The phase-field method is a thermodynamics-based approach most often employed to model phase changes and evolving microstructures in materials. It is a mesoscopic method, in which the variables may be abstract non-conserved quantities measuring whether a system is in a given phase (e.g., solid, liquid, etc.) or a conserved quantity, such as a concentration. Interfaces are described by the smooth variation of those quantities from one phase to another and are diffuse, not sharp.

The phase-field method is increasingly being used in materials science and engineering because of its flexibility and utility. We discuss the basic method here, but researchers are continually creating new features and new approaches within the basic phase-field framework.

We first introduce the basic mathematical formalism, followed by some simple examples of the phase field in one and two dimensions. Implementation of the phase field requires some new computational methods, which will be discussed in the regular text and an appendix. Finally, we will discuss some applications of the phase-field method in materials research.

CONSERVED AND NON-CONSERVED ORDER PARAMETERS

In phase-field modeling, the state of a system is described by a function of position and time. This function could be a specific property of the system such as concentration or it could be a parameter that indicates what phase the system is in, e.g., solid or liquid. This function is generally referred to as an order parameter.

The behavior of a material can be related to the types of bonding between the atoms, whether it be metallic, covalent, ionic, etc. That bonding represents the distribution of electrons around the nuclei. Covalent bonds have a localized electronic distribution between atoms and are generally strong and directional. Materials with strongly covalent bonds include important semiconductors, such as silicon, gallium, and diamond. Metallic systems, in contrast, may have a degree of directionality to their bonding, but the dominant feature is a delocalized sea of electrons. Ionic bonds are dominated by the strong electrostatic interactions between the ions. Fundamentally, the properties of each material start with its bonding.

A fundamental description of bonding requires a calculation of the electronic distributions. The class of methods that yield such information are called electronic structure methods. In this chapter, we shall briefly review the basics of these methods, pointing out their inherent approximations. There are numerous books devoted to the fundamental theories behind these methods – embodied in quantum mechanics – as well as many texts devoted to electronic structure methods themselves [167, 219, 251, 254]. We can at best give a brief guide to this topic needed for discussions later in the text and as well as for a basis for understanding and evaluating this fascinating field.

Not so many years ago, practitioners of electronic structure calculations typically used homegrown computer codes, which often required heroic efforts on the parts of the programmers.

Engineered designs are generally based on the use of a constrained, and fixed, set of materials. Because materials development is slow, the role of the materials engineer is generally one of materials selection, i.e., choosing a material from a restricted list to fit a specific need in a product design process. Traditionally, the optimal material was a balance between best meeting the product performance goals and minimizing the cost of the material. In recent years, an increased focus has been on the life cycle of the material, with an eye towards recycling and reuse.

The selection of the best material for an application begins with an understanding of the properties needed for the design as well as a way to display and access the properties of candidate materials. If the design is based on a single criterion for the material, such as density, for example, then the choice of a material is usually pretty simple. If multiple criteria must be met, then a way to compare multiple properties of a set of materials with each other is needed. A common way to do that is through an “Ashby plot”, a scatter plot that displays one or more properties of many materials or classes of materials [13, 14]. For example, suppose one needs a material that is both stiff and light. Stiffness is measured in Young's modulus, while knowing the density of a material will enable one to pick the lightest material for a specific volume.

In this chapter, we discuss how to extend the methods introduced in the previous chapters from atomic to macromolecular systems. The basic ideas are the same, but there are additional complexities that arise from the molecular shapes. The simulation of molecular systems, especially polymeric and biological materials, is a very active field and we barely touch the surface here. For more information, please see the texts in the Suggested reading section.

After a review of the basic properties of macromolecules, the chapter continues with a discussion of some of the common approaches to model the interaction between the molecules, followed by descriptions of how molecular dynamics and Monte Carlo methods can be applied to molecular systems. When discussing systems of large molecules, such as polymers or proteins, however, it becomes challenging to include the full complexity of the molecules within a calculation. Thus, various models that approximate the physics have been developed. The chapter ends with a discussion of some of these approximate methods.

INTRODUCTION

Polymers (macromolecules) are large molecules made up of long chains of monomer units. In some biological molecules, the number of monomers (N) can be quite high, e.g., in DNA N ˜ 108 in some cases. In other systems, N can be of the order of a few hundred. The identity of the monomer units defines the overall properties of the polymer – DNA and RNA are made up of nucleotides, proteins are made up of amino acids, etc.

Overview

Legal drafters seek to give effective written expression to instructions from clients. The skills needed to undertake that task are sometimes formidable. And the skills are exercised in a context where the ultimate arbiters of meaning are not the clients or the drafters, but the courts. This is not to say that drafters should write defensively, tailoring words and concepts for the judge alone. On the contrary, they must write for their clients, who are the readers and users of the documents. But they must also appreciate that a document may end up before a judge, and so must strive to ensure that the judge will interpret their draft as they (the drafters) intended. This, therefore, means that competent drafters must understand the principles which courts apply when interpreting legal documents – the so-called ‘principles of interpretation’.

This chapter examines the ways in which judges interpret legal documents. It begins by reviewing cases where judges have decried aspects of the ‘traditional’ style of legal drafting. Our purpose in this section is to emphasise that traditional legal drafting has no inherent superiority over the modern, plainer style which we recommend in this book.

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.