Materials Science, Volume 1: Structure 9783110495126, 9783110495348, 9783110492729

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Gengxiang Hu, Xun Cai, Yonghua Rong Materials Science

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Gengxiang Hu, Xun Cai, Yonghua Rong

Materials Science

Volume 1: Structure Revised and translated by Zhuguo Li, Jie Dong, Qiang Guo, and Guozhen Zhu

Authors Prof. Gengxiang Hu School of Material Science and Engineering Shanghai Jiao Tong University No. 800 Dongchuan Road Shanghai 200240, China Prof. Xun Cai Shanghai Key Laboratory of Materials Laser Processing and Modification School of Material Science and Engineering Shanghai Jiao Tong University No. 800 Dongchuan Road Shanghai 200240, China Prof. Yonghua Rong School of Material Science and Engineering Shanghai Jiao Tong University No. 800 Dongchuan Road Shanghai 200240, China

ISBN 978-3-11-049512-6 e-ISBN (PDF) 978-3-11-049534-8 e-ISBN (EPUB) 978-3-11-049272-9 Library of Congress Control Number: 2020949178 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Shanghai Jiao Tong University Press, Shanghai and Walter de Gruyter GmbH, Berlin/Boston Cover image: P. DUMAS/EURELIOS/SCIENCE PHOTO LIBRARY Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

Foreword Shanghai Jiao Tong University pioneered in setting up the Department of Materials Science and Engineering in China in 1988, which began with the Department of Metallurgy in 1958, and renamed the School of Materials Science and Engineering in 1997. In the background of discipline development, Materials Science textbook, written by Professor Gengxiang Hu, Professor Xun Cai, and Professor Yonghua Rong, was published by Shanghai Jiao Tong University Press in 2000; then the second and third editions were successively published in 2006 and 2010, respectively. This textbook has been used in more than 30 universities or colleges in China so far. In the textbook, a trial has been performed, that is, the knowledge of metals, ceramics, and polymers is not individually described in chapters, but is distributed in knowledge structure of materials science: atomic structure and interatomic bonding, the structure of solids, crystal defects, deformation and recrystallization, phase transformation and phase diagrams, and diffusion. Such a consideration is to let students better understand the difference among metals, ceramics, and polymers. This textbook focuses on the knowledge of structural materials based on mechanical properties and also refers to functional materials based on physical properties. Although it is devoted to materials science, the basic knowledge of the synthesis, processing, and treatment of various materials in practical applications is also introduced because both the theoretical background and engineering common sense are a prerequisite for any engineer to be successful. As a part of China–Germany cultural exchange, in 2015, De Gruyter invited Shanghai Jiao Tong University Press to publish the Materials Science as English edition in Germany. For this reason, Professor Zhuguo Li built a team in the School of Materials Science and Engineering; its members were selected from professors and associate professors who have been teaching this course for many years. This English edition of Materials Science adds several sections, cancels some content, corrects some mistakes, and rearranges chapters based on the Chinese edition textbook according to the development of materials science and teaching experience. I read and revised the whole manuscript of Materials Science (English edition), and I hope that this textbook is also welcomed by the European undergraduates. Yonghua Rong Professor School of Materials Science and Engineering Shanghai Jiao Tong University December, 2018

https://doi.org/10.1515/9783110495348-202

Contents Foreword

V

About the authors About the translators

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Chapter 1 Atomic structure and interatomic bonding 1 1.1 Atomic structure 1 1.1.1 Substance construction 1 1.1.2 Structures of atoms 2 1.1.3 Electronic Structures of atoms 2 1.1.4 Natures of the elements in the periodic table 6 1.2 Interatomic bonding 7 1.2.1 Metallic bonding 7 1.2.2 Ionic bonding 8 1.2.3 Covalent bonding 9 1.2.4 Van der Waals bonding 10 1.2.5 Hydrogen bonding 12 1.3 Polymer chain structure 13 1.3.1 Short-range structure of the polymer chains 14 1.3.2 Long-range structure of polymer chains 22 Chapter 2 The structure of solids 30 2.1 Fundamentals of crystallography 30 2.1.1 Space lattice and unit cells 30 2.1.2 Miller indices for directions and planes 36 2.1.3 Stereographic projections 44 2.1.4 Crystal symmetry 48 2.2 Metallic crystal structures 57 2.2.1 Three typical metallic crystal structures 58 2.2.2 Close-packed crystal structures and interstitial sites 2.2.3 Polymorphism and allotropy 67 2.3 Phase structures of alloys 68 2.3.1 Solid solution 69 2.3.2 Intermediate phases 77

63

VIII

2.4 2.4.1 2.4.2 2.4.3 2.5 2.6 2.6.1 2.6.2 2.6.3 2.7 2.8 2.8.1 2.8.2 2.9

Contents

Ionic crystal structure 92 Structural rules of ionic crystals 93 Typical ionic crystal structures 95 Crystal structure of silicate 103 Covalent crystal structure 108 Crystal structures of the polymers 110 Crystalline forms of polymers 110 The models of the polymer crystal structures 113 The cell structures of the crystalline polymers 115 Quasicrystal structure 119 Liquid crystalline structure 121 Molecular structure and classification of liquid crystals Structure of liquid crystal 122 Amorphous structure 124

121

Chapter 3 Crystal defects 129 3.1 Point defects 129 3.1.1 The formation of point defects 129 3.1.2 Equilibrium concentration of point defects 132 3.1.3 Movement of point defects 134 3.2 Dislocations 134 3.2.1 Basic types and characteristics of dislocations 135 3.2.2 Burgers vector 139 3.2.3 The movement of dislocation 143 3.2.4 Elastic properties of dislocations 147 3.2.5 Generation and multiplication of dislocations 156 3.2.6 Dislocations in real crystals 159 3.3 Surface and interface 171 3.3.1 Surfaces 171 3.3.2 Grain boundaries and subgrain boundaries 173 3.3.3 Twin boundaries 182 Chapter 4 Deformation and recrystallization 186 4.1 Elasticity and viscoelasticity 186 4.1.1 Nature of elastic deformation 186 4.1.2 Elastic deformation characteristics and elastic modulus 4.1.3 Imperfect elasticity 191 4.1.4 Viscoelasticity 193 4.2 Plastic deformation of crystals 195 4.2.1 Plastic deformation of single crystals 195

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4.2.2 4.2.3 4.2.4 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.6

Plastic deformation of polycrystals 210 Plastic deformation of alloys 214 Effect of plastic deformation on microstructures and properties 224 Recovery and recrystallization 232 Changes in the microstructures and properties of the cold-deformed metals during annealing 233 Recovery 235 Recrystallization 238 Grain growth 249 Microstructure after recrystallization 256 Dynamic recovery and recrystallization during hot working 260 Dynamic recovery and dynamic recrystallization 260 Effect of hot forming on microstructures and properties 265 Creep 268 Superplasticity 271 Deformation characteristics of ceramics 275 Deformation characteristics of polymers 277

Chapter 5 Diffusion in solids 282 5.1 Phenomenological theory 282 5.1.1 Fick’s first law 282 5.1.2 Fick’s second law 283 5.1.3 Solutions to diffusion equations 285 5.1.4 Diffusion in substitutional solid solutions 295 5.1.5 Solution to diffusion equations when diffusion coefficient is a function of concentration 299 5.2 Thermodynamic analysis of diffusion 302 5.3 Atomic theory of diffusion 304 5.3.1 Diffusion mechanisms 304 5.3.2 Atomic jump and diffusion coefficient 307 5.4 The activation energy of diffusion 313 5.5 Random walk and diffusion length 314 5.6 Factors that affect diffusion 316 5.7 Reaction diffusion 320 5.8 Diffusion in ionic solids 321 5.9 Molecular motion in polymers 325 5.9.1 The origin of molecular chain motion and its compliance 325

IX

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Contents

5.9.2

The molecular motion of polymers and the influencing structural parameters 327 The molecular motion of polymers at different mechanical states 331

5.9.3

References Index

339

337

About the authors Professor Gengxiang Hu graduated from the Mechanical Engineering Department of Shanghai Jiao Tong University (SJTU) in 1952 and worked as assistant professor at the Mechanical Engineering Department. From 1956 to 1979, he was a lecturer at the Department of Metallurgy at SJTU, and from 1979 to1980, he was an associate professor in Materials Science Department at SJTU, and deputy director of the Institute of Superalloy. From 1980 to 1982, he was a visiting scholar at the University of California, Berkeley, and Lawrence National Laboratory for further study on “embrittlement of iron-based superalloys.” In 1985, he was promoted to professor at the Department of Materials Science. Professor Hu coauthored the textbook Metal Science, which has been widely praised by teachers and students since it was published in 1980. It is used as the undergraduate textbook in more than 50 universities in China. In 1987, it won the first prize as an excellent textbook of China National Machinery Commission, and in 1988, it won the National Excellent Textbook award of China National Higher Education Commission. He also received a number of awards for excellence in teaching at SJTU. In terms of scientific research, Professor Hu is mainly engaged in the research of “superalloys” and “superplasticity” of metal materials. He has published more than 100 scientific research papers in domestic and foreign journals. In order to commend Professor Hu for his outstanding contribution to the development of higher education in China, he is entitled to a special allowance from the State Council.

Professor Xun Cai Born on 29 November 1943 in Zhejiang, China, he graduated from the Department of Materials Science and Engineering at the Harbin Institute of Technology in 1965 and received PhD in physics at the Technical University of Vienna in 1988. After college, he was employed at first as an engineer and then as a senior engineer at Shanghai Electric Micromotors Research Institute. In 1988, he was appointed as an associate professor and then promoted to university professor at the Department of Materials Science and Engineering of Shanghai Jiao Tong University. In 1992, he received the title of “Outstanding Scientist Who Made Notable Contributions to China” from the Chinese government. Professor Cai’s research concentrated on the areas of electric contact alloys, thin-film materials, and surface modification technologies. He has published 10 books and more than 300 research papers on materials science.

Professor Yonghua Rong graduated from the University of Science and Technology, Beijing, in 1976, majoring in metal physics, then become an assistant professor at Shanghai Jiao Tong University (SJTU). From 1994 to 1995, he, as a senior visiting scholar, studied 718 superalloys using transmission electron microscope at Lehigh University, USA. In 1999, he was promoted to professor in the School of Materials Science and Engineering at SJTU. He wrote the textbook Introduction to Analytical Electron Microscopy (Chinese version) for postgraduates and the monograph

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About the authors

Characterization of Microstructures by Analytical Electron Microscopy (AEM) (English version). He also presided and wrote Microstructural Characterization of Materials for undergraduates. Professor Rong was a director of “Phase Transformation Theories and Their Applications” group from 1998 to 2010. He is interested in phase transformation in nanomaterials and shapememory alloys, especially advanced high strength steels, which reflect in his National Natural Science Foundation projects and one key project as well as in his monograph Advanced High Strength Steels and Their Process Development. He published more than 200 scientific research papers in domestic and foreign journals. He won the second prize of Natural Science Award of Chinese Universities in 2000, the second prize of National Teaching Achievement Award in 2001, the first prize of Shanghai Teaching Achievement Award in 2000, and the second prize of Shanghai Science and Technology Progress Award in 2004. He also received six awards for excellence in teaching at SJTU.

About the translators Professor Zhuguo Li School of Materials Science and Engineering Shanghai Jiao Tong University Professor Zhuguo Li has obtained his bachelor degree in welding and automation in 1994 from Shanghai Jiao Tong University, master’s degree in welding engineering in 1997 from the same university, and PhD in materials processing science from Osaka University in 2004. Since then, he joined Shanghai Jiao Tong University. He became a full-time professor since 2010 and a distinguished professor since 2020. Now he is the vice dean of School of Materials Science and Engineering, Shanghai Jiao Tong University, and the director of Shanghai Key Laboratory of Materials Laser Processing and Modification. He also serves as the deputy secretary general of China Welding Society. His research interests include laser welding, laser cladding, and laser additive manufacturing, and he is the coauthor of more than 150 SCI papers published on peer-review journals. He won the first prize of Science and Technology Award from CMES in 2016, and the first prize of Science and Technology Progress Award from Shanghai Municipality in 2015 and 2019.

Jie Dong received his BSc from Northeastern University, Shenyang, China, in 1998, and PhD in materials forming from Northeastern University, China, in 2004. He joined Shanghai Jiao Tong University, Shanghai, China, as a postdoctoral fellow, and then became a lecturer, a vice professor, and a professor. From 2006 to 2007, he served as a visiting scholar with the Research Center of Dresden Rossendorf, Dresden, Germany. He has taught two courses, Fundamentals of Materials Science and Principles of Materials Processing, for 10 years. He was voted as the most popular teacher by students three times in the School of Materials Science and Engineering. He focused his research on plastic forming and application study of advanced light alloys. He and his group have developed some high-performance magnesium and aluminum alloys by rare earth alloying and some large-scale parts by some advantaged forming technology such as subcurrent semicontinuous casting, isothermal hot rolling and forging, differential temperature drawing, and hot spinning. More than 100 parts of high-performance Mg and Al alloys have been applied. He has authored or coauthored over 100 peer-reviewed journal papers and holds over 30 patents, and some of which have been converted into commercial products.

Qiang Guo State Key Lab of Metal Matrix Composites, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China; Tel/fax: +86-21-54742392; Email: [email protected] Qiang Guo is an associate professor at the State Key Lab of Metal Matrix Composites, Shanghai Jiao Tong University (SJTU). He received his BSc degree in microelectronics from Peking University in 2005, and MEng in materials science and engineering from Massachusetts Institute of Technology (MIT) in 2006. He obtained his PhD in advanced materials in 2010 from the National University of Singapore under the

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About the translators

Singapore-MIT Alliance program. From 2010 to 2012, he was a postdoc in the Division of Engineering and Applied Sciences in California Institute of Technology. His current research focus is on the fabrication and mechanical behavior of metal matrix composites. He has been teaching the SJTU undergraduate core module of “Fundamentals of Materials Science and Engineering” since 2014 fall. Guozhen Zhu, the Canada Research Chair Tier II (in mechanical and functional design of nanostructured materials), now is an assistant professor at the University of Manitoba, Canada. She worked as a research professor from 2014 to 2017 at Shanghai Jiao Tong University, China. She attended Tsinghua University (Beijing) for her undergraduate studies at Materials Science and Engineering (2007) Department and followed her interest in materials science at McMaster University, Canada, where she completed her master’s degree in 2009 and her PhD in 2012. She has published 34 papers in prestigious journals such as Nature, J Phys Chem C, and Scientific Reports. She won national and international awards such as Gerard T. Simon awards from Microscopical Society of Canada and awards from European Microscopy Congress for her research.

Chapter 1 Atomic structure and interatomic bonding Material is referred to a substance with present or expected future applications for human beings and has been considered as the substantial basis of the national economy. The development of industrial and agricultural production, the science and technology progress, and the improvement of people’s living standards are relying on an extensive sort of materials, such as metals, ceramics, and polymers, having various properties that meet the different requirements. For a long time, people have been studying and investigating the factors that influence the properties of the materials and the ways to improve their performance when using the materials. The researches have shown that the atomic structure of the elements within the material is the fundamental factor that determines the material performance. It includes interactions and combinations among atoms, the spatial distribution and movement of atoms or molecules, and the morphology of atomic clusters and so on. Therefore, the first topic we need to understand is the microscopic structure of the material and its internal construction and microstructure state, so that we can find out ways to improve and develop new materials from its internal contradictions. A substance is made up of atoms, and an atoms consist of a positively charged nucleus is located in the center of the atom and negative electrons outside the nucleus. The electronic structure of atoms determines the interatomic bonding of the atoms with each other. Therefore, the understanding of the electronic structure of atoms is not only helpful to classify the materials, but also useful to find out the fundamental principles of physical, chemical, and mechanical properties of materials.

1.1 Atomic structure 1.1.1 Substance construction As is known to all material is composed of numerous fine particles gathered together in some certain way. These particles may be considered as molecules, atoms, or ions. The molecule is a kind of particle that can exist alone and maintain its chemical properties. The volume of a molecule is small, for example, the diameter of a water molecule is about 0.2 nm. However, the mass of different molecules is

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Chapter 1 Atomic structure and interatomic bonding

different. For instance, H2 is the smallest molecule in the world and its relative molecular mass is only 2; however, a natural polymer compound, such as proteins, has an average relative molecular weight of up to a few millions. Further analysis showed that the molecule is composed of a number of smaller particles, i.e., atoms. In a chemical reaction, the molecules can be further broken down into the atoms; whereas, the atoms are indivisible. Therefore, an atom is the smallest particle in a chemical reaction. However, the atoms are not the basic particles of the material in quantum mechanics. They have complex structures. The atomic structure directly affects the interatomic bonding.

1.1.2 Structures of atoms The recent scientific experiments show that the atom is composed of the protons and neutrons, as well as the electrons. The neutron in the nucleus is electrically neutral and the proton carries a positive charge, which possesses exactly the same charge as that of an electron, whose charge is equal to –e (e = 1.6022 × 10−19 C). The electrostatic attraction shows that the negatively charged electrons are tightly bound around the positively charged nucleus. As a whole, an atom is electrically neutral because in an atom the number of protons and electrons is the same. An atom is volumetrically small. The atomic diameter is about the order of 10−10 m, while the diameter of nucleus is only about the order of 10−14 m. Besides, the atom mass is mainly undertaken by the nucleus, while the surrounded electrons are located within a relatively enormously large space in an atom. The mass of an neutron or an proton is approximately 1.67 × 10−24 g, while the mass of electron is about 9.11 × 10−28 g; thus a proton is almost 1,833 times an electron.

1.1.3 Electronic Structures of atoms The electrons revolve around the nucleus of an atom, which is just like a “cloud” with a negative charge around the nucleus, so it is called the electronic cloud. The electrons have the duality of the wave and particle. The electronic motion doesn’t have fixed orbits but the position probability of its motion outside the nucleus can be determined through the statistical methods according to the energy levels of electrons. The electrons with a lowest energy usually appear near the nucleus, while the electrons with the highest energy move far away from the nucleus. In quantum mechanics, the basic equation reflecting the motion of the microscopic particles is the Schrodinger’s equation. The wave function obtained by solving the equation describes the state of the electronic movement and the occurrence probability in

1.1 Atomic structure

3

somewhere outside the nucleus. Videlicet, the spatial position and electron energy in an atom can be determined by the four quantum numbers shown as follows: (1) Principal quantum number n It determines the electron energy and its average distance with the nucleus of an atom, i.e., the quantum shell of the electron, as shown in Fig. 1.1. n can only be a positive integer, such as 1, 2, 3, 4,. . ., and it explicitly represents the quantization of the electron energy in an atom, thus being called the principal quantum number. The principal quantum number (or quantum shell) is represented by a capital letter. For example, n=1 means that the quantum shell has the lowest energy, and it is the closest orbit to the nucleus, named as K-shell, according to the old quantum theory. The numbers 2, 3, 4, . . . indicate higher energy levels named L-, M-, N-, . . ., respectively.

K shell (n = 1) L shell (n = 2)

11 protons 12 neutrons

M shell (n = 3)

Fig. 1.1: The electron distribution of K-, L- and M-shells in the Na atomic structure.

(2) Azimuthal quantum number li It indicates the energy level of an electron in the same quantum shell, i.e., the electron subshell. The azimuthal quantum number (also referred to as the secondary quantum number) l can be 0, 1, 2, 3, . . .. For example, when n equals 2, it means that two orbital azimuthal quantum numbers exist, l2 = 0 and l2 = 1, that is, L-shell consists of two electronic subshells according to the electronic energy difference. For convenience, the electron energycorresponding to the orbit with an azimuthal quantum number of li is commonly labeled as lowercase letters as follows:

4

Chapter 1 Atomic structure and interatomic bonding

In the same quantum shell, the electron subshell energy is increased by the order of s, p, d, f, and g. The electronic cloud pattern varies in different electronic subshells. For instance, the electronic cloud of s subshell is spherical with the atomic nucleus sitting at the center, and the electronic cloud of p subshell is spindle. (3) Magnetic quantum number mi It gives the energy series or orbital number of each azimuthal quantum number. In each electron subshell, the total number of a magnetic quantum number is 2li + 1. If li equals 2, the magnetic quantum number is 2 × 2 + 1 = 5, with the values of −2, −1, 0, + 1, + 2. Magnetic quantum number determines the spatial orientation of the electron cloud. If the space occupied by the electron cloud has a certain pattern and extension direction in a certain quantum shell, we name it as an orbital, then the four subshells, s, p, d, and f, will have 1, 3, 5, and 7 tracks, respectively. (4) Spin quantum number si It reflects different electron spin directions Si is specified as +1/2 and −1/2, which illustrate the spinning clockwise and counterclockwise and are commonly represented by “↑” and “↓”, respectively. In the nucleus with no less than one electron, the arrangement of electrons outside the nucleus follows the three principles, i.e., the minimum energy principle, the Pauli exclusion principle, and the Hund’s rule. (a) Minimum energy principle The electrons are arranged in a way to achieve the minimum energy of the system. Specifically, the electrons will always fill the orbitals with the lowest energy firstly and subsequently move up to the higher energy orbitals after the orbitals with the lower energy are fully occupied. In other words, as the K-shell is filled up with electrons, the L shell will then be filled with the remaining electrons, and then comes to the M-shell . . ., shells will be filled from inside out. In the same shell, the electrons are arranged by the order of s, p, d, and f, accordingly. (b) Pauli exclusion principle The same set of quantum numbers within the same system cannot be shared by two electrons. In other words, any two electrons would not possess the same four quantum numbers mentioned earlier. Therefore, for a shell with the principal quantum number of n, the maximum of electrons is 2n2. The Pauli exclusion principle allows either a configuration that consisting of a pair of electrons in one single p orbital, but with subsequent opposite spin; or each of the two electrons should exist in a separate p orbital, but with either same (parallel) or opposite (antiparallel) spin.

1.1 Atomic structure

5

(c) Hund’s rule Any orbital in a sublevel is singly occupied before it is doubly occupied. Meanwhile, all the electrons in the singly occupied orbitals possess the same spin (to maximize total spin). A relative stable system containing the atoms with the lowest energy is obtained when the orbitals are fully occupied, half occupied or completely empty. For instance, considering the ground state of carbon, nitrogen, and oxygen, the electron configurations are shown in Fig. 1.2.

Fig. 1.2: The electronic configuration of carbon, nitrogen, and oxygen atoms.

However, it should be noticed that the electron configuration does not always follow the order of the rules mentioned above. Especially in cases where atoms have relatively large atomic number, and d and f levels start to be occupied neighboring levels would have some overlaps. For instance, the 4s sublevel possesses lower energy level than the 3d sublevel. Also, the energy level of the 5s is lower than those of the 4d and 4f levels. Thus, it is possible that the inner layer has not yet to be occupied before the next shell begins to assign electrons. Theoretically, the electronic configuration of the iron atom seems to be:

However, its actual electron configuration of iron is

It deviates from the theoretical electron configuration with an unfilled 3d energy level which makes iron exhibit magnetic behavior. It should be stated that in a single atom, the electrons are located at different energy levels whose motion state can describe through four quantum numbers (ni, li, mi, and si) obtained by solving the Schrodinger’s equation. However, a solid consists of a large number of atoms being brought together and bonded, thus, the ordered atomic structure in crystalline material can be formed. Since the atoms possess close proximity, electrons from one atom are acted by the

6

Chapter 1 Atomic structure and interatomic bonding

electrons and nuclei of the adjacent atoms. Such phenomenon causes each distinct atomic state to split into closely spaced electron states in the solid, thus forming an electronic energy band. Within each electron energy band, the energy states are discrete, yet the difference between adjacent states is quite small. Moreover, gaps would probably exist between the adjacent bands; in most of the cases, energy lying within the band gaps would not be available for electron occupancy. Solid energy band theory is the foundation of condensed matter physics, which is the basic theory especially in the study of the motion of an electron in the solid. Some properties of solids, such as electrical and magnetic properties, are closely related to the electronic structure of the solids. Relevant information will be described in Chapter 10.

1.1.4 Natures of the elements in the periodic table Element is defined as a specie of atoms possessing a same nuclear change number (i.e., the atomic number, Z). The periodic law illustrates the periodicity of the outer electrons around the nucleus when the elements are arranged in the order of increasing atomic numbers. The periodic table tabularly organizes the chemical elements in the order of the periodic law, in which the location of an element represents its atomic structure, as well as the properties. The elements in the same row of period share the same trend in radius, electron affinity, ionization energy, and electronegativity. When moving from the left to the right side along a period, the atomic number increases, however, the atomic radius usually decreases. Also, the ionization energy tends to increase. Elements with the tendency to donate valence electrons are defined electropositive and elements acquire additional electrons are defined electronegative. Apparently, from left to right in a periodic table, the electronegativity tendency increases and the electropositivity tendency decreases. The lower electron affinity, ionization energy, and electronegativity indicate more metallic character of an element. Conversely, the nonmetallic property increases along with the higher values of such properties. Each vertical column in the periodic table refers to a group or a family, in which the atomic radius, as well as the ionization energy, increases, whereas the electronegativity decreases from the top to the bottom. Therefore, the metallic character increases as the atomic number increases in the same group. Similarly, since the isotopes of the same element occupy the same position in the periodic table, despite their different mass, they share exactly the same chemical properties. The periodic table can derive relationships between the element properties and predict the reaction ability of atoms with other elements. The chemical valence of an element is closely related to the electronic structure of the atoms, especially the

1.2 Interatomic bonding

7

number of the outermost electrons (valence), which can be determined according to its location in the periodic table. For example, the outermost layer of the argon atoms, 3s + 3p, is completely filled by eight electrons, which means that the valence electron number of the argon atom is zero. Consequently, it is stable when reacting with other atoms, and is classified as an inert element. However, as an alkali metal, potassium has only one electron in the outermost layer of each atom, and thus the valence is one. The only electron is easy to lose and makes the 4s level empty. The electronic distribution of the transition elements is very complex. For example, the electrons of sublevels s and d, and even those of the sublevel f may become valence electrons, leading to a variety of valence for transitions elements. In short, the element properties, atomic structure, and the location of an element in the periodic table have a close relationship with each other. Therefore, we can figure out the atomic structure and the certain properties of an element according to its location in the periodic table and vice versa.

1.2 Interatomic bonding The interatomic bonding is an issue to find out how to bond two or more atoms to form molecules or solids. A molecule is composed of atoms by the interatomic bonding, and then the solid is composed of atoms or molecules. The interatomic bonding can be divided into two types: the chemical bond and the physical bond. A chemical bond is the primary bonding, which includes the metallic bond, ionic bond, and covalent bond. However, the physical bond is a secondary bonding, which is also known as the Van der Waals bond. In addition, there is another kind of bond called the hydrogen bond, which is a mixture of the chemical bond and the Van der Waals bond. All the bondings are all described in detail further.

1.2.1 Metallic bonding A typical characteristic of the metallic atom is its small valence electron number: one, two, or at most three. The fewer electrons are easy to bond of the atomic nucleus and move throughout the whole metal. As a result, metallic atoms would “share” the valence electrons with each other in a metallic aggregate (usually solid) to form electron clouds. The bond is formed by the free electron–positive ion interaction, which is called the metallic bonding, as shown in Fig. 1.3. A vast majority of the metals are bonded by the metallic bond, and its basic characteristic is the sharing of the electrons. The metallic bond has neither saturation nor orientation, thus a close-packed structure with a lower energy is easy to form due to a higher probability of the combination among the atoms. The metallic bond will not be destroyed when the metal

8

Chapter 1 Atomic structure and interatomic bonding

is deformed by an external force and only the mutual position among atoms is changed leading to good ductility properties for metals. Besides, due to the presence of free electrons, the metals have a good electrical and thermal conductivity.

Fig. 1.3: The schematic diagram of a metallic bonding.

1.2.2 Ionic bonding Most of the salts, alkalis, and metal oxides are mainly composed of ionic bond. The nature of this kind of bond is that the metallic atom donates its valence electrons to the nonmetallic atom, which after that transforms into a positively charged ion. Then the nonmetal atom becomes a negative ion by accepting the valence electrons. In this way, the positive and the negative ions are combined together by the electrostatic force between them. The basic combination unit is ions rather than atoms. The positive and negative ions arrange alternately when they combine through the ionic bond, in which the attractive force reaches the maximum between the opposite charges; however, the repulsive force reaches the minimum between the same charges. Therefore, the ionic bonding has neither saturation nor orientation and the factors that determine the crystal structure of ions are the charge of positive and negative ions, as well as the geometrical features. In addition, the coordination number of the ion in the ionic structure is usually high (Fig. 1.4). In general, the melting point and the hardness of the ionic crystals are higher than those of metal crystals due to their strong and solid electrostatic force between the positive and negative ions. In addition, the ionic crystals have a good electrical insulation because it is difficult to generate free electrons. However, when they are at the molten state in the high temperature, the positive and negative ions can move freely under the external electric field, exhibiting the ionic conductivity.

1.2 Interatomic bonding

9

Fig. 1.4: The schematic diagram of the ionic bonding in NaCl.

1.2.3 Covalent bonding Covalent bonding is the chemical bond involving the sharing of the electron pairs between two or more atoms that have the similar electro-negativities. The covalent bonds are divided into the polar and nonpolar bonds depending on the deviation of the sharing electron pairs with respect to the two bonding atoms. In the H2 molecule, the hydrogen atoms share the electron pair via the nonpolar covalent bonding. The covalent bond plays an important role in semimetals (carbon, silicon, tin, germanium, etc.), polymers, as well as the inorganic nonmetallic materials. Figure 1.5 depicts a schematic of the covalent bonding between oxygen and silicon atoms in SiO2.

Fig. 1.5: The schematic of the covalent bonding between oxygen and silicon atoms in SiO2.

10

Chapter 1 Atomic structure and interatomic bonding

Except that the electron cloud of sub-level s is of the spherical symmetry, the atomic structure theory shows that the electron clouds of other sublevels, such as those of p and d, have certain degrees of orientations. When a covalent bond is formed, it naturally has an orientation and its distribution strictly follows such orientation due to the maximum overlap of the electron clouds. As a result, a significant characteristic of the covalent bonding is the directionality i.e., the bonding occurs in a certain direction in the space, which is quite different from the ionic bonding that is free of orientations. Besides, when an electron forms an electron pair with a specific electron, this electron will no longer form a pair with other electrons. Moreover the number of sharing electron pairs allows all atoms to have eight electrons in the outermost layer, which reflects the saturation of the covalent bond. Furthermore, each bond has a relatively small coordination number. The strength of the covalent bonding is high, thus the covalent crystals usually have stable structures with high values in the melting point, hardness, and brittleness. However, the materials formed via the covalent bonding are always insulators and have a weak electrical conductivity because of the poor mobility of the bonding pairs between adjacent atoms.

1.2.4 Van der Waals bonding Due to the electrostatic forces acting among atoms and molecules, their bondings are invariable. One may wonder what causes neutral molecules such as CO2 and F2, as well as “inert” gas atoms, such as He, Ne, and Ar atoms, to condense and eventually reach the solid state at an adequately low temperature. It has been proposed that the weak attractions that lead to the bonding of the atoms and molecules as mentioned earlier can be conceived as the consequence of dipole interactions. This kind of bonding is named Van der Waals bonding. A dipole forms if the gravity centers containing positive and negative charges included in a molecular body are separated. Two dipoles can interact with each other and their combined energy would be reduced when positive end of one dipole is oriented and thus becoming closer to the negative end of the other dipole. With this kind of relatively weak and instantaneous induction effect of the electrical dipole moment, the atoms or molecules of the stable atomic structure are bonded together by the Van der Waals forces (as shown in Fig. 1.6), which depicts the forces between the polar molecules. Van der Waals forces include the electrostatic force, induced force, and dispersion force. The electrostatic force is caused by the electrostatic interaction of the permanent dipoles between the polar atoms or molecules, and the magnitude of this kind of force is inversely proportional to both the absolute temperature and the polar distance. The induced force is a kind of interaction between a permanent dipole and the induced dipole in polarless molecules since a molecule possessing a permanent dipole can induce another dipole in an intrinsically dipole-less molecule/atom with a spherically symmetric distribution of electrical charges when the polar atom (or molecule) and

1.2 Interatomic bonding

11

the nonpolar atom (or molecule) interact with each other. The magnitude of the induced force is independent of the temperature and inversely proportional to the polar distance. The dispersion force is the interaction force of atomic instantaneous dipole induced by the motion of some electrons. This dipole can induce a dipole in a neighboring atom instantly, and the electrical charge distribution fluctuations in both neighboring atoms are correlated with each other. Such correlation performs the basis for a Van der Waals force with weak attraction, thus inert gases can solidify. The magnitude of the dispersion force is also independent to the temperature and is inversely proportional to the polar distance. In the ordinary polar-less polymer materials, the dispersion force can be as high as 80–100% of the Van der Waals force between the molecules.

Fig. 1.6: The schematic of the Van der Waals forces between the polar molecules.

As a kind of physical bond, Van der Waals bond is a second valence bond with no saturation or orientation. It is generally found in a variety of the molecules and has a significant effect on the materials properties, such as melting point, boiling point, and the solubility. Its bonding energy is usually less than the chemical bonding energy by 1–2 orders in magnitude and is far less stable than the chemical bonding. For example, the water becomes vapor when is heated to the boiling point at which Van der Waals bonding in the water is destroyed. However, it requires an extremely high temperature to split the covalent bond between the hydrogen and oxygen atoms. The bonding energies of some materials are listed in Table 1.1. It should be noted that since the relative molecular mass of the polymer materials is very high, the total Van der Waals forces are larger than the chemical bonding forces although the energy of an isolated Van der Waals bond is much less than that of the chemical bonding. Therefore, the chemical Table 1.1: The bonding energy and melting point of some materials. Material

Bonding type

Bond energy (kJ mol–)

Melting point/°C

(ev per atom, ion, molecule)



.

−

Al



.



Fe



.

,

W



.

,

Hg

Metal bond

12

Chapter 1 Atomic structure and interatomic bonding

Table 1.1 (continued ) Material

Bonding type

Bond energy –

(kJ mol ) NaCl

Ionic bond

MgO Si

Covalent bond

C (diamond) Ar

Van der Waals force

Cl NH

Hydrogen bond

HO

Melting point/°C

(ev per atom, ion, molecule)



.



,

.

,



.

,



.

>,

.

.

−



.

−



.

−



.



bonds may be broken before all the Van der Waals forces are destroyed. Consequently, the polymer is often exist in the solid or the liquid state instead of the gaseous state. The Van der Waals force may have a significant effect on the nature of the materials. For instance, one important reason why different polymers have distinct properties is that the Van der Waals forces between the polymer molecules are different.

1.2.5 Hydrogen bonding The hydrogen bonding is a special polar-molecule bonding, and it is caused by the combination of one hydrogen atom with two atoms with large electronegativity and small radius at the same time, which exists in HF, H2O, NF3, etc. Since there is only one electron outside the nucleus of a hydrogen atom, the electrons in these molecules are shared by other atoms, and the combined hydrogen-end is exposed with a positive charged atomic nucleus. Thus, it will be attracted by the negative end of the neighboring molecules and forms an intermediate bridge, called the hydrogen bridge (Fig. 1.7). The hydrogen bonding has the features of saturation and orientation.

Fig. 1.7: The schematic of the HF hydrogen bonding.

1.3 Polymer chain structure

13

The hydrogen bond also belongs to the secondary valence bond because it is also combined with the dipole attraction of the atoms (molecular or atomic groups). Its bonding energy is between those of the chemical bonding and the Van der Waals force that exists in the molecule or inter-molecule. The hydrogen bonding is very important in the polymer materials, such as cellulose, nylon, protein, and so on, which have strong hydrogen bondings and the very special crystal structures and properties. It is worth mentioning that materials with a single bonding in the world are rare, and most of the internal atomic bondings are often a mixture of various bonds. For instance, the metallic bonding is dominant in the metallic materials; however, a small amount of the covalent bonding appears in the transition metals, such as W, Mo, and other atomic bonding. That is why these materials have high melting points. The intermetallic compound has a certain ionization tendency, as well as a metal-ion mixture bonding due to the difference in the electronegativity between the metal and the ion. The phenomenon of the mixed covalent and the ionic bonds is more common in the ceramic compound. The ratio of the ionic bonding in the compound AB depends on the difference in the electronegativity of the elements A and B, in which the larger the difference of the electronegativity is, the higher the proportion of the ionic bonding will be. The proportion of the ionic bonds, IC, in the compound AB can be estimated as follows: IC = ½1 − e − 0.25ðxA − xB Þ2  × 100% Where xA and xB represent the electronegativities of A and B, respectively. Another example is the diamond, which only has a covalent bonding, and the same condition happens to the group of the elements Si, Ge, Sn, and Pb. There is a certain proportion of free electrons remain when the covalent bonding is formed, which means that a part of the metallic bonding exists in these elements and the ratio of the metallic bonding is sequentially increasing from the top to the bottom in the same family of the periodic table. Meanwhile, all of the bonds in the materials form with Pb are metallic bonds. The structure recognition of the polymers and many organic molecules is rather complex, since they are combined by the covalent bonding in the long molecules; meanwhile, the Van der Waals force or hydrogen bonding exists between the chains. Section 1.3 is devoted to the structure of the polymer chains.

1.3 Polymer chain structure The basic element of the polymers is the organic macromolecule compound. Although the chemical composition and the structural units of the polymers are generally simple, the relative molecular mass of the polymers can reach tens of thousands or even millions. The structural units of the polymer may contain more than one type, in which each structural unit (also called mer) may have different configurations when hundreds of the units bond together in different ways. Besides, polymers exhibit the

14

Chapter 1 Atomic structure and interatomic bonding

heterogeneity of structure and the imperfection of the crystallization, as a result, the structures of polymers are quite complicated. The structure of polymer contains two parts: the polymer chain structure and the aggregation structure. The chain structure includes a short-range structure and a long-range structure. The short-range structure contains the construction and configuration. The construction is the study of the type and the arrangement of the atoms in the molecular chain, the classification of the chemical structure of the polymer chains, the bond sequence of the structural unit, the constituent of the chain structure and the branch, cross-link, and the end group of polymer. While the configuration is the study of the arrangement law of the substituents around the particular atoms in the space. The short-range structure belongs to the chemical structure, which is also known as the first-layer structure. The long-range structure is also called the secondlayer structure, which shows the size and the shape of a single polymer, the flexibility of the chain and the conformation of a molecule in different environments. Several conformations of a single polymer are shown in Fig. 1.8. The aggregation structure refers to the internal structure of a whole polymer material, including the crystalline structure, amorphous structure, orientation structure, liquid crystal structure, and the texture structure. The former four structures are also called the third-layer structures, describing how the molecules pile up in the polymeric micelles, while the texture structure refers to the arrangement and packing of the structure between different molecules or polymer additive molecules, known as the higher order structure.

Fig. 1.8: The schematic diagram of a single polymer conformation.

1.3.1 Short-range structure of the polymer chains 1. The chemical composition of the chain structural unit The previous researches proved that the polymer has a chain structure. The ordinary synthetic polymer is a structure of molecular chain called a polymer chain, which is polymerized through the polymerization and the polycondensation reactions of the monomers. The number of the repeating structural units in the polymer chain is called the polymerization degree. If the chemical compositions of the polymer chains are different, the chemical and the physical properties of these polymers are also

1.3 Polymer chain structure

15

different. According to the different chemical compositions of the structural unit, the polymer can be divided into some groups: the carbon chain polymer, the miscellaneous chain polymers, the element polymer, the ladder-like polymer, the double helical polymer, and other types of the polymer. Some common structural units of the polymer chains are depicted in Fig. 1.9.

Fig. 1.9: Common structural units of the polymer chains.

The above polymers belong to the type of the carbon chain polymer. The polymers are linked by covalent bonds between the carbon atoms, and they only differ in structure with different side groups. They are mostly obtained by additional polymerization reaction, thus not easy to hydrolyze. Besides, they are typical thermoplastics except for the polytetrafluoroethylene and can be used in the production of films, sheets, spinning, and so on. CH2

CH

CH

CH2

n

1,4-Poly butadiene

CH3 CH2

C

CH

CH2

n

Polyisoprene

CH3 Si O

n

Polydimethylsilane

CH3

Fig. 1.10: Rubber polymer structure.

The polymers in Fig. 1.10 are rubbers. The first two belong to the carbon chain polymer, in which 1, 4-polybutadiene is the synthetic rubber, also known as the butadiene styrene rubber, while polyisoprene is the natural rubber. The third polymer–polydimethylsiloxane (PDMS) is the silicone rubber, which belongs to the element polymer. Its

16

Chapter 1 Atomic structure and interatomic bonding

main chain contains the silicon and oxygen atoms, instead of carbon atoms, and its side groups contain the organic groups. The polymers in Fig. 1.11 are synthetic fiber materials. The polyacrylonitrile polymer is a carbon chain polymer, while nylon 66 is a heterochain polymer, in which the main chain contains not only the carbon atoms but also the nitrogen atoms. The polyethylene is a typical carbon chain structure, as shown in Fig. 1.12.

Fig. 1.11: Synthetic fiber materials.

Fig. 1.12: The schematic diagram of the repeat unit of (a) the chain structure and (b) the zigzag main chain structure of a polyethylene.

2. Geometrical morphology of a polymer chain The geometrical morphology of a polymer chain is determined by the functionality of the monomer. The functionality refers to the number of the positions in a monomer that can bond with other monomers. Generally, the geometrical morphology of a polymer is linear (Fig. 1.13(a)). The polymer chain can twist into a group, or stretch to form a straight line. This is determined by the flexibility of the polymer and the external conditions. For instance, a polyethylene has a typical linear chain structure. There are two positions in the chain links where two new monomers can attach to, so it is bifunctional. The bifunctional monomer can only form the chain structure. There is no chemical bonding between

1.3 Polymer chain structure

17

different linear polymers, so the polymers can slip when heated or under stress. Therefore, the linear polymer can be dissolved, and it can be melted when heated and is easy to be processed. If there are some monomers or impurities which have three or more functionalities during the polycondensation process of a linear polymer, or there is a radical chain transfer reaction in an addition polymerization process, or the second double bonding in the diene monomer is activated, a branch or cross-link polymer can be obtained. The branch polymer can be dissolved in a suitable solvent, or melted when they are heated, however, its aggregation structure and properties are significantly affected by the preexisting branch. Polymer chains can transform into a three-dimensional net-like polymer, called the network polymer, by the formation of branch chains in between. The cross-link polymer has a qualitative difference with the branch polymer, and it is insoluble and infusible. Only if the degree of cross-linking is not too large, it can swell in the solvent. Thermosetting resins, vulcanized rubber, wool, hair, etc. are the cross-link polymer structures.

Fig. 1.13: The schematic diagram of a (a) linear, (b) branch, (c) cross-link, and (d) three-dimensional network structure.

3. Bonding of the structural unit If the chemical compositions of the different polymers are the same, but the connection types of the structure unit are different, the properties of the polymers can vary significantly. In the polycondensation reaction and the ring-opening polymerization, the connection of structure units is unique. However, in an addition polymerization, the bonding type of the monomers is different.

18

Chapter 1 Atomic structure and interatomic bonding

a. Bonding of a homopolymer structural unit Except for the ethylene, whose structural units only have one bonding way in its polymer chain due to its symmetry, in general, the monoolefinic monomer has three bonding ways of the structural units according to the different substituent R: the head-to-head, the tail-to-tail, and the head-to-tail bondings, as shown in Fig. 1.14.

Fig. 1.14: Three bonding ways of the structural units according to the different substituent R.

Among all bondings mentioned above, the head-to-tail structure covers the majority situations, and its intensity is higher. The bonded structure of a diene monomer to form a polymer is more complex. In addition to the above three, extra bonding ways are possible according to the different positions where the double-bond opens. For instance, during the polymerization process of an isoprene, there are 1,2,3,4additional polymerization and 1,4-additional polymerization. The aforementioned products are illustrated in Fig. 1.15.

Fig. 1.15: Illustration of the polymerization process of an isoprene.

1.3 Polymer chain structure

19

b. Sequence structure of a polymer The polymer that is made of two or more kinds of the monomers is called copolymer. For a copolymer, in addition to the structural factors of the homopolymer, a series of the complex structural problems exist. For example, the biopolymer can be divided into the alternating copolymers, random copolymers, graft copolymers, and block copolymers on the basis of the bonding way. The schematic diagram is shown in Fig. 1.16 where the solid circles and open circles represent two different monomers, respectively. The block copolymers and graft copolymers are obtained by the continuous and two-stage polymerization reactions. Hence, they are the called multistep polymer.

Fig. 1.16: The schematic diagram of the (a) random, (b) alternating, (c) graft, and (d) block copolymers.

Different structures of the polymer have different influences on the properties of the materials. The random sequence of two monomers in the random copolymers not only changes the interaction among the structure units but also changes the interaction among the polymers. Therefore, the solution properties, the crystallization properties, and the mechanical properties of the copolymers are quite different from those of the homopolymer. For instance, while the polyethylene and the polypropylene are plastic, the random copolymer of ethylene–propylene with high content of propylene is rubber. In order to improve the properties of the polymers, several types of monomers are commonly used for the copolymerization to obtain a product which combines the advantages of these homopolymers. For instance, acrylonitrile butadiene styrene (ABS) resin is a terpolymer of the acrylonitrile, butadiene, and styrene, which exhibits the characteristics of all components. Among them, the acrylonitrile is a cyanogen radical that improves the chemical resistance of the polymer, the tensile strength, as well as the hardness of the product. While butadiene makes polymer exhibit rubberlike behavior and improves the impact toughness of the product. Also, styrene has a good influence on the high-temperature fluidity of the product, and thus it can be

20

Chapter 1 Atomic structure and interatomic bonding

easily hot formed and improve the surface finish of a product. Therefore, ABS resin is a kind of a high-performance thermoplastics. 4. Configuration of the polymer chain The configuration of the polymer chain is a kind of the stable geometric arrangement that is fixed by chemical bonds of the molecules. The change of such configurations should be performed through the broken and recombination of the chemical bonds. Isomers with a different configurations are divided into two kinds: the optical isomers and the geometric isomers. a. Optical isomers Four covalent bonds of the carbon atoms in the hydrocarbon molecules form a tetrahedral cone with the bond angle of 109°28ʹ. When the four groups of the carbon atoms are different with each other, the carbon atoms are called the asymmetric carbon atoms. Meanwhile, the asymmetric carbon atom can form two kinds of structures mirrored with each other with different optical properties, which are called as the optical isomers. In a polymer with the structure unit of -CH2-CHR- type, two types of the optical isomer units will exist as long as there is an asymmetric carbon atom because two chain ends are not exactly the same. The structure unit in the polymer chain has three types of arrangements. Figure 1.17 shows the polypropylene chain as an example. When all CH3 substituents stay in the same side of the main chain, the polymer is formed by a single kind of the optical isomer unit, and thus the produced polymer is called an isotactic configuration. When the polymer is alternately connected by two optical isomer units where the CH3 substituents locate alternately on both sides of the main chain, the produced polymer is called the syndiotactic configuration.

C

C

C C

C

C C

C

C C

C

C C

C C

C C

C

(a) C

C

C

C C

C

C

C C

C

C

C

C

C

C C

C

C

Atactic

C

Isotactic (d)

(b) C

C C

C C

C

C

C C

C

C

C C

C

C C

C

(c) C

Carbon

Hydrogen

CH3

Fig. 1.17: Three-dimensional configurations of polypropylene: (a) isotactic, (b) syndiotactic, (c) atactic, and (d) visualization diagram. M refers to monomer.

1.3 Polymer chain structure

21

When the substituent is irregularly arranged on both sides of the main chain, the polymer that is completely randomly connected by two kinds of the optical isomer units is called the atactic configuration. The schematic of the isotactic configuration and also the atactic configuration are pictorially shown in Fig. 1.17(d), where M represents the monomer unit. Figure 1.17 is a simplified representation of the main chain of the molecules in the polypropylene, and in fact the main chain of the molecules in the polypropylene has a spiral arrangement, and thus the substituents are also arranged around the spiral chain with their spatial rotation, as shown in Fig. 1.18. The isotactic configuration and the syndiotactic configuration can be called the isotactic polymer and the syndiotactic polymer, respectively. The substituents in the isotactic polymer chains are regularly arranged, therefore crystals can be formed by the tightly gathered molecular chains. Moreover, the isotactic polymer has a higher degree of crystallinity, as well as the melting point. The dissolution of isotactic polymers is difficult. For instance, the melting points of the isotactic and syndiotactic polypropylene are 180 and 134 °C, respectively. They can be spun to form the polypropylene fiber. However, a random polypropylene is a kind of elastic body, exhibiting the characteristics of rubber with poor strength.

Fig. 1.18: Arrangement of polypropylene spiral chain.

b. Geometric isomers When the addition reaction of the diene monomer produces 1,4 occurs, the internal double bond in each unit of the polymer chain can form a cis- and a trans-configuration, which is called geometric isomers. The polymer of this kind can be divided into trans, cis, or mixed configurations. Consider 1,4-polybutadiene as an example. Its cis- and trans-structures are shown in Fig. 1.19. C is structure:

Trans structure:

Fig. 1.19: Cis- and trans-structures of 1,4-polybutadiene.

22

Chapter 1 Atomic structure and interatomic bonding

The above polybutadiene structures have different performances due to their structure difference. For example, the polybutadiene with the isotactic or syndiotactic structure formed by 1,2-addition reaction can only be used as plastics because it has a regular structure, an easier crystallization behavior, and poor flexibility. Cis-1,4-polybutadiene is a good elastic rubber at room temperature due to the large gap between the molecular chains. The trans-1,4-polybutadiene has a poor elastic behavior at room temperature because it is easy to be crystallized due to the regular structure. The effect of geometry on the performance of 1,4-polyisoprene is similar with the influence of structure. The natural rubber contains more than 98% cis-1,4-polyisoprene and roughly 2% 3,4-polyisoprene, which makes it soft and flexible. Besides, Guta rubber is trans-polyisoprene with hard and tough properties and exhibits two kinds of crystalline states at room temperature. Their melting point and glass transition temperature are shown in Table 1.2.

