So far, Part III has described melting, precipitation, and unmixing phase transformations. For alloys, these all require the diffusion of atoms over moderate distances, for which continuum diffusion equations provide much of the essential behavior. Ordering transformations also require the movements of individual atoms, but during ordering the atom movements are over such short distances that a diffusion equation is not appropriate. Nevertheless, the atoms move independently (often by a vacancy mechanism), and the configurational entropy can undergo large changes with only a few jumps per atom.

This chapter describes diffusionless transformations, in which the atoms in a crystal move cooperatively, and the crystal is distorted into a new shape. The atoms may not move at exactly the same time, but the transformation is very fast, and does not require a vacancy mechanism for the motions of individual atoms. Diffusionless transformations include “twinning,” in which a crystal transforms into a different variant of the same type of crystal. Martensitic transformations are changes in crystal structure that occur by shears and dilatations, but again without long-range diffusion. Vacancy migration is not important for either twinning or martensitic transformations. Because the atoms do not move with individual independence, the change in configurational entropy is small. The entropy of martensitic transitions is primarily vibrational, sometimes with magnetic entropy as in the case of iron.

Most phase transformations in materials occur by nucleation and growth, where a small particle of the new phase nucleates in the parent phase, and then grows as atoms diffuse to its surface. Especially at low temperatures, the first new particles can be quite small, and their crystal structures may differ from the equilibrium phases that form with increased temperature or longer times. A nucleus of a distinct phase must have an interface to the surrounding parent phase. This interface between the precipitate and matrix has an atomic structure and chemical composition that is not simply a termination of bulk crystal of the precipitate and the matrix. For free surfaces and for interfaces between crystals, this chapter explains important aspects of surface energy, surface thermodynamics, and kinetic processes at surfaces. Most solid-solid phase transformations require some consideration of elastic energy, and the balance between surface energy, elastic energy, and volume free energy is altered as a precipitate grows and changes its shape.

Chapter 4 presented concepts of nucleation in a homogeneous medium or at a generic surface, but additional microstructural aspects of nucleation are presented here. Diffusional transport to the new phase often controls the rate of growth, and the rate of the phase transformation. Some examples are given, but numerous nucleation and growth transformations in different materials have received significant study. The eutectoid transformation is described, and its role in steel metallurgy is discussed in brief.

What is a phase transition?

A phase transition is an abrupt change in a system that occurs over a small range in a control variable. For thermodynamic phase transitions, typical control variables are the “intensive variables” of temperature, pressure, or magnetic field. Thermodynamic phase transitions in materials and condensed matter, the subject of this book, occur when there is a singularity in the free energy function of the material, or in one of the derivatives of the free energy function. Accompanying a phase transition are changes in some physical properties and structure of the material, and changes in properties or structure are the usual way that a phase transition is discovered. There is a very broad range of systems that can exhibit phase transitions, extending from atomic nuclei to traffic flow or politics. For many systems it is a challenge to find reliable models of the free energy, however, so thermodynamic analyses are not available.

Our focus is on thermodynamic phase transitions in assemblages of many atoms. How and why do these groups of atoms undergo changes in their structures with temperature and pressure? In more detail, we often find it useful to consider separately:

1. • nuclei, which have charges that define the chemical elements,

2. • nuclear spins and their orientations,

3. • electrons that occupy states around the nuclei, and

4. • electron spins, which may have preferred orientations with respect to other spins.

Since its very beginning, quantum mechanics has been developed to deal with systems on the atomic or sub-atomic scale. For many decades, there has been no reason to think about its application to macroscopic systems. Actually, macroscopic objects have even been used to show how bizarre quantum effects would appear if quantum mechanics were applied beyond its original realm. This is, for example, the essence of the so-called Schrödinger cat paradox (Schrödinger, 1935). How-ever, due to recent advances in the design of systems on the meso- and nanoscopic scales, as well as in cryogenic techniques, this situation has changed drastically. It is now quite natural to ask whether a specific quantum effect, collectively involving a macroscopic number of particles, could occur in these systems.

In this book it is our intention to address the quantum mechanical effects that take place in properly chosen or built “macroscopic” systems. Starting from a very naïve point of view, we could always ask what happens to systems whose classical dynamics can be described by equations of motion equivalent to those of particles (or fields) in a given potential (or potential energy density). These can be represented by a generalized “coordinate” φ(r, t) which could either describe a field variable or a “point particle” if it is not position dependent, φ(r, t) = φ(t).