Material Science - Unit I - R2 PDF

MM1201: Material Science (3-0-0) for Department of Metallurgical and Materials Engineering Dr. Anushree Dutta, Assistant Professor Dept. of Metallurgical and Materials Engineering Email Id: [email protected] UNIT-I NIT Jamsh...

MM1201: Material Science (3-0-0) for Department of Metallurgical and Materials Engineering Dr. Anushree Dutta, Assistant Professor Dept. of Metallurgical and Materials Engineering Email Id: [email protected] UNIT-I NIT Jamshedpur  Syllabus UNIT-I Introduction: Types of materials from structure to property, Crystal structure: Bravias lattices, Lattice direction and planes. Crystal Imperfections: point, line and planar defect. Classification of Engineering Materials: Crystalline and non-crystalline, polymer, ceramics, composites, metal and alloys, glass, Classification on the basis of structure. Crystal vs lattice, BCC, FCC, HCP, 3D packing, ABC, AB packing density, calculation of theoretical density, planes, closed packed plane and direction concept, related numerical problems 7 system, 14 Bravais lattices, Miller indices / Miller Bravais Planar density, Packing fraction, Voids in (SC, FCC, BCC, HCP) Crystal Imperfections: point, line and planar defect. Point defect: (Vacancy, Interstitial, Substitutional, Defect in ionic solid, Frenkel, Schottky) Line Defect: (Dislocation. Edge, screw, Burger circuit, Burger Vector, glide, climb, cross-slip, energy of dislocation) Planar Defect: Grain boundary, Twin, Stacking faults. UNIT-II: Deformation of material, Mechanical properties of materials: Hardness (types of hardness measurements), Tensile (stress-strain diagram, all properties derived from stress-strain curve: true stress-strain, elastic-plastic, conversion from engineering to true stress-strain curve, yield point phenomenon, proof stress, measure of ductility, effect of temperature on mechanical properties) Impact (Charpy and Izod test, energy absorbed vs. temperature curve, effect of variables on this transition curve) Fatigue (Concept of fatigue, Failure process of fatigue, crack initiation, crack propagation and growth, different cycle of loading, S-N curve, fatigue life concept) Creep of metals (Creep curve, different stages of creep, dependence on stress and temperature, creep rate, variety of creep tests at constant load and constant stress, concept of creep mechanisms (dislocation, grain boundary sliding, coble, NH creep) UNIT-III: Electron theory of Metals: Bond theory, Uncertainty principle, Free electron theory, Zone theory, The dependence of the energies on the wave number, The density of state curves, Conductors and insulators, Semiconductors, Dielectric behavior, Ferroelectricity, Piezoelectricity, Magnetism. UNIT-IV: Principles of solidification: Nucleation and growth, Homogeneous and heterogeneous nucleation, Phase Diagrams: Phase rule, Isomorphous, eutectic, peritectic, eutectoid and peritectoid transformation, Fe-cementite diagram. Heat Treatment of Steel: TTT diagram, different heat treatment process: Annealing (Recovery recrystallization and grain growth), Normalizing and Hardening, Hardenability. UNIT-V: Selection of Engineering Materials: Common engineering materials including metals and alloys (Copper and Aluminium alloys), Ceramic materials, Composite Materials, Polymer materials. Building materials, Transformer materials Text Books: 1. Material Science and Engineering by William D. Callister 2. Material Science and Engineering by V. Raghavan 3. Physical Metallurgy Principles by Abbaschian.R, Abbaschian.L, Reed Hill R.E. Reference Book: 1. Modern Physical Metallurgy by R. E Smallman and R. J. Bishop Materials Materials are substances whose properties make them useful in structures, machines, devices or products to serve the purpose. Material Science involves study of relationships between synthesis, processing, structure, properties and performance of materials that enables engineering function. It also involves discovery and design of new materials. Importance We all are surrounded by different types of materials Materials drive our society and without materials, there is no engineering To select a material for a given application considering its properties, cost and performance. To understand the limits of materials and change their properties depending on application. Able to create a new material that will have some desirable and superior properties. Materials science involves investigating the relationship between structures & properties of materials.  Materials scientists make the materials that make everything better! Everything is made of something. Materials scientists investigate how materials perform and why they sometimes fail. By understanding the structure of matter (from atomic scale to millimeter scale), they invent new ways to combine chemical elements into materials with unprecedented functional properties. Other branches of engineering rely heavily on materials scientists and engineers for the advanced materials used to design and manufacture products such as safer cars with better gas mileage, faster computers with larger hard drive capacities, smaller electronics, threat-detecting sensors, renewable energy harvesting devices and better medical devices. Classification of materials Engineering materials can broadly be classified based on their nature : 1.Metals (Approx. 