Table 1.2: Melting point and glass transition temperature of several polymers. Polymer

Melting point Tm (°C)

Glass transition temperature Tg (°C)

Cis-,

Trans-,

Cis-,

Trans-,

Polyisoprene





−

−

Polybutadiene





−

−

1.3.2 Long-range structure of polymer chains 1. Molecular weight The molecular weight of polymers with very long chains is extremely high. When such large macromolecules are synthesized from smaller molecules during the polymerization process, the polymer chains will not grow to the same length, thus resulting in distribution in the chain length or molecular weight. Therefore, it is meaningless to focus on the relative molecular weight of a single polymer molecule. Instead, an average molecular weight can be determined by the various physical properties measurements. The average relative molecular mass of polymer is characterized by the statistical relative molecular weight of different polymers. The number-average molecular weight is defined as X Mn = xi Mi where Mi is the relative molecular weight of ith polymer, and xi stands for the fraction of the total number of chains within the corresponding size range.

1.3 Polymer chain structure

23

The weight-average molecular weight is defined as X Mw = wi Mi where wi denotes the molecules weight fraction within the same size interval. An alternative way of expressing the average chain size of a polymer is the polymerization degree n, which represents the average mer unit number in a chain. Therefore, the polymerization degree of polymer is determined by
n = M n =m
is the mer molecular weight. where m For a copolymer which possesses two or more different mer units, m is determined by X
= m f j mj where fj and mj are referred to the chain fraction and molecular weight of mer j, respectively. A multi-dispersion polymer cannot be depicted by only the average value of a single relative molecular weight. The ideal method is to calculate the distribution curve of the relative molecular weight for a polymer. The curve provides the relationship between the relative content and the relative molecular weight of each component in a polymer. Figure 1.20 shows the differential distribution curve of the relative
n and weight molecular weight for a typical polymer when the number average is M
w . According to this figure, if the distribution of molecular weight is average is M wide, the relative molecular weight will not be quite uniform, and vice versa.

Fig. 1.20: Molecular weight distribution for a typical polymer.

It is worth mentioning that the relative molecular weight of a polymer not only affects the rheological properties of the polymer solution and its melt but also plays a decisive role on the mechanical properties such as strength, elasticity, and toughness. With the increase of relative molecular mass, the Van der Waals force increases

24

Chapter 1 Atomic structure and interatomic bonding

and the slip between molecules is not easy to occur. Thus, there is a physical cross-link with respect to the molecule is formed and the strength of polymer increases regularly as an oligomer turns to a polymer. When the relative molecular mass is increased to a certain value, such dependence becomes less obvious, and the strength gradually approaches to a limit value. The critical relative molecular weight for such performance transition, Mc, has various values for the different polymer, indicated by the arrows in Fig. 1.21(a). At the same time, a given polymer has different Mc for different properties, as shown by the arrows in Fig. 1.21(b).

5

500 400 PS 300

1,200 1,000

PS

4 PS

800

3 2

200 100

0

100,000 200,000 300,000 Relative molecular mass (a)

600

0 100,000 200,000 300,000 Relative molecular mass (b)

Impact toughness

PC

600 Tensile strength

1,400 Bending strength

700

Relative molecular mass 10,000 20,000

1

Fig. 1.21: Relationship between mechanical property and relative molecular mass of (a) polystyrene (PS) and (b) carbonate.

The distribution of the relative molecular weight has a great influence on the processing properties and performance. For synthetic fibers that have a small average relative molecular weight, if the distribution is wide, the constituent content with small relative molecular weight will be high, degrading the spinning performance and mechanical strength. The general distribution of the relative molecular weight is narrow in plastics, which is favorable for the control of processing and the improvement of the performance of the products. For the rubbers, however, since their average relative molecular weights are very large, owing to the difficulties in processing, plasticizing is necessary to decrease the relative molecular weight and widen its distribution. Therefore, the resultant part with lower relative molecular weight has a lower viscosity and can be used as a plasticizer, which facilitates the processing and forming of rubbers.

1.3 Polymer chain structure

25

2. Internal rotation conformation of the polymer chain A single bond is composed of sigma (σ) -electrons, and the linear polymer chains contain tens of thousands of σ bonds. If the internal rotation of every single bond in a main chain is completely free, this kind of polymer chain is called a free link chain. It has an infinite number of conformations and each conformation is constantly changing, which is the ideal state of a flexible polymer chain. In a real polymer chain, the bond angle is a constant, which is 109°28ʹ for the carbon chain. Even though a single bond could rotate freely, each bond can only exists on the surface of the cone whose axis is the previous bond and the cone angle is 2θ (θ = π – 109°28ʹ) (Fig. 1.22(a)). Internal rotations are allowed for each single bond, leading to numerous morphologies of polymers. Supposing the number of positions associated with the internal rotation within each single bond is m, then the number of possible conformations of polymer containing n single bonds is mn−1. When n is large enough, the value of mn−1 is undoubtedly an enormous number. In addition, it can be seen from the statistical law that the probability of an extended (straight chain) conformation within the polymer chain is extremely small, and the probability of the formation of the twist conformation is relatively high due to the high configuration entropy. In fact, the internal rotation is generally hindered by atoms or groups on the chain and it consumes a certain energy when it rotates. The internal rotation curve reflects the potential energy. The internal rotation angle of the ethane molecules is shown in Fig. 1.22(b), where the potential barrier, ΔE, explains the difference between the cis- and trans-conformations. The conformation that the hydrocarbon bonds of two carbon atoms coincide with each other along the carbon–carbon bond direction is called the cis-conformation, whose potential energy can reaches the maximum value. The conformation where the difference of the hydrocarbon bond of two atoms is 60° is called the trans-conformation, whose potential energy has the lowest value on the potential energy curve. Meanwhile, the arrangement of

Fig. 1.22: Internal rotation of polymer chain with fixed bond angle and internal rotation potential energy diagram of ethane molecule.

26

Chapter 1 Atomic structure and interatomic bonding

atoms in the molecules under this condition is most stable. The transition from the trans-conformation to cis-conformation needs to overcome the barrier of the potential energy. The molecules with different conformations resulting from the internal rotations of single bonds are called rotational isomers. The internal rotation of the polymer chain, like that of the small molecules, is not completely free due to the influence of atoms or groups on the chain. Therefore, it shows both flexibility and rigidity. Its softness is attributed to the independent movements of chain segments consisting of several dozens or even hundreds of links in the whole chain. Furthermore, under a given temperature, for molecules with different structures, shorter chain segment with independent movements or a greater number of such independent chain segment will result in better flexibility. For the same kind of large molecule, the higher the temperature is on the shorter the chain segment is the better the flexibility of the molecule chain will be. Therefore, in addition to the molecular structure, the length of chain segment is also related to the environmental conditions, such as the temperature, the external force, the medium, and the interactions between molecules in the polymer aggregation. 3. Main factors affecting the flexibility of polymer chain The flexibility of the polymer chain means the ability of changing its conformation. The influence of molecular structure on the flexibility of the chain will be discussed further. a. Effect of the main chain structure The main chain structure has an important effect on the rigidity and flexibility of the polymer chain. For instance, the energy barriers of internal rotation around C–O, C–N, and Si–O single bonds are lower than those around the C–C single bond. Taking the C–O and C–C bonds as the examples, since there are no atoms or atomic groups around the oxygen atom, and the distance between the nonbonded atoms in the C–O–C chain is larger than that in the C–C–C chain, the interaction and the hindering effect on the internal rotation is small in the C–O–C chain. Besides, the Si–O bond not only has the characteristics of the C–O bond but also has longer bond length and lager bond angle, companied with those of both C–O–C and C–C–C bonds. It enlarges the distance between the nonbonded atoms, decreasing their interaction force. Thus, polyester, polyamide, and PDMS are all flexible polymer chains.

Fig. 1.23: Flexible polymer chains.

1.3 Polymer chain structure

27

Especially for PDMS with the structure of Fig. 1.23, it has excellent flexibility and is a kind of special rubber that can be used under low-temperature condition. If the main chain contains the aromatic heterocyclic structure, the polymer of this kind has a better rigidity because the aromatic heterocyclic cannot rotate internally. Therefore, they possess good high-temperature properties and are used as the high-temperature-resistant engineering plastics, such as polycarbonate, polysulfone, and polyphenylene ether (PPO). The structure of the PPO is shown in Fig. 1.24.

Fig. 1.24: The structure of the polyphenylene ether (PPO).

The main chain structure of PPO contains aromatic rings, which make rigid and resistant to the high temperature. At the same time, the PPO has C–O bond inside, which makes it flexible and able to through the injection molding. The main chain of diene polymer contains a double bond. Although the double bond itself does not rotate, it reduces the internal rotation barrier for the neighboring single bonds and also decreases the repulsive force between them due to the enlargement of the distance between the nonbonded atoms. Consequently, they are all flexible and can be used as rubbers, such as polybutadiene and polyisoprene. However, polymer chains with a conjugated double bond cannot rotate because its π-type electron cloud has no axial symmetry and the energy of the electron clouds is the lowest when they overlap with each other. Meanwhile, the internal rotation will break and deform the π-type electron cloud. For example, polyphenyl (Fig. 1.25), polyacetylene (Fig. 1.26), and certain heterocyclic polymers are typically rigid molecular chains.

Fig. 1.25: Polyphenyl structure.

Fig. 1.26: Polyacetylene structure.

b. Effect of the substituents The polarity of the substituent groups, the distance between substituents arranged along the molecular chains, the symmetry of substituents on the main chain, and the volume of the substituents will all affect the flexibility of polymer chains.

28

Chapter 1 Atomic structure and interatomic bonding

The polarity intensity of the substituents determines the attractive force, the potential barrier inside the molecule, and the magnitude of the intermolecular forces. The greater the polarity of the substituents is, the stronger is the interaction between nonbonded atoms will be. However, the larger the internal rotation resistance inside the molecules is, the worse is the flexibility of the molecular chain, and vice versa. For instance, for three polymers, polypropylene, PVC, and polyacrylonitrile, the polarity of the methyl group in the polypropylene is the weakest while the polarity of the chlorine atom inside the PVC, which belongs to the polar group, is not as strong as that of the –CH group in the polyacrylonitrile. Thus, the polarity of three groups increases in sequence but their flexibility in sequence decreases. Generally speaking, fewer polar groups means larger interval distances in the chain. Meanwhile, both the interaction force between the polar groups and the hindering effect of the spatial position are reduced, which lead to the relatively easier internal rotation and better flexibility. For example, the polar substituents of the chlorinated polyethylene are chlorine atoms and the number of chlorine atoms in the main chain is smaller when compared with that in the PVC. Thus, the flexibility of the molecular chain of the former is better, but decreases with the increasing degree of chlorination. The position of substituents has a certain influence on the flexibility of the molecular chain when two different substituents are connected to one carbon atom, the flexibility of the molecular chain will be reduced. For instance, the rigidity of polymethyl methacrylate (PMMA) with two polar groups connected to one carbon atom is stronger than that of PMMA with only one polar group. The volume of the substituent group determines the level of steric hindrance. For example, the volumes of substituent groups in polyethylene, polypropylene, and polystyrene increase, together with the increase in the spatial steric hindrance effect; therefore leading to the decrease in the flexibility of corresponding molecular chains. c. Effect of the cross-link The internal rotation of single bonds around the cross-linking point is greatly hindered when the polymer molecules are cross-linked through chemical bonds. When the degree of cross-link is relatively low, the chain length of the molecules between the cross-link points is far longer than that of the chain segment. Therefore, it is possible for the chain segment to move as an unit. For instance, for the rubber with a low degree of vulcanization, on one hand its main chain has a good flexibility. On the other hand, the larger distance between the cross-link sulfur bridges allows the internal rotation of the cross-link points within the chain segments. Thus, the good flexibility is maintained. If the degree of cross-link is larger, the internal rotation inside the single bond is impossible and then the flexibility is almost gone. For example, when the cross-linking degree of the rubber is more than 30%, it becomes a hard rubber.

1.3 Polymer chain structure

29

4. Configuration statistics of the polymer chain The flexibility of the polymer is measured by the number of conformations. However, the number of conformations in a polymer chain can be numerous since a long chain has tens of thousands of single bonds such and each of them may have several internal rotation isomers. In general, the longer the molecule chain is, the larger the number of the conformation will be. And then it is easier for the polymer chain to curl up. The curl-up degree of the polymer chain is measured by the straight-line distance between its endpoints is enhanced, namely, the end-to-end distance. In this way, the flexibility of the polymer can be estimated by measuring the length of the end-to-end distance instead of calculating the total number of conformations. The average endto-end distance has to be calculated since the momentary end to end distance varies with time and the type of molecules. For a polymer changing instantly and irregularly, the vector operation is used to calculate the average value of the square of the end-to-end distance, i.e., the mean square end-to-end distance. Considering a free binding chain with neither the limitation of bond angle nor the energy barrier of the internal rotation, the mean square end-to-end distances can be obtained by both the geometric method and the statistical method as follows: h2f .j = nl2 where n is the number of the bond, and l is the length of the bond. For a freely rotating chain which is restricted to the bond angle (θ) according to the geometric method, we have: h2f .r = nl2

1 + cosθ 1 − cosθ

In fact, any polymer chain can be neither a freely rotating chain nor a free bonding chain. The single bonds are restricted by each other when they are rotating. It means a single bond will drive a section of chain nearby to move when it rotates. Thus, each bond cannot be a unit of independent movement. However, we can treat a length of chain consisting of several bonds as an independent unit, which is called a chain segment. If a chain is freely bonded by chain segments and rotate irregularly, it is called the equivalent freely joined chain. Since the distribution of the segment chain of the equivalent freely joined chain obeys the Gauss distribution function, it is also called a Gauss chain. Gauss chain is an embodiment of the common character of a large number of flexible polymers that exist in the real polymer molecules. Its mean square endto-end distance is as follows: h20 = ne l2e where ne is the number of the chain segment in the polymer chain, and le is the length of each segment.

Chapter 2 The structure of solids Solid materials can be classified as crystalline materials and noncrystalline materials categorized by the regularity in the arrangements of atoms or ions. A crystalline material is referred to one in which the atoms are located in a repeating or periodic array over macroatomic distances. That is to indicate, long-range order exists in three-dimensional (3D) space, in which each atom is bonded to the nearest neighboring atoms. All metals, most of the ceramic materials, and some certain polymers form crystalline structures under general solidification conditions. In some materials, the long-range atomic order is absent, which are called the noncrystalline or amorphous materials. The arrangement of atoms has an important influence on the microstructure and properties of the solid materials. The properties of metals, ceramics, and polymers are related to with the mentioned arrangements of atoms. For instance, the metals with face-centered cubic crystal structure, such as Cu and Al, have an excellent ductility properties, however, the metals with a hexagonal closepacked crystal structure, such as Zn and Cd, are relative brittle. The rubber with a linear molecular chain has combined properties of good elasticity, toughness, and wear resistance. While for the thermosetting resin with 3D network of molecular chain, once it is cured after heating, its shape will not change; whereas, it possesses high hardness and good heat and corrosion resistance properties. Therefore, to study the internal structure of the solid materials, that is, the arrangement and distribution of the atoms, is the basis to understand the properties of materials. which is the only way to find approaches in improving and developing new materials. It should be pointed out that the existence of a crystalline or noncrystalline substance is also dependent on the external environment conditions and processing methods because crystals or noncrystals can be transformed from each other.

2.1 Fundamentals of crystallography The long-range order, which means a regular and a periodic arrangement of atoms, molecules, or ions is the basic characteristic of a crystal. There are two main differences between the characteristics of non-crystalline materials and crystalline materials. First, crystalline materials have a definite melting point compared with non-crystalline materials that have a softening temperature range. Second, the crystalline materials are anisotropic; however, the noncrystalline materials are isotropic. In order to better understand the spatial distribution of atoms, ions, molecules, or atomic groups in the crystal and also analyze the crystalline structure, the basics of crystallography are first introduced in this chapter.

https://doi.org/10.1515/9783110495348-002

2.1 Fundamentals of crystallography

31

2.1.1 Space lattice and unit cells In crystalline materials, the arrangement of particles such as atoms, molecules, ions, or atomic groups in the 3D space has various types. In order to study the arrangement of particles in the crystal, the real crystals can be treated as perfect crystals, where each particle is abstracted as a geometric point, called lattice point, and is arranged in a regular way in space. These lattice points are arranged periodically and regularly in space and have exactly the same environment. Meanwhile, such arrays of lattice points are arranged regularly in three-dimensional (3D) space, which is called a space lattice or briefly a lattice. To describe the space lattice graphically, the concept of the space grid is introduced using a 3D geometric framework formed by connecting all the lattice points using parallel lines, as shown in Fig. 2.1.

Fig. 2.1: Geometry of the space grid.

A unit cell is a good illustration of the characteristics of lattice and is a representative basic unit (namely the minimum parallelepiped) taken from the lattice. When the unit cells are stacked periodically in 3D space, the space lattice is formed. The different unit cells can be obtained by various selection methods operated in the same space lattice. Figure 2.2 shows a variety of unit cells in the same twodimensional (2D) lattice. However, to obtain a unit cell that fully represents the symmetry of the crystal lattice, the selecting method of the unit cell should follow the following principles: a. The parallelepiped should express the highest symmetry of the lattice, that is, (1) the number of identical edges and identical corners of the parallelepiped shall be the largest; (2) the number of the right angle shall be the most if the right angle exists in the parallelepiped. b. The unit cells have the smallest volume after meeting the above conditions.

32

Chapter 2 The structure of solids

Fig. 2.2: Selection of unit cells in the lattice.

The shape and size of the unit cells are usually expressed by six lattice parameters. The length of three edges a, b, c (called lattice constant) and the edge angles α, β, γ of the parallelepiped are shown in Fig. 2.3. In fact, it is more convenient to describe the unit cells using three lattice vectors, namely a, b, and c, which not only determines the shape and size of the unit cells but also the space lattice according to the additivity of vectors.

Fig. 2.3: Unit cell, crystal axis, and lattice vector.

According to the relationship between the six lattice parameters, all the space lattice can be categorized into seven types, namely seven crystal systems, as shown in Table 2.1. Table 2.1: Crystal system. Crystal system

Edge length and angle relationship

Example

Triangle

a ≠ b ≠ c, α ≠ β ≠ γ ≠ °

KCrO

Monoclinic

a ≠ b ≠ c, α = γ = ° ≠ β

β-S, CaSO• HO

Orthogonal

a ≠ b ≠ c, α = β = γ = °

α-S, Ga, FeC

Hexagonal

a = a = a ≠ c, α = β = °, γ = °

Zn, Cd, Mg, NiAs

Rhomboid

a = b = c, α = β = γ ≠ °

As, Sb, Bi

Square

a = b ≠ c, α = β = γ = °

β-Sn, TiO

Cubic

a = b = c, α = β = γ = °

Fe, Cr, Cu, Ag, Au

33

2.1 Fundamentals of crystallography

According to the nature of the lattice point mentioned earlier, that is, the surrounding environment of every lattice point is identical, 14 types of units were derived from the mathematical method by A. Bravais, which reflect all the characteristics of the space lattice. After that, all these 14 types of space lattice and are called Bravais lattice, as listed in Table 2.2.

Table 2.2: Bravais lattice. Bravais lattice

Crystal system

Fig. . Bravais lattice

Crystal system

Fig. .

Simple triangle

Triangle

(a)

Simple hexagonal

Hexagonal

(h)

Simple monoclinic

Monoclinic

(b)

Simple rhomboid

Rhomboid

(i)

(c)

Simple square

Square

(j)

Monocentric monoclinic Simple orthogonal

Orthogonal

(d)

Body-centered square

(k)

Monocentric orthogonal

Orthogonal

(e)

Simple cubic

Body-centered orthogonal

(f)

Body-centered cubic

(m)

Face-centered orthogonal

(g)

Face-centered cubic

(n)

Cubic

(l)

Figure 2.4 shows the 14 types of Bravais lattices. The different unit cells can be obtained by various selection methods in the same space lattice, as the same 2D plane Fig. 2.2. For example, it is possible to represent the body-centered cubic (bcc) lattice by the simple triclinic unit cells (the solid lines of Fig. 2.5(a)). Besides, it is possible to show the face-centered cubic (fcc) lattice by the simple rhombohedral unit cells (solid lines of Fig. 2.5(b)). However, the new unit cells can not fully reflect the highest symmetry of the cubic space lattice, and thus such selection methods not acceptable. It should be noted that there is a difference between the crystal structure and the space lattice. The space lattice is a geometric abstraction of the particle arrangement inside the crystal and is used to describe the symmetry of the crystal structure Since the surrounding environment of each lattice must be identical, there are only 14 types of space lattice. The crystal structure refers to the specific arrangement of the real particles in the crystal, such as an atom, an ion, or a molecule. The combination of lattice points consisting of different atoms, ions, or molecules with its 14 Bravais lattices leads to infinite crystal structures.

34

Chapter 2 The structure of solids

Fig. 2.4: 14 Types of Bravais lattices.

2.1 Fundamentals of crystallography

(a)

35

(b)

Fig. 2.5: Selection of other unit cells from BCC lattice (a) or FCC lattice (b).

Fig. 2.6: Close-packed hexagonal structure.

Figure 2.6 shows a schematic of the hexagonal close-packed (hcp) crystal structure, which cannot be considered as a kind of space lattice because the atoms inside the unit cells have different environment with atoms located at the corners. If both an atom at the corner of a unit cell and an atom inside the unit cell are considered as the lattice points (such as the 0,0,0 lattice point can be regarded as the atom pair consisting of atoms at 0,0,0, and 2/3, 1/3, 1/2 lattice positions), the resultant hcp crystal will be a simple hexagonal lattice. Figure 2.7 depicts three crystal structures of Cu, NaCl, and CaF2, which belongs to different crystal structures; but the same lattice, i.e., the FCC lattice For Cu crystal structure, one atom represents one lattice point because every Cu atom has the same environment, so Cu-fcc crystal structure also called fcc lattice; for NaCl crystal structure, one Na or Cl ion cannot represent one lattice point because they have different atomic environment, but one NaCl molecule can represent one lattice point, so NaCl crystal structure is called fcc lattice; In a similar way, for CaF2 crystal structure, one CaF2 molecule represents one lattice point, so it is also called fcc lattice. Another example is CsCl and Cr, both of which belong to the BCC crystal structure as shown in Fig. 2.8. While Cr belongs to the body-centered lattice; CsCl belongs to the simple cubic lattice because CsCl molecule represents one lattice point.

36

Chapter 2 The structure of solids

Fig. 2.7: Crystal structures with same lattice.

Fig. 2.8: Similar crystal structure with different lattices.

2.1.2 Miller indices for directions and planes When considering problem associated with the growth, deformation, phase transformation and properties it is necessary to specify the position of atoms, the directions of atomic arrays and the crystallographic planes of atoms. Labeling convention system have been established in which and numbers or indices have been used to designate point locations, directions, and planes. Unit cell is the determination basis for the determination of indices, where a right-handed coordinate system is adopted, consisting of three (x, y, and z) axes situating at one of the corners and coinciding with the unit cell edges. A convenient method, called the Miller index, is a unified description for the crystallographic direction indices and also the crystallographic plane indices. 1. Crystallographic direction index It can be understood from Fig. 2.9 that the spatial position of any lattice point P can be calculated using the vector ruvw or the coordinates of the lattice point, u, v, and w: ruvw = OP = ua + vb + wc 

(2:1)

the crystallographic directions is only different in the values of u, v, and w. Therefore, the crystallographic direction can be expressed by the reduced index

2.1 Fundamentals of crystallography

37

as [uvw]. The steps to determine the crystallographic direction index are as follows: a. Set one certain lattice point as the origin of the coordinate and the crystal axes as the coordinate axes x, y, and z. The unit length equals to the lattice vector of the unit cell. b. Drawing a straight line OP over the origin O and make it parallel to the crystal direction needs to be calibrated. c. Selecting a lattice point, P, which is the nearest to the origin point O in the line OP, and determining three coordinate values for the point P. d. Changing three coordinate values into the ratio of smallest integer numbers, u, v, and w, along with the adding a square bracket [uvw], which is reduced index. For each of the three axes, both positive and negative coordinates will exist, therefore, negative indices are possible as well, which are referred to as a bar over the corresponding index, such as [1
10], [
100]. Figure 2.10 lists some important crystallographic direction indexes of the orthogonal system.

Fig. 2.9: Lattice vector index of orthorhombic system.

Fig. 2.10: Important crystal indices in the orthogonal system.

It is worthy to point out that the Miller direction index indicates a group of the crystallographic directions with the same orientation. That is, when the crystallographic directions are parallel to each other, the crystallographic direction indexes are same, such as [110] group including [
1
10 ], [220], and [
2
20]. For some crystal structures, several nonparallel directions with different indices are actually equivalent based on their symmetry; this means that the spacing of atoms along each direction is the same. For example, in cubic crystals, all the directions represented by the following indices are equivalent: [111], [
111], [1
11], [
1
11], [
1
1
1], [1
1
1], [
1
1
1], and [11
1]. As a convenience, equivalent directions are grouped together into a family, which are enclosed in angle

38

Chapter 2 The structure of solids

brackets, like . Furthermore, directions in cubic crystals having the same indices regardless of order or sign. For example, [120] and ½2
10, are equivalent. However, this is, in general, not true for other crystal systems. For example, for crystals of tetragonal symmetry, [100] and [010] directions are equivalent, whereas [100] and [001] are not. 2. Crystallographic planes index The orientations of planes for a crystal structure are represented by indices in a similar manner of crystallographic direction. Again, the unit cell is the basis, with the three-axis coordinate system represented in Fig. 2.11. In all but the hexagonal crystal system, crystallographic planes are specified by three Miller indices as (hkl), which represent a group of parallel planes. In all but the hexagonal crystal system, crystallographic planes with identical indices are equivalent. The procedure of determining the h, k, and l index numbers is as follows:

Fig. 2.11: Representation method of crystal plane index. Crystal plane index of some crystal planes in orthogonal lattice.

a. Set the reference coordinate in the lattice as have been done in the determination of crystal direction index and also putting the original point outside the planes in order to avoid zero interception. b. Take three intercepts for the planes with the coordinate axes and the reciprocals of these intercepts. A plane that parallels to an axis can be considered having an infinite intercept in the corresponding axe, and, therefore in another way, a zero index. c. If necessary, the three numbers can be changed to a set of a smallest integers by multiplication or division by a common factor. For example, Fig. 2.11 indicates the intercepts 1/2, 1/3, and 2/3 calibrated as a1 n b1 and c1 of the plane are 1/2, 1/3, and 2/3, with their reciprocals being 2,3, and 3/2. The resulting smallest integers numbers are 4, 6, and 3, and then the crystallographic plane

2.1 Fundamentals of crystallography

39

index is obtained by bracketing the above integers, i.e., (463). A negative intercept is indicated using a bar or minus positioned over the corresponding index, such as (
110) and (11
2). Figure 2.12 depicts some crystallographic plane indices in the orthogonal system.

Fig. 2.12: Some crystal plane index in hexagonal system.

A plane “family” contains all the crystallographically equivalent planes – planes possessing the same atomic packing; and a family is designated by indices {h k l} enclosed in braces. It represents a set of equivalent planes associated with their symmetry. For instance, in the cubic system: f110g = ð110Þ + ð101Þ + ð011Þ + ð110Þ + ð101Þ + ð011Þ + ð110Þ + ð101Þ + ð011Þ + ð110Þ + ð101Þ + ð011Þ where the six indices in the first row and the six indices the second row are parallel to each other correspondingly. The planes represented by these twelve indices together constitute a dodecahedron. Therefore, the family of {110} crystallographic planes is also called the face of the dodecahedron. For {111} family, f111g = ð111Þ + ð111Þ + ð111Þ + ð111Þ + ð111Þ + ð111Þ + ð111Þ + ð111Þ where, the former four planes and the latter four planes are parallel to each other correspendingly, There eight planes which together constitute an octahedron. Therefore, the family of {111} crystallographic planes is called the face of the octahedron. In addition, there is an important nature in the cubic system, namely, the crystallographic direction and the plane with the same index should be perpendicular to each other. For example, [110] direction is perpendicular to (110) plane, and [111] is perpendicular to (111).

40

Chapter 2 The structure of solids

3. Index of hexagonal system An issue arises for crystals with hexagonal symmetry is that some equivalent crystallographically equivalent directions or planes will not possess the same set of three indices. For example, in the hexagonal system, the axes are a1, a2, and c and the angle between a1 and a2 is 120°. Moreover, the axis c is perpendicular to the both a1 and a2 as shown in Fig. 2.12. However, the resultant index can not fully express the symmetry of the hexagonal system. Also, these equivalent crystal directions or planes, however do not have indices belonging to the same family, For example, the six primistic faces of a hexagonal crystal are crystallographically equivalent, but their plane indices are (100), (010), (110), (100), (010) and (110), which do not belong to a same plane family. This is circumvented utilizing a four-axis, also called Miller–Bravais, coordinate system shown in Fig. 2.13. These three a1, a2, a3 axes are included within a single spatial plane (called the basal plane) and form 120° angles with one another. The c axis is perpendicular to the basal plane. Oriental indices (or plane indices), which are obtained above, will be represented by four indices, as [uvtw] (or (hkil)). As well known, there are no more than three independent coordinate axes in the 3D space. Therefore, only two free indexes of the first three crystal plane indices exist with the limited condition of i = –(h + k). Figure 2.13 lists some crystal indices of the hexagonal system. The specific calibration method is the same as the one mentioned earlier, by which the equivalent crystal planes can be reflected from four indices. For instance, the above [100], [010], and [110] correspond to ½2
1
10, ½
12
10, and ½11
20, respectively, and they belong to the {1120} family and exhibit equivalently crystallographic equivalent, as shown in Fig. 2.13.

Fig. 2.13: Representation method of hexagonal crystal orientation index (c-axis vertical to figure plane).

2.1 Fundamentals of crystallography

41

Conversion from the three-index system to the four-index system for crystallographic direction must satisfy the limited constraint of u + v =‒t. Therefore, the conversion of ½UVW ! ½uvtw is accomplished by the following formulas: U = u − t, V = u − t, W = w; 1 1 u = ð2U − VÞ, v = ð2V − UÞ, t = − ðu + vÞ, w = W 3 3

(2:2)

4. Weiss zone law In crystallography those planes which contain a common direction [uvw] are also called the “[uvw] crystal zone”. The common direction is called “zone axis.” The zone axis [uvw] and the plane (hkl) with the crystal zone follow the Weiss zone law (or crystal zone law), that is, hu + kv + kw = 0 

(2:3)

If the crystallographic plane indices of two planes, (h1 k1 l1) and (h2 k2 l2), which are not parallel to each other, are known, their zone axis index can be calculated from the following formula:        k1 l1   l1 h1   h1 k1        u:v:w= : : ,  k2 l2   l2 h2   h2 k2  or :  u   u : v : w =  h1  h 2

v k1 k2

 w   l1   l 

(2:4)

2

Similarly, two crystallographic direction indices of [u1 v1 w1] and [u2 v2 w2] are given, the index of the plane containing the above tow directions can be determined by        v1 w1   w1 u1   u1 v1        h:k:l= : :   v2 w2   w2 u2   u2 v2  or :  h   h : k : l =  u1  u 2

k v1 v2

 l   w1   w 

(2:5)

2

5. Interplanar distance The difference between the crystallographic planes with different indices mainly lies in their directions and interplanar distance (spacing) of the planes. Once a

42

Chapter 2 The structure of solids

plane index is given, its normal direction and the interplanar distance can be determined. The direction of a crystallographic plane is expressed by the normal direction of the plane. Meanwhile, the orientation of any straight line in the space is specified by its direction cosines. In the cubic system the index components of a crystallographic plane, h, k, and l have the following relations to each other:  h : k : l = cos α : cos β : cos γ (2:6) cos2 α + cos2 β + cos2 γ = 1 The interplanar distance, dhkl, for (hkl) plane can be derived from this equation. In fact, the plane index has an inverse relationship with the interplanar distance. Figure 2.14 shows planes of the simple cubic lattice with different interplanar distance, along which the largest and the smallest interplanar distances are those between (100) planes and those between (320) planes, respectively. In addition, a large interplanar distance means a denser arrangement of atoms on the plane and vice versa. The relationship between the distance dhkl and index (h k l) of a crystal plane can be derived according to the geometry in Fig. 2.15. Suppose that ABC is the nearest plane to the origin point O, and the angles of α, β, and γ are introduced to characterize the angles the normal N and a, b, and c axes, we get a b c cosα = cosβ = cosγ h k l "      # h 2 k 2 l 2 + + d2hkl = cos2 α + cos2 β + cos2 γ a b c dhkl =

(2:7)

Fig. 2.15: Formula derivation of crystal plane spacing. Fig. 2.14: Interplanar spacing.

2.1 Fundamentals of crystallography

43

Since in the Cartesian coordinates, we have: cos2 α + cos2 β + cos2 γ = 1 the interplanar distance of planes in the orthogonal system is calculated using the following relationship: 1 dhkl = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 k2  l 2ffi ° a + b + c For the cubic system, since the a = b = c, the above equation can be simplified as a dhkl = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h + k 2 + l2 ° For the hexagonal system, the interplanar distance is calculated as 1 dhkl = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2  2 ° 4 ðh + hk + k Þ + cl 3 a2

6. Interference planes and Miller index planes As mentioned earlier, Miller index planes are reduced index planes, such as For instance, planes with indices of (100),ð
100Þ,(200), ð
200Þ, etc. they are all denoted by (100) the Miller index of (100). It implies that the Miller index does not distinguish the interplanar distances and the normal directions of parallel planes. However, the concept of Miller index planes cannot be employed in X-ray diffraction or electron diffraction. As is well known, the Bragg equation is the geometry condition of diffraction in real space and its expression is 2dhkl sin θ = nλ; where θ is the incident angle, and here is defined as an angle between (hkl) plane and incident wave; n is diffraction order number and n = 0, ± 1, ± 2, ± 3 · · · integers. If n = 0, the diffraction is called zero order diffraction, indicating that the incident wave will not be reflected by a set of (hkl) planes, to form a transmitted wave. If n = 1, the diffraction is called the first-order diffraction, indicating that incident wave can be reflected by the (hkl) planes, and form a first-order diffracting wave. Similar conditions happen for case of n = ± 2, ± 3 · · · . In analysis of X-ray diffraction or electron diffraction, the Bragg equation is always used in the form of the first-order diffraction: 2

dhkl sin θ = λ n

2dnhnknl sin θ = λ

44

Chapter 2 The structure of solids

Where dhkl/n = dnhnknl. The above equations express such a physical sense: the n order diffraction of any (hkl) lattice planes is equivalent to the first-order diffraction of (nhnknl) planes. For example, the second-order diffraction of (100) planes is equivalent to the first-order diffraction of (200) planes. It is worthy to point out that (nhnknl) planes are here are named as interference planes which are distinguished from Miller index planes. For example, the (100) planes in interference planes are different from (200) or ð
100Þ since the spacings or the normal direction of their planes are different, through these planes in Miller index planes all belong to {100} plane family and are not discriminated. Besides, in the primitive cubic lattice, (100), (200) . . . ð
100Þ, ð
200Þ, . . . interference planes have corresponding diffraction spots in a single crystal electron diffraction pattern. Another physical concept is that interference planes may not correspond to a real interatomic plane. For example, in the simple cubic lattice, no any lattice points (atoms or molecules) exist on (200) planes. Based on the earlier mentioned concept, the Bragg equation can be rewritten in a form of the first-order diffraction: 2dhkl sin θ = λ

(2:8)

Keeping in mind, (hkl) planes are interference planes, and n is implicit in dhkl in eq. (2.8).

2.1.3 Stereographic projections In order to analyze the crystal structure, special directions and planes within a crystal are usually referred. The stereographic projection is a 2D graphical technique for conveniently measuring angles and directions in 3D crystal structures. The stereographic projection provides us a useful tool and it is widely used in the study of materials science. 1. The principle of the stereographic projection The crystal to be studied is imagined to locate at the center (C) of a reference sphere, as shown in Fig. 2.16. Any crystal plane of interest is projected by the following procedure: 1) The normal to a certain (hkl) plane is extended until it intersects the reference sphere at a point P, which is called the pole of this plane. 2) The projection plane is placed tangent to the reference sphere; the point (A) of tangency is at one end of a diameter, the other end of which is the origin (B) of projection. 3) Suppose P is on the left half of the reference sphere, extent line segment BP and intersect with the project plane at point P’, as shown from the sectional view of the sphere in the Fig 2.16, P’ is called the stereographic projection of P.

2.1 Fundamentals of crystallography

45

Draw a circle NESW perpendicular diameter AB with C being the center, and project it to the project it to the projection plane as N’E’S’W’, which is named as the base circle. For all the poles on the left half of the reference sphere their stereographic projections will fall somewhere inside the base circle. 4) For planes with poles lying on the right half of the reference sphere, by moving the projection plane form A to B, the stereographic projection can be obtained following the same procedure as mentioned in (3), except that minus signs are imposed on the indices of (hkl).

Fig. 2.16: Schematic diagram of polar projection.

2. Wulff net A significant property of the stereographic projection comparing with the gnomonic projection (the projection point at the sphere center) and the orthographic projection (projection point at infinity) is that the angle between the poles on the projection are always the true angles between the normal planes represented by poles. The angle between plane normal is also the angle between the planes represented by these normal ones. As a result, by determining angles between poles we obtain the angles between planes. In order to able to easily project the sphere stereographically and also measure the angles on the projection, the following technique of Wulff net is used, as shown in Fig. 2.17. The latitude and longitude lines are ruled onto a sphere (usually at 2° interval). These latitude and longitude lines are then projected stereographically onto the plane to Wulff net. Because of our familiarity with the globe, the top of the net will be called the north pole, and the horizontal axis (west-east axis) will be called the equator. Then, the vertically running lines are longitudes and the horizontal lines are latitudes. With the use of this grid we can locate any point on the projection if we know its longitude and latitude coordinates.

46

Chapter 2 The structure of solids

Fig. 2.17: Wolff net (index value 2°).

Angles can be measured by counting degrees along a latitude or along a longitude, that is, true angles between planes represented by poles must be measured along great circles (longitude and equator) but not along small circle (latitudes). For example, consider the three points A,B,C shown in Fig. 2.18(a), and they are located on the 0°long., 40°N lat., and 60°E long, respectively. Figure 2.18(b) shows a 3D view of the upper right-hand quadrant of Fig. 2.18(a) with the same three points located. With the Wulff net in the position of Fig. 2.18(a) we can easily measure an angle between B and C by moving along the 60°E long. And counting the degree marks on the net as 30° (40° − 10°), which corresponds to the angle β in Fig. 2.18(b). This angle also corresponds to the true angle between the planes of

Fig. 2.18: Relationship between Wolff net and the reference ball.

2.1 Fundamentals of crystallography

47

these two poles because it is the angle between their normal ones. However, with the Wulff net we should not measure the angle between A and B by moving along the 40°N long. and counting the degree as 60°, because this angle corresponds to α′ in Fig. 2.18(b), rather than α as a true angle. To measure the angle α between A and B with a Wulff net, we now pass a plane through the sphere of Fig. 2.18(b), which contains the origin and point A and point B. The trace of this plane on the sphere would be a great circle because the center of this trace circle would also be the center of the sphere. Then by measuring the angle between A and B along the trace of this great circle, we can obtain α. 3. Standard projection The simple, symmetrical positions of the (001) plane in the stereographic projection of Fig. 2.19 characterizes the standard stereographic projection of a cubic crystal, in which crystal plane（001）is parallels to the projection plane. When any (hkl) plane in cubic crystals parallels to projection plane, the standard stereographic projection will be called the standard (hkl) stereographic projection. Because of the high symmetry of a cubic crystal, all significantly different orientations are represented by the points within a single reference stereographic triangle formed by three adjacent (100), (110), and (111) planes in Fig. 2.19, and the triangle is usually called standard triangle. It often refers to the trace of a plane on the stereographic projection in practice. If any plane, (hkl), is extended out to the reference sphere, it will intersect the sphere along a great circle. This great circle is the trace of the plane and it may simply be defined as the locus of point 90°away from the pole of the plane. For example in Fig. 2.19, the great circle, consisting of poles from
100pole to 100 pole through 0
11pole, is the trace of (011) plane because the 011 pole is 90° away from the great circle. As mentioned earlier, planes that intersect along a common direction are said to belong to the same zone and are called zone planes (planes of a zone). The common line is called a zone axis. If you want to determine the zone axis of some two planes, such as (100) and ð0
11Þ, find the traces of these two plane on the standard stereographic projection, and the intersection point of these traces is the 011 pole, and thus the [011] direction is the zone axis of these two planes. In fact, the all planes corresponding to poles at the above great circle belong to the planes of [011] zone. In addition to standard (001) stereographic projection for cubic crystals mentioned earlier, standard (110), (112), or (111) stereographic projections are also given in some professional books, such as transmission electron microscopy for a convenient use.

48

Chapter 2 The structure of solids

Fig. 2.19: Detailed (001) standard projection of the cubic crystal.

2.1.4 Crystal symmetry Symmetry is one of the basic characteristics of crystals. Most of crystals such as natural diamonds, crystals, and snow flakes have the regular geometric shapes. The macroscopic symmetry of the crystal shape is the manifestation of the internal symmetry of the crystal structure. Some physical parameters such as thermal expansion, elastic modulus, and the optical constants are related to the properties of crystal symmetry. Therefore, it is significant to understand the symmetry of crystals. 1. Macrosymmetry elements and their corresponding operation matrices In crystallographic macrosymmetry, there are only eight fundamental symmetry elements: i-fold rotation axis with 1 (identity element), 2, 3, 4, 6; mirror (m); inversion
as shown in Fig. 2.20. In the figure, the big center (i); rotation-inversion axis (4), circles represent the plane z = 0. Small solid circles represent the objects above the

2.1 Fundamentals of crystallography

49

Fig. 2.20: Schematic diagram of eight fundamental symmetry elements in point groups.

plane z = 0 and open circles the objects below the plane z = 0. Different symmetry element operations can be expressed by their corresponding transition (transformation) matrices, respectively.

Fig. 2.21: Symmetry axis of (a) cubic crystal and (b) hexagonal crystal.

Fig. 2.22: Rotation of symmetry axis along x axis.

50

Chapter 2 The structure of solids

Fig. 2.23: Schematic drawing of symmetrical plane.

Fig. 2.24: Schematic drawing of symmetric center.

Fig. 2.25: Schematic drawing of rotation-inversion axis.

(1) Rotation axes Assume that the crystal rotates about a certain axis within the body. If a rotation angle is θ = 2π/n degrees, the rotation about the axis gives rise to a disposition indistinguishable from the original, then this axis is defined as an n-fold rotation axis, where n is the rotation times. Considering the effect of lattice periodicity, the possible elements in proper rotations are only five elements, namely, n = 1, 2, 3, 4, or 6, as shown in Figs. 2.21. An n-fold rotation operation can be expressed in a matrix form. In the coordinate system Oxyz, when a given point M can be repeated by rotating anticlockwise around the origin O at an angular interval θ, a new coordinate system Ox’y’z’ can be

2.1 Fundamentals of crystallography

51

obtained as shown in Fig. 2.22. Thus, the transition matrix between the two coordinate systems can be expressed as follows: 2 3 2 3 2 3 α11 α12 α13 x x′ 6 ′7 6 7 6 7 4 y 5 = 4 α21 α22 α23 5 4 y 5 2

α11 where ðαijÞ = 4 α21 α31

z′ 3

α12 α22 α32

α31

α32

α33

z

α13 α23 5 (i, j = 1, 2, 3) is the transition matrix: α33 2

1

6 ðαijÞ = 4 0 0

0

3

0

7 sin θ 5

cos θ − sin θ

cos θ

where the rotation axis is x-axis. If x-axis is the rotation axis of twofold, fourfold, threefold, or sixfold proper rotations, the corresponding rotation angle is 180°, 90°, 120°, or 60°, respectively. Thus, the corresponding symmetry transition matrices are, respectively: 2 3 2 3 1 0 0 1 0 0 6 7 6 7 60 −1 6 07 0 1 7 4 5, 4 0 5 2

0

0

−1

1

0

0 pffiffi 7 3 7 2 7, 5 1 −2

6 60 6 4 0

− −

1 2 pffiffi 3 2

3

2

0

−1

1

0

6 60 6 4 0

−

0

3 0 pffiffi 7 37 1 2 2 7 5 pffiffi 3 2

1 2

(2) Mirror If the disposition of a given point to one side of a plane is the mirror image of this point on the other side, as shown in Fig. 2.23, this plane is a mirror plane. And the corresponding symmetry element is designated as a mirror (or a reflection) which is referred to as m. In the coordinate system Oxyz, if a given point M(a, b, c) carries out a mirror operation with the yOz plane as a mirror plane, the new point M after the operation can be derived, that is, the x-axis coordinates of the new point is – a, and the other two remain unchanged. Thus, the transition matrix of this symmetry operation is

52

Chapter 2 The structure of solids

2 6 4

for plane yOz:

−1

0

0

1

0

3

7 05

0 0

1

0

0

The others are 2

1

6 60 4

for plane xOy:

2 for plane zOx:

7 07 5

1

−1

0 0 1

3

0

6 60 4

−1

0

0

0

3

7 07 5 1

(3) Inversion center In crystals, if two equidistance points in a line which goes through the origin of the crystal coordinate, the operation of one point to another in the crystal is called an inversion center (Fig. 2.24), symbolized as i. Thus, the transition matrix of this operation is 2 3 −1 0 0 6 7 05 4 0 −1 0

0

−1

(4) Rotation-inversion axes A rotation-inversion axis combines rotation about an n-fold rotation axis with inversion, as shown Fig. 2.25. It should be noted that rotation-inversion operations
can be decomposed into elements involving i or m. For example, (
1,
2,
3, 6)
= 3=m . Thus, the rotation-inversion axes except 4
1 = i,
2 = m,
3 = 3 + i; 6
are not in
may be dependent and cannot be fundamental symmetry elements. In crystals, 4 usually confused. However, it is a fundamental symmetry element, and its transition matrix is 2 3 2 3 2 3 1 0 0 −1 0 0 −1 0 0 6 7 6 7 6 7 60 6 6 7 0 17 0 7 4 5×4 0 −1 5=4 0 0 −15 0

−1 4

0

0 ×

0 i

−1

0 =

1

0

4

In conclusion, eight fundamental symmetry elements and their corresponding transition matrices are deduced in crystallographic macrosymmetry. In order to

2.1 Fundamentals of crystallography

53

directly show the meaning of fundamental symmetry elements and their combinations, some simple figural labels are selected for convenience. Table 2.3 lists the international notations of the symmetry elements and their corresponding figural labels.

Table 2.3: The international notations of the symmetry elements and their corresponding figural labels.

2. Microsymmetry elements There are two microsymmetry elements in a space lattice: screw axis and glide plane, and they are called the translational symmetry elements. (1) Screw axis A screw operation is the combination of a rotation by a certain angle 2π/n about an axis (called the screw axis) with a translation by some distance T/n along the axis, where n is an integer and T is a periodic vector along the screw axis. The permissible n is 1, 2, 3, 4, 6. When a onefold screw operation occurs to a motif, the motif is periodically repeated and no new symmetry element can be produced. When a twofold rotation combines with a parallel translation T/2, a screw 21 is produced. Note that when the translation is T, 22 = 2. Similarly, the combination of a threefold rotation with parallel translations produces screws 31, 32, 33 (=3), respectively. The new symmetry elements generated by the combination of a fourfold rotation with parallel translations are 41, 42, 43, respectively, as shown in Fig. 2.26. Similarly, 61, 62, 63, 64, 65 can be generated by combination of a sixfold rotation and parallel translations. (2) Glide plane In crystallography, a glide plane is a symmetry operation by a translation parallel to this plane, which may leave the crystal unchanged. The permissible glides in crystals are a glade (t = a/2), as shown in Fig. 2.27, b glide (t = b/2), c glide (t = c/2), n glide [t = 1/2(a + b), 1/2(b + c), 1/2(c + a)] and d glide [t = 1/4(a + b), 1/4(b + c), 1/4(c + a)]. The notations of symmetry operations and their corresponding figural labels are listed in Table 2.4.

54

Chapter 2 The structure of solids

Fig. 2.26: The operation of fourfold screw axis.

Fig. 2.27: The operation of a glade glide.

3. The 32 point groups and space groups The combination of 8 fundamental symmetry elements will generate 32 point groups. The point group is a set of the symmetry elements operated around a fix point. By the combination laws of 8 fundamental symmetry elements the 32 point groups can be derived. The international notation of one point group is represented by the combination of certain symmetry elements in sequence, such as the rotations shown in Table 2.5. Usually, the number of symmetry elements is three. The orientation of these three symmetry elements in different crystal systems is represented by their sequence. The rotation axis and the rotation-inversion axis are represented that they are parallel to the crystallographic direction in a crystal system. And the reflection in certain crystallographic direction means that the reflection is normal to this direction. If a proper or improper rotation axis and a reflection of some direction are represented simultaneously, the corresponding symbol can be written as a fraction where the numerator is the proper or improper axis and the denominator is the reflection normal to the axis. For example, 2/m represents a twofold rotation with a reflection (mirror) normal to the rotation axis. Table 2.5 lists the sequence of symmetry elements in international notations related to directions in 7 crystal systems. For instance, in cubic

2.1 Fundamentals of crystallography

55

Table 2.4: The notations of symmetry operations and their corresponding figural labels.

Table 2.5: The sequence of symmetry elements in international notations of one point group related to directions in seven crystal systems. Crystal systems Cubic Hexagonal Tetragonal Trigonal Orthorhombic Monoclinic Triclinic

Three directions in international notations c c c c a b a

a+b+c a a a b

a+b a + b a+b

Notice Hexagonal also

c

crystal system, the c axis (i.e., [001]) is the direction of the first symmetry element, and a + b + c (i.e., [111]) is the second direction as well as a + b (i.e., [110]) is the third one. Table 2.6 lists the distribution of 32 point groups in seven crystal systems.

