In our discussion of the equilibrium concentration of point defects in Chapter 5, we have assumed that for a given crystal there is only one type of defect with non-negligible concentration. We now discuss the scenario in which multiple types of point defects coexist in the same crystal. In Section 6.1, we show that the open crystal structure of silicon allows both vacancies and self-interstitials to exist with appreciable concentrations at thermal equilibrium. In Section 6.2, we discuss strongly ionic solids, in which all point defects are charged, and the charge neutrality condition requires point defects to be created in pairs. The equilibrium point defect concentration can be obtained by minimizing the Gibbs free energy of the crystal, or, alternatively, by the method of chemical equilibrium, each subjected to the charge neutrality condition. In Section 6.3, we discuss nonstoichiometric ionic solids in which atomic point defects can exist in both charged or neutral states, and electronic defects can also be present. The concentration of various atomic and electronic defects can be controlled by the partial pressures in the vapor phase.

In Section 6.4, we show that the type of the dominant point defect in intermetallic compounds, such as Ni–Al alloys, can change with composition. In Section 6.5, we discuss the formation of vacancy clusters. When a crystal is quenched from a high temperature to a low temperature, there is not sufficient time for vacancies to reach the true equilibrium concentration at the low temperature, and they tend to form clusters instead. We show that the thermodynamic principles can be applied to predict the concentration of vacancy clusters in the state where the total number of vacant sites is constrained.

Vacancies and self-interstitials in Si

Most of our discussion so far has concerned close-packed metals where vacancies can form but where self-interstitials are so energetic that they are essentially never present (except for metals irradiated by energetic particles like neutrons). On the other hand, the open crystal structure of Si (see Chapter 1) allows self-interstitials to form relatively easily. There is enough room in some interstitial positions to allow Si atoms (treated as spheres) to fit into the interstitial sites with no distortion at all (see Exercise problem 4.2).

In the previous two chapters we have studied the concentration of point defects in crystals after thermal equilibrium has been reached. We now discuss the motion of point defects, which is necessary for the equilibrium concentration to be reached in the first place. In fact, the motion of individual point defects never stops even after thermal equilibrium is reached. It is only when we take a coarse-grained view, do we find that the continuum concentration field of the point defects stops changing once equilibrium is reached.

In this chapter, we consider the motion of point defects both at the individual (discrete) level and at the collective (continuum) level. In Section 7.1, we consider a single vacancy and discuss the mechanism of its motion. We use the principles of statistical mechanics to show that the rate at which a vacancy jumps to a neighboring site is determined by a migration free energy (barrier) through a Boltzmann factor, and hence is strongly sensitive to temperature. In Section 7.2, we extend this result to the motion of interstitial and substitutional solute atoms. Because a neighboring vacancy is often required for a substitutional solute atom to jump, the jump rate of a substitutional solute atom is determined by a Boltzmann factor containing the sum of the migration free energy and the vacancy formation free energy.

In Section 7.3, we consider a large collection of point defects, each making random jumps to neighboring sites at a constant probability rate, and show that the evolution of their concentration field with time can be described by the diffusion equation. In Section 7.4, we show that if the crystal is subjected to an inhomogeneous stress field, then the equilibrium point defect concentration, if it exists, is not necessarily uniform. In this case, it would be necessary to use a generalized diffusion equation using the chemical potential defined in Section 5.4.

Under certain boundary conditions, no equilibrium solution exists for the diffusion equation, although a steady-state solutionmay exist. This is particularly common for vacancies, which can be constantly created at vacancy sources, travel across the crystal lattice, and be annihilated at vacancy sinks. As a simple model for this scenario, in Section 7.5 we consider vacancies in crystals subjected to a uniform deviatoric stress, to understand diffusional creep of crystals at high temperatures.

The perfect crystal structure is an idealization of the atomic arrangements in real crystalline materials. After a brief introduction of several common perfect crystal structures, we start our study of imperfections in crystals with some remarks about why so much attention is focused on these defects. The central reason is that perfect crystals, without imperfections, would be relatively uninteresting materials, without most of the useful properties with which we are all familiar.We consider some of the physical properties that crystals would have or not have if they were perfect. Through this thought experiment, we show that most of the useful engineering properties of crystalline materials are defect controlled and thus depend on the properties and behavior of imperfections.

Perfect crystal structures

Single crystals and polycrystals

The word “crystal” usually brings to mind large mineral (e.g. quartz) blocks on display in museums, or the shiny diamond on a wedding ring. Their faceted surfaces and often distinct geometric shape give rise to a sense of beauty not found in other more “common” materials. As an example, Fig. 1.1a shows a photograph of a ruby crystal. However, crystalline materials are easily found in our everyday life. In fact, most engineering materials are crystalline. Metals, semiconductors, and ceramics are all crystalline materials, even though they may not have faceted surfaces.

