بسم هللا الرحمن الرحیم. Materials Science. Chapter 3 Structures of Metals & Ceramics

Transcription

1 بسم هللا الرحمن الرحیم Materials Science Chapter 3 Structures of Metals & Ceramics 1

2 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? 2

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4 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. 4

5 Materials and Packing Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers crystalline SiO2 Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures -rapid cooling Si Oxygen "Amorphous" = Noncrystalline noncrystalline SiO2 5

6 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 6

7 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... 7

8 Simple Cubic Structure (SC) Rare due to low packing density (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) 8

9 Atomic Packing Factor (APF) APF = Volume of atoms in unit cell* Volume of unit cell APF for a simple cubic structure = 0.52 a *assume hard spheres close-packed directions contains 8 x 1/8 = 1 atom/unit cell R=0.5a atoms unit cell APF = 1 volume 4 3 p (0.5a) 3 atom a 3 volume unit cell 9

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11 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 ( ), Tantalum, Molybdenum Coordination # = 8 (Courtesy P.M. Anderson) 2 atoms/unit cell: 1 center + 8 corners x 1/8 11

12 Atomic Packing Factor: BCC APF for a body-centered cubic structure = a a 2 a R a Close-packed directions: length = 4R = 3 a atoms unit cell APF = p ( 3a/4) 3 a 3 volume unit cell volume atom 12

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14 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 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 14

15 Atomic Packing Factor: FCC APF for a face-centered cubic structure = 0.74 maximum achievable APF 2 a a Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell atoms unit cell APF = p ( 2a/4) 3 a 3 volume atom volume unit cell 15

16 ABCABC... Stacking Sequence 2D Projection A sites B sites C sites FCC Stacking Sequence A B B C B C B B C B B FCC Unit Cell A B C 16

17 Theoretical Density, r Density = r = r = Mass of Atoms Total n A V C N A 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 = x atoms/mol 17

18 Theoretical Density, r Ex: Cr (BCC) A = g/mol R = nm n = 2 atoms/unit cell R a a = 4R/ 3 = nm atoms unit cell r = volume unit cell a x g mol r theoretical r actual atoms mol = 7.18 g/cm 3 = 7.19 g/cm 3 18

19 The five fundamental two-dimensional Bravais lattices: 1 oblique, 2 rectangular, 3 centered rectangular (rhombic), 4 hexagonal, and 5 square 19

20 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 20

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22 Lattice System Possible Variations Axial Distances (edge lengths) Axial Angles Cubic Primitive, Body-centred, Facecentred a = b = c α = β = γ = 90 Tetragonal Primitive, Body-centred a = b c α = β = γ = 90 Orthorhombic Primitive, Body-centred, Facecentred, Base-centred a b c α = β = γ = 90 Hexagonal Primitive a = b c α = β = 90, γ = 120 Rhombohedral Primitive a = b = c α = β = γ 90 Monoclinic Primitive, Base-centred a b c α = γ = 90, β 90 Triclinic Primitive a b c α β γ 90 22

23 P (pcc) I (bcc) F (fcc) Cubic 23

24 P I Tetragonal 24

25 Orthorhombic P C I F 25

26 P Hexagonal 26

27 P Rhombohedral 27

28 P C Monoclinic 28

29 P Triclinic 29

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31 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 identical position in another unit cell b 31

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35 Crystallographic Directions x z 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 ] where overbar represents a negative index families of directions <uvw> 35

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39 Linear Density Linear Density of Atoms LD = Number of atoms Unit length of direction vector [110] ex: linear density of Al in [110] direction a = nm a # atoms length LD = 2 = 2a 3.5 nm -1 39

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41 Crystallographic Planes 41

42 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) 42

43 Crystallographic Planes z example a b c 1. Intercepts Reciprocals 1/1 1/1 1/ Reduction Miller Indices (110) example a b c 1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/ Reduction Miller Indices (100) a x a c c z b b y y x 43

44 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 Miller Indices (634) x a c b y Family of Planes {hkl} Ex: {100} = (100), (010), (001), (100), (010), (001) 44

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48 Planar Density of (100) Iron Solution: At T < 912 C iron has the BCC structure. 2D repeat unit (100) a = 4 3 R 3 Radius of iron R = nm atoms 2D repeat unit 1 Planar Density = a 2 area 2D repeat unit = R 2 = atoms 12.1 nm 2 atoms = 1.2 x m 2 48

49 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 area = 2 ah = 3 a = 3 R = R atoms nm 2 = 7.0 = h 2 = 0.70 x a R atoms m

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58 Single vs Polycrystals Single Crystals -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. (E poly iron = 210 GPa) -If grains are textured, anisotropic. E (diagonal) = 273 GPa E (edge) = 125 GPa 200 mm 58

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60 X-Ray Diffraction Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation. Can t resolve spacings Spacing is the distance between parallel planes of atoms. 60

61 First medical X-ray by Wilhelm Röntgen of his wife Anna Bertha Ludwig's hand

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65 X-Rays to Determine Crystal Structure Incoming X-rays diffract from crystal planes. extra distance travelled by wave 2 q q d reflections must be in phase for a detectable signal spacing between planes Measurement of critical angle, q c, allows computation of planar spacing, d. X-ray intensity (from detector) d = n 2 sin q c q q c 65

66 Intensity (relative) z c X-Ray Diffraction Pattern z c z c a x b y (110) a x b y a x (211) b y (200) Diffraction angle 2q Diffraction pattern for polycrystalline -iron (BCC) 66

67 Materials and Packing Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers crystalline SiO2 Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures -rapid cooling Si Oxygen "Amorphous" = Noncrystalline noncrystalline SiO2 67

68 Densities of Material Classes In general r metals > r ceramics > r polymers Why? Metals have... close-packing (metallic bonding) often large atomic masses r (g/cm 3 ) Ceramics have... less dense packing often lighter elements Polymers have... low packing density (often amorphous) lighter elements (C,H,O) Composites have... intermediate values Metals/ Alloys Platinum Gold, W Tantalum Silver, Mo Cu,Ni Steels Tin, Zinc Titanium Aluminum Magnesium Graphite/ Ceramics/ Semicond Polymers Composites/ fibers B ased 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 G raphite PTFE Silicone PVC PET PC HDPE, PS PP, LDPE Glass fibers GFRE* Carbon fibers CFRE * A ramid fibers AFRE * Wood 68

69 Glass Structure Basic Unit: 4- Si0 4 tetrahedron Si 4+ O 2- 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+ Quartz is crystalline SiO2: Na + Si 4+ O 2- (soda glass) 69