-𝑐 vacantsite. The DFT and ACE relaxed vacancy formation energy for this site is 0.36 and 0.84 eV,respectively.For the final example (see Figures 7.14(e) and (f)) , an Au atom from a SU2-b site movesinto a SU2-𝑐 vacant site. A slight displacement by the gold atom from a SU2-𝑏’ site towardsthe recently vacant SU2-𝑏 site. Additionally, another gold atom from SU2-𝑐 site slightly movestowards the SU2-𝑏 sites. Only gold host atoms tend to move in the above structures comparedto platinum host sites. This preference in movement could be due to the lower cohesive energyof the Au atom compared to Pt, and similar behavior is seen in another study. In the paper byWang and others [83], vacancy formation energy at GBs computed by EAM potential is studiedin TiTaV equiatomic alloy. The Ti atoms show a considerable displacement around a vacancycompared to the other two species in the alloy.Suzuki and Mishin [72] studied the point defect formation energy (vacancy and interstitial)in FCC Al and Cu GBs. They found that vacancies tend to be localized and delocalized invarious GBs. It was noted that for FCC Al, the vacancy moved from site 𝑏 to site 𝑐 (seeFigure 7.9 for notation) in a Σ5(210) GB. This delocalization of vacancies is also observed inanother study by Tanguy [203]. In this article, the author tries to study the motion of single anddouble vacancies in four GBs in FCC Al and BCC Fe. It is noted that in some cases, clusteredvacancies are encountered. In a study by Chesser et al. [204], the kinetic behavior of Σ3 GBwhen vacancies are introduced is studied for the Al- 8 at. % Si alloy at 700 K. A collectionof diffusive events is observed where vacancies facilitate reconfiguration (up to a magnitudeof two times the first nearest neighbor distance) of atoms inside the GB core region. It waspointed out that these collections of diffusive events only occur when the GB area is larger inthe simulation domain. In the current scenario, the small GB area in the simulation leads tointermittent and local rearrangement of atoms. Additionally, only one vacancy is introducedhere and hence no collection of events is observed. Furthermore, it is important to note thatall the above cited works are done using semi-empirical calculations and performing DFT onlarge system size is not tractable.Another question is why all the ITER7 ACE predictions consistently overestimate the va-cancy formation energy. A recent study by Mosquera-Lois and colleagues [205] explicitly showsthat surrogate models under-predict the transition paths and migration barriers for point de-fects. A schematic of this underlying problem is shown in Figure 7.15. They show that samplingof structures around the minima is insufficient to determine the PES’s absolute minima.134Figure 7.15: Schematic of potential energy surface (PES) shown as a function of configurationalcoordinates predicted by DFT (green semi-transparent line) and Surrogate model (orange semi-transparent line). The two minima highlight the energies of two different configurations (blue-colored atoms shown with bonds) and a possible relaxation path between the two configurations.Red dots represent sampling structures to build the surrogate model. The final structures forsurrogate model and DFT are shown in orange and red solid circles. The schematic is takenwith permission from the open access article of Mosquera-Lois and co-authors [205].There is interest in the literature for this specific aspect: surrogate models show systematicPES softening. Recent work by Deng and co-authors [206] identified issues in surrogate modelssuch as M3GNet, CHGnet, and MACE. Softening occurs on a biased sampling of close-to-equilibrium structures, affecting the curvature of the PES. They propose a linear correction tothe surrogate model predictions to improve their forecasts, especially augmenting high-energystructures’ energies in the surrogate PES.7.4.3 Comparison of the computational cost of ACE models withDFTAfter establishing the prediction accuracies for a property of interest between DFT and ACEsurrogate models, it is important to analyze the computational costs of these models. For thecase of DFT calculation, a single geometry relaxation run of a vacancy in a GB supercell takes6000 s on four compute nodes (total of 160 cores or 320 cores using hyper-threading), eachusing 40 Intel “Skylake” cores. This calculation is done on the Niagara compute cluster in theDigital Research Alliance of Canada [207]. For the same calculation using the ACE surrogatemodel, the simulation takes 6 s on one compute node in the above-mentioned compute cluster.This leads to a speedup of 4000 for the same calculation done using the ACE model.1357.5 Summary1. The vacancy formation energy is used as a quantity of interest in this chapter to comparepredictions from DFT and ACE surrogate models. Predictions of formation energy inthe bulk alloy are captured well using iterative ACE surrogate model fitting. Vacancyformation energies at GB are smaller than bulk vacancy formation energy, even withsome less than 0 eV. Vacancy delocalization and structural changes are observed for DFTcalculations where atomic shuffling/rearrangement occurs in the GB region.2. Using the iterative fitting of ACE surrogate models, vacancy formation energy at GBs isnot captured well by ACE models. This trend is because the final structure predicted bythe ACE model does not match DFT predictions. The uncertainties or predicted errorsdo not scale with true errors when switching from EAM to DFT training data. Thisscaling implies there is a need to fix the methodology behind obtaining the uncertainties.Conformal predictions and pointwise optimal parameter set regression are required toscale the uncertainties since data from atomistic simulations are in the low-noise regime.Additionally, softening of the surrogate model’s potential energy surface needs to be testedand fixed to get better matching with DFT data.136Chapter 8Summary and future work8.1 SummaryThis work puts forward a framework for studying defect interactions in binary alloys usingDFT and developing an ACE surrogate model based on DFT. The main conclusions of thisthesis are the following:• Solutes in bulk Ti- 25 at. % Mo tend to dissolve which is the opposite of the trendseen in GB segregation in Ti- 25 at. % Mo alloy, i.e., Y>Zr>Nb. The volume differencebetween before and after solute substitution captures most of the segregation tendency.Additionally, solutes at the Mo site seem to be more strongly segregated than at the Tisite. Furthermore, the segregation of Ti to Mo GB sites is stronger than that of Mo to TiGB sites. This trend is predominantly explained by the volumes of the host Ti and Mosites.• Eshelby-type model predictions for Y work well compared to Zr and Nb. Using the DFTGB segregation energy data from Y, Zr and Nb, predictions of segregation energy aremade for other transition metal solute elements. This prediction provides an estimate ofsegregation for important solute elements in Ti-Mo alloys. Most elements in the earlytransition metal elements series, such as Sc, Y, Zr, Cd, La, Hf and Hg, strongly trend tosegregate with segregation energy to be -0.5 eV or lower.• The other two compositions in Ti-Mo such as 50 and 75 at. % Mo, more GB structureswith GB plane sites are used to study segregation of Y. It is more readily seen that the Ysegregation energies exhibit a distribution in these cases. A strong tendency to segregateto GB in Ti- 50 at. % Mo alloy with mean segregation energy of -0.75 and -1.1 eV forTi and Mo sites, respectively, is observed. The mean segregation energy increases to -1.1and -1.45 eV for Ti and Mo GB sites in the 75 at. % Mo alloy. The trend of segregation137energy with volume difference before and after substitution is observed for Ti sites ratherthan for Mo sites.