The distinction between a large ruby crystal and an engineering metallic alloy is that the former is a single crystal and the latter is usually a polycrystal. A polycrystal is an aggregate of many small single crystals (called grains), each with a different orientation. As an example, Fig. 1.1b shows a micrograph of a nickel-based superalloy (where the word “super” refers to its superior mechanical properties). The size of each single crystal grain in this superalloy is on the order of 10 to 100 micrometers (μm), too small to be seen by the naked eye. That is why the shape of a piece of metal does not seem faceted to the eye; the facets can be observed with the aid of a microscope.

Point defects are imperfections in a crystal that are confined to atomic dimensions in all three directions. Depending on the chemical species involved, point defects can be considered as either intrinsic or extrinsic. Intrinsic point defects include vacancies, i.e. missing atoms, and self-interstitials, i.e. extra atoms having the same chemical species as the host crystal. Extrinsic point defects, on the other hand, are atoms having a different chemical species from the host crystal that they enter. Such point defects are often called impurity or solute atoms. Impurities usually refer to “unwanted” foreign atoms in a crystal, while solute atoms are often intentionally introduced into the crystal to alter its properties.

Point defects have a profound effect on the properties of engineering crystalline materials, either by themselves, or through their interactions with dislocations (line defects) and grain boundaries (planar defects). An example of the former situation is the semiconductor industry, whose success hinges on their ability to control the electronic properties of silicon by selective doping, through which transistors and integrated circuits can be made. An example of the latter situation is solid solution hardening, in which the elastic distortion around point defects allow them to interact with dislocations and alter the mechanical strength of the crystalline material.

In the following four chapters, we focus on the fundamental mechanics and thermodynamic principles that are needed to understand how point defects influence material properties. In Chapter 4, we study the stress and strain fields around point defects, using the elasticity theory introduced in Chapter 2. These results lead to an estimate of the energy cost of introducing point defects, as well as how point defects interact with each other and with other types of defects (e.g. dislocations) to be introduced later. In Chapter 5, the energy cost of introducing point defects is combined with the statistical thermodynamic principles of Chapter 3, to predict the concentration of point defects in a crystal at thermal equilibrium. It will be seen that, due to the entropic gain in allowing point defects, the equilibrium concentration of point defects in a crystal at finite temperature is never zero.

The following five chapters constitute Part III of this book and focus on dislocations, the major line defects in crystals. Chapter 8 on dislocation geometry introduces the variables needed to properly define the dislocation line, specifically the Burgers vector and line sense vector. It also describes how dislocationmotion produces plastic strain in the crystal. Chapter 9 on dislocation mechanics discusses the stress field and energy of dislocations. The discussion is based on the linear elasticity theory, which is an accurate description of the solid only at sufficiently small strain, i.e. in regions sufficiently far away from the dislocation center. The interaction between dislocations, and effect of interfaces on the dislocation stress field, are studied in Chapter 10, which also discusses several applications of dislocation mechanics, such as the strain relaxation between misfit layers. In Chapter 11, we examine the structure of the dislocation core in closepacked crystals, in which a perfect dislocation dissociates into partial dislocations bounding a stacking fault area. The atomistic structures of the dislocation in several non-close-packed crystal structures are discussed in Chapter 12. The dislocation core structure is the result of non-linear interatomic interactions, and strongly influences the dislocation mobility.

It is instructive to compare the layout of Part III on dislocations with Part II on point defects. For example, Chapter 9 (dislocation mechanics) is the counterpart of Chapter 4 (point defect mechanics); in these chapters the stress fields of individual defects are discussed. However, Chapters 8 (dislocation geometry), 11 (partial dislocations) and 12 (dislocation core structure) have no corresponding chapters in Part II. This is because the geometry and atomistic core structure of dislocations are much more complex than those of the point defects; the latter are mostly covered in Sections 4.1 and 4.2.

At the same time, there is no chapter on “dislocation thermodynamics” that corresponds to Chapter 5 on point defect thermodynamics. This is because the energy of a dislocation line is so large that they should not exist if the crystal were in a truly thermodynamic equilibrium state. In comparison, a finite concentration of point defects should always exist at thermal equilibrium. Therefore, when discussing dislocations, we are necessarily dealing with non-equilibrium (perhaps meta-stable) states. For example, dislocations are usually generated in great quantities during plastic deformation, which is a highly dissipative, non-equilibrium process.