①

/m

 /m



One  or m Three mutual perpendicular  or two mutual perpendicular m

/m



2/m means the symmetry plane of the structure is vertical to 2-fold axis, and so forth

Feature symmetry element N/A

/m

①






One 4 or 4

/m /m /m




One 3 or 3

/m


3

m

 m





m




4



m



m


One 6 or 6

/m /m /m





 

 m

Four 


/m /m 3

 m


6




/m 3
4


3



Cubic

/m





m


6

Rhombohedral Hexagonal

/m






1

m

m



Symmetrical elements m


4



Tetragonal

Triclinic Monoclinic Orthogonal

Crystal system

Table 2.6: The distribution of 32 point groups in seven crystal systems.

56 Chapter 2 The structure of solids

2.2 Metallic crystal structures

57

So far, only the considered groups of operations are those repeating an object about a fixed point, and thus this is equivalent to referring to groups of operations that repeating all space points about a point and yet leave the points unmoved, which gives rise to the designation point groups. The group of operations containing translations does not leave a point unmoved. More general kinds of repetition involve not only rotations, reflections, and inversions, but also translations. It is possible for an object to be repeated by translations as well as by rotations, to form a more complex and extended pattern. The group of operations containing translations does not leave a point unmoved. In a sense, however, it leaves space unmoved if a translation requires a duplication of space at periodic intervals. A complete consideration of combinations of translations with all kinds of rotations yields complex symmetry groups, which are called space groups. Since the symmetry elements in the same point group must intersect at one fixed point (center of the point group) or more fixed points, it is evident that only when a lattice point of Bravais lattice coincides with the center of the corresponding point group, the rest lattice points conform to the symmetry operations in the point group. Thus every single spot of Bravais lattice can be the center of the point group. In this way, it is self-consistency. For example, the structure bearing a fourfold axis should be placed at a lattice point of the primitive cubic lattice to maintain the rotational symmetry. The space lattice consists of translational (periodic) equivalent points (lattice points). Therefore, when the lattice points are placed with structure motifs so as to establish a crystal structure, the symmetry of crystal should be described by the combination of the symmetry elements in the point group and the translational symmetry elements in the space lattice. In this way, in derivation of space groups we should give priority to considering the two new kinds of symmetry elements produced by combination of the rotations (or reflection) in point groups and the translational symmetry elements in space lattices (i.e., glide planes and screw axes). In sum, the combination of the 32 point groups with the translational symmetry elements generates 230 space groups. The international notation of space group is composed of one capital English letter and following international notation of point group, in which the capital English letter stands for designations of the general space-lattice types, as shown in Table 2.7. For example, Pð4=mÞnm space group indicates that lattice type is primitive lattice (P) and the point group is ð4=mÞnm.

2.2 Metallic crystal structures Metals are generally crystals in the solid state. The internal factors in determining the crystal structure are both the bonding type and the strength of atoms, ion or molecules. Since the metallic bond of a metallic crystal is nondirectional and unsaturated, so the atoms inside the metal trend to arrange closely, and then a simple crystal structure with a high degree of symmetry can be formed.

58

Chapter 2 The structure of solids

Table 2.7: Designations of the general space-lattice types. Symbol

Name

Locations of additional points

Total number of lattice points per cell

P

Primitive

I

Body-centered

Center of cell



A

A-centered

Center of A, or （）face



B

B-centered

Center of B, or （）face



C

C-centered

Center of C, or （）face



F

Face-centered

Center of A, B, and C faces



R

Rhombohedral

At 23 31 31 and 31 23 23 , i.e., two points along the long body diagonal of cell





2.2.1 Three typical metallic crystal structures The crystal structures of all elements in the periodic table have been experimentally measured. Most of metals are fcc structure (notation: A1), bcc structure (notation: A2), and hcp structure ( notation: A3), as shown in Figs. 2.28–2.30, respectively. In crystalline structure description, atoms (or ions) are regarded as solid spheres possessing well-defined diameters. This is termed the atomic hard sphere model where spheres denoting the nearest-neighbor atoms touching one another. The atomic order in solid crystals indicates that small atom groups form a repetitive pattern, thus in crystal structure descriptions, it is convenient to subdivide the large structures into small repeating entities named unit cells. For most crystal structures the unit cells are parallelepipeds or prisms containing three sets of parallel plains. A unit cell is chosen based on the crystal structure symmetry, wherein all the atom positions in the crystal can be generated by extending the unit cell integral distances translations along each of the edges. Therefore, extending the unit cell concept is the basic structural crystal structure unit and defines the geometric crystal structure and the positions of atoms inside the crystal. The different features among fcc, bcc, and hcp crystals can be characterized by several aspects: the arrangement of atoms, the number of atoms in a unit cell, the lattice constant, the atomic radius, the coordination number, the atomic packing factor, and the atomic gap size.

2.2 Metallic crystal structures

Fig. 2.28: Face-center cubic structure.

Fig. 2.29: Body-centered cubic structure.

Fig. 2.30: Close-packed hexagonal structure.

59

60

Chapter 2 The structure of solids

Table 2.8: Crystallographic characteristics of three typical metal structures. Structural characteristics

Crystal structure type Face-centered cubic (A)

Body-centered cubic (A)

Close-packed hexagonal (A)

a pffiffiffi 2 a 4

c pffiffiffi 3 a 4

a,c(c/a = .) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! a 1 a2 c2 + 4 2 2 3

Number of atoms in the unit cell n







Coordination number (CN) Density K







.

.

.





. R

. R

. R

Quantity







Size

. R

. R()

. R

Lattice constant Atom radius R

Tetrahedral gap Octahedral gap

Quantity  Size

. R()

1. The number of atoms in a unit cell The crystal structure has been found for many metallic materials containing a cubic unit cell, with atoms at the corners of the cube and the center of the cube or the centers of the cube faces. Figures 2.28–2.30 depict that that the each of corner atoms of a unit cell is shared by eight unit cells; whereas, the atoms in the cube face are shared by two adjacent unit cells, and the atoms locate in the interior of a unit cell only belong to this unit cell. Therefore, the number of atoms were occupied by a unit cell in three typical structures are: 1 1 +6× =4 8 2 1 Body − centered cubic structure: n=8× +1=2 8 1 1 Close − packed hexagonal structure: n = 12 × + 2 × + 3 = 6 6 2 Face − centered cubic structure:

n=8×

2.2 Metallic crystal structures

61

2. Lattice constant and atomic radius The size of a unit cell is generally measured by the edge length of the unit cell a, b, and c, which are called the lattice constants (lattice parameters). They are the important parameters to characterize the crystal structures. The lattice constant is obtained by the X-ray diffraction (XRD) analysis. The different metals may have the same lattice types, but their lattice constants would be different because the electronic structure of each element that determines the bond condition of atoms is different. In addition, the lattice constant varies with the temperature. If the metallic atoms are considered as the hard (rigid) spheres with sphere radius, R, the relationship between the lattice constant and R in three typical crystal structures can be determined as follows: pffiffiffi FCC structure: lattice constant is a, and p2ffiffiffia = 4 R; BCC structure: lattice constant is a, and 3a = 4 R; hcp structure: lattice constants are denoted by a and c. In the ideal case, where c/a = 1.633, a = 2 R. However, the actual measured axial ratio often deviates from  1 = 2 1.633, and in this case: a2 =3 + c2 =4 =2 R. Lattice constants and the atomic radius of common metals are listed in Table 2.9.

Table 2.9: Lattice constant and atomic radius of common metals. Metal Lattice Lattice constant type (nm) (room temperature)

Atomic Metal Lattice radius type (nm) (CN = )

Lattice constant (nm) (room temperature)

Atomic radius (nm) (CN = )

Al

A

.

.

Cr

A

.

.

Cu

A

.

.

V

A

.

. ( °C)

Ni

A

.

.

Mo

A

.

.

γ-Fe

A

.( °C)

.

α-Fe

A

.

.

β-Co

A

.

.

β-Ti

A

. ( °C)

.( °C)

Au

A

.

.

Nb

A

.

.

Ag

A

.

.

W

A

.

.

Rh

A

.

.

β-Zr

A

. ( °C)

. ( °C)

Pt

A

.

.

Cs

A

. (− °C)

. (− °C)

Ta

A

.

.

α-Co

A

. . .

.

62

Chapter 2 The structure of solids

Table 2.9 (continued ) Metal Lattice Lattice constant type (nm) (room temperature)

Atomic Metal Lattice radius type (nm) (CN = )

Lattice constant (nm) (room temperature)

Atomic radius (nm) (CN = )

Be

A

a . c/a . c .

.

α-Zr

A

. . .

.

Mg

A

. . .

.

Ru

A

. . .

.

Zn

A

. . .

.

Re

A

. . .

.

Cd

A

. . .

.

Os

A

. . .

.

α-Ti

A

. . .

.

Note: Atomic radius is not a constant, it is not only related to the extent conditions such as the temperature and pressure, but it is also closely related to the crystal structure (eg. coordination number) and bond changes. According to V.M. Goldschmidt’s work, atom radii decreases along with coordination number, thus it is only meaningful to compare atom radii when the coordination number is the same. Therefore, the atom radius of the elements in Table 2.6 are calculated when coordination number is 12.

3. Coordination number and atomic packing factor The other two significant characteristics of the crystal structure are the coordination number and the atomic packing factor (APF). For metals, each atom has the same number of nearest-neighbor or touching atoms, which is named the coordination number. The APF is the sum of the sphere volumes of all atoms within a unit cell divided by the unit cell volume (V), that is, APF (Kv) can be described as: Kv =

nv V

where n is the number of atomic in the unit cell, and v is the volume of an atom. Since the metallic atoms are considered as the rigid spheres with the same diameter, v = 4πR3 /3. For example, APF of fcc crystal is: pffiffi 3 2 4 4 3 π ×4 πR × 4 3 4 a = = 74% Kv = 3 3 3 a a

2.2 Metallic crystal structures

63

and APF of bcc crystal:

Kv =

4 3 3 πR × 2 a3

=

4 3π

pffiffi 3 3 ×2 4 a a3

= 68%

The coordination number and atomic packing factor of three typical crystal structures are listed in Table 2.10. For the fcc structure, the APF is 0.74, which is the maximum packing factor for spheres possessing the same diameter. Metals typically have relatively large atomic packing factors to maximize the shielding provided by the free electron cloud. Table 2.10: Coordination number and APF of typical metal crystal structure. Crystal structure type

Coordination number (CN)

Density K

A



.

A

( + )

.

A

( + )

.

Note: 1. Coordination number of body-centered cubic structure is 8, nearest second nearest neighbor atom distance is xxxα, Moreover there are 6 second nearest neighbor atoms with distance of α, which it would also be included in the coordination number and result in CN=(8 + 6). 2. In hexagonal close-packed structure, only when c/a = 1.663 the coordination number is 12, if c/a ≠ 1.633, then there are 6 nearest neighbor atoms (in the same layer) and 6 second nearest neighbor atoms (3 atoms in the upper larger and 3 in the lower layers), therefore the coordination number should be (6 + 6).

2.2.2 Close-packed crystal structures and interstitial sites Figures 2.28–2.30 indicate that there is a group of close-packed atomic planes and directions in all three crystal structures. They are {111} in the FCC structure, {110} in the BCC structure and {0001} in the HCP structure. These closepacked atomic planes stack up layer by layer in space, then three different crystal structures formed. From the last section, the atomic packing factor of the fcc structure, and hcp structure are both 0.74. They are the densest structures in the pure metal. In fcc structure and hcp structure, the atoms in the close-packed atomic plane are tangential to the nearest atom, while in the bcc structure, except the atom in the body-center which is tangential to all the 8 atoms located at the top corners, the atoms located at the top corner are not tangential to each other. Therefore, the atomic packing factor

64

Chapter 2 The structure of solids

is smaller than those of the face-centered cubic structure and the hexagonal closedpacked structure. Further observation showed that the arrangement of atoms on the {111} plane in the fcc structure, as well as the {0001} plane in the hcp structure, are identical as shown in Fig. 2.31. If we locate the centers of the atoms on the close-packed atomic plane to form at a hexagonal grid, the hexagonal grid can then be divided into six equilateral triangles, and, the centers of the six equilateral triangles coincide with the centers of the six interstices along the atoms. Figure 2.32 illustrates that the six interstices are divide into the B, and C groups, each of which group constitutes an equilateral triangle. In order to get the tightest stacking, each atom of the second layer of the close-packed plane should be located at the interstices (low ebb) of adjacent three atoms on the first layer of the close-packed plane (A layer). It is easy to see that there are two kinds of stacking types of the close-packed plane, one is the ABAB . . . or ACAC . . . stacking sequence, which constitutes the hcp structure (Fig. 2.30); while the other is ABCABC . . . or ACBACB . . . stacking order, which is a fcc structure (Fig. 2.28).

Fig. 2.31: Atom arrangement on closely packed planes of close-packed hexagonal structure and face-centered cubic structure.

Fig. 2.32: Analysis of close-packed faces in face-centered cubic and close-packed hexagonal structures.

2.2 Metallic crystal structures

65

From the analysis of rigid sphere model for the arrangement of atoms in the crystal, as well as the atomic packing factor, it is obvious that there are many interstices (voids) in the metallic crystal. These interstices have a significant impact on the properties of metals, phase structure of alloys, diffusion, and phase transition. Figures 2.33–2.35 are the schematic diagrams of interstitial sites of three typical metallic crystal structures. The interstice is located at the center of the octahedron which is constituted by six atoms and is called a octahedral site, while the interstice is located at the center of the tetrahedron which is constituted by four atoms and is called a tetrahedral interstice. In Fig. 2.36, a solid circle represents a metallic atom, assuming its radius is rA ; while the open circle represents a interstice, assuming its radius is rB . Essentially, rB represents the radius of the ball that can be placed in the interstice (Fig. 2.36).

Fig. 2.33: Interstices in the face-centered cubic structure.

Fig. 2.34: Interstices in the body-centered cubic structure.

66

Chapter 2 The structure of solids

Fig. 2.35: Interstices in the hexagonal close-packed structure.

The radius of the ball that can be placed in the interstice rB Octahedral

The radius of the ball that can be Tetrahedral placed in the interstice rB

rA

r

B ― rA = 0.414

r

B ― rA = 0.225

Fig. 2.36: Rigid sphere model of the interstices in the face-centered cubic crystal.

The number and size of tetrahedral and octahedral interstices in three kinds of crystal structures are obtained by using the geometrical relationship. For example, for the size of octahedral interstice in fcc crystal r=

4 a − 2R pffiffi2 R − 2R = 0.414R, r=R = 0.414 = 2 2

for the size of tetrahedral interstice in fcc crystal pffiffiffi pffiffiffi  3 3 4 r= a−R= . pffiffiffi − 1 R = 0.225R, r=R = 0.225 4 4 2

(2:9)

(2:10)

for the size of octahedral interstice in bcc crystal along r=

4 a − 2R pffiffi3 R − 2R = 0.154R, r=R = 0.154 = 2 2

and for the size of octahedral interstice in bcc crystal along

(2:11)

2.2 Metallic crystal structures

pffiffiffi pffiffiffi 2 2 4R r pffiffiffi − R = 0.633R, r= a−R= = 0.633 2 2 3 R

67

(2:12)

for the size of tetrahedral interstice in bcc crystal sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi  a 2 a2 5 4 + · pffiffiffi − 1 R − 0.291R, r=R − 0.291 −R− r= 4 2 4 3

(2:13)

Table 2.11 lists the number and size of tetrahedral and octahedral interstices in three kinds of crystal structures.

Table 2.11: Interstices in three typical crystals. Crystal structure

Interstices type

Face-centered cubic

Body-centered cubic

Closely packed hexagonal

Interstices number

Interstices size

Tetrahedral interstices



.

Octahedral interstices



.

Tetrahedral interstices



.

Octahedral interstices



. () . ()

Tetrahedral interstices



.

Octahedral interstices



.

Note: Tetrahedral and octahedral sites in bcc structure are asymmetric. The edge length is not all equal. It has a significant effect on the solid solution of the interstitial atom and the related distortion.

2.2.3 Polymorphism and allotropy A solid metal has different crystal structure under different temperatures and/or pressures, which is named polymorphism. The transformed product is called the allotropy. A famous example of allotropy of great technological significance is the case of iron. At normal pressure (= 1 atm) and room temperature iron is bcc crystal structure (also named as α-iron or ferrite). At 912 °C (Fig. 4.29) iron performs a solid–solid phase transition thus above the temperature of 912°C iron becomes fcc crystal structure (also named as γ -iron). This γ -iron would transform into (again) a bcc crystal structure at 1,394 °C. Such high-temperature bcc modification of iron is called δ-iron phase, which continues to exist up to the iron melting temperature at 1,538 °C. The allotrope to allotrope phase transitions are reversible, i.e. they happen upon both heating and cooling processes. Thus, what kind of allotrope occurs (α, γ, or δ) is only temperature (at normal pressure) dependent. However, it is a δ-Fe between 1,394 °C and the melting point,

68

Chapter 2 The structure of solids

which is to bcc structure. Due to the different atomic packing factors (APF) in the different crystal structures, it is accompanied with the saltation of the mass volume when such a metal transforms from one kind of crystal structure to another. Figure 2.37 shows the expansion curve of iron when it is heated. There are apparent turning points in the curve due to the saltation of volume when the α-Fe transforms to the γ-Fe and then the γ-Fe transforms to the δ-Fe with the raising temperature. Some metals such as Mn, Ti, Co, Sn, Zr, U, and Pu also show the features of polymorphism. The properties of these metals vary with temperature or pressure, and understanding this point is important in their applications. 1.032

Linear expansion coefficient L/L0

1.028 1.024 1.020 1.016 0.012 1.008 1.004

α

γ

󰛿

1.000 0.995

–273 0

400 800 1,200 1,600 2,000 Temperature (°C)

Fig. 2.37: Expansion curve of pure iron during heating.

2.3 Phase structures of alloys Although pure metals have been significantly applied in industry, due to their low strength the most widely used metallic materials in the industry are alloys. An alloy is a kind of substance with metallic properties, and it is composed of two or more elements, including metallic elements and nonmetallic elements that mixed by smelting, sintering or other methods. For instance, the most commonly used carbon steels, as well as the cast irons, are alloys mainly composed of Fe and C. Besides, brass is an alloy consisting Cu and Zn. Alloying is the main approach to change and improve the properties of metallic materials. In alloying the concept of “phase” is defined as a homogeneous portion of a microstructure that possesses uniform physical and chemical characteristics. Two

2.3 Phase structures of alloys

69

phases are separated by the interfaces. A single-phase alloy is composed of one kind of phase, while the multiphase alloy is composed of more than two kinds of phases. Different atoms can exist together in a single crystal structure. If atoms of more than one element can be sit randomly at the atomic sites in the same crystal structure, within a certain composition range, the corresponding substance is named a solid solution. For such crystal structure if different types of atoms occupy preferably elementspecific sites, for a certain composition range, the corresponding substance is named an ordered solid solution. within a crystal structure of a compound holding different atom types all atoms (ideally) reside on only element-specific sites of the crystal structure. In other words, a solid solution is composed of the solvent and other solute atoms uniformly dissolved in it, and the crystal structure of the solid solution depends on the crystal structure of the solvent. For instance, at ambient temperature and at 1 atm pressure pure solid Fe possesses a bcc crystal structure and pure solid Al has a fcc crystal structure. When under sufficiently high temperature and at 1 atm pressure, we add a small amount of Al to a pure Fe melt thus a homogeneous liquid mixture of Fe and Al atoms is generated. Upon cooling, solidification of the liquid would occur. The solid obtained (at room temperature and at 1 atm) does not consist of a small volume fraction of fcc Al and a large volume fraction of bcc Fe. Instead, it is composed of one single bcc solid solution with the same structure of the solvent Fe. While the crystal structure of a compound is different from that of any single element in the compound. Take the mixing of Fe and Al as an example again. When we add enough amount of Al to a melt of pure Fe such that a superlattice Fe3Al compound may form during solidfication, and the crystal structure of this compound is different from that of Fe or Al. The phase structures of alloys are controlled by their atomic size, electronegativity and valence concentration, respectively.

2.3.1 Solid solution The main feature of the solid solution structure is that it keeps the crystal structure of the initial solvent. Based on the location of a solute atom in the solvent lattice, a solid solution is divided to the substitutional solid solution and also the interstitial solid solution. They will be discussed as follows, respectively. 1. Substitutional solid solution When the solute atoms dissolve into a solvent, they reside at the same sites of a solvent lattice so that the solute atoms may replace the part of solvent atoms in the solvent lattice accompanying with the variation of lattice parameters. This kind of solid solution is called substitutional solid solution.

70

Chapter 2 The structure of solids

Although the combination of metallic elements generally forms the substitutional solid solution, the solubility of solute atoms in the solvent lattice is usually limited. The solubility is mainly determined by the following factors. a. Crystal structure It is essential for elements to have the same crystal structure if they want to form a infinite solid solution. Only when elements A and B have a same structure type, an atom B may be continuously replace atom A if the size difference of the atoms of A and B elements is less than 15%, as shown in Fig. 2.38. Obviously, if the crystal structure of two elements is different, the solubility between elements will be finite. During the formation of the finite solid solution, the solubility of the solute element in the solvent element with the same structure is usually larger than that of the solute in a solvent with a different structure. Table 2.12 lists the solubility range of some alloy elements in Fe h, which can be used to well explain the above rule. For example, the solubility of bcc Mo in fcc γ − Fe is only about 3%, but the solubility of bcc Mo in bcc α − Feis high up to about 37.5%.

Fig. 2.38: A schematic diagram of the atomic substitution of two elements in infinite solid solution.

b. Atomic size The experiments indicate that in the same structure condition, it is possible for a solid solution to form a solid solution with a big solubility when the difference of atomic radii of the solute and solvent, Δr, is less than 15%, moreover, the larger Δr is, the less the solubility will is. The effect of the atomic size on solubility is mainly related to the lattice distortion caused by the solute atoms. The larger Δr means the larger lattice distortion accompanying with larger distortion energy, which will lower the structural stability and solubility.

2.3 Phase structures of alloys

71

Table 2.12: Solubility of alloying elements in iron. Element Structure type

Solubility in γ-Fe

Solubility in α-Fe under room temperature

Solubility in α-Fe

.

.

.( °C)

.

.

.( °C)

Orthogonal

. ~ .

~.

 °C) α-Ti closely packed hexagonal ( °C) α-Zr closely packed hexagonal ( °C) γ-Mn Face-centered cubic ( ~  °C) α,β-Mn Complex cubic ( °C) α-Co closely packed hexagonal (41%, the solute atoms can enter the interstitial positions to form interstitial a solid solution. These solute atoms are the nonmetallic elements with an atomic radius less than 0.1 nm, such elements include H, B, C, N, O, and their atomic radii are 0.046, 0.097, 0.077, 0.071, and 0.060 nm, respectively. The formation of interstitial solid solutions usually leads to the solvent lattice distortion due to the larger size of solute atoms than the interstitial size accompanying with a higher distortion energy , which increases with the increase of solute atom size. Therefore, the solubility limit of the interstitial solid solutions is very small but such a little solubility limit exhibits a strong strengthening effect. The solubility of the interstitial solid solution is related to the size of the solute atom, as well as the shape and size of the interstitial sites in the solvent crystal structures. For instance, the maximal solubility of C in the γ-Fe is w(C) = 2.11 wt.%, while the maximal solubility of C in the α-Fe is only w(C) = 0.0218 wt.%. The reason is that the carbon atoms in the γ-Fe and α-Fe are positioned in the octahedral sites, but, the

2.3 Phase structures of alloys

75

size of octahedral sites in the γ-Fe occupied by C is much larger than that in the α-Fe. It is worth to pointing out that the tetrahedral and octahedral sites in the bcc α-Fe structure are asymmetric. Although the size of octahedral sites is 0.154 R in the direction, which is less than 0.291 R of the tetrahedral sites, the size of octahedral sites is 0.633 R in the direction, which is much larger than that of the tetrahedral sites. Therefore, when a C atom is squeezed into octahedral sites of α-Fe, it only needs to push away the Fe atoms above and below the octahedral sites. Obviously, this way is much easier than that to squeeze into the tetrahedral sites, because the latter needs to push away four Fe atoms. Even though, the real solubility of C in α-Fe is still very low. 3. Ordering and disordering in solid solution Figure 2.42 shows the schematic of the distribution of solute atoms in the solid solutions. Indeed, there is no any solid solution with the complete random distribution. It is usually recognized that under the thermodynamic equilibrium state, the distribution of solute atoms in the disordered solid solution is uniform in the macroscopic scale, but it is not uniform in the microscopic scale. They may have a regular distribution and then form an ordered solid solution some the certain conditions. At these conditions, the solute atoms are positioned in the fixed locations of the lattice. In addition, the ratio of the solute and the solvent atoms in each unit cell is constant. The lattice structure of the ordered solid solution is called superstructure or superlattice, which will be in detail described in the following subsection. The distribution of solute atoms in the solid solution depends on the relative value of the binding energy (EAA, EBB) of like pairs of atoms of the same type and the binding energy EAB of atoms of different types. If EAA ≈ EBB ≈ EAB, the solute atoms tend to be distributed randomly. If (EAA + EBB)/2 and b2 = a/n < υ2 ν2 w2 >, then a a a ½u1 v1 w1  + ½u2 v2 w2  = ½u1 + u2 v1 + v2 w1 + w2  (3:10) n n n ffi a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Usually, the vector module jbj u2 + v2 + w2 stands for the strength of the dislocation. n In the same crystal, the larger the Burgers vector is, the larger distortion caused by the dislocation is, and the higher the energy has. Dislocations with high energy trend to decompose into two or more dislocations with low energies: b1→b2 + b3. Such process satisfies |b1|2 >|b2|2 +|b3|2 in order to reduce the total energy of the system. b = b1 + b2 =

3.2 Dislocations

143

3.2.3 The movement of dislocation A most important characteristic of a dislocation is its mobility, which is directly related to the plastic deformation of the crystal. The mechanical properties of the crystal, such as the strength, plasticity, and fracture, are all linked to the movement of the dislocation. Thus, the understanding of dislocation movement can improve or control the mechanical properties of crystals. The movement of dislocations can be classified into the slip and climb. 1. Slip of the dislocation The slip of the dislocation occurs under the external shear stress. This process carries forward through the movement of atoms around the dislocations step by step (each step is less than an atomic distance) along the direction of Burgers vector on the slip plane. Figure 3.13 illustrates the slip process of the edge dislocation. Under the external shear stress τ, the atoms around the dislocation move the distance of less than one atomic spacing from position “.” into position “。.” Accordingly, the dislocation moves one atomic distance to the left on the slip plane. As shown in Fig. 3.13(b), the dislocation continues moving to the left under the external shear stress. The dislocation line passes through the entire crystal on the slip plane, leaving a step on the surface of the crystal. The width of the step equals to the magnitude of the Burges vector. The movement of the dislocation causes the plastic deformation of the crystal. With the movement of the dislocation, the passing region ABCD by the dislocation (or say the slipped region) expands, and the non-slipped region continues shrinking. The boundary between the slipped and non-slipped regions is the dislocation line. In addition, the movement direction of the edge dislocation is always vertical to the dislocation line and parallel to the Burgers vector during its slip. The slip plane of the edge dislocation is determined by the dislocation line and its Burgers vector, thus the slip of edge dislocation is on a unique slip plane.

(a)

(b)

Fig. 3.13: Slip of an edge dislocation: (a) slip processes and (b) atomic displacement around edge dislocations.

144

Chapter 3 Crystal defects

Figure 3.14 illustrates the slip process of the screw dislocation, in which “。” stands for the atoms under the slip plane, “.” represents the atoms above the slip plane. Similar to those around the edge dislocation, atoms around the screw dislocation (Fig. 3.14(a)) have a small displacement (Fig. 3.14(b)), indicating a small force for the movement of screw dislocation. When the dislocation slips over the entire crystal, a step, with the Burgers vector b in width, appears on the surface of the crystal. More remarkably, the slip direction of the screw dislocation is perpendicular to the dislocation line and its Burgers vector. Since the dislocation line of the screw dislocation is parallel to its Burgers vector (Fig. 3.14(c)), the slip of screw dislocation is not limited on a single slip plane.

(a)

(b)

(c)

Fig. 3.14: Slip of the screw dislocation: (a) the slip processes, (b) the original atomic position, and (c) atomic displacement around the screw dislocation.

Figure 3.15 illustrates the slip process of a mixed dislocation. Since we all know that any mixed dislocations can decompose into edge-type and screw-type components, we can describe the slip process of the mixed dislocations through defining the slip processes of their components. According to the right-hand rule of determining the moving direction of a dislocation, the thumb stands for the crystal moving along the Burgers vector b, the index finger points to the dislocation line, while the middle finger represents the moving direction of the dislocation. Under the external shear stress τ, the mixed dislocation expands along the normal direction at each position on the slip plane. Eventually, the crystal moves a |b| distance along the direction of its Burgers vector.

Fig. 3.15: Slip of the mixed dislocation.

3.2 Dislocations

145

As mentioned above, any planes including the dislocation line of a screw dislocation can be the slip plane. Thus, a screw dislocation may transfer from its original slip plane into another slip plane when hindered on the original one. Such process is called the cross-slip. After that, the screw dislocation can also slip back to the original plane after its cross-slip. This is called the double cross-slip, as shown in Fig. 3.16.

Screw dislocation

b

Cross-slip plane

Main slip plane

Fig. 3.16: Double cross-slip of the screw dislocation.

2. Climb of the dislocation Edge dislocations, in addition to slip on the slip plane can climb perpendicular to the slip plane. We define the positive climb as the process of moving the extra atomic plane up, and the negative climb as the reverse process, as shown in Fig. 3.17. The climb of edge dislocations is the expansion or shrinking of extra plane, thus, it occurs through the mass migration (the diffusion of atoms or vacancies). The positive climb occurs when vacancies migrate to the bottom of the extra plane or atoms at the bottom of the extra plane diffuse to somewhere else (Fig. 3.17(b)); on the other hand, the negative climb occurs when atoms diffuse to the bottom of the extra plane (Fig. 3.17(c)). Screw dislocations do not have extra atomic planes, thus, they cannot climb.

(a)

(b)

(c)

Fig. 3.17: Climb of the edge dislocation: (a) unclimbed dislocation, (b) positive climb from vacancies and (c) negative climb from interstitial atoms.

Dislocation climb changes the number of atoms around dislocations, or saying, involves mass migration. Thus, dislocation climb occurs with diffusion. The climb is also called “nonconservation motion” while the slip is called “conservation

146

Chapter 3 Crystal defects

motion.” Dislocation climb requires thermal activation, which conveys higher energy than that needed for the dislocation slip. Dislocation climb can hardly occur at the room temperature for most materials but easier at a high temperature. The oversaturated point defects, for example, vacancies and interstitial atoms from high-temperature quenching, cold working, and high-energy particle irradiation, benefit the dislocation climb. 3. The intersection of moving dislocations When a dislocation moves on its slip plane, it intersects with other dislocations (usually we define other dislocations passing the slip plane as forest dislocations). The interaction from dislocation intersection plays an important role in material strengthening and the creation of point defects. a. Jogs and kinks Usually, it is hard for the entire dislocation line to move on the slip plane. Part of the dislocation line may slip while the rest is pinned. Such process leads to the dislocation line with a zigzag shape, defined as kinks. If such zigzag dislocations are vertical to the slip plane, they are defined as jogs, jogs and kinks can also form during the intersection of dislocations. As we have know, the climb of edge dislocation occurs through the diffusion of vacancies and atoms. Atoms (or vacancies) gradually migrate into the dislocation line, probably leaving steps between the climbed and nonclimbed regions. Such step is another type of jogs. We may treat dislocation climb as the migration of jogs along dislocation lines, leading to the movement of dislocation lines forward-up or forward-down. Thus, the climb process is related to the formation energy of jogs and their migration rate. Figure 3.18 shows the jogs and kinks of screw and edge dislocations. It can also be noted that the jogs of edge dislocations are edge dislocations while their kinks are screw dislocations. All jogs and kinks of screw dislocations are edge dislocations.

(a)

(b)

Fig. 3.18: Jogs and kinks in: (a) edge dislocation and (b) screw dislocation.

3.2 Dislocations

147

3.2.4 Elastic properties of dislocations Dislocations have local lattice distortion and corresponding elastic fields. The strain field of the dislocation is important to estimate the dislocation energy, the interaction forces between dislocations, and so on. 1. Stress fields of dislocations It is difficult and complex to precisely compute the elastic stress field around a dislocation. In order to simplify the case, we apply the elastic continuum model with the following three assumptions. First, the crystal is elastic and obeys the Hooke’s law. Second, the crystal is isotropic. Third, the crystal is continuous medium, without any voids. The stress, strain, and displacement of the crystal are continuous. It should be noted that this model ignores the severe lattice distortion within dislocation core. Thus, the calculated results are not valid for dislocation cores. This fact has been verified experimentally. From the mechanics of materials, any point inside a solid needs nine stress components to describe its stress states. Figures 3.19(a) and (b) shows the stress components of units in Cartesian and cylindrical coordinates, respectively. σxx, σyy, and σzz (σrr, σθθ, and σzz) are three normal stress components, τxy, τyx, τxz, τzx, τyz, and τzy (τrθ, τθr, τzr, τrz, τzθ, and τθz) are six shear stress components. The first subscript of these components stands for the normal direction of the acting surface, and the second subscript is for the stress direction.

(a)

(b)

Fig. 3.19: Stress components in: (a) Cartesian coordinates and (b) cylinder coordinates.

At equilibrium, τij = τji, that is τxy = τyx, τxz = τzx, τyz = τzy (τrθ = τθr, τzr = τrz, τzθ = τθz). Thus, six components are enough to determine the stress state. The normal components are εxx, εyy, and εzz, and the shear components are γxy, γyz, and γzx.

148

Chapter 3 Crystal defects

a. Stress field of screw dislocation A screw dislocation can be constructed by employing a hollow cylinder in an isotropic material as follows: cut the cylinder along the xz plane, slip the cut surfaces by a displacement of b, then glue the cut surfaces together. As shown in Fig. 3.20, OO’ is the dislocation line, and MNOO’ is the slip plane.

Fig. 3.20: Model for a screw dislocation within a continuous medium.

Since the cylinder has only the z-displacement, there is only a shear strain: γθz = b/ (2πr). Thus, the shear stress is τzθ = τθz = Gγθz = Gb/(2πr). The rest stress components are zero, that is, σrr = σθθ = σzz = τrθ = τθr = τrz = τzr = 0. In Cartesian coordinates, we have τyz = τzy =

Gb x · 2π x2 + y2

τz = τxz =

Gb y · 2π x2 + y2

σxx = σyy = σzz = τxy = τyx = 0

(3:11)

Thus, the stress field of screw dislocations has the following characteristics: (1) There are only shear components. Normal stress components are zero, which indicates screw dislocations will not lead to the expansion and compression of the crystal. (2) The shear stress is inversely proportional to r, regardless θ and z. At any fixed r, τθz is a constant. Thus, the stress field of the screw dislocation is axial symmetry. It should be noted that when r!0, τθz!∞, which does not agree with the reality. This suggests this model is not suitable for the dislocation core.

149

3.2 Dislocations

b. Stress fields of edge dislocation The stress fields of edge dislocations are more complicated than those of screw dislocation. Similarly, the edge dislocations can also be constructed by employing a hollow cylinder in an isotropic material: cut the cylinder along xz plane, move the cut surfaces by a displacement of b along the x direction, then glue the cut surfaces together. Figure 3.21 illustrates the stress field of an edge dislocation.

Fig. 3.21: Model for edge dislocation.

Accordingly, stress components can be solved based on the elasticity theory as σxx = − D · σyy = D ·

yð3x2 + y2 Þ ðx2 + y2 Þ2

yðx2 − y2 Þ

ðx2 + y2 Þ2   σzz = v σxx + σyy τxy = τyx = D ·

xðx2 − y2 Þ ðx2 + y2 Þ2

τxz = τzx = τyz = τzy = 0

(3:12)

If in cylindrical coordinates, the stress components are σrr = σθθ = − D ·

sinθ r

σzz = vðσrr + σθθ Þ τrθ = τθr = D ·

cosθ r

τrz = τzr = τθz = τzθ = 0

(3:13)

where D = Gb=ð2πð1 − vÞÞ, G is the shear modulus, v is Poisson’s ratio, and b is the magnitude of the Burgers vector.

150

Chapter 3 Crystal defects

The stress field of edge dislocations has the following characteristics: (1) There are normal stress components and shear components. They are proportional to G and b, and inversely proportional to r. The farther it is away from the dislocation, the smaller the stress is. (2) All stress components are functions of x and y, regardless of z. This suggests that the stress is the same for any point in the line parallel to the dislocation. (3) The stress field of edge dislocation is symmetrical with respect to the extra half plane, or the y-axis. (4) If y = 0, σxx = σyy = σzz = 0. This suggests that there is only shear stress, no normal 1 stress, on the slip plane. The shear stress reaches its maximum of 2πðGb 1 − vÞ · x (5) If y > 0, σxx < 0. If y < 0, σxx > 0. The compressive stress is on the top of the slip plane and the expansive stress is below the slip plane. (6) At any point of the stress field, | σxx | >| σyy |. (7) If x = ± y, σyy, or σxy equals to zero. There is only σxx along the diagonal line in the x–y coordinates, and τxy, (τyz), and σyy have reverse signs beside the diagonal lines. Figure 3.22 shows the stress field around an edge dislocation. Note: the above equations are not applicable for dislocation cores.

Fig. 3.22: Stress field around an edge dislocation within a continuous medium.

2. The strain energy of dislocations The lattice distortion around dislocation leads to the elastic stress field and increase the energy of the crystal. Such increment in energy is defined as the strain energy of a dislocation.

3.2 Dislocations

151

The dislocation energy includes two parts: the distortion energy of dislocation cores Ec and the elastic strain energy from the stress field Ee. The dislocation cores have very large lattice distortion, thus, the Hooke’s law is not applicable for the dislocation cores. The distortion energy can only be estimated through the lattice model. The distortion energy is roughly 1/10–1/15 of the total energy, and thus can be ignored. The dislocation energy can be represented by the elastic strain energy outside the dislocation cores, which can be estimated through the work being done to create a unit dislocation. Assume that the edge dislocation shown in Fig.3.21 is of a unit length. Since during the formation of the dislocation, the displacement along the slip direction is changing from 0 to b, the displacement is a variable, changing from 0 to b. The force for the slip plane MN changes with r. At displacement x, the shear stress is τθr = D · ðcosθ=rÞ, in which θ = 0. Thus, the work from the shear stress τθr is ðR ðb

ðR ðb τθr dxdr =

W= r0 0

r0 0

Gx 1 Gb2 R · ln · dxdr = 4πð1 − vÞ 2πð1 − vÞ r r0

(3:14)

This is the elastic energy Eee for a unit length edge dislocation. Similarly, the elastic energy for a unit length screw dislocation is Ees =

Gb2 R · ln 4π r0

Regarding the mixed dislocation with φ angle between the dislocation line and the Burgers vector, it can be decomposed into the edge component b sinφ and screw component b cosφ. Since there is no interaction between the edge component and screw component that are perpendicular to each other, the elastic strain energy can be calculated independently for two components. The elastic strain energy of a mixed dislocation is the sum of the strain energies of these two components, i.e., Eem = Eee + Ees =

Gb2 sin2 φ R Gb2 cos2 φ R Gb2 R · ln · ln + · ln = 4πð1 − vÞ r0 4π r0 4πK r0

(3:15)

  where K = 1 − v 1 − vcos2 φ is the angle factor for the mixed dislocation, and K ≈ 1–0.75. In fact, the energies of all straight dislocations can be estimated based on the above equation. Apparently, K = 1 is for screw dislocations, K = 1–ν is for edge dislo   cations and K = 1 − v 1 − vcos2 φ is for mixed dislocations. The elastic strain energy is related to r0 and R. In general, r0 is close to b, which is roughly 10−10 m. R is the maximum operation radius of the elastic strain field. In fact, there are substructures and dislocation networks in real crystals. Therefore, R ~ 10−6 m. Thus, the total elastic strain energy can be simplified as E = αGb2 where α is a geometrical factor with a value of 0.5–1.

(3:16)

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Chapter 3 Crystal defects

In summary, we have the following conclusions: (1) The energy of dislocation includes two parts: Ec and Ee. The energy for dislocation cores Ec is less than 1/10 of total energy, and thus can be ignored. The elastic strain energy of the dislocation Ee ∝ ln(R/r0), thus dislocations have longrange strain fields. (2) The elastic strain energy is proportional to b2. Thus, the dislocation with smallest Burgers vector b is the most stable in the crystal. The dislocation with a large b can decompose into dislocations with smaller b, in order to reduce the total energy. This is the underlying reason for the fact that the slip direction is always along the close-parked direction. (3) Ese/Eee = 1−ν, and ν ~ 1/3 for most metals. Thus, the elastic strain energy of a screw dislocation is 2/3 of the one for of an edge dislocation. (4) The dislocation energy is defined per unit length; thus, the dislocation energy is related to the shape of the dislocation. Since the straight line has the shortest distance, the straight dislocation has a smaller energy than the curved dislocation. Thus, straight dislocations are stable. The curved dislocations trend to become straight and reduce their length. (5) The existence of dislocations increases the internal energy of the system. Although dislocations may increase the entropy of the crystal, such increment is limited and can be ignored. Thus, the existence of dislocations leads to the higher energy for the crystal. The dislocations are unstable defects from the thermodynamic viewpoint. 3. Linear tension of dislocation The total elastic strain energy of dislocations is proportional to their length. Dislocations trend to shrink to reduce their energy. Thus, there is a linear tension T to straighten the dislocation. Linear tension is the configuration force, similar to the surface tension of the liquid. Linear tension can be defined as the energy increment for increasing a unit length of dislocation. Thus, linear tension T is just the elastic strain energy of the dislocation of a unit length in magnitude: T ≈ αGb2

(3:17)

where α is constant, and is roughly 0.5–1.0. It should be noted that the linear tension of the dislocation is not only to straighten the dislocation, but also the underlying reason for the formation of threedimensional dislocation network in crystals. Because all dislocations converging at each point have the balanced linear tension in the dislocation networks, these dislocations are stable in crystals.

3.2 Dislocations

153

4. Interacting force of dislocations Dislocations in crystals produce the strain fields around them. Since there are many dislocations inside a real crystal, a dislocations will be subjected to an acting force from the strain fields of other dislocations and give a counter-acting force. The interacting force changes with the dislocation type, Burgers vector, and dislocation line. a. Interaction between two parallel screw dislocations As shown in Fig. 3.23, two parallel screw dislocations, s1, s2, have Burgers vectors b1, b2. The dislocation lines are parallel to z-axis. Dislocation s1 is at the origin, and s2 is at (r, θ). Because of the fact that the stress field of the screw dislocation includes only shear components with axial symmetry, dislocation s2 is subjected to the radial forces of dislocation s1: fr = τθz · b2 =

Gb1 b2 2πð1 − vÞ

(3:18)

fr has the same direction with vector r. Similarly, dislocation s1 exerts the force with the same magnitude and reverse direction from dislocation s2. Thus, the interacting force of two parallel screw dislocations, is proportional to the project of the dislocation strength and inversely proportional to the distance between dislocations. The force direction is along the radial direction r and perpendicular to the dislocation line. When b1 is parallel to b2, fr > 0, that is, the two parallel screw dislocations repel each other. When b1 has the appositive direction with b2, fr < 0, indicating the two parallel screw dislocations attract each other when they have opposite signs (Fig. 3.23(b)). b2

b1

S2 b2

fr

r

b1 fr

fr r

S1 θ O b1

r (a)

b2 fr r

b1 and b2 same sign

b1 and b2 contrary sign (b)

Fig. 3.23: Interaction between two parallel screw dislocations.

b. Interaction between two parallel edge dislocations As shown in Fig. 3.24, two edge dislocations e1, e2, are parallel to z-axis with a interspacing of r(x,y). Their Burgers vectors b1 and b2 are parallel to x-axis. Set e1 at the origin, and set the slip plane of e2 parallel to that of e1, which is parallel to x–z plane. Thus, the shear stress τyz and normal shear σxx are the two components of

154

Chapter 3 Crystal defects

the e1 strain field interacting with e2, leading to the x-slip and y-climb of e2, respectively. These two interacting forces are fx = τθz · b2 = fy = − σxx · b2 =

Gb1 b2 xðx2 − y2 Þ 2πð1 − vÞ ðx2 + y2 Þ2 Gb1 b2 yð3x2 + y2 Þ 2πð1 − vÞ ðx2 + y2 Þ2

(3:19)

With regard to two parallel edge dislocations with the same signs, the slip force fx changes with the position of dislocation e2. Figure 3.25(a) illustrates their interaction forces as following:

Fig. 3.24: Interaction between two parallel edge dislocations.

When |x| > |y|, if x > 0, then fx > 0; if x < 0, then fx < 0. This means dislocation e2 at the region (1) and (2) in Fig. 3.25(a). The two dislocations repel each other. When |x| < |y|, if x > 0, then fx < 0; if x < 0, then fx > 0. This means dislocation e2 at the region (3) and (4) in Fig. 3.25(a). The two dislocations attract each other. When |x| = |y|, fx = 0. Dislocation e2 is at the metastable position. Once e2 deviates from its metastable position, it will suffer an attracting or repelling force from e1, which makes e2 far away from e1. When x = 0, dislocation e2 is at y-axis, and fx = 0. Dislocation e2 is at the equilibrium position, which is stable. Dislocation e2 suffers the attracting force of dislocation e1 and will be pulled back to the equilibrium condition if e2 has any deviation from the equilibrium condition. As a result, dislocations are vertically arranged, forming a dislocation wall, which can form small-angle grain boundaries. When y = 0, if x > 0, fx > 0; if x < 0, fx < 0. The module of fx is reversely proportional to x. Edge dislocations repel each other with the same sign on the same slip plane. The repulsion force increases with the decrement in their distance. The climbing force fy has the same sign with y. When dislocation e2 is on the top of the slip plane of dislocation e1, e2 suffers the positive climbing force fy. When dislocation e2 is below the slip plane, e2 suffers the negative force. Thus, two dislocations, along y-axis, repel each other. The interacting force fx, fy of two edge dislocations with different signs have the reverse sign of that in the case of two edge dislocations with the same signs as

3.2 Dislocations

y

y

3

3 2

155

π/4 1

O

2

x

π/4 1

O

4

4

(a)

(b)

x

fx 0.3

y x

0.2

Same b

0.1 0 –7y –6y –5y –4y –3y –2y –y

y 2y

3y

4y

5y

6y

7y x

–0.1 –0.2 y –0.3

x

Different b

(c) Fig. 3.25: Interaction of two edge dislocations along x-axis: (a) two dislocations with the same sign, (b) two dislocations with opposite signs, and (c) the interacting force fx of two parallel dislocations.

mentioned above. In addition, the metastable and equilibrium positions of dislocation e2 exchange. When |x| = |y|, e2 is at the equilibrium position, as shown in Fig. 3.25(b). Figure 3.25(c) comprehensively display the relationship between the interacting force fx of two parallel dislocations and their distance x. y in the figure stands for the vertical distance of two dislocations (i.e. the interspacing of atomic planes), x stands for the horizontal distance (measured by the multiple of y), the unit of fx is Gb1 b2 =ð2πð1 − vÞyÞ. As shown in Fig. 3.25(c), the interacting force of two same-sign dislocations (the solid line) has the same magnitude but a reverse direction with that dislocations with reverse signs (the dotted line). Regarding the fy of dislocations with reverse signs, namely, it has opposite sign with y. These two dislocations along y-axis with opposite signs always attract each other, and will probably disappear because of the recombination of dislocations.

156

Chapter 3 Crystal defects

Except those cases, no interaction force exists between parallel screw and edge dislocations because they have perpendicular Burgers vectors, leading no interaction of the individual strain fields. If one or both of the two parallel dislocations are mixed-type dislocations, their interaction force can be estimated through calculating the individual interaction forces of edge-type and screw-type components of dislocations.

3.2.5 Generation and multiplication of dislocations 1. Density of dislocations Except these fine whistlers, there are a large number of dislocations in real crystals. Dislocation density represents the number of dislocations. Dislocation density is defined as the total length of dislocation lines a in unit volume of crystal. The mathematical equation is ρ=

L  − 2 cm V

(3:20)

where L is the total length of the dislocation and V is the volume of the crystal. However, the total length of the dislocation cannot be practically measured in a real crystal. In order to simplify the problem, dislocations are treated as straight lines crossing the crystal. Thus, the dislocation density means the total number of the dislocation passing the unit area, ρ=

nl n = lA A

(3:21)

where l is the length of each dislocation and n is the number of the dislocation shown in unit area A. Apparently, not all dislocations intersect the observing surface. Thus, the measured dislocation density is smaller than the real value. The experimental results suggest that the dislocation density is 106–108 cm−2 in typical fully annealed metals. The dislocation density can be lower than 103 cm−2 in the single-crystal metal with special preparation. The dislocation density is 1010–1012 cm−2 in metals after severe cold deformation. 2. Dislocation generation Dislocation density is high in most crystals, even in those single crystals carefully processed. How to create so many dislocations? The sources of dislocations can be: (1) Dislocations generated during the crystal growth: a. Dopants or inclusions are nonhomogenously distributed during the solidification process, leading to nonuniform compositions of the solidified crystals

3.2 Dislocations

157

with different lattice constants. Thus, dislocations are introduced to accompany such nonuniformity. b. Because of the temperature gradient, composition gradient, and mechanical vibration, the grown crystal may deviate or bend, leading to the formation of misoriented crystal domains and corresponding dislocations in the crystal. c. During the crystal growth, neighboring grains collide or knock because of the liquid flow or thermal stress during cooling. Such processes cause steps on the surface or dislocations. (2) A large number of oversaturated vacancies are generated during the fast solidification process. Dislocations form when vacancies assemble. (3) Stress concentration usually appears at the interfaces (second-phase particles, twins, grain boundaries, etc.) and fine cracks. Dislocations appear when the stress is high enough to cause slips. 3. Dislocation multiplication As we know, some dislocations exist inside crystals. Their movement, under the external stress, causes the crystal deformation. Accordingly, the number of dislocation should decrease after deformation. However, dislocation density increases four to five orders in magnitude after severe deformation. This implies the dislocation multiplication during deformation processes. There are a few mechanisms describing the dislocation multiplication. The important one is Frank–Read dislocation sources. Figure 3.26 shows the mechanism of Frank–Read dislocation sources. The edge dislocation AB on the slip plane cannot move when its ends are pinned. The dislocation is supposed to slip under the shear stress with direction b. However, the dislocation line bends since AB are pinned (see Fig. 3.26(b)). The slip force Fd = τb, per unit length dislocation line, is always perpendicular to the dislocation line. Thus, each part of the dislocation expands toward its normal direction under Fd, and both ends rotate around A and B (see Fig. 3.26(c)). When the bent dislocation lines are close enough to cancel since they have the same Burgers vector but reverse dislocation directions (or saying that one is a right-hand screw dislocation while the other is a left-hand screw

Fig. 3.26: Frank–Read dislocation sources.