• Eshelby models for Y at Ti and Mo sites are tested with the help of pristine Mo and TiGB site Voronoi volumes. The model works more well for the larger host site, i.e. Ti,than Mo, a smaller host atom. With scaling constants of Ti and Mo sites from the Ti -25 at. % Mo alloy, reasonable predictions can be made for mean segregation Y energiesin Ti - 50 and 75 at. % Mo alloys.• Using surrogate models such as ACE to develop EAM-accurate predictions on bulk Ptand Au seems to work reasonably well. For the case of Pt and Au GBs, by adding Σ5(012) and (013) GBs into the training set, all the GBs studied in this work are predictedin good agreement. In bulk Pt-Au alloys, just adding Pt- 50 at. % Au alloys in thetraining set cannot to capture mixing enthalpy in the full composition range. Finally, thevacancy formation energy distribution for Pt – 25 at. % Au alloy is not well capturedusing the training sets in this work. Although reasonable predictions within +/- 0.1 eVare seen for the other two compositions – 50 and 75 at. % Au alloys.• Since ACE models are linear, uncertainties or predicted errors can be extracted withoutmuch computational overhead. For all quantities of interest studied here, uncertainties donot scale linearly with true error. Conformal predictions and Pointwise Optimal Param-eter Set (POPS) regression will be needed to correct this behavior and provide reliableuncertainties.• With DFT simulations, vacancy formation energy distribution for a ~ 50 at. % Au alloy isstudied for bulk and Σ5 (012) GB. It is seen that there is a decrease of ~ 25 % in the meanvacancy formation energy in GB compared to bulk. With the help of iterative fitting ofsurrogate ACE models, this quantity of interest is reproduced to +/- 0.1 eV for the caseof bulk Pt-Au alloy. In the GB case, vacancy delocalization occurs for a few sites, andthis effect is not captured in the structural relaxation with ACE models.1388.2 Recommendations for future workThe main improvements and further recommendations concluded from the present work are:• Studying grain boundaries (GBs) in metallic systems using DFT always suffered fromdomain-size limitations, i.e., exploring GB structures up to a few hundred atoms. Semi-empirical models always suffered from not being electronically accurate. Now, using aregression model that takes input data, it is possible to study low-symmetry boundarieswith DFT accuracy. Further, the kinetics of these GBs can now be explored, and mech-anisms of GB migration and GB interactions with solute atoms can now be examined indetail with DFT accuracy. Also, observations on a particular GB from TEM/APT can becompared with kinetic behavior of the same GB from ab-initio informed surrogate model.• In this thesis, only the interactions of dilute concentrations of solute (third species) arestudied in a binary random alloy. A natural extension of this work would be to considerthe multi solute interactions in the binary alloy. For example, an interesting extension ofPt-Au is the system of Pt-Pd-Au alloys, where Au solute interactions at the GBs can beinvestigated. This investigation can be done by DFT simulations, although the resultswould suffer from spurious solute-solute periodic image interactions due to small domainsize limitations in DFT. A way forward is to train a surrogate model by systematicallyincluding small size training structures of pure elements, binary systems and the ternaryalloy. Then, the surrogate model can be used to evaluate solute interactions in ternarysystems and further in more complex multi-component systems.• Vacancy cluster formation (multiple vacancies) and void coalescence at GBs can be anissue for stabilizing metallic nanocrystalline alloys, e.g. Pt-Au alloys, and be detrimentalto the their mechanical properties. Studying these mechanisms in further detail wouldbe possible by systematically including these configurations in the training set. Further,building such a surrogate model and using active learning strategies to explore the phase-space for studying such mechanisms is the next step in identifying the possible causes forthese atomistic events to occur and building better materials.