Our study of dislocations up to this point has focused on their geometry and their role in accommodating plastic deformation through their motion.We now turn to another important aspect of dislocations: their elastic fields. The remarkable thing about (perfect) dislocations is that while they leave a crystal internally perfect after they have passed through the crystal, they produce elastic distortions in the crystal as long as they are present. Their elastic fields determine, to a large extent, how they interact with each other and with other structural defects in the crystal.

Our study of the elastic properties of dislocations will assume that the material in which they are found is elastically isotropic.While most crystals are not elastically isotropic, the framework that emerges by assuming elastic isotropy is still quite useful and even reasonably accurate for most crystals. In this chapter, we will use the Volterra dislocation model, in which the dislocation line is a singularity that needs to be avoided. The atomistic structure of the dislocation core will be discussed in Chapter 12.

In Section 9.1 we discuss the stress, strain, and displacement fields of infinitely long, straight dislocations, and in Section 9.2 we obtain the elastic energy of these dislocations. Based on the elastic energy results, we describe in Section 9.3 the line tension model, in which a curved dislocation line is approximated as a taut string with a certain resistance to stretching. In Section 9.4 we derive the Peach–Koehler formula that determines the force per unit length on the dislocation exerted by the local stress. Given that dislocations generate their own stress fields, the Peach–Koehler formula can be used to predict the interaction between dislocations, which will be discussed in Chapter 10.

Elastic fields of isolated dislocations

The goal of this section is to obtain the stress, strain, and displacement fields of an infinitely long screw, edge, or mixed dislocation in an infinite medium, or at the center of an infinitely long cylinder of finite radius.

Currently there are essentially two methods in use for the first-principles calculations of phonon frequencies: the linear response theory and the direct approach. The linear response theory evaluates the dynamical matrix through the density functional perturbation theory. In comparison, one advantage of the direct or mixed-space approach over the linear-response method is that it can be applied with the use of any code capable of computing forces. The direct approach is also referred to as the small displacement approach, the supercell method, or the frozen phonon approach.

However, none of the previous implementations of the supercell approach are able to accurately handle long-range dipole–dipole interactions when calculating phonon properties of polar materials. The problem has been solved by the parameter-free mixed-space approach, which makes full use of the accurate force constants from the supercell approach in real space and the dipole–dipole interactions from the linear response theory in reciprocal space. The mixed-space approach is the only existing method that can accurately calculate the phonon properties of polar materials within the framework of the supercell or small displacement approach.

The program YPHON is written in C++ and can be downloaded at http://cpc.cs.qub.ac.uk/summaries/AETS_v1_0.html. The precompiled executable binaries should work for most Linux and Windows systems. Recompiling YPHON requires the GNU Scientific Library (GSL), which is a numerical library for C and C++ programmers. If one is just interested in phonon dispersions, the phonon density-of-states (PDOS), or the neutron scattering cross-section weighted PDOS the so-called generalized phonon density-of-states (GPDOS), this is enough. YPHON also makes it a lot easier to plot phonon dispersions and PDOS. In this case, it is required that Gnuplot be installed.

The static energy and force constants from first-principles calculations need to be formatted to the YPHON input formats (text formats as detailed later). At present, YPHON works closely with VASP.5 or later. The mixed-space approach has built up a unique base of the supercell approach to polar materials and has been adopted in a number of software tools such as CRYSTAL14 by R. Dovesi, ShengBTE (a solver of the Boltzmann transport equation for phonons) by W. Li, J. Carrete, N. A. Katcho, and N. Mingo, the Phonopy package by Atsushi Togo, and the Phonon Transport Simulator (PhonTS) by Chernatynskiy and Phillpot.

The CALPHAD modeling of thermodynamics was pioneered by Kaufman and Bernstein [47] and has been reviewed in detail by Saunders and Miodownik [48] and Lukas, Fries, and Sundman [49]. Information on features of software tools for CALPHAD modeling can be found in two series of publications in the CALPHAD journal [50, 51]. The key feature of the CALPHAD method is the modeling of the Gibbs energy of individual phases using both thermodynamic and phase equilibrium data. The main significance of the CALPHAD method is the following.

1. i. It enabled the development of the concept of lattice stability, i.e. the energy difference between the stable and non-stable crystal structures of a pure element.

2. ii.The Gibbs energy expression of each phase covers the full temperature, pressure, and composition spaces including both stable and non-stable regions of the phase. This enables the evaluation of the Gibbs energy of a system as a function of non-equilibrium state, i.e. with ξ as an independent variable.