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Chapter 3 Crystal defects

dislocation). Finally, they form a closed dislocation loop and a bent dislocation line. Under the external stress, the dislocation loop continues expanding and the bent dislocation line is straightened due to linear tension and then repeats the previous process to create new dislocation loops. In such a dislocation multiplication way, the deformation of crystal can reach an evident slip amount. In order to start the Frank–Read dislocation source, the external force must overcome the resistance of bending the dislocation line. The external force τ has the relationship with the curvature r of τ = Gb/2 r. That is, the smaller the curvature is, the larger the external force is. As shown in Fig. 3.34, the shear stress is maximum, r = L/2 (L is the distance between A and B), when AB becomes semicircle with the minimum curvature. Thus, the critical shear stress is τc =

Gb L

(3:22)

Frank–Read dislocation sources have been experimentally confirmed in Si, Ge, Al–Cu, Al–Mg, austenitic steels, and KCl crystals. Dislocation multiplication can occur through the double cross-slip mechanism and the climb mechanism. As mentioned before, screw dislocations result in edgestep after double cross-slip. Such edge-steps cannot stay on the original slip plane (such as ð11
1Þ in Fig. 3.27(a)), or move with the original dislocation line, and thus become the pining points for the original dislocation. Such edge steps can be the Frank–Read source for the original dislocation on the slip plane. Figure 3.27 illustrates the dislocation multiplication through the double cross-slip. The two edge steps AC and BD, forming during the double cross-slip of a screw dislocation, become a Frank–Read dislocation source on the new slip plane (111). Sometimes, the dislocation loop on the second (111) plane can cross-slip to the third {111} plane to slide, leading to a rapid dislocation multiplication. Such process is more effective compared to the Frank–Read multiplication.

(111) Slip plane

– (111) Cross-slip plane Edge jog

– (111) Edge jog C

C D

D

A B b (111) Slip plane

b Screw dislocation

b

(a)

A B

(b)

Fig. 3.27: Double cross-slip mechanism for dislocation multiplication.

(c)

3.2 Dislocations

159

3.2.6 Dislocations in real crystals In the above sections, we discussed the dislocation structure and their properties, particularly in simple cubic crystals. However, dislocations can be very complicate in real crystals. They can have complex configurations and special properties. 1. Burgers vector in real crystals The Burgers vector b of simple cubic crystals always equals to the lattice vector. In the real crystal, the Burgers vector of dislocations can be smaller than the lattice vector. We usually define a perfect dislocation when its Burgers vector is equal to a lattice vector, and a partial dislocation when its Burgers vector is less than a lattice vector. Therefore the Burgers vectors of perfect dislocations do not cause the change in the stacking sequence of atoms, while partial dislocations introduce the change in the stacking sequence of atoms in the crystal. In a real crystal, dislocation glide occurs in such a way that the distortion associated with it is minimal. This implies that dislocation glide normally occurs along the most closely packed direction in the most densely packed planes. In other words, the Burgers vectors cannot be any vectors. The Burgers vectors must satisfy the structural and energetical conditions of the crystals. The structural condition means that the Burgers vector must link one atomic equilibrium position to anther equilibrium position. The energy condition means that the unit Burgers vectors should be stable, which indicates that it should have a small magnitude since the energy of the dislocation is proportional to the b2.

Table 3.1: Burgers vector of unit dislocation in typical crystal structure. Structure type

Burgers vector

Simple cubic

a a

2 a

2 a


2

Face-centered cubic Body-centered cubic Closely packed hexagonal

Orientation




|b|

Number

a 1 pffiffiffi 2a 2 1 pffiffiffi 3a 2



a



 

2. Stacking faults The partial dislocations are usually related to the change in the stacking sequence of atoms in real crystals. As mentioned in Chapter 2, close-packed structures can be treated as the stacking of the close-packed planes in a desirable sequence: facecentered cubic (fcc) structures have the sequence of ABCABC . . . . of {111} planes; hexagonal clos e-packed (hcp) structures have the sequence of ABAB . . . .of {0001}

160

Chapter 3 Crystal defects

planes. In order to simplify the process, we use Δ to stand for the sequence of AB, BC, CA, . . ., ∇ to represent the reverse order of BA, AC, CB, . . .. Thus, the stacking sequence of fcc structure is ΔΔΔΔ . . . (Fig. 3.28(a)), while the stacking sequence of hcp structure is Δ∇Δ∇ . . .(see Fig. 3.28(b)).

(a)

(b)

Fig. 3.28: Stacking sequences of close-packed planes: (a) fcc and (b) hcp.

In the real crystals, the normal stacking sequence of close-packed planes may break or arrange with error, which is called stacking faults. For example, fcc structure may ̌ Δ . . .), in change its normal stacking sequence into ABC#BCA . . . (this is Δ Δ ∇Δ which the pointing arrow stands for one atomic plane that has been removed (i.e. the atomic plane). This is called the intrinsic stacking fault, as shown in Fig. 3.29(a). On the other hand, one atomic plane (plane B) may be insert into the normal stacking ̌ Δ . . .). The pointing arrows represent sequence, that is, ABC#B#ABCA . . . (Δ Δ∇̌ ∇Δ the stacking faults caused by inserting the plane B. This is called the extrinsic stacking fault, as shown in Fig. 3.29(b). When compared the above two types, one extrinsic stacking fault equals to two intrinsic stacking faults. It should be mentioned that a thin layer of hcp structure forms at the stacking faults of fcc crystals.

(a)

(b)

Fig. 3.29: Stacking faults in the fcc crystal: (a) the intrinsic stacking fault and (b) the extrinsic stacking fault.

hcp crystals have their stacking faults, similar to those in fcc crystals. The intrinsic stacking faults have the sequence of . . .∇Δ∇∇Δ∇ . . ., or . . . BABACAC . . . . While the extrinsic stacking faults have the sequence of . . .∇Δ∇∇∇Δ∇ . . . or . . . BABACBCB . . . . Body-centered cubic (bcc) crystals have the close-packed planes {110} and {100}, both of which have the stacking sequence of ABABAB . . . . Thus, it is not possible to form stacking faults for the {110} and {100} planes. On the other hand, the {112} planes stack with a fixed periodicity, as shown in Fig. 3.30. Because of the

3.2 Dislocations

161

Fig. 3.30: {112} planes in bcc crystals.

fact that planes are vertical to the vector with the same index in cubic crystals, the stacking sequence of {112} planes can be observed along the direction. Their stacking sequence is ABCDEFAB . . . . When the stacking sequence changes, the ABCDCDEFA . . . stacking fault will form. Forming stacking faults usually does not cause lattice distortions. Stacking faults break the symmetry and periodicity of the lattice, resulting in abnormal reflections. Stacking faults increase the energy of the crystal, referring the stacking fault energy (SFE) γ (J/m2). SFE can be estimated experimentally. Table 3.2 lists the SFE for selected fcc crystals. Apparently, the probability of finding stacking faults is strongly dependent on the SFE. The higher energy is, the lower the chance of finding a stacking fault is. For example, we can find a huge density of stacking faults in austenitic steels because of their low SFE while almost no stacking faults is found in Al due to their high SFE.

Table 3.2: Stacking fault energy and equilibrium distance of some metals. Metal

Ag Au Cu

Stacking fault energy γ (J/m)

Equilibrium distance of partial dislocation (atomic distance)

Metal

Stacking fault energy γ (J/m)

Equilibrium distance of partial dislocation (atomic distance)

. . .

. . .

Al Ni Co

. . .

. . .

3. Partial dislocations If the stacking faults involve a part of (not the entire) atomic an plane, the boundary between the stacking fault and perfect crystal contains a Burgers vector b, whose magnitude is less than that of the Burgers vector in the perfect crystal (see Fig. 3.31). In fcc crystals, there are two important partial dislocations: i.e., the Shockley partial dislocation and the Frank partial dislocation.

162

Chapter 3 Crystal defects

Fig. 3.31: Dislocations at the boundaries of stacking faults: Shockley partial dislocation.

a. Shockley partial dislocations Figure 3.32 shows the structure of the Shockley partial dislocation. The closepacked planes (111) are perpendicular to the paper surface, which stands for the (10–1) plane. The right part of the crystal has the normal stacking sequence of ABCABC . . ., while the left part stacks in the order of ABCBCAB . . . . The boundary between the stacking fault and the perfect crystal is the Shockley partial dislocation. This process equals to the process of shifting the A plane in the left part to the position of the B plane along [1–21] direction to form a dislocation. The Burgers vector of the dislocation is b = a/6[1–21], perpendicular to the dislocation line. Thus, this Shockley partial dislocation is of the edge-type. – [101]

[121]

[111]

B A

A

– [121]

C

C

B

L Slip direction b B

M

C

A C

B

B A

A

Fig. 3.32: Shockley partial dislocation.

According to the relationship of Burgers vectors and dislocation lines, Shockley partial dislocations can be pure edge type, pure screw type or mixed type. Shockley partial dislocations can slip along {111} plane, as a result, the associated stacking fault expands or shrinks. However, pure edge-type Shockley partial dislocations cannot climb due to the fact that its stacking fault cannot move while climbing. b. Frank partial dislocation Fig. 3.33 illustrates the Frank partial dislocation by removing a half plane. The moved half plane is defined as a negative Frank partial dislocation. Otherwise, the one with half extra plane is a positive Frank partial dislocation. Their Burgers vectors

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163

Fig. 3.33: Frank partial dislocations.

are a/3 < 111 >, vertical to the stacking fault plane {111}, but with the reverse directions. Frank dislocation is a pure edge dislocation. Since the Burgers vector of a Frank partial dislocation does not lie in a close-packed plane, glide of the dislocation is impossible. Such a dislocation is called a sessile dislocation. However, it can climb through the movement of point defects, expanding or shrinking the half atomic plane. In summary, the Frank partial dislocations are nonslip dislocations, while Shockley partial dislocations are movable dislocations. Similar to perfect dislocations, the Burgers vector can stand for the corresponding partial dislocation. Please note that the starting point of the Burgers circuit of partial dislocations must locate on the stacking fault. The hcp crystals are similar to the fcc crystals. There are Frank partial dislocations and Shockley partial dislocations in hcp too. Regarding bcc crystals, partial dislocations are possible at the edge of stacking faults on {112} plane. Since the SFE of bcc crystals is very high, so far the stacking faults in bcc crystals have not been observed. 4. Dislocation reactions In real crystals, unstable dislocations can transfer into stable dislocations. Dislocation lines with different Burger vectors can combine into one dislocation line. On the other hand, one dislocation line can decompose (dissociate) into dislocations with different Burgers vectors. Usually, the transfer among different dislocations (decomposition or combination) is defined as dislocation reactions. The occurring of dislocation reactions depends on the following two conditions: (1) Geometrical constraint: Burgers vectors keep constant during the dislocation reactions, or saying, the sum of burgers vectors for all initial dislocations equal to the sum of Burgers vectors for all dissociated dislocations: Σbb = Σba

(3:23)

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Chapter 3 Crystal defects

(2) Energetical constraint: Dislocation reactions must be the process of reducing energy. This is, the total energy (jαba j2 ) of all dislocation after reactions should be less than the value (jαbb j2 Þ that before reactions. Since the energy for each dislocation is proportional to b2, their total energy can be estimated as Σjbi j2 . Σjbb j2 > Σjba j2

(3:24)

We will discuss the dislocation configuration in real crystals in the following sections. 5. Dislocations in fcc crystals a. Thompson tetrahedron All important dislocations and dislocation reactions can be labeled through the tetrahedron and the associated indications proposed by Thompson. As shown in Fig. 3.34(a), A, B, C, and D are the face centers of three neighboring {001} planes {001} of fcc crystal and the origin, respectively. The four apex A, B, C, and D form a tetrahedron with four {111} planes and edges. This is Thompson tetrahedron, in where α, β, γ, and δ stand for the centers of the opposite planes of A, B, C, and D vertexes, respectively (see Fig. 3.34(b)). Figure 3.34(c) is the spreading of all four {111} planes with the base of the ABC plane. Regarding the Thompson tetrahedron: (1) Four planes stand for four possible slip planes: (111), (-111), (1-11), and (11-1) planes. (2) Six edges stand for 12 slip directions, or saying 12 possible perfect dislocations within fcc crystals. (3) The lines linking the apex of each plane and its center stand for 24 slip vectors of 1/6 , corresponding to 24 Shockley partial dislocations.

(a) Fig. 3.34: Thompson tetrahedron.

(b)

(c)

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165

(4) The lines linking four apex and the centers of their opposite triangles stand for eight slip vectors of 1/3[111], related to eight possible Frank partial dislocations in fcc. (5) The lines linking four face centers, αβ, αγ, αδ, βγ, γδ, βδ, stand for six Lomer–Cottrell dislocations. All dislocation reactions, particularly those complicate reactions, can be explained though Thompson tetrahedron. For example, the decomposition of a a/2[-110] dislocation can be described as BC ! Bδ + δC

(3:25)

b. Extended dislocation In fcc crystals, perfect dislocations with lowest energies are those on the {111} plane with Burgers vectors of a/2 . We will discuss its slip on {111} planes. As learned from Chapter 2, fcc crystals have the stacking sequence of ABCABC . . . . If b = a/2[1–10] slips along (111)[1–10] under the shear force on the atomic plane A, atoms on planes B move from position B1 to B2, crossing the “peak” of A atomic planes, which requires a high energy (Fig. 3.35). However, such process can be easier if it is divided into two step: first, starting from position B1, the atoms move across the “troughs” (valley) of A planes to the neighboring position C, with b1 = 1/6[-12-1]; second, atoms move from position C to the position B2, or saying b2 = 1/6 [-211]. Apparently, the first step of atoms B moving into position C, causes the change in staking sequences of {111} planes, which changes from the normal sequence of ABCABC . . . into the sequence of ABCACB . . . .. The second step of moving from position C to position B, involves two partial dislocations will form between the stacking sequence back to normal. Since the first slip causes the stacking fault, two partial dislocations will form between the stacking fault region and normal region. Thus, b1 and b2 are Shockley partial dislocations. This process is the decomposition of one perfect dislocation b into two partial Shockley partial dislocations b1 and b2, or saying b = b1 + b2. Such a dislocation including two partial dislocations and a stacking fault between them is called an extended dislocation.

Fig. 3.35: Decomposition of one perfect dislocation on with b=a/2[-110] (111)fcc.

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Chapter 3 Crystal defects

Since these two partial dislocations are on the same slip, plane and they have the same sign but θ = 60° rotation between their Burgers vector. Since θ < π/2, they must repel each other. Figure 3.36 illustrates a a/2[1–10] extended dislocation in fcc crystals.

Fig. 3.36: A/2[1–10] extended dislocations in fcc crystals.

6. Dislocations in other crystals a. Dislocations in bcc crystals As mentioned earlier, the bcc crystals have dislocations with Burgers vectors a/2, and slip direction of . However, the slip planes of bcc systems are not well defined, which can be {110}, {112} and {123}. Their slip planes change with composition, temperature and deformation rate. Because of many possible slip planes, slip traces usually have a wave shape as a result of cross-slips from one slip plane to another. The fact of cross-slips also indicates the high SFE because extended dislocations seldom form (or the width of stacking fault is very narrow). In fact, the extended dislocations and stacking faults have not being detected experimentally under TEM. The above analysis on the dislocation decomposition is just a deduction. Recent theoretical analysis suggests that dislocation cores of a/2 screw dislocation can be extended among the width of a few Burgers vectors, resulting in the limited movement. The displacement around ½[111] screw dislocation cores is calculated, suggesting the parallel threefold symmetrical axis (in Fig. 3.37). b. Dislocations in hcp crystals The shortest lattice vector is along in hcp crystals, and the second shortest lattice vector is , thus, the unit dislocation is a/2 , c , and 1/3 . The slip plane is related to the a/c ratio. When c/a is larger than 1.633, the slip plane is (0001); when c/a is less than 1.633, the slip plane is {10–10} or {10–11}.

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167

Fig. 3.37: Screw dislocation in α-Fe.

Regarding Co, Zn, and Gd crystals with small SFE, 1/3 < 11–20 > dislocation was observed to decompose according to the following reaction on the basal planes {0001}: 1
1 1 ½1120 ! ½10
10 + ½01
10 3 3 3

(3:26)

where the right part is the Shockley partial dislocations, with the stacking faults inbetween. Such dislocations are called extended dislocations, as shown in Fig. 3.38.

Fig. 3.38: Extended dislocations in the hcp crystal.

c. Dislocations in NaCl crystal NaCl is the ionic crystal, in fact, it is fcc lattice, where each lattice site is corresponding to a pair of Na+ and Cl– ions. The slip system of NaCl crystal is {110} . Regarding the close-packed plane {100}, pairs of cations and anions have to cross the slip plane together, resulting in a strong resistance of slip from the lattice. Such a slip process hardly occurs. Within NaCl crystals, the Burgers vector is b = a/2 , which is the shortest distance of neighboring cations or anions. Figure 3.39 illustrates an edge dislocation with a slip system of (110)[1–10], which has two half extra-atomic planes. Such

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Chapter 3 Crystal defects

Fig. 3.39: Dislocations in NaCl crystal.

a structure is neutral, however, may lead to negative or positive charges at the surface (Fig. 3.40)

Fig. 3.40: Charged dislocation in NaCl crystal.

d. Dislocations in α-Al2O3 crystal α-Al2O3 has a trigonal lattice. Specifically, O2− ions are nearly hcp, Al3+ stays at 2/3 of all octahedral interstitial sites. The ionic arrangements and dislocation reactions are shown in Fig. 3.41. In order to reduce energy, dislocations in α-Al2O3 (with a Burger vector of 1/3 < 11–20 >) can decompose into two partial dislocations with a Burgers vector of 1/3 < 10–10 >, that is 1 1 1 ½1120 ! ½1010 + ½0110 3 3 3

(3:27)

The partial dislocations with the Burgers vector of 1/3 < 10-10 > is the unit vector of O2− ion lattice. Thus, b1-dislocations do not cause any changes in the stacking

3.2 Dislocations

169

Fig. 3.41: Ionic arrangement on (0001) Al2O3.

sequence of O2− ions but they change the stacking sequences of Al3+. The stacking fault forms between two b1 dislocations, and thus the above reaction is also the decomposition of extended dislocation b0. e. Dislocations in diamond-type crystals Diamond-type crystals (e.g., diamond, silicon, and germanium) are covalent crystals with complex cubic structures. Such a crystal structure can be treated as the interlacement of two fcc lattices with ¼ shift along the diagonal direction of the crystal (as shown in Fig. 3.42).

Fig. 3.42: (1–10) projections of the diamond structure.

Within the diamond-type crystals, the densely packed atomic planes are {111}planes, with the stacking sequence of AaBbCcAaBbCc . . ., as shown in Fig. 3.42. Aand a-planes overlap if they are projected to {111} planes. Accordingly, we can simplify the stacking sequence of {111} planes to be ABCABC . . . since no difference exists between projected A and a.

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Chapter 3 Crystal defects

The slip plane is {111} plane in the diamond structure, and the Burgers vector is ½ . Because the covalent bonds have directions, dislocation cores along have a low energy, and dislocations prefer to align parallel to the direction. Thus, most dislocations are pure screw dislocations or 60° dislocations, as shown in Fig. 3.43. Due to the large resistance of the lattice, dislocations can hardly move. When the SFE is low, perfect dislocations on {111} planes can decompose into extended dislocations.

(a)

(b)

Fig. 3.43: Dislocations in the diamond: (a) screw dislocations and (b) 60° dislocations.

f. Dislocations in polymer crystals As mentioned earlier, it is hard for fine, soft, and complex polymer chain to form a perfect crystal. In fact, even within those crystals under ideal growth conditions, crystal defects are inevitable. As shown in Fig. 3.44(a), slender cavities exist within the crystal, usually at the end of polymer chains. Such slender cavities can be treated as the intersection of two dislocations. When the nearby polymer molecules move toward the slender cavities, the positions of cavities change. Due to the continuous movement of nearby polymer molecules, a pair of screw dislocations form, as indicated in Fig. 3.44(b), and may eventually form the mosaic structure.

Fig. 3.44: Dislocations in polymers.

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171

Holland V. F. discovered brink dislocations and dislocation networks in the polyethylene single crystal, most of which are screw dislocations.

3.3 Surface and interface Strictly speaking, interfaces include external surfaces (free surfaces) and internal surfaces. Surfaces stand for the boundaries between the bulk and the gas or liquid. Surfaces are related to the friction, abrasion, oxidation, corrosion, segregation, catalysis, and absorption, as well as optical and microelectronic properties. The internal interfaces include grain boundaries, subgrain boundaries, twin boundaries, stacking faults, and phase boundaries. Interfaces usually are the regions of a few atomic width, which have different atomic arrangements and chemical compositions compared to the bulk. Because interfaces have two-dimensional structures, they are the plane defects. The existence of interfaces is important to the physical, chemical, and mechanical properties of the materials.

3.3.1 Surfaces The atomic arrangements are different on the surface compared to those within the bulk. On the surface, each atom is only partially surrounded by other atoms. Every surface atom has fewer neighboring atoms. In addition, the surface composition can be different because of segregation and surface absorption effect. Atoms in a surface usually have different bondings compared to those within the bulk. Thus, atoms in the surface usually deviate from their normal lattice position, affect the surrounding atoms, lead to local lattice distortion, and thus have higher energy. These atomic layers with higher energy at the surface are so-called surfaces. The increment in energy on the surface is the surface energy γ (J/m2). Surface energy can also be described as the work being done to create a surface of a unit area: γ=

dW dS

(3:28)

where dW is the work being done to create the surface dS. The surface energy can also be defined as the surface tension (N/m). Because the surface is the terminal atomic plane, surface atoms are only bonded to the crystal side. Thus, the surface energy can also be estimated through the number of broken bonds: γ=

total number of broken bonds × ½energy=bond new surface area

(3:29)

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Chapter 3 Crystal defects

Surface energy is related to the atomic density of surface plane. The close-packed surface has smallest surface energy. Thus, the exposure surfaces of the crystal are usually the close-packed atomic planes, which have low surface energies. Figure 3.45 is the pole figure of Au fcc crystal surface energy, in which the radial vector stands for the magnitude of the surface tension vertical to the surface. As shown in Fig. 3.45, the close-packed planes {111} have the smallest surface energy. In case of the crystal surface deviating from the close-packed atomic planes, the surface usually has a few steps for the purpose of reducing its energy. The flat surfaces of these steps are the close-packed planes. The density of these steps is related to the angle between the surfaces and the close-packed plane. Because of the high energetic states of the surfaces and the residual broken bonds, impurities can be more easily attached to the surfaces and thus reduce the surface energy. In addition, the surface steps also benefit the surface diffusion.

Fig. 3.45: Au surfaces in H2 at 1,030 °C.

Fig. 3.46: Low-index plane.

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173

The surface energy is related to the surface curvature, in addition to the density degree of the surface atoms. Under the same condition, the larger the surface curvature is, the larger the surface energy is. The surface properties play an important role in the crystal growth and the formation of new phase during the solid-state phase transformation.

3.3.2 Grain boundaries and subgrain boundaries Most materials are polycrystalline ones. The grain boundary exists between two grains of the same crystal structure and composition but different crystal orientations. Grain boundaries are internal interfaces. In addition, every grain can be consisting of a few subgrains, with very slightly misorientation between each other. The boundaries between such subgrains are subgrain boundaries. The average diameter of grains is around 0.015~0.25 mm, and the average diameter of subgrains is ~0.001 mm. The geometrical properties of grain boundaries and subgrain boundaries can be described through the orientation of grain boundaries and the misorientation of adjacent grains. Starting from simple, two-dimensional case (Fig. 3.47), the grain boundaries can be defined as the misorientation θ and the angle ϕ of grain boundaries with respect to a defined plane of the selected lattice. Regarding the threedimensional grain boundaries, the geometrical description of the boundaries has five orientation angles. The picture can be simplified as following: cut the crystal in Fig. 3.48(a) along xOz plane, and then rotate the right part of the crystal around xaxis, and thus create misorientation of two grains. Similarly, the right part of the crystal can also rotate around y- or z-axes. Thus, three angles are necessary to define the misorientation of two adjacent crystals. In addition to three angles defining the crystal misorientation, another two angles are necessary to describe the boundaries. As shown in Fig. 3.48(b), the grain boundary can rotate around x- or z-axis, resulting in a different plane. If the grain boundary rotates on y-axis, the grain boundary does not change. In conclusion, three-dimensional grain boundaries have five independent variables, that is, five degrees of freedom.

Fig. 3.47: Grain boundaries in a two-dimensional lattice.

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Chapter 3 Crystal defects

(a)

(b)

Fig. 3.48: Grain boundaries in a three-dimensional lattice.

According to the value of misorientation angle θ of the adjacent crystals, there are two types of grain boundaries: (1) small (low)-angle grain boundaries – the misorientation between adjacent crystals is less than 10°. The misorientations of subgrain boundaries are usually smaller than 2°, thus also belonging to small-angle grain boundaries. (2) Large-angle grain boundaries – the misorientation between adjacent crystals is larger than 10°. Most grain boundaries are large-angle grain boundaries. 1. Small-angle grain boundaries According to the types of misorientations between adjacent crystals, we can classify small-angle grain boundaries into tilt grain boundaries, twist boundaries, mixed grain boundaries, and so on. We can describe their structures with different models. a. Symmetrical tilt grain boundaries The symmetrical grain boundaries are formed when the two adjacent crystals tilt symmetrically (Fig. 3.49). When the misorientation angle θ is small, the boundary is consisting of a series of parallel edge dislocations (Fig. 3.50). The dislocation interspacing D is related to the burgers vector b: D=

b 2sin θ2

(3:30)

When θ is very small, D ≈ ðb=θÞ The symmetrical tilt grain boundary provides an indirect evidence for the existence of dislocations. It can be found from eq. (3.30) that if reasonable values of b (0.3 nm) and sufficiently small values of θ (0.05°) are used, D will equal to about 0.34 μm, which is larger than the resolution of light optical microscopy (which is about 0.2 μm). Because the lattice distortion leads to the emergence of “pits” in the cross section of specimen, these pits will occur at the small-angle symmetrical tilt boundary at distances which can be resolved by light optical microscopy. In fact, on this basis, Vogel et al. performed a famous experiment in 1953, and they were able to demonstrate that the small-angle symmetrical tilt boundary as a wall of edge dislocations, as shown in Fig. 3.50.

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175

(a)

(b) Fig. 3.49: Formation of symmetrical tilt grain boundary: (a) before formation and (b) after formation.

Fig. 3.50: Symmetrical tilt grain boundaries.

b. Asymmetrical tilt grain boundaries as shown in Fig. 3.51, the symmetrical grain boundaries rotate an angle of ϕ around x-axis, resulting in the formation of asymmetrical tilt grain boundary. These tilt grain boundaries have the same misorientation θ, however the boundaries are not symmetrical regarding the adjacent crystals. The degree of freedom is 2, θ, and ϕ.

176

Chapter 3 Crystal defects

Fig. 3.51: An asymmetrical tilt grain boundary.

Such structure is consisting of two groups of dislocations, which have burgers vectors, b(?) and b(‘), perpendicular to each other. The interspacing of each dislocation group, D(?) and D(‘), can be solved according to their geometrical relationship. D? =

b? b‘ , D‘ = θsinϕ θcosϕ

(3:31)

c. Twist grain boundaries Twist grain boundary is another type of small-angle grain boundaries. It can be considered by relatively rotating two adjacent crystals by angle of θ around an axis perpendicular to one of their common planes, as shown in Fig. 3.52. Its degree of freedom is 1. As shown in Fig. 3.53, these grain boundaries have intercepting screw dislocations. The pure twist grain boundaries and tilt grain boundaries are the simple cases of small-angle grain boundaries. The difference is that the rotation axis is inside grain boundaries for tilt grain boundaries while the rotation axis is perpendicular

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177

Fig. 3.52: A twist grain boundary.

Fig. 3.53: Dislocations in a twist grain boundary.

to grain boundaries for twist grain boundaries. In general, small-angle grain boundaries can be treated as the boundaries of two crystals, which are rotating at some angles around a fixed axis. Such rotation axis is not parallel or perpendicular to grain boundaries. Such general small-angle grain boundaries are consisting of a series of edge, screw, and/or mixed dislocations. 2. Large-angle grain boundaries Large (high)-angle grain boundaries are common boundary structures in polycrystals. Large-angle grain boundaries have complicate structures, in which the atomic arrangement is irregular and cannot be explained through dislocation models. Our understanding on the structure of large-angle grain boundaries is limited within a few proposals. One of the proposal treats large-angle grain boundaries as irregular steps, instead of smooth boundaries among the adjacent grains, as shown in Fig. 3.54.

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Chapter 3 Crystal defects

Fig. 3.54: A large-angle grain boundary.

At these boundaries, there are atoms D belonging to both crystals, atom A belonging to neither of the crystals, compression region B, and expansion region C. This complicate structure is formed under the complicate interaction between atoms within both crystals. In summary, the atoms at the large-angle grain boundaries are messy, probably with some neat regions. The large-angle grain boundaries are consisting of good matching regions and bad matching regions. The bad matching regions occupy a larger area when the misorientation θ increases. The width of large-angle grain boundaries is usually less than three atomic interspacing. According to the results from ion microscopy, the model of coincidence site lattices (CSL) was proposed. As shown in Fig. 3.55, some atoms are overlapped at the grain boundaries when the adjacent cubic crystals are misorientated 37° (or saying crystal 2 was rotated 37° with respective to crystal 1). These coincidence sites have

Fig. 3.55: One-fifth coincidence sites at grain boundaries with a misorientation of 37°.

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179

their own periodicity, which is usually longer than the one of original lattices. In the particular example in Fig. 3.67, one of every five atoms are the coincidence sites, or saying the density of coincidence sites is 1/5. The density of coincidence sites Σ in a CSL is defined as Σ=

number of coincidence sites total number of lattice sites

Apparently, different densities of coincidence sites can be found within systems with different crystal structures and different rotation angles and axes. Table 3.3 lists some important coincidence sites of cubic metals. Table 3.3: The important coincidence position lattice in cubic system metals. Crystal structure

Rotation axis Rotation angle (°) Coincidence position density

Body-centered cubic

() () () () () ()

. . . . . .

/ / / / / /

Face-centered cubic

() () () ()

. . . .

/ / / /

According to this model, there are a certain percentage of coincidence sites amongst the large-angle grain boundaries. Apparently, the more the coincidence sites are, the more atoms are shared by both crystals, the smaller the lattice distortion is, the smaller the grain boundary energy is. Therefore, a (coherent) twin boundary (Σ = 3) has the lowest grain boundary energy. However, there are only limited structures of coincidence sites at grain boundaries with different misorientations. The model of coincidence lattice sites cannot be fully applicable for any large-angle grain boundaries. In summary, the discussion about the structure of large-angle grain boundaries is still going on. 3. Grain boundary energy Because of the irregular arrangement of atoms at grain boundaries, grain boundaries usually have higher energies. Their energy can be defined as the energy change of

180

Chapter 3 Crystal defects

unit area of grain boundaries. This energy also equals to the energy difference per unit area between the regions with or without grain boundaries. The small-angle grain boundary energy is mainly contributed by the dislocation energy (forming dislocations and arranging such misfit dislocations). The dislocation density is determined by the crystallographic orientation difference of adjacent grains. Thus, small-angle grain boundary energy is related to the orientation difference. In 1952, Read and Shockley derived the following equation for the energy per unit area, γ, of small-angle tilt and twist grain boundaries γ = γ0 θðA − lnθÞ

(3:32)

where γ0 = Gb=ð4πð1 − vÞÞ is a constant, defined by the shear modulus G, the Poisson ratio v, and burgers vector b. A is an integrated constant, determined by the atomic misarranged energy at the dislocation core. As the dislocation density in the small-angle grain boundary is proportional to the misorientation angle θ based on eq. (3.30), the first term in the eq. (3.32), γ0θA, simply expresses that the total energy in the grain boundary increases upon the dislocation density in the boundary increases. The second term in this equation, –γ0θ ln θ, accounts for the effect of the interaction between the stress fields of the dislocations. Upon arrangement of the dislocations in these small-angle boundaries, the long-range nature of the dislocation strain fields of the individual dislocations becomes annihilated. This is the driving force for a polygonization effect during the recover annealing. As a result, although the total grain-boundary energy increases upon increasing θ, the energy per dislocation in the grain boundary decreases upon increasing θ. Accordingly, the energy of small-angle grain boundary energy increases with the increment of orientation. Note, such equation only validates for small-angle grain boundaries, not for large-angle grain boundaries. In fact, most grain boundaries are large-angle grain boundaries, with the misorientations of 30~40°. Experimental measurement suggests that most large-angle boundary energies are around 0.25~1.0 J/m2, regardless their misorientations, as shown in Fig. 3.56. Grain boundary energy can be expressed in the form of the interfacial tension, which can be measured through the contact angles. Figure 3.57 indicates the triple junction of three grains, with grain boundaries between any two grains. At equilibrium, the interfacial tensions at point O, γ1-2, γ2-3, γ3-1, reach the force balance, for example, the sum of all vectors is zero, thus γ1 − 2 + γ2 − 3 cosφ2 + γ3 − 1 cosφ1 = 0

Interface energy/J ‧ m–2

3.3 Surface and interface

1.6

γs Copper surface in H2

0.5

γg

0.4

181

Large-angle grain boundaries

1 7 coherent grain boundaries 1 coherent grain 5 around [111] boundaries around [100]

0.3

Small-angle grain boundaries γ = γ0θ(A–Inθ)

0.2 0.1

Twin plane γr

0

10

20 30 40 Misorientations/(° )

70

Fig. 3.56: Interface energy of different types of grain boundaries within copper.

Fig. 3.57: Triple junction of three grains.

Or γ γ γ1 − 2 = 2−3 = 3−1 sinφ3 sinφ1 sinφ2

(3:33)

Thus, choosing a certain grain boundary as a standard one, through measuring Φ we can calculate the boundary energy with respect to others. Under the equilibrium condition, the triple junctions are likely to maintain the most stable angle of 120°. This also suggests the equal boundary energy of the involved grains. 4. Characteristics of grain boundaries (1) Grain boundaries have large lattice distortions and the corresponding boundary energy. Thus, the grain growth and the straightening of grain boundaries can reduce the boundary area, leading to the decrease of the total grain boundary

182

(2)

(3)

(4)

(5)

(6)

Chapter 3 Crystal defects

energy. Such processes are spontaneous process. In addition, the grain growth and the straightening of grain boundaries must involve the diffusion process. Thus, the higher annealing temperature and longer annealing time will facilitate the above two processes. Because of the irregular arrangement of atoms at grain boundaries, they can impede the movement of dislocations at room temperature, leading to the increase in the resistance of plastic deformation. Macroscopically, grain boundaries have higher strength and hardness. The finer the grains are, the stronger the material is. This phenomenon is the so-called grain refining. At a high temperature, grain boundaries can cause the slip of adjacent grains, thus weaken the material. Atoms at grain boundaries have higher energy since they deviate from their equilibrium positions. There are a lot of defects at boundaries, such as vacancies, impurities, dislocations, and so on. Thus, the diffusion rate is much larger at grain boundaries compared to the one within the bulk. During the solid-state transformation process, new phases usually nucleate at grain boundaries because of the higher energies grain boundaries and higher mobility of boundary atoms. The finer the grains are, the more the grain boundaries are, the higher the nucleation rate of new phases is. Because of the segregation and internal absorption, grain boundaries usually have a higher concentration of impurities, and therefore have lower melting points. Thus, during the heating process, high temperatures will lead to the melting and oxidation of the grain boundaries, and such a phenomenon is called “overheating.” The corrosion rate is higher at grain boundaries because of their higher energy from unstable atoms and the enrichment of impurities. This is also the base for the optical observation, since grain boundaries are etched faster than grains.

3.3.3 Twin boundaries Twins are defined as the mirror-symmetric relationship of two crystals (or two parts of one crystal). Twin planes are the mirror plane, which are the shared plane of two adjacent crystals. Twins can have coherent twin boundaries and incoherent twin boundaries, as shown in Fig. 3.58. Coherent twin boundaries are the twin planes (Fig. 3.58(a)). The boundary atoms belong to both the adjacent crystals. Atoms are perfectly matched at the twin boundaries. As the twin boundary all lattice sites at the boundary are sites of the CSL. No lattice distortion is involved. Thus, coherent twin boundaries have a low boundary energy (roughly 1/10 of the normal boundary energy). They are stable, and are usually straight lines. Twins are often observed in annealed fcc metals with a low SFE, and such twins then are called annealing twins.

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183

C A Twin plane

B C A

θ

C B A C

Noncoherent twin plane (a)

(b)

Fig. 3.58: Twins in fcc: (a) the coherent twin boundary, (b) the non-coherent twin boundary.

The incoherent twin boundaries involve a rotation of the twin boundaries around the twin planes (Fig. 3.58(b)). Thus, some atoms are shared by the adjacent crystals. There is a large mismatch at the boundaries, leading to a high twin boundary energy (roughly ½ normal boundary energy). The formation of twins is related to the stacking faults. For example, fcc crystals have the sequence order of ABCABC . . . for {111}-planes. If the stacking sequence order reverses at some planes, becoming ABCACBACBA . . . , CAC is the stacking fault, following by the reverse stacking sequence. Then these crystals have a mirror relationship, that is, the twin relationship (Fig. 3.58(a)). According to their formation conditions, twins can be classified as deformation twins, growth twins, and annealing twins. Because of the stacking fault involved in twins, the materials with high stacking faults energy usually have fewer twins. 1. Coherent phase boundary The coherency means that all atoms at the boundaries belong to the both adjacent lattice, or saying, that the lattices of adjacent phases are connected. As shown in Fig. 3.59(a), the fully coherent phase boundaries have very low energy. However, the only ideally coherent boundaries found are twin boundaries. With regard to phase boundaries, the adjacent crystals are two different phases, for example, with two different crystal structures and different lattice parameters. Therefore, the lattice distortion exists when forming a coherent phase boundary. In order to match each other at the interface, the crystal with a smaller interplanar spacing suffers extension, the one with a larger inter-planar spacing suffers compressions (Fig. 3.59(b)). Thus, such constrained coherent phase boundaries have higher energies than those ideally coherent boundaries (e.g., twin boundaries).

184

Chapter 3 Crystal defects

(a)

(b)

(c)

(d)

Fig. 3.59: Phase boundaries: (a) perfect coherent phase boundary, (b) coherent phase boundary with elastic distortion, (c) semicoherent phase boundary, and (d) incoherent phase boundary.

2. Semicoherent phase boundary When the adjacent crystals have a comparably large difference in lattice spacing, the one-to-one matching at the phase boundary is destroyed, and therefore, misfit dislocations appear at the phase boundary (Fig. 3.59(c)) in order to decrease the lattice distortion. These phase boundaries maintain partially matched of atoms, and can be named as semicoherent phase boundaries or partial coherent phase boundaries. The misfit dislocations have an identified interspacing, determined by the mismatch of two lattices. The mismatch δ is defined as δ=

aα − aβ aα

(3:34)

where aα and aβ are the lattice parameters of the adjacent phases, and aα > aβ. Therefore, the interspacing of misfit dislocations is D=

aβ δ

(3:35)

When δ is very small, D is very large, indicating that α and β phases are nearly coherent at the interface, forming coherent phase boundaries; When δ is very large, D is very small, indicating that α and β phases are nearly incoherent at the interface, forming noncoherent phase boundaries.

3.3 Surface and interface

185

3. Incoherent phase boundary When the two phases besides a phase boundary have very different atomic arrangements, that is, with a large δ, they form an incoherent phase boundary (Fig. 3.59(d)). Such phase boundaries are similar to large-angle grain boundaries, which usually include thin transition areas of irregularly arranged atoms. The energy of a phase boundary can be measured through similar methods for grain boundaries. Theoretically, the phase boundary energy includes the elastic distortion energy (elastic strain energy) and the chemical energy. Distortion energy is determined by the lattice mismatch δ, while the chemical energy is estimated through the changes in chemical bonding at the interfaces. Different phase boundaries have different structures and different contributions of these two types of energies. Coherent phase boundaries keep a good match of atoms at the interface without the change of the chemical bonding, i.e., the distortion energy is the dominent energy for coherent phase boundaries. On the other hand, incoherent phase boundaries have significant changes in both the number of bonds and the bonding type; thus, the chemical energy is the main contributor. Noncoherent phase boundaries have a significant higher energy. In general, the phase boundary energy decreases for different boundaries in the order of incoherent, semicoherent, and coherent phase boundaries.

Chapter 4 Deformation and recrystallization Structural components often form various shapes by deformation. Such deformation is generally intense with exceeding elastic limit. In metals, stress–strain curve commonly exhibits a linear relation in elastic deformation stage. As strain is increased, many materials eventually deviate from this linear proportionality, and the point of the departure being termed the proportional limit. This nonlinearity is usually associated with stress-induced plastic deformation. Here the material is undergoing a rearrangement of microscopic structure, in which atoms are being moved to new equilibrium positions. Figure 4.1 shows the stress–strain curve of low-carbon steel (soft steel) under tension. σe , σs , and σb marked in the curve, as important parameters of mechanical properties, are defined as elastic limit, yield strength, and tensile strength, respectively. Plastic deformation not only changes the shape and dimension of a material, but also the microstructures and macroproperties of a material. This plasticity in crystalline materials can arise from dislocation motion. Accompanying with the rise of free enthalpy due to nucleation and multiplication, motion, reaction, and accumulation of dislocations, the material could become thermodynamically unstable. Thus, recovery and recrystallization will arise locally associated with local heating or in the whole specimen when the deformed material is heated. Therefore, it is of important theoretical and practical significance to understand the deformation behaviors, the recovery/ recrystallization process, and the related micromechanisms as well as their influencing factors.

4.1 Elasticity and viscoelasticity As shown in Fig. 4.1, a material generally undergoes three stages during deformation: elastic deformation, plastic deformation, and fracture. Therefore, first, the elastic behavior should be studied. 4.1.1 Nature of elastic deformation Elastic deformation is defined as the amount of deformation that can completely recover after the external force is removed. Its nature can be understood from the point of view of the interatomic bonding force.

https://doi.org/10.1515/9783110495348-004

4.1 Elasticity and viscoelasticity

b

𝜎b s

k

e

Stress

𝜎s 𝜎e

187

εb

O

Strain

εk

Fig. 4.1: Stress–strain curve of low-carbon steel (soft steel) under tension.

When there is no external force, the interatomic bonding energy or bonding force between two neighboring atoms in a crystal can be theoretically described as a function of their interatomic distance (as shown in Fig. 4.2). Slope Attractive force

s0

U

F

0 Repulsion force dU =0 dr O

r0

r (a)

O

r0

r (b)

Fig. 4.2: Bonding energy and bonding force as a function of interatomic distance: (a) relationship between bonding energy and interatomic distance, and (b) relationship between bonding force and interatomic distance.

When the atoms are in their equilibrium positions, the interatomic distance is r0 and the corresponding potential energy U is the lowest, or the interatomic interaction force is zero, that is to say, the atoms are in the most stable state. When the crystal is subjected to a external force, atoms will deviate slightly from their equilibrium positions. If the interatomic distance is larger than the equilibrium distance r0 , an attractive force will be generated; and if the interatomic distance is smaller than the equilibrium distance r0 , a repulsive force will be generated. As soon as the external force is released, the atoms will recover back to their original equilibrium positions, so the deformation will recover completely, which is termed as elastic deformation. On an atomic scale, macroscopic elastic strain is manifested as small

188

Chapter 4 Deformation and recrystallization

changes in the interatomic spacing and the stretching of interatomic bonds. As a result, the magnitude of the modulus of elasticity is a measure of the resistance to separation of adjacent atoms/ions/molecules, namely, the interatomic bonding forces. As for most typical metals, Young’s modulus magnitude ranges between 45 GPa for magnesium and 407 GPa for tungsten. 4.1.2 Elastic deformation characteristics and elastic modulus The main characteristics of elastic deformation are: (1) The ideal elastic deformation is reversible, namely, a deformation will occur when a material is loaded; and the deformation will disappear and its original shape recovers when the load is removed. (2) Metals, ceramics, and some polymers usually keep a linear relation between stress and strain in their elastic range during loading or unloading. Such an elastic behavior is described by Hooke’s law: σ = Eε under uniaxial tensile or compressiveðnormalÞ stress τ = Gγ under shear stress

(4:1)

where σ, τ are normal stress and shear stress, respectively. ε, γ are normal strain and shear strain, respectively. E, G are elastic modulus (or Young’s modulus) and shear modulus, respectively. The relationship between elastic modulus and shear modulus is G=

E 2ð1 + νÞ

(4:2)

for elastic isotropic material where ν is the Poisson’s ratio. Poisson’s ratio is defined as the ratio of the lateral and axial strains. For metals, the Poisson’s ratios are between 0.25 and 0.35, while the values for polymers are relatively larger. As mentioned in Chapter 3, one of the characteristics of the crystals is being elastically anisotropic; thus, the elastic modulus varies with crystallographic direction. The stress–strain relationship of an elastic material can be expressed as generalized Hooke’s law in a matrix form: 9 8 98 9 8 σx > > C11 C11 C11 C11 C11 C11 >> εx > > > > > > > > > > >> > > > > > > > > > > > εy > σy > C21 C22 C23 C24 C25 C26 > > > > > > > > > > > > > > > > > > > > < >< > > > > = = < εz = σz C31 C32 C33 C34 C35 C36 = (4:3) > > > γxy > τxy > C41 C42 C43 C44 C45 C46 > > > > > > > > > > > > > > > >> > > > > > > > > > > >> > > > > > > > > > τxz > > C51 C52 C53 C54 C55 C56 > > γxz > > > > > > > ; : ;: ; : γ τyz C61 C62 C63 C64 C65 C66 yz

4.1 Elasticity and viscoelasticity

where there are 36 elastic constants Cij , or called stiffness coefficients. The formula can also be rewritten as 9 8 98 9 8 εx > > S11 S11 S11 S11 S11 S11 >> σx > > > > > > > > > > > > > > >> > > > > > > > εy > σy > S21 S22 S23 S24 S25 S26 > > > > > > > > > > > > > > > >> > > > > > > > = < σz = S32 S33 S34 S35 S36 31 = > > > γxy > τxy > S41 S42 S43 S44 S45 S46 > > > > > > > > > > > > > > > > > > > > > > > > > > >> > >γ > > S51 S52 S53 S54 S55 S56 > > τxz > > > xz > > > > > > > > > > ; > ;> ; : : : γyz τyz S61 S62 S63 S64 S65 S66

189

(4:4)

where there are 36 flexibility coefficients Sij . In most cases, the stiffness matrix and flexibility matrix are mutually inverse matrices: C = S − 1, S = C − 1 Since the matrices are symmetric, Cij = Cji , Sij = Sji , the numbers of the independent stiffness coefficients and flexibility coefficients are both reduced to 21. Since there are various symmetries in crystals, the number of independent elastic coefficients will be further reduced with the increasing symmetry. Only three independent elastic coefficients exist for cubic crystals with the highest symmetry, while five for hexagonal crystals and nine for orthorhombic crystals. There are three main ways in which a load may be applied，that is, tension, compression, and shear. Therefore, in addition to E and G, the compressive modulus or bulk elastic modulus (K) is introduced. K has the following relationship with E and ν: K=

E 3ð1 − 2νÞ

(4:5)

As mentioned earlier, the magnitude of elastic modulus is a measure of the interatomic bonding forces; as a result, covalent crystals exhibit the highest elastic modulus values, such as diamond; metallic or ionic crystals exhibit relatively low elastic moduli, such as steels and NaCl; and molecular solids have the lowest moduli, such as plastics and rubber. The elastic modulus reflects the bonding force between atoms; therefore, it is not sensitive to the microstructures. The addition of small amounts of alloying elements, or the change of processing and treatment methods will have no obvious influences on the elastic modulus. For example, the ultimate tensile strength of highstrength steels can be ten times higher than that of low-carbon steels, but their elastic moduli are almost the same. However, the elastic modulus of a crystalline material is anisotropic. In single crystals, the elastic moduli along different crystallographic directions are very different: the value along the close packing direction of atoms is the

190

Chapter 4 Deformation and recrystallization

highest, and the values along other directions are relatively low. While in polycrystals, the elastic modulus is isotropic as the polycrystalline aggregate does not have a strong texture. The elastic moduli and material constants of some commonly used materials are listed in Tables 4.1 and 4.2. Table 4.1: Elastic moduli and material constants of various materials. Materials Cast iron α iron, steel Copper Aluminum Nickel Brass/ Titanium Tungsten Plumbum Diamond Ceramics Sintered AlO Quartz glass Flint glass Organic glass Hard rubber Rubber Nylon Silk Polystyrene Polyethylene

E (× MPa)

G (× MPa)

Poisson’s ratio

 – – – –    – ,       . . . . .