• Finally, the cost of training such surrogate models needs to be considered. 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For k-point convergence, wesee that the energy converges to a constant value at 4x4x4 k-mesh. This corresponds to 0.02−˚1 of k-mesh spacing, which is adopted in the study. For energy cutoff convergence, the energydifference, ΔE at 360 eV is ~ 5 meV off when compared to the constant value at higher energycutoffs. Hence, we chose to use 360 eV as a plane wave energy cutoff.Figure A.1: The detailed supercell setup along with the results for convergence with respect tok-point sampling and plane wave energy cutoff is shown in (a) and (b), respectively.163A.2 Procedure to create ten different configurations ofalloy GB structures in Ti - 25 at.% Mo alloyThe GB structures were constructed by mirroring a random bulk slab at the (310) plane (y-direction in Figure A.2) to increase the symmetry at the interface. This leads to a stabilizationof GB and reduces the potential for local reconstructions. It is important to note that the GBsupercells always exhibit a fixed global alloy composition. Only locally (e.g. in the GB region),a deviation from the fixed alloy composition is possible. The mentioned 10 configurations(obtained by mirroring and joining bulk supercells at 0 to 9 y-planes labelled in Figure A.2)represent available configurations within the scope of the present study, where a single 60-atombulk slab was utilized and mirrored at various (310) planes.Figure A.2: Numbering of planes in the y-direction of the bulk slab of the Σ5 GB, 10 differentplanes lead to 10 different GBs. Grey and red atoms refer to Ti and Mo sites, respectively.A.3 Atomic structures and GB properties of five stablealloy GB configurations in Ti -25 at.% Mo alloyGB energy and GB excess volume are calculating using the following expressions, respec-tively:𝛾GB = [EGB(90, 30, 0) − 90 · ETi,alloy − 30 · EMo,alloy]/(2 · AGB) (A.1)𝛿𝑉 =(VGB,equlibrium − VGB,as-created) /(2 · AGB) (A.2)In Equation (A.2), VGB,as-created refers to the initial volume of the GB before allowing for ad-ditional relaxation in the direction perpendicular to the GB plane. Some of the above quantitiesare listed below:164VGB,as-created = 1988.9396 Å3AGB = 97.8945Å2Table A.1: GB energy (𝛾𝐺𝐵), GB excess volume (𝛿𝑉), Mo content at GB region and GB planefor each of the five stable Ti - 25 at.% Mo alloy GB configurationsConfigurationGB energy,𝛾GB (mJm−2)GB excessvolume,𝛿𝑉 (Å3Å2)Mo content inthe GB region(at.% Mo, outof 60 atoms)Mo content atGB plane(at.% Mo, outof 12 atoms)1 895 0.3194 20 16.62 (GB-I) 1100 0.3156 23.3 33.33 (GB-II) 994 0.3541 23.3 504 1083 0.3084 26.7 33.35 1011 0.3312 30 33.3165The GB atomic structures visualized with OVITO are shown in the figure below:Figure A.3: Atomic structures of all five stable GB structures of configuration 1, 2, 3, 4 and 5(from Table A.1) shown in (a), (b), (c), (d) and (e), respectively. Grey and red atoms refer toTi and Mo sites, respectively.166Appendix BStructure of GBs for Ti - 50 and 75at.% Mo alloysThis section covers the procedure to construct all ten GBs that are used to study Y segregation.As mentioned before, a 60-atom alloy supercell is first generated using ATAT program and arepresentative alloy structure for 50 and 75 at.% Mo alloys is created. Those bulk structuresare shown in Figure B.1(a) and (b) for 50 and 75 at.% Mo alloy. Each plane of atoms in they-direction are used to create a mirrored-bulk structure for creating a symmetric tilt GB. Thereare ten such planes and thus there are ten GBs containing 120 atoms for each structure.Figure B.1: Alloy structure containing 60 atoms for (a) 50 at.