1. Chapter 3 -1 ISSUES TO ADDRESS... How do atoms assemble into solid structures? How does the density of a material depend on its structure? When do material properties vary with the sample (i.e., part) orientation? Chapter 3 Metallic and Ceramic Structures How do the crystal structures of ceramic materials differ from those for metals?

2. Chapter 3 -2 Non dense, random packing Dense, ordered packing Dense, ordered packed structures tend to have lower energies. Energy and Packing Energy r typical neighbor bond length typical neighbor bond energy Energy r typical neighbor bond length typical neighbor bond energy

3. Chapter 3 -3 atoms pack in periodic, 3D arrays Crystalline materials... -metals -many ceramics -some polymers atoms have no periodic packing Noncrystalline materials... -complex structures -rapid cooling crystalline SiO 2 noncrystalline SiO 2 "Amorphous" = Noncrystalline Adapted from Fig. 3.40(b), Callister & Rethwisch 3e. Adapted from Fig. 3.40(a), Callister & Rethwisch 3e. Materials and Packing SiOxygen typical of: occurs for:

4. 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

5. Chapter 3 -5 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... Metallic Crystal Structures

6. Chapter 3 -6 Rare due to low packing density (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson) Simple Cubic Structure (SC)

7. Chapter 3 -7 APF for a simple cubic structure = 0.52 APF = a 3 4 3  (0.5a) 3 1 atoms unit cell atom volume unit cell volume Atomic Packing Factor (APF) APF = Volume of atoms in unit cell* Volume of unit cell *assume hard spheres Adapted from Fig. 3.42, Callister & Rethwisch 3e. close-packed directions a R=0.5a contains 8 x 1/8 = 1atom/unit cell

8. Chapter 3 -8 Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 3e. (Courtesy P.M. Anderson) Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. Body Centered Cubic Structure (BCC) ex: Cr, W, Fe (  ), Tantalum, Molybdenum 2 atoms/unit cell: 1 center + 8 corners x 1/8

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

10. Chapter 3 -10 Coordination # = 12 Adapted from Fig. 3.1, Callister & Rethwisch 3e. (Courtesy P.M. Anderson) Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. Face Centered Cubic Structure (FCC) ex: Al, Cu, Au, Pb, Ni, Pt, Ag 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

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

12. Chapter 3 -12 A sites B B B B B BB C sites C C C A B B ABCABC... Stacking Sequence 2D Projection FCC Unit Cell FCC Stacking Sequence B B B B B BB B sites C C C A C C C A

13. Chapter 3 -13 Coordination # = 12 ABAB... Stacking Sequence APF = 0.74 3D Projection 2D Projection Adapted from Fig. 3.3(a), Callister & Rethwisch 3e. Hexagonal Close-Packed Structure (HCP) 6 atoms/unit cell ex: Cd, Mg, Ti, Zn c/a = 1.633 c a A sites B sites A sites Bottom layer Middle layer Top layer

14. Chapter 3 -14 Theoretical Density,  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 Density =  = VC NAVC NA n An A  = Cell Unit of VolumeTotal Cell Unit in Atomsof Mass

15. Chapter 3 -15 Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2 atoms/unit cell  theoretical a = 4R/ 3 = 0.2887 nm  actual a R  = a 3 52.002 atoms unit cell mol g unit cell volume atoms mol 6.022 x 10 23 Theoretical Density,  = 7.18 g/cm 3 = 7.19 g/cm 3 Adapted from Fig. 3.2(a), Callister & Rethwisch 3e.

16. Chapter 3 -16 Bonding: -- Can be ionic and/or covalent in character. -- % ionic character increases with difference in electronegativity of atoms. 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. Degree of ionic character may be large or small: Atomic Bonding in Ceramics SiC: small CaF 2 : large

17. Chapter 3 -17 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

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

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

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

21. Chapter 3 -21 Bond Hybridization Bond Hybridization is possible when there is significant covalent bonding –hybrid electron orbitals form –For example for SiC X Si = 1.8 and X C = 2.5 ~ 89% covalent bonding Both Si and C prefer sp 3 hybridization Therefore, for SiC, Si atoms occupy tetrahedral sites

22. Chapter 3 -22 On the basis of ionic radii, what crystal structure would you predict for FeO? Answer: based on this ratio, -- coord # = 6 because 0.414 < 0.550 < 0.732 -- crystal structure is NaCl Data from Table 3.4, Callister & Rethwisch 3e. Example Problem: Predicting the Crystal Structure of FeO Ionic radius (nm) 0.053 0.077 0.069 0.100 0.140 0.181 0.133 Cation Anion Al 3+ Fe 2+ 3+ Ca 2+ O 2- Cl - F -

23. Chapter 3 -23 Rock Salt Structure Same concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure r Na = 0.102 nm r Na /r Cl = 0.564  cations (Na + ) prefer octahedral sites Adapted from Fig. 3.5, Callister & Rethwisch 3e. r Cl = 0.181 nm

