Discovery and Implementation of fast, accurate, and transferable Many-body Interatomic Potentials by Adarsh Balasubramanian A thesis submitted to The Johns Hopkins University in conformity with the requirements for the degree of Master of Science and Engineering Baltimore, Maryland December, 2019 © 2019 by Adarsh Balasubramanian All rights reserved Abstract This thesis deals with discussions on the motivation and approach for discovering new interatomic potential models, and methods of implementing new interatomic potentials in LAMMPS, followed by results obtained from caulculating physical properties like stacking fault energies, surface energies etc. The length and time scales of atomistic modeling approaches are limited by the com- putational cost of the methods used to predict material properties. In recent years there has been great progress in the use of machine learning algorithms to develop fast and accurate interatomic potential models, but it remains a challenge to come up with models that generalize well and have good performance capabilities. In this thesis, I shall discuss my contributions towards addressing this challenge. First, I shall provide an overview of molecular dynamics and the role played by inter- atomic potential models in the same for accurately determining physical properties like energies and forces. Following this, I shall provide an outline of the symbolic regression approach, the software developed for carrying out the interatomic potential evolution- ary search, and the methodologies that have been used for discovering the interatomic potential models. Then, I shall present the forms of the discovered models for copper, take a look at their computational aspects and walk through at the assessments of their speed and overall complexity by detailing their LAMMPS implementations. I also discuss in-depth about the LAMMPS internals and the various considerations to look at while implementing potential models in LAMMPS. Finally, with the model implementations in hand, I make attempts at validating the same by computing stacking fault energies, surface energies and present other relevant properties, and compare them against other popular interatomic potential models. Advisor: Dr. Tim Mueller ii Acknowledgments I take this opportunity to thank all the people who have supported and helped me both in this work and on my path to this point. I am profoundly grateful to Dr. Tim Mueller for his continuous encouragement and expert guidance throughout the project to ensure that this project meets its milestones right from its commencement to its completion. His exceptional insight and broad view led me to the wonderful field of atomistic modeling and helped me avoid so many detours during the progress of my research. I would also like to express deepest appreciation towards Professor Mueller for having provided support and accommodations throughout the duration of my masters programme, which helped me drive the program to fruition. Next, I would like to express my deepest gratitude to my research colleague Alberto Hernandez who spearheaded this project on discovery of interatomic potentials and validated the work in this thesis by calculating the required properties and comparing them against previous work. His advice and suggestions have always helped me find the right direction and go through research difficulties, and without his help, I would not have achieved what I have today. I also take this opportunity to thank all my research colleagues at the Mueller group for having provided ideas and feedback throughout the project. I also would like to acknowledge the high-performance computing resources from the Maryland Advanced Research Computing Cluster (MARCC) and from the Homewood High-Performance Cluster (HHPC), which helped me perform voluminous amounts of simulations and obtain results for research. There are so many other great people at Johns Hopkins who have helped me in the past few months that I cannot list all of them. Special mention to the faculty and staff of Materials Science and Engineering who, through the months of my education, provided a stimulating atmosphere for leaning. Adarsh Balasubramanian iii Preface Scientific Articles Fast, accurate, and transferable many-body interatomic potentials by symbolic regres- sion Alberto Hernandez, Adarsh Balasubramanian, Fenglin Yuan, Simon A. M. Mason & Tim Mueller npj Computational Materials volume 5, Article number: 112 (2019) URL: https://www.nature.com/articles/s41524-019-0249-1 Contribution to paper: I worked on implementing the newly discovered models in LAMMPS, assisted in open-sourcing the software and generated data for obtaining a portion of the results. Scientific Work not part of the thesis Algorithms and Code for the Generation of Generalized Monkhorst-Pack Grids Yunzhe Wang, Pandu Wisesa, Adarsh Balasubramanian & Tim Mueller Submitted to