Chapter 3: Structures of Metals & Ceramics

Chapter 3: Structures of Metals & Ceramics School of Mechanical Engineering Professor Choi, Hae-Jin Chapter 3-1

Chapter 3: Structures of Metals & Ceramics ISSUES TO ADDRESS... How do atoms assemble into solid structures? How does the density of a material depend on its structure? How do the crystal structures of ceramic materials differ from those for metals? When do material properties vary with the sample (i.e., part) orientation? Chapter 3-2

Energy and Packing Non dense, random packing Energy typical neighbor bond length typical neighbor bond energy r Dense, ordered packing Energy typical neighbor bond length typical neighbor bond energy r Dense, ordered packed structures tend to have lower energies. Chapter 3-3

Materials and Packing Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures -rapid cooling crystalline SiO2 Adapted from Fig. 3.40(a), Callister & Rethwisch 3e. Si Oxygen "Amorphous" = Noncrystalline noncrystalline SiO2 Adapted from Fig. 3.40(b), Callister & Rethwisch 3e. Chapter 3-4

Metallic Crystal Structures How can we stack metal atoms to minimize empty space? 2-dimensions vs. Now stack these 2-D layers to make 3-D structures Chapter 3-5

Metallic Crystal Structures Tend to be densely packed. 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. - Electron cloud shields cores from each other Have the simplest crystal structures. We will examine three such structures... Chapter 3-6

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 (a), Tantalum, Molybdenum Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 3e. 2 atoms/unit cell: 1 center + 8 corners x 1/8 Chapter 3-7

Atomic Packing Factor (APF) APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres Chapter 3-8

Atomic Packing Factor: BCC APF for a body-centered cubic structure = 0.68 3 a a 2 a Adapted from Fig. 3.2(a), Callister & Rethwisch 3e. atoms unit cell APF = R 2 a 4 3 p ( 3a/4)3 a 3 Close-packed directions: length = 4R = 3 a volume unit cell volume atom Chapter 3-9

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 & Rethwisch 3e. 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 Chapter 3-10

Atomic Packing Factor: FCC APF for a face-centered cubic structure = 0.74 maximum achievable APF 2 a a Adapted from Fig. 3.1(a), Callister & Rethwisch 3e. atoms unit cell APF = Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell 4 4 3 p ( 2a/4) 3 a 3 volume atom volume unit cell Chapter 3-11

Hexagonal Close-Packed Structure (HCP) ABAB... Stacking Sequence 3D Projection A sites 2D Projection Top layer c B sites Middle layer a Coordination # = 12 APF = 0.74 c/a = 1.633 A sites Adapted from Fig. 3.3(a), Callister & Rethwisch 3e. Bottom layer 6 atoms/unit cell ex: Cd, Mg, Ti, Zn Chapter 3-12

Hexagonal Close-Packed Structure (HCP) Chapter 3-13

Theoretical Density, r Density = r = r = Mass of Total n A V C N A Atoms Volume of in Unit Unit Cell Cell where n = number of atoms/unit cell A = atomic weight V C = Volume of unit cell = a 3 for cubic N A = Avogadro s number = 6.022 x 10 23 atoms/mol Chapter 3-14

Theoretical Density, r Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2 atoms/unit cell Adapted from Fig. 3.2(a), Callister & Rethwisch 3e. volume unit cell atoms unit cell r = R a 2 52.00 a 3 6.022 x 10 23 a = 4R/ 3 = 0.2887 nm g mol r theoretical r actual atoms mol = 7.18 g/cm 3 = 7.19 g/cm 3 Chapter 3-15

Atomic Bonding in Ceramics Bonding: -- Can be ionic and/or covalent in character. -- % ionic character increases with difference in electronegativity of atoms. Degree of ionic character may be large or small: CaF 2 : large SiC: small Adapted from Fig. 2.7, Callister & Rethwisch 3e. (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. Chapter 3-16

Ceramic Crystal Structures Oxide structures oxygen anions larger than metal cations close packed oxygen in a lattice (usually FCC) cations fit into interstitial sites among oxygen ions Chapter 3-17

Factors that Determine Crystal Structure 1. Relative sizes of ions Formation of stable structures: --maximize the # of oppositely charged ion neighbors. - - + - - - - + - - - - + - - unstable stable stable 2. Maintenance of Charge Neutrality : --Net charge in ceramic should be zero. --Reflected in chemical formula: AmXp CaF2: Ca 2+ cation + Adapted from Fig. 3.4, Callister & Rethwisch 3e. F - anions F - m, p values to achieve charge neutrality Chapter 3-18

