The goal of this book is to introduce the basic methods used in the computational modeling of materials. The text reflects many tradeoffs: breadth versus depth, pedagogy versus detail, topic versus topic. The intent was to provide a sufficient background in the theory of these methods that the student can begin to apply them to the study of materials. That said, it is not a “computation” book – details of how to implement these methods in specific computer languages are not discussed in the text itself, though they are available from an online resource, which will be described a bit later in this preface.

Modeling and simulation are becoming critical tools in the materials researcher's tool box. My hope is that this text will help attract and prepare the next generation of materials modelers, whether modeling is their principal focus or not.

Structure of the book

This book is intended to be used by upper-level undergraduates (having taken statistical thermodynamics and at least some classical and quantum mechanics) and graduate students. Reflecting the nature of materials research, this text covers a wide range of topics. It is thus broad, but not deep. References to more detailed texts and discussions are given so that the interested reader can probe more deeply. For those without a materials science background, a brief introduction to crystallography, defects, etc. is given in Appendix B.

Molecular dynamics provides a way to model the dynamical motion of atoms and molecules by calculating the force on each atom and solving the equations of motion. In this chapter, we apply the same approach to the motion of entities other than atoms. These entities will typically be collected groups of atoms, such as dislocations or other extended defects. The first step will be to identify the entities of interest, to determine their properties, and then to calculate the forces acting on them. By following similar procedures as in molecular dynamics, the equations of motion can then be solved and the dynamics of the entities determined.

The principal focus of these types of simulations is the mesoscale, that region between atomistics and the continuum, and the goal is often the determination of the microstructure. These extended defect structures are typically many μm in scale and are thus beyond what can generally be studied atomistically. It is not just the length scale that limits the applicability of atomistic simulations to microstructural evolution. The time scales for microstructural evolution are also much much longer than the nanoseconds of typical molecular dynamics simulations. The defects in question could be grains and the questions of interest could be the growth of those grains and their final morphology. One could also be interested in determining the development of dislocation microstructure and its relation to deformation properties. There, the dislocations might be the entities of interest.

This text is focused on the modeling of materials structure and properties. The language and choice of problems and methods reflects the interests of the materials science and engineering (MSE) community. We realize, however, that there is increased interest in these problems from people in fields outside MSE. The purpose of this chapter is to give a rapid overview of materials science strictly from the point of view of what is covered elsewhere in the text. It is certainly not a comprehensive introduction to materials.

INTRODUCTION

Materials in use are solids and most, but certainly not all, are crystals, by which we mean systems of atoms that have a regular, periodic structure. Few materials in actual use, however, are perfect crystals. Most have defects, imperfections in their lattices that have a profound effect on the overall properties of those materials. These defects may be point defects, such as vacancies, line defects (typically dislocations), or planar defects, such as surfaces or interfaces between two crystals. The distribution of those defects is referred to as a materials microstructure. Understanding the evolution of the microstructure as well as its role in determining overall properties is a major thrust of materials modeling and simulation.

In this chapter, we introduce basic crystallography of simple crystals, as well as how to represent that crystallography in calculations. We then discuss the defects of those materials and the ramification of those defects on materials properties. We also emphasize the role of dynamic processes, such as diffusion, on materials.

In this appendix we review some of the basic ideas and methods behind quantum mechanics. This brief treatment is meant only to introduce the reader to this important subject. A number of elementary texts are listed in the Suggested reading for those who would like to go further into this fascinating field.

HISTORY

Quantum mechanics arose from an attempt to understand discrepancies between predictions of classical mechanics and observed (experimental) behavior. Around 1900, there was increasing recognition that some phenomena could not be understood based on classical physics. One of these problems was blackbody radiation, i.e., the glow that is given off by a heated object which is an indicator of its temperature. Planck came up with an explanation for blackbody radiation in a cavity, but had to describe the energetics of the system as consisting of oscillators whose energy was quantized (i.e., integer multiples of some quantity). In 1905, Einstein took that idea one step farther and proposed that electromagnetic radiation (i.e., light) is itself quantized as an explanation of the photoelectric effect. We now call these quanta of light photons.

One of the other main failures of classical theory was its inability to explain the spectrum of hydrogen, which has distinct lines. One of the most important results from the quantum mechanical description of the H atom, and of all matter, is that quantum systems have states with discrete energy levels (not continuous as in classical mechanics). Transitions of electrons between these discrete levels lead to the observed spectra of the H atom and other atoms and molecules.

An automaton is defined as a “a mechanism that is relatively self-operating; especially: robot” or as a “machine or control mechanism designed to follow automatically a predetermined sequence of operations or respond to encoded instructions” [226]. A classic cellular automaton is like an algorithmic robot. The cellular automata method describes the evolution of a discrete system of variables by applying a set of deterministic rules that depend on the values of the variables as well as those in the nearby cells of a regular lattice. Despite this simplicity, cellular automata show a remarkable complexity in their behavior.

Cellular automata have been used to model a number of effects in materials, mostly recrystallization, corrosion, and surface phenomena, with other applications ranging from hydration in cement to friction and wear, many of these applications being discussed below. Numerous applications extend classic cellular automata to include probabilistic rules, more complex lattice geometries, and longer-ranged rules. With the use of probabilistic rules, the distinction between cellular automata methods and Monte Carlo methods become a bit blurred, as will be discussed below.