  – –   –  .–. –  –   . . . –

. .–. .–. .–. .–. – – . .–. . . . . . . . . . . .

In engineering, the elastic modulus of a material is a measure of its stiffness. If the external force is the same, the higher the value of E of the material is, the higher the stiffness is, while the smaller the elastic deformation will be. For instance, the value of E for steels is three times bigger than that of aluminum, thus the elastic deformations of the steels are only 1/3 to that of aluminum under the same stress. (3) The maximum amounts of elastic deformations are varied with the materials. Most of metals can be elastically deformed and the stress–strain curve exhibits linear relationship obeying Hooke’s law, and their elastic deformations are usually not more than 0.5%. While for polymers, like rubber, the elastic deformations can be as high as 1,000%; moreover, the elastic deformation is nonlinear.

191

4.1 Elasticity and viscoelasticity

Table 4.2: Elastic and shear moduli of some metallic single crystals and polycrystalline (room temperature). Metal

E (GPa) Single crystal

Aluminum (Al) Copper (Cu) Gold (Au) Silver (Ag) Plumbum (Pb) Iron (Fe) Tungsten (W) Magnesium (Mg) Zinc (Zn) Titanium (Ti) Beryllium (Be) Nickel (Ni)

Max

Min

. . . . . . . . . – – –

. . . . . . . . . – – –

G (GPa) Polycrystal

. . . . . . . . . . . .

Single crystal Max

Min

. . . . . . . . . – – –

. . . . . . . . . – – –

Polycrystal

. . . . . . . . . . – .

4.1.3 Imperfect elasticity In the description of elastic behavior mentioned earlier, an applied load will induce a strain of the body instantaneously; while upon unload the strain will be removed instantaneously. In such a case, the effect of loading time on strain cannot be considered, and the body is called an ideal elastic one. In fact, most materials used in engineering are crystalline or even amorphous, or both, and there are various types of defects in materials. Therefore, in tensile curve the linear relationship of stress–strain in elastic stage for loading and unloading cycles is destroyed due to its dependence of time on strain. The phenomenon is called imperfect elasticity. Imperfect elasticity phenomena include Bauschinger effect, anelasticity, elastic hysteresis, and cyclic toughness. 1. Bauschinger effect A material is predeformed by a small amount of plastic deformation (less than 4%). If it is reloaded in the same direction, the elastic limit σe will be increased; while if it is reloaded in the reverse direction, the elastic limit will be decreased. This phenomenon was named as Bauschinger effect, and it is a general phenomenon for most polycrystalline metals. Bauschinger effect is very important for the fatigue of the components bearing cyclic strains. During fatigue test, the plastic deformation of the component will be created in each cycle, in which the reverse loading will make σe decrease, showing a phenomenon of cyclic softening.

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Chapter 4 Deformation and recrystallization

2. Anelasticity Ideal elastic deformation is time-independent, that is, an applied stress produces an instantaneous elastic strain, and upon release of the load the strain would totally and immediately return to zero. In most engineering materials, however, there also exists a time-dependent elastic strain component. This time-dependent elastic behavior is called anelasticity, which is related to time-dependent microscopic and atomistic processes. Figure 4.3 illustrates anelasticity. After loading, elastic strain reaches Oa from zero (point O) instantaneously for a given stress, and at the constant stress strain gradually increases to point b with time, then upon unloading strain does not immediately return to zero, rather than to point c, moreover, bc = Oa. With time extension, strain gradually returns from point c to zero (point d); moreover, c′d = a′b. In this figure, both elastic strains a′b and c′d are called anelastic strains due to timedependent elastic behaviors.

Fig. 4.3: Strain relaxation under a constant stress.

The anelasticity is related to composition and microstructure of materials, also to the test conditions. More inhomogeneous microstructures, higher temperature and higher shear stress would lead to a more obvious anelasticity. For metals, the anelastic component is normally small and is often neglected. However, for some polymeric materials, its magnitude is significant. 3. Elastic hysteresis Since the strain cannot response to the stress instantaneously, the unloading curve and the loading curve will not overlap and a hysteresis loop of strain–stress curve will be formed. Such a curve is called elastic hysteresis, as shown in Fig. 4.4. Different loading modes will produce different hysteresis loops. Elastic hysteresis indicates that the deformation work consumed in a material during loading is larger than that in the material during unloading. This phenomenon is due to internal friction whose dissipation energy can be measured by the area of elastic hysteresis loop.

4.1 Elasticity and viscoelasticity

193

Fig. 4.4: Elastic hysteresis (loop) and cyclic toughness: (a) elastic hysteresis (loop) under unidirectional loading; (b) elastic hysteresis under alternative loading (low strain rate); (c) elastic hysteresis under alternative loading (high strain rate); (d) plastic hysteresis under alternative loading (loop).

4.1.4 Viscoelasticity In addition to elastic deformation and plastic deformation, viscous flow is also recognized as one type of deformation. Viscous flow is referred to be that amorphous solid and liquid would flow under a small external force, and the deformation cannot completely recover after removing external force. Pure viscous flow obeys Newton’s law of viscous flow: σ=η

dε dt

(4:6)

where σ is stress, dε=dt is strain rate, η is called viscosity coefficient, which reflects fluid friction, namely fluidity, and its unit is Pa · s. Some amorphous materials, sometimes even polycrystalline materials, can show both elasticity and viscosity under a small stress. This phenomenon is referred to as viscoelasticity. Viscoelastic deformation not only relates to time, but also has the recoverability of elastic deformation. That is to say, it has the characteristics of both elastic and viscous deformation. Moreover, the external conditions (such as temperature) may have a significant impact on the viscoelasticity behavior of a material (especially for polymers). Viscoelasticity is one of the important mechanical properties of polymer materials, so polymer materials are often referred to as viscoelastic materials. This is mainly related to their molecular chain structures. When an external force is applied on a polymer material, the intramolecular bond angle and the bond length will change, also the structure of the segment will spread along the direction of the external force. On the other hand, there will be a relatively slip between molecular chains that results in viscous deformation. When the external force is small enough, the former deformation is reversible and the latter is irreversible. In general, the feature

194

Chapter 4 Deformation and recrystallization

of viscoelastic behavior is that the imposition of a stress results in an instantaneous elastic strain, which is followed by a viscous, time-dependent strain in a form of anelasticity, then upon unloading elastic strain does not immediately return to zero. To study the representation of law of viscoelastic deformation, elastic deformation can be represented by a spring and a viscous dashpot. Different combinations of these two modules are used to describe different models of viscoelastic deformation. Figure 4.5 shows two most typical models: the Maxwell model and the Voigt model. The former is in series form, while the latter is the parallel form. Here, the deformation of the spring module is time independent, and the relationship between stress and strain obeys Hooke’s law. When the stress is released, the strain returns to zero. The dashpot in the model is composed of a cylinder with viscous fluid and a piston. The piston movement results from the viscous flow, thus the deformation is in accordance with Newton’s law of viscous flow.

(a)

(b)

Fig. 4.5: Viscoelastic deformation models: (a) the Maxwell model; (b) the Voigt model.

Maxwell model is especially useful for explaining the stress relaxation. The relationship between stress and time can be calculated as follows:     Et t = σ0 exp − (4:7) σðtÞ = σ0 exp − η τ′ where τ′ = η=E is called relaxation parameter. Voigt model can be used to describe the creep recovery, anelasticity and elastic memory processes. The time-dependent stress can be calculated by σðtÞ = Eε + η

dε dt

(4:8)

Viscoelastic deformation is characterized by the strain lag model. When a cyclic stress is applied, the stress–strain curve becomes a loop. The loop area is the energy loss, called the internal friction, which is caused by one single stress cycle. It is similar to the stress–strain loop in elastic hysteresis, as shown in Fig. 4.4.

4.2 Plastic deformation of crystals

195

4.2 Plastic deformation of crystals When the stress exceeds the elastic limit of a material, plastic deformation takes place, leading to an irreversible permanent deformation. Engineering materials are mostly polycrystalline. The deformation of polycrystals is related to the deformation behavior of each grain and their microstructure. This chapter discusses the plastic deformation of single crystals and the plastic deformation of polycrystals.

4.2.1 Plastic deformation of single crystals Plastic deformation of single crystals is realized mainly by slip and partially by twinning and kinking at low temperature and conventional strain rate. From an atomic perspective, plastic deformation corresponds to the breaking of bonds with original neighboring atoms and then reforming bonds with new neighbors; atoms are being moved to new equilibrium positions and upon removal of the stress they do not return to their original positions. Under high-temperature conditions, other deformation modes are activated, such as diffusional creep, grain boundary slip, or grain boundary rotation. 1. Slip system a. Slip line and slip band The slip can be defined as the parallel movement of two adjacent crystal regions relative to each other across a plane. The plane is called slip plane. The relative displacement is called slip vector, which is a crystal translation vector. In order to observe slip phenomena, a well-polished single-crystal metal bar is stretched to produce plastic deformation. Many parallel lines can be seen on the surface of the metal bar, which is called slip traces or slip lines, sometimes forming slip bands, as shown in Fig. 4.6. These parallel sets of lines are actually surface relief due to slips. The distance between these slip lines is only about 100 interatomic distances, and the total slip distance along each slip line can reach about 1,000 interatomic distances, as shown in Fig. 4.7. The distribution of the slip bands also shows the inhomogeneity of plastic deformation, that is, the region between slip bands or the slip lines does not plastically deform. Dislocations would only move on certain plane and direction during plastic deformation. The crystal plane and direction are called “slip plane” and “slip direction,” respectively. The slip direction is lying in the slip plane. A system consisted of a slip plane, and in its plane a slip direction is called one-slip system. The number of slip systems is related to crystal structures, and the number of easy slip system varies with material. Table 4.3 lists the slip systems in several common metals.

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Chapter 4 Deformation and recrystallization

Fig. 4.6: Surface photo of single-crystal metal after tension.

~1,000 Interatomic distance Slip band ~100 Interatomic distance Slip line ~10,000 Interatomic distance

Fig. 4.7: Schematic diagram of the formation of the slip bands.

b. Slip system The slip planes and slip directions are usually the most densely packed planes and the most closely packed directions of atoms. This is because the most densely packed planes have the largest interplanar spacing and the lowest lattice resistance; therefore, slips tend to occur along these planes. Along the most closely packed directions, atoms can slide in a shortest distance, namely, minimum Burgers vector (b) of a dislocation. For example, the slip planes in fcc crystals are {111} plane family, and the slip directions are direction family. In a bcc crystal, its most dense plane is not like the most densely packed plane in an fcc crystal. Slip occurs in the closely packed directions, but glide can take place on several closely packed planes {110}, {112} or/and {123} that contain the closely packed direction. The dominant slip plane is related to both material and temperature. For hcp crystal structure, the slip direction is usually < 11
20 > and the slip planes are usually basal plane {0001}. When c/a < 1120

Zn

{}


> < 1120


g f1122


3
> < 112

Be, Re, Zr

f10
10g


> < 1120

Mg

{0001}
g f1122 f10
11g


> < 1120 < 10
10 >
> < 1120

Ti, Zr, Hf

f10
10g f10
11g {0001}


> < 1120
> < 1120
> < 1120

the {0001} family is no longer unique densely packed plane family, slip can happen on prismatic plane f10
10g or pyramidal planef10
11g, and so on. Slip system represents a spatial orientation that a crystal may take in slip process. Under the same conditions, the more slip systems a crystal has, the more spatial orientations it may take in slip process, the easier the slip, and the better plasticity the crystal will be. The number of the slip systems in fcc crystals is {111}4 3 = 12; for bcc crystals, the number of slip systems is {110}6 2 +{112}12 1 +{123}24 1 = 48; and for hcp crystals, the number of slip systems is only (0001)1 3 = 3. Therefore, the hcp crystals have less slip systems, thus it can be inferred that the plasticity of hcp crystals should be the worst comparing with fcc or bcc crystals. However, compared with fcc crystals, bcc crystals have more slip systems, but always have a lower plasticity because they have only one slip direction of .

198

Chapter 4 Deformation and recrystallization

c. Critical resolved shear stress A slip happens when a certain minimum stress, the critical resolved shear stress (CRSS), has been attained. The concept of a CRSS can be understood with the aid of Fig. 4.8.

Fig. 4.8: Calculation of the resolved shear stress.

When a single-crystal cylinder with a cross-sectional area, A, is loaded by an axial force F, the force component of F along the slip direction is F⋅cos λ. When it is divided by the area of the slip plane, A/cosφ, the corresponding resolved shear stress τ is attained by τ=

F cos φ cos λ A

(4:9)

where σ = F = A is the normal stress. The φ is the angle between the normal of the slip plane and the axial force F. The λ is the angle between slip direction and the axial force F. Plastic deformation will occur when the resolved shear stress of the slip system reaches the critical shear stress value, τc . The cos φ cos λ is called orientation factor or Schmid factor. Larger orientation factor means the larger resolved shear stress, and the corresponding slip system is more easy to be operated. Therefore, the CRSS for onset of slip is τC = σs cos φ cos λ where σs is the normal stress when slip just begins, that is, yield stress. For a given φ, if the slip direction is located on the plane consisting of F and the normal of slip plane, namely φ + λ = 90°, the τ value along the direction is larger than other τ values with other λ values. At this moment, orientation factor cos φ cos λ = cos φ cosð90 − φÞ = 21 sin 2φ. So when φ is 45°, orientation factor reaches its maximum value of 21. Figure 4.9 shows the influence of the orientation factor on the yield stress, σs , for the magnesium single crystal with a hcp structure. In Fig. 4.9 the small circles indicate experimental values, and the curve represents calculated values. Both are well consistent with the change in

4.2 Plastic deformation of crystals

199

12

Yield stress (MPa)

10 8 6 4 2

0 0.1 ϕ = 90°

0.3

0.5 0.3 cos ϕ cos λ

0.1 0 ϕ = 0°

Fig. 4.9: Relationship of yield stress and crystal orientation in magnesium crystal under tension.

orientation factor. When φ = 90 or = 90 , σs is infinite, that is, when the slip plane is parallel to the applied force, or slip direction is vertical with the applied force, slip can not occur. When the slip direction is located in the plane composed of the applied force and the normal of slip plane, meanwhile φ = 45 , the orientation factor will reach the maximum (0.5), and σs will reach the minimum, that is, the required resolved shear stress value of slip can be reached under a minimum normal stress. Usually, large orientation factor that is favorable for slip is called soft orientation; and small orientation factor that is unfavorable for slip is called hard orientation. In summary, the CRSS of slip is a true physical quantity, which demonstrates the onset of the yielding in a single crystal. Its value is related to not only crystalline structure, pureness, and temperature, but also the processing, treatment process, deformation velocity and type of slip system, and so on. Table 4.4 lists the CRSS of some metals. d. Rotation of the crystal plane due to slip In addition to the relative sliding in the slip direction across the slip plane in a loaded single crystal, the rotations of both the slip direction and slip plane occur to the direction of the loading axis. This phenomenon is particularly obvious for hcp crystals because they have only a set of slip plane. Fig. 4.10 is the schematic diagram of rotations of both the slip direction and slip plane in a single crystal during tensile test. Assume that the deformation is unconstrained, it could lead to a shape change of the crystal and a deviation of the axial direction when the sample is stretched by the external force F in axial direction of the crystal, as shown in Fig. 4.10(b). However, in a tensile test the sample is fixed by two chucks that keep axial direction unchanged. The orientation of the single crystal must be rotated accordingly, and in turn, the sliding planes tend to

200

Chapter 4 Deformation and recrystallization

Table 4.4: Critical resolved shear stress of slip of some metal crystals. Metal

Temperature

Purity (%)

Slip plane

.

Slip direction

Critical shear stress

Ag

Room temperature

{}

.

Al

Room temperature

–

{}

.

Cu

Room temperature

.

{}

.

Ni

Room temperature

.

{}

.

.

{}

.

{}

.

.

{1010}

.

Room temperature

.

{}

.

Room temperature

.

{}

.

Mg

 °C

.

{}

.

Mg

 °C

.

{1011}

.

Fe

Room temperature

Nb

Room temperature

Ti

Room temperature

Mg Mg

–

F Chuck F ϕʹ > ϕ ϕʹ

ϕ

Crystal plane bending and rotation

Slip direction

Rotation Crystal plane bending and rotation

F

(a)

(b)

F

(c)

Fig. 4.10: Tensile deformation of a single crystal: (a) original specimen; (b) free slip deformation; (c) deformation restricted by the grips.

rotate toward the axial direction as shown in Fig. 4.10(c). The slip planes at two ends of the sample will bend due to the geometry constraint imposed by the chucks. Figure 4.11 shows the rotation origin of the slip planes. This figure gives the components of the axial force on one middle layer among slip planes in the single crystal, as shown in 4.11 (a). σ1 and σ2 are the normal stresses on the upper and lower slip planes of the layer upon an external force, respectively. Under the action

4.2 Plastic deformation of crystals

(a)

201

(b)

Fig. 4.11: Couple effect of rotating crystal during uniaxial tensile.

of this couple, the slip plane will rotate and tend to be parallel to the axial direction. Figure 4.11(b) shows the maximum shear stresses τ1 and τ2 acting on the upper and lower slip planes of the middle layer, which can be, respectively, divided into two shear stresses, τ1′, τ2′, parallel to the slip direction, and two stresses τ1″, τ2″, perpendicular to the slip direction. The former shear stresses are the effective resolved shear stress associated with the slip, and the latter shear stresses act as a couple to cause the crystal rotation, striving to make the slip direction rotate to the direction of the maximum shear stress. As for the compression deformation, the rotation of the crystals will also occur, but the slip planes rotate to the direction perpendicular to the axis of the external force, as shown in Fig. 4.12.

(a)

(b)

Fig. 4.12: Rotation of the crystal during uniaxial compression test (a) before compression and (b) after compression.

From the above-mentioned analysis, it is known that both the slip planes and the slip direction will rotate gradually during the slip process of the crystals, finally resulting in the change in the resolved shear stress on the slip plane. Because the resolved shear stress of a slip system reaches maximum value as φ = 45°. If φ is gradually close to 45°, the resolved shear stress of the slip system will increase, being beneficial to the further slip; conversely, if φ is gradually away from 45°, the resolved shear stress will decrease, meaning that further slip tends to be more difficult.

202

Chapter 4 Deformation and recrystallization

e. Multiple slips For the crystals with many sets of slip systems, those possible slip systems will be activated first in the most favorable orientation (the maximum resolved shear stress) of the slip systems, but due to the rotation of the crystals during deformation, the resolved shear stress of some other slip systems may also be gradually increased to a critical value, which is sufficient to activate their movements. These slips may occur simultaneously or alternately on two or more slip systems, thereby generating multiple slips. For the crystals with a larger number of slip systems, besides multiple slips, cross-slip may occur, namely, a screw dislocation glides alternately onto two or more preferred slip planes with the same slip direction. The cross-slip can make the slip of the crystals more flexible. It is worth pointing out that in the case of multiple slips, interactions between dislocations in different slip systems will hinder the continuous movement of dislocations, which is also an important strengthening mechanism. f. Resistance of dislocation movement As pointed out in Chapter 3, the estimated shear force for initializing plastic deformation of crystals is roughly G/30 (G is the shear modulus) based on the sliding model of the rigid body. However, the experimental measurement of a real crystal is 3–4 orders of magnitude lower than the theoretical value. In order to explain the above deviation, the concept of dislocation was proposed. The dislocation line can be defined as the boundary between the slipped and nonslipped areas. When a dislocation moves out to the surface of the crystal, the displacement produced is equal to the Burgers vector b along the slip plane. While a large number (n) of the dislocations move along the same slip plane out to the surface, slip bands (Δ = nb) can be observed experimentally. Slip in a crystal is activated by an external force. The resistance to dislocation motion arises from the lattice resistance. Due to the periodicity of the lattice structure, the central energy of the dislocation occurs periodically when the dislocation moves along the slip plane, as shown in Fig. 4.13. Position 1 and position 2 are equal. When the dislocation stays in these equilibrium positions, the energy is the minimum, equivalent to the energy in a valley. When the dislocation moves from position 1 to position 2, it requires to cross a peak (barrier), which means that a moving dislocation will encounter a lattice resistance. The force that is required to move a dislocation is called Peierls– Nabarro (P-N) force, because the early models were proposed by Nabarro and Peierls. Fig. 4.13: Central energy change of a dislocation during slip.

4.2 Plastic deformation of crystals

The P-N force can be calculated by the following equation:   2G 2πd 2G 2πW = τP − N = exp − exp − 1−ν ð1 − νÞb 1−ν b

203

(4:10)

It is equivalent to a CRSS for activating an edge dislocation in an ideal simple cubic crystal (as shown in Fig. 4.14). In this formula, d is the spacing of the slip planes, b is the atomic spacing in the slip direction (i.e., Burgers vector), ν is the Poisson’s ratio, and W = ðd=ð1 − νÞÞ represents the width of the dislocation core.

Fig. 4.14: Edge dislocation in a simple cubic lattice.

For a simple cubic structure, d = b, τP− N = 3.6 × 10 − 4 G as ν=0.3; τP− N = 2 × 10 − 4 G as ν= 0.35. The values are much smaller than the theoretical shear modulus (τ ≈ G=30), while the measured value of the CRSS has the same order of magnitude. Moreover, the calculated values of P-N force agree reasonably well with measured values of the CRSS in ionic crystals, but are found to be somewhat higher than the measured CRSS in metals. According to the P-N force formula, the bigger the dislocation width is, the smaller the P-N force will be. This is because the core width of a dislocation indicates the scope of the seriously distorted lattice caused by the dislocation. If the width is larger, the atoms around the dislocation can be relatively close to their equilibrium positions, and the elastic distortion of the lattice can be lower. This conclusion is consistent with the experimental results. For example, fcc metals have a larger dislocation width, so their P-N forces are smaller accompanying with a lower yield stress. While the dislocation width of the bcc metals is narrower, the P-N force is larger accompanying with a higher yield stress. As for the covalent and ionic crystals with strong bonding, their dislocation widths are very narrow accompanying with a very high yield stress. As a result, the movement of dislocations is difficult, and the covalent and ionic crystals show a hard and brittle characteristic. In addition, τP− N has an exponential relationship with (–d/b). When the d value is biggest and the b value is smallest, the P-N force will be smallest. This explains why the slip of dislocations always occurs on the close-packed plane with the largest plane spacing along the close-packed direction.

204

Chapter 4 Deformation and recrystallization

It is worthy of pointing out that in eq. (4.10) the shear modulus (G) increases with lowering temperature. Therefore, the movement of dislocations is more difficult at low temperature than high temperature. In real crystals, when the dislocation line moves from a valley position to its adjacent valley position under a certain temperature, the whole dislocation line cannot cross the energy peak at the same time. Two analogies to dislocations have been pointed out by Orowon: a worm propels itself by stretching out a section of its body and thus moving its head forward; and its entire body is propelled forward by moving some tension sections along its body to the tail. Therefore, during the movement of a dislocation, it is likely to be that with the help of the thermal activation, a small part of the dislocation first crosses the energy peak as shown in Fig. 4.15, forming a dislocation kink. As a result, the dislocation line lies in two energy valleys only with a small dislocation kink crossing the energy peak. The small dislocation kink can easily move over the peak, moving the whole dislocation forward. By this way, the CRSS required by the movement of dislocations can be further reduced. Dislocation line Energy valley Energy peak

2π

Energy valley Energy peak Fig. 4.15: Torsion movement of dislocation.

In addition to the lattice resistance, there are some other resistances. For instance, the interaction between dislocations will bring a resistance to block dislocation movement. The kinks and jogs that are generated during dislocation intersection, especially the latter, will play a role in pinning dislocations, leading to the increase of the resistance of the dislocation movement. The interactions between dislocations and other crystal defects, such as point defects, grain boundaries, and second-phase particles and so on, also will hinder the movement of dislocations, resulting in the strengthening of the crystals. 2. Twin Twinning is another important type of plastic deformation, which is often recognized as a supplementary deformation mechanism when the slip is difficult to be activated.

4.2 Plastic deformation of crystals

(a)

205

(b)

Fig. 4.16: Schematic diagram of the deformation process of a twin of face centered cubic crystals (a) twinning planes and twinning directions; (b) movement of the atoms during twinning deformation.

a. Deformation process of twins The schematic diagram of deformation twinning process is shown in Fig. 4.16. Based on the crystallography, fcc crystals can be seen as an ideal stacking order of the closed packed {111} planes in the [111] direction according to the ABCABC. . .. Upon an external shear stress, the twinning of crystal during deformation maybe occur, that is, in a local deformed area, all the atoms in (111) plane will move in a relative distance of a6 ½11
2, a uniform shear, along the direction of ½11
2 (AC’ direction), as shown in Fig. 4.16(b). In the figure, the paper plane is ð
110Þ, and (111) plane is perpendicular to the paper plane; AB is the intersection of (111) plane and paper surface, which is equivalent to [112] direction. As shown in the figure, a uniform shear occurs in the middle part of the crystal, where every (111) plane between the line AB and the line GH moves a distance of a6 ½11
2 with respect to its adjacent (111) plane in ½11
2 direction. Such shear does not change the crystal structure, but changes the crystal orientation, forming a mirror image with that in the nonshear region. The region with mirror symmetry is called a twin, and the boundary between them is called twin boundary. The process of twin formation is called twinning. A twinning system consists of a twinning plane, such as (111) plane, and twinning direction on the plane, such as ½11
2 direction. Figure 4.17 shows the formation and growth of twins in hcp Zn metal. b. Characteristics of twins According to the above-mentioned analysis on twinning deformation, twins have the following characteristics:

206

Chapter 4 Deformation and recrystallization

Fig. 4.17: The growth of the twin in Zn during tension: (a) before deformation; (b) after deformation.

(1) Twinning deformation occurs under the shear stress and it usually occurs in the stress concentration zone caused by the pileup of dislocations. Thus, the CRSS of a twinning system is much larger than that of dislocation slip. (2) Twinning is a kind of uniform shear deformation. In the shear zone every layer of atoms that are parallel to twinning plane will move the same distance with respect to its adjacent crystal along the twinning direction, and the shear amount of every layer of atoms is proportional to their distances with respect to the twinning plane. (3) Twins exhibit mirror symmetry with respect to the undeformed crystal. c. Formation of twins There are three main ways to form twins in crystals: the first type is produced by the mechanical deformation, which can be called “deformation twin” or “mechanical twin.” Such a twin has the feature of a lenticular shape or flake. The second type is called “growth twin.” It comes from the vapor–solid transformation (vapor deposition), liquid–solid transformation (solidification), or solid-state transformation. The third type comes from the recrystallization during annealing of metals, which is called “annealing twin.” The growth of deformation twins also includes two stages: nucleation and growth. A thin twin sheet first breaks out at a very fast speed as “nucleation,” and then the twin is widened by the migration of twinning boundaries as “growth.” The twin nucleation needs a much greater CRSS than that of slip. For instance, the CRSS required by twinning in Mg crystal is measured as 4.9–34.3 MPa while that required by slip is only 0.49 MPa. Therefore, the stress can be accumulated to its required value of twin nucleation only when the slip systems are blocked. Twin nucleation usually occurs in the areas where stress is highly concentrated, such as grain boundaries, but the stress required by twin growth after the twin nucleation is relatively small. For example, the local stress required by twin nucleation in Zn single crystal must exceed 10−1 G (G is shear modulus), but it is only about 10−4 G

4.2 Plastic deformation of crystals

207

for the twin growth. Therefore, the velocity of twin growth is very fast comparable to the speed of shock waves. Because a considerable amount of energy is released when twins form in a very short time, clear sound is sometimes heard. Figure 4.18 shows the tensile curve of a single-crystal Cu at 4.2 K. At the beginning of the deformation, the stress–strain curve is smooth and the stress increases gradually, which is related to the slip of dislocations, and the stress (load) reaches a highest value and then suddenly drops, which means that the slip of dislocation is blocked. After that, the stress fluctuates with the increase of strain, and the stress–strain curve exhibits saw teeth, which is caused by twinning deformation. Because the CRSS required by twin nucleation is much higher than that required by twin growth, a sudden drop in load is accompanied with the emergence of the first twin plate. Therefore, it is reasonably believed that the nucleation of the first twin plate needs a very great CRSS, but the subsequent twin nucleation can be triggered by stress field from the first or more twin nucleation, in which the stress is much less the CRSS of the first twin plate nucleation. Moreover, every saw tooth corresponds to nucleation of a new twin. In such a way, the twins nucleate and grow up continuously, resulting in the occurrence of saw tooth-like stress–strain curve. At the end of saw tooth-like deformation stage, stress–strain curve demonstrates that the stress smoothly increases with strain again, as shown in Fig. 4.18, which indicates deformation returns from twinning dominated to the slip dominated. Some slip systems are activated again in the twinned domains because twinning changes the crystal orientation. 10.0

Load (kN)

7.5 5.0

Slip

Twin

Slip

2.5

0

0.25

0.50 0.75 1.00 Strain (%)

1.25

1.50

Fig. 4.18: Tensile curve of single-crystal Cu at 4.2 K.

In general, hcp metals such as Cd, Zn, and Mg with low symmetry and less slip systems tend to deform by twinning. The common twinning system of hcp metals has twinning plane and twinning direction f10
12g and , respectively. Since bcc metals have high stacking fault energy proportional to twinning energy, twinning in some bcc metals can occur only in special conditions, such as at low temperature and high strain rate. The twinning plane and twinning direction of bcc metals are {112} and respectively. While for most fcc metals, twinning is rarely activated

208

Chapter 4 Deformation and recrystallization

since the slip of dislocation cannot be blocked at room temperature. At low temperature (4–78 K), the slip of dislocations becomes difficult during deformation, so twinning will occur. It is worthy to point out the twinning in fcc metals can occur at low temperature in low deformation rate, which is different from that in bcc metals, which is attributed to low-stacking fault energy for fcc metals. The twinning plane and twinning direction of fcc metals are {111} and , respectively. The direct contribution of twins on the amount of the plastic deformation is much smaller than that of dislocation slips. For example, if the deformation of a hexagonal close-packed Zn crystal depends only on twinning, and then its elongation is only 7.2%. However, since twinning deformation changes the orientation of the crystals, some slip systems with an unfavorable orientation may turn to be a favorable orientation, which can stimulate further slip. In this way, twinning and slip can be activated alternately to achieve a larger deformation. d. Dislocation mechanism of twins The shear is uniform in the deformed part, so the relative displacement of every layer of atoms parallel to twinning plane can be achieved by a partial dislocation (Shockley dislocations) with respect to its adjacent plane. In fcc metals (Fig. 4.19), if a perfect dislocation of a/2 pslips ffiffiffi along {111} plane, it will result in a relative movement of an atomic distance of 2=2a between the two sides of the slip plane. And the stacking order of {111} plane will not change, but maintain the order of ABCABC. But if two Shockley dislocations slip respectively along two neighboring {111} planes, it will result pffiffiffi in a relative movement of 6=6a, not an atomic distance. The stacking order will change from ABCABC to ABCACBACB due to the stacking fault in the crystal plane, which results in a piece of twin in the upper crystal.

(a)

(b)

Fig. 4.19: Twinning in fcc crystals.

But how does this process proceed? A. H. Cottrell and B. A. Bilby proposed that twins can form a pole mechanism of dislocation multiplication. Figure 4.20 shows the schematic diagram of the pole mechanism of dislocation multiplication. Three dislocation lines OA, OB, and OC intersect at point O. OA and OB do not intersect

4.2 Plastic deformation of crystals

209

Fig. 4.20: Polar mechanism of dislocation multiplication.

on their slip planes, which can be defined as sessile dislocations (here called pole dislocations). Both OC and its Burgers vector b3 are on the slip plane, rotating around the point O, which can be defined as a sweeping dislocation. And the slip planes can be defined as sweeping plane. If the sweeping dislocation OC is a partial dislocation and OA, OB Burgers vector of b1, b2 have a sweeping component perpendicular to the surface, respectively, with their values being the plane spacing of the sweeping planes (slip planes), therefore, the sweeping plane is not a plane but a continuous curve surface (i.e. a helical surface). On this condition, a twin with a single atomic layer can be produced when the sweeping dislocation OC rotates in one cycle. At the same time, OC will climb up to the adjacent crystal plane by an atomic distance. Sweeping dislocation sweeps continuously, which makes the dislocation line OC and point O climb up constantly. That is to say, the sweepings of the partial dislocations in many sweeping planes result in uniform shear deformations in a relatively wide area, and thus the deformation twin is produced. 3. Kink In some special case, both twins and slips cannot be activated, the crystals will deform in other form. For instance, upon uniaxial compression of a single hcp crystal Cd, if the applied force direction is parallel to the (0001) plane (slip plane), the resolved shear stress on the slip plane is zero and slips cannot be activated for Φ = 90°，that is, cos Φ = 0°. If the resistance of the twinning is very large, the twins cannot be activated either. On this condition, if the stress is increased continuously, the crystals will be bent locally in order to make the crystal shape adapt to the external force when it exceeds to a threshold value. Figure 4.21 shows that this deformation is in a kinking way and the deformation zone forms a kink band. Figure 4.21(a) shows that kinking deformation is different from twinning, and the former is an asymmetrical deformation. The lattice kinks in zone ABCD. The top and bottom interfaces of the kinking zone (AB, CD) are consisted of two columns of edge dislocations with opposite signs. Both sides (AD, BC) of zone ABCD have almost the same number of dislocations but opposite signs. Each bend zone is consisted of some dislocations with the same sign. Strong deformation produced by kinking results in such

210

Chapter 4 Deformation and recrystallization

(a)

(b)

Fig. 4.21: Kinking of a single-crystal Cd under compression: (a) schematic diagram of kinking; (b) kinking band in a crystal.

a distribution of dislocations, which is favorable for the relaxation of their stress field. After kinking, the crystal orientation is changed, which is favorable for the activation of slip systems. In addition to the above-mentioned cases, kinking band also occurs with the emergence of twinning. When twinning is activated, it forces the surrounding crystal to produce a larger strain to accommodate the shear deformation in twinning zone, especially when the ends of the crystal are constrained (such as by the limitation of tensile grips). In order to make it compatible to the restriction, the strain within the contiguous regions become larger, and kinking bands will be formed in the contiguous zones, as shown in Fig. 4.22.

4.2.2 Plastic deformation of polycrystals Engineering materials are usually polycrystalline aggregates. Grain boundary separates grain from grain. At room temperature, basic deformation mechanism of each grain in a polycrystal is similar to the single crystal. Due to the orientation difference (misorientation) between the neighboring grains, plastic deformation of polycrystals needs not only to overcome the obstruction of grain boundaries, but also to adapt itself to the neighboring grains during deformation. Thus, the deformation of polycrystals is more complex comparing with that of single crystal and will be discussed in the following sections.

4.2 Plastic deformation of crystals

Twin

Shear direction

211

Twin Kinking band

(a)

(b)

Fig. 4.22: The kinking band accompanied by twinning deformation.

1. Effect of grain orientation The effect of grain orientation on plastic deformation of polycrystals is mainly reflected on the deformation restriction and deformation compatibility. With an applied force on a polycrystal, the shear stress subjected by different crystals is different due to the misorientation of neighboring grains. Thus grains are not deformed at the same time. The grains with favorable orientation will plastically deform associated with dislocation slips initially, while slip systems in the grains with unfavorable orientation cannot be activated. Since every grain is surrounded by other grains, plastic deformation in each grain needs to be compatible with its neighboring grains. Otherwise, crack initiates. In order to achieve deformation compatibility between grains in polycrystals, slip systems in grains need to be activated including both of favorable orientations and unfavorable orientations. Theoretical studies show that at least five independent slip systems need to be activated to accommodate arbitrary deformation of polycrystals. An arbitrary plastic deformation can be expressed by six strain components: εxx, εyy, εzz, γxy, γyz, and γxz. Since plas tic deformation does not change the volume of materials (ΔV V = εxx + εyy + εzz = 0), only five independent strain components are needed to satisfy deformation compatibility. Therefore, deformation compatibility can be achieved by multiple slip systems. This is relevant to crystal structure: fcc crystals and bcc crystals with enough slip systems can satisfy this requirement, so they have good plasticity; in contrast, the deformation compatibility of hcp crystals is poor due to less slip systems. 2. Effect of grain boundaries Atomic arrangement of grain boundary is irregular accompanying with serious lattice distortion. Besides, grain orientations are different across grain boundary, so a given slip direction and slip plane of dislocation are discontinuous in the two grains; in other words, the slip of dislocations cannot be easily transferred from one grain into its neighboring grain, which indicates blocking effect imposed by grain boundary. From the tensile test results of a sample with only two to three grains, we can observe that grain boundary is bamboo like (as shown in Fig. 4.23). This indicates

212

Chapter 4 Deformation and recrystallization

Fig. 4.23: Bamboo-like grain boundary after tensile tests.

that slips are blocked nearby grain boundary so that the amount of intergranular deformation is less than that of intragranular deformation. After tensile test, slip bands in each grain will end at its grain boundary. Transmission electron microscope (TEM) observation reveals that a large number of dislocations are blocked at grain boundary, as shown in Fig. 4.24. These pile-up dislocations at the grain boundary will generate an opposite force for dislocation source (generator). This opposite force will be increased with the number (n) of the blocked dislocations: n=

kπτ0 L Gb

(4:11)

where τ0 is the resolved shear stress on the slip plane; L is the distance from dislocation sources to the grain boundary; k is a coefficient: k = 1 for screw dislocation and k = 1−ν for edge dislocation. When n increases to a certain value, the corresponding opposite force will prevent dislocation sources from generating dislocations, which strengthens the crystals.

Dislocation resource

Stress concentration and stress field

τ0 τ0

Pile-up dislocations Stress field

Fig. 4.24: Schematic diagram of pile-up dislocations at the grain boundary.

Because the number of grain boundaries depends on grain size, the effect of grain boundaries on initial plastic deformation of polycrystals can be embodied by grain size. Experiments have proved that the strength of the polycrystals will increase with the decreasing grain size. The relationship between the yield strength σs and the average grain size d can be expressed by Hall–Petch equation: 1

σs = σ0 + Kd − 2

(4:12)

where σ0 is the yield strength of the single crystal and K is a coefficient related to the nature of the crystal. Figure 4.25 shows the relationship between the lower yield

4.2 Plastic deformation of crystals

Low yield point (MPa)

500

50

Average grain size (μm) 20 10 5 4 3

2

213

1.5

600 500 400 300 200 100 0

4

8

12

16

1 – 2

1 –– 2

d

20

24

28

mm

ω(C) = 0.005%

ω(C) = 0.12%

ω(C) = 0.05% ω(C) = 0.09%

ω(C) = 0.15% ω(C) = 0.20%

Fig. 4.25: Relationship between lower yield point and grain size in low-carbon steels.

point and the grain size in low-carbon steels. These experimental results are well consistent with eq. (4.12). Although Hall–Petch equation is an empirical formula, it also can be derived from dislocation pile-up model based on dislocation theory. A large number of experiments indicate that Hall–Petch equation can also be employed in other cases. For instance, the effect of the subgrain size or the lamellar spacing on the yield strength, as shown in Fig. 4.26; the relationship between fracture stress and grain size in brittle materials; the relationship between fatigue strength or hardness and the grain size in metals, and so on. As fine-grained materials have high strength, hardness, and excellent toughness, structural materials used at room temperature are desired to be comprised of fine grains. However, when materials deform at temperature higher than 0.5 Tm (Tm corresponds to the melting point of the materials), the diffusion rate of atoms is accelerated along the grain boundary. Under a high deformation temperature, grain boundaries show a certain viscosity characteristic and their deformation resistance is greatly diminished. Even upon a small stress, grains may slide along grain boundaries for a long time. This becomes an important deformation mode for polycrystalline aggregates at a high temperature (see Section 4.4.3). In addition, there may occur another deformation mechanism, called diffusion creep in polycrystals, especially in fine-grained polycrystals at high temperature. The process is related to the diffusion of vacancies due to its low formation energy and low migration energy

214

Chapter 4 Deformation and recrystallization

Fig. 4.26: Relationship between yield strength and subgrain size in Cu and Al.

barrier. Since grain boundaries are the source and the annihilation trap of vacancies, the finer polycrystalline grains are, the greater the rate of diffusion creep is, which is unfavorable for high-temperature strength of polycrystals. Accordingly, an equal strength temperature TE in polycrystalline materials is introduced. When the temperature is below TE, the strength of grain boundary is higher than that of grain inner. Otherwise, opposite effect will occur when temperature is higher than TE (as shown in Fig. 4.27).

Fig. 4.27: Scheme of an equal strength temperature TE.

4.2.3 Plastic deformation of alloys The metallic materials used in engineering are mostly alloys. Generally speaking, the deformation of alloys is similar to that of the pure metals, but exhibits some new features due to the presence of alloying elements.

4.2 Plastic deformation of crystals

215

According to the difference in the constituent phases, the alloys can be divided into single-phase solid solution alloy and multiphase alloy. Both have different behaviors of plastic deformation. 1. Plastic deformation of single-phase solid solution alloys The difference between single-phase solution alloys and pure metals is the existence of solute atoms in alloys. Solute atoms play a strengthening role in the plastic deformation of the alloy, which is called solid solution strengthening. Besides, the phenomena of obvious yield point and strain aging can be found in some solid solution alloys, as described further: a. Solid solution strengthening The deformation resistance of solid solution alloys is increased by the content of solute atoms. Figure 4.28 shows that the variation of the strength, hardness, and elongation of Cu–Ni solid solution with solute content. The strength and hardness of the alloy are enhanced accompanying with the decrease of ductility. Such a solid solution strengthening effect can be described by the following equation: τ=

dτ x x or Δσs = A 2 dx a0 b

(4:13)

where dτ=dx is the increment of the CRSS caused by the lattice distortion of a unit solute atoms, x is the atomic fraction of the solute atoms, a0 is the lattice constant of the solvent crystal, b is Burgers vector of the dislocation, A is a constant.

Fig. 4.28: Relationship between mechanical properties and composition of Cu–Ni solid solution.

216

Chapter 4 Deformation and recrystallization

Comparing the stress–strain curve of pure metals with that of their solid solution alloys (as shown in Fig. 4.29), it can be seen that the addition of solute atoms not only increases flow strength, but also increases work-hardening rate.

Fig. 4.29: Stress–strain curve of Al–Mg solid solution.

Solid solution strengthening effects caused by different solute atoms are quite different. Figure 4.30 shows the change of the CRSS with different content type of alloying elements added into a single-crystal copper. There are main factors affecting the solid solution strengthening effect as follows. (1) The higher the atomic fraction of solute atoms is, the stronger the strengthening effect would be. Especially the initial strengthening effect is more obvious. (2) The greater the atomic size difference between the solute atoms and the solvent atoms in metals is, the greater the strengthening effect would be. (3) The strengthening effect of interstitial solute atoms is stronger than that of substitution solute atoms because the lattice distortion caused by interstitial atoms is much greater than that caused by substitution solute atoms. (4) The greater is the difference of valence electron number between solute atoms and solvent atoms, the more obvious is the solid solution strengthening effect. Namely, the yield strength is enhanced with the increase in the electron concentration of the solid solution alloys. It is generally believed that solid solution strengthening is related to many factors: elastic interaction, chemical interaction, and electrostatic interaction between solute atoms and dislocations. Furthermore, when the plastic deformation proceeds in the solid solution alloys, the dislocation movement will change the distribution of solute atoms to a short-range order or segregation, which raises system energy and increases the slip resistance of dislocations.

4.2 Plastic deformation of crystals

217

Fig. 4.30: Influence of solid solution elements on the critical resolved shear stress of a single-crystal copper.

b. Yield phenomenon and strain aging Figure 4.31 shows a typical stress–strain curve of the low-carbon steels. There is an obvious yield point, which is different from the tensile curves of medium-carbon and high-carbon steels or other alloys. When the tensile sample begins yielding, the stress falls down suddenly. The sample continues to yield and is elongated under a substantially constant stress, called yield-point elongation, or called Lüders strain and the tensile curve exhibits a stress platform. The stress value corresponding to yield beginning and stress finishing is called the upper and lower yield point, respectively. Beyond the upper yield point and until the end of the range at the lower yield stress, the plastic deformation is not uniform in the specimen: the instantaneous dislocation multiplication is restricted to one band (or more bands) of material, the so-called Lüders band,

Fig. 4.31: Engineering stress–strain curve of annealed low-carbon steel and yield phenomenon.

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Chapter 4 Deformation and recrystallization

which would propagate along the whole (length of the) polycrystal sample, inducing the same plastic strain at every position that it passes. Upon continued deformation, that is, beyond the yield-point elongation zone, macroscopically homogeneous deformation and strain/work hardening occurs. The angle between Lüders band and tensile axis is usually about 45°. Also each small fluctuation of the stress corresponds to the formation of a new deformation band, as shown in Fig. 4.31. The yield phenomenon was for the first time observed in low-carbon steels. Under appropriate conditions, the stress difference between the upper and lower yield point can reach 10%−20% and the yield-point elongation can be over 10%. After that, in many other metals and alloys, such as Mo, Ti and Al alloys and Cd, Zn single crystals, α and β brass, and so on, the yield phenomenon was also found if these metal materials contain a proper amount of solute atoms. The pronounced yield drop can be ascribed to the pronounced increase in the density of mobile dislocations once the upper yield point is passed. Initially in a material, many dislocations are recognized to be immobilized because of their interaction with point defects, such as dissolved (interstitial) atoms (e.g., carbon and nitrogen in iron). Generally, it is believed that the interaction between solute or impurity atoms and dislocations can form solute atoms atmosphere, called Cottrell or Cottrell-Bilby atmosphere in solid solution alloys. From the stress field of an edge dislocation, it can be found that the extra-half-plane region above the slip plane of the dislocation suffers a compressive stress and the region below the slip plane suffers a tensile stress. If there exist interstitial atoms, such as C, N, they will interact with the dislocations and segregate in the tensile stress region of an edge dislocation so that the elastic strain energy of the dislocation can be reduced. When the dislocation is in a lower energy state, it tends to be stable and not easy to move, namely, the dislocation is pinned. Concentrations/rows of many interstitial atoms develop along dislocation lines, which are named as “Cottrell–Bilby atmospheres” or “Cottrell–Bilby clouds.” Especially in bcc crystals, the interaction between interstitial solute atoms and dislocations is very strong accompanying a strong pinning effect of dislocation. The movement of dislocations requires a greater applied stress to break the pinning effect caused by Cottrell atmosphere, which is the origin of the upper yield point appearance. The stress required will be evidently reduced once the dislocations break the pinning effect, which corresponds to lower yield point and under this stress a horizontal platform in the stress–strain curve generates. Cottrell–Bilby atmospheres are widely used to explain the yield phenomenon. But after the 1960s, Gilman and Johnston found that there also existed discontinuous yield phenomenon in dislocation-free copper whiskers, low dislocation density covalent crystal Si, Ge, and ionic crystals LiF. For the sake, another mechanism was proposed as follows. It can be known from dislocation theory that the strain rate of plastic deformation ε_ p is proportional to the movable dislocation density ρm , the average velocity of the dislocation movement v and Burgers vector of dislocation b: ε_ p ∝ ρm ν b

(4:14)

4.2 Plastic deformation of crystals

219

The average velocity of the dislocation movement is closely related to the stress: v=

 m′ τ τ0

where τ0 is the stress required for the dislocation movement in a unit velocity, τ is the effective shear stress of the dislocations, m′ is called the stress sensitivity index, which is associated with materials. In the tensile test, ε_ p is determined by the speed of the machine grips and is close to a constant value. At the beginning of plastic deformation, the dislocation density in crystals is low or a large number of dislocations are pinned. Thus the movable dislocation density ρm is low. In order to maintain a constant value of ε_ p , τ needs to be increased to obtain a larger value of v. This is the reason why the stress value of the upper yield point is high. However, once the plastic deformation starts, ρm is raised by dislocation multiplication. In order to maintain a constant value of ε_ p , the rapid increase of ρm will inevitably lead to a sudden drop of v, corresponding to a sudden drop of the yield stress, which is the origin of the lower yield point appearance. These two mechanisms are not mutually exclusive but complementary to each other. The yield phenomenon of low-carbon steels can be better explained by combining the two mechanisms. The precondition of the dislocation multiplication theory is that the movable dislocation density in the original crystals is very low. The original dislocation density in low-carbon steels is 108cm−2, but ρm of movable dislocation density is only 103cm−2. Therefore, the low density of movable dislocations in low-carbon steels is attributed that most of the dislocations are pinned by carbon atoms, forming Cottrell-Bilby atmosphere. There exists a phenomenon of strain aging, which is associated with the yield phenomenon of low-carbon steels, as shown in Fig. 4.32. When the tensile test of an annealed low-carbon steel tensile sample is carried out above the yield point, the load is removed. Then the tensile sample is immediately reloaded. It can be found

Fig. 4.32: Strain aging of low-carbon steels during tensile test (1) pre-deformation; (2) reloading immediately after unloading; (3) reloading after being placed for a few days.