% Mo and (b) 75 at.% Mo alloy.Each y-plane is highlighted using a blue arrow and a number that will used to distinguishdifferent GB that are constructed. Red colored and grey colored atoms correspond to Mo andTi species, respectively.167Table B.1: Number of Mo (N𝑀𝑜) and Ti sites (N𝑇𝑖) at GB plane for each of the ten , denotedby GB#, in Ti-50 at.% Mo alloy GBsGB# 𝑁Mo 𝑁TiGB_0 3 3GB_1 4 2GB_2 4 2GB_3 4 2GB_4 1 5GB_5 2 4GB_6 2 4GB_7 2 4GB_8 4 2GB_9 4 2B.1 Information about GB and its structure for Ti - 50at.% Mo alloysUsing Figure B.1(a) and the mirror approach to construct alloy GBs, all ten GBs for 50 at.% Moalloy are shown in Figure B.2. The number of Mo and Ti at each GB for this alloy compositionis tabulated in Table B.1.168Figure B.2: Visualization of atomic structure of 50 at.% Mo alloy GBs that are labelled inthe text as (a) GB_0, (b) GB_1, (c) GB_2, (d) GB_3, (e) GB_4, (f) GB_5, (g) GB_6, (h)GB_7, (i) GB_8 and (j) GB_9. Red colored and grey colored atoms correspond to Mo and Tispecies, respectively.B.2 Information about GB and its structure for Ti - 75at.% Mo alloysWith the structure from Figure B.1(b), ten alloy GBs are constructed for 75 at.% Mo alloyand shown in Figure B.3. The number of Mo and Ti at each GB for this alloy composition istabulated in Table B.2.169Table B.2: Number of Mo (N𝑀𝑜) and Ti sites (N𝑇𝑖) at GB plane for each of the ten , denotedby GB#, in Ti-75 at.% Mo alloy GBsGB# 𝑁Mo 𝑁TiGB_0 5 1GB_1 3 3GB_2 4 2GB_3 4 2GB_4 6 0GB_5 5 1GB_6 5 1GB_7 4 2GB_8 5 1GB_9 4 2Figure B.3: Visualization of atomic structure of 75 at.% Mo alloy GBs that are labelled inthe text as (a) GB_0, (b) GB_1, (c) GB_2, (d) GB_3, (e) GB_4, (f) GB_5, (g) GB_6, (h)GB_7, (i) GB_8 and (j) GB_9. Red colored and grey colored atoms correspond to Mo and Tispecies, respectively.170Appendix CSimulation settings for constructingtraining set in VASPSince the VASP code is subject to different versions, it is noted that all calculations wereperformed with version 5.4.4, which was released on 18 April 2017. Four choices need to bemade: (i) Energy cutoff (ENCUT tag in INCAR [208]) for truncating the plane wave basis, acutoff of 350 eV is used, (ii) the stopping criterion for electronic convergence, this is an energy-based cutoff (as ions are not moved) and set to 10−6 eV. Third, the smearing width parameter,𝜎, helps to speed convergence without losing accuracy, see here [209] for more details . In thischapter, this is chosen to be 0.1 eV instead of the default 0.2 eV, this is because the two outputs,“F” and “E0” (when sigma->0), are reasonably close to each other. Typically, E0 is used tocalculate thermodynamic quantities, but this is not true for training surrogate models usingVASP. This choice of 𝜎 means “F” can be taken as the energy observable, which is required sinceonly F is differentiable with respect to atomic positions to give accurate Hellmann-Feynmanforces. The fourth choice is the spacing of k-mesh in each direction, this is KSPACING = 0.2Å−1, as described in the VASP manual [210] and another recent MLIP study [211, 212]. It isimportant to note that this is different from the k-point density reported (a factor of 2𝜋) inChapter 4 and Chapter 5. The final choice for creating DFT data for surrogate models is thechoice of XC functional, which is PBEsol for Pt, Au and Pt-Au alloys.171Appendix DJulia programming scripts for ACEmodelsThis section provides an example script written in Julia programming language to create anACE model and a script to use the generated ACE model to perform geometry relaxation.D.1 A Julia script using ACEpotentials.jl for buildingregression modelsThe main steps behind this script are:(i) Activate the Julia environment which has the appropriate packages to build ACE models(ii) Define the weights and load the training data(iii) Set committee size and single atom reference energies(iv) Construct the ACE model basis by