24. Chapter 3 -24 MgO and FeO O 2- r O = 0.140 nm Mg 2+ r Mg = 0.072 nm r Mg /r O = 0.514  cations prefer octahedral sites So each Mg 2+ (or Fe 2+ ) has 6 neighbor oxygen atoms Adapted from Fig. 3.5, Callister & Rethwisch 3e. MgO and FeO also have the NaCl structure

25. Chapter 3 -25 AX Crystal Structures Adapted from Fig. 3.6, Callister & Rethwisch 3e. Cesium Chloride structure:  Since 0.732 < 0.939 < 1.0, cubic sites preferred So each Cs + has 8 neighbor Cl - AX–Type Crystal Structures include NaCl, CsCl, and zinc blende

26. Chapter 3 -26 AX 2 Crystal Structures 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. Fluorite structure

27. Chapter 3 -27 ABX 3 Crystal Structures Adapted from Fig. 3.9, Callister & Rethwisch 3e. Perovskite structure Ex: complex oxide BaTiO 3

28. Chapter 3 -28 Density Computations for Ceramics Number of formula units/unit cell Volume of unit cell Avogadro’s number = sum of atomic weights of all anions in formula unit = sum of atomic weights of all cations in formula unit

29. Chapter 3 -29 Densities of Material Classes  metals >  ceramics >  polymers Why? Data from Table B.1, Callister & Rethwisch, 3e.  (g/cm ) 3 Graphite/ Ceramics/ Semicond Metals/ Alloys Composites/ fibers Polymers 1 2 20 30 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). 10 3 4 5 0.3 0.4 0.5 Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum Graphite Silicon Glass-soda Concrete Si nitride Diamond Al oxide Zirconia HDPE, PS PP, LDPE PC PTFE PET PVC Silicone Wood AFRE* CFRE* GFRE* Glass fibers Carbonfibers Aramid fibers 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 In general

30. Chapter 3 -30 Silicate Ceramics Most common elements on earth are Si & O SiO 2 (silica) polymorphic forms are quartz, crystobalite, & tridymite The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material Si 4+ O 2- Adapted from Figs. 3.10-11, Callister & Rethwisch 3e crystobalite

31. Chapter 3 -31 Bonding of adjacent SiO 4 4- accomplished by the sharing of common corners, edges, or faces Silicates Mg 2 SiO 4 Ca 2 MgSi 2 O 7 Adapted from Fig. 3.12, Callister & Rethwisch 3e. Presence of cations such as Ca 2+, Mg 2+, & Al 3+ 1. maintain charge neutrality, and 2. ionically bond SiO 4 4- to one another

32. Chapter 3 -32 Quartz is crystalline SiO 2 : Basic Unit: 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+ (soda glass) Adapted from Fig. 3.41, Callister & Rethwisch 3e. Glass Structure Si0 4 tetrahedron 4- Si 4+ O 2- Si 4+ Na + O 2-

33. Chapter 3 -33 Layered Silicates Layered silicates (e.g., clays, mica, talc) –SiO 4 tetrahedra connected together to form 2-D plane A net negative charge is associated with each (Si 2 O 5 ) 2- unit Negative charge balanced by adjacent plane rich in positively charged cations Adapted from Fig. 3.13, Callister & Rethwisch 3e.

34. Chapter 3 -34 Kaolinite clay alternates (Si 2 O 5 ) 2- layer with Al 2 (OH) 4 2+ layer Layered Silicates (cont) Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces. Adapted from Fig. 3.14, Callister & Rethwisch 3e.

35. Chapter 3 -35 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.

36. Chapter 3 -36 Polymorphic Forms of Carbon (cont) Graphite –layered structure – parallel hexagonal arrays of carbon atoms –weak van der Waal’s forces between layers –planes slide easily over one another -- good lubricant Adapted from Fig. 3.17, Callister & Rethwisch 3e.

37. Chapter 3 -37 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.

38. Chapter 3 -38 Some engineering applications require single crystals: Properties of crystalline materials often related to crystal structure. (Courtesy P.M. Anderson) -- Ex: Quartz fractures more easily along some crystal planes than others. -- diamond single crystals for abrasives -- turbine blades Fig. 9.40(c), Callister & Rethwisch 3e. (Fig. 9.40(c) courtesy of Pratt and Whitney). (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) Crystals as Building Blocks

39. Chapter 3 -39 Most engineering materials are polycrystals. 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). Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) 1 mm Polycrystals Isotropic Anisotropic

40. Chapter 3 -40 Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: 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.) Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (E poly iron = 210 GPa) -If grains are textured, anisotropic. 200  m 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].) Single vs Polycrystals E (diagonal) = 273 GPa E (edge) = 125 GPa

41. Chapter 3 -41 Polymorphism Two or more distinct crystal structures for the same material (allotropy/polymorphism) titanium ,  -Ti carbon diamond, graphite BCC FCC BCC 1538ºC 1394ºC 912ºC  - Fe  - Fe  - Fe liquid iron system

42. Chapter 3 -42 Fig. 3.20, Callister & Rethwisch 3e. Crystal Systems 7 crystal systems 14 crystal lattices Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. a, b, and c are the lattice constants