220

Chapter 4 Deformation and recrystallization

that the yield point no longer appears in the tensile curve (curve 2). Yield phenomenon does not occur in the specimen under this condition. If the pre-deformed samples at room temperature for a few days or heated at 200 °C for a very short time are reloaded, the yield phenomenon reappears and the yield stress is further raised (curve 3). This phenomenon is called strain aging. Similarly, the Cottrell atmosphere mechanism can be used to explain the strain aging of low-carbon steels. When the sample is reloading immediately after unloading, dislocations have already broken out of the pinning by Cottrell atmosphere consisting of carbon atoms, moreover, carbon atoms do not have enough time to return to the tensile regions of edge dislocations during reload stage, so the yield point will not appear. If the pre-deformed sample is placed for a long time or by heated for a very short time after unloading, the carbon atoms will have enough time to segregate around dislocations by diffusion and form Cottrell-Bilby atmosphere, so the yield phenomenon will appear again. 2. Plastic deformation of multiphase alloys Metal materials used in engineering are mainly two or multiphase alloys. In comparison with single-phase solid solution alloys, the existence of the second phase makes the plastic deformation of multiphase alloy more complex than that of single-phase solid solution alloys, which refers to the effects of the amount, size, shape, and distribution of the second phases as well as the difference between the second phase and matrix phase in deformation behavior. According to the average size of the second-phase particles, the alloys can be divided into two categories, that is, when the average sizes of the second-phase particles and the matrix grain are of the same order, the alloys are called convergent two-phase alloys; if the second-phase particles are smaller and dispersed in the matrix grains, the alloys are called dispersive two-phase alloys. The plastic deformation and strengthening behaviors of these two kinds of alloys are different. a. Plastic deformation of convergent alloys If the grain sizes of the two phases are of the same order and they are both plastic phases, the deformability of the alloys will depend on the volume fraction of the two phases. As a first order approximation, it can be assumed that the two phases have the same magnitude of stress and strain during deformation. Thus, the average flow stress σ at a certain strain or the average strain ε under a certain stress can be expressed by mixing law, respectively: σ = φ1 σ1 + φ2 σ2 ε = φ1 ε1 + φ2 ε2

4.2 Plastic deformation of crystals

221

where φ1 and φ2 are the volume fractions of the two phases, respectively, φ1 + φ2 = 1, σ1 and σ2 are the flow stresses of the two phases at a certain strain, respectively, ε1 and ε2 are the strains of the two phases under a certain stress, respectively. Figure 4.33 shows the stress–strain curves of two phase alloys where each phase undertakes the same strain or the same stress.

(a)

(b)

Fig. 4.33: Stress–strain curves of convergent alloys where each phase has the same strain (a) and the same stress (b).

In fact, the stresses or the strains cannot be equal in two phases. The abovementioned assumption and the mixing law only can be used to qualitatively estimate the influence of the volume fraction of the second phases. Experimental results show that in this kind of alloys slips often first occur in the relative soft phase during plastic deformation. If the amount of hard phases is little, plastic deformation is almost conducted in the soft phase. If the second phase is a hard phase and its volume fraction φ is greater than 30%, the second phase can play a significant strengthening effect. If one phase is a plastic phase and the other is a brittle phase in the convergent alloys, the plastic deformation behaviors depend not only on the relative amount of the second phases, but also on their shape, size, and distribution. Taking cementite (Fe3C, hard and brittle) and ferrite (α-Fe-based solid solution) in carbon steels as an example, the influence of shape and size of the cementite on the mechanical behaviors of high-carbon steels is shown in Table 4.5. Where d represents the interlamellar spacing.

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Chapter 4 Deformation and recrystallization

Table 4.5: Influence of cementites on the mechanical properties of carbon steels. Material and Pure microstructure iron

σb (MPa) δ (%)

Eutectoid steel (ω(C) = .%)

ω(C) = .%

Lamellar Sorbite Troostite Globular pearlite (d ≈  nm) (d ≈  nm) pearlite (d ≈  nm)

Quenching + tempering at  °C

Network cementite





,

,



,













.



b. Plastic deformation of dispersive alloys When the second-phase particles are dispersed and uniformly distributed in the matrix phase, it will produce a significant reinforcement effect. The second-phase particle can obstruct dislocation motion, and thus strengthen the alloys. Usually the second-phase particles can be divided into “nondeformable” and “deformable” ones. These two types of particles have different interaction modes with mobile dislocations and different strengthening mechanisms. In general, the secondphase particles added by powder metallurgy into the dispersion strengthened alloys are nondeformable; the second-phase particles precipitated from the supersaturated solid solution by aging treatment are mostly deformable. But if the precipitation phase particles grow up to a certain size in the process of aging, they can also play a role of the nondeformable particles. (1) Strengthening effect of impenetrable particles When a moving dislocation encounters “strong” particles, the dislocation line first bends around the particles. If the particles are able to resist penetration, then the externally applied stress forces the dislocation to pass between adjacent particles. The hindering effect on the dislocations of the impenetrable particles is shown in Fig. 4.34. With the increase of the stress, the bending dislocation bows out and encircles the particle. Around the particle, the positive and negative dislocation segments will counteract and form a dislocation loop around each particle, thus freeing the main dislocation line to continue its passage through the alloy. This mechanism is known as Orowan strengthening after the man who first described it. The applied shear stress necessary to force dislocations between particles is τ. According to dislocation theory, the required shearing stress of a bent dislocation line with a radius of R is τ=

Gb 2R

where R is radius of curvature, and R =λ=2, so the maximum shear stress for a dislocation to pass two adjacent particles at distance λ by bowing out is given by

4.2 Plastic deformation of crystals

223

Fig. 4.34: Schematic diagram of dislocation bowing out and encircling impenetrable particles.

τ=

Gb λ

This is a critical shear stress value, and only when the stress exceeds this value, the dislocation can bypass the particle. According to the above-mentioned formula, the strengthening effect of the impenetrable particles is inversely proportional to the distance of particles (i.e., λ). In other words, the more particles and the smaller the spacing of the particles are, the stronger the strengthening effect will be. As a result, the decrease of the size and/or the increase of the volume fraction of the particles are helpful to strengthen the alloys. The Orowan strengthening mechanism mentioned earlier has been verified by many experiments. (2) Strengthening effect of deformable particles If the second-phase particles are deformable, the dislocations would cut and shear them, namely, the second phase will be deformed together with the matrix as shown in Fig. 4.35. In this situation, strengthening effect mainly depends on the intrinsic characteristics of the particles and their relationship with the matrix. The strengthening mechanism is very complex and varies with different alloys. The main effects are described as follows.

Fig. 4.35: Schematic diagram of dislocation shearing particles.

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Chapter 4 Deformation and recrystallization

① When a dislocation shears across a particle, there will produce a step with a width of b in the surface of the particle. New surface of the step will increase the interfacial energy. ② If the particle is an ordered structure, when the dislocations shearing across particles, the ordered structure will be disturbed in the upper and lower part of the slip plane, resulting in the appearance of antiphase domain boundary and causing the increase of the energy. ③ As the second-phase particle have different crystal structures, at least different lattice constants with the matrix, so when the dislocations shear across the particles the atomic misalignment and distortion on slip plane will assume extra energy and obstruct the dislocation motion. ④ Because of the difference of specific volume, there will be some coherent or semicoherent boundaries between the particles and matrix, thus the elastic stress fields around particles will interact with the dislocations and hinder their motion. ⑤ Because the orientations of the slip systems in the particles and in the matrix are inconsistent, there will be a jog after the dislocation sheared across, being an obstacle for the slip of the dislocations. ⑥ Because of the difference of the stacking fault energy between the particles and the matrix, the width of the extended dislocations will be changed after the dislocation sheares across, causing the increase of the energy. Owing to all those strengthening factors mentioned earlier, the strength of the alloys will be improved. In short, the two mechanisms mentioned earlier can be used not only to explain the strengthening effect of the second particles in alloys, but also to explain the plasticity of the multiphase alloys. No matter which mechanism is dominant, it would be controlled by the intrinsic characteristics, such as sizes, and distribution of the particles. Therefore, the strength and plasticity of the precipitation strengthening alloys and dispersion strengthening alloys can be adjusted by changing those factors.

4.2.4 Effect of plastic deformation on microstructures and properties 1. Changes in the microstructures Accompanied with plastic deformation, the microstructures of the alloys change obviously. Besides a large amount of slip bands or twin bands produced in grains, the preexisting equiaxed grains will be gradually extended along the deform direction with the increase of the deformation, as shown in Fig. 4.36 When the deformation is heavy, grains will become blurred and indistinguishable and appear as some fibrous stripes, which are called fiber structure. The extending direction of the fiber structure is the direction that the materials flow. Note: the structures of the cold-deformed alloys are related to the observed cross-section of the

4.2 Plastic deformation of crystals

225

Fig. 4.36: Optical and TEM images of cold rolled coppers with different reductions. (a) 30% compression ratio 300×; (b) 30% compression ratio 30,000×; (c) 50% compression ratio 300×; (d) 50% compression ratio 30,000×; (e) 90% compression ratio 300×; and (f) 90% compression ratio 30,000×.

sample. If the sample is cut along the normal to deformation direction, the microstructures cannot truly reflect the above characteristics of the cold-deformed grains.

2. Changes in the substructures As described previously, upon an applied stress the plastic deformation of the polycrystals occurs by the movement and multiplication of the dislocations. With the increase of the deformation, the dislocation density in the crystals will increase rapidly. After a serious cold deformation, the dislocation density can be increased from 106–107 cm−2 at initial annealing state to 1011–1012 cm−2. The substructures, such as dislocations, and their distribution in the deforming polycrystals can be characterized by transmission electron microscope (TEM). With the increase of deformation, the substructures exhibit dislocation tangles (Fig. 4.36(b)),

226

Chapter 4 Deformation and recrystallization

dislocation cells (Fig. 4.36(d)), cell elongation, in which high density of dislocations form a cell wall, while within a cell there are low density of dislocations. Between cells there is small misorientation. The fiber structures observed in optical microscope actually consist of considerable slender dislocation cells observed in TEM (Fig. 4.36(f)). The formation of the dislocation cells is also related to the type of the materials in addition to the degree of deformation. Metals and alloys with high stacking fault energy (such as Al, Fe) have not or have narrower extended dislocations, thus dislocations easily cross-slip, and in turn form obvious dislocation cells, as show in Fig. 4.37. If alloys with low-stacking fault energy (such as stainless steel and α brass) have wider extended dislocations, and such extended dislocation are difficult to cross-slip, and in turn form complex networks instead of cells, as shown in Fig. 4.38.

Fig. 4.37: Cellar structure of dislocations in pure Fe deformed by a strain of 20% at room temperature.

Fig. 4.38: Complex dislocation networks of stainless steel cold rolled by a reduction of 2%.

3. Changes in the properties The mechanical, physical, and chemical properties of the materials will be altered greatly with the changed microstructures and structures during the plastic deformation. a. Work hardening Figure 4.39 shows the strength and elongation of a cold rolled copper. Table 4.6 shows the influence of the cold drawing on the mechanical properties of a lowcarbon steel (0.16 wt.% C). It is obvious that the strength of the cold-deformed

4.2 Plastic deformation of crystals

227

Fig. 4.39: Influence of cold rolling on the tensile mechanical properties of copper and copper alloy.

Table 4.6: Influence of cold drawing on the mechanical properties of low-carbon steel. Reduction ratio of cold drawing (%)      

Yield strength Ultimate strength (MPa) (MPa)      

     

Elongation Reduction of area (%) (%)      

     

metal materials will increase with the increase of deformation accompanying with the decrease of ductility, which is called work hardening. Work hardening is an important property of the metal materials and can be used as a strengthening method, especially for the pure metals and some alloys that cannot be strengthened by heat treatment, such as austenitic stainless steel. These metals and alloys will be strengthened mainly based on work hardening by cold rolling or cold drawing. Figure 4.40 shows a typical stress–strain curve (also called work-hardening curve) of a single-crystal metal. Its plastic deformation can be divided into three stages: I. Easy slip stage After τ reaches to its critical stress τc , there is a relatively large strain with a small increase of the stress. The stress–strain curve is approximate to a straight line with

228

Chapter 4 Deformation and recrystallization

a low slope of θ1 ðθ = ðdτ=dγÞ or θ = ðdσ=dεÞÞ, namely, a low work-hardening rate. Its magnitude is about 10 − 4 G (G is the shear modulus of materials). In this stage only single slip occurs.

Fig. 4.40: Stress–strain curve of single crystals.

II. Linear hardening stage The stress is increasing linearly with the increase of the strain. The stress–strain curve is also approximate to a straight line, but with a bigger slope, presenting a higher work-hardening rate, θ2 ≈ G=300, which is close to a constant. In this stage multislip is predominant. III. Parabolic hardening stage The stress increases slowly with the increase of the strain, presenting a parabolic line. Its work hardening θ3 decreases gradually. In this stage cross-slip appears. The actual stress–strain curves of various single crystals vary with their crystal structures, crystal orientations relative to tensile direction, impurity contents, test temperatures, and so on, including the length change of every stage, even a certain stage disappears. Figure 4.41 shows the stress–strain curves of three typical metal structures. Face-centered cubic Cu and body-centered cubic Nb single crystals

Fig. 4.41: Stress–strain curves of FCC, BCC, and HCP single-crystal metal.

4.2 Plastic deformation of crystals

229

present the typical three stage characteristics. The body-centered cubic crystal will present a weak yield phenomenon in curve, indicating that Nb single crystal has a trace of impurities and Cottrell atmospheres form. The first stage of close-packed hexagonal Mg single crystals usually is longer than that of fcc and bcc single crystals, and the second stage is usually not fully developed at the fracture of the sample. As for polycrystals, because of the hindering effect of the grain boundaries and the compatible deformation required among the neighboring grains, during deformation several slip systems will be simultaneously activated. So their stress–strain curves will not show the first stage as single crystals and their slopes are usually steeper (higher work-hardening rate), especially for refined grain polycrystals at the beginning of the deformation, as shown in Fig. 4.42.

(a)

(b)

Fig. 4.42: Stress–strain curves of single crystals and polycrystals.

Work hardening is closely related to dislocations, and the relationship between the flow stress and the dislocation density was proposed: pffiffiffi τ = τ0 + αGb ρ

(4:15)

where τ is the required shearing stress after work hardening; τ0 is the required shear stress without work hardening; α is a constant concerning materials that is usually taken as 0.3–0.5; G is shear modulus; b is Burgers vector of dislocations, and ρ is dislocation density. The above-mentioned formula has been validated by many experiments. It can be known from eq. (4.15) that the increase of dislocation density and their pinning effect are the determining factors for the work hardening. b. Change in other properties After plastic deformation, the physical and chemical properties of metal materials will be changed because of the increase of the defects, such as lattice distortion, vacancies and dislocations, and so on. For example, plastic deformation can increase

230

Chapter 4 Deformation and recrystallization

the resistivity due to the increase of scattering of defects to electrons; moreover, the increased value is proportional to the deformation amount. The resistivity of pure copper with a cold drawing deformation rate of 82% will increase by 2% and that of H70 brass with the same deformation rate will increase by 20% while the resistivity of tungsten filament with a deformation rate of 99% will increases by 50%. Besides, after plastic deformation, temperature coefficient of resistance, magnetic conductivity, and thermal conductivity of the metals will all decrease while magnetic hysteresis loss and coercivity of ferromagnetic materials will increase. Since plastic deformation can result in the increase of the structural defects and free enthalpy, diffusion process will be accelerated, chemical activity and corrosive rate will also increase. 4. Deformation texture When a single crystal is plastically deformed, the crystal rotates and its slip direction tends to the tensile axis. Similarly, all of the individual grains will rotate when a polycrystalline metal is plastically deformed. However, the rotation of each grain is restricted by neighboring grains because of the constraints of coherency (coordination) at grain boundaries. Grains elongate and a preferred orientation of the grains is evident in fcc and bcc metals at strains exceeding 40%, and in hcp metals at strains exceeding 10%. These preferred orientations are called textures. Based on deformation methods, textures can be classified into two types. Drawing wire method will lead to the formation of fiber texture. In wires the grains tend to rotate so as to align in a specific crystallographic direction parallel to the wire axis. The bcc metals have a [110] fiber texture, and the fcc metals have a double fiber texture with some grains having the [111] direction and the others having [100] direction parallel to the axis. For rolling plate method, deformation causes grains to develop a preferred crystallographic direction parallel to the rolling direction and a preferred crystallographic direction parallel to the rolling plane. In some fcc metals the principal rolling texture is ð110Þ½
112, which means the {110} planes parallel to the rolling plane and direction locating on the rolling planes parallel to the rolling direction. Some typical metal fiber textures and plate textures are shown in Table 4.7. Table 4.7: Typical fiber and plate textures. Lattice structure

Metals or alloys

Fiber texture

Plate texture

Face-centered cubic

α-Fe, Mo, W Ferrite steel

{} +{} +{}

Body-centered cubic

Al, Cu, Au, Ni, Cu–Ni Cu + same with deformation texture f111Þ < 21
1 > ;f001g + f112g and < 1
10 > lies 15° to rolling direction {} after a two-stage control rolling and annealing; {} , {} after annealing at high temperature (>, °C)
> f111g < 211 same with deformation texture f001g, <
110 > lies 12° to rolling direction same with deformation texture

3. Annealing twin Annealing twin can be found in some fcc metals and alloys, such as Cu and its alloys, Ni and its alloys, and austenitic stainless steels, etc., after cold deformation and recrystallization annealing. Figure 4.64 indicates three typical twin types. Twin A lies on the corner of the grain boundaries; twin B grew across the whole grain; imperfect twin C grew but stopped inside the grain. The twin boundaries consisting of (111) planes in fcc metals are parallel to each other and coherent boundaries. But for the twins that stop their growth inside the grains, steps on their coherent twin boundaries are incoherent. The formation of annealing twins leads to a stacking fault on {111} plane, mak
BACBACBA C
AB C . . . as ing the stacking order change from ABCABC . . . to AB C
planes are coherent twin boundaries, and the crystal shown in Fig. 4.65. The two C
between the two C planes consists of an annealing twin band.

4.3 Recovery and recrystallization

(a)

259

(b)

Fig. 4.64: Annealing twin (a) illustration of three typical types of twins and (b) annealing twins in Cu.

Fig. 4.65: Stacking sequence of (111) plane in the formation of annealing twins in fcc metals.

About the forming mechanism of annealing twin, it is generally believed that annealing twin is formed in the process of grain growth. As shown in Fig. 4.66, when the grain grows via grain boundary migration, the stacking sequence of (111) planes in a grain boundary junction may accidentally become wrong, triggering the formation of a twin boundary in the grain boundary junction. The annealing twin will grow up by the movement of the high-angle grain boundary. In the process of growing up, if a new wrong stacking sequence of (111) planes accidentally happens again and the stacking sequence will return to the original stacking sequence, and the second coherent twin boundary will form and between these two twin boundaries a twin band will form. Similarly, the formation of the annealing twins must meet energy condition. Annealing twins tend to form in crystals with low-stacking fault energy.

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Chapter 4 Deformation and recrystallization

(a)

(b)

(c)

Fig. 4.66: Formation and growth of annealing twins in the grain boundary junction during grain growth.

4.4 Dynamic recovery and recrystallization during hot working In engineering, the processing above the recrystallization temperature is usually called “hot working,” while the processing below the recrystallization temperature and without heating is known as “cold working.” As for “warm working” between them, its deformation temperature is lower than the recrystallization temperature, but is higher than room temperature. For example, the recrystallization temperature of Sn is −3 °C. Therefore, the processing of Sn at room temperature is hot working. However, the recrystallization temperature of W is 1,200 °C; the drawing of tungsten wire below 1,000 °C belongs to warm working. Therefore, recrystallization temperature is the dividing line to distinguish cold working from hot working. During hot working, because the deformation temperature is above the recrystallization temperature, during deformation recovery and recrystallization process will occur, such recovery and recrystallization are called dynamic recovery and dynamic recrystallization, respectively, which distinguish from the recovery and recrystallization of the cold-deformed metals during annealing. Therefore, during the hot working, both the work hardening due to plastic deformation and the softening caused by dynamic recovery and recrystallization will exist at the same time. After hot working, the microstructures and the properties of the metals will depend on their offset.

4.4.1 Dynamic recovery and dynamic recrystallization Recovery and recrystallization during hot working are complex due to the existence of external forces. According to their characteristics, they can be divided into the following four forms: dynamic recovery, dynamic recrystallization, static recovery, and static recrystallization. The So-called dynamic recovery and dynamic recrystallization happen during hot working, their occurrences are under the action of both

4.4 Dynamic recovery and recrystallization during hot working

261

external force and temperature, which distinguish from recovery and recrystallization of cold deformed metal during annealing that is only under the effect of the temperatures. During cooling, after hot working and removal of external force, recovery and recrystallization occur since their driving force is derived from strain energy by hot working. Such recovery and recrystallization are called static recovery and static recrystallization since no external are involved, which is similar to recovery and recrystallization of cold deformed metal during annealing. Therefore, only dynamic recovery and dynamic recrystallization will be described. 1. Dynamic recovery Usually, extended dislocation in metals with high stacking fault energy (such as Al, αFe, Zr, Mo, and W) is very narrow in distance between partial dislocations. Both crossslip of screw dislocations and climb of edge dislocations can be carried out easily. These dislocations tend to leave from nodes and dislocation nets and then counteract with dislocations of opposite signs. Therefore, the dislocation density is very low so that the storage energy is not enough to trigger the dynamic recrystallization. Dynamic recovery is a dominant softening mechanism in hot working for this kind of metals. a. Stress–strain curves of dynamic recovery Figure. 4. 67 shows the true stress–true strain curve of the dynamic recovery. Dynamic recovery can be divided into three different stages: Stage I – microstrain stage. Stress increases rapidly, and work hardening appears with the total strain less than 1%. Stage II – homogeneous strain stage. The slope of the stress-strain gradually declines, and uniform plastic deformation, as well as dynamic recovery, occur at the same time in metals, the “work-hardening” effect is counteracted by the “softening” effect caused by the dynamic recovery.

Fig. 4.67: The true stress–true strain curve of the dynamic recovery.

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Chapter 4 Deformation and recrystallization

Stage III – a steady flow stage. The work hardening and dynamic recovery almost balance, and the work-hardening rate tends to be zero accompanying the formation of a steady state where stress does not increase with the strain. The steady flow stress is evidently affected by the temperature and strain rate. b. Mechanism of dynamic recovery With the increase of strain, the density of dislocations increases by proliferation (multiplication) accompanying the appearance of dislocation tangles as well as cellular structures. But high hot deformation temperature provides thermal activation energy for recovery process, during which the climbing of edge dislocations, cross-slip of screw dislocations, and the counteracting of dislocations with opposite signs will occur, and thus the dislocation density will decrease continuously. Also when proliferation rate and elimination rate of dislocations reach a balance, work hardening cannot appear, and the stress–strain curve turns to steady flow stress stage. c. The microstructure in dynamic recovery In the steady flow stress stage caused by dynamic recovery, with increasing strain, grains are stretched into fibrous shape along the deformation direction, but the interior of grains remains equiaxial subgrain structure without any strain, as shown in Fig. 4.68.

(a)

(b)

Fig. 4.68: Dynamic recovery subgrains after extrusion for aluminum at 400 °C: (a) optical microscopy photograph (polarized light 430×); (b) transmission electron microscopy photograph.

The completeness and size of subgrains and their misorientation during dynamic recovery mainly depend on the deformation temperature and deformation rate, and their relation can be described by

4.4 Dynamic recovery and recrystallization during hot working

d − 1 = a + lg z

263

(4:38)

where d is the average diameter of subgrains; a and b are constants; Z = ε_ expðQ=ðRT ÞÞ is strain rate modified with temperature, in which Q is activate activation energy for the process; R is gas constant. 2. Dynamic recrystallization For metals with low-stacking fault energy (such as Cu, Ni, γ-Fe, and stainless steel) due to the widely extended dislocation, it is difficult to carry out dynamic recovery by cross-slip and climb of edge dislocations. Therefore, subsequent dynamic recrystallization tends to happen. a. Stress–strain curve of dynamic recrystallization The true stress–true strain curve of dynamic recrystallization in metal is shown in Fig. 4.69. The dynamic recrystallization process contains three stages under high strain rate: I – Work-hardening stage with small strain. ε < εc (critical strain when dynamic recrystallization begins). Stress increases rapidly with the increase of strain in which no dynamic recrystallization occurs. II – Beginning of dynamic recrystallization. ε> εc, although at this point softening effect caused by dynamic recrystallization has emerged, the work hardening is still dominant. When σ = σmax, recrystallization is accelerated, and the stress will decline with the increase of strain, and softening is dominant. III – steady flow stage. ε> εs (strain corresponding to uniform deformation) work hardening and dynamic recrystallization softening keep dynamic balance.

True stress

σmax

High strain rate

σc Low strain rate

O

εc

εs True strain

Fig. 4.69: The true stress–true strain curve of the dynamic recrystallization.

Under the condition of low strain rate, the steady flow curve is fluctuated, which is mainly related to the alternative action of work hardening caused by deformation and softening caused by dynamic recrystallization.

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Chapter 4 Deformation and recrystallization

Note: When t(°C) = constant, with the increase of ε_ , the stress–strain curve of dynamic recrystallization will move upward from left to right, and the ε corresponding to σmax increases. When ε = constant, with the increase of t(°C), the stress–strain curve of dynamic recrystallization will move downward from right to left, and the ε corresponding to σmax decreases. b. Mechanism of dynamic recrystallization During hot working, dynamic recrystallization also includes nucleation and growth. The nucleation mode of dynamic recrystallization is related to ε_ and the resulting change of dislocation configuration. When ε_ is low, the dynamic recrystallization nucleates through the bowing out mechanism of the original grain boundary. When ε_ is higher, the dynamic recrystallization carries out through the gathering and growth of sub-grains. These two mechanisms have in detail been described in mechanisms of static recrystallization nucleation. c. Microstructure of dynamic recrystallization In the steady state of deformation for metals, grains are equiaxed and grain boundaries are serrated. The observation of transmission electron microscopy indicates that a grain also contains subgrains separated by dislocations (see Fig. 4.70). This is obviously different from grains with low dislocation density generated by static recrystallization during annealing. So the strength and hardness of dynamic recrystallized microstructure is higher than that of the static recrystallization microstructure for the same grain size.

Fig. 4.70: Substructures separated by dislocations in the dynamic recrystallized grains in the deformation of Ni at 934 °C (ε_ = 1.63 × 10 − 2 s − 1 , ε = 7.0).

4.4 Dynamic recovery and recrystallization during hot working

265

After dynamic recrystallization, the grain size is inversely proportional to flow stress (see Fig. 4.71). In addition, when the strain rate is lower and the deformation temperature is higher, the grains after dynamic recrystallization are larger, and more complete. Therefore, if we control strain rate, temperature, strain of deformation per pass, interval time, as well as cooling velocity, we can adjust the grain size and strength of the material in hot working.

Recrystallized grain size (μm)

1,000 500

Static recrystallization

100 50 Dynamic recrystallization 10 5 10

20

40

Flow sress (MPa)

60 80 100

Fig. 4.71: Relationships between recrystallized grain size and flow stress in Ni.

Moreover, solute atoms often act as barriers for dynamic recovery, which promotes dynamic recrystallization. Dispersively distributed precipitates formed during hot working can stabilize the subgrain and hinder the grain boundary migration and dynamic recrystallization, which is favorable to refining grains.

4.4.2 Effect of hot forming on microstructures and properties Except for the castings and sintered components, almost all the metal materials need hot working before they are transformed into some products. Either in the intermediate hot working or the final one, microstructures will evidently affect the properties of the final products. 1. Influence of hot working on mechanical properties at room temperature Hot working will not trigger obvious work hardening in the metal materials, but can eliminate some defects from casting process. For example, hot working can reduce or eliminate pores and voids from casting process and change the shape, size, and distribution of the inclusions; or partly eliminate some segregations; or turn coarse columnar grains and dendrites into fine and uniform equiaxed grains accompanying

266

Chapter 4 Deformation and recrystallization

with the improvement of density and mechanical properties of the metal material. Therefore, metal materials after hot working always have better mechanical properties than as-cast metals. During hot working (or forming) of the metals, the subgrains can be refined by controlling the process of dynamic recovery, and such substructures can be kept at room temperature in an appropriate cooling velocity, resulting in a higher strength than that of the dynamic recrystallized metals. The strengthening due to the formation of substructures is usually called “substructure strengthening,” which is an effective way to improve the strengths of the metals. For example, substructure strengthening of aluminum and its alloys, the thermomechanical treatment of steels and high-temperature alloys, and the rolling control of high-strength low-alloy steels are all related to the refining of the subgrains. At room temperature, the yield strength of a metal, σs, and the average diameter of the subgrains, d, has a similar Hall–Petch relationship: σs = σ0 + kd − ρ

(4:39)

where σ0 is yield strength of a single crystal without subgrain boundaries, k is a constant, the index, ρ is about 1–2 for most metals. 2. Microstructure characteristics of hot forming or working materials a. Processing streamlines During hot forming, the inclusions, segregation, second phase, grain boundaries, and phase boundaries will be elongated along the main deformation direction with the increase of the strain. So some processing streamlines (as shown in Fig. 4.72) or

Fig. 4.72: Processing streamlines in a forged steel.

4.4 Dynamic recovery and recrystallization during hot working

267

fibrous textures can be seen on a grinded and corroded surface. The existence of this fibrous texture will bring an anisotropy of the mechanical properties of metals, that is, the mechanical properties parallel to the streamline is always better than those perpendicular to the streamline, especially the ductility and toughness. Therefore, when hot forming some components, a correct distribution of the processing streamlines should be fully considered, namely, their streamlines should be parallel to the direction of the maximum tensile stress and perpendicular to the direction of the shear stress or impact stress. b. Lamellar structure After hot working of dual-phase and/or multiphase alloys, each phase is elongated, like some bands appearing alternatively along the deformation direction. This kind of structure is called “lamellar structure or band structure.” For example, after hot rolling, low-carbon steel presents a lamellar or band distribution of pearlite and ferrite along the rolling direction, forming “lamellar structure” (shown in Fig. 4.73). The dendrites and inclusions from casting during hot working often become lamellar or band structures due to their segregation and the elongation. Moreover, during cooling, the lamellar structures transform to different phases because of different compositions in segregation zone.

Fig. 4.73: Lamellar structure of hot rolled low-carbon steel plates (100×).

The existence of lamellar structures will also bring the anisotropy of the mechanical properties of metals. The plasticity and impact toughness in the vertical direction will decrease remarkably, especially for fibrous inclusions. The methods to prevent or eliminate lamellar structures are considered as follows: 1. Deforming in single field, rather than dual-phase field 2. Reducing the amount of inclusions 3. Using normalizing treatment or annealing before hot working

268

Chapter 4 Deformation and recrystallization

4.4.3 Creep In high-pressure steam boiler, steam turbine, chemical refining equipment, and aircraft engine, many metal components and refractory materials would be applied in a high-temperature environment for a long period. As for them, it is not enough only to take the mechanical properties under short-term static load at room temperature into account. Therefore, a conception of “creep” is adopted to study the effects of the high temperature and the duration of the load. In materials science, creep is referred to a very slow and continuous plastic deformation of a solid material under a stable stress (usually 0.3Tm. Hence, the study of creep is significant for high-temperature materials. 1. Creep curve The creep process can be schematically demonstrated by a creep curve. A typical creep curve is shown in Fig. 4.74. The slope of any point on the creep curve represents its creep rate. The whole creep process can be divided into three stages.

Fracture I

II

III

ε1 εc =

a O

dε t dt 1

Slope

dε dt

t1

Initial strain Time

I.

Fracture time

ε0

Strain

ε = ε0+ dε t dt

Fig. 4.74: Schematic of typical creep curve.

Transient or deceleration creep stage. Oa is the primary strain induced by the load, and then the creep begins from a point with a big creep rate, then slowing down with increasing time. This is a process of work hardening. II. Steady-state creep stage. The strain rate eventually reaches a minimum value and becomes nearly stable. This is due to the balance between work hardening and annealing (thermal softening). The commonly called “creep strain rate” re· fers to the rate εs at this stage. III. Accelerated creep stage. The strain rate increases exponentially because of necking phenomena. Fracture always occurs at this stage.

4.4 Dynamic recovery and recrystallization during hot working

269

The creep curves of different materials are different under different conditions. The secondary stage of a certain material will shorten until diminishing with the increase of temperature and stress. The service life at high temperature will be decreased greatly. Creep rate (ε_ ) is the most important parameter, as it determines the creep life and elongation of materials. Experiments show that it has an exponential relationship with the stress σ. Considering that creep is a thermal activation like the process of recrystallization, the relationship can be represented by   Q ε_ = Cσn exp − RT ε_

Q=

R ln ε_ 1

2

1 T2

−

1 T1

(4:40)

where Q is activation energy of creep; C is a constant of materials; ε_ 1 and ε_ 2 are the creep rates at T1 and T2, respectively; n is stress index, 1–2 for polymers and 3–7 for metals. Apparently, Q can be calculated by two creep rates at two temperatures for a given stress. As for most metals and ceramics, when T = 0.5Tm, creep activation energy is often of the order of the activation energy of self-diffusion. This means that creep can be regarded as diffusion of atoms under stress, determined by the process of diffusion. 2. Mechanisms of creep When the crystal deforms under room temperature or a temperature T 0.3

0.9

m

0.6 0.3 0 10–5

10–4

10–3

10–2 . ε (min−1) (b)

10–1

1

10

Fig. 4.76: The relationship among σ, m, and ε_ of Mg–Al alloy during deformation at 350 °C (grain size: 10.6 µm).

Strain rate sensitivity index m

1

10–1

10–2

10–3 1

10

103 102 Total elongation (%)

Fe-1.2Cr-1.2Mo-0.2V Fe-1.3Cr-1.2Mo Ni Mg-0.5Zr Ti-6A1-4V

104

Pb-Sn Zircalloy4 Ti-5A1-2.5Sn Pu

Fig. 4.77: The relationship between the elongation and strain rate sensitivity index of some metals.

4.4 Dynamic recovery and recrystallization during hot working

273

which can cause recrystallization of the eutectic structures; for eutectoid alloys, the fine grains can be obtained by hot deformation or quenching; for precipitated alloys, the fine grains can be obtained by precipitation of second phases during hot deformation or cooling deformation. The m value reflects the material resistance to necking during tensile process, which is an important parameter for assessing the potential superplasticity of this material. Generally, the elongation increases with the increase of m. In order to obtain a higher superplasticity, the required m value of the material is generally not less than 0.5. The larger the value of m, the more sensitive the stress is to the strain rate, and the more remarkable the superplasticity is. The value of m can be obtained from the following formula:   ∂lgσ Δlgσ lgσ2 − lgσ1 lgðσ2 =σ1 Þ ≈ = (4:43) = m= · ∂lg ε ε, T Δlg ε· lgε· 2 − lgε· 1 lgðε· 2 =ε· 1 Þ 2. The nature of superplastic About the nature of superplastic deformation, the popular viewpoint is that the superplasticity is caused by the grain boundary rotation and grain rotation. Figure 4.78 diagrammatizes well why the superplastic material can retain the equiaxed grains after a large strain. As shown in Fig. 4.78, when a tensile stress is applied along the longitudinal direction to the whole group of four hexagonal grains, a transverse compression stress in the grain will occur. Under the effect of those stresses, the grains change from the initial state (I) through the intermediate state, (II) to reach the final state, and (III) by grain boundary sliding and directional diffusion of atoms. The shape of grains at the initial state and the final state are identical in an ideal case but their locations change, leading to an elongation along the longitudinal direction accompanying with a deformation of the specimen.

(a)

(b)

Fig. 4.78: Mechanism of superplastic deformation of fine grains (the dash line stands for bulk diffusion direction): (a) two-dimensional grain transfer mechanism and (b) grain boundary sliding mechanism with directional spreading.

274

Chapter 4 Deformation and recrystallization

Many experimental results indicated that the variation of the structures in the superplasticity deformation has the following features: (1) There is no slip of dislocations and thus the dislocation density does not increase in grains. (2) The grains will grow up because the superplasticity occurs at a high temperature for a long time. (3) The grains always keep equiaxed shape although the deformation is heavy. (4) The original laminar dual phases will become uniform after superplastic deformation. (5) When cold deformation and recrystallization are adopted to prepare ultra-fine grains, if there exists a texture, the texture will disappear after superplastic deformation. In addition to fine-grain superplasticity mentioned earlier, phase transformation superplasticity should also be noted. For the material with solid-phase transformation, cyclic heating and cooling can be conducted at the starting temperature of phase transformation to induce the phase transformation repeatedly and violent result in the movement of the atoms without any external load thus may lead to superplasticity. 3. Applications of superplasticity The elongation of superplastic alloys is dramatically larger when T and ε_ have similar process parameters of polymers or glasses, so these alloys can be formed by the processing technologies in glass and plastics industry, such as blow forming, and in such a way any complex shapes can be formed by one step. Moreover, the products will not rebound since no elastic deformation is involved, resulting in precise and smooth products. For stamping of a metal sheet, only a female die can be used to form products by using air pressure or vacuum. For bulk metals, a closed mold can be used to form products just by a equipment with very small tonnage. In addition, the quality of mold material is not required to be of high grade due to a very low deformation rate. The superplasticity process also has some disadvantages, such as requiring a long time, repeated deformation iteration of heat treatment, and other complex processes. In addition, this technique requires isothermal forming and the deformation rate is very low, so the molds are easy to oxidize. Recently, the superplasticity has been obtained in many types of alloys and steels, such as Sn-based, Zn-based, Albased, Ti-based, Cu-based, Mg-based, and Ni based, and they have been applied in many industries.

4.5 Deformation characteristics of ceramics

275

4.5 Deformation characteristics of ceramics In comparison with metals and polymers, ceramics exhibit brittleness and hardness, which are closely related to their atomic bonding and crystal structure. The atomic bonds of ceramics are usually composed of ionic bonds or/and covalent bonds. In covalent-bonded ceramics, the atoms are bonded by means of sharing electron pairs, so the bonds of ceramics have a very strong direction and saturation. During plastic deformation, since the dislocation movement will destroy the strong bonding, the dislocation movement will encounter a quite large lattice resistance (P-N force), which is markedly different from the dislocation movement in the metal crystals. For ionic-bonded ceramic materials, positive and negative ions are arranged one by one. Under the external force, a dislocation is difficult to move an atomic spacing along the direction vertical or parallel of to the ionic bond direction because the huge electrostatic repulsion will be generated between the same charged ions, as shown in Fig. 4.79(a). If the dislocation moves along the direction 45° away from the ionic bond direction, the adjacent planes will be always attracted by electrostatic forces (as shown in Fig. 4.79) during the sliding process. Therefore, single crystal, such as NaCl and MgO, can have a large plastic deformation under a compressive stress at room temperature. However, polycrystal ceramics, even the fcc NaCl-type ceramics, are very hard to deform because they cannot meet the abovementioned conditions of the mutual coordination for adjacent grains, and such mutual coordination must have five independent slip systems at least. In NaCl single crystal, the slip systems are {110} and the number of independent slip system is six; but in polycrystal NaCl, there are only two independent slip systems (as shown in the Table 4.11). Therefore, the ion-bond polycrystals are very brittle, and the crack forms easily along grain boundary, leading to a final brittle fracture.

(a)

(b)

Fig. 4.79: The effect of bonds on movements of dislocations: (a) covalent bonds (b) ionic bonds.

The brittleness of ceramics is also related to their processing methods. Since there are many micropores in the sintered ceramics, in the cooling process these micropores will lead to microcracks easily, moreover, the oxidation erosion leads to cracks on the surface. Under external force, serious stress concentration may be generated at the tips of microcracks. The maximum stress at the tip of cracks is estimated by the theory of elastic mechanics to reach the theoretical fracture strength. Besides, there are few movable dislocations in the ceramic crystals, and the

276

Chapter 4 Deformation and recrystallization

Table 4.11: The independent slip systems of some kinds of materials. Temperature range

Crystalline compound

Low

MgO

High

Slip system

Number of independent systems

Mechanical properties

{110}



CaF, UO

{}



Diamond

{111}



AlO, BeO, Graphite


{0001}



Partially brittle

MgO

{110} {001} {111}



High-temperature ductility

CaF, UO

{001} {110} {111}



High-temperature ductility

TiO

{101} {110}



Partially brittle

MgAlO

{111} {110}



Partially brittle

dislocation movement is extremely difficult, so the ceramic materials may undergo brittle failure below yield stress. As well known, the tensile and compressive mechanical behaviors are obviously different. For example, the ultimate tensile stress of the sintered polycrystals Al2O3 is 280 MPa and the ultimate compressive stress is 2,100 MPa. The tensile strength of ceramics is usually determined by the maximum size of cracks in the crystal because when the cracks reach the critical size, these cracks will be instable and extend accompanying with rupture immediately. While for compression, the crack will be closed or extend slowly, and turns parallel to the compression axial direction, so the compressive strength is determined by the average size of the cracks. Some cases should be noted: (1) There are distinctions between amorphous ceramics and crystalline ceramics. For the amorphous ceramics, below the glass transition temperature Tg, they may generate elastic deformation. On the contrary, as the temperature is above Tg, their deformation is similar to that of the viscous flow of liquids, and the mechanical behavior can be described by eq. (4.6). (2) The deformation temperature also has a significant effect on the mechanical behavior of ceramic materials. As shown in Fig. 4.80, the stress–strain curves of polycrystalline MgO indicate that the material exhibits almost brittle failure at room temperature. However, with increasing the temperature, the brittleness of these

277

4.6 Deformation characteristics of polymers

materials decreases gradually, and they become easier for plastic deformation. Besides, at high temperature, the creep and viscous flow phenomenon are also observed. (3) In order to improve the brittleness of ceramics, to reduce the grain size to submicron or nanometer scale is usually adopted to prepare the high plastic and toughness ceramics. Also, zirconia toughening, transformation toughening, fiber reinforcement, and in situ particles reinforcement, are some effective methods to improve the brittleness of ceramic materials.

25 °C

500

Stress (MPa)

400

300 1,000 °C 200

100

1,250 °C

0

0.005 0.010 0.015 0.020 Strain (%)

Fig. 4.80: The stress–strain curves of MgO polycrystals.

4.6 Deformation characteristics of polymers Under external force, polymer materials will have elastic and plastic deformation, and the total strain is εt εt = εe + εp

(4:44)

where εe is the elastic strain, the εp is the plastic strain. In order to understand the deformation mechanisms, the best model materials are semicrystalline polymers. The mechanism of elastic deformation in semicrystalline polymers in response to tensile stresses is the elongation of the chain molecules in the direction of the applied stress, by the bending and stretching of the strong chain covalent bonds. In addition, there may be some slight displacements of adjacent molecules, which are resisted by relatively weak secondary or van der Waals bonds. Moreover, polymers are composed of both crystalline and amorphous regions. Therefore, the elastic modulus may be attributed to some combination of the moduli of crystalline and amorphous phases. In elastic deformation stage, the entire molecular chains exhibit the recoverable motion.

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Chapter 4 Deformation and recrystallization

As indicated in Section 4.1, most polymer materials exhibit the feature of high elasticity and viscoelasticity. The plastic deformation of polymers is not caused by slip but by viscous flow of molecular chains. Viscous flow occurs when the chains slide against each other. When the force is removed, these chains stay in the new position and thus plastic deformation occurrs. The extent of plastic deformation of polymers is related to the viscosity of the material. As shown in Fig. 4.81, the viscosity, η, can be described by η=

τ Δv=Δx

(4:45)

where τ is shear stress to make chains sliding against each other; Δv=Δx stands for sliding velocity gradient of the chains. If the viscosity is high, higher applied force is needed to produce a required displacement. Therefore, the deformation amount of a high viscosity polymer is not large.

104 1

η (Pa·s)

4 5

103

2

6 3

102

104

105

106 τ (Pa)

200 400 800 1,600 Load on plunger (×10–4 Pa)

Fig. 4.81: The relationship of apparent viscosity and shear stress: 1, polycarbonate(280 °C); 2, polyethylene (200 °C); 3, polyoxymethylene (200 °C); 4, polymethylmethacrylate (200 °C); 5, acetate fiber (180 °C); 6, nylon (230 °C).

It must be pointed out that, in comparison with metallic materials, the mechanical properties of polymers are more dependent on temperature and time. Besides, the deformation characteristics of polymers are greatly different when they have different degrees of crystallinity and cross-linking. For example, amorphous linear polymers can only generate elastic deformation below the temperature of Tg. and viscous flow is generated above Tg, like that in glasses while the deformation of crystalline polymers is like those of metallic materials. Tension test is a common method for studying the mechanical behaviors of polymers. Using stress–strain curves, some parameters, such as elastic modulus, yield strength, fracture strength, ultimate elongation, and so on, can be obtained to evaluate the polymers. Different polymers have different stress–strain curve, as shown in the Fig. 4.82. Line 1 stands for the stress–strain curve of brittle polymers, showing a

4.6 Deformation characteristics of polymers

279

60

Strain (MPa)

50

1

40 2 30 20 3 10

0

1

2

3

4 5 Strain (%)

6

7

8

Fig. 4.82: The stress–strain curves of polymers: 1, brittle polymers; 2, glassy polymers; 3, elastomers.

brittle fracture before yield point. In this situation, these materials only deform slightly before they fracture. Line 2 represents the stress–strain curves of glassy polymers. They at first deform elastically, then deform plastically over the yield point. If the external force is removed, the polymer will not recover to its original length but keep the deformed shape. Line 3 shows the stress–strain curve of elastomers. They exhibit low yield strength and high elongation. Many polymers show instability at the stage of uniform plastic deformation. The local strain increases faster than the whole strain, leading to uneven deformation and various instability of plasticity. The most common and important phenomenon is a neck formed in tensile process. Figure 4.83 shows a typical stress–strain curve of semicrystalline polymers under uniaxial tensile. The entire curve can be divided into three stages. The first stage is elastic deformation, in which the stress is firstly increased linearly with strain, here the elongation should be 0–20%, then a yield point is emerging. The strain of polymers is usually larger than that of metals, and polymers show a softening phenomenon when passing the yield point.

Fig. 4.83: The stress–strain curve of semicrystalline polymer and the outline of deformation sample.

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Chapter 4 Deformation and recrystallization

Then the material enters the second stage, in which the cross-section of the sample suddenly becomes uneven, emerging one or more necks, which is different from metals and alloys whose necks only emerge after uniform deformation. First, the cross-sections of necking location and nonnecking location remain constant, respectively; then, the necking region is extended continuously, and the nonnecking region is shortened gradually; finally, the entire deformation region of the sample becomes quite narrow. In the third stage, the stress increases gradually with the increase of strain until the sample fractures. Under an applied tensile load，the plastic deformation for semicrystalline polymers can be described by the interactions between crystalline chain-folded lamellae and amorphous tie chains. This process occurs in several stages, as shown in Fig. 4.84.

Fig. 4.84: Deformation process of a semicrystalline polymer. (a) the Initial structure: two adjacent crystalline lamellar and interlamellar amorphous chain; (b) the first stage: elongation of amorphous tie chains; (c) the second stage: tilting of lamellar chains; (d) the third stage: separation of crystalline segments; (e) the final stage: orientation of crystalline segments and tie chains along the tensile axis.

4.6 Deformation characteristics of polymers

281

Before deformation，two adjacent chain-folded lamellae and the interlamellar amorphous tie chain are shown in Fig. 4.84(a). During the first stage of deformation (Fig. 4.84(b)) the chains in the amorphous regions slip past each other and align in the loading direction, which leads to the elongation of amorphous tie chains. The deformation in the second stage occurs by tilting of lamellar chain (Fig. 4.84(c)). Then, in the crystalline segments, the lamellae separate from each other，but the segments remain attached to one another by tie chains (Fig. 4.84(d)). In the final stage (Fig. 4.84(e)), crystalline segments and tie chains become oriented in the direction of the tensile axis. As a result, a highly oriented structure of semicrystalline polymers is produced.

Chapter 5 Diffusion in solids The transport of matters can be carried out by both convection and diffusion in gas and liquid, while diffusion is the only mass transport mechanism in solids. Diffusion refers to the migration of thermally activated atoms or molecules, and is deeply involved in many important processes such as solidification and homogenization annealing of cast metals, recovery and recrystallization of coldworked metals, sintering of ceramic or metal powders, solid–solid phase transformations, high-temperature creep, and various surface treatments. To understand and subsequently control these processes, a comprehensive understanding of the principles governing diffusion is required. In general, there are two approaches to study diffusion: (1) phenomenological theory – to describe the rate and amount of mass transport using experimentally determined quantities; (2) atomic theory – how atoms migrate during diffusion. In this chapter, we mainly discuss the diffusion laws, the factors that affect diffusion, and the diffusion mechanisms of solid materials. Solid materials include metals, ceramics, and polymers, where the interatomic bonds are dominated by metallic bonds for metals and ionic bonds for ceramics. In comparison, polymeric materials are characterized by a chained structure with covalent bonds and/or hydrogen bonds. Consequently, the diffusion mechanisms in these three classes of solids are different, which will be treated separately in this chapter.

5.1 Phenomenological theory 5.1.1 Fick’s first law When there is concentration of nonuniformity in solids, atoms would diffuse from regions of high concentration to regions of low concentration. Adolf Fick studied the rate of atomic transport and in year 1855, and he pointed out that the diffusion flux of atoms is proportional to the mass concentration gradient: J= −D

dρ dx

(5:1)

Equation (5.1) is called Fick’s first law or the first law of diffusion. J is called the diffusion flux, and is defined as the mass of the diffusing species flowing across unit area cross section perpendicular to the x-axis (the diffusion direction) per unit time, and has a unit of kg/(m2·s). D is the diffusion coefficient and has a unit of m2/s,

https://doi.org/10.1515/9783110495348-005

5.1 Phenomenological theory

283

and ρ is the mass concentration (kg/m3). The minus sign in the equation indicates that the diffusion direction is opposite to the direction of concentration gradient (dρ=dx), that is, the atoms would migrate from regions of high concentration to those of low concentration. In 1953, R. P. Smith reported the determination of carbon diffusion coefficient in γ-Fe，where he used a pure Fe shell with a radius r and length l, and kept the inside of the shell under a carburization atmosphere and the outside of the shell a decarburization atmosphere. The entire system was kept at a constant temperature of 1,000 °C. After a sufficiently long time, the carbon concentration in the shell no longer varies with time and the steady-state condition was fulfilled. Then, the amount of carbon atoms diffusing across the shell per unit time, q=t, becomes a constant, and the diffusion flux can be written as J=

q q = At 2πrlt

−D

dρ q = dr 2πrlt

From Fick’s first law, we have

so q = − Dð2πltÞ

dρ d ln r

where q, l, t can be measured experimentally. Thus, once the concentration profile in the shell is determined as a function of position (r), the diffusion coefficient (D) can then be obtained from ρ versus lnr correlation. In particular, if D is independent of r, ρ and lnr should be linearly related. However, as shown in Fig. 5.1, it was found in the experiment that the ρ versus lnr curve was highly nonlinear, suggesting that D itself was a function of ρ. Specifically, in the high concentration regime (high ρ), dρ=d ln r was low and thus D was accordingly high. Conversely, in the low concentration regime (dρ=d ln r is high), D was low as well. For example, when the mass fraction ρ = 0.15% at 1,000 °C, D = 2.5 × 10−11 m2/s; when the mass fraction ρ = 1.4%, D = 7.7 × 10−11 m2/s.