3/4 of the elements in periodic table is metal) 2.Ceramics 3.Polymers (Their chemistries are different, and their mechanical and physical properties are different) 4.Composites (nonhomogeneous mixture of the other three types, rather than a unique category) E.g. Metal-matrix composite, ceramic-matrix composites, polymer matrix composites  Classification of Materials : Metals ▪ Metals have these typical physical properties: ▪ Lustrous (shiny) ▪ Capable of changing their shape permanently ▪ Hard ▪ High density (are heavy for their size) ▪ High tensile strength (resist being stretched) ▪ High melting and boiling points ▪ Good conductors of heat and electricity E.g. Steels, aluminium, copper, silver, gold, Brasses, bronzes, manganin, invar, Superalloys, Boron rare earth, magnetic alloys Classification of Materials: Ceramics ▪ Nonmetallic inorganic substances which are brittle and have good thermal and electrical insulating properties ▪ High melting points (so they're heat resistant). ▪ Great hardness and strength. ▪ Considerable durability (they're long-lasting and hard-wearing). ▪ Low electrical and thermal conductivity (they're good insulators). ▪ Chemical inertness (they're unreactive with other chemicals). E.g. MgO, CdS, Silica, soda-lime-glass, concrete, cement, Ferrites and garnets Classification of Materials: Polymers ▪ Corrosion resistance and resistivity to chemicals, ▪ Low electrical & Thermal conductivity, ▪ Low density, ▪ High strength to weight ratio, particularly when reinforced, ▪ Noise reduction, ▪ Ease of manufacturing and complexity of design possibilities ▪ Relatively low cost. E.g. Plastics: PVC, PTFE, polyethylene, Fibres: terylene, nylon, cotton Natural and synthetic Rubbers, Leather  COMPOSITES – Light, strong, flexible – High costs Eg. Metal-matrix composite, ceramic-matrix composites, polymer matrix composites  ADVANCED MATERIALS Materials that are utilized in high-tech applications Semiconductors Have electrical conductivities intermediate between conductors and insulators Biomaterials Must be compatible with body tissues Smart materials Could sense and respond to changes in their environments in predetermined manners Nanomaterials Have structural features on the order of a nanometer, some of which may be designed on the atomic/molecular level  Crystal geometry Crystal Lattice Motif/Basis 7 crystal system 14 Bravais lattice Miller indices of directions and planes Crystal A 3D periodic arrangement of atoms in space in termed as crystal. Lattice A 3D periodic arrangement of points in space in termed as lattice. Crystal lattice A 3D periodic arrangement of A 3D periodic arrangement of atoms in space points in space Physical object Geometrical concept It has some physical properties It has only geometrical properties such as weight, density, electrical and thermal conductivity Relationship between crystal and lattice Crystal = lattice + Motif or basis Motif or basis: An atom or a group of atoms associated with each lattice point is called a motif or basis of the crystal Crystal = lattice (Underlying periodicity of crystal (How to repeat)) + Motif or basis (atom or group of atoms which is periodically repeated (what to repeat)) Crystal = lattice + Motif or basis (in 2D) Crystal Structure: Unit Cell Materials can be broadly classified as crystalline and noncrystalline solids. In a crystal, the arrangement of atoms is in a periodically repeating pattern, whereas no such regularity of arrangement is found in a noncrystalline material. A region of space which can generate entire lattice by repetition through lattice translation. Crystal structure tells us the details of the atomic arrangement within a crystal. It is usually sufficient to describe the arrangement of a few atoms within what is called a unit cell. The crystal consists of a very large number of unit cells forming The unit cell is the smallest unit, regularly repeating patterns in space. The which, when repeated in space main technique employed for determining indefinitely, will generate the the crystal structure is the X-ray space lattice. diffraction. A crystalline solid can be either a single crystal, where the entire solid consists of only one crystal, or an aggregate of many crystals separated by well-defined boundaries. In the latter form, the solid is said to be polycrystalline.  Crystal Systems ▪ Only 7 crystal systems have been identified. ▪ These 7 basic crystal systems are called Primitive lattices. ▪ Unit cell of a primitive lattice contains atoms only the corners. Crystal Structure: Unit Cell Parameters of a Unit Cell ❑ Unit cell is smallest repeatable entity that can be used to completely represent a crystal structure. ❑ It can be considered that a unit cell is the building block of the crystal structure and defines the crystal structure by virtue of its geometry and the atom positions within.  Parameters of a Unit Cell ❑ The type of atoms and their radii R, ❑ Cell dimensions (Lattice spacing a, b and c) ❑ Angle between the axis α, β, γ. Important Parameters of a Unit Cell Number of atoms per unit cell (n). For an atom that is shared with m adjacent unit cells, we only count a fraction of the atom, 1/m. n = (1/8 x 8) + 1 = 2 Important Parameters of a Unit Cell CN, the coordination number, which is the number of nearest neighbours to given atom. CN = 6 Important Parameters of a Unit Cell APF, the atomic packing factor, which is the fraction of the volume of the cell actually occupied by the hard spheres. APF = Sum of atomic volumes/Volume of cell.  Bravais Lattice Bravais showed that there are 14 possible arrangement of points (atoms) in the space known as Bravais lattices.  Crystal Structure of Metals ❑Most of the metals crystallize into three forms of crystal systems: ❑ 1) Face-Centered Cubic Structure (FCC) ❑ 2) Body-Centered Cubic Structure (BCC) ❑ 3) Hexagonal Close Paced Structure (HCP) Simple Cubic Cell Number of atoms (n) = 1/8 x 8 = 1 R Effective length of unit cell (a) = 2R Co-ordination Number (CN) = 6 Volume of unit cell (Vc) = a a3 = (2R)3 = 8R3 Volume of all atoms in the unit cell ( Vs) = n x 4/3 πR3 = 4/3 πR3 Atomic Packing Factor (Efficiency) of the cell ()= Vs/Vc = 52.4% Void = 100 -  = 47.6 % Body Centered Cubic Cell R Number of atoms (n) = (1/8 x 8) + 1 = 2 4R Effective length of unit cell (a) = 4/3½ R Co-ordination Number (CN) = 8 Volume of unit cell (Vc) = a3 = (4/3½ R)3 a a Volume of all atoms in the unit cell ( Vs) = n x 4/3 πR3 = 8/3 πR3 Atomic Packing Factor (Efficiency) of the cell () = Vs/Vc = 68 % Void = 100 -  = 32 % Face Centered Cubic Cell Number of atoms (n) =  (1/8 x 8) + (1/2 x 6 ) = 4 Effective length of unit cell (a) = 2 2½ R Co-ordination Number (CN) = 12 Volume of unit cell (Vc) = a3 = (2 2½ R)3 Volume of all atoms in the unit cell ( Vs) = n x 4/3 πR3 = 16/3 πR3 Atomic Packing Factor (Efficiency) of the cell () = Vs/Vc = 74 %  Void = 100 -  = 26 % Hexagonal Closed Packed cell Number of atoms (n) = Effective length of unit cell (a) = Co-ordination Number (CN) = Volume of unit cell (Vc) = Volume of all atoms in the unit cell ( Vs) = R R Atomic Packing Factor (Efficiency) of the cell () = Void = 100 -  = Hexagonal Closed Packed Cell Volume of unit cell (Vc) =  Base consists of 6 triangles.  