Chapter 03 C R Y S T A L S

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Chapter 03 C R Y S T A L S

  1. 1. Chapter 3 Structures of Metals and Ceramics • How do atoms assemble into solid structures? • How do the structures of ceramic materials differ from those of metals? • How does the density of a material depend on its structure? • When do material properties vary with the sample (i.e., part) orientation?
  2. 2. ENERGY AND PACKING Now, bonding energy is not only between two atoms, its from many atoms. Dense, regular-packed structures tend to have lower energy. • Non dense, random packing Energy typical neighbor bond length typical neighbor r bond energy average • Dense, regular packing Energy typical neighbor bond length typical neighbor r bond energy
  3. 3. Building 3D ‘ordered’ array of atoms for Dummies (i) Construct lattice (ii) Filling the lattice with atoms or molecules or group of atoms/molecules You could choose many number of different unit cells for the same building process.
  4. 4. 7 Crystal Systems & 14 Crystal Lattices Any crystalline structure (3D ordered array of atoms/molecules) must fall into one of the systems and one of the crystal lattices.
  5. 5. Often called ‘lattice constants’ Unit cells
  6. 6. 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 non-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...
  7. 7. A B B C A B B B C C B B
  8. 8. SIMPLE CUBIC (SC) STRUCTURE • Unit cell (Bravais lattice): Simple cubic • Rare due to poor packing (only Po has this structure) • Close-packed directions are cube edges. • Coordination # (CN) = 6 (# of nearest neighboring atoms) 1/8 CN is the one way to tell how much the structure is packed with atoms.
  9. 9. Here’s the better way to tell about packing. ATOMIC PACKING FACTOR (APF) Volume of atoms* in unit cell APF = Volume of unit cell *assume hard spheres Close-packed direction: a= 2R a volume R=0.5a atoms atom 4 unit cell 1 π (0.5a)3 3 APF = = 0.52 There are 8 of 1/8 atoms. 3 a volume 1 atom/unit cell unit cell
  10. 10. FACE-CENTERED CUBIC (FCC) Structure • Unit cell (Bravais lattice): FCC • Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. • γ-Fe, Al, Ni, Cu, Ag, Pt, and Au • Coordination # = 12 Grey and red atoms are same.
  11. 11. ATOMIC PACKING FACTOR: FCC Structure Close-packed directions: length = 4R = 2a Unit cell contains: 6 x 1/2 + 8 x 1/8 a = 4 atoms/unit cell atoms volume 4 3 unit cell 4 π ( 2a/4) 3 atom APF = = 0.74 3 volume a unit cell
  12. 12. BODY-CENTERED CUBIC (BCC) Structure • Unit cell (Bravais lattice): BCC • Close packed directions are cube diagonals. • α-Fe, Cr, Mo, W, and V • Coordination # = 8
  13. 13. ATOMIC PACKING FACTOR: BCC Close-packed directions: length = 4R = 3a Unit cell contains: 1 + 8 x 1/8 R = 2 atoms/unit cell a atoms volume 4 3 unit cell 2 π ( 3a/4) 3 atom APF = 3 volume = 0.68 a unit cell
  14. 14. Summary (Metal Cubic System + HCP) Unit Cell Name of Structure (Bravais lattice) CN APF SC SC 6 0.52 FCC FCC 12 0.74 BCC BCC 8 0.68 HCP hexagonal 12 0.74 Next slide
  15. 15. HEXAGONAL CLOSE-PACKED (HCP) STRUCTURE • Unit cell (Bravais lattice): Hexagonal • ABAB... Stacking Sequence • 3D Projection A sites • 2D Projection B sites Top layer A sites Middle layer Bottom layer • Coordination # = 12 • APF = 0.74 • Be, Mg, α-Ti, Zn, and Zr Unit cell: 1/3 of it
  16. 16. Closed Packed Planes (metals) FCC – ABCABC HCP – ABABAB A B B C A B B B C C B B A A sites B B sites C C sites
  17. 17. THEORETICAL DENSITY, ρ # atoms/unit cell Atomic weight (g/mol) ρ= nA Volume/unit cell VcNA Avogadro's number (cm3/unit cell) (6.023 x 10 23 atoms/mol) Example: Copper Data from Table inside front cover of texbook • crystal structure = FCC: 4 atoms/unit cell • atomic weight = 63.55 g/mol (1 amu = 1 g/mol) • atomic radius R = 0.128 nm (1 nm = 10-7cm) Vc = a3 ; For FCC, a = 4R/ 2 ; Vc = 4.75 x 10-23cm3 Result: theoretical ρCu = 8.89 g/cm3
  18. 18. Before we study crystal structure of ceramics, We need to learn crystallographic notations
  19. 19. Crystallographic Points, Directions, and Planes Points (Example - cubic system) No parenthesis ! No comma ! In fact, we’ll only deal with cubic in this course. a, b, c : lattice constant q r s : multiple or fraction of Point Coordinates? lattice constant
  20. 20. Examples Fraction possible
  21. 21. Crystallographic Points, Directions, and Planes Directions (Cubic) [uvw] & <uvw> Miller Indices
  22. 22. Family: <111> [111] [111] [111] [111] Cubic system [111] [111] [111] [111] How about tetragonal system?
  23. 23. [112] [111] [111]
  24. 24. Crystallographic Points, Directions, and Planes Planes (Cubic) (hkl) & {hkl} Miller Indices
  25. 25. Crystallographic Points, Directions, and Planes (hkl) & {hkl} Miller Indices Planes (Cubic)
  26. 26. Linear and Planar Densities FCC crystal structure (metal) Closed packed direction Closed packed plane LD = # of atoms centered on direction vector/length of direction vector PD = # of atoms centered on a plane/area of plane
  27. 27. Closed Packed Planes (metals) FCC – (111) : ABCABC A B B C A B B B C C B B A A sites B B sites C C sites
  28. 28. Closed Packed Planes (metals) HCP – (0001): ABABAB A B B C A B B B C C B B Unit cell: hexagonal
  29. 29. Now we learn crystal structure of ceramics.
  30. 30. CERAMIC CRYSTALS • Bonding: --Mostly ionic, some covalent. --% ionic character increases with difference in electronegativity. • Large vs small ionic bond character: H 2.1 CaF2: large He - Li Be C F Ne 1.0 1.5 SiC: small 2.5 4.0 - Na Mg Si Cl Ar 0.9 1.2 1.8 3.0 - K Ca Ti Cr Fe Ni Zn As Br Kr 0.8 1.0 1.5 1.6 1.8 1.8 1.8 2.0 2.8 - Rb Sr I Xe 0.8 1.0 2.5 - Cs Ba At Rn 0.7 0.9 2.2 - Fr Ra 0.7 0.9 Table of Electronegativities
  31. 31. IONIC BONDING & STRUCTURE • Charge Neutrality: F- --Net charge in the CaF2: Ca2+ + cation anions structure should be zero. F- --General form: AmXp # of atoms m, p determined by charge neutrality • Rcation/Ranion (Ratio of ionic radii) ⇒ determines CN (next slide) --maximize the # of nearest oppositely charged neighbors (while maintaining charge neutrality and stability) - - - - - - + + + - - - - - - unstable stable stable
  32. 32. COORDINATION # AND IONIC RADII Q: How many anions can you arrange around a cation? rcation • Coordination # increases with r anion rcation ZnS Coord # ranion (zincblende) < .155 2 .155-.225 3 NaCl (sodium .225-.414 4 chloride) .414-.732 6 CsCl (cesium .732-1.0 8 chloride)
  33. 33. Different crystal structures with the same Bravais lattice (unit cell) FCC Bravais lattice (Metal vs. Ionic Material) • Structure of NaCl • Structure of FCC metals Bravais lattice: FCC Bravais lattice: FCC Coordination #: 6 Coordination #: 12
  34. 34. APF (or Ionic pakcing factor (IPF)) metals vs ionic material • Structure of NaCl • Structure of FCC metals Bravais lattice: FCC Bravais lattice: FCC Coordination #: 6 Coordination #: 12 Note the difference in closed-packed direction. a = 2r Na+ + 2rCl- a = 2r√2
  35. 35. Semiconducting Materials (Covalent bonding)
  36. 36. Allotropes & Polymorphs Different stable (or metastable) Allotropes of carbon crystal structures of the same compounds Graphite Different stable (and metastable) crystal structures of single element Diamond Fullerene (C60) Carbon nanotube
  37. 37. Crystalline vs. Amorphous Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Si Oxygen Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2
  38. 38. Single-crystalline vs. Polycrystalline Grain boundaries
  39. 39. POLYCRYSTALS • Most engineering materials are polycrystals. 1 mm • Nb-Hf-W plate with an electron beam weld. • Each "grain" is a single crystal. • If crystals are randomly oriented, overall component properties are not directional. • Crystal sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).
  40. 40. SINGLE VS POLYCRYSTALS • Single Crystals E (diagonal) = 273 GPa -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: E (edge) = 125 GPa • Polycrystals -Properties may/may not 200 µm vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.
  41. 41. X-ray Diffraction to determine Crystal Structure X-ray Beams 1 & 2 have to be in phase to be diffracted. Detector Source (next slide) variables spacing Extra distance travelled by wave 2 between planes • Incoming X-rays diffract from crystal planes.
  42. 42. n: order of reflection Bragg’s law n λ = 2 d sin θ Extra distance travelled by beam 2 have to be an integer multiple of λ. • Bragg’s law is a necessary but not sufficient condition for diffraction.
  43. 43. θ-2θ scan θ Typically X-ray source and detector X-ray are both rotating. Detector source If sample S is polycrystalline, X-ray data will resemble the date below.
  44. 44. DENSITIES OF MATERIAL CLASSES Graphite/ ρmetals ρceramics ρpolymers Metals/ Ceