Our study of grain boundaries to this point has focused on their geometry and special misorientations that lead to periodic patterns in the GB structure.We now turn to another important aspect of grain boundaries: their energies and possible elastic fields. A planar grain boundary usually does not have a long range stress field by itself. However, certain grain boundaries contain periodic dislocation arrays as part of their structures. In such cases, there is an appreciable stress field around the GB at distances comparable to the inter-dislocation spacing in the GB. The GB model based on dislocation arrays, combined with the theory of coincidence site and DSC lattices, provides a way to understand the GB energy as a function of its misorientation angle.

We have seen that the GB energy as a function of the misorientation angle has a complex structure, as shown in Fig. 13.2 and Fig. 13.8. Nonetheless, such plots suggest a classification of grain boundaries broadly into three types: singular, vicinal, and general [129]. The singular GBs correspond to the sharp minima on the energy plots. Their misorientations usually correspond to low-Σ CSLs. The singular GBs are usually special in other properties as well, such as mobility and point defect segregation. The vicinal GBs have both misorientation and GB plane direction sufficiently close to the singular GBs, and they can be considered as singular GBs superimposed with one or more GB dislocation arrays. The spacing between the nearest dislocations in the array reduces as the misorientation deviates further away from that of the singular GB. The general GBs are those boundaries that are sufficiently different from the singular GBs that the dislocation array is no longer a useful model as the necessary dislocation density would be so large that the dislocation cores would overlap.

In this classification of GBs, the case of zero misorientation and its vicinal range deserves extra attention. On the one hand, when the misorientation angle θ equals zero, the grain boundary disappears and the GB energy is zero, because the two crystals are perfectly aligned with each other.

A qualitative understanding of the behaviors of point defects can be established by considering atoms as hard spheres packed together to form the crystal. Crude as the hard sphere model may seem, it can be used to explain many of the observations made about point defects. In Section 4.1, we define the hard sphere radius of an atom and show its influence on the site preference of solute atoms. In Section 4.2, we use the hard sphere model to show the type of the distortions (spherically symmetric or not) in the host crystal around a solute atom. This allows us to explain why certain solutes have a much stronger solid solution hardening effect than others.

We then need to go beyond the hard sphere model in order to be more quantitative. In Section 4.3, we define the Seitz radius, which is more useful than the hard sphere radius for keeping track of the volume occupied by atoms of different kinds in solid solutions. We will see that atoms often appear to take on a different radius as a solute atom in another crystal compared to the radius it takes in its own crystal. In Section 4.4, we apply elasticity theory to predict the elastic fields around a solute atom. For simplicity, the size of the point defect is shrunk to zero and is modeled as force dipoles acting on a point in an elastic medium. In Section 4.5, a more realistic model is developed, in which the solute atom is modeled as an elastic sphere to be inserted into a hole inside an elastic medium. Elastic fields arise because the initial size of the sphere is larger than the initial size of the hole. Even though many atomistic and electronic details concerning point defects are ignored, the models developed in this chapter are increasingly more quantitative and can be used to explain a large number of behaviors of point defects.

Hard sphere model

Hard sphere radius

It is common to treat atoms in a crystal as undeformable spheres and to calculate the atomic sizes from the lattice parameters (measured using X-ray diffraction).We call this the hard sphere approach.

In our treatment of dislocations thus far, we have avoided the dislocation core. For example, in Volterra's dislocation model, the stress–strain fields diverge on the dislocation line, so that a cylindrical region of material is usually removed around the dislocation line to avoid the singularity. In the line tension model, the dislocation is modeled as a string that carries a line energy per unit length, but is otherwise featureless. In Chapter 11, we have seen that perfect dislocations in close-packed metals tend to dissociate into partial dislocations, but the partial dislocations were still treated as Volterra's dislocation lines. In reality, every (perfect or partial) crystal dislocation has a core region, which possesses a specific atomistic structure, called the core structure. The core structure is determined by non-linear interatomic interactions and the crystal structure, and, in turn, strongly influences the energetics and mobility of the dislocations. In this chapter, we discuss typical dislocation core structures and their effects on dislocation properties in several crystal structures.

In Section 12.1, we start our discussion with the classical Peierls–Nabarro (PN) model, which was the first physical model for the dislocation core and naturally predicts that the dislocation core should have a finite width. In Section 12.2, we generalize the original PN model to account for the presence of stacking faults in FCC metals. Consistent with the hard sphere model in Chapter 11, the generalized PN model also predicts dissociation of perfect dislocations into partials, except that each partial now has a finite width.

For crystals whose structures are sufficiently different from close-packed, hard spheres are no longer a good model for the atoms. Nonetheless, the geometry of the stacking of atomic layers is still useful for understanding the dislocation core structures in these crystals, as discussed in Section 12.3 (diamond cubic crystals) and Section 12.4 (BCC crystals). Finally, in Section 12.5 we discuss the interaction between dislocations and point defects, which usually leads to segregation of point defects around the dislocation core.

Peierls–Nabarro model

The classical model by Peierls and Nabarro [111, 112] considers the spreading of the dislocation over the glide plane.

Having discussed the elastic field around a single point defect, we now apply the thermodynamics principles (Chapter 3) to obtain the equilibrium concentration of point defects in crystals under a given temperature and pressure. The fundamental principle used repeatedly is that the Gibbs free energy of the crystal is minimized when the point defects reach the equilibrium concentration.

We start by discussing the equilibrium concentration of extrinsic point defects, i.e. substitutional and interstitial solutes, in Section 5.1. The approach is then applied, in Section 5.2, to vacancies, which are intrinsic point defects. In Section 5.3 we discuss the experimental methods to measure the equilibrium concentration and thermodynamic properties of vacancies, and compare the experimental data with theoretical estimates. The chemical potential of point defects is defined in Section 5.4.

Equilibrium concentration of solutes

We consider a dilute substitutional solution of B atoms in an A matrix. Let NA (which is fixed) be the number of A atoms in the system and let NB be the number of B atoms dissolved in the A-rich crystal. The total number of atomic sites in the A-rich crystal is N = NA + NB. Thus χ = NB/N is the fraction of atomic sites where the “wrong” kind of atom is located. χ is also the molar fraction of B atoms in the crystal.We follow the regular solution/quasi-chemical approach in which the formation energy of the point defect is dominated by the energies of the chemical bonds associated with the impurity defect (quasi-chemical, Eq. (5.5)) and where the mixing entropy is that for an ideal solution (regular solution, Eq. (5.19)).

Let the A-rich crystal be in contact with a large B crystal, which acts as an infinite supply of B atoms. For simplicity, we only allow B atoms to enter the A-rich crystal as solutes, but forbid A atoms to enter the B crystal as solutes. Each time a B atom is dissolved in the lattice, the A atom it replaces takes up a site at the A/B interface and extends the A lattice by one atomic volume, as shown in Fig. 5.1a. Note that the number of A atoms NA is conserved, while the number of B solute atoms NB and the total number of atomic sites N for the A-rich crystal are not conserved.

Our treatment of dislocations in previous chapters has focused on perfect dislocations, line defects that are surrounded by perfect crystal and have Burgers vectors equal to the shortest complete lattice translation vector. When perfect dislocations glide in a crystal they cause the atoms on either side of the slip plane to be displaced relative to each other by exactly a lattice translation vector, so that the crystal is perfect both ahead of and behind the gliding dislocation. Here we study the atomic motions associated with slip in more detail and observe that the sliding of atomic planes relative to each other often does not go immediately from one perfect state to the next. Instead, the slipping of atomic planes from one perfect state to the next may be broken up into two or more steps by the movement of partial dislocations separated by faults in the atomic stacking. We will see that partial dislocations and stacking faults in different crystal structures can be anticipated from the atomic packing arrangements in the crystal by treating the atoms as hard spheres and taking account of the bonding between them.

In Section 11.1, we consider the dissociation of perfect dislocations in FCC metals into Shockley partials, and obtain the equilibrium separation between the partials from a forcebalance analysis. We introduce Thompson's notation to conveniently label the Burgers vectors of various perfect and partial dislocations in FCC metals. We also examine non-planar dislocation structures such as the transient core structure during cross-slip of a screw dislocation and the stable core structure of a Lomer–Cottrell dislocation. Finally, we consider the Frank partial dislocation loop formed from the condensation of vacancies, and the transformation of the Frank partial loop into a perfect dislocation loop or a stacking fault tetrahedron.

In Section 11.2, we discuss the types of dislocations in HCP metals. Because the atomic arrangements on the basal planes of HCPmetals are very similar to those on the ﹛1 1 1﹜ planes in FCCmetals, perfect dislocations on the basal planes are also dissociated into Shockley partials in HCPmetals. In Sections 11.3 and 11.4, we discuss dislocations inCrCl3 andNi3Al, respectively, which are crystals formed by more than one chemical species.

In Table B.1 we present elastic constants, coefficient of thermal expansion, and melting temperature for a set of common crystalline solids of pure elements. Because for most materials each crystal grain is elastically anisotropic, isotropic elasticity is only an approximation. The values listed here correspond to the averaged elastic properties of a polycrystal consisting of a large number of randomly oriented single-crystalline grains.

There is a significant scatter in the averaged isotropic elastic constants in the literature. The scatter in the anisotropic elastic constants of single crystals is comparably less. Therefore, the values in the table are computed values from anisotropic elastic constants of single crystals. Specifically, the bulk modulus B and shear modulus μ are computed from the self-consistent averaging [151, 152] of single crystal values. Given B and μ, the Young's modulus is computed as E = 9Bμ/(3B + μ) and the Poisson's ratio as ν = (3B - 2μ)/(6B + 2μ).

The anisotropic elastic constants of single crystals, together with the thermal expansion coefficients and melting temperature are obtained from [153]. Both the elastic constants and coefficients of thermal expansion are room-temperature values. Given the experimental scatter and temperature dependence of these properties, an uncertainty of a few percent is to be expected for the values reported here. This should be sufficiently accurate for most calculations and estimates within the isotropic approximation. If very precise calculations are required, the original literature should be consulted and the temperature dependence of these properties should be taken into account. Three or more significant digits are often reported to provide self-consistency among B, E, μ, and ν, even though the data themselves are probably not accurate down to the last significant digit. The melting points are accurate to within 1 K.

A grain boundary (GB) is the interface between two single crystals (i.e. grains) of the same material with different orientations. We mentioned in Chapter 1 that most engineering metals and alloys are polycrystals, which are aggregates of a large number of single crystal grains separated by grain boundaries. Grain boundaries play an important role in a wide range of material behaviors and properties. For example, grain boundaries can increase the strength of metals by blocking the motion of lattice dislocations, leading to the Hall–Petch behavior, in which the strength of crystalline materials increases with decreasing grain size. We have mentioned in Chapter 7 that grain boundaries act as sources and sinks of vacancies, and that vacancy diffusion along GBs is the mechanism of Coble creep. Because impurities tend to segregate at grain boundaries, the chemical environment is often different at grain boundaries, which can be preferential sites for crack initiation or chemical attack. Grain boundary engineering [127], i.e. the keeping of “good” GBs and removal of “bad” GBs through processing, has led to significant improvements in material strength and corrosion resistance.

In Chapter 13 (Grain boundary geometry), we first define the five orientation variables that specify the misorientation between the two grains and the direction of the boundary plane normal.We then introduce the coincidence site lattice (CSL) and the associated Σ number that are very useful to characterize special (low energy) grain boundaries. We will see that special grain boundaries usually have low Σ numbers but low Σ numbers do not necessarily mean the boundaries are special. The displacement shift complete (DSC) lattice, of which the CSL is a sublattice, and prescribes the allowable Burgers vectors of GB dislocations, which can be much shorter than those of lattice dislocations.

In Chapter 14 (Grain boundary mechanics), we explore the relationship between grain boundaries and dislocations. First, low angle grain boundaries are equivalent to arrays of lattice dislocations, as described by the famous Read–Shockley model. Second, a grain boundary with misorientation vicinal to a low- Σ GB can be considered as a superposition of the low- Σ GB and an array of GB dislocations. Finally, we discuss disconnections, which are steps on the GBs with a non-zero Burgers vector content, i.e. they are simultaneously GB steps and GB dislocations.

This book is mainly written for senior undergraduate and junior graduate students wanting to gain an understanding of the behavior of defects in crystalline materials using the fundamental principles of mechanics and thermodynamics. We choose the word imperfections to emphasize that the crystalline materials in which these defects are found are nearly perfect. In other words, the densities of these defects are usually very low. Yet, they can greatly alter and even control the properties of the host crystal. It can be said that the main purpose of the entire field of materials science is to modify the properties of materials through the control of their defects.

The book is written based on a set of lecture notes of a course (MSE206 Imperfections in Crystalline Solids) that one of us (WDN) taught in the Materials Science and Engineering Department of Stanford University for more than 50 years. This course is now taught in the Mechanical Engineering Department (as ME209 by WC). We wanted to turn these lecture notes into a textbook so that it can be used by students and instructors in other universities who are interested in learning/teaching this subject.

The scope of this book has significant overlap with two important books in this area: Introduction to Dislocations by Hull and Bacon, and Theory of Dislocations by Hirth and Lothe. The book by Hull and Bacon provides a clear introduction for junior undergraduate students to the field of defects in crystals, while the book by Hirth and Lothe is a monograph and a valuable reference to experienced researchers in this field. It has long been recognized by the community that what we lack is a textbook that bridges the gap between these two books, a textbook that can be used in the teaching of core senior undergraduate/junior graduate courses on defects in crystals.

To this end, we can share the personal experience of one of us (WC) while he was a graduate student himself (at MIT). Realizing that his PhD thesis research would deal with the modeling of crystalline defects, he first read through Introduction to Dislocations by Hull and Bacon. The book provided a very useful introduction, but after reading it, he still did not feel quite “ready” for his research tasks. So he realized it was necessary to delve into Theory of Dislocations by Hirth and Lothe.

In Chapter 9, we have seen that dislocations produce stress fields in the crystal that contains them. We have also seen that stresses produce Peach–Koehler forces on dislocations. Therefore, dislocations exert forces on each other through the stress fields they produce. In this chapter, we discuss dislocation–dislocation interactions, as well as the interaction between dislocations and other defects in the crystal, and the consequence of these interactions on the strength of bulk crystalline materials. We also consider the applications of these interactions to the mechanical properties of thin films.

In Section 10.1, we consider the interaction between two dislocations in an infinite medium, starting with two infinitely long parallel screw or edge dislocations. A few examples of the interaction between two non-parallel dislocation lines are also considered. In Section 10.2, we consider arrays formed by more than two dislocations of the same sign. When the dislocation interactions are attractive, the dislocation array corresponds to low angle grain boundaries. When the dislocation interactions are repulsive, the dislocation array is called a pile-up, because it can be found in front of an obstacle which blocks the dislocation motion.

In Section 10.3, we discuss two dislocation mechanisms which increase the strength of crystals. In the Taylor hardening mechanism, the strength is controlled by the interaction between dislocations themselves. In the Orowan bowing mechanism, the strength is associated with the presence of non-shearable particles, between which the gliding dislocation must bow for plastic deformation to occur. In Section 10.4, we consider several models in which the kinetics of dislocation motion, multiplication, and annihilation are used to explain the plastic deformation behavior of single crystals, such as the phenomena of yield point drop and sigmoidal creep.

The last two sections of this chapter deal with dislocations near interfaces. In Section 10.5, we discuss the conditions under which it becomes energetically favorable for dislocations to form at the interface between two materials to relieve the misfit strain. In Section 10.6, we discuss the stress and strain fields of dislocations near free surfaces and interfaces between twomaterials, and the forces exerted on these dislocations by the interfaces. For straight screw dislocations parallel to the interfaces, the effect of the interfaces can be modeled by sets of image dislocations in infinite, homogeneous elastic media.

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.