In this chapter we will introduce the basic ideas behind cellular automata, using as examples some of the classic applications of the method. We will then go through a few applications of the methods to materials issues, highlighting the power of the method to model complex behavior. Much more detail about a range of applications, both in materials research and elsewhere, can be found elsewhere [68, 268, 269].

In almost all methods used to model materials, the system will be described by a set of discrete objects of some sort. Those objects might be atoms and the goal may be to calculate the cohesive energy by summing interatomic interaction potentials. The objects do not have to be atoms, however. We may want to sum the interactions between spins or dislocations or order parameters or whatever. Learning how to calculate these sums is thus fundamental to essentially all materials modeling and simulation.

In modeling a material we typically face a rather major complication – we are trying to model a macroscopic system that contains large numbers of objects. For example, a bulk sample of a material may include many moles of atoms. Modeling the behavior of all those atoms would be computationally impossible. To approximate the (effectively) infinite systems, we use various boundary conditions, mostly based on introducing a repeating lattice. How one sums the interactions between the objects within the framework of these boundary conditions is the focus of this chapter.

SUMS OF INTERACTING PAIRS OF OBJECTS

We will often encounter systems that consist of objects that interact with each other in some way. The classic example is the cohesive energy of a solid, which is determined from the sum of the interactions between the constituent atoms and molecules. The simplest case is when the interactions occur only between pairs of objects and depend only on the distance between the pairs.

Before introducing more complex methods, we start with a model of a fundamental materials process, the random-walk model of diffusion. The random-walk model is one of the simplest computational models in materials research and thus can help us introduce many of the basic ideas behind computer simulations. Moreover, despite its simplicity, the random-walk model is a good starting place for describing one of the most important processes in materials, the diffusion of atoms through a solid.

RANDOM-WALK MODEL OF DIFFUSION

Diffusion involves atoms moving from site to site under the influence of the interactions with the other atoms in the system. An atom typically sits at a site for a time long compared with its vibrational period and then has a rapid transit to another site, which we will refer to as a “jump”. To describe that process properly requires much more detail than we now have in hand (though we shall rectify that situation somewhat in the forthcoming chapters). Thus, we will take a very simple model that ignores all atomic-level details and that focuses just on the jumps.

Consider the simple example of a single atom moving along a surface, which we will assume consists of a square lattice of sites with a nearest-neighbor distance of a. Diffusion occurs by a series of random jumps from site to site in the lattice, as shown in Figure 2.1.We can understand the basic physics by considering the energy of the interaction between the diffusing atom and the underlying solid, which we show schematically in Figure 2.2a.

In this chapter we introduce the Monte Carlo method, a remarkably powerful approach that is the basis for three chapters in this text. For the purposes of this chapter, Monte Carlo provides an alternative to molecular dynamics for providing thermodynamic information about a material. It differs from molecular dynamics in that it is based on a direct evaluation of the ensemble average, as discussed in Appendix G, and thus cannot yield direct dynamical information, at least as described for the version of Monte Carlo in this chapter.

The Monte Carlo method was devised at Los Alamos in the 1940s to solve multidimensional integrals and other rather intractable numerical problems [227]. The method is based on statistical sampling and is called Monte Carlo in recognition of the very famous casinos there. It is not called Monte Carlo because of gambling – at least not entirely – it is named Monte Carlo at least in part because of its remarkable ability to solve intractable problems.

INTRODUCTION

What is the Monte Carlo method? As first employed, it was a way to solve complicated integrals. As a simple example, Monte Carlo is used to evaluate the one-dimensional integral 1n(x)dx in Figure 7.1a. First, a region that includes the function to be integrated is defined. Random points in that region are chosen via a random-number generator. The integrated function is just the fraction of the points that fall below the curve multiplied by the area of the sampled region.

In Chapter 7, the basic ideas of the Monte Carlo method are presented and demonstrated on two types of problems, the Ising model and simulations of atomic systems. In Chapter 8, applications to molecular systems, including polymers and biomolecules, are discussed. The Ising model does not really fit into the theme of Chapter 7, which is included in the part of the text focused on atomic and molecular scale simulations. The Ising model really belongs in the current chapter, which focuses on the use of the Monte Carlo method to simulate models of materials behavior at the mesoscale. However, the Ising model is just too good an example of the Metropolis Monte Carlo method, so we included it in Chapter 7. Consider its natural home, however, here.

There are many types of Monte Carlo applications that have been developed to model material properties and response. We shall examine only one of them in detail, the Q-state Potts model as applied to grain growth. Not only has the Potts model had wide applicability and impact in materials research, it reflects most of the issues faced in any mesoscale modeling based on the Monte Carlo method.

With the development of inexpensive, yet very fast, computers and the availability of software for many applications, computational modeling and simulation of materials has moved from being entirely in the hands of specialists to being accessible to those who use modeling not as their principal activity, but as an adjunct to their primary interests. With that change in accessibility of materials modeling and simulation come exciting new opportunities for using computational modeling to greatly advance the development and refinement of materials and materials processing.

The goal of this text is not to make experts – there are entire books on subjects that are treated in a few pages here. The text is, by design, introductory and we leave out many, if not most, details about implementation. We will present the key features and possibilities of computational materials science and engineering and discuss how to use them to advance the discovery, development, and application of materials.

MODELING AND SIMULATION

Before we start discussing materials modeling and simulation, it is appropriate to consider those words a bit more carefully. What do we mean by a “model” or a “simulation”? How are they different? Not to be overly pedantic, but it may help our discussion if we are a bit more precise in our definitions of these terms.