Area of Base = 6 x ½ x a x L  = 3a2 sin 60o Volume of HCP cell = Area of base x height  = 3a2 sin 60o x c  ⇒ c/a = 1.633. Volume of HCP cell (Vc) =  = 3a2 sin 60o x c  = 4.2426 a3 = 4.2426 (2R) 3 a L a Hexagonal Closed Packed Cell ❑ Number of atoms (n) = ❑ (1/6 x 12) + (1/2 x 2) + 3 = 6 ❑ Effective length of unit cell (a) = 2R ❑ Co-ordination Number (CN) = 12 ❑ Volume of unit cell (Vc) = 4.2426 (2R) 3 ❑ Volume of all atoms in the unit cell ( Vs) = n x 4/3 πR3 = 6 x 4/3 πR3 ❑ Atomic Packing Factor (Efficiency) of the cell () = Vs/Vc = 74 % ❑ Void = 100 -  = 26 % Coordination number For metals, each atom has the same number of nearest-neighbor or touching atoms, which is the coordination number (CN). For Simple Cubic, CN = 6, For body centered cubic, CN = 8, and For face-centered cubic and Hexagonal closed packed lattice the CN=12. Crystal Structure of Metals Miller Indices of direction Put in square brackets [1 0 0] Miller Indices of direction Miller Indices of a family of symmetry related direction Miller Indices of a family of symmetry related direction Miller Indices of planes Miller Indices of planes  Miller Indices of planes X Miller Indices of planes Miller Indices of planes  Miller Indices for direction and planes A technique to denote various directions and planes in crystal (discussed in class) Steps to be followed to determine direction and planes (discussed) Miller Indices of family of symmetry related directions and planes for cubic crystal and tetragonal crystal (discussed) Weiss Zone law (discussed)  Miller-Bravais Indices of HCP crystals In HCP crystal, Miller-Bravais indices are of a 4-axis coordinate system for 3-dimensional crystals. This coordinate system is based on the 3-axis Miller index, but with an extra axis which is used for hexagonal crystals. The system can indicate directions or planes, and are often written as (hkil) Convert [u’v’w’] to [uvtw] and (hkl) index to (hkil) (next slides) For Miller-Bravais indices, we need to label 4 axes in the hexagonal crystal. In the basal plane, we have 3 axes of equal length each separated by 120º, which we call a1, a2, and a3 (they are each the same as the lattice parameter “a”). Then there is the c axis, perpendicular to those three.  Three-index to four-index system for direction in hexagonal crystal Three axes are all contained within a single plane (called the basal plane) and are at 120 degree angles to one another. The z axis is perpendicular to this basal plane. To convert [u’v’w’] to [uvtw] 2𝑢′ −𝑣 ′ u= 3 2𝑣 ′ −𝑢′ v= 3 t = -(u+v) w = 𝑤′ ത Therefore, becomes Ref: Callister & Rethwisch  Crystallographic planes for HCP For crystals having hexagonal symmetry, it is desirable that equivalent planes have the same indices; as with directions, this is accomplished by the Miller–Bravais system This convention leads to the four-index (hkil) scheme. i is determined by the sum of h and k through i = -(h+k) We can convert the (hkl) plane to the (hkil) four index system plane  Hexagonal Miller-Bravais Coordinate System for Planes Planes in the 4-axis system are very similar to the 3-axis system as “h,” “k,” and “l” are the same in both systems. “i” is simply defined by the formula h + k = -i. Therefore, (112ത 0) becomes (110), (101ത 0) becomes (100), and (23ത 11) becomes (23ത 1) Ref: Callister & Rethwisch Linear and planar density Linear and planar densities are important considerations relative to the process of slip- that is, the mechanism by which metals plastically deform Slip occurs on the most densely packed crystallographic planes and, in those planes, along directions having the greatest atomic packing. Directional equivalency is related to linear density (LD) Equivalent direction have identical linear density Linear density is defined as the number of atoms per unit length whose centers lie on the direction vector for a specific crystallographic direction i.e. number of atoms centered on direction vector LD = length of direction vector units of linear density are reciprocal length (e.g., nm-1, m-1).  Linear density Determine the linear density of the direction for the FCC crystal structure An FCC unit cell (reduced sphere) and the direction therein are shown in Figure a. Represented in Figure b are the five atoms that lie on the bottom face of this unit cell; The direction vector passes from the center of atom X, through atom Y, and finally to the center of atom Z. With regard to the numbers of atoms, it is necessary to take into account the sharing of atoms with adjacent unit cells. Each of the X and Z corner atoms is also shared with one other adjacent unit cell along this direction (i.e., one-half of each of these atoms belongs to the unit cell being considered), while atom Y lies entirely within the unit cell. Thus, there is an equivalence of two atoms along the direction vector in the unit cell. Now, the direction vector length is equal to 4R (Figure b); thus, the linear density (LD) for FCC: (a) Reduced-sphere FCC unit cell with the direction indicated. (b) The bottom face-plane of the FCC unit cell in (a) on which is shown the atomic spacing in the direction, through atoms Ref: Callister & Rethwisch labeled X, Y, and Z.  