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.