5.1.2 Fick’s second law Using Fick’s first law together with the mass conservation condition, we can derive the Fick’s second law to solve time-dependent diffusion problems. In particular, as demonstrated in Fig. 5.2, if we take a volume element along the direction of the diffusion flux

284

Chapter 5 Diffusion in solids

1.6

0.12

1.4 1.2

0.08

1.0 0.8

0.06

ω(C) (%)

ρ(C)/kg·L–1 (1,000 °C)

0.10

0.6

0.04

0.4 0.02 0 0.24

0.2 0.26

0.28 0.30 0.32 –lnr(1,000 °C)

0.34

0 0.36

Fig. 5.1: The relationship between lnr and ρ at 1,000 °C.

(x-axis), with a length dx and a cross-sectional area A, and assume the diffusing fluxes flowing in and out of the element as J1 and J2, respectively, then from mass conservation condition, we will have: [Mass flowing into the element − Mass flowing out of the element = Mass accumulated inside the element], or [Rate of mass flowing in − Rate of mass flowing out = Rate of mass accumulation]. Since the rate of mass flowing into and out of the volume element are J1A and J2 A = J1 A + ð∂ðJAÞ=∂xÞdx, respectively, the accumulation rate is then − ð∂J =∂xÞAdx. Such mass accumulation rate can also be calculated by the time-dependent mass concentration (ρ) as ð∂ρ=∂tÞAdx. Therefore, ∂ρ ∂J ∂ρ ∂J Adx = − Adx and then =− ∂t ∂x ∂t ∂x By applying Fick’s first law into the above-mentioned equation, we obtain   ∂ρ ∂ ∂ρ = D ∂t ∂x ∂x

(5:2)

Equation (5.2) is called Fick’s second law or the second law of diffusion. Moreover, if D is independent of ρ, eq. (5.2) can be further simplified to ∂ρ ∂2 ρ =D 2 ∂t ∂x

(5:3)

5.1 Phenomenological theory

285

More generally, if three-dimensional diffusion is taken into consideration, and if we assume an isotropic diffusion coefficient (such as the case for cubic crystals), Fick’s second law can be written as  2  ∂ρ ∂ ρ ∂2 ρ ∂2 ρ + + =D ∂t ∂x2 ∂y2 ∂z2

(5:4)

When diffusion is caused by concentration gradient, it can be regarded as “chemical diffusion”; on the other hand, when diffusion is driven by thermal vibration instead of concentration gradient, it is called “self-diffusion,” and its diffusion coefficient, Ds, can be derived from eq. (5.1): −J Ds =  lim ð ∂ρ Þ ∂ρ ∂x

!0

(5:5)

∂x

In other words, in an alloy system, the self-diffusion coefficient, Ds, is the diffusion coefficient when the mass concentration gradient of the corresponding constituent is approaching zero.

Fig. 5.2: The concentration changing rate of the diffusing species in a volume element: (a) transient variation of concentration and distance; (b) transient relationship between diffusing fluxes and distance; and (c) the change of mass diffusion flux, J1, through a volume element.

5.1.3 Solutions to diffusion equations For actual nonsteady-state diffusion problems, one has to solve the Fick’s second law with particular initial and boundary conditions. Obviously, different initial and

286

Chapter 5 Diffusion in solids

boundary conditions would lead to different solutions to the diffusion equations. In this section, several simple but useful solutions will be discussed. 1. Diffusion couple with fixed concentrations at two ends If rod A with a mass concentration ρ2 is welded with rod B with a mass concentration of ρ1, where the weld plane is perpendicular to the x-axis, the mass concentration close to the weld plane will change over time if the diffusion couple is annealed at elevated temperature, as demonstrated in Fig. 5.3.

Weld plane ρ2

A

t0

Diffusion direction, ρ1 –x

ρs =

ρ1 + ρ2 2

ρ1

ρ2

ρ

t1 t2

B

+x

0

0

x

Fig. 5.3: The composition versus distance curves of the diffusion couple.

Assuming the rods are sufficiently long so that the concentrations at the two ends are not affected by diffusion and are fixed at any time, and such a condition is also called “infinite solid,” then the initial and boundary conditions would be initial conditions:  x > 0, ρ = ρ1 t=0 x < 0, ρ = ρ2 and boundary conditions

 t≥0

x = ∞, ρ = ρ1 x = − ∞, ρ = ρ2

There are quite a few methods to solve partial differential equations, and here we will use the intermediate variable substitution method. In particular, we define an

 pffiffiffiffiffi intermediate variable β = x 2 Dt , then   ∂ρ dρ ∂β β dρ ∂2 ρ ∂2 ρ ∂β 2 ∂2 ρ 1 d2 ρ 1 = = = =− , and = ∂t dβ ∂t 2t dβ ∂x2 ∂β2 ∂x ∂β2 4Dt dβ2 4Dt

287

5.1 Phenomenological theory

Therefore, the Fick’s second law (eq. (5.3)) becomes −

β dρ 1 d2 ρ =D 2t dβ 4Dt dβ2

so

d2 ρ dρ + 2β =0 dβ dβ2

and then dρ dβ = A1 expð − β2 Þ. By integration, we will finally have the general solution as ðβ ρ = A1 expð − β2 Þdβ + A2

(5:6)

0

where A1 and A2 are constants to be determined. According to the definition of error function, ðβ 2 erfðβÞ = pffiffiffi expð − β2 Þdβ π 0

It can be shown that erf(∞) = 1, and erf(–β) = –erf(β). The values of error functions with different β values are tabulated in Table 5.1. Table 5.1: β values and the values of error functions at different β values (β is 0–2.7). β . . . . . . . . . . . . . . . .



















. . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . .  . . . . . . .  . . .  .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  .  . . .

β . erf(β) .

. .



288

Chapter 5 Diffusion in solids

For the error function, it can also be shown that ∞ ð

−ð∞ pffiffiffi pffiffiffi π π expð− β Þdβ = expð− β2 Þdβ = − and 2 2 2

0

0

Applying the earlier equations into eq. (5.6), together with the initial and boundary conditions, we will then obtain the expressions for constants A1 and A2: A1 =

ρ1 − ρ2 2 ρ +ρ pffiffiffi , A2 = 1 2 2 2 π

Therefore, the mass concentration ρ varies with position x and time t as follows:   ðβ ρ1 + ρ2 ρ1 − ρ2 2 ρ1 + ρ2 ρ1 − ρ2 x 2 pffiffiffi expð − β Þdβ = + + erf pffiffiffiffiffi . ρðx, tÞ = 2 2 2 2 π 2 Dt

(5:7)

0

When x = 0 (at the interface), erf(0) = 0, and the mass concentration at the interface, ρs, satisfies ρs = ðρ1 + ρ2 Þ=2, and does not change over time. This is a natural result since we assume that the diffusion coefficient is independent of concentration, and thus the concentration reduction on the left of the interface is identical to the concentration increase on the right of the interface. Furthermore, if the initial concentration on the right side of the diffusion couple is zero (ρ1 = 0), eq. (5.7) is simplified to    ρ x (5:8) ρðx, tÞ = 2 1 − erf pffiffiffiffiffi 2 2 Dt and the concentration at the interface is then ρ2 =2. 2. Diffusion cell with fixed concentration at one end High-temperature austenite carburization is an important approach to improve the surface properties and lower the processing cost of low-carbon steels. In such cases, before the carburization treatment, component with initial carbon concentration ρ0 can be considered as a semi-infinite diffusion cell, or a semi-infinite solid, where far from the carbon source the carbon concentration is not affected by the carburization and keeps at ρ0. Subsequently, we will have the following initial and boundary conditions: initial conditions:

t = 0 x ≥ 0 ρ = ρ0 , and 

boundary conditions: t ≥ 0

x = 0, ρ = ρs x = ∞, ρ = ρ0

In other words, it is assumed that the surface concentration of the diffusion cell becomes ρs as soon as carburization takes place. From eq. (5.6), we have

5.1 Phenomenological theory

 x ρðx, tÞ = ρs − ðρs − ρ0 Þerf pffiffiffiffiffi 2 Dt

289



(5:9)

If the component to be carburized is initially pure Fe (ρ0 = 0), eq. (5.9) will be simplified to    x (5:10) ρðx, tÞ = ρs 1 − erf pffiffiffiffiffi 2 Dt In actual problems, it is usually required to calculate the diffusion time needed to achieve a certain thickness of carbon-rich surface layer. This can be done using eq. (5.9) and is illustrated in the following example. Example: A piece of low-carbon steel with a initial carbon weight fraction 0.1% is put in a carburization atmosphere with a carbon fraction of 1.2% at 920 °C. To achieve a carbon fraction of 0.45% at 0.002 m beneath the sample surface, how much carburization time is needed? Solution: Since the carbon diffusion coefficient in γ-Fe at 920 °C is D = 2 × 10−11 m/s2, from eq. (5.9), we have   ρs − ρðx, tÞ x = erf pffiffiffiffiffi ρs − ρ0 2 Dt To convert mass concentration to mass fraction, we should divide both the denominator and numerator of the left-hand side of the above-mentioned equation by the density of the low-carbon steel:   ws − wðx, tÞ x = erf pffiffiffiffiffi ws − w0 2 Dt Putting in ws = 1.2%, w(0.002 m, t) = 0.45%, w0 = 0.1%, and D = 2 × 10−11 m/s2, we have   224 erf pffiffi ≈ 0.68 t From the error function table (Table 5.1), we have 224 pffiffi ≈ 0.71, t ≈ 27.6 h t Therefore, when the mass concentration at a particular carburization depth x is fixed, the error   pffiffiffiffiffi function erf x 2 Dt should be a fixed value, indicating that the carburization depth x and diffusion time t satisfy the following relation: pffiffiffiffiffi x = A Dt or x 2 = BDt (5:11) where A and B are constants. Hence, if we want to double the carburization depth, the diffusion time should be quadrupled.

3. Thin-film decay source If we deposit a thin layer of metal A at the end of a long rod made of metal B, and weld two such rods together, a thin-film source of metal A is then formed between the two

290

Chapter 5 Diffusion in solids

metal B rods. Then if the diffusion couple is annealed at an elevated temperature, the concentration of metal A would evolve in the B rods as a function of time. If the rod axis is defined as x-axis, and the metal A source is located at the origin of x-axis, the initial concentration of metal A has a δ function format: ρðx = 0, t = 0Þ = ρ, ρðx≠0, t = 0Þ = 0. When the diffusion coefficient of metal A is independent of its concentration, the solution to the thin-film decay source problem can be derived from Fick’s second law (eq. (5.3)):   k x2 (5:12) ρðx, tÞ = pffiffi exp − 4Dt t where k is a constant to be determined. From eq. (5.12), it can be easily concluded that the solute (metal A in this case) concentration is symmetric relative to the origin. Assume that the diffusing species has a mass per unit area of M, then ∞ ð

ðt ∞ ð ρðx, tÞdxdt =

M= 0 −∞

ρðx, 0Þdx

(5:13)

−∞

 We define x2 4DtÞ = β2 , then pffiffiffiffiffi dx = 2 Dtdβ

(5:14)

Putting equations (5.12) and (5.14) into (5.13), we will have ∞ pffiffiffiffi ð pffiffiffiffiffiffi expð − β2 Þdβ = 2k πD M = 2k D −∞

From Gaussian error functions we know that ∞ ð

−ð∞ pffiffiffi pffiffiffi π π expð − β Þdβ = expð − β2 Þdβ = − and 2 2 2

0

0

so the constant to be determined is M k = pffiffiffiffiffiffi 2 πD

(5:15)

The correlation between Gaussian solutions and Gaussian error functions can be obtained. Specifically, inserting eqs. (5.15) into (5.12), we will then have the concentration of thin-film decay source as a function of position and diffusion time:   M x2 (5:16) ρðx, tÞ = pffiffiffiffiffiffiffiffi exp − 4Dt 2 πDt

291

5.1 Phenomenological theory

Figure 5.4 demonstrates the solute concentration profile at different Dt values  1 1 

 pffiffiffiffiffiffiffiffi = 16, 4 , 1 calculated from eq. (5.16), where M 2 πDt is the modulation amplitude of the concentration profile, which decays over diffusion time. When t = 0, the width of the profile is zero while its modulation amplitude is infinity. In other words, for a thin-film diffusion source with a finite initial width (w), the Gaussian solution discussed earlier is only an approximate solution. The longer the diffusion time and the narrower the initial distribution of the diffusion source, the Gaussian solution would become more accurate. A validity check for the accuracy of the Gaussian solution is t>

w2 2D

(5:17)

1.25

ρ·M–1 (m–3)

1.00 1/16 0.75 0.50

1/4

0.25 0 –5

1 –4

–3

–2

–1

0

1

2

3

4

5

Fig. 5.4: The solute concentration profile of a thin-film decay source as a function of time after diffusion (the numbers represent different Dt values).

If material is deposited onto one end of the rod made from metal B to achieve a unit area mass of M, after diffusion annealing, the mass concentration of metal A is twice the value obtained for the diffusion couple discussed earlier, that is,   M x2 (5:18) ρðx, tÞ = pffiffiffiffiffiffiffiffi exp − 4Dt πDt This is because material A would diffuse in only one direction here instead of two. From the equipartition theorem of statistical physics, the average diffusion length, d, of atoms at a particular time t can be calculated: ∞ ð

x ρðx, tÞdx 2

d = 2

=

−∞ ∞ ð

M pffiffiffiffiffiffiffiffi 2 πDt

ρðx, tÞdx −∞

∞ ð pffiffiffi  2 Since e − αx x2 dx = π 4 α − ð3=2Þ , therefore, 0

∞ ð

  x2 dx x exp − 4Dt 2

−∞

M

.

292

Chapter 5 Diffusion in solids

pffiffiffi   3 pffiffiffiffiffiffiffi 1 π 1 −2 d = pffiffiffiffiffiffiffiffi = 2Dt and d = 2Dt 2 πDt 2 4Dt 2

Consequently, the diffusion length is also proportional to the square root of diffusion time in this Gaussian solution. The above-mentioned thin-film decay source is often adopted to measure the self-diffusion coefficient of metals using tracer atoms. Because there is no concentration gradient in a homogeneous pure metal, a typical way to study the atomic migration in pure metals is to deposit a thin layer of radioactive isotope A* as the tracer onto the surface of base metal A. After diffusion annealing, the concentration profile of the tracer isotope A* is measured. As A* has same chemical properties as A, the diffusion coefficient measured in the absence of concentration gradient is the self-diffusion coefficient of A. 4. Homogenization of compositional segregation Solid solution alloys would have the dendritic segregation in the grains when solidified under nonequilibrium conditions, which is usually detrimental to the properties of the alloys. Conventionally, homogenization diffusion annealing is used to reduce such an effect. Again, here we will use variable partition method to solve the Fick’s second law to describe the homogenization problems. Suppose the solute concentration across a second-order dendrite has a sinusoidal dependence on positions along the x-axis (Fig. 5.5(a)): ρðxÞ = ρ0 + A0 sin

(a)

πx λ

(5:19)

(b)

Fig. 5.5: The diagrams of a second-order dendrite and the solute concentration: (a) the diagram of a second-order dendrite; (b) the solute concentration across a second-order dendrite (the dendrite segregation is treated by the sinusoidal wave).

5.1 Phenomenological theory

293

where ρ0 is the average mass concentration, A0 is the initial modulation amplitude of the concentration profile, that is, A0 = ρmax – ρ0 (Fig. 5.5(b)), and λ is the spacing between the maximum concentration ρmax and minimum concentration ρmin, that is, half the distance between the axis of the second-order dendrites. Since the solute atoms would diffuse from regions of high concentration to regions of low concentration, the concentration would eventually approach the average concentration ρ0 over time. Therefore, it is considered that upon homogenization diffusion annealing, the wavelength, λ, does not change, while the amplitude of the sinusoidal modulation is gradually reduced. We will then have the following boundary conditions (Fig. 5.5(b)): ρðx = 0, tÞ = ρ0   dρ λ x= ,t =0 dx 2

(5:20) (5:21)

Equation (5.20) indicates that at x = 0 the concentration (ρ0) is unchanged, and eq. (5.21) indicates that the peak of the modulation is kept at x = ðλ=2Þ at all times. Subsequently, we set ρðx, tÞ = XðxÞTðtÞ and use the variable partitioning method to solve Fick’s second law: ∂ρ ∂2 ρ =D 2 ∂t ∂x and we can get a couple of ordinary differential equations: d2 X π 2 + X=0 dx2 λ π 2 dT T=0 +D dt λ The general solutions are XðxÞ = A cos

πx πx + B sin λ λ

  Dπ2 t TðtÞ = exp − 2 λ Therefore,   πx πx

Dπ2 t ρðx, tÞ = A cos + B sin exp − 2 λ λ λ

(5:22)

where A and B are constants to be determined. As ρðx = 0, t = 0Þ = ρ0 , eq. (5.22) is then simplified to

294

Chapter 5 Diffusion in solids

  πx πx

Dπ2 t ρðx, tÞ = ρ0 + A cos + B sin exp − 2 λ λ λ

(5:23)

Substituting eqs. (5.21) into (5.23), we have A = 0, and   πx Dπ2 t ρðx, tÞ − ρ0 = B sin exp − 2 λ λ When t = 0, ρðx, tÞ − ρ0 = B sinðπx=λÞ Compared with the initial condition, eq. (5.19), it is shown that B = A0, and the final solution is   πx Dπ2 t (5:24) ρðx, tÞ − ρ0 = A0 sin exp − 2 λ λ Since only the concentration at x = λ=2 is considered during homogenization and λ sin πx λ = 1 at x = 2     λ πx Dπ2 t ρ , t − ρ0 = A0 sin exp − 2 2 λ λ Because A0 = ρmax – ρ0, we have     ρ λ2 , t − ρ0 Dπ2 t = exp − 2 ρmax − ρ0 λ

(5:25)

where the right-hand side is called the “decay function.” In particular, if we want the modulation amplitude to be reduced to 1%, that is,   ρ λ2 , t − ρ0 1 = ρmax − ρ0 100 we then have t = 0.467

λ2 D

(5:26)

Equation (5.26) indicates that at a certain temperature, that is, D is a constant, the smaller the spacing between the dendrites, the less time is needed for homogenization. Therefore, to reduce the homogenization diffusion time, rapid solidification can be used as it can suppress the dendritic growth. Hot forging or hot rolling can also be applied since they may break the dendrites. On the other hand, if λ is a constant, one may reduce the homogenization diffusion time by raising the diffusion temperature (however, recall that it should always be lower than the solidus line), making D higher. In any case, complete elimination of dendrites is impossible since it requires t ! ∞.

5.1 Phenomenological theory

295

5.1.4 Diffusion in substitutional solid solutions In a C–Fe solid solution, C atoms occupy the interstitial sites in the Fe lattice because of their smaller size and easier migration. Therefore, when considering diffusion in C–Fe alloys, the diffusion of Fe atoms is negligible when compared with that of C atoms. However, for substitutional solid solutions, as the atomic sizes of the solute and solvent atoms are comparable, diffusion has to occur by the exchange of neighboring atoms. Moreover, since the mobility of the solute and solvent atoms are also similar in substitutional solid solutions, the different diffusion rates of the constituents need to be taken into account. This was first verified by E. O. Kirkendall et al. in the year 1947, where they electroplated a layer of pure Cu onto a 30 wt.% Zn brass block, and placed two rows of Mo wires at the interface between the pure Cu and the brass. The schematic experimental setup is illustrated in Fig. 5.6. After diffusion annealing at 785 °C for 56 days, the spacing between the two rows of Mo wires was found to reduce by 0.25 mm, and tiny holes were also observed in the brass. If Cu and Zn have an equal diffusion coefficient, then during diffusion there would be an equal number of Zn atoms and Cu atoms exchanging with each other across the interface. Considering Zn atoms are larger in size than Cu atoms, the outward movement of Zn atoms would make the Mo wires (the “marker plane”) move to the brass side. Unfortunately, the as-calculated movement of the Mo markers only accounts for about 1/10 of the observed value. In other words, the difference in the atomic sizes is not the cause for the movement of Mo wires. Instead, the only possible mechanism is that the Zn atoms have larger diffusion flux than that of Cu atoms during thermal annealing as a result of their different diffusion coefficients. Such a movement of Mo wires due to the unequal diffusion fluxes is called “Kirkendall effect,” which not only exists in the Cu–Zn system, but was also found later in various substitutional-type diffusion couples such as Ag–Au, Ag–Cu, Au–Ni, Cu–Al, Cu–Sn, and Ti–Mo. For example, as shown in Fig. 5.7, the Kirkendall effect was observed in 12 wt. % Al–Cu and pure Cu diffusion couple after diffusion annealing for 1 h, where the black line in the middle is the original welding plane and the interface on the left is the marker plane (the sample surface was etched by 8 wt. % CuCl2–ammonia).

Fig. 5.6: Kirkendall experiment.

296

Chapter 5 Diffusion in solids

Fig. 5.7: About 12 wt. % Al–Cu and pure Cu diffusion couple (500×).

L. S. Darken developed the phenomenological explanation for the Kirkendall effect in 1948, where he considered the migration of the marker plane as the movement of fluids, that is, the marker plane “flowed” across the reference plane (the welding plane) as a whole. The so-called Darken’s equations can be derived as follows. Assume we have a diffusion couple made of pure species A and B [Fig. 5.8(a)], and their intrinsic diffusion coefficients are DA and DB, respectively. Then the diffusion fluxes relative to the lattice are JA = − DA

∂ρA ∂x

(5:27)

JB = − DB

∂ρB ∂x

(5:28)

where JA and JB are the fluxes of A and B atoms across a given lattice plane, respectively. If the diffusion couple is annealed at a sufficiently high temperature, a concentration profile would develop as demonstrated in Fig. 5.8(b). If we make the simple assumption that the total number of atoms per unit volume keeps a constant, ρ0, regardless of interdiffusion and the associated change in the concentration profile, then we have ρ0 = ρA + ρB and ∂ρA =∂x = − ∂ρB =∂x. Therefore, at a given position, the concentration gradients that drive the diffusion of A and B atoms are equal but have opposite signs, and the fluxes of A and B relative to the lattice can be derived as. JA = − DA

JB = DB

∂ρA ∂x

∂ρA ∂x

(5:29)

297

5.1 Phenomenological theory

Fig. 5.8: Interdiffusion and vacancy flow in a diffusion couple A and B. (a) Comparison of concentration profiles after interdiffusion of parts A and B; (b) the corresponding fluxes of atoms and vacancies as a function of position x; (c) the rate at which the vacancy concentration would increase or decrease if vacancies were not created or destroyed by dislocation climb.

These fluxes are schematically manifested in Fig. 5.8 for the case DA > DB, that is, jJA j > jJB j When atoms migrate by the vacancy process, the jump of an atom into a vacant site can be considered equivalently as the jumping of the vacancy onto the atom, as illustrated in Fig. 5.9. In other words, if there is a net flux of atoms in one direction, there would be an equal flux of vacancies in the opposite direction. Therefore, in Fig. 5.8(a) there is a flux of vacancies –JA due to the migration of A atoms plus a flux of vacancies –JB due to the diffusion of B atoms, that is, JV = − JA − JB . This is indicated in vector notations in Fig. 5.8(a). In terms of DA and DB (eq. (5.29)), then we have JV = ðDA − DB Þ

∂ρA ∂x

(5:30)

This results in a variation in JV across the diffusion couple as illustrated in Fig. 5.8(b). To keep the vacancy concentration everywhere near the equilibrium, additional vacancies must be created on the B-rich side and destroyed on the A-rich side. The rate at which vacancies are created or destroyed at any position is given by ∂ρV =∂t = − ∂JV =∂x (Fick’s second law) and this would vary across the diffusion couple as shown in Fig. 5.8(c).

298

Chapter 5 Diffusion in solids

Fig. 5.9: The jumps of atoms in one direction can be regarded as the jumping of vacancies in the opposite direction.

The velocity at which any given lattice plane moves, v, can be related to the flux of vacancies crossing it. If the plane has a cross-sectional area A, during a small time interval δt, the plane would then sweep out a volume of Avδt containing Avδtρ0 atoms. This number of atoms is then removed by the total number of vacancies crossing the plane during the same time interval, that is, JVAδt, giving Jv = ρ0 v. Subsequently, the velocity of the moving lattice planes would vary across the couple in the same way as JV (see Fig. 5.8(b)). Substituting eq. (5.3) gives v = ðDA − DB Þ

∂XA ∂x

(5:31)

where the mole fraction of A, XA = ρA/ρ0 and XB = ρB/ρ0. The total flux of A atoms across a stationary plane relative to the specimen (or the laboratory frame) is then the sum of two contributions: (i) a diffusive flux JA = − DA ∂ρA =∂x due to diffusion relative to the lattice; and (ii) a flux vρA due to the velocity of the lattice in which diffusion occurs. Therefore, JA ′ = − DA ∂ρA =∂x + vρA . By combining this equation with eq. (5.31), we would then obtain the equivalent equation of Fick’s first law for the flux relative to the specimen: JA ′ = − ðXB DA + XA DB Þ

∂ρA ∂x

(5:32) ~

~

This can be simplified by defining an interdiffusion coefficient D as D = XB DA + XA DB , so that Fick’s first law becomes ~

JA ′ = − D

∂ρA = − JB ′ ∂x

(5:33)

5.1 Phenomenological theory

And Fick’s second law for diffusion in substitutional alloys becomes   ∂ρA ∂ e ∂ρA = D ∂t ∂x ∂x

299

(5:34)

From the above-mentioned discussion, we know that diffusion in substitutional solid solutions still follow the format of Fick’s first law, where the differences are ~ that the interdiffusion coefficient D replaces the intrinsic diffusion coefficients (DA and DB) of the constituents, and the diffusion occurs in opposite directions of the two constituents. ~ If the interdiffusion coefficient D, the migration velocity of the marker v, and the concentration gradient are measured at a certain temperature, DA and DB can then be calculated from eq. (5.31). In particular, for a diffusion couple made with 30 wt.% Zn brass and pure Cu, Darken calculated the intrinsic diffusion coefficients of Cu and Zn when the Zn concentration at the marker plane was 22.5 wt.%: DCu = 2.2 × 10−13 m2/s, DZn = 5.1 × 10−13 m2/s, and DZn/DCu ≈ 2.3. From eq. (5.28) we know that ~ ~ when xB ! 0 (xA ! 1), D ≈ DB . Similarly, when xA ! 0 (xB ! 1), D ≈ DA . In other ~ words, in a dilute solid solution, the interdiffusion coefficient D is close to the intrinsic diffusion coefficient DA or DB. Along with the increase in the concentration of the solute, the difference between the interdiffusion coefficient and the intrinsic diffusion coefficients would become more significant. For instance, earlier studies showed ~ that when the concentration of Zn is approaching zero, DZn ≈ DZn = 0.3 × 10 − 13 m2/s, in~ dicating that when the Zn concentration increases from 0 to 22.5 wt.%, DZn increases by about 17 times.

5.1.5 Solution to diffusion equations when diffusion coefficient is a function of concentration In the earlier discussions, it is assumed that the diffusion coefficient D is independent of the concentration. However, this is not always the case. For example, in austenite, the diffusion coefficient of the interstitial C atoms would increase if their concentration is increased. Also, in diffusion couples such as Ni–Cu, Au–Pt, and Au–Ni, when ~ the concentration of substitutional atoms increases, the interdiffusion coefficient, D, would increase as well. Therefore, strictly, the Fick’s second law should follow the expression of eq. (5.2):   ∂ρ ∂ ∂ρ = D ∂t ∂x ∂x It is extremely difficult to solve this partial differential equation. Fortunately, Boltzmann and Matano gave the method to determine D(ρ) as a function of ρ from experimentally measured ρ(x) relationships.

300

Chapter 5 Diffusion in solids

Assume we have an infinite long diffusion couple where the initial conditions are:  x > 0, ρ = ρ0 when t = 0, x < 0, ρ = 0 Boltzmann introduced a parameter, η, so that the partial differential equations are converted to ordinary differential equations:  pffiffi If η = x t , then ∂ρ dρ ∂η x dρ = =− 3 ∂t dη ∂t 2t 2 dη ∂ρ dρ ∂η 1 dρ = = pffiffi ∂x dη ∂x t dη         ∂ ∂ρ ∂ D dρ 1 ∂ dρ ∂η 1 d dρ pffiffi = = pffiffi D D = D ∂x ∂x ∂x dη ∂x t dη dη t dη t ∂η If we put the above-mentioned equation into eq. (5.2), we will have   η dρ d dρ − = D 2 dη dη dη thus −

  η dρ dρ = d D 2 dη

(5:35)

Now the initial conditions become  when t = 0,

η > + ∞, ρ = ρ0 η < − ∞, ρ = 0

The concentration profile of Cu–brass diffusion couple after diffusion time, t, is shown as the solid line in Fig. 5.10. If we want to know the diffusion coefficient when the concentration is ρ1 where 0 < ρ1 < ρ0, we should do integration on both sides of eq. (5.35): 1 − 2

ρð1

ρð1

ηdρ = 0

0

        dρ dρ ρ = ρ1 dρ dρ = D d D = D − D dη dη ρ = 0 dη ρ = ρ dη ρ = 0

(5:36)

1

where D

  pffiffi dρ dρ dρ dx =D t = D dη dx dη dx

When ρ = 0, dρ=dx = 0, then   dρ =0 D dη ρ = 0

(5:37)

5.1 Phenomenological theory

301

x' = 0 (The original interface before diffusion) ρ0

ρo

Tangent line

ρZn

Cu

Cu–Zn

ρ1

0 x

x = 0 (Matano plane) Fig. 5.10: Cu–brass diffusion couple.

So eq. (5.36) is simplified to −

1 2

ρð1

0

  dρ ηdρ = D dη ρ = ρ1

(5:38)

pffiffi Substituting η = x t and eqs. (5.37) into (5.38), we have 1 − 2

ρð1

0

  pffiffi dρ x pffiffi dρ = Dðρ1 Þ t dx ρ = ρ1 t

so ρð1   1 dx Dðρ1 Þ = − xdρ 2t dρ ρ = ρ1

(5:39)

0

 Ðρ where dx dρρ = ρ1 is the reciprocal of the slope of ρ–x at ρ = ρ1, and 0 1 xdρis the area underneath the curve. Therefore, in principle, now we are able to calculate D(ρ1). Ðρ However, in 0 1 xdρ, we need to define the origin of the integration. Should it be defined at the initial welding plane? Matano provided such a definition, where he defined the x = 0 plane (“Matano plane”) using eq. (5.36). In particular, as dρ=dx = 0when ρ = ρ0 and ρ = 0: 1 − 2

ρð0

ρð0

ηdρ = 0

0

        dρ dρ ρ = ρ0 dρ dρ = D d D = D − D =0 dη dη ρ = 0 dη ρ = ρ dη ρ = 0 0

Therefore: ρð0

0

1 ηdρ = pffiffi t

ρð0

0

0ρ 1 ρðc ρð0 ð0 xdρ = 0 . @ xdρ = xdρ + xdρ = 0A 0

0

ρc

302

Chapter 5 Diffusion in solids

This indicates that, on either side of the x = 0 plane, the diffusion fluxes are equivalent in magnitude but opposite in direction. As shown in Fig. 5.10, we use ρ0 to represent the mass concentration at x = 0, and use x′ = 0 to represent the initial welding plane. It should be noted that the Matano plane would coincide with the initial welding plane only if the density of the diffusion couple does not change over time. In other words, Matano plane is the plane in the diffusion couple across which the diffusion fluxes on either side of the plane are equivalent in magnitude but opposite in direction.

5.2 Thermodynamic analysis of diffusion Fick’s first law indicates that the diffusion flux is in the direction from the high concentration region to the low concentration region in the system, and the net result of diffusion is to reduce the concentration gradient and subsequently homogenize the system. However, as a matter of fact, not all diffusion processes follow this rule, and in some cases, matters would diffuse from the low concentration region to the high concentration region, leading to an increase in the concentration gradient. For example, during early ageing of Al–Cu alloys, Cu-rich clusters would develop. Also, in some solid solution alloys, spinodal decomposition would occur, causing solute-rich regions. Such diffusion is called “uphill diffusion” or “reverse diffusion.” From thermodynamic analysis, we know that the driving force for diffusion is the chemical potential gradient, ∂μ=∂x, instead of the concentration gradient ∂ρ=∂x. Subsequently, both conventional (“downhill”) and unconventional (“uphill”) can be rationalized. According to thermodynamics, the chemical potential, µi, represents the Gibb’s free energy of the i atoms, that is, μi = ð∂G=∂ni Þ, where ni is the total number of atoms of the i constituent. The driving force of the atoms can be calculated as follows: F= −

∂μi ∂x

(5:40)

where the minus sign indicates that the driving force and subsequently the diffusion flux are in the direction of chemical potential reduction. In other words, at a constant temperature and pressure, as long as there is a chemical potential gradient in the system, diffusion would continue to proceed until the chemical potential is homogenized in the system. Driven by chemical potential gradient, the directional flow of atoms in the solids would encounter the resistance from solvent atoms, and the resistance force is proportional to the diffusion velocity. When the diffusion velocity increases to the value where the resistance from matrix solvent atoms is equivalent to the diffusion driving force, the diffusion rate of the solute atoms would achieve their velocity limit,

5.2 Thermodynamic analysis of diffusion

303

that is, the average diffusion velocity. In particular, the average diffusion velocity, v, is proportional to the diffusion driving force F: v = BF where the parameter B represents the velocity of the diffusing atoms per unit driving force and is called the “mobility.” The diffusion flux can be calculated as the product of the mass concentration and average velocity of the diffusing atoms: J = ρi v i Therefore, J = ρi Bi Fi = − ρi Bi

∂μi ∂x

From Fick’s first law: J= −D

∂ρi ∂x

and if the two equations above are combined, we have D = ρi Bi

∂μi ∂μi ∂μi = Bi = Bi ∂ρi ∂ ln ρi ∂ ln xi

where xi = ρi/ρ. Since from thermodynamics we know that ∂μi = kT∂ ln ai , where ai is the activity of component i in the solid solution: ai = rixi, and ri is the activity coefficient. Therefore, the earlier equation can be written as   ∂ ln ai ∂ ln ri (5:41) = kTBi 1 + D = kTBi ∂ ln xi ∂ ln xi For an ideal solid solution (ri = 1) and the dilute solid solution (ri = constant), the expression in the brackets (called “thermodynamic factor”) equals to 1, so D = kTBi.

(5:42)

Therefore, in ideal or dilute solid solutions, the diffusion velocity of different components only depends on their respective mobilities. Equation (5.42) is called Nernst–Einstein equation. For actual solid solutions, this is also true and can be justified as follows: In binary systems, from the Gibbs–Duhem relation: x1 dμ1 + x2 dμ2 = 0 where x1 and x2 are the molar fraction of component 1 and component 2, respectively. As µi is a function of xi, from dµi = RTdlnai, we have xi dμi = RTðdxi + xi d ln ri Þ

304

Chapter 5 Diffusion in solids

Combining the two above-mentioned equations and considering dx1 = –dx2, and we have x1 d ln r1 = − x2 d ln r2 which is then divided by dx1, and dx1 = –dx2 is considered: d ln r1 d ln r2 = d ln x1 d ln x2

(5:43)

From eqs. (5.43) and (5.41), we know that the thermodynamic factors for component 1 and component 2 are identical, and the difference in their diffusion velocities (D1 and D2) is caused by their different mobilities of B1 and B2. According to eq. (5.41), when ð1 + ð∂ ln ri =∂ ln xi ÞÞ > 0, D > 0, indicating that the diffusion proceeds from the high concentration region to low concentration region (downhill); when ð1 + ð∂ ln ri =∂ ln xi ÞÞ < 0, D < 0, indicating that the diffusion proceeds from the low concentration region to high concentration region (uphill). Therefore, the factor that determines the diffusion direction is the chemical potential gradient, and the net result of diffusion (no matter it is downhill or uphill) is to reduce the chemical potential gradient until it becomes zero. There are several other situations where uphill diffusion can happen: (1) When an elastic stress field is present. When there is elastic stress gradient in a crystal, atoms with a larger atomic size tend to diffuse to elongated regions of the materials, and atoms with a smaller atomic size would diffuse to compressed regions of the materials, leading to a nonuniform distribution of solute atoms in the solid solution. (2) The absorption of atoms at the grain boundaries. Grain boundaries have higher energies than the bulk of the crystal, and the atoms at the grain boundaries are disorderly arranged. If the segregation of solute atoms to the grain boundaries can reduce the total energy of the system, they would diffuse preferentially to the grain boundaries, resulting in a higher solute concentration at the grain boundaries than in the bulk. (3) A high electric or temperature field may also cause the diffusion of atoms in particular directions, giving rise to inhomogeneity in diffusion.

5.3 Atomic theory of diffusion 5.3.1 Diffusion mechanisms In crystals, atoms are vibrating relative to their equilibrium positions, and may jump from one equilibrium position to another (diffusion). A few possible diffusion mechanisms are illustrated in Fig. 5.11.

5.3 Atomic theory of diffusion

305

Fig. 5.11: Diffusion mechanisms in crystal. 1 – direct exchange mechanism; 2 – circular exchange mechanism; 3 – vacancy; 4 – interstitial; 5 – push; 6 – crowdion.

1. Exchange mechanism The exchange mechanism of neighboring atoms is schematically demonstrated in Fig. 5.11–1, where two neighboring atoms are shown to have exchanged their positions. This mechanism is unlikely to take place in closely packed structures as it is associated with a big lattice distortion and high activation energy. In 1951, Zener proposed the circular exchange mechanism, as shown in Fig. 5.11–2, four atoms would exchange their positions simultaneously. The energy cost of the circular exchange is much lower than that of the direct exchange, but its possibility is still low because of the confinement for the collective movement of atoms. In both direct exchange and circular exchange mechanisms, the net diffusion flux passing through the crosssection area is zero, that is, the diffusing atoms are exchanging by equal numbers, so Kirkendall effect should not show up. Up to now, the exchange mechanism has not been observed in metals and alloys, although in metallic liquids or amorphous materials, such cooperative movement of atoms may be easier to happen. 2. Interstitial mechanism In the interstitial diffusion mechanism, as shown in Fig. 5.11–4, atoms migrates from one interstitial site in the lattice to another interstitial site. Small solute atoms such as H, C, and N are likely to adopt this mechanism to diffuse in crystals. However, if a larger substitutional solute atom occupies the interstitial site forming a Frenkel defect, diffusion of that atom following the interstitial diffusion mechanism would be difficult as it would cause a significant lattice distortion. Accordingly, people proposed the “interstitialcy” mechanism, where an interstitialcy atom may “push” its neighboring

306

Chapter 5 Diffusion in solids

lattice atom into a nearby interstitial site and then occupy the vacant lattice site itself, as shown in Fig. 5.11–5. Moreover, some other researchers proposed a “crowdion” diffusion mechanism similar to the interstitialcy mechanism, where an interstitial atom “squeezes” into the diagonal (i.e., the closely packed direction) of an bcc crystal, making a row of atoms deviate from their equilibrium positions and form a “crowdion,” as illustrated in Fig. 5.11–6. In the crowdion diffusion mechanism, atoms would diffuse along the diagonal of the crystal structure. 3. Vacancy mechanism As discussed in previous sections, there are vacancies in crystal structures. At a certain temperature, there exists an equilibrium concentration of vacancies, and such equilibrium concentration increases with increasing temperature. Vacancies would make atomic migration easier to take place, and in most cases, diffusion of atoms adopts the vacancy mechanism, as shown in Fig. 5.11–3. This is corroborated by the Kirkendall effect discussed previously. In particular, since the diffusion rate of Zn atoms is higher than that of Cu atoms, vacancies have to form continuously on the Cu side, so that a net flux of Zn atoms migrate across the marker plane to the Cu side and a net flux of vacancies migrate across the marker plane to the brass side. The excess vacancies on the brass side would then either agglomerate or get annihilated. As shown in Fig. 5.12, the vacancy mechanism allows for unequal diffusion fluxes across the marker plane, causing a subsequent movement of the marker plane toward the brass side. Similarly, as demonstrated by the pure Cu/12 wt.% Al–Cu alloy diffusion couple shown in Fig. 5.7, since the diffusion coefficient of Cu atoms is higher than that of Al atoms, a net flux of Cu atoms would migrate from the left of the marker plane to the right, making the marker that initially coincides with the welding plane separate with the

Fig. 5.12: The diagram of marker plane’s movement (black spots: atoms; blocks: vacancies; dashed lines: marker plane) (a) initial state; (b) vacancy generation; (c), (d), (e) vacancy plane migrates to the right; (f) vacancy annihilation (the marker plane migrates to the right through comparing (a) and (b)).

5.3 Atomic theory of diffusion

307

welding plane and move to the left. Meanwhile, a net of flux of vacancies diffuse to the Cu side, which would then agglomerate to form small voids (Fig. 5.7). As the plateshaped β-Cu3Al phase form upon quenching the Cu–Al alloy, the interface between the β phase and pure Cu can be considered as the marker plane. 4. Grain boundary diffusion and surface diffusion For polycrystalline materials, diffusion can proceed in three mechanisms: diffusion inside the grains (bulk diffusion), grain boundary diffusion, and diffusion across the free surfaces, whose diffusion coefficients are denoted as Dl, DB, and Ds, respectively. Figure 5.13 demonstrates the difusion in a bicrystal, where radioactive isotope M is evaporated onto a plane normal to the grain boundary plane (y = 0). After diffusion annealing treatment, the diffusion depth of M atoms into the grains is found to be the highest through the surface diffusion mechanism, followed by grain boundary mechanism and subsequently the bulk diffusion mechanism, indicating that the diffusion coefficient for different mechanisms satisfies Dl < DB < Ds. Because crystal defects such as grain boundaries, free surfaces, and dislocations are associated with significant lattice distortion and ensuing easier migration of atoms, the diffusion rates through these defects are substantially higher than that in perfect crystals. Accordingly, diffusion along crystal defects is usually called “short circuit” diffusion. M O

x

y

DS

DB

Free surfaces

DL

A Grain boundary

Fig. 5.13: The diffusion in a bi-crystal.

5.3.2 Atomic jump and diffusion coefficient 1. Atomic jump frequency Taking interstitial diffusion as an example, the diffusion of solute atoms proceeds via the jump of the atoms from one interstitial site to another. Figure 5.14(a) schematically

308

Chapter 5 Diffusion in solids

(a)

(b)

Fig. 5.14: The octahedral interstitial sites in fcc crystal structure and (100) plane.

shows the octahedral interstitial sites in fcc crystal structure, and Fig. 5.14(b) demonstrates the atomic packing scheme in fcc {100} planes. In the figure, number 1 represents the initial position of the interstitial atom, and number 2 represents its position after the jump. Upon atomic jump, atoms 3 and 4 have to be “pushed” away in order for the diffusion to take place, causing an instantaneous local lattice distortion. This local lattice distortion is actually the resistance to atomic jumps, that is, the energy barrier for diffusion. As shown in Fig. 5.15, the energy barrier for the atoms to jump from position 1 to position 2 is ΔG = G2–G1, so only the atoms with free energies higher than G2 can make successful jumps.

Fig. 5.15: The relationship between free energy of the atoms and its position.

According to the Maxwell–Boltzmann statistics, among a total number of N solute atoms, the number of atoms having free energy higher than G2 is   − G2 nðG > G2 Þ = N exp kT where k is the Boltzmann constant (k=1.380649 × 10‒23 J/K) and T is the temperature (in Kelvin). Similarly, the number of atoms having free energy higher than G1 is   − G1 nðG > G1 Þ = N exp kT

5.3 Atomic theory of diffusion

309

so   nðG > G2 Þ G2 − G1 = exp − kT nðG > G1 Þ As G1 is the minimum free energy and corresponds to the equilibrium position, so n (G > G1) ≈ N. Therefore, the above-mentioned equations become     nðG > G2 Þ G2 − G1 ΔG = exp − (5:44) = exp − kT N kT where expð − ðΔG=kT ÞÞ represents the fraction (probability) of atoms that are able to jump at a certain temperature T. If we assume a crystal is comprised of n atoms that jump for m times within a time interval dt, then the jump frequency (i.e., the average number of jumps per unit time of the atoms) can be expressed as Γ=

m ndt

(5:45)

Figure 5.16 shows two neighboring parallel planes (Plane 1 and Plane 2) containing interstitial atoms, and the two planes are both vertical to the plane of the page. Suppose both planes have unit areas and contain n1 and n2 interstitial atoms, respectively. At a certain temperature, the jump frequency of atoms is Γ, and the probability for the atoms to jump from Plane 1 to Plane 2 and from Plane 2 to Plane 1 is P. Then, during time interval Δt, the number of atoms jumping from Plane 1 to Plane 2 and from Plane 2 to Plane 1 are, respectively: N1 − 2 = n1 PΓΔt

Fig. 5.16: The jump of interstitial atoms between two neighboring planes.

310

Chapter 5 Diffusion in solids

and N2 − 1 = n2 PΓΔt If n1 > n2, the net number of interstitial solute atoms obtained by Plane 2 is N1 − 2 − N2 − 1 = ðn1 − n2 ÞPΓΔt so ðN1 − 2 − N2 − 1 ÞAr ðn1 − n2 ÞPΓΔtAr = = JΔt NA NA where J = (n1–n2)PΓAr/NA is obtained from the definition of the diffusion flux, NA is the Avogadro’s number, and Ar is the atomic mass. Suppose the spacing between Plane 1 and Plane 2 is d, the mass concentration can be calculated as ρ1 =

n1 A r n2 A r , ρ2 = NA d NA d

(5:46)

The concentration on Plane 2 can also be written as ρ2 = ρ1 +

dρ d dx

(5:47)

From eqs (5.46) and (5.47), we have ρ2 − ρ1 =

1 Ar dρ and ρ2 − ρ1 = ðn2 − n1 Þ d NA d dx

From the above-mentioned equations, we have n 2 − n1 =

dρ 2 NA d dx Ar

so J = ðn1 − n2 ÞPΓ

Ar dρ = − d2 PΓ NA dx

Comparing the above-mentioned equation with Fick’s first law, we have: D = Pd2 Γ

(5:48)

The first two terms on the right-hand side of eq. (5.48) are determined by the crystal structure of the solid solution, and Γ is strongly dependent on both the intrinsic properties of the matter and temperature. For example, from the diffusion coefficient of interstitial C atoms dissolved in γ-Fe at 1,198 K, we can obtain its jump frequency Γ = 1.7 × 109 s–1, while the jump frequency of C atoms at room temperature

5.3 Atomic theory of diffusion

311

austenite γ is 18 orders of magnitudes smaller, only 2.1 × 10−9 s–1, indicating the significant of diffusion temperature on the jump frequency. Furthermore, eq. (5.48) is also valid for substitutional diffusion. 2. Diffusion coefficient For interstitial diffusion, if we assume the vibration frequency of the atoms is ν, and the number of neighboring interstitial sites of the solute atoms (i.e., the coordination number of the interstitial sites) is z, then Γ should be the product of ν, z, and the fraction of atoms that are energetically favorable for the jump, exp (–ΔG/kT):   ΔG Γ = νz exp − kT Because ΔG = ΔH–TΔS ≈ ΔU–TΔS, we have     ΔS ΔU exp − Γ = νz exp k kT Substituting the above-mentioned equation to eq. (5.48), we have     ΔS ΔU exp − D = d2 Pνz exp k kT Define D0 = d2 Pνz expðΔS=kÞ, then we have     ΔU Q = D0 exp − D = D0 exp − kT kT

(5:49)

where D0 is the diffusion constant, ΔU is the thermodynamic internal energy needed for the interstitial diffusion of solute atoms, which can also be considered as the diffusion activation energy of the interstitial atoms (Q). In solid solutions, the substitutional diffusion and self-diffusion are mostly carried out by the vacancy diffusion mechanism. Compared to interstitial diffusion, in addition to the migration energy required for the atoms to jump from one equilibrium position to another, formation energy for the adjacent vacancy site is also needed. In the previous chapters, we already know that there is an equilibrium vacancy molar fraction in a crystal at temperature T:   ΔUV ΔSV + XV = exp − kT k where ΔUV is the formation energy of the vacancies, and ΔSV is the corresponding increment in entropy. In substitutional alloys or pure metals, if the coordination number is Z0, then the fraction of lattice sites occupied by atoms around a vacancy can be calculated as

312

Chapter 5 Diffusion in solids

  ΔUV ΔSV Z0 XV = Z0 exp − + kT k

(5:50)

Assume the free energy needed for an atom to jump into a vacancy site is ΔG ≈ ΔU–TΔS, then the atomic jump frequency Γ should be the product of atomic vibration frequency (ν), the fraction of lattice sites occupied by atoms around a vacancy (Z0XV), and the fraction of atoms that are energetically favorable for the jump [exp (–ΔG/kT)]:     ΔUV ΔSV ΔU ΔS exp − + Γ = νZ0 exp − + kT k kT k Inserting the above-mentioned equation into eq. (5.48), we have     ΔSV + ΔS ΔUV + ΔU exp − D = d2 PνZ0 exp k kT If we define D0 = d2 PνZ0 expððΔSV + ΔSÞ=kÞ, then     ΔUV + ΔU Q = D0 exp − D = D0 exp − kT kT

(5:51)

where the activation energy for diffusion Q = ΔUV + ΔU. As demonstrated in Table 5.2, experiments have consistently shown that the activation energy for substitutional or self-diffusion is much higher than that of interstitial diffusion. Table 5.2: The D0 and Q values in selected diffusion systems (approximation). Diffusing species

Matrix

C C Fe Fe Ni

γ-Fe α-Fe α-Fe γ-Fe γ-Fe

D (×− (m/s))

Q (× (J/mol))

. .  . .