supplying correlation order, totaldegree and cutoffradius(v) Choose a solver and solve for coefficients(vi) Print out a training and testing error table(vii) Export the resulting model for predictions on other structuresA typical script used for building ACE models is provided below:,1 using Pkg2 Pkg.activate("/home/ACE_run")1723 using Distributed4 # running on 8-cores5 addprocs(7, exeflags="--project =$( Base.active_project ())")6 @everywhere using ACEpotentials7 using ACEpotentials8 using ExtXYZ9 train = read_extxyz("training_set.xyz")10 weights = Dict("default" => Dict("E" => 20.0, "F" => 4.0, "V" => 1.0))11 c_size = 10012 tdeg = 1613 # use "F" (free_energy label) instead of "E0" from VASP14 data_keys = (energy_key = "free_energy", force_key = "force",virial_key = "virial")15 # single atom energies computed using PBEsol16 au_single = -.15106143E-0117 pt_single = -.32912649E+0018 # define model hyperparameters19 model = acemodel(elements =[:Pt, :Au],20 order = 3,21 totaldegree=tdeg ,22 rcut =6.5,23 Eref =[:Pt => pt_single , :Au => au_single ])24 # set a smoothness prior25 P = smoothness_prior(model; p = 4)26 # set solver as BLR27 solver = ACEfit.BLR(committee_size = c_size , factorization =:svd);28 # perform ACE fit29 acefit !(model , train , prior=P; solver=solver , weights=weights ,data_keys ...);30 # save the model in json and LAMMPS format31 save_potential("model_save.json",model)32 export2lammps("PtAu.yace",model)33 # compute training set errors34 error_tab = ACEpotentials.linear_errors(train , model ,weights=weights);35 # save training set errors in a txt file36 open("training_errors.txt", "w") do file37 println(file , "Training Error Table")38 println(file , "$error_tab.")39 end173D.2 A Julia script for geometry minimization in JuLIP.jlusing a ACE modelThis script loads a ACE model and creates vacancies at different sites in a GB. It alsocontains a workflow on creating directories and storing relevant information from the geometryoptimization for post-processing and is shown below:,1 using Pkg2 Pkg.activate("/home/ACE_run")3 using JuLIP4 using ACEpotentials5 using ExtXYZ6 using Statistics7 using CSV8 using DataFrames9 # load ACE model10 mdl = read_dict(load_dict("model.json")["IP"])11 # read initial structure12 struct1 = read_extxyz("gb.xyz")[1]13 # set the calculator for the structure14 set_calculator !(struct1 , mdl)15 # define gtol values and corresponding labels16 gtols = [5e-1, 3e-1, 1e-1, 5e-2, 1e-2]17 labels = ["pos_relaxed_5e -1", "pos_relaxed_3e -1", "pos_relaxed_1e -1","pos_relaxed_5e -2", "pos_relaxed_1e -2"]18 # perform vacancy creation , minimization , and save the structures19 for site in 1: length(struct1)20 # create a vacancy21 struct_vac = deepcopy(struct1)22 deleteat !(struct_vac , site)23 # set the calculator24 set_calculator !( struct_vac , mdl)25 # initialize dataframe26 df = DataFrame(Filename=String[], StdDev=Float64[], Energy=Float64[])27 structures = JuLIP.Atoms{Float64 }[] # Ensure correct type28 # perform the minimization and save the structures29 for (i, gtol) in enumerate(gtols)30 minimise !(struct_vac , gtol=gtol)17431 energy = ACE1.co_energy(mdl , struct_vac)[1]32 stddev = std(ACE1.co_energy(mdl , struct_vac)[2])33 push!(df , (Filename=labels[i], StdDev=stddev , Energy=energy))34 push!( structures , deepcopy(struct_vac))35 end36 # create a directory37 dir_name = "site_$site"38 mkdir(dir_name)39 # convert structures to an Array of JuLIP.Atoms{Float64}40 structures_array = Array{JuLIP.Atoms{Float64}, 1}( structures)41 # save all the structures to a single file42 write_extxyz("$dir_name/iter1.extxyz", structures_array)43 # save the stddev , filenames , and energies to a CSV file44 CSV.write("$dir_name/stddevs.csv", df)45 end175
Thesis/Dissertation
eng
Materials Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Militzer, Matthias
Atomistic simulations of crystalline defect interactions in metallic alloys
Text