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].

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.

The need for change

We begin with a proposition that underlies all we have said so far in this book: legal documents should be written in modern, standard English – that is, in standard English as currently used and understood. Identifying modern, standard English is not difficult. It can be found in articles in the more serious newspapers, in popular and academic books on many subjects, and in reports of governments and public authorities. Its hallmark is a style that is direct, informative and readable.

This exhortation to write in modern, standard English is hardly new. For many traditional lawyers, however, the change seems intimidating. It forces them to rethink, and rethinking takes time. At the most basic level, they must switch from stilted, archaic words and phrases to modern, idiomatic equivalents. Instead of such or the said, they must write the; instead of in the event that, they must write if; instead of notwithstanding the fact that, they must write despite.

In earlier chapters we discussed the many benefits of drafting legal documents in modern, standard English. We also discussed the many techniques for drafting legal documents in that style. In the course of those chapters we considered before-and-after examples to illustrate particular points. Panel 13 lists the examples and where they can be found.

The literature on plain language contains many other before-and-after examples.

In this final chapter we draw together the leading techniques in a more expanded treatment of before-and-after examples, giving step-bystep examples of drafting in the modern style. We illustrate with clauses drawn from five types of private legal documents, all commonly encountered in legal practice: leases, company constitutions, wills, conveyances, and building contracts.

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.