Coordination # and Ionic Radii Coordination # increases with r cation Coord r anion # < 0.155 2 linear r cation r anion To form a stable structure, how many anions can surround around a cation? ZnS (zinc sulfide) Adapted from Fig. 3.7, Callister & Rethwisch 3e. 0.155-0.225 0.225-0.414 0.414-0.732 0.732-1.0 3 4 6 8 Adapted from Table 3.3, Callister & Rethwisch 3e. triangular tetrahedral octahedral cubic NaCl (sodium chloride) Adapted from Fig. 3.5, Callister & Rethwisch 3e. CsCl (cesium chloride) Adapted from Fig. 3.6, Callister & Rethwisch 3e. Chapter 3-19

Computation of Minimum Cation-Anion Radius Ratio Determine minimum r cation /r anion for an octahedral site (C.N. = 6) 2 anion cation = r + 2r 2a a = 2r anion 2r anion + 2r cation = 2 2r anion r anion + r cation = 2r anion r cation = ( 2-1)r anion rcation = 2-1= r anion 0.414 Chapter 3-20

Example: Predicting the Crystal Structure of FeO On the basis of ionic radii, what crystal structure would you predict for FeO? Cation Al 3+ Fe 2+ Fe 3+ Ca 2+ Anion O 2- Cl - F - Ionic radius (nm) 0.053 0.077 0.069 0.100 0.140 0.181 0.133 Data from Table 3.4, Callister & Rethwisch 3e. Answer: rcation = r anion 0. 077 0. 140 = 0.550 based on this ratio, -- coord # = 6 because 0.414 < 0.550 < 0.732 -- crystal structure is NaCl Chapter 3-21

Rock Salt Structure Same concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure r Na = 0.102 nm r Cl = 0.181 nm r Na /r Cl = 0.564 \ cations (Na + ) prefer octahedral sites Adapted from Fig. 3.5, Callister & Rethwisch 3e. Chapter 3-22

MgO and FeO MgO and FeO also have the NaCl structure O 2- Mg 2+ r O = 0.140 nm r Mg = 0.072 nm r Mg /r O = 0.514 \ cations prefer octahedral sites Adapted from Fig. 3.5, Callister & Rethwisch 3e. So each Mg 2+ (or Fe 2+ ) has 6 neighbor oxygen atoms Chapter 3-23

AX Crystal Structures AX Type Crystal Structures include NaCl, CsCl, and zinc blende Cesium Chloride structure: r Cs + = = r Cl - 0.170 0.181 0.939 Adapted from Fig. 3.6, Callister & Rethwisch 3e. \ Since 0.732 < 0.939 < 1.0, cubic sites preferred So each Cs + has 8 neighbor Cl - Chapter 3-24

AX 2 Crystal Structures Fluorite structure Calcium Fluorite (CaF 2 ) Cations in cubic sites UO 2, ThO 2, ZrO 2, CeO 2 Antifluorite structure positions of cations and anions reversed Adapted from Fig. 3.8, Callister & Rethwisch 3e. Chapter 3-25

ABX 3 Crystal Structures Perovskite structure Ex: complex oxide BaTiO 3 Adapted from Fig. 3.9, Callister & Rethwisch 3e. Chapter 3-26

Density Computations for Ceramics Number of formula units/unit cell r = n ( SA V C + SAA ) C N A Volume of unit cell Avogadro s number SA C = sum of atomic weights of all cations in formula unit SA A = sum of atomic weights of all anions in formula unit Chapter 3-27

r(g/cm ) 3 ) Densities of Material Classes In general r metals > r ceramics > r polymers Why? Metals have... close-packing (metallic bonding) often large atomic masses Ceramics have... less dense packing often lighter elements Polymers have... low packing density (often amorphous) lighter elements (C,H,O) Composites have... intermediate values 30 20 10 5 4 3 2 1 0.5 0.4 0.3 Metals/ Alloys Platinum Gold, W Tantalum Silver, Mo Cu,Ni Steels Tin, Zinc 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 PTFE Silicone PVC PET PC HDPE, PS PP, LDPE Data from Table B.1, Callister & Rethwisch, 3e. Glass fibers GFRE* Carbon fibers CFRE* Aramid fibers AFRE* Wood Chapter 3-28

Silicate Ceramics Most common elements on earth are Si & O Si 4+ O 2- Adapted from Figs. 3.10-11, Callister & Rethwisch 3e crystobalite SiO 2 (silica) polymorphic forms are quartz, crystobalite, & tridymite The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material Chapter 3-29

Glass Structure Basic Unit: 4- Si0 4 tetrahedron Si 4+ O 2- Quartz is crystalline SiO2: Glass is noncrystalline (amorphous) Fused silica is SiO 2 to which no impurities have been added Other common glasses contain impurity ions such as Na +, Ca 2+, Al 3+, and B 3+ Na + Si 4+ O 2- (soda glass) Adapted from Fig. 3.41, Callister & Rethwisch 3e. Chapter 3-30

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. Adapted from Fig. 3.16, Callister & Rethwisch 3e. Chapter 3-31

Polymorphic Forms of Carbon (cont) Graphite layered structure parallel hexagonal arrays of carbon atoms Adapted from Fig. 3.17, Callister & Rethwisch 3e. weak van der Waal s forces between layers planes slide easily over one another -- good lubricant Chapter 3-32

Polymorphic Forms of Carbon (cont) Fullerenes and Nanotubes Fullerenes spherical cluster of 60 carbon atoms, C 60 Like a soccer ball Carbon nanotubes sheet of graphite rolled into a tube Ends capped with fullerene hemispheres Adapted from Figs. 3.18 & 3.19, Callister & Rethwisch 3e. Chapter 3-33

Crystal Systems Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems 14 crystal lattices a, b, and c are the lattice constants Fig. 3.20, Callister & Rethwisch 3e. Chapter 3-34

c z Point Coordinates 111 Point coordinates for unit cell center are a/2, b/2, c/2 ½ ½ ½ x a z 000 b 2c y Point coordinates for unit cell corner are 111 b y Translation: integer multiple of lattice constants à identical position in another unit cell b Chapter 3-35

x z Crystallographic Directions y Algorithm 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvw] ex: 1, 0, ½ => 2, 0, 1 => [ 201 ] -1, 1, 1 => [ 111 ] families of directions <uvw> where overbar represents a negative index Chapter 3-36

Linear Density Linear Density of Atoms º LD = Number of atoms Unit length of direction vector [110] ex: linear density of Al in [110] direction a = 0.405 nm a # atoms length LD = 2 2a = 3.5 nm -1 Chapter 3-37

a 3 HCP Crystallographic Directions z a 2 Adapted from Fig. 3.24(a), Callister & Rethwisch 3e. a 1 Algorithm ex: ½, ½, -1, 0 => - 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a 1, a 2, a 3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvtw] [ 1120 ] a 3 a 2 2 a 1 2 dashed red lines indicate projections onto a 1 and a 2 axes a 1 a 2 -a 3 Chapter 3-38

HCP Crystallographic Directions Hexagonal Crystals a 3 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows. z a 2 a 1 Fig. 3.24(a), Callister & Rethwisch 3e. - [ u'v'w '] u v t w = = = = w ' [uvtw ] 1 (2u'-v') 3 1 (2v'-u') 3 -( u + v) Chapter 3-39

Crystallographic Planes Adapted from Fig. 3.25, Callister & Rethwisch 3e. Chapter 3-40

Crystallographic Planes Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices. Algorithm 1. Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl) Chapter 3-41

Crystallographic Planes z example a b c 1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/ 1 1 0 3. Reduction 1 1 0 4. Miller Indices (110) example a b c 1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/ 2 0 0 3. Reduction 2 0 0 4. Miller Indices (100) x x a a c c z b b y y Chapter 3-42

Crystallographic Planes z example a b c 1. Intercepts 1/2 1 3/4 2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction 6 3 4 4. Miller Indices (634) x a c b y Family of Planes {hkl} Ex: {100} = (100), (010), (001), (100), (010), (001) Chapter 3-43

Crystallographic Planes (HCP) In hexagonal unit cells the same idea is used z example a 1 a 2 a 3 c 1. Intercepts 1-1 1 2. Reciprocals 1 1/ -1 1 1 0-1 1 3. Reduction 1 0-1 1 a 2 a 3 4. Miller-Bravais Indices (1011) a 1 Adapted from Fig. 3.24(b), Callister & Rethwisch 3e. Chapter 3-44