Planar density Planar density (PD) is taken as the number of atoms per unit area that are centered on a particular crystallographic plane, or number of atoms centered on plane PD = area of plane The units for planar density are reciprocal area (e.g., nm-2, m-2). For example, consider the section of a (110) plane within an FCC unit cell as represented in Figures a and b. Although six atoms have centers that lie on this plane (Figure b), only one-quarter of each of atoms A, C, D, and F, and one-half of atoms B and E, for a total equivalence of just 2 atoms, are on that plane. The area of this rectangular section is equal to the product of its length and width. From Figure b, the length (horizontal dimension) is equal to 4R, whereas the width (vertical dimension) is equal to 2R 2 (corresponds to the FCC unit cell edge length). Thus, the area of this planar region is (4R)(2R 2 ) , and the planar density is determined as follows Figure (a) Reduced phere FCC unit cell with the (110) plane. (b) Atomic packing of an FCC (110) plane. Corresponding atom positions from (a) are Ref: Callister & Rethwisch indicated. Voids in closed packed structure Tetrahedral voids Octahedral voids Size of the voids Radius of the largest sphere that can fit inside the void without displacing the spheres at the corners defining the void. i. The largest sphere that can fit inside the tetrahedral void is 0.225 R ii. The largest sphere that can fit inside the octahedral voids is 0.414 R Closed-packed crystal structures a) A portion of a close-packed plane of atoms; A, B, and C positions are (b) The AB stacking sequence for close- indicated. packed atomic planes. Ref: Callister & Rethwisch HCP ❑ The real distinction between FCC and HCP lies in where the third close-packed layer is positioned. ❑ For HCP, the centers of this layer are aligned directly above the original A positions. This stacking sequence, ABABAB… is repeated over and over. ❑ Of course, the ACACAC... arrangement would be equivalent. ❑ These close-packed planes for HCP are (0001) type planes Close-packed plane stacking sequence for hexagonal close-packed. Ref: Callister & Rethwisch FCC ❑ In face-centered crystal structure, the centers of the third plane are situated over the C sites of the first plane. ❑ This yields an ABCABCABC... stacking sequence; that is, the atomic alignment repeats every third plane. ❑ More difficult to correlate the stacking of close-packed planes to the FCC unit cell. These planes are of the (111) type; an FCC unit cell is outlined on the upper left-hand front face of Figure b, (a) Close-packed stacking sequence for face-centered cubic. (b) A corner has been removed to show the relation between the stacking of close-packed planes of atoms and the FCC crystal structure; the heavy triangle outlines a (111) plane. Ref: Callister & Rethwisch  Crystalline and Noncrystalline Materials Single crystals In a crystalline solid, if the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal. All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature. They may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled. If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume a regular geometric shape having flat faces, as with some of the gemstones; A photograph of a garnet single crystal the shape is indicative of the crystal structure (See Figure). Ref: Callister & Rethwisch Polycrystalline Materials ❑ Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline. ❑ Various stages in the solidification of a polycrystalline specimen are represented schematically in Figure. ❑ First small crystals or nuclei form at various positions. These have random crystallographic orientations, as indicated by the square grids. ❑ The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each. The extremities of adjacent grains impinge on one another as the solidification process approaches completion. ❑ the crystallographic orientation varies from grain to grain. Also, there exists some atomic mismatch within the region where two grains meet; are called a grain boundary Ref: Callister & Rethwisch  Noncrystalline materials Noncrystalline solids lack a systematic and regular arrangement of atoms over relatively large atomic distances. Sometimes such materials are also called amorphous (meaning literally “without form”), or supercooled liquids, as their atomic structure resembles that of a liquid. An amorphous condition may be illustrated by comparison of the crystalline and noncrystalline structures of the ceramic compound silicon dioxide (SiO2), which may exist in both states. Figures a and b present two-dimensional schematic diagrams for both structures of SiO2. Even though each silicon ion bonds to three oxygen ions for both states, but the structure is much more disordered and irregular for the noncrystalline structure. Two-dimensional schemes of the structure of (a) crystalline silicon dioxide and (b) noncrystalline silicon dioxide. Ref: Callister & Rethwisch Noncrystalline materials Whether a crystalline or amorphous solid forms depends on the ease with which a random atomic structure in the liquid can transform to an ordered state during solidification. Amorphous materials, therefore, are characterized by atomic or molecular structures that are relatively complex and become ordered only with some difficulty. Rapidly cooling through the freezing temperature favors the formation of a noncrystalline solid, because little time is allowed for the ordering process. Metals normally form crystalline solids, some ceramic materials are crystalline, the inorganic glasses are amorphous. Polymers may be completely noncrystalline and semicrystalline consisting of varying degrees of crystallinity. Ref: Callister & Rethwisch Tetrahedral and Octahedral voids Position of the voids in FCC crystal Position of octahedral voids in FCC crystal, cube edges (0,0, ½), (0, ½,0), (½,0,0) and body centre (½, ½, ½) Position of tetrahedral voids are located in body diagonals at (¼, ¼, ¼), (¾, ¾, ¾) In HCP crystal same types of octahedral and tetrahedral voids are present Position of the voids in BCC crystal Coordinates of octahedral voids in BCC crystal, cube edges (0,0, ½), (0, ½,0), (½,0,0) Position of tetrahedral voids are located in body diagonals at (½, ¼, 0)  Structure of alloys When the molten metal are melted together and crystalized, a single crystal structure may form. In the unit cell of this crystal both the metal atoms are present in proportion to their concentration. This structure is called as solid solution It may be of three types 1. Random substitutional solid solution 2. Ordered substitutional solid solution 3. Interstitial solid solution Structure of alloys 1. Solid solution Variation in composition Usually the crystal structure of the solution is that of one of the components 2. Intermetallic compound Fixed composition Crystal structure different from either of the components Structure of alloys Examples: Interstitial solid solution: Austenite: Solid solution of C in -Fe (FCC) Ferrite: solid solution of C in -Fe (BCC) Substitutional solid solution: -brass: Solid solution of Cu (FCC) and Zn (HCP) (Limited solubility) Solid solution -brass has FCC structure Cu and Ni exhibit the complete solubility Interstitial solid solution: Geometrical limit of solid solubility in interstitial solid solution: all voids occupied Interstitial solid solution cannot exhibit unlimited solid solubility Substitutional solid solution:  Polymorphism and allotropy Some metals, as well as nonmetals, may have more than one crystal structure, a phenomenon known as polymorphism When found in elemental solids, the condition is often termed allotropy Example pure iron has a BCC crystal structure at room temperature, which changes to FCC iron at 912 °C. Most often a modification of the density and other physical properties accompanies a polymorphic transformation Imperfections/defects in crystal Based on the dimensionality of defects (on the order of an atomic diameter), classification of defects in crystal are as follows: Zero dimensional defect/Point defect – Ex. Vacancy, Interstitial defect, substitutional defect One dimensional defect/line defect – Ex. Dislocation Two dimensional defect/planar defect – Ex. Free surface, grain boundary, twin boundary, stacking fault Three dimensional defect – Ex. Voids, Inclusion, crack Point defects (Vacancy, Substitutional impurity and interstitial impurity) ❑ A vacancy refers to an atomic site from where the atom is missing (Fig. (a)) ❑ A substitutional impurity (or solute) is a point imperfection. It refers to a foreign atom that substitutes for or replaces a parent atom in the crystal (Fig. (b)) ❑ An interstitial impurity is also a point imperfection. It is a small sized atom occupying the void space in the parent crystal, without dislodging any of the parent atoms from their sites (Fig. (c)) ❑ An atom can enter the interstitial void space only when it is substantially smaller than the parent atom. Point imperfections in an elemental crystal: (a) vacancy; (b) substitutional impurity; (c) interstitial impurity; and (d) field ion micrograph of platinum showing a vacancy. [(d) Courtesy: E.W. Mueller.]  Point defects a. Vacancy b. Interstitialcy c. Interstitial impurity d. Substitutional impurity  Point defects (cont’d) The presence of a point imperfection introduces distortions in the crystal. If the imperfection is a vacancy, the bonds that the missing atom would have formed with its neighbours are not there. In the case of an impurity atom, as a result of the size difference, elastic strains are created in the region of the crystal immediately surrounding the impurity atom. The elastic strains are present irrespective of whether the impurity atom is larger or smaller than the parent atom. A larger atom introduces compressive stresses and corresponding strains around it, while a smaller atom creates a tensile stress-strain field. Similarly, an interstitial atom produces strains around the void it is occupying. All these factors tend to increase the enthalpy (or the potential energy) of the crystal. The work required to be done for creating a point imperfection is called the enthalpy of formation (Hf) of the point imperfection. It is expressed in kJ mol–l or eV/point imperfection. The enthalpy of formation of vacancies in a few crystals is shown in following Table  Point defects (cont’d) In close packed structures, the largest atom that can fit the octahedral and the tetrahedral voids have radii 0.414r and 0.225r, respectively, where r is the radius of the parent atom. Carbon is an interstitial solute in iron. It occupies the octahedral voids in the high temperature FCC form of iron. The iron atom in the FCC crystal has a radius of 1.29 Å, whereas the carbon atom has a radius of 0.71 Å (covalent radius in graphite). The carbon radius is clearly larger than 0.414 X 1.29 = 0.53 Å, which is the size of the octahedral void. Therefore, there are strains around the carbon atom in the FCC iron, and the solubility is limited to 2 wt.%. In the room temperature BCC form of iron, the voids are still smaller and hence the solubility of carbon is very limited, that is, only 0.008 wt.%. Point defects: vacancies Point defects: vacancies The fractions of vacancies n/N in a crystal at temperature T is given by: Where, n is the number of vacant sites N is the number of atomic sites ΔHf is the enthalpy of formation of vacancy k is the Boltzmann constant There is equilibrium number of vacancies at a given temperature Vacancies Vacancies Contributions of vacancies to the thermal expansion Contributions of vacancies to the thermal expansion  Motion of vacancy Diffusion in crystals is explained, in terms of vacancies, by assuming that the vacancies move through the lattice, thereby producing random shifts of the atoms from one lattice position to another. The basic principle of vacancy diffusion is illustrated in the below Figure, where three successive steps in the movement of a vacancy from position I to II are shown. In each case, it can be seen that the vacancy moves as a result of an atom jumping into a hole from a lattice position bordering the hole. In order to make the jump, the atom must overcome the net attractive force of its neighbors on the side opposite the hole. Work is therefore required to make the jump into the hole, or, as it may also be stated, an energy barrier must be overcome. Energy sufficient to overcome the barrier is furnished by the thermal or heat vibrations of the crystal lattice. The higher the temperature, the more intense the thermal vibrations, and the more frequently are the energy barriers overcome. Vacancy motion at high temperatures is very rapid and, as a consequence, the rate of diffusion increases rapidly with increasing temperature. Three steps in the motion of a vacancy through a crystal  Defects in