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.

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.

Introduction

The traditional style of legal writing is the product of many influences. Some influences are constant, some are sporadic. They rarely exist in isolation; usually, many operate together. This chapter reviews the main influences on traditional legal drafting, namely: