Introduction to Materials Science and Engineering

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(1)2020 Fall. Introduction to Materials Science and Engineering. 09. 15. 2020. Eun Soo Park Office: 33-313 Telephone: 880-7221 Email: [email protected] Office hours: by appointment 1.

(2) Materials Science and Engineering Processing • • • • •. Structure • • • •. Sintering Heat treatment Thin Film Melt process Mechanical. Atomic Crystal Microstructure Macrostructure. Properties Theory & Design. • • • • •. Mechanical Electrical Magnetic Thermal Optical. 2.

(3) Contents for previous class. CHAPTER 3: The Structure of Crystalline Solids I. Crystal Structures - Lattice, Unit Cells, Crystal system II. Crystallographic Points, Directions, and Planes - Point coordinates, Crystallographic directions, Crystallographic planes III. Crystalline and Noncrystalline Materials - Single crystals, Polycrystalline materials, Anisotropy, Noncrystalline solids 3.

(4) Stacking of atoms in solid Finding stable position. 2 1. • Minimize energy configuration – Related to the bonding nature 4.

(5) I. Crystal structure Lattice. : 결정 공간상에서 점들의 규칙적인 기하학적 배열. – 3D point array in space, such that each point has identical surroundings. These points may or may not coincide with atom positions. – Simplest case : each atom  its center of gravity  point or space lattice  pure mathematical concept example: sodium (Na) ; body centered cubic. Hard-sphere unit cell. Reduced sphere unit cell. Aggregate of many atoms.

(6) (4) 7 crystal systems. Unit cell. 6.

(7) (4) 7 crystal systems (continued). Unit cell. 7.

(8) I. Crystal structure : Unit cell 14 Bravais Lattice - Only 14 different types of unit cells are required to describe all lattices using symmetry cubic. hexagonal. rhombohedral (trigonal). tetragonal. orthorhombic. monoclinic. triclinic. P. I. F. C 8.

(9) II. Crystallographic points, directions and planes Chapter 3.5 Point coordinates. • position: fractional multiples of the unit cell edge lengths – ex) P: q,r,s. cubic unit cell. 9.

(10) Crystallographic Directions • a line between two points or a vector • [uvw] square bracket, smallest integer • families of directions: <uvw> angle bracket cubic. 10.

(11) Chapter 3.7 Crystallographic Planes. Lattice plane (Miller indices). z. m00, 0n0, 00p: define lattice plane. p. m, n, ∞ : no intercepts with axes. . c. . a. x. . b. n y. m. Plane (hkl) Family of planes {hkl}. Miller indicies ; defined as the smallest integral multiples of the reciprocals of the plane intercepts on the axes. Intercepts @ (mnp). 2. 1. 3. Reciprocals. ½. 1. 1/3. Miller indicies. 3. 6. 2. (362) plane. (362). 11.

(12) Directions, Planes, and Family • line, direction – [111] square bracket – <111> angular bracket - family. • Plane – (111) round bracket (Parentheses) – {111} braces - family. 12.

(13) Contents for today’s class. CHAPTER 3: The Structure of Crystalline Solids I. Crystal Structures - Lattice, Unit Cells, Crystal system II. Crystallographic Points, Directions, and Planes - Point coordinates, Crystallographic directions, Crystallographic planes III. Crystalline and Noncrystalline Materials - Single crystals, Polycrystalline materials, Anisotropy, Noncrystalline solids - Quasicrystals - Noncrystalline solids : Amorphous solid. 13.

(14) 7 crystal systems. Unit cell. 14.

(15) 1. Hexagonal Crystal • Miller-Bravais scheme. [uvtw] t= −(u + v). (hkil ) i= −( h + k ) 15.

(16) Miller-Bravais vs. Miller index system in Hexagonal system Directions. Conversion of 4 index system (Miller-Bravais) to 3 index (Miller).         t = u ' a1 + v' a2 + w' c = ua1 + va2 + ta3 + wc. Miller-Bravais to Miller 4 to 3 axis. u ' = u − t = 2u + v v ' = v − t = 2v + u w' = w. Miller to Miller-Bravais 3axis to 4 axis system. 1 u = (2u '−v' ) 3 1 v = (2v'−u ' ) 3 w = w' t=-(u+v). Ex. M [100] u=(1/3)(2*1-0)=2/3 v=(1/3)(2*0-1)=-1/3 w=0 => 1/3[2 -1 -1 0] Ex. M-B [1 0 -1 0] u’=2*1+0=2 v’=2*0+1=1 w’=0 => [2 1 0]. 16.

(17) HCP Crystallographic Directions • Hexagonal Crystals – 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows. z. [ u 'v 'w ' ] → [ uvtw ]. a2. -. a3 a1. 1 u = (2 u ' - v ' ) 3 1 v = (2 v ' - u ' ) 3 t = - (u + v ) w = w'. Fig. 3.8(a), Callister 7e.. 17.

(18) Miller index. 1 u = (2u '−v' ) 3 1 v = (2v'−u ' ) 3 w = w'. 𝑎𝑎1. 21� 1� 0 𝑎𝑎2. 𝑎𝑎3. 100. 2 3 −1 v= 3 w=0. u=. 𝑐𝑐⃗. Miller-Bravais index i=-(h+k). 1 21� 1� 0 3. 𝑐𝑐⃗. hkil. 21� 1� 0. 𝑎𝑎1. 1 21� 1� 0 3. 𝑎𝑎3. 𝑎𝑎2 18.

(19) Miller index. 1 u = (2u '−v' ) 3 1 v = (2v'−u ' ) 3 w = w'. 𝑎𝑎1. 101. 2 u= 3 −1 v= 3 w =1. 21� 1� 3 𝑎𝑎2. Miller-Bravais index. 𝑎𝑎3. 1 21� 1� 3 3. 21� 1� 3 𝑐𝑐⃗. 21� 1� 0. 1 21� 1� 3 3 𝑎𝑎1. 𝑐𝑐⃗. 𝑎𝑎3 𝑎𝑎2.

(20) Miller index. 𝑥𝑥. 101 𝑦𝑦. Miller-Bravais index. 1� 21� 1� 3 3. 𝑧𝑧⃗. 𝑧𝑧. 𝑎𝑎1. 𝑐𝑐⃗. 001. 𝑥𝑥⃗. 𝑎𝑎1. 100. 𝑎𝑎3 𝑎𝑎2. 𝑦𝑦⃗. 𝑎𝑎2. 𝑎𝑎3 𝑐𝑐⃗.

(21) Crystallographic Planes (HCP) • In hexagonal unit cells the same idea is z used example 1. Intercepts 2. Reciprocals 3.. Reduction. a1 1 1 1 1. a2 ∞ 1/∞ 0 0. a3 -1 -1 -1 -1. c 1 1 1 1. a2. a3. 4.. Miller-Bravais Indices. (1011). a1 Adapted from Fig. 3.8(a), Callister 7e.. 21.

(22) Miller index. Miller-Bravais index. 𝑐𝑐⃗. 𝑐𝑐⃗. 𝑎𝑎2. 𝑎𝑎2 𝑎𝑎3. 101� 1. 𝑎𝑎1. 𝑎𝑎3. Inverse of 1 =1 Position at 𝑎𝑎1 =1. 101. Does not Meet with 𝑎𝑎2. 𝑎𝑎1 Inverse of 1 =1 Position at 𝑐𝑐⃗ =1 22.

(23) Miller index. Miller-Bravais index. 1� 11. 𝑧𝑧⃗. 𝑎𝑎1. 𝑐𝑐⃗. 001. 𝑥𝑥⃗. 𝑎𝑎1. 100. 𝑥𝑥. 𝑎𝑎3. 1� 101 𝑎𝑎2. 𝑦𝑦. 𝑎𝑎3. 1� 11. 𝑐𝑐⃗. 𝑧𝑧. Miller index 𝑎𝑎2. 𝑦𝑦⃗. 23.

(24) Miller index. 𝑥𝑥. 1� 11 𝑦𝑦. Miller-Bravais index. 𝑧𝑧⃗. 𝑧𝑧. 𝑎𝑎1. 𝑐𝑐⃗. 001. 1� 101. 𝑎𝑎2. 𝑎𝑎3 𝑐𝑐⃗. u ' = u − t = 2u + v v ' = v − t = 2v + u w' = w. 𝑥𝑥⃗. 𝑎𝑎1. 100. 𝑎𝑎3. Miller index 𝑎𝑎2. 𝑦𝑦⃗. 1� 11. 24.

(25) Miller index. 𝑥𝑥. 1� 11 𝑦𝑦. Miller-Bravais index. 𝑧𝑧⃗. 𝑧𝑧. 𝑐𝑐⃗. 001. u ' = u − t = 2u + v v ' = v − t = 2v + u w' = w. 𝑥𝑥⃗. � vs (210) 1010. 𝑎𝑎1. 𝑎𝑎1. 100. 1� 101 𝑎𝑎2. 𝑎𝑎3 𝑐𝑐⃗. 1 u = (2u '−v' ) 3 1 v = (2v'−u ' ) 3 w = w'. 𝑎𝑎3. t=-(u+v). 𝑎𝑎2. 𝑦𝑦⃗. 25.

(26) Schematic view of planes 1. c 𝟏𝟏 𝟐𝟐. � 𝟏𝟏 1. a 𝟏𝟏. Take Reciprocal. b 𝟏𝟏. (𝟏𝟏𝟏𝟏𝟏𝟏). 𝟏𝟏 𝟏𝟏. �𝟏𝟏) (𝟏𝟏𝟏𝟏. 𝟏𝟏 𝟐𝟐. Inter-planar spacing 26.

(27) Schematic view of (111) plane c. b. a. 27.

(28) 2. Inter-planar distance (면간거리). d (hkl) plane. β. α. 𝑛𝑛.

(29) Interplanar spacing of the (hkl) plane The value of d which characterizes the distance between adjacent planes in the set of planes with Miller indices (hkl) is given by the following relations. The cell edges and the angles are a, b, c and α, β ,γ. 1 h2 + k 2 + l 2 = d2 a2. Cubic :. . 1 h2 + k 2 l 2 + 2 = Tetragonal : c a2 d2 . 1 h2 k 2 l 2 = + + Orthorhombic : d 2 a2 b2 c2 . 1 4  h 2 + hk + k 2  l 2  + 2 =  Hexagonal : d 2 3  a2  c . 1 1  h 2 k 2 sin 2 β l 2 2hl cos β   +  = + 2− Rhombohedral : ac d 2 sin 2 β  a 2 b2 c  . 1 (h 2 + k 2 + l 2 ) sin 2 α + 2(hk + kl + hl )(cos 2 α − cosα ) = Monoclinic : d2 a 2 (1 − 3 cos 2 α + 2 cos 3 α ) . 1 1 = 2 ( S11h 2 + S 22 k 2 + S 33l 2 + 2 S12 hk + 2 S 23 kl + 2 S 31hl ) Triclinic : 2 d V Where :. V = a b c (1 − cos α − cos β − cos γ + 2 cos α cos β cos γ ) 2. 2. 2 3. 2. 2. 2. S11 = b 2 c 2 sin 2 α S 22 = a 2 c 2 sin 2 β S 33 = a 2 b 2 sin 2 γ S12 = abc 2 (cos α cos β − cos γ ) S23 = a 2 bc(cos β cos γ − cos α ) S31 = ab 2 c(cos γ cos α − cos β ). 29.

(30) III. Crystalline ↔ Noncrystalline Materials. CRYSTALS AS BUILDING BLOCKS • Some engineering applications require single crystals: --diamond single crystals Natural and artificial. --turbine blades Fig. 8.30(c), Callister 6e. (Fig. 8.30(c) courtesy of Pratt and Whitney).. (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.). • Crystal properties reveal features of atomic structure. --Ex: Certain crystal planes in quartz fracture more easily than others.. (Courtesy P.M. Anderson). 30.

(31) Solidification:. Liquid. Solid. 4 Fold Anisotropic Surface Energy/2 Fold Kinetics, Many Seeds. 31.

(32) Grain Boundaries. 32.

(33) Grains. Deformed. Annealed. Atomic view of grain boundaries.

(34) 3. Single vs Polycrystals • Single Crystals. E (diagonal) = 273 GPa Data from Table 3.3, Callister 7e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.). -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron:. • Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa). -If grains are textured, anisotropic.. E (edge) = 125 GPa. 200 µm. Adapted from Fig. 4.14(b), Callister 7e. (Fig. 4.14(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].). 34.

(35) Polycrystals. Anisotropic. • Most engineering materials are polycrystals.. Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany). 1 mm. • Nb-Hf-W plate with an electron beam weld. • Each "grain" is a single crystal. • If grains are randomly oriented,. Isotropic. overall component properties are not directional.. • Grain sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).. 35.

(36) 4. Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism) iron system titanium liquid α, β-Ti 1538ºC δ-Fe BCC carbon 1394ºC diamond, graphite FCC γ-Fe 912ºC BCC. α-Fe 36.

(37) DEMO: HEATING AND COOLING OF AN IRON WIRE • Demonstrates "polymorphism". The same atoms can have more than one crystal structure.. 37.

(38) 5. Non-crystalline Materials Amorphous materials a wide diversity of materials can be rendered amorphous indeed almost all materials can. - metal, ceramic, polymer ex) amorphous metallic alloy (1960) - glassy/non-crystalline material cf) amorphous vs glass - random atomic structure (short range order) - showing glass transition.. Amorphous Glass. - retain liquid structure - rapid solidification from liquid state 38.

(39) Obsidian is a naturally occurring volcanic glass formed as an extrusive igneous rock. It is produced when felsic lava extruded from a volcano cools rapidly with minimum crystal growth. Obsidian is commonly found within the margins of rhyolitic lava flows known as obsidian flows, where the chemical composition (high silica content) induces a high viscosity and polymerization degree of the lava. The inhibition of atomic diffusion through this highly viscous and polymerized lava explains the lack of crystal growth. Because of this lack of crystal structure, abcde obsidian blade edges can reach almost molecular thinness, leading to its ancient use as projectile points and blades, and its modern use as surgical scalpel blades..

(40) Glass formation: stabilizing the liquid phase First metallic glass (Au80Si20) produced by splat quenching at Caltech by Pol Duwez in 1957. 1500. 1414 Liquid. Temperature (℃). 1300. Tmmix. laser mix trigger T − Tm ∆T * = m ≥ 0.2 mix Tm. 1064.4. 1100 900. deep eutectic ΔT* = 0.679. 700. -Rapid Relative decrease of splat quenching melting temperature liquid metal droplet. metal anvil. in most of glass forming alloys. 500. (where,. 363. Tmmix = ∑ xiTmi. 18.6. 300. xi = mole. Tm. ,. metal fraction, piston. Tmi = melting point). 100 0. 10. Au. 20. 30. 40. 50. 60. 70. 80. 90. 100. Si. W. Klement, R.H. Willens, P. Duwez, Nature 1960; 187: 869.. by I.W. Donald l1978;30:77. = 3 cm. t = 20 μm w = 2 cm et al, J. Non-Cryst. Solids, 40.

(41) Bulk glass formation in the Pd-/Ni-/Cu-/Zr- system. 41.

(42) Recent BMGs with critical size ≥ 10 mm. 42.

(43) Strength vs. Elastic Limit for Several Classes of Materials. 2500. Strength (MPa). 2000. Steels. 1500. Metallic glass. Titanium alloys. 1000. 500 Woods. Polymers. Silica 0 0. 1. 2. 3. Elastic Limit (%). : Metallic Glasses Offer a Unique Combination of High Strength and High Elastic Limit 43.

(44) Processing metals as efficiently as plastics: net-shape forming!. Seamaster Planet Ocean Liquidmetal® Limited Edition. Superior thermo-plastic formability. : possible to fabricate complex structure without joints Multistep processing can be solved by simple casting. Ideal for small expensive IT equipment manufacturing.

(45) 6. What are Quasicrystals?. 45.

(46) Crystals can only exhibit certain symmetries In crystals, atoms or atomic clusters repeat periodically, analogous to a tesselation in 2D constructed from a single type of tile. Try tiling the plane with identical units … only certain symmetries are possible. A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps.. 46.

(47) Parallelogram : 2 fold. YES.

(48) Rectangle : 2 fold. YES.

(49) Triangle : 3 fold. YES.

(50) Square 4 fold. YES.

(51) Hexagon : 6 fold. YES.

(52) So far so good … but what about five-fold, seven-fold or other symmetries??.

(53) Pentagon : 5 fold. ?. No!.

(54) Heptagon : 7 fold. ?. No!.

(55) Symmetry axes compatible with periodicity. According to the well-known theorems of crystallography, only certain symmetries are allowed: the symmetry of a square, rectangle, parallelogram, triangle or hexagon, but not others, such as pentagons.. 55.

(56) Crystals can only exhibit certain symmetries Crystals can only exhibit these same rotational symmetries* ..and the symmetries determine many of their physical properties and applications. * In 3D, there can be different rotational symmetries along different axes, but they are restricted to the same set (2-, 3-, 4-, and 6-fold). 56.

(57) Which leads us to….

(58) Quasicrystals (Impossible Crystals) were first discovered in the laboratory by Daniel Shechtman, Ilan Blech, Denis Gratias and John Cahn. in a beautiful study of an alloy of Al and Mn. 58.

(59) D. Shechtman, I. Blech, D. Gratias, J.W. Cahn (1984). 1 µm Al6Mn. 59.

(60) Their surprising claim:. “Diffracts electrons like a crystal . . . But with a symmetry strictly forbidden for crystals”. Al6Mn. 60.

(61) QUASICRYSTALS Similar to crystals, BUT… • Orderly arrangement . . . But QUASIPERIODIC instead of PERIODIC • Rotational Symmetry . . . But with FORBIDDEN symmetry • Structure can be reduced to a finite number of repeating units. D. Levine and P.J. Steinhardt (1984).

(62) Discovery of a Natural Quasicrystal. L Bindi, P. Steinhardt, N. Yao and P. Lu Science 324, 1306 (2009). LEFT: Fig. 1 (A) The original khatyrkite-bearing sample used in the study. The lighter-colored material on the exterior contains a mixture of spinel, augite, and olivine. The dark material consists predominantly of khatyrkite (CuAl2) and cupalite (CuAl) but also includes granules, like the one in (B), with composition Al63Cu24Fe13. The diffraction patterns in Fig. 4 were obtained from the thin region of this granule indicated by the red dashed circle, an area 0.1 µm across. (C) The inverted Fourier transform of the HRTEM image taken from a subregion about 15 nm across displays a homogeneous, quasiperiodically ordered, fivefold symmetric, real space pattern characteristic of quasicrystals. RIGHT: Diffraction patterns obtained from natural quasicrystal grain. 62.

(63) 2011 Nobel Prize in Chemistry: Quasicrystal. A new ordered phase showing the apparent fivefold symmetry was observed by Sastry et al. [Mater. Res. Bull. 13: 1065-1070] in 1978 in a rapidly solidified Al-Pd alloy, but was interpreted to arise from a microstructure Consisting of a series of fine twins. This was later shown to be a two-dimensional (or decagonal) qasicrystal..

(64) Quasicrystals Mathematically impossible but exist Crystal with 5 fold symmetry 1984 Al86Mn14 alloy : rapidly solidified ribbon_Shectman et al.. : materials whose structure cannot be understood within classical crystallography rules. “Quasiperiodic lattices”, with long-range order but without periodic translations in three dimensions. • long range order: quasiperiodic • no 3-D translational symmetry • sharp diffraction patterns http://www.youtube.com/watch?v=k_VSpBI5EGM 64.

(65) Atomic arrangement in the solid state  Solid materials are classified according to the regularity with which atoms and ions are arranged with respect to one another.  So, how are they arranged ? (a) periodically – having long range order in 3-D Crystal (b) quasi-periodically. Quasicrystal. (c) randomly – having short range order with the characteristics of bonding type but losing the long range order  Crystal: Perfection → Imperfection. Amorphous 65.

(66) 2020 Fall. Introduction to Materials Science and Engineering. 09. 17. 2020. Eun Soo Park Office: 33-313 Telephone: 880-7221 Email: [email protected] Office hours: by appointment 1.

(67) Contents for previous class. CHAPTER 3: The Structure of Crystalline Solids I. Crystal Structures - Lattice, Unit Cells, Crystal system II. Crystallographic Points, Directions, and Planes - Point coordinates, Crystallographic directions, Crystallographic planes III. Crystalline and Noncrystalline Materials - Single crystals, Polycrystalline materials, Anisotropy, Noncrystalline solids - Quasicrystals - Noncrystalline solids : Amorphous solid. 2.

(68) Contents for previous class. * Atomic arrangement in the solid state  Solid materials are classified according to the regularity with which atoms and ions are arranged with respect to one another.  So, how are they arranged ? (a) periodically – having long range order in 3-D Crystal (b) quasi-periodically. Quasicrystal. (c) randomly – having short range order with the characteristics of bonding type but losing the long range order  Crystal: Perfection → Imperfection. Amorphous. 3.

(69) Crystal structure : Unit cell 14 Bravais Lattice - Only 14 different types of unit cells are required to describe all lattices using symmetry cubic. hexagonal. rhombohedral (trigonal). tetragonal. orthorhombic. monoclinic. triclinic. P. I. F. C 4.

(70) 1. Miller-Bravais vs. Miller index system in Hexagonal system Directions. Conversion of 4 index system (Miller-Bravais) to 3 index (Miller).         t = u ' a1 + v' a2 + w' c = ua1 + va2 + ta3 + wc. Miller-Bravais to Miller 4 to 3 axis. u ' = u − t = 2u + v v ' = v − t = 2v + u w' = w. Miller to Miller-Bravais 3axis to 4 axis system. 1 u = (2u '−v' ) 3 1 v = (2v'−u ' ) 3 w = w' t=-(u+v). Ex. M [100] u=(1/3)(2*1-0)=2/3 v=(1/3)(2*0-1)=-1/3 w=0 => 1/3[2 -1 -1 0] Ex. M-B [1 0 -1 0] u’=2*1+0=2 v’=2*0+1=1 w’=0 => [2 1 0]. 5.

(71) 2. Interplanar spacing of the (hkl) plane The value of d which characterizes the distance between adjacent planes in the set of planes with Miller indices (hkl) is given by the following relations. The cell edges and the angles are a, b, c and α, β ,γ. 1 h2 + k 2 + l 2 = d2 a2. Cubic :. . 1 h2 + k 2 l 2 + 2 = Tetragonal : c a2 d2 . 1 h2 k 2 l 2 = + + Orthorhombic : d 2 a2 b2 c2 . 1 4  h 2 + hk + k 2  l 2  + 2 =  Hexagonal : d 2 3  a2  c . 1 1  h 2 k 2 sin 2 β l 2 2hl cos β   +  = + 2− Rhombohedral : ac d 2 sin 2 β  a 2 b2 c  . 1 (h 2 + k 2 + l 2 ) sin 2 α + 2(hk + kl + hl )(cos 2 α − cosα ) = Monoclinic : d2 a 2 (1 − 3 cos 2 α + 2 cos 3 α ) . 1 1 = 2 ( S11h 2 + S 22 k 2 + S 33l 2 + 2 S12 hk + 2 S 23 kl + 2 S 31hl ) Triclinic : 2 d V Where :. V = a b c (1 − cos α − cos β − cos γ + 2 cos α cos β cos γ ) 2. 2. 2 3. 2. 2. 2. S11 = b 2 c 2 sin 2 α S 22 = a 2 c 2 sin 2 β S 33 = a 2 b 2 sin 2 γ S12 = abc 2 (cos α cos β − cos γ ) S23 = a 2 bc(cos β cos γ − cos α ) S31 = ab 2 c(cos γ cos α − cos β ). 6.

(72) 3. Single vs Polycrystals • Single Crystals. E (diagonal) = 273 GPa Data from Table 3.3, Callister 7e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.). -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron:. • Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa). -If grains are textured, anisotropic.. E (edge) = 125 GPa. 200 µm. Adapted from Fig. 4.14(b), Callister 7e. (Fig. 4.14(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].). 7.

(73) 4. Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism) iron system titanium liquid α, β-Ti 1538ºC δ-Fe BCC carbon 1394ºC diamond, graphite FCC γ-Fe 912ºC BCC. α-Fe 8.

(74) 5. Non-crystalline Materials Amorphous materials a wide diversity of materials can be rendered amorphous indeed almost all materials can. - metal, ceramic, polymer ex) amorphous metallic alloy (1960) - glassy/non-crystalline material cf) amorphous vs glass - random atomic structure (short range order) - showing glass transition.. Amorphous Glass. - retain liquid structure - rapid solidification from liquid state 9.

(75) 6. QUASICRYSTALS. Similar to crystals, BUT… • Orderly arrangement . . . But QUASIPERIODIC instead of PERIODIC • Rotational Symmetry . . . But with FORBIDDEN symmetry • Structure can be reduced to a finite number of repeating units. D. Levine and P.J. Steinhardt (1984).

(76) Materials Science and Engineering Processing • • • • •. Structure • • • •. Sintering Heat treatment Thin Film Melt process Mechanical. Atomic Crystal Microstructure Macrostructure. Properties Theory & Design. • • • • •. Mechanical Electrical Magnetic Thermal Optical. 11.

(77) Contents for today’s class. CHAPTER 4: The Structure of Crystalline Solids (9th edition) I. METALLIC CRYSTALS _ APF , CN … - The face-centered cubic crystal structure (FCC) - The body-centered cubic crystal structure (BCC) - The hexagonal closed-packed crystal structure (HCP) II. Ceramic Crystal Structure (Chapter 12, 10th edition) - Ionic arrangement geometries - Some common ceramic crystal structure/ Density computation - Silicate ceramics/ Carbon. 12.

(78) CHAPTER 3: The Structure of Crystalline Solids. I. METALLIC CRYSTALS • tend to be densely packed. • have several reasons for dense packing: - Typically, only one element is present, so all atomic radii are the same. - Metallic bonding is not directional. - Nearest neighbor distances tend to be small in order to lower bond energy.. • have the simplest crystal structures. We will look at three such structures... 13.

(79) 1. SIMPLE CUBIC STRUCTURE (SC) • Rare due to poor packing (only 84Polonium has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors). (Courtesy P.M. Anderson). 14.

(80) 2. Coordination Number In chemistry and crystallography, the coordination number (CN) of a central atom in a molecule or crystal is the number of its nearest neighbor. In chemistry the emphasis is on bonding structures in molecules or ions and the CN of an atom is determined by simply counting the other of atoms to which it is bonded (by either single or multiple bonds). The solid-state structure of crystals often have less clearly defined bonds, so a simpler model is used, in which the atoms are represented by touching spheres. In this model the CN of an atom is the number of other atoms which it touches. For an atom in the interior of a crystal lattice, the number of atoms touching the given atom is the bulk coordination number, for an atom at a surface of a crystal this is the surface coordination number.. 15.

(81) Coordination Number. 16.

(82) Coordination Number. 17.

(83) 3. ATOMIC PACKING FACTOR : simple cubic. • APF for a simple cubic structure = 0.52. Adapted from Fig. 3.19,. Callister 6e.. 18.

(84) 4. Body Centered Cubic Structure (BCC) • Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.. ex: Cr, W, Fe (α), Tantalum, Molybdenum. • Coordination # = 8. Adapted from Fig. 3.2, Callister 7e.. 2 atoms/unit cell: 1 center + 8 corners x 1/8 (Courtesy P.M. Anderson). 19.

(85) ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68 3a Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell. R. a. a 2a. Close-packed directions <111> length = 4R =. 3a. Highest density planes {110} Adapted from. Callister 6e.. 20.

(86) {110} Family. 21.

(87) {110} Family. 22.

(88) ATOMIC PACKING FACTOR: BCC • APF for a body-centered cubic structure = 0.68 3a Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell. R. a. a 2a. Close-packed directions <111> length = 4R =. 3a. Highest density planes {110} Adapted from. Callister 6e.. 23.

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(90) Planar Density of (100) Iron At T < 912°C iron has the BCC structure. 2D repeat unit a=. (100). Radius of iron R = 0.1241 nm. Adapted from Fig. 3.2(c), Callister 7e.. atoms 2D repeat unit. Planar Density = area 2D repeat unit. 1 a2. 4 3 R 3. 1. = 4 3 R 3. 2. = 12.1. atoms 19 atoms = 1.2 x 10 nm2 m2 25.

(91) Planar Density of (111) Iron (111) plane. 1 atom in plane/ unit surface cell. 2a. atoms in plane atoms above plane atoms below plane. 3 a 2. h= 2. area = 2 ah = 3 a 2 = 3 atoms 2D repeat unit. =. 16 3 2 R 3. 1. atoms = = 7.0 2. Planar Density = area 2D repeat unit. 4 3 R 3. 16 3 3. R. 2. nm. 0.70 x 1019. atoms m2 26.

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(93) Metallic crystal system • Stacking sequence A A B C C A A B B C A A B C C A A B B C A A. A A B B C C A A B B C C A A B B C C A A B B C C A A. B A C B A C. A. A. A 28.

(94) Metallic crystal system • A – B – C – A stacking sequence  FCC A A B C C A A B B C A A B C C A A B B B B C A A. B A. A C C B A C C. B A C C C B B A A B B C C C A. A C C B A C B B A. B A. A. C C A B A C A. B A C B A C B A 29.

(95) Metallic crystal system • A – B – A – B stacking sequence  HCP. A A. A. A A A A B C C A A B B C A A A A B C C A A B B B B C A A A A. B A A. A C B A C. B A A. C B A B B C C A A. C B A. A A C B A A C B B A A. B A A. A A. C A A B A C. A A. A B A B A B A.

(96) 5. Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.. ex: Al, Cu, Au, Pb, Ni, Pt, Ag. • Coordination # = 12. Adapted from Fig. 3.1, Callister 7e.. 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 (Courtesy P.M. Anderson). 31.

(97) FCC stacking.

(98) ATOMIC PACKING FACTOR: FCC • APF for a face-centered cubic structure = 0.74 <110> {111}. a. Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell. Adapted from Fig. 3.1(a),. Callister 6e.. 33.

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(100)

(101)

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(103) Two closely packed stackings A position B position C position. FCC A-B-C-A-B-C-…... HCP A-B-A-B-A-…... FCC : Al, Ni, Cu, Ag, Ir, Pt, Au HCP : Be, Mg, Sc, Ti, Co, Zn, Y, Zr, Se, Te, Ru.

(104) 6. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP) • ABAB... Stacking Sequence • 3D Projection. • 2D Projection A sites B sites A sites. Adapted from Fig. 3.3,. Callister 6e.. • Coordination # = 12 • APF = 0.74. Close-packed direction : <1120> plane : (0001) 39.

(105) I. Metallic crystal system.

(106) Ceramic Crystal Structures. Materials-Bonding Classification. 금속-비금속 원소간 화합물 ‘구운 것 (firing)’. < 실제 많은 재료는 2개 혹은 그 이상의 결합에 혼합 >. 41.

(107) Covalent Bonding H2 H 2.1 Li 1.0 Na 0.9 K 0.8. Be 1.5 Mg 1.2 Ca 1.0. Rb 0.8 Cs 0.7. Sr 1.0 Ba 0.9. Fr 0.7. Ra 0.9. column IVA. H2O C(diamond) SiC Ti 1.5. Cr 1.6. Fe 1.8. Ni 1.8. Zn 1.8. Ga 1.6. C 2.5 Si 1.8 Ge 1.8. F2 He O 2.0. As 2.0. Sn 1.8 Pb 1.8. F 4.0 Cl 3.0 Br 2.8 I 2.5 At 2.2. Ne -. Cl2. Ar Kr Xe Rn -. GaAs • • • •. molecules with nonmetals molecules with metals and nonmetals elemental solids (RHS of Periodic Table) compound solids (about column IVA) 42.

(108) Ionic Bonding • Predominant bonding in Ceramics NaCl MgO CaF 2 CsCl. Give up electrons. Acquire electrons. Adapted from Fig. 2.7, Callister 7e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition, Copyright 1939 and 1940, 3rd edition. Copyright 1960 by Cornell University. 43.

(109) Ionic bond = metal. +. nonmetal. Accepts electrons. Donates electrons. (양이온). (음이온). Dissimilar electronegativities ex: MgO. Mg. 1s2 2s2 2p6 3s2 [Ne]. Mg2+ 1s2 2s2 2p6. O. 1s2 2s2 2p4. 3s2 O2- 1s2 2s2 2p6. [Ne]. [Ne]. ex) Oxide structures – oxygen anions larger than metal cations – close packed oxygen in a lattice (usually FCC) – cations fit into interstitial sites among oxygen ions. 44.

(110) Chapter 4.6. 7. Factors that Determine Ceramic Crystal Structure 1. Relative sizes of ions – Formation of stable structures: --maximize the # of oppositely charged ion neighbors.. -. +. -. -. -. unstable 2. Maintenance of Charge Neutrality :. +. -. stable. --Net charge in ceramic CaF 2 : should be zero. --Reflected in chemical formula:. -. Adapted from Fig. 4.4, Callister & Rethwisch 9e.. + -. -. stable Ca 2+ + cation. Fanions F-. A m Xp. m, p values to achieve charge neutrality. 45.

(111) Chapter 4.6. Coordination Number and Ionic Radii r cation • Coordination Number increases with r anion To form a stable structure, how many anions can surround around a cation? r cation r anion < 0.155. Coord. Number linear 2 triangular. 0.155 - 0.225. 3. 0.225 - 0.414. 4 tetrahedral. 0.414 - 0.732. 6 octahedral. 0.732 - 1.0. 8. Adapted from Table 4.3, Callister & Rethwisch 9e.. cubic. ZnS (zinc blende) Adapted from Fig. 4.7, Callister & Rethwisch 9e.. NaCl (sodium chloride). Adapted from Fig. 4.5, Callister & Rethwisch 9e.. CsCl (cesium chloride) Adapted from Fig. 4.6, Callister & Rethwisch 9e. 46.

(112) Interstitial Position & Coordination Number • Interstitial sites in FCC structure. Octahedral sites ; 4. Tetrahedral sites ; 8.

(113) Open space in FCC 1. 48.

(114) Open space in FCC 1. a) Octahedral sites 1 +1/4 +1/4 +1/4+1/4 +4*(1/4) +4*(1/4) =4. 49.

(115) Open space in FCC 2. Visible Atoms Atoms behind. Tetrahedral sites = 8 50.

(116) Interstitial Position & Coordination Number • Interstitial sites in HCP structure. Tetrahedral sites ; 4. 3 ( 0, 0, ) 8. 5 ( 0, 0, ) 8. 1 2 1 1 2 7 ( , , ) ( , , ) 3 3 8 3 3 8.

(117) Interstitial Position & Coordination Number • Interstitial sites in HCP structure. Octahedral sites ; 2. 2 1 1 2 1 3 ( , , ) ( , , ) 3 3 4 3 3 4.

(118) Computation of Minimum Cation-Anion Radius Ratio • Determine minimum rcation/ranion for an octahedral site (C.N. = 6). a = 2ranion. 53.

(119) Example Problem: Predicting the Crystal Structure of FeO • On the basis of ionic radii, what crystal structure would you predict for FeO? Cation Ionic radius (nm) Al 3+ 0.053 Fe 2+ 0.077 Fe 3+ 0.069 Ca 2+ 0.100 Anion O2Cl F-. • Answer:. based on this ratio, -- coord # = 6 because. 0.140 0.181 0.133. 0.414 < 0.550 < 0.732 -- crystal structure is NaCl Data from Table 4.4, Callister & Rethwisch 9e. 54.

(120) Chapter 4.6. Coordination Number and Ionic Radii r cation • Coordination Number increases with r anion To form a stable structure, how many anions can surround around a cation? r cation r anion < 0.155. Coord. Number linear 2 triangular. 0.155 - 0.225. 3. 0.225 - 0.414. 4 tetrahedral. 0.414 - 0.732. 6 octahedral. 0.732 - 1.0. 8. Adapted from Table 4.3, Callister & Rethwisch 9e.. cubic. ZnS (zinc blende) Adapted from Fig. 4.7, Callister & Rethwisch 9e.. NaCl (sodium chloride). Adapted from Fig. 4.5, Callister & Rethwisch 9e.. CsCl (cesium chloride) Adapted from Fig. 4.6, Callister & Rethwisch 9e. 55.

(121) Chapter 4.7, 4.8, 4.9. 8. Ceramic Structure. 56.

(122) a) AX Crystal Structures AX–Type Crystal Structures include NaCl, CsCl, and zinc blende. Same concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure rNa = 0.102 nm rCl = 0.181 nm rNa/rCl = 0.564 ∴ cations (Na+) prefer octahedral sites. Adapted from Fig. 4.5, Callister & Rethwisch 9e.. 0.414 - 0.732. 6 octahedral 57.

(123) Show why NaCl is FCC. Atom B Atom A. =. 58.