43. Chapter 3 -43 Point Coordinates Point coordinates for unit cell center are a/2, b/2, c/2 ½ ½ ½ Point coordinates for unit cell corner are 111 Translation: integer multiple of lattice constants  identical position in another unit cell z x y a b c 000 111 y z 2c2c b b

44. Chapter 3 -44 Crystallographic Directions 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 families of directions z x Algorithm where overbar represents a negative index [ 111 ] => y

45. Chapter 3 -45 ex: linear density of Al in [110] direction a = 0.405 nm Linear Density Linear Density of Atoms  LD = a [110] Unit length of direction vector Number of atoms # atoms length 1 3.5 nm a2 2 LD  

46. Chapter 3 -46 HCP Crystallographic Directions 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 ]ex: ½, ½, -1, 0 => Adapted from Fig. 3.24(a), Callister & Rethwisch 3e. dashed red lines indicate projections onto a 1 and a 2 axes a1a1 a2a2 a3a3 -a3-a3 2 a 2 2 a 1 - a3a3 a1a1 a2a2 z Algorithm

47. Chapter 3 -47 HCP Crystallographic Directions Hexagonal Crystals –4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows.    'ww t v u )vu( +- )'u'v2( 3 1 - )'v'u2( 3 1 -  ]uvtw[]'w'v'u[  Fig. 3.24(a), Callister & Rethwisch 3e. - a3a3 a1a1 a2a2 z

48. Chapter 3 -48 Crystallographic Planes Adapted from Fig. 3.25, Callister & Rethwisch 3e.

49. Chapter 3 -49 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)

50. Chapter 3 -50 Crystallographic Planes z x y a b c 4. Miller Indices (110) examplea b c z x y a b c 4. Miller Indices (100) 1. Intercepts 1 1  2. Reciprocals 1/1 1/1 1/  1 1 0 3. Reduction 1 1 0 1. Intercepts 1/2   2. Reciprocals 1/½ 1/  1/  2 0 0 3. Reduction 2 0 0 examplea b c

51. Chapter 3 -51 Crystallographic Planes z x y a b c 4. Miller Indices (634) example 1. Intercepts 1/2 1 3/4 a b c 2. Reciprocals 1/½ 1/1 1/¾ 21 4/3 3. Reduction 63 4 (001)(010), Family of Planes {hkl} (100),(010),(001),Ex: {100} = (100),

52. Chapter 3 -52 Crystallographic Planes (HCP) In hexagonal unit cells the same idea is used example a 1 a 2 a 3 c 4. Miller-Bravais Indices(1011) 1. Intercepts 1  1 2. Reciprocals 1 1/  1 0 1 1 3. Reduction1 0 1 a2a2 a3a3 a1a1 z Adapted from Fig. 3.24(b), Callister & Rethwisch 3e.

53. Chapter 3 -53 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.

54. Chapter 3 -54 Planar Density of (100) Iron Solution: At T < 912  C iron has the BCC structure. (100) Radius of iron R = 0.1241 nm R 3 34 a  Adapted from Fig. 3.2(c), Callister & Rethwisch 3e. 2D repeat unit = Planar Density = a 2 1 atoms 2D repeat unit = nm 2 atoms 12.1 m2m2 atoms = 1.2 x 10 19 1 2 R 3 34 area 2D repeat unit

55. Chapter 3 -55 Planar Density of (111) Iron Solution (cont): (111) plane 1 atom in plane/ unit surface cell 33 3 2 2 R 3 16 R 3 4 2 a3ah2area           atoms in plane atoms above plane atoms below plane ah 2 3  a 2 2D repeat unit 1 = = nm 2 atoms 7.0 m2m2 atoms 0.70 x 10 19 3 2 R 3 16 Planar Density = atoms 2D repeat unit area 2D repeat unit

56. Chapter 3 -56 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.

57. Chapter 3 -57 X-Rays to Determine Crystal Structure X-ray intensity (from detector)   c d d  n 2 sin  c Measurement of critical angle,  c, allows computation of planar spacing, d. Incoming X-rays diffract from crystal planes. Adapted from Fig. 3.37, Callister & Rethwisch 3e. reflections must be in phase for a detectable signal spacing between planes d incoming X-rays outgoing X-rays detector   extra distance travelled by wave “2” “1” “2” “1” “2”

58. Chapter 3 -58 X-Ray Diffraction Pattern Adapted from Fig. 3.20, Callister 5e. (110) (200) (211) z x y a b c Diffraction angle 2  Diffraction pattern for polycrystalline  -iron (BCC) Intensity (relative) z x y a b c z x y a b c

59. Chapter 3 -59 Atoms may assemble into crystalline or amorphous 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). SUMMARY 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. 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. Ceramic crystal structures are based on: -- maintaining charge neutrality -- cation-anion radii ratios. Interatomic bonding in ceramics is ionic and/or covalent.

60. Chapter 3 -60 Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy). 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. X-ray diffraction is used for crystal structure and interplanar spacing determinations.

61. Chapter 3 -61 Core Problems: Self-help Problems: ANNOUNCEMENTS Reading: