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.

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.

Background and types of antifoamers and defoamers

Although foams are thermodynamically unstable, under practical conditions they can remain fairly stable for a considerable period of time, and it is often necessary to add chemicals to prevent foaming or to destroy the foam. Early definitions of antifoamers referred to the chemicals or materials pre-dispersed in the liquid phase prior to processing to prevent foam formation (produce low foamability) while defoamers were added to eliminate existing stable foams (produce low foam stability) by a shock effect. However, today this distinction is confusing since most chemical additives cover several roles and the nomenclature varies according to the industry where they are used. In fact, they are often referred to as foam control agents, foam inhibitors, foam suppressants and air release agents.

Foaming causes problems throughout a range of industrial processes, for example, in the production and processing of paper, pharmaceuticals, materials, textiles, coatings, crude oil, washing, leather, paints, adhesives, lubrication, fuels, heat transfer fluids and so on. In the processing of food and beverages such as sugar beet, orange and tomato juice, beer, wine and mashed potatoes, foaming problems caused by soluble proteins and starch are commonly encountered. Food containers are washed and recycled and again foaming must be prevented during these processes. It is also frequently necessary to break foam in storage vessels to increase the capacity (such as beer), and foam breaking is necessary to increase the efficiency of distillation or evaporation processes. There are numerous reviews of the antifoaming/defoaming area and a comprehensive book by Garrett (1) in 1993 covers the basic physical chemistry and most of the industrial uses of antifoamers. A more recent publication by Garrett (2) in 2015 summarizes further developments associated with the mode of action and also the mechanical aspects of defoaming are reviewed. Early publications by Owen (3) classify different products, and Kerner (4) lists the antifoaming products supplied by major companies. There are over 100 suppliers, if smaller companies are included, and many international suppliers have manufacturing capacity whereas smaller companies specialize in formulations for particular industries or processes.

The aim of this book is to provide a comprehensive, well-structured insight into the physical chemistry of liquid foams which can be used by both academics and industrialists. Liquid foams may occur naturally or by design and may be desirable or undesirable. Generally, there is a multitude of complex causes of foaming and antifoaming and the text is structured to give clarity to the field by providing an up-to-date, state-of-the-art guide explaining the chemistry of real foam systems. It is hoped that the reader will achieve a reasonably clear understanding of why foaming occurs, how it can be measured and how it can be prevented. As the use of foams spans different disciplines, some introductory aspects of physics, chemical engineering and material science of foams are included but this is relatively easy to follow. This book is orientated toward the descriptive rather than the theoretical and contains many diagrams. It is also a rich source of information and references, arranged in a way which the reader should find useful and also provides an historical prospect to the area of foams and foaming.

The most popular academic books dealing solely with foams include the classics Foams by J. J. Bikerman (1973), published by Springer-Verlag, Berlin and The Physics of Foams by D. Weaire and S. Hutzler (1999), published by Clarendon Press, Oxford. Both of these books ran into several updated editions but considerable advancements in the field have been made since their publication. Other early texts are Foams and Biliquid Foams-Aphrons by F. Sebba (1987), published by Wiley and the two books – Antifoaming (edited by P. Garrett, 1993) and Foams (edited by R. K. Prud'homme and S. A. Kahn, 1996) – published in the Surfactant Science Series (Marcel Dekker). These are fairly well-read books but are essentially a collection of viewpoints which describe many varied aspects of foaming and antifoaming science. Foam and Foam Films by D. Exerowa and P. M. Kriglyako (1997), published by Elsevier in the Studies in Interfacial Science Series, has been well received but presents a strongly fundamental text with the main emphasis on thin films. More recently is the book Foam Engineering, edited by P. Stevenson (2012) and published by Wiley, covers rheology, flow and foam processing and is aimed toward the chemical engineering community.

Introduction

There are a number of methods for generating foams, and these have been reasonably well documented throughout the literature. They may be classified into two groups; the first involves the entrapment of air bubbles from the atmosphere and this can be achieved by relatively simple techniques such as shaking, pouring, circulation, sparging (introduction of gas using frits), etc. The second method involves artificially producing gas bubbles by physical methods (e.g. by nucleation or electrolysis) or chemical methods, which are commonly exploited in the production of polymer foams and involve the use of so-called blowing agents. These are chemical compounds that decompose or react to produce gas bubbles. While it is difficult to control the bubble size using physical methods of bubble formation, with chemical methods it is much easier to achieve a narrow-size distribution along with a high generation rate. Many different types of gases are used for foam generation, but it is important to recognize that foams generated with less soluble gases, such as N2 or air, will coarsen more slowly than foams produced with more soluble gases such as CO2 since gas diffusion through the soap films is largely determined by the gas solubility and the diffusion coefficient. In traditional foaming processes such as froth flotation, mechanical air entrapment methods are frequently used since they are relatively inexpensive, whereas in the production of material foams more sophisticated chemical processes have been developed.

The adsorption of surfactant on the freshly generated bubbles

The initial step in the generation of bubbles and foams involves the formation of a gas/liquid interface. This process involves work which can be quantified as the product of the interfacial tension and the increase in area of the interface; it be expressed by the equation

where ΔA is the created interfacial area and γ is the surface tension of the freshly produced bubbles. In water, bubbles have a high interfacial energy and become instantly unstable. Therefore, it is essential that surfactant adsorbs at the interface and reduces the surface tension and stabilizes the bubble. The adsorption kinetics plays an important role in the stabilization of the bubble, and the surfactant molecules need to rapidly diffuse from the bulk solution to the bubble interface.

Introduction

Foaming is less commonly encountered in non-aqueous fluids than in water-based media, but it is generally accepted that similar physico-chemical principles are applicable, such as the adsorption to the bubble interface. This enables some analogies to be drawn with aqueous systems but there are problems which need to be resolved, since it was originally thought that the degree of dissociation of solubilized chemical additives was limited in non-aqueous systems and free ions were almost absent. The physical properties of the liquid also play a more important role in determining the stability of the non-aqueous foam, and if one considers the many different types of non-aqueous liquids, for example liquid metals, liquid polymers, crude oils, drilling fluids, lubricants and solvents (base cleaners), then it is clear that wide variations in physical properties need to be considered. With these liquids, distinct differences in, for example, viscosity, conductivity, dielectric constant and density are evident.

This makes it difficult to generalize the prediction of foaming behavior in non-aqueous systems, but usually some type of surface activity is needed as in aqueous systems. For example, in the generation of aluminum foams, silicon carbide particles are added which are captured by the bubbles and act to stabilize the gas/liquid metal interface (1). In mixed hydrocarbon media, lyotropic liquid crystals which are surface-active self-organized assemblies play a role in foaming performance (2, 3, 4). It has also been shown that the addition of inorganic electrolytes to organic liquids inhibits bubble coalescence by reducing drainage, but it is only recently that an improved understanding of bubble stability and interfacial drainage in non-aqueous liquids has been achieved. In these systems, it has been shown that drainage rates may be influenced by the particular arrangement of ions in the interfacial region (5, 6). For non-aqueous liquids with high bulk viscosity such as molten polymers, thick oils and metal foams, the drainage can be relatively slow, and this will reduce the foam decay rate to some degree.

The formation of self-assemblies from pre-micellar surfactant species

The adsorption of amphiphilic surfactant molecules at the bubble interface is not the only important phenomenon occurring during foam formation. Another extremely important process also occurs in bulk solution at high surfactant concentrations. This involves a spontaneous self-assembly process in which higher molecular structural aggregates or units of surfactant are formed from lower molecular weight pre-micellar species (monomer, dimer and trimer units). In the simplest case, this corresponds to the formation of a spherical micelle, and the transition concentration (of monomer) at which this occurs is called the “critical micelle concentration” (CMC). Marked changes in foaming behavior, as well as changes in electrical conductivity, surface tension, turbidity and uptake of organic dyes, occur in bulk solution above the CMC, but the molecular concentration of the surfactant in the water remains constant, with the surplus molecules forming additional micelles. Fig. 2.1 depicts the successive steps involved in the growth of the micelle from monomeric species, with monomers initially aggregating to form dimeric and trimeric species. As these complexes grow in size, an increasing proportion of the interface of the added monomer molecules achieves contact with the micellar hydrocarbon segments until the maximum degree of hydrocarbon/hydrocarbon interaction is reached.

For many simple long-chain linear amphiphilic surfactants, this results in the formation of a perfectly spherical complex which produces the maximum surfactant packing density. In this case, the micellar structure is complete, but difficulties may occur with some types of charged surfactants due to the repulsive charge on the head groups, and these interactions must be counterbalanced with the structure, which will result in different types of molecular arrangements within the micelle. A more detailed theoretical examination of the origins of the free energy changes which occur on eliminating the hydrocarbon/water interaction is described in considerable detail in an early classic text The Hydrophobic Effect by Tanford (1).

Several different models have been used to describe the overall process of micellization, and these have been well documented in the literature (2).