3. iii.Thermodynamic data are usually obtained by measurements of heat such as the enthalpy of transition and heat capacity, as discussed in Section 4.2, which have large uncertainties typically in the range of kilojoules per mole-of-atoms. On the other hand, phase equilibrium data as discussed in Section 4.1, though more accurate, only contain information on compositions of phases at equilibria, i.e., the relative Gibbs energy of phases at equilibrium. The combination of these two sets of data is foundational in CALPHAD modeling and allows for the accurate modeling of thermodynamic properties of individual phases and reliable calculations of phase stability and driving forces.

4. iv. CALPHAD provides a framework to model thermodynamic properties of multi-component systems of industrial importance, enabling computational materials design. It has also been extended to model a range of properties of individual phases in multi-component systems such as diffusion coefficients, elastic coefficients, and thermal expansion, supplying input data for computational simulations of phase transformations during materials processing.

In this chapter, the basics of CALPHAD modeling of the Gibbs energy of individual phases are presented. For detailed implementations in various software packages and modeling procedures, readers are referred to the references given above.

Importance of lattice stability

For modeling of the Gibbs energy of individual phases, it is necessary to define the values of °Gi in Eq. 2.48.

General condition for equilibrium

A system is heterogeneous if some properties have different values in different portions of the system when the system is at equilibrium. Two scenarios may exist, where variations of the properties can be either continuous or discontinuous. In the scenario of continuous variations, the gradients of the variations must be coupled so that the system remains at equilibrium. The number of independent variables is thus reduced. These gradients must also be constrained along the boundaries between the system and the surroundings. This type of constrained equilibrium is not discussed in the book as it involves heterogeneous boundary conditions between the system and the surroundings and depends on the morphology of the system.

In the second scenario, with discontinuous variations, the properties have different values in different portions of the system, but remain homogenous within each portion. The system is in equilibrium as each portion is in equilibrium with all other portions of the system. The homogeneous portions represent different phases in the system, with the properties in each phase being homogeneous at equilibrium. In the previous chapter, it was been shown that all potentials are homogeneous in a homogeneous system.

For a heterogeneous system, the same conclusion can be obtained. If the temperature is inhomogeneous, heat can be conducted from high temperature locations to low temperature locations, and this process is irreversible based on the second law of thermodynamics because it increases the internal entropy of the system. If the pressure is inhomogeneous, the amounts of lower molar volume phases will increase to reduce the internal energy of the system. If the chemical potential of a component is inhomogeneous, the chemical potential difference of the component will drive that component to locations with a lower chemical potential in order to decrease the internal energy of the system. Therefore, it can be concluded that all potentials are homogeneous in a heterogeneous system at equilibrium, and the variables that are not homogeneous are thus their conjugate molar quantities. Under certain special circumstances, to be discussed later in this book, some molar quantities may also have the same values in difference phases.

In the previous chapter, the experimental techniques used to obtain the thermochemical and phase equilibrium data that were the inputs for the thermodynamic modeling of a system were summarized. However, experimental data are not always available. This is due to the fact that (i) the experiments are expensive, especially when they involve developing new materials, and (ii) the experiments cannot reliably access the non-stable phases in most cases. The alternative approach is to predict the thermochemical data by first-principles calculations. The prediction of material properties, without using phenomenological parameters, is the basic spirit of first-principles calculations. In particular, the steady increase of both computer power and the efficiency of computational methods have made the first-principles predictions of most thermodynamic properties possible, including both enthalpy and entropy as a function of temperature, volume, and/or pressure.

By definition, the term “first-principles” represents a philosophy that the prediction is to be based on a basic, fundamental proposition or assumption that cannot be deduced from any other proposition or assumption. This implies that the computational formulations are based on the most fundamental theory of quantum mechanics, the Schrödinger equation or density functional theory, and the inputs to the calculations must be based on well-defined physical constants – the nuclear and electronic masses and charges. In other words, once the atomic species of an assigned material are known, the theory should predict the energies of all possible crystalline structures, without invoking any phenomenological fitting parameters.

This chapter is organized in sequence from thermodynamic calculations to fundamental theory, to help those readers who are more interested in realistic calculations using existing computer codes. Detailed theoretical discussions follow the subsections on thermodynamic calculations for those readers who are also interested in the derivation of the formulations used in the thermodynamic calculations. The subsections are arranged accordingly in the order: (i) examples of the commonly adopted calculation procedures for thermodynamic properties using the elemental metal nickel as the main prototype; (ii) derivation of the Helmholtz energy expression under the first-principles framework; (iii) introduction of the solution to the electronic Schrödinger equation within two well-developed frameworks – the quantum chemistry approach and the density functional theory; (iv) detailed description of the procedure on how to solve the Schrödinger equation for the motions of atomic nuclei by means of lattice dynamics; and (v) First-principles approaches to disordered alloys.