Crystallographic Planes We want to examine the atomic packing of crystallographic planes Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important. a) Draw (100) and (111) crystallographic planes for Fe. b) Calculate the planar density for each of these planes. Chapter 3-45

Planar Density of (100) Iron Solution: At T < 912 C iron has the BCC structure. 2D repeat unit 4 3 3 (100) a = R Adapted from Fig. 3.2(c), Callister & Rethwisch 3e. atoms 2D repeat unit 1 Planar Density = a 2 area 2D repeat unit = 4 3 1 3 Radius of iron R = 0.1241 nm R 2 = atoms 12.1 nm 2 = 1.2 x 10 19 atoms m 2 Chapter 3-46

Planar Density of (111) Iron Solution (cont): (111) plane 1 atom in plane/ unit surface cell 2 a atoms in plane atoms above plane atoms below plane atoms 2D repeat unit Planar Density = area 2D repeat unit h = æ 4 3 ö 2 area = 2 ah = 3 a = 3 ç R = 3 è ø 16 3 1 3 2 R atoms nm 2 = 7.0 = 2 0.70 x 10 19 3 a 2 16 3 3 R atoms m 2 2 Chapter 3-47

FCC Stacking Sequence ABCABC... Stacking Sequence 2D Projection A sites B sites C sites A B B C B C B B C B B FCC Unit Cell A B C Chapter 3-48

Crystals as Building Blocks Some engineering applications require single crystals: -- diamond single crystals for abrasives (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) Properties of crystalline materials often related to crystal structure. -- Ex: Quartz fractures more easily along some crystal planes than others. -- turbine blades Fig. 9.40(c), Callister & Rethwisch 3e. (Fig. 9.40(c) courtesy of Pratt and Whitney). (Courtesy P.M. Anderson) Chapter 3-49

Polycrystalline Materials Small crystallite nuclei Growth of crystallites Grain boundary Completion of solidification Grain structure under microscope Chapter 3-50

Polycrystals Most engineering materials are polycrystals. Anisotropic 1 mm Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) Nb-Hf-W plate with an electron beam weld. Each "grain" is a single crystal. If grains are randomly oriented, 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). Isotropic Chapter 3-51

Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: Polycrystals Single vs Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (E poly iron = 210 GPa) -If grains are textured, anisotropic. E (diagonal) = 273 GPa E (edge) = 125 GPa Data from Table 3.7, Callister & Rethwisch 3e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.) 200 mm Adapted from Fig. 5.19(b), Callister & Rethwisch 3e. (Fig. 5.19(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].) Chapter 3-52

Polymorphism Two or more distinct crystal structures for the same material (allotropy/polymorphism) iron system titanium liquid a, b-ti 1538ºC carbon BCC d-fe diamond, graphite 1394ºC FCC g-fe 912ºC BCC a-fe Chapter 3-53

X-Ray Diffraction Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation. Can t resolve spacings < l Spacing is the distance between parallel planes of atoms. Chapter 3-54

X-Rays to Determine Crystal Structure Incoming X-rays diffract from crystal planes. extra distance travelled by wave 2 q q l d reflections must be in phase for a detectable signal Adapted from Fig. 3.37, Callister & Rethwisch 3e. spacing between planes Measurement of critical angle, q c, allows computation of planar spacing, d. X-ray intensity (from detector) d = nl 2 sin qc q q c Chapter 3-55

(relative) Intensity z c X-Ray Diffraction Pattern z c z c a x b y (110) a x b (200) y a x (211) b y Adapted from Fig. 3.20, Callister 5e. Diffraction angle 2q Diffraction pattern for polycrystalline a-iron (BCC) Chapter 3-56

SUMMARY Atoms may assemble into crystalline or amorphous structures. Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures. We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). Interatomic bonding in ceramics is ionic and/or covalent. Ceramic crystal structures are based on: -- maintaining charge neutrality -- cation-anion radii ratios. Crystallographic points, directions and planes are specified in terms of indexing schemes. Crystallographic directions and planes are related to atomic linear densities and planar densities. Chapter 3-57

SUMMARY Materials can be single crystals or polycrystalline. Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains. Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy). X-ray diffraction is used for crystal structure and interplanar spacing determinations. Chapter 3-58