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.

Wordiness and redundancy

Wordiness is the legal profession's most recognisable trait, redundancy its chief characteristic. Lawyers really do go on. Their motto might be: ‘Never use one word where you can use two; and the more you use, the better.’ As an American judge has put it: ‘The legal mind finds magnetic attraction in redundancy and overkill.’

Wordiness and redundancy are seen most often in common pairings like null and void, goods and chattels, fit and proper, storm and tempest, well and sufficiently, agreed and declared. (We discussed the tautological nature of some common pairings in chapter 2.) But they are also seen in more lengthy and ambitious forms of repetition. Consider the following typical lease provision, setting out some of the tenant's rights:

Traditional legal language

The English language of today is still recognisably the language of Chaucer and Shakespeare, of Abraham Lincoln and Winston Churchill, of the Book of Common Prayer and the Authorised Version of the Bible. It is also the language of lawyers in many countries: the United Kingdom, Ireland, the United States of America, Canada, India, Australia, New Zealand and Singapore, to name but a few. In English, lawyers draft documents and compose letters, formulate statutes and propagate regulations, prepare pleadings and argue their cases.

Legal English, however, has traditionally been a special variety of English. Mysterious in form and expression, it is larded with law-Latin and Norman-French, heavily dependent on the past, and unashamedly archaic. Antiquated words flourish – words such as aforementioned, herein, therein and whereas, which are rarely now heard in everyday language. Habitual jargon and stilted formalism conjure a spurious sense of precision – the said, the aforesaid, the same. Oddities abound: oath-swearers do not believe something, they verily believe it; parties do not wish something, they are desirous of it; the clearest photocopy only purports to be a copy; and so on. All this, and much more, from a profession that regards itself as learned.

In the preceding chapters we examined the influences that tend to perpetuate the traditional style of legal drafting. We also considered the ways in which legal documents are interpreted. We traced the move towards plain legal language and explored some of the benefits of using modern, standard English in legal documents. Now we move to discuss rather more closely the techniques of drafting in modern, standard English. This discussion we divide into three parts: issues of structure and form (the subject of this chapter); issues of particular significance or difficulty for legal drafters (chapter 6); and issues with words and phrases (chapter 7).

In this chapter, then, we deal with structure and form. Legal drafters traditionally pay little attention to this topic.

Document structure

Structure and form are crucial to an effective, readable legal document. The contents of a legal document should be consciously ordered to enable it to be read as quickly and efficiently as the subject-matter will allow. To achieve this, the document must be ordered logically – by which we mean logically fromthe reader’s perspective. Each clause and paragraph should be presented in a way that is both sensible and comprehensible to the reader.

Three logical structures

To require all legal documents to adopt a common structure would be illogical. Each transaction has its own key elements, and each client is different.However,we venture to suggest that most transactional documents could logically followone of three structures.They are: telescoping (or frontloading); thematic; or chronological. Let us consider each in turn.

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].