    

Diffusing species

Matrix

Mn Cu Zn Ag Ag

γ-Fe Al Cu Ag (bulk) Ag (g-b)

Both eqs. (5.49) and (5.51) follow the Arrhenius-type relation:   Q D = D0 exp − kT

D (×– (m/s))

Q (× (J/mol))

. . . . .

    

(5:52)

where k is the Boltzmann constant [1.38×10–23 J/K], and Q is the molar activation energy. In other words, different diffusion mechanisms have the same general expression of diffusion coefficient, while the particular values of D0 and Q are different.

5.4 The activation energy of diffusion

313

5.4 The activation energy of diffusion Now we know that for different diffusion mechanisms, Q is usually different. In particular, in interstitial diffusion, Q = ΔU, while in vacancy diffusion Q = ΔUV + ΔU. Moreover, there are other diffusion mechanisms such as grain boundary diffusion, surface diffusion, and dislocation diffusion, which have different activation energies. Therefore, in practice, it is critical to know the activation energy to deal with diffusion problems under certain conditions. In the following, the experimental approach to obtain the activation energy for diffusion is discussed. As shown previously, the expression for diffusion coefficient can be generally written as   Q D = D0 exp − kT so if we do a logarithmic manipulation to the earlier equation, we have ln D = ln D0 −

Q . kT

(5:53)

Therefore, we can measure the lnD versus 1/T from experiments, and if they are linearly related (as shown in Fig. 5.17), the slope is then –Q/k, and the intercept of the extrapolated lnD versus 1/T curve with the vertical axis is lnD0.

Fig. 5.17: The relationship diagram between lnD and 1/T.

Generally, D0 and Q are considered to be correlated with the type of the diffusing species and the diffusion mechanism, but are independent on temperature. Consequently, ln D versus 1/T curve should be a straight line. Obviously, if the diffusion mechanism is different at low temperatures and high temperatures, a kink should be observed on the lnD versus 1/T curve, corresponding to two different slopes and subsequently two different activation energies. In addition, it should be noted that, when calculating the slope using the equation Q = –ktanα, the α angle directly measured from the curve (Fig. 5.17) should not be used as the vertical

314

Chapter 5 Diffusion in solids

and horizontal axis of the curve possess different units. Instead, tanα should be calculated using Δ ln D=Δð1=TÞ.

5.5 Random walk and diffusion length If the movement of diffusing atoms follows a straight line, then the diffusion length of atoms should be proportional to time. However, previous discussions showed that the diffusion length is proportional to the square root of time instead. Therefore, it can be concluded that the diffusion of atoms is similar to the Brownian motion of pollen grains in water, where the atoms jump to all directions in a random manner, that is, the diffusion is a type of random walk Suppose an atom makes n times of jumps, and ri is the vector corresponding to the ith jump relative to the origin. After the n jumps, the displacement of the atom relative to the origin is defined as Rn, so Rn = r1 + r2 + r3 + · · · + rn =

n X

ri

(5:54)

i=1

To get the magnitude of Rn, we do scalar product of eq. (5.54), that is, R2n = r1 · r1 + r1 · r2 + r1 · r3 + · · · + r1 · rn + r2 · r1 + r2 · r2 + r2 · r3 + · · · + r2 · rn + · · · +

(5:55)

rn · r1 + rn · r2 + rn · r3 + · · · + rn · rn Equation (5.55) can be considered as the summation of a few types of polynomials, P where the first type is ni= 1 ri ri (n terms); the second type includes all terms of ri · ri + 1 and ri + 1 · ri , where each set has (n – 1) terms. Since ri · ri + 1 =ri + 1 · ri , so there are totally 2(n – 1) terms for this type. Therefore, we have R2n =

n X i=1

ri · ri + 2

n−1 X

ri · ri + 1 + 2

i=1

n−2 X

ri · ri + 2 + · · · =

i=1

n X i=1

ri2 + 2

n−j n−1 X X

ri · ri + j .

(5:56)

j=1 i=1

Since ri · ri + j = jri jjri + j j cos θi, i + j , where θi, i + j is the angle between the two vectors. Therefore, R2n =

n X i=1

ri2 + 2

n−j n−1 X X

jri j · jri + j j cos θi, i + j

(5:57)

j=1 i=1

For cubic crystals, it can be assumed that all jump vectors have the same magnitude, so eq. (5.57) can be simplified to

5.5 Random walk and diffusion length

n−j n−1 X X

R2n = nr2 + 2r2

cos θi, i + j

315

(5:58)

j=1 i=1

Equation (5.58) gives the square of the distance between the atom and the origin after n jumps. However, since diffusion is the collective effect of a lot of atoms jumping for n times, the meaningful term describing− the diffusion length should be the average jumping distance of the atoms, that is, R2n , so ! n−j n−1 X − 2X 2 2 Rn = nr 1 + cos θi, i + j (5:59) n j=1 i=1 Since the atomic jumps are random, the direction of a particular jump is independent of previous jumps, that is, the jump frequency to all directions is the same and for each jump vector there is a jump vector acting in the opposite direction. P P Therefore, nj =−11 ni =−1j cos θi, i + j =0, and eq. (5.59) becomes −

R2n = nr2

(5:60)

qffiffiffiffiffi − pffiffiffi R2n = nr

(5:61)

or

From eq. (5.61), it is clear the average migration distance of the atoms is proportional to the square root of the number of jumps (n). As D = Pd2Γ, if three-dimensional jump is taken into account, P = 1/6, the jump distance d = r, and the jump frequency Γ = n/t. Combining with eq. (5.60), we have −

R2n =

6n D = 6Dt Γ

(5:62)

or qffiffiffiffiffi − pffiffiffiffiffi R2n = 2.45 Dt

(5:63)

Therefore, it is clear that diffusion can be considered a kind of random walk, and the diffusion length derived from the theoretical analysis is consistent with that obqffiffiffiffiffi 2 tained from empirical diffusion equations, that is, Rn is proportional to the square root of diffusion time. From eq. (5.61), we know that the average displacement (mean square root displacement) of a diffusing atom is proportional to the square root of the number of its random jumps. Since n = Γt, the average displacement is very sensitive to temperature. For example, if γ-Fe goes through C diffusion annealing at 925 °C for 4 h,

316

Chapter 5 Diffusion in solids

the jump frequency of C atoms is 1.7 × 109 s–1, and the jump distance of C atoms from one octahedral interstitial site to another is 0.253 nm, the total migration distance of C atoms (nr) is 6.2 km. However, the penetration depth of C atoms into the material (mean square root displacement) is approximately 1.3 mm, as a result of the random jump of the atoms. On comparison, if the diffusion annealing is carried out at 20 °C instead, the jump frequency of C atoms becomes only 2.1 × 10−9 s–1, the total migration distance of C atoms is reduced to 1.25 × 10−6 km, and the penetration depth of C atoms into the material is reduced to 1.4 × 10−9 mm. Therefore, at 20 °C, the C diffusion length is actually negligible.

5.6 Factors that affect diffusion 1. Temperature Temperature is the single most important factor that affects the diffusion rate. Higher temperature would lead to more enhanced thermal activation and subsequently easier atomic migration and higher diffusion coefficient. For example, Table 5.2 indicates that the diffusion constant and activation energy of C atoms diffusing in γ-Fe are D0 = 2.0 × 10−5 m2/s and Q = 140 × 103J/mol, so we can calculate the diffusion coefficients of C at 1,200 and 1,300 K as:  − 140 × 103 = 1.61 × 10 − 11 m2 =s D1, 200 = 2.0 × 10 − 5 exp 8.314 × 1, 200 D1, 300 = 2.0 × 10 − 5 exp



− 140 × 103 = 4.74 × 10 − 11 m2 =s 8.314 × 1, 300

Therefore, when temperature increases from 1,200 to 1,300 K, the diffusion coefficient would increase by three times, that is, the carburization rate would increase by three times. Therefore, when dealing with any diffusion-dominated processes, we need to consider the effect of temperature. 2. The type of solid solution As discussed previously, the particular diffusion mechanism depends on the type of the solid solution. Solutes in interstitial solid solutions usually have lower activation energies. For example, interstitial solutes such as C and N in Fe have drastically lower activation energies than those of Cr and Al in Fe. Therefore, to achieve the same penetration depth in the steel component, the diffusion takes much less time for C and N than for Cr and Al.

317

5.6 Factors that affect diffusion

3. Crystal structure Diffusion process can be affected by the crystal structure, for example, some metals can go through allotropic transformation, which would lead to a considerable variation in the diffusion coefficient. For example, at 912 °C, Fe would have γ-Fe↔α-Fe transformation, and the self-diffusion coefficient of Fe in α-Fe is 240 times than that in γ-Fe. Also, the same alloy element may have different diffusion coefficients in solid solutions of different crystal structures, for instance, at 900 °C, the diffusion coefficient of Ni in α-Fe is 1,400 times higher than that in γ-Fe. Similarly, in an interstitial solid solution, the diffusion coefficient of N at 527 °C in α-Fe is 1,500 times higher than that in γ-Fe. In general, all elements possess much higher diffusion coefficients in α-Fe than in γ-Fe because of the less closely packed structure in bcc structure than that of the fcc structure and subsequently easier atomic migration in bcc crystal. Figure 5.18 demonstrates the melting point diffusivities for various classes of materials and crystal structures.

bcc rare earth bcc alkali Metal carbides bcc transition metals hcp fcc metals Graphite Alkali halides Indium Oxide Trigonal Ice Diamond cubic

10–16

10–15

10–14 10–13 10–12 10–11 Melting point diffusivity, D(Tm) (m2/s)

10–10

10–9

Fig. 5.18: The melting point diffusivities for various classes of materials and crystal structures.

In addition, the solubility of diffusion elements may be different in crystals of different structures, causing different concentration gradients and subsequently different diffusion rates. For example, carburization of steels is often carried out at the high temperature austenitic state. The high temperature would enhance diffusion, moreover, the C solubility in γ-Fe is much higher than that in α-Fe, giving a higher concentration gradient in the austenite, an accelerated diffusion of C, and consequently a deeper penetration of C into the sample.

318

Chapter 5 Diffusion in solids

The anisotropy can also affect diffusion. In general, the lower the structural symmetry, the more anisotropic diffusion behavior in different crystallographic orientations will be. In the highly symmetric cubic crystals, there is little anisotropy for diffusion, however, in the less symmetric rhombohedral crystals (such as Bi), D may differ greatly in different crystallographic orientations, where the difference may be as high as 1,000 times. 4. Crystal defects Engineering materials practically in use are mostly polycrystalline. For polycrystalline materials, as discussed previously, diffusion can be carried out by three different mechanisms: lattice (bulk) diffusion, grain boundary diffusion, and surface diffusion. If we use QL, QS, and QB to represent the activation energies for lattice, surface, and grain boundary diffusions respectively, and use DL, DS, and DB to represent the diffusion coefficients for lattice, surface and grain boundary diffusions respectively, then generally we have: QL > QB > QS, and DS > DB > DL. Figure 5.19 shows the temperature dependence of the self-diffusion coefficient of polycrystalline and single crystalline Ag, where the self-diffusion coefficient of single crystalline Ag represents the lattice diffusion coefficient while the self-diffusion coefficient of polycrystalline Ag is determined by the combined behavior of both lattice diffusion and grain boundary diffusion. As clearly demonstrated in the figure, for temperatures higher than 700 °C, the self-diffusion coefficients of single crystalline and polycrystalline Ag are almost the same, when for temperatures lower than 700 °C, the self-diffusion coefficient of polycrystalline Ag would become much higher than that of single crystalline Ag, showing a significant effect of the grain boundaries. It is worth mentioning that even for grain boundary diffusion itself of Ag, there may be anisotropy as well, as indicated by the anisotropic grain boundary self-diffusion with different grain boundary misorientations up to a misorientation angle of 45 °.

Temperature (°C) 900 800 700 600 500 450 400 350

lgD

–8

–10

–12

–14 0.8

1.0

1.2 1.4 1.6 (1/T) (× 10–3 K–1) Polycrystalline Single crystalline

Fig. 5.19: The relationship between the selfdiffusion coefficients of Ag and 1/T.

5.6 Factors that affect diffusion

319

It is generally believed that dislocations have a similar effect on diffusion as grain boundaries, that is, to facilitate the atomic diffusion. However, in some cases, dislocations may also restrict diffusion because of the interaction between dislocations and interstitial atoms. In brief, crystal defects such as grain boundaries, free surfaces, and dislocations usually serve as faster channels for diffusion, owing to the lattice distortions around such defects, which raise the energy of the diffusing atoms and make their jumps easier. Consequently, the activation energies associated with crystal defects are substantially lower than that of lattice diffusion, enhancing atomic diffusion. 5. Chemical composition From the microscopic diffusion mechanisms, we know that the atomic jump over the energy barrier is associated with local distortion of the crystal lattice and subsequently the breakage of chemical bonds. Therefore, it is reasonable to think that the self-diffusion coefficient of metals is closely related to their atomic bonding strengths and should scale with other physical parameters that are determined by the bonding strength such as melting temperature, latent heat of melting, and thermal expansion coefficient. In general, the higher the melting temperature, the higher the activation energy for self-diffusion. For both substitutional and interstitial solid solutions, in addition to the abovementioned intrinsic properties of the constituent phases, diffusion coefficient may also depend on the concentration of the solutes. However, as long as the concentration and/or concentration gradient are low, we can assume the diffusion coefficient is independent of concentration when solving the diffusion equations. The influence of a third constituent (or contaminant) on the diffusion behavior of binary alloys may be complex, where the introduction of the additional constituent may improve the diffusion rate, reduce the diffusion rate, or have no effect on diffusion kinetics. However, it is worth mentioning that, for some third constituents, not only the diffusion rate but also the diffusion direction can be affected. For example, using a diffusion couple made of two types of single phase austenite alloys with similar initial C concentrations [alloy 1: Fe–C alloy with w(C) = 0.441%; alloy 2: Fe–C–Si alloy with w(C) = 0.478%; w(C) = 3.8%], Darken showed that after diffusion annealing at 1,050 °C for 13 days, a C concentration gradient was developed with a concentration profile demonstrated in Fig. 5.20. Since the addition of Si into Fe-C alloy raises the chemical potential of C, C atoms can have uphill diffusion into the side without Si content. 6. Stress If there is a stress gradient inside the alloy, the stress may provide the atoms with the driving force for diffusion, so that chemical diffusion may still take place even if the solute concentration is uniform in the alloy. On the other hand, if an external stress

320

Chapter 5 Diffusion in solids

0.6 w(C) (%)

w(Si) = 3.80% 0.586% 0.5 0.4

0.441%

0.478% 0.315%

0.3 –1.0

–0.5

0 Distance (mm)

0.5

1.0

Fig. 5.20: The carbon (C) concentration profile of diffusion couple after diffusion annealing for 13 days.

is applied to the alloys so that a stress gradient is developed, diffusion would occur where atoms would migrate toward the elongated part of the lattice.

5.7 Reaction diffusion If an element penetrates into a sample surface via diffusion, its concentration may eventually exceed the solubility limit of the matrix metal, so that an intermetallic phase (or another type of solid solution) may form near the surface of the matrix metal. Such phenomenon (the formation of new phases) is often called reaction diffusion or transformation diffusion. 900

Temperature (°C)

800 (γ-Fe) 700 600

680 ± 5

650

2.8

4.55

(α-Fe)

ε 500 400 Fe

450 5.7 1

2

3

4

5

γʹ 6

450 6.1 7

8

9

10

w(N) (%) Fig. 5.21: The Fe–N phase diagram.

The newly formed phase can be analyzed according to the equilibrium phase diagram. For example, the new phases formed during nitridation of pure Fe at 520 °C can be determined from the Fe–N phase diagram (Fig. 5.21). As the N concentration at the surface of the metal is higher than that in the bulk, N-rich intermetallic phases would form at the surface of the metal. In particular, when the N concentration is higher

5.8 Diffusion in ionic solids

321

than 7.8 wt.%, closely packed hexagonal structured ε phase would form (depending on the N concentration, Fe3N, Fe2−3N or Fe2N may form), which may have a N concentration ranging from 7.8 wt.% to 11.0 wt.%, and the N atoms would regularly occupy the interstitial sites of the closely packed hexagonal structure comprised by Fe atoms. The further from the metal surface, the lower the N concentration. Eventually, the γ´ phase (Fe4N) would form, with a N concentration ranging from 5.7 wt.% to 6.1 wt.%, and the N atoms would regularly occupy the interstitial sites of the fcc structure comprised by Fe atoms. Deeper into the metal matrix, α solid solution with a bcc lattice and an even lower N concentration would form. The N concentration profile in pure Fe after nitridation and the corresponding phases are illustrated in Fig. 5.22.

ε

w(N) (%)

γʹ

α x

O (a) ε

γʹ

α (b)

Fig. 5.22: The N concentration profile (a) in pure Fe after nitridation and the corresponding phases (b).

Experimental observations indicate that the microstructure as a result of reaction diffusion does not contain a mixture of two phases, and the concentration at the phase boundaries is discontinuous and corresponds to the solubility limit of that phase at a certain temperature. This can be explained by the thermodynamic equilibrium condition: if the microstructure as a result of reaction diffusion contains a mixture of two phases, then the chemical potentials of the two phases, μi (i represents the ith compo nent), should be identical, leading to a zero chemical potential gradient (∂μi ∂x=0) and subsequently a zero driving force for diffusion in this region. Similarly, in a ternary system, there can be two-phase mixtures in the microstructure as a result of reaction diffusion, while three-phase mixture cannot be developed.

5.8 Diffusion in ionic solids In metals and alloys, atoms can jump into any vacant substitutional or interstitial sites. However, in ionic solids, the diffusing ions can only migrate into sites with

322

Chapter 5 Diffusion in solids

the same electric charges, and the migration into sites with different electric charges is forbidden. The diffusion of ions relies on vacancies, and the distribution of vacancies in ionic solids is different from that in metals. In particular, since the separation of a couple of ions with opposite electric charges would result in significant increase in the electrostatic energy, to maintain local electroneutrality, two types of defects with different electric charges would form, for example, a cation vacancy and an anion vacancy. If a random distribution of equal number of cation vacancies and anion vacancies is formed, such as the Na+ vacancy and Cl– vacancy pairs formed in NaCl crystals as demonstrated in Fig. 3.3, this kind of defect pair is called “Schottky-type vacancies.” Using the same method to calculate the equilibrium vacancy molar fraction in metals, we can also obtain the equilibrium molar fractions of cation vacancies (xvc) and anion vacancies (xva). At equilibrium,     − Gva − Gvc − ΔGs = A exp (5:64) xva xvc = A exp kT kT where ΔGs is the formation energy needed for a Schottky pair, A is a coefficient determined by the vibration entropy, which is usually assumed to be 1. When the formation energy of an interstitial cation (ΔGic) is much lower than that of a cation vacancy (ΔGvc), the electric charges associated with the formation of cation vacancy can be compensated by interstitial cations. Such a defect configuration is called “Frenkel vacancies” disorder state, as demonstrated in Fig. 5.23. Similarly, when the formation energy of an interstitial anion (ΔGia) is much lower than that for an anion vacancy (ΔGva), the electric charges associated with the formation of anion vacancies can be compensated by interstitial anions. Such defect configuration is another Frenkel vacancies disorder state. If xic is the equilibrium molar fraction of interstitial cations, at equilibrium (where the defects are randomly distributed), we have   − ΔGF (5:65) xic xvc = exp kT

Fig. 5.23: Frenkel vacancies.

5.8 Diffusion in ionic solids

323

where ΔGf is the formation energy needed for a Frenkel pair (an interstitial ion and an ion vacancy). When ΔGic ≈ ΔGva, interstitial cations and anion vacancies would coexist, and to maintain electroneutrality, the condition of xvc = xva + xic should be satisfied. Similarly, when ΔGia ≈ ΔGvc, interstitial anions and cation vacancies would coexist, and to maintain electroneutrality, xva = xia + xvc should be satisfied. Similar situation would exist if the chemical valence of the ions changes, and Fig. 5.24 shows two such examples. In Fig. 5.24(a), in wustite (FeO), some Fe2+ can be oxidized to Fe3+. To maintain electroneutrality, cation vacancies have to form. Then, compared with pure FeO, the actual compound would be nonstoichiometric. The inverse situation may also show up, for example, some Ti4+ are reduced to Ti3+ in TiO2, and O vacancies are formed to maintain electroneutrality. Moreover, when the ions are replaced by another type of ions with different valence state, extra defects would also appear. As demonstrated in Fig. 5.24(b), when CaO is used as the stabilizer for ZrO2, the low valence Ca2+ replaced Zr4+, and O vacancies are formed to maintain electroneutrality.

(a)

(b)

Fig. 5.24: The schematic drawing of nonstoichiometric compound: (a) FeO and (b) ZrO2 stabled by CaO.

In solids, the directional migration of electrons and ions under electric field would generate current. In metals and semiconductors, electric conduction is carried out by the flow of electrons. In comparison, in ionic solids, the mobility of ions is higher than that of the electrons at high temperatures, and electric conduction is carried out by the directional diffusion of ions. When measuring the ion diffusion coefficient (DT) using isotopes, we can obtain the relation between DT and the electroconductivity (σ): for the interstitial diffusion of ions, cq2 σ = i DT kT

(5:66)

324

Chapter 5 Diffusion in solids

for the vacancy diffusion of ions, cq2 σ = i DT fkT

(5:67)

where c is the number of ions per unit volume, qi is the charge of the ion, and f is a parameter relevant to the vacancy diffusion mechanism (f < 1). Figure 5.25 shows the temperature dependence of Na+ diffusion coefficient experimentally determined from Na isotope, and calculated from eq. (5.67). In NaCl crystal, Na+ is the diffusing species and the diffusion proceeds by a vacancy mechanism (f = 0.78). It is clearly shown that the two curves are consistent with each other above 550 °C. Below 550 °C, considerable deviation appears because of the existence of contaminants having different valence states of Na atoms.

Diffusion coefficient (m2/s/°C)

10–12

Temperature (°C) 700 600 500 400 350

10–13

10–14

10–15

10–16

10–17 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 (1/T ) (×10–3 K–1) Measured by Na isotope Calculated by eq.(4.67)

Fig. 5.25: The temperature dependence of Na+ diffusion coefficient in NaCl.

In ionic crystals, since the cohesion energy (Table 5.3) of ionic bonds is generally higher than that of metals, the energy barrier for diffusion is much higher in ionic crystals. Also, to maintain local electroneutrality, pairs of defects have to form, which is associated with an extra formation energy. Furthermore, the diffusing ions can only occupy sites with the same electric charges that would give rise to a longer migration distance. Therefore, the diffusion rate of ions is in general notably lower than that of metallic atoms. It should also be noted that the diffusion coefficient of cations is usually higher than that of anions. This may be caused by the smaller size of cations and their loss of valence electrons. For example, in NaCl, the activation energy for the diffusion of Cl ions is about twice that of Na.

5.9 Molecular motion in polymers

325

Table 5.3: The activation energy for diffusion in some ionic crystals. Diffusing atoms Fe in FeO Na in NaCl O in UO U in UO Co in CoO Fe in FeO

Q (kJ/mol)      

Diffusing atoms Cr in NiCrO Ni in NiCrO O in NiCrO Mg in MgO Ca in CaO

Q (kJ/mol)     

5.9 Molecular motion in polymers Polymers are made of molecular chains. These long molecules are composed of structural entities called mer units, which are successively repeated along the chain. The interaction between the C atoms on the chains (“backbone” of the polymer chain) is the strong covalent bonding, which is called the primary bond, and the primary bond strength is determined by the chemical composition of the polymer. The interaction between the chains is the much weaker Van der Waals force or hydrogen bonds. These interactions are called the secondary bonds, whose strengths are about 1–10% of the primary bonds. However, since the polymer chains are long, the total strength of secondary bonds usually exceeds that of the primary bonds. Due to the different structure of polymers as compared with metals and ceramics, the factor that affects the mechanical behavior of polymers is the molecular motion, rather than the motion of atoms and/or ions in metals and ceramics. Why different polymers may have different mechanical behaviors at room temperature (or at any other certain temperature)? Why the same polymer may behave differently at different temperatures? To answer these questions and understand the microstructure and macroscopic properties of polymers, we need to gain a thorough and deep understanding of the molecular motion in polymers. 5.9.1 The origin of molecular chain motion and its compliance The long polymer chains are usually twisted rather than straight. Upon external loading, these chains may become straight as a result of molecular motion and the associated internal rotation of the single bond on the backbones. The distribution of σ electron cloud of single chemical bond is axially symmetric, so during molecular motion these single bonds can rotate relative to their axis, which is called “internal rotation.” The internal rotation of the C–C bonds is completely free if there are no other atoms or functional groups connected to the backbone, but the bond angle is kept at 109°28’during the bond rotation. The internal rotation of the single bonds would cause the variation in the atomic arrangement and subsequently different morphologies of the molecules, which

326

Chapter 5 Diffusion in solids

are called “conformations.” There are thousands of chemical bonds in the polymer chains, and the bonds go through high frequency internal rotations (e.g., the internal rotation frequency of ethane molecules at 27 °C can reach as high as 1011–1012 times per second). Such an abundant of conformations may either straighten or twist the polymer chains. In addition, the conformation of polymer chains follows statistical rules, thus the probability of having a straight conformation is low because of its low configurational entropy. On the other hand, the probability of having a twisted conformation is high because of its high configurational entropy. Therefore, the morphology of polymer chains is usually twisted, which is caused by the internal rotation of the single bonds. The more freedom the internal rotation has, the higher tendency of having a twisted and compliant conformation of the polymer chains. In practice, because of the atoms and/or functional groups connected to the chain unit (mer unit), the internal rotation of the bonds is not completely free. In particular, when the atoms and functional groups are very close to each other, repulsive forces from the outer electron cloud would be generated, restricting the rotation of the bonds. Therefore, we should not consider the chain unit in the polymer chains as the basic unit for independent motion, and need to introduce a concept of “chain segment.” Specifically, if the polymer chain is regarded as a compliant chain made of a lot of stiff chain segments, then the chain segments are actually the basic units for independent motion, where the movement of each chain segment is not related to the movement of other chain segments, that is, the movement of each chain segment is completely independent. The length of the chain segment can serve as a measure of the mobility (compliance) of the polymer chains. If the length of the chain segment is lp, and the length of the chain unit is l, we have   Δε (5:68) lp = l exp kT where lp is also called the sustained length, and Δε is the difference between the energy barriers of different conformations. When Δε!0, lp!l, that is, the length of the chain segment is approaching that of the chain unit, the polymer chain is then the most compliant; with increasing Δε, especially when Δε/kT≫1, lp increases as well, and eventually, if lp increases to the total length of the polymer chain L = nl (n is the degree of polymerization), the entire polymer chain would be comprised by a stiff segment and become a single stiff rod-shaped molecule, that is, the polymer chain would have no compliance at all. Therefore, the static compliance of the chains can be measured by the ratio between the length of the chain segment and the total length of the chain: Δε   lp l exp kT 1 Δε (5:69) = exp x= = L nl n kT Obviously, the polymer chain would be compliant only if x is small.

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5.9.2 The molecular motion of polymers and the influencing structural parameters The molecular motion of polymers can be divided into motion units of two different length scales, that is, the large scale unit and the small scale unit. The former refers to the motion of the entire polymer chain while the latter refers to the motion unit of chain segment or the units smaller than the chain segments. Alternatively, using the terminology of low molecules, the movement of the entire polymer chain can be called Brownian motion, and the movement of all kinds of small scale units can be called micro Brownian motion. At low temperatures where the thermal energy is insufficient to activate the movement of the entire chain and/or the chain segments, the motion would be limited to motion units smaller than the chain segments. Figure 5.26 illustrates the motion of the chain units along the backbone where there is small variation in the bond angle and bond length. The rotation of the side groups and motion inside the side groups are also demonstrated.

Fig. 5.26: Several small-scale movement units in polymer: 1, the motion of the chain units along the backbone; 2, the rotation of the side groups; 3, motion inside the side groups.

With increasing temperature, movement of part of the chain segments may be activated, although the entire polymer chain is frozen. At this point, various conformations of the polymer chain (such as twisting or straightening) can appear in response to the external loading. Figure 5.27 represents a chain segment containing four carbon atoms where there are both enough free space and thermal energy to accommodate the motion of the four carbon atoms, so that the chain segment may diffuse by the internal rotation of the two carbon atoms at either end of the segment.

Fig. 5.27: The schematic drawing of the chain segment movement.

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If the temperature is further increased, the molecules would have even higher kinetic energy, so that the center of mass may start to displace, that is, flow of the polymer chain. However, unlike the low molecule compounds where the migration unit is the entire molecule, polymers flow in a manner similar to the movement of snakes, that is, the displacement of the polymer chain is achieved by the progressive migration of the chain segments, as demonstrated in Fig. 5.28.

(a)

(b)

Fig. 5.28: The schematic drawing of molecular conformations variation before and after the displacement of barycenter (a) before the center of mass displacement, (b) after the displacement of barycenter.

The compliance of the single bond rotation and the chain segments is determined by the structure of the polymer as well as the environmental parameters (such as temperature, pressure, and the ambient medium). The structural parameters that affect the compliance of the polymer chains include the following. 1. The structure of the backbone If the backbone is comprised exclusively by single bonds, the polymer chain would show a good compliance because the single bonds can go through internal rotation. The difficulty for internal rotation is affected by the structure of the backbone. In the three types of conventional backbone structures, the internal rotation of Si–O bonds is the easiest, followed by C–O bonds and then C–C bonds. This is because O atoms are not connected to any atoms or functional groups rather than the backbone, and the existence of C–O bonds provides more space between the nonbonded atoms than that in C–C bonds, making the internal rotation easier. For example, the O O compliance of polyethyleneglycol adipate (—[O—(CH2)6 —O—C—(CH2)4—C—C]— ) is n better than that of polyethylene, so the former can be used as a paint. Compared to the C–O bond, the Si–O bond is characterized by a longer bond length and larger Si–O–Si bond angle than C–O–C, making the internal rotation CH3 easier. For example, the compliance of polydimethylsiloxane ( —[Si—O—]n ) is excelCH3

lent and thus can be used as a type of synthetic rubber.

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When the backbone contains aromatic heterocycle, since aromatic heterocycle cannot go through internal rotation, the polymer chain would have low compliance but high stiffness and heat-resistance. For example, because of the benzene on the backbone, polycarbonate is both stiff and heat resistant, and can be used as an engineering plastic. Because the π electron cloud of double bond does not have axial symmetry, the polymer chains cannot go through internal rotation when the backbone contains double bonds. Therefore, polyphenyl and polyacetylene are stiff polymers. However, when the backbone contains separated double bonds, the compliance of the polymer chain would actually improve, as the absence of side groups or C–H bond on the carbon atoms as compared with single bonds would increase the distance between the atoms or the functional groups, making the repulsive force between the atoms or functional groups lower. For example, polychloroprene ( —[CH2)—C— ] ) contains separated double bonds, so its compliance is consid—CH2 — n

C1 ( —[ CH2—CH—]n ) . The former is used as erably better than that of polyvinyl chloride C1 a typical rubber, and the latter is a type of stiff and hard plastic.

2. The properties of substituent groups The polarity of substituent groups has a strong effect on the compliance of the polymer chains. The higher the polarity of the substituent groups, the stronger their interactions, the more difficult the internal rotation becomes, and subsequently the lower the compliance. For example, the substituent groups of polyacrylonitrile, polyvinyl chloride, and polypropylene are −CN, −Cl, and −CH3, respectively, where the polarity follows −CN > −Cl > −CH3, so the compliance increases from polyacrylonitrile to polyvinyl chloride and polypropylene. For nonpolarity substituent groups, the volume and symmetry are the main structural parameters. The larger the volume of the substituent groups, the more difficult the internal rotation, and the lower the compliance of the polymer chain. For example, H H H H polypropylene ( —[ C—C—]n ) has lower compliance than polyethylene ( —[ C—C—]n ). H CH3

H H

The symmetry of the substituent groups also has a significant effect on the compliance. Symmetric distribution of the substituent groups would increase the distance between the backbones, lower the force between the chains, and thus more favorable for internal rotation and improving the compliance. For example, the side

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H CH3

substituent groups in polyisobutene ( —[ C—C—]n ) are symmetric, while the side subH CH3

stituent groups in polypropylene are nonsymmetric, thus the compliance of polyisobutene is better than that of polypropylene, so polyisobutene can be used as rubber.

3. The length of the chains The length of the polymer chains is related to its relative molecular weight. The larger the relative molecular weight, the longer the polymer chain. If the polymer chain is short, the number of single bonds that can go through internal rotation would be small, and subsequently the number of conformations would also be limited. Therefore, low molecule materials are always stiff and lack of compliance. On the other hand, if the polymer chain is long, the number of single bonds would be large, thus the compliance of chain would be better. However, when the relative molecular weight is higher than some threshold value (e.g., 104), the conformation of the molecules would follow statistical rules, and the influence of relative molecular weight on compliance would become minimal. The discussions above deal with a few main structural parameters that may affect the compliance of the polymer chains. When multiple parameters are acting at the same time, we should analyze and determine which parameter(s) is (are) the dominant one(s). 4. Degree of cross-linking When the polymer backbones are cross-linked, the internal rotation of the single bonds close to the cross-linking would be restricted. When the degree of crosslinking is low, the molecular chain length between the cross-linking points would be much longer than that of the chain segment, so the chain segments (the motion units) would still be mobile and thus the compliance of the polymer would still be high. On the other hand, if the degree of cross-linking is high, the motion of the chain segments would be restricted, and the number of conformations would become low, leading to a reduced or even diminished compliance of the polymer. 5. Crystallinity When the polymer chains are crystalline, the interaction force between the chains is high because of the restriction from the lattice energy, so the compliance is low. When the semicrystalline polymers contain microcrystals, the crystalline regions serve as physical cross-linking, making the polymer chain movement close to amorphous regions more difficult. Therefore, the higher the crystallinity, the lower the compliance of the polymer.

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5.9.3 The molecular motion of polymers at different mechanical states Polymers can be divided into linear, branched, and space (3D-network) types according to their geometry. On the other hand, the state of aggregation includes crystalline structure, amorphous structure, orientation structure, liquid crystal structure, and textured structure. In this part, we will describe the mechanical states of crystalline and amorphous polymer chains of different geometries, and discuss the corresponding mechanisms of molecular motion.

Deformation (%)

1. The three mechanical states of linear amorphous polymers If a constant force is applied to a sample of amorphous polymer, a deformationtemperature (or thermal mechanical) curve can be obtained, as illustrated in Fig. 5.29.

Rubber-like state

Glassy state

O

Tb

Tg

Viscous flow state

Tf

Td

Temperature (°C) Fig. 5.29: The deformation–temperature curve of linear amorphous polymers.

From the figure, we know that the amorphous polymer is stiff at low temperatures, and the deformation upon loading is small. With increasing temperature, the sample would enter into a region where the sample becomes a soft elastomer, with significant but relatively stable deformation. If the temperature is further increased, the deformation would increase rapidly and the sample would finally become a complete viscous fluid. Therefore, according to the temperature dependence of the mechanical properties, we may classify the mechanical states of linear amorphous polymers into three categories: the glassy state, the rubber-like state, and the viscous flow state. The transition temperature between the glassy state and the rubber state is called the glass-transition temperature (Tg), and the transition temperature between the rubber state and the viscous flow state is called the viscous flow temperature or softening temperature (Tf). The three mechanical states of linear amorphous polymers are the macroscopic representation of the different states of internal molecular motion at different temperatures.

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a. The glassy state Below Tg, the molecules do not have sufficient energy to overcome the energy barrier for internal rotation, so the motion of the chain segments cannot be activated and the chain segments are “frozen.” In this situation, only motion units smaller than the chain segments such as chain units and/or side groups can move, as demonstrated by Fig. 5.23, thus the polymer chains are not able to transform from one conformation to another. Upon external loading, as the motion of the chain segments is frozen, the bond length and bond angle of the backbone can only vary slightly (the covalent bonds would be broken if the variation in bond length and/or angle is too significant). Therefore, the deformation would immediately recover once the external load is released. In other words, the deformation is fairly small and the stress-strain relationship follows Hooke’s law. The deformation in this region is thus called “universal elasticity,” and the amorphous polymers in this universal elasticity region are considered to be under the glassy state. Amorphous polymers in the universal elasticity are characterized by high elastic modulus, ranging from 1010–1011 Pa, with little reversible deformation on the order of about 0.01–0.1%. b. High elastic state When the temperature is raised above Tg, the kinetic energy of the molecules increases, and the extra volume expansion also increases the free volume (the space in the random packing structure), making the free rotation of the chain segments possible, as shown in Fig. 5.24. Meanwhile, various conformations can be achieved by the internal rotation of the single bonds along the backbone. However, the kinetic energy of the molecules is still not sufficient to cause the collective movement of the polymer chains and the relative motion between the chains. At high elastic (or rubber-like) state, the polymer chains would react in response to the external loading by single bond internal rotation and the change in the conformations of the chain segments: when loaded, the polymer chains can transform from a twisted state to a straightened state, which would cause considerable deformation macroscopically; upon release of the load, the polymer chains would return to their initial twisted state by single bond internal rotation, resulting in the macroscopic elastic contraction. Obviously, as the motion of the chain segments is time-dependent, such elastic deformation is also time-dependent (relaxation behavior). Moreover, this type of deformation is a process of backbone internal rotation triggered by external loads, the external load needed is much lower than that required for the deformation of polymers under the glassy state (with changes in the bond length and angle), and the deformation amount is also substantially higher, on the order of 100–1,000%. Meanwhile, the elastic modulus would reduce to 105–107 Pa. This kind of mechanical behavior is a rubber-like behavior. The high elastic state mainly involves the motion of the chain segments, which is unique for polymers. Therefore, the high elastic state is a unique mechanical behavior for

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polymers. The higher the relative molecular mass, the higher number of chain segments, and the wider the range of the high elastic region. On the other hand, the lower the relative molecular mass, the range of the high elastic region would become smaller or even diminished. c. Viscous flow state When the temperature exceeds Tf, the kinetic energy of the molecules would further increase, and the collective or sequential directional movement of the chain segments, that is, the relative displacement of the center of mass of the polymer chains, may become possible. In this case, as demonstrated in Fig. 5.25, the relative sliding between the polymer chains would cause a large irreversible deformation, that is, viscous flow, and the polymer becomes a viscous fluid. Because the viscous flow state is mainly correlated with the polymer chain motion, the longer the polymer chains, the higher the sliding resistance between the chains, the higher the viscosity, and subsequently the higher Tf. Because of the dispersed nature of the polymer chains in length, the transition from the rubber-like state to the viscous flow state covers a wide range of temperature. In practice, the molding and processing temperature of polymers should fall in the range between Tf andTd, where Td is the polymer decomposition temperature. Above Td, the polymer would decompose before gasification because of the melt of the backbone. 2. The mechanical behavior of amorphous space polymers Amorphous space polymers are a type of three-dimensional network structure where the polymer chains are connected by branched chains or chemical bonds. The characteristic of molecular motion is related to the density of the cross-linking. The deformation–temperature curve of space phenolic resin added with a different amount of cross-linking agent hexamethylenetetramine is shown in Fig. 5.30. When the amount of cross-linking agent added is low (as shown in Fig. 5.30, Curve 1), the degree of cross-linking is also low, so there is little resistance to molecular motion and a large number of chain segments can be thermally activated, leading to macroscopic glassylike or rubber-like behavior. However, cross-linking restricts the motion of large molecular chains, making the displacement of their center of mass difficult, so the viscous flow state is absent. With increasing addition of cross-linking agent, the density of crosslinking increases, and the distance between the cross-linking points becomes shorter. This would give a higher resistance to motion, a higher Tg, and a smaller high-elastic region, as shown by Curve 2 in Fig. 5.30. When the amount of crosslinking agent added is increased above a threshold value (the density of cross-linking is increased above a threshold value), the motion of chain segments would become diminished and only glassy-state of the polymers remains. In that case, as demonstrated by Curve 3 in Fig. 5.30, the mechanical behavior would become the same with that of low molecule compounds.

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3. Mechanical behavior of crystalline polymers Crystalline polymers have a fixed and clearly defined melting temperature (Tm). Below Tm, completely crystalline polymers are strong and hard, as a result of the closely packed molecular structure, the strong interaction force between the molecules, the high resistance to the motion of chain segments, and subsequently the absence of the rubber-like state. Above Tm, if the relative molecular mass of the polymer is low, the molecular motion would make the polymer a fluid, entering the viscous flow region, because the crystalline structure is destroyed by the molecular motion and is converted to amorphous structure. In this case, Tm is the viscous flow temperature. On the other hand, if the relative molecular mass of the polymer is high, at T > Tm, the high elastic state would appear, as the kinetic energy of the molecules is still not sufficient to activate the collective motion of the long polymer chains, and only the chain segment motion is possible. When T > Tf, the center of mass of the polymer would displace, causing relative sliding between the molecules and making the polymer into the viscous flow region. The deformation–temperature curve of crystalline polymers is demonstrated in Fig. 5.31.

Deformation (%)

30 20 1 10

2

0 50

75

100

125

150

Temperature (°C)

Deformation (%)

1

O

Tg

Tm Temperature (°C)

Tf

3 175

Fig. 5.30: The deformation–temperature curve of space phenolic resin: 1, adding 3 wt.% hexamethylenetetramine; 2, adding 5 wt.% hexamethylenetetramine; 3, adding 11 wt.% hexamethylenetetramine.

2

Fig. 5.31: The deformation–temperature curve of crystalline polymers: 1, the relative molecular mass of the polymer is low and Tm = Tf ; 2, the relative molecular mass of the polymer is high and Tm = Tf.

As a matter of fact, completely crystalline polymers do not exist, and there is always a substantial fraction of amorphous zones in crystalline polymers. These amorphous zones go through glass transition and viscous flow transition as discussed

5.9 Molecular motion in polymers

335

previously, and thus possess the glassy state, the high elastic state and the viscous flow state in different temperature ranges. Therefore, the mechanical behavior of crystalline polymers is determined by the combined effect of crystalline zones and amorphous zones. In addition, the mechanical behavior of crystalline polymers also depends on the relative molecular mass of the polymers. The high elastic region can be further divided into a leather state and a rubber state. When Tg < T < Tm, the crystalline regions are still at the strong and hard crystalline state, while the amorphous regions have already transformed into the compliant rubber-like state. The combined effect would make the polymer both hard and tough at the macroscopic scale, and such a state is called the leather state. The correlation between the mechanical behavior and temperature as well as the relative molecular mass for partially crystalline polymers is illustrated in Fig. 5.32. Transition region

Temperature

Viscous flow state

Rubber state Tm Leather state Tg

Hard crystalline state

O

Relative molecular mass

Fig. 5.32: The correlation between the mechanical behavior and temperature as well as the relative molecular mass for partially crystalline polymers.

In summary, the different mechanical behaviors of different polymers at room temperature (or any certain temperature) can be rationalized by the microscopic molecular motion. In particular, the difficulty of the molecular motion depends on the molecular structure of the polymers. When the thermal energy provided at room temperature is not sufficient to activate the motion of the chain segments, the polymer would be at the strong and hard glassy state or the crystalline state. On the other hand, if the thermal energy provided at room temperature is able to “unfreeze” the motion of the chain segments, but is still insufficient to activate the collective movement of the entire polymer chain (the displacement of the center of mass), the polymer would be at the high elastic state. Finally, if the thermal energy provided at room temperature is sufficient to activate the collective movement of the entire polymer chain (the displacement of the center of mass), the polymer would be at the viscous flow state. Similarly, for a given polymer, different temperatures would result in different states of the molecular motion in the polymer, and subsequently the different mechanical behaviors.

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W. D. Callister Jr. and D. G. Rethwisch, Materials Science & Engineering – An Introduction, 8th, John Wiley & Sons, Inc, 2009. E. J. Mittemeijer, Fundamentals of Materials Science, Springer Heidelberg Dordrecht, London New York, 2010. E. J. Hearn, Mechanics of Materials 2, An Introduction to the Mechanics of Elastic and Plastic Deformation of Solids and Structural Materials, 3th, Butterworth-Heinemann, 2001. R. J. Arsenault, Treatise on Materials Science and Technology Volume 6: Plastic deformation of Materials, Academic press INC, 1975. A. S. Argon, Strengthening Mechanisms in Crystal Plasticity, Oxford University Press, 2008. P. R. Rios, F. Siciliano Jr, H. Ricardo, Z. Sandimc, R. L. Plaut, A. F. Padilha and, A. F. Padilha, Nucleation and growth during recrystallization, Materials Research, 8, 3, 225–238, 2005. G. Gottstein and L. S. Shvindlerman, Grain Boundary Migration in Metals 2rd Edition, Thermodynamics, Kinetics, Applications, CRC Press, 2010. T. Sakai, A. Belyakov, R. Kaibyshev, H. Miura and J. J. Jonas, Dynamic and post-dynamic recrystallization under hot, cold and severe plastic deformation conditions, Progress in Materials Science, 60, 2014, 130–207. R. P. Smith, The diffusivity of carbon in iron by the steady state method, Acta Metallurgica, 1, 1953, 578–587. A. Kumar, H. Barda, L. Klinger, M. W. Finnis, V. Lordi, E. Rabkin and D. J. Srolovitz, Anomalous diffusion along metal/ceramic interfaces, Nature Communications, 9, 2018, 5251. K. Cheng, H. Xu, B. Ma, J. Zhou, S. Tang, Y. Liu, C. Sun, N. Wang, M. Wang, L. Zhang and Y. Du, An in-situ study on the diffusion growth of intermetallic compounds in the Al–Mg diffusion couple, Journal of Alloys and Compounds, 810, 2019, 151878. D. V. Regone, Thermodynamics of Materials, I, John Wiley & Sons, New York, 1995. D. A. Porter, K. E. Easterling and M. Sherif, Phase Transformations in Metals & Alloys, 3rd, CRC Press, Boca Raton, 2009. D. R. Askeland, The Science and Engineering of Materials, Springer Nature, 1996. Lee., Solar Cells, Thermal Design Heat Sinks Thermoelectrics Heat Pipes Compact Heat Exchangers and Solar Cells, 2010. J. G. Speight, Inorganic Chemistry, Elsevier BV, 2017. T. B. Massalski, Structure and Stability of Alloys, Elsevier BV, 1996. R. Ferro and A. Saccone, Structure of Intermetallic Compounds and Phases, Elsevier BV, 1996. Z. L. Wang and Z. C. Kang, Functional and Smart Materials, Springer Nature, 1998. U. Messerschmidt, Dislocation Dynamics During Plastic Deformation, Springer Science and Business Media LLC, 2010. A. Bhaduri, Mechanical Properties and Working of Metals and Alloys, Springer Nature, 2018. T. T. Fang, Line Defects in Crystalline Solids, Elsevier BV, 2018. Y. Rong, Characterization of Microstructures by Analytical Electron Microscopy (AEM), Higher Education Press-Springer, 2012. C. W. Robert and H. Peter, editors, Physical Metallurgy, 1, North-Holland, 1999. Brown and Ashby, Acta Metallurgica, 28, 1980, 1085.

https://doi.org/10.1515/9783110495348-006

Index abnormal growth 257–258 activation energy 307, 313–316, 318, 320, 321, 326, 327 atomic 3–31

kink 197, 206, 211–213 nucleation 208–209, 240–245, 248–250

bonding 3–31 boundary 137, 138, 140, 145, 156, 159, 163, 164, 173, 175–187 burgers vector 141–146, 151, 153–155, 157, 159, 161, 163–165, 167–170, 172, 176, 178, 182

periodic table 8–9, 15 phase 67, 69–94, 107, 112, 115, 126–128 plastic deformation 188, 193, 195, 197–234, 239, 262, 263, 270, 272, 273, 277–282 point defects 131–136, 148, 165 polymer 110, 112–121, 123, 124, 126–130 polymer chain 15–31

carburization 285, 290, 291, 318, 319 concentration gradient 284, 285, 287, 294, 298, 301, 304, 319, 321 crystallography 32–59

recovery 196, 234–276 recrystallization 188–283

defects 131–187 dislocation 136–173, 176–179, 182, 184, 186 elasticity 188–196, 280 energy 132, 134–136, 138, 144, 148, 149, 152–154, 161, 163, 166, 167, 170, 172–175, 181–185, 187 grain Boundary Migration 241, 243, 244, 249, 251, 253, 254, 256, 257, 261, 267 growth 207–209, 235, 236, 240, 243, 244, 248–262, 266

slip 195, 197–206, 208–214, 218, 220, 223, 226, 229–232, 239–240, 249, 271, 276–278, 280, 283 stacking faults 161–165, 167–169, 171, 173, 185 steady-state diffusion 285 strain aging 217, 219–222 strengthening mechanism 204, 217, 218, 224–226 structure 32–130 substitutional diffusion 297–301, 313, 314 twin 197, 206–213, 226, 256, 260–262, 271

homogenization 294–296 viscoelasticity 188–196, 280 imperfect Elasticity 193–195 interatomic 3–31 interface 132, 159, 173–187 interstitial diffusion 307–309, 313–315, 325 ionic 79, 80, 94–110

https://doi.org/10.1515/9783110495348-007

yield phenomenon 219–222, 231