Modelling Simul. Mater. Sci. Eng. Preprint: https://arxiv.org/abs/1907.13610 Contribution to paper: I worked on the initial release of kpLib: the C++ implementa- tion of the algorithm for efficient generation of Monkhorst-Pack grids, and provided integrations with spglib. iv Table of Contents Table of Contentsv List of Tables vii List of Figures viii 1 Introduction1 2 An Overview of Molecular Dynamics3 2.1 Introduction . .3 2.2 Interatomic Potentials . .4 2.2.1 Lennard-Jones Potential . .5 2.2.2 Embedded Atom Method . .5 2.3 Integration Schemes . .6 3 An Overview of the Methodology for Interatomic Potential Discovery8 3.1 Motivation for our approach . .8 3.2 A brief overview of Genetic Programming . .9 3.3 Description of the Symbolic Regression approach . 10 4 Computational Details and Methods of Implementation 12 4.1 LAMMPS and the manybody package . 12 4.2 A peek into the core LAMMPS internals . 12 4.2.1 Setup of the MD environment . 12 4.2.2 Potential Model Design . 14 4.2.3 LAMMPS Optimizations for minimizing unnecessary computations 16 v 4.3 Details of GP1 and GP2 implementations in LAMMPS . 17 5 Results 19 5.1 Discovering new models for Copper . 19 5.2 Stacking Fault Energy Calculations . 21 5.2.1 Mechanistic insights of stacking fault formation in FCC metals . 22 5.2.2 Calculating Stacking Fault Energy with LAMMPS . 23 5.2.3 Stacking fault results for GP1 and GP2 . 24 5.3 Surface Energy Calculations . 25 5.3.1 Calculating Surface Energies with LAMMPS . 26 5.3.2 Surface Energy Results for GP1 and GP2 . 26 5.4 Other Property Calculations for Copper using GP1 and GP2 . 27 5.5 Benchmarking results . 29 5.5.1 Serial Performance . 29 5.5.2 Parallel Performance . 30 6 Discussions and Future Work 32 6.1 Other implementations . 32 6.2 Future Work . 33 6.3 Code Availability . 34 Appendix 35 A Useful information on LAMMPS 35 A.1 Anatomy of the LAMMPS EAM funcfl file . 35 A.2 Tips and tricks on creating potential models in LAMMPS . 36 A.3 Enabling GP1, GP2 and GP3 in LAMMPS . 37 B Useful information on POET 38 B.1 Details of the training data for running POET on Cu . 38 B.2 Description of the potential models . 39 vi List of Tables 5.1 Prediction errors for the intrinsic stacking fault (gISF) energy and the unsta- ble stacking fault (gUSF) energy. (a) ab initio target data. (b) experimental target data. 25 5.2 Prediction errors for surface energies. (a) ab initio target data. (b) experi- mental target data. 26 5.3 Error of the values predicted by interatomic potentials for copper relative to the respective reference. Details about EAM, ABCHM, CuNi, EAM1 are available in the Appendix. Cij are elastic constants, a0 is the lattice parameter, DE (bcc-fcc) is the energy difference between bcc and fcc phases, Ev is the fcc bulk vacancy formation energy, Ev (unrelaxed, 2*2*2) is the unrelaxed vacancy formation energy computed on a 2*2*2 supercell, Em is the migration energy for fcc bulk vacancy diffusion, Ea is the activation energy for fcc bulk vacancy diffusion, Edumbbell is the dumbbell <100> forma- tion energy, g is the phonon frequency, and gISF and gUSF are the intrinsic and unstable stacking fault energies, respectively. a SC was fit to the bulk modulus. b Elastic constants were used to select GP1, GP2, and GP3 from c d the convex hull. Fit to ensure that Ef cc < Ebcc and Ef cc < Ehcp. Fit to vacancy formation energy. 28 A.1 Line-by-line description of a funcfl file. 35 B.1 Acronyms used for the Interatomic Potential Models . 39 vii List of Figures 2.1 Simplified Schematic of the molecular dynamics algorithm . .7 3.1 Tree graphs of a) Lennard-Jones potential parametrized for argon, equation , b) Sutton-Chen EAM potential parametrized for copper, equation , c) GP1 and d) GP2. .9 3.2 Example of a crossover operation involving combining the pink subtree with the blue subtree to form a new individual. 11 4.1 A pictorial representation of the LAMMPS internal treatment of atoms. The blue circles denote the primary atoms, and the gray atoms denote the ghost atoms. Atoms inside the red circle denote the neighbors of the central red border atom, and these atoms contribute primarily to the calculations of the energies and forces of the atom in consideration. 13 4.2 A pictorial explanation of the LAMMPS interpolation procedure. Splines are interpolated from the EAM data file which are then used for calculating salient properties for the given system of atoms. 15 4.3 A pictorial explanation of the terms involved in the force calculation. As- suming the red outline atom corresponds to atom i and the pink to atom j, Fi will have contributions from dEi/drj as well as dEj/dri........... 16 4.4 A pictorial representation of Forward Communication which involves com- municating values computed in the primary atoms to the ghost atoms to avoid repetitive calculations. 17 5.1 The three-dimensional convex hull of models found by the machine-learning algorithm.