Ionic Crystal (Frenkel defect and Schottky defect) In ionic crystals, the formation of point imperfections is subject to the requirement that the overall electrical neutrality is maintained. An ion displaced from a regular site to an interstitial site is called a Frenkel imperfection. As cations are generally the smaller ions, it is possible for them to get displaced into the void space. Anions do not get displaced like this, as the void space is just too small for their size. A Frenkel imperfection does not change the overall electrical neutrality of the crystal. The point imperfections in silver halides and CaF2 are of the Frenkel type. A pair of one cation and one anion can be missing from an ionic crystal as shown in Fig. The valency of the missing pair of ions should be equal to maintain electrical neutrality. Such a pair of vacant ion sites is called a Schottky imperfection. This type is dominant in alkali halides Ref: Callister & Rethwisch Point defects in ionic solids  Defects in Ionic Crystal The ratio of cations to anions is not altered by the formation of either a Frenkel or a Schottky defect. If no other defects are present, the material is said to be stoichiometric. Stoichiometry may be defined as a state for ionic compounds wherein there is the exact ratio of cations to anions as predicted by the chemical formula. For example, NaCl is stoichiometric if the ratio of Na ions to Cl ions is exactly 1:1. Trivalent cations such as Fe3+ and Cr3+ can substitute for trivalent parent Al3+ cations in the Al2O3 crystal. Whereas, if the valency of the substitutional impurity is not equal to the parent cation, additional point defects may be created due to such substitution. For example, a divalent cation such as Cd2+ substituting for a univalent parent ion such as Na+ will, at the same time, create a vacant cation site in the crystal so that electrical neutrality is maintained.  One dimensional/Line defect (Dislocation) A dislocation is a linear or one-dimensional defect around which some of the atoms are misaligned. Different types of dislocation - Edge dislocation - Screw dislocation - Mixed dislocation One type of dislocation an extra portion of a plane of atoms, or half-plane, the edge of which terminates within the crystal. This is termed an edge dislocation; Edge dislocation is at the edge of a extra half plane Only the bottom edge of the half plane is defect, not the entire half plane Edge Dislocation This is reflected in the slight curvature for the vertical planes of atoms as they bend around this extra half-plane. The magnitude of this distortion decreases with distance away from the dislocation line; at positions far removed, the crystal lattice is virtually perfect. Sometimes the edge dislocation in above Figure is represented by the symbol ⊥ which also indicates the position of the dislocation line. An edge dislocation may also be formed by an extra half-plane of atoms that is included in the bottom portion of the crystal; its designation is T. Within the region around the dislocation line there is some localized lattice distortion. The atoms above the dislocation line in Figure are squeezed together (compressive stress), and those below are pulled apart (tensile stress) Edge Dislocation Edge dislocation is a type of line defect in crystal lattices in which the defect occurs either due to the presence of an extra plane of atoms or due to the loss of a half of a plane of atoms in the middle of the lattice. This defect causes the nearby planes of atoms to bend towards the dislocation. Therefore, the adjacent planes of atoms are not straight. The region in which the defect occurs is the dislocation core or area. The Burgers vector b is the vector which defines the magnitude and direction of slip. Therefore, it is the most characteristic feature of a dislocation Edge dislocation A dislocation line is a boundary between slip and no slip region on a slip plane One-dimensional or line defects: dislocation  Edge dislocation Extra half plane 2a a Missing half plane Edge dislocation  Edge dislocation 1. Defect is concentrated in the marked region Edge dislocation line 2. Abruptly ending plane created the dislocation. 3. Only the bottom edge of the half plane is defect not the entire half plane  Edge dislocation: slip approach 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8  Edge dislocation 1 2 3 4 5 6 7 8 Magnitude and Slip No Slip direction of slip No Slip Slip A dislocation line is a boundary between slip and no slip region on a slip plane 1 2 3 4 5 6 7 8 Characteristics vector of dislocation Slip No Slip 𝑡Ƹ 𝑏 Tangent vector (𝒕ො) (line vector) parallel or tangent to the dislocation line Burgers vector (𝒃): magnitude and direction of slip  Screw dislocations screw dislocation line Parallel planes perpendicular to the dislocation line join to form a continuous helical surface.  Screw dislocation Another type of dislocation, called a screw dislocation, may be thought of as being formed by a shear stress that is applied to produce the distortion shown in below Figure a: the upper front region of the crystal is shifted one atomic distance to the right relative to the bottom portion. The atomic distortion associated with a screw dislocation is also linear and along a dislocation line, line AB in Figure b. The screw dislocation derives its name from the spiral or helical path or ramp that is traced around the dislocation line by the atomic planes of atoms. Sometimes the symbol is used to designate a screw dislocation. The Burgers circuit of the screw dislocation form a spiral or helical path or ramp (like screw), thus the name screw dislocation Edge dislocation Screw dislocation  Burgers Circuit S. F 1 2 3 4 5 6 1 2 3 4 5 6 1 1 𝑏 1 2 2 2 3 3 3 4 4 4 5 5 5 6 5 4 3 2 1 6 5 4 3 2 1 Burgers vector b of a dislocation in a crystal is the Closed burgers circuit in Burgers circuit construction using finish-start/right-hand perfect crystal (FS/RH) convention (closure failure is the burgers vector) Mixed dislocation Most dislocations are not pure screw or edge, but a combination of both (have both edge and screw component) Transition region exhibits varying degrees of screw and edge character Mixed dislocation  Edge, Screw and mixed dislocations Classification based on the angular relation between burger vector (𝑏)and tangent vector (𝑡). Ƹ Slip No Slip Slip No Slip 𝑡Ƹ 𝑡Ƹ 𝑏 𝑏 If 𝑏 is perpendicular to 𝑡Ƹ then it is called edge dislocation 𝑏 is parallel to 𝑡Ƹ then it is called screw dislocation Slip No Slip 𝑡Ƹ 𝑏 If 𝑏 is neither parallel or nor perpendicular to the 𝑡Ƹ then it is called as mixed dislocation ❑ Dislocation play dominant role in plastic deformation – slip of a large number of dislocations ❑ Plastic deformation is irreversible changes in shape due to applied stress ❑ Elastic deformation is reversible, stretching of atomic bonds ❑ Strength of materials is lower than theoretical values due to the presence of dislocations, only a small number of bonds is broken at any given time during slip ❑ Dislocations provide ductility in metals ❑ Dislocation control mechanical properties of metals and alloys by interfering with dislocation movement (obstacles stronger material) ❑ Dislocation density = the total length of dislocations per unit volume; units: m/m3 = m-2  Dislocation motion ❑ Dislocation are important in crystal because they can move under applied stress ❑ Different types of dislocation motion are there ❑ Dislocation glide ( for edge, screw and mixed dislocation) ❑ Dislocation Climb (for edge dislocation only) – Only at higher temperature ❑ Cross-slip (for screw dislocation only) ❑ Glide is motion of dislocation in its own slip plane ❑ All types of dislocations ( for edge, screw and mixed dislocation) can glide Dislocation glide for edge dislocation τCRSS = critical resolved shear stress on the slip plane in the direction of b  Motion of Edge Dislocation Slip approach of edge dislocation Dislocation line is a boundary between slip and no-slip regions on a slip plane Slip plane – The plane in which dislocation moves through a crystal Dislocation Climb  Dislocation Climb ❑ The motion of an edge dislocation on a plane perpendicular to the glide plane is called climb motion. ❑ As the edge dislocation moves above or below the slip plane in a perpendicular direction ❑ The incomplete plane either shrinks or increases in extent. This kind of shifting of the edge of the incomplete plane is possible only by subtracting or adding rows of atoms to the extra plane. ❑ Climb motion is said to be nonconservative. This is in contrast to the glide motion which is conservative and does not require either addition or subtraction of atoms from the incomplete plane. ❑ During climbing up of an edge dislocation, the incomplete plane shrinks. Atoms move away from the incomplete plane to other parts of the crystal. During climbing down, the incomplete plane increases in extent. Atoms move into the plane from other parts of the crystal. ❑ This results in an interesting interaction between the climb process and the vacancies in the crystal. Cross-slip of screw dislocation A screw dislocation cross-slips from one slip plane onto another nonparallel slip plane As there is no unique glide plane defined for a screw dislocation, it can change its slip plane during its motion. Elastic energy of dislocation line Elastic strain energy associated with Compression dislocation line due to strain field around the Slip plane dislocation line. The elastic strain energy Tension E = ½ Gb2 Where G is the share modulus b is the magnitude of burgers vector 1 plane missing  Energy of dislocations ❑ Dislocations have distortion energy associated with them ❑ Elastic strain energy E per unit length of a dislocation of Burgers vector b 1 2 E  Gb 2 G → () shear modulus b → |b| ❑Edge → Compressive and tensile stress fields ❑Screw → Shear strains 2D defects: Surfaces and interfaces Homophase interface (same phase) a) Grain boundary b) Twin boundary c) Stacking faults Heterophase interface (different phase) a) Free surfaces (solid/gas interface) b) Solid/liquid interface c) Crystal 1/crystal 2 (interphase interface)  Two-dimensional defect Surface imperfections are two dimensional in the mathematical sense. They refer to regions of distortions that lie about a surface having a thickness of a few atomic diameters. The external surface/Free surface of a crystal is an imperfection in itself as the atomic bonds do not extend beyond the surface. External surfaces have surface energies that are related to the number of bonds broken at the surface. For example, consider a close packed plane as the surface of a close packed crystal. An atom on the surface of this crystal has six nearest bonding neighbours on the surface plane, three below it, and none above. Therefore, three out of twelve neighbours of an atom are missing at the surface. The surface energy of a crystal bears a relationship to this number. Ref: Material Science and Engineering by V. Raghavan  Free surface Free surface or external surface Bonds broken at the free surface Energy associated with the number of broken bonds will be stored in the surface Surface energy per unit area (energy stored in the surface) Surface energy in terms of bond breaking model ε = bond energy Number of A bonds broken A Number of per atom (nB) atoms present in the unit area on the surface (nA) γ = (A nA nB ε)/2A = (nA nB ε)/2 (surface energy)  Grain boundary In addition to the external surface, crystals may have surface imperfections inside. A piece of iron or copper is usually not a single crystal. It consists of a number of crystals and is said to be polycrystalline. During solidification or during a process in the solid state called recrystallization, new crystals form in different parts of the material. They are randomly oriented with respect to one another. They grow by the addition of atoms from the adjacent regions and eventually impinge on each other. When two crystals impinge in this manner, the atoms caught in between the two are being pulled by each of the two crystals to join its own configuration. They can join neither crystal due to the opposing forces and, therefore, take up a compromise position. These positions at the boundary region between two