Computational Materials

My research work is in computational materials at the University of Florida. Rather than keeping technical notes in a notebook, I've taken electronic notes and posted some of my notes online. None of the notes are particularly formal. Some of the information on this site is theoretical, some is technical, and some is procedural. None of it is particularly organized.

My research work is primarily in computational materials. Some of the information in this section is from Dr. Susan Sinnott's class in Computational Materials taken in Fall 2013. Some of it is from Dr. Richard Hennig's class taken in Fall 2016. I will incorporate notes which will eventually form my dissertation. However, this section is more of a philosophy of computational materials. I'm hard-working, but in a way which allows me to be lazy later. I try to automate a good portion of my work, the bulk of this work is currently in pyflamestk, which I developed at the University of Florida. However, upon graduation, I will probably fork the code to support the research I like to do, as well as automate tasks.

Note on Preliminaries

The notes contained here are more theoretical in nature, in as it explains the formulas and concepts in computational materials science, as it was written predominantly as notes for my dissertation, and provided here as a public reference.

Computational materials require some familiarity with technical computing.  If you do computational materials, much of your frustration will not be related to the problems with with the theory of DFT or molecular dynamics, but with the technical side of things.

In addition, other topics such as MPI clusters, programming, and compilation will be an issue and often much of the information which can be found on the internet, which is written for people who control their computing environment, a situation not applicable in most computing environments.

I've organized a lot of this information that I collected and added some information to make this a computational materials science self-study bootcamp, which assumes very little on your part so that you can become productive fairly quickly in using computational materials science.

Computational Materials Science Bootcamp

Introduction

The increase in computational power available the past decades has led to a birth of a new way of doing science which involves the intersection of theory with experiment. Referred to collectively as computational materials science, they allow for computer experiments under perfectly controllable and reproducible conditions. Through the use of numerical routines to extend the reach of theory, these computational experiments can explain physical phenomena and can be useful in guiding experimental work.

In addition it is even possible to predict new phenomena by conducting experiments in silico that otherwise would be too difficult, expensive, or simply impossible to perform. Most recently, these computational techniques have been used to predict properties of materials by exhaustive search of different combinations of chemical species into a finite set of space groups. This has been extended to searches in 2D materials for specific applications. These techniques are exclusively the domain of Density Functional Theory.

However, the by far most rewarding outcome of computer simulations is the invaluable insight they provide into the behavior and the dynamics of a system. The high computational costs of ab-initio molecular dynamics is punishing due to the cost of solving the Kohn-Sham equations, the length of simulation time, and the size of features that are often most interesting.

The development of interatomic potentials has been stymied over the past decade due to the long research times required to develop an interatomic potentials. Without an adequate methodology to develop new potentials, software to automate tasks required to develop, select, and test an interatomic potential, the use of molecular dynamics to predict kinetic behavior in silico will not reach it's full potential.

Structure of this Document

Crystallography
Molecular dynamics
Density Functional Theory
Conventional Process for Developing Potentials
Lattice Dynamics
Phase Field Models

Introduction to Crystallography

The work contained here breaks the problem from theoretical building blocks necessary to understand the process of developing new potentials. This starts rather pedantically from a rather simple approach to crystallography in developing the crystal structures used in computational materials from an applied mathematical perspective where one is interested in for making computational simulations. This material is provided first to establish a consistent mathematical notation throughout this work, but will be reintroduced as necessary. This work is subject to some mathematical notation abuse which will be explained when necessary.

The most fundamental concept of materials science is that the structure of the material affects the property of the material. In computational materials, particularly atomistic simulations, this is especially true. Most material properties can be described by the relationship between crystal structure and energy.

Building crystal structures
Building bulk structures

Density Functional Theory (DFT)

Although the discussion of Molecular Dynamics typically precedes Density Functional Theory, our discussion of computational materials will first start with the first principals starting with quantum mechanics and the solution of the Schroedinger's equation and different computational approaches to solving the Schroedinger's equation. I will try to keep the discussion of quantum mechanics at a conversational level, and will forego proofs and experimental consequences of statements for the purposes of compactness to motivate the discussion.

The explaination density functional theory (DFT) starting from the application of the Born-Oppenheimer approximation to the Schoedinger's equation to turn a multi-atom system in a multi-electron system and describe the evolution of quantum mechanical approaches which leads to the primary ab-initio engine used throughout this work, plane augmented wave density functional theory. In this approach, the one-to-one representation of the ground state wavefunctions of the Hamiltonian given by the Born-Oppenheimer approximation is established through the Hohenberg-Kohn theorems, and solved numerically by solving the Kohn-Sham equations using a plane wave basis set.

Molecular Dynamics (MD)

We then motivate the discussion through molecular dynamics and the strength in molecular dynamics in solving dynamical systems by sampling of the Boltzmann distribution through a variety of thermodynamic ensembles by looking at the Carr-Parrinello method. Due to the computational cost of a DFT calculation between the updates in the Carr-Parrinello method, the need for empirical interatomic potentials becomes motivated. We discuss different classes of interatomic potentials.

This is moved to a different section. Here.