(124) MgO and FeO MgO and FeO also have the NaCl structure O2-. rO = 0.140 nm. Mg2+. rMg = 0.072 nm. rMg/rO = 0.514 ∴ cations prefer octahedral sites Adapted from Fig. 4.5, Callister & Rethwisch 9e.. 0.414 - 0.732. 6 octahedral. So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms 59.

(125) a) AX Crystal Structures AX–Type Crystal Structures include NaCl, CsCl, and zinc blende Cesium Chloride structure:. ∴ Since 0.732 < 0.939 < 1.0, cubic sites preferred Fig. 4.6, Callister & Rethwisch 9e.. So each Cs+ has 8 neighbor Cl0.732 - 1.0. 8. cubic. 60.

(126) b) AX2 Crystal Structures Fluorite structure • Calcium Fluorite (CaF2) • Cations in cubic sites • UO2, ThO2, ZrO2, CeO2 • Antifluorite structure – positions of cations and anions reversed Fig. 4.8, Callister & Rethwisch 9e.. 61.

(127) c) ABX3 Crystal Structures • Perovskite structure Ex: complex oxide BaTiO3. Fig. 4.9, Callister & Rethwisch 9e.. 62.

(128) Chapter 4.10 이론밀도. 9. Density Computations for Ceramics Number of formula units/unit cell. Avogadro’s number Volume of unit cell = sum of atomic weights of all cations in formula unit = sum of atomic weights of all anions in formula unit. Formular unit: 화학식 단위에 포함되어 있는 모든 이온 ex) BaTiO3 Ba 이온 1개, Ti 이온 1개, 산소이온 3개. 63.

(129) Densities of Material Classes In general ρ metals > ρ ceramics > ρ polymers 30. Why? Metals have.... Ceramics have... • less dense packing • often lighter elements. ρ (g/cm3 ). • close-packing (metallic bonding) • often large atomic masses. 20. Platinum Gold, W Tantalum. 10. Silver, Mo Cu,Ni Steels Tin, Zinc. 5 4 3 2. Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O). Composites have... • intermediate values. Metals/ Alloys. 1. 0.5 0.4 0.3. Titanium Aluminum Magnesium. Graphite/ Ceramics/ Semicond. Polymers. Composites/ fibers. Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix). Zirconia Al oxide Diamond Si nitride Glass -soda Concrete Silicon Graphite. Glass fibers PTFE Silicone PVC PET PC HDPE, PS PP, LDPE. GFRE* Carbon fibers CFRE* Aramid fibers AFRE*. Wood. Data from Table B.1, Callister & Rethwisch, 9e.. 64.

(130) Chapter 4.11. 10. Silicate Ceramics a) Most common elements on earth are Si & O. Si4+ O2-. Figs. 4.10 & 4.11, Callister & Rethwisch 9e. crystobalite. • SiO2 (silica) polymorphic forms are quartz, crystobalite, & tridymite • The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material 65.

(131) b) Silicates Bonding of adjacent SiO44- accomplished by the sharing of common corners, edges, or faces. Mg2SiO4. Ca2MgSi2O7. Adapted from Fig. 4.13, Callister & Rethwisch 9e.. Presence of cations such as Ca2+, Mg2+, & Al3+ 1. maintain charge neutrality, and 2. ionically bond SiO44- to one another. 66.

(132) c) Glass Structure • Basic Unit:. Glass is noncrystalline (amorphous). 4Si0 4 tetrahedron Si 4+ O2-. • Quartz is crystalline SiO2:. • Fused silica is SiO2 to which no impurities have been added • Other common glasses contain impurity ions such as Na+, Ca2+, Al3+, and B3+ Na + Si 4+ O2-. (soda glass) Adapted from Fig. 4.12, Callister & Rethwisch 9e. 67.

(133) d) Layered Silicates • Layered silicates (e.g., clays, mica, talc) – SiO4 tetrahedra connected together to form 2-D plane. • A net negative charge is associated with each (Si2O5)2- unit • Negative charge balanced by adjacent plane rich in positively charged cations. Fig. 4.14, Callister & Rethwisch 9e.. 68.

(134) d) Layered Silicates (cont) • Kaolinite clay alternates (Si2O5)2- layer with Al2(OH)42+ layer. Fig. 4.15, Callister & Rethwisch 9e.. Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces.. 69.

(135) Chapter 4.12. 11. Polymorphic Forms of Carbon Diamond – tetrahedral bonding of carbon • hardest material known • very high thermal conductivity. – large single crystals – gem stones – small crystals – used to grind/cut other materials – diamond thin films • hard surface coatings – used for cutting tools, medical devices, etc.. Fig. 4.17, Callister & Rethwisch 9e.. 70.

(136) Polymorphic Forms of Carbon (cont) Graphite – layered structure – parallel hexagonal arrays of carbon atoms. Fig. 4.18, Callister & Rethwisch 9e.. – weak van der Waal’s forces between layers – planes slide easily over one another -- good lubricant 71.

(137) Contents for today’s class. CHAPTER 4: The Structure of Crystalline Solids (9th edition) I. METALLIC CRYSTALS _ APF , CN … - The face-centered cubic crystal structure (FCC) - The body-centered cubic crystal structure (BCC) - The hexagonal closed-packed crystal structure (HCP) II. Ceramic Crystal Structure (Chapter 12, 10th edition) - Ionic arrangement geometries - Some common ceramic crystal structure/ Density computation - Silicate ceramics/ Carbon. 72.

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