crystals are so distorted and unrelated to one another that we can compare the boundary region to a noncrystalline material. The thickness of this region is only a few atomic diameters, because the opposing forces from neighbouring crystals are felt by the intervening atoms only at such short distances. The boundary region is called a crystal boundary or a grain boundary Ref: Material Science and Engineering by V. Raghavan  Crystal Defects Surface Defects: Grain Boundaries Grain boundary is a narrow region between two grains of about two to few atomic diameters in width, and is the region of mismatch between adjacent grains. ❑ Grain boundary is a narrow region between two grains and is the region of mismatch between adjacent grains ❑ Atoms are arranged less regularly at the grain boundary. This produce less efficient packing of the atoms at the boundary. ❑ Thus, the atoms along the grain boundary have higher energy than those within the grains.  The crystal orientation changes sharply at the grain boundary. The orientation difference is usually greater than 10–15°. For this reason, the grain boundaries are also known as high angle boundaries. The average number of nearest neighbours for an atom in the boundary of a close packed crystal is 11, as compared to 12 in the interior of the crystal. On an average, one bond out of the 12 bonds of an atom is broken at the boundary. The grain boundary between two crystals, which have different crystalline arrangements or differ in composition, is given a special name, viz. an interphase boundary or an interface. In an Fe–C alloy, the energies of grain boundaries and interfaces are compared in the following manner: Grain boundary between BCC crystals 0.89 J m-2 Grain boundary between FCC crystals 0.85 J m-2 Interface between BCC and FCC crystals 0.63 J m-2 Ref: Material Science and Engineering by V. Raghavan  If the mismatch i.e., orientation difference between two grains is more than 10-15o, the grain boundary is known as high angle grain boundary. 136  When the orientation difference between two crystals is less than 10°, the distortion in the boundary is not so drastic. Such boundaries have a structure that can be described by means of arrays of dislocations. They are called low angle boundaries. Figure shows a low angle tilt boundary, where neighbouring crystalline regions are tilted with respect to each other by only a small angle. The tilt boundary can be described as a set of parallel, equally-spaced edge dislocations of the same sign located one above the other. Similar low angle boundaries formed by screw dislocations are called twist boundaries Ref: Material Science and Engineering by V. Raghavan  Stacking faults Stacking faults are also planar surface imperfections created by a fault (or error) in the stacking sequence of atomic planes in crystals. Consider the stacking arrangement in an FCC crystal …ABCABCABCABC….. If an A plane indicated by an arrow above is missing, the stacking sequence becomes ….ABCABCBCABC…. The stacking in the missing region is BCBC which is HCP stacking. This thin region is a surface imperfection and is called a stacking fault. The number of nearest neighbours in the faulted region remains 12 as in the perfect regions of the crystal, but the second nearest neighbour bonds in the faulted region are not of the correct type for the FCC crystal. Hence, a small surface energy is associated with the stacking fault, in the range 0.01–0.05 J m-2. Similarly, we can define a stacking fault in an HCP crystal as a thin region of FCC stacking Ref: Material Science and Engineering by V. Raghavan  Twin boundary ❑ Another planar surface imperfection is a twin boundary. ❑ The atomic arrangement on one side of a twin boundary is a mirror reflection of the arrangement on the other side, as illustrated in Fig. ❑ Twin boundaries occur in pairs such that the orientation change introduced by one boundary is restored by the other, as shown in below Fig. ❑ The region between the pair of boundaries is called the twinned region. Twin boundaries are easily identified under an optical microscope. Ref: Material Science and Engineering by V. Raghavan  Stacking faults Fault in a stacking sequence of a crystal Stacking sequence in a cubic closed packed structure C C B B HCP like sequence near fault plane A A c-plane missing C B fault B A A C C  Twin boundary cubic closed packed structure cubic closed packed structure with twin boundary C C B B A A C C Mirror plane (twin plane) B A A B C C Twin boundary: boundary in a crystal such that crystal on either side are mirror image of each other  Volume imperfections ❑ Volume imperfections can be foreign-particle inclusions, large voids or pores, or noncrystalline regions which have the dimensions of at least a few tens of Å. ❑ The accumulation of vacancies produce voids. The foreign atoms produce dissymmetry within crystals. These defects affect properties of metal. Atomic Vibrations At any temperature above absolute zero every atom in a solid material is vibrating very rapidly about its lattice position. This behavior is considered as defect/ imperfection. At any given instant of time, not all the atoms vibrate with same frequency and amplitude nor with the same energy. With the rise in temperature, there will be rise in average energy. Temperature of solid is really just a measure of the average vibrational activity of atoms and molecules. 143 Atomic Vibrations: At room temperature Vibration has a frequency of ~1013/sec, amplitude of few thousands nanometers. Most of the properties and processes in solids are manifestations of this vibrational atomic motion. Eg: Melting occurs when the vibrations are vigorous and large enough to rupture large number of atomic bonds. 144  Summary of defects in crystal ❑ Point, Line, Area, and volume defects exist in real solids ❑ The number and type of defects can be varied and controlled (e.g. T increases vacancy conc.) ❑ Defects affects materials properties ❑ Defects may be are desirable or undesirable (e.g. dislocations may have good and bad role depending on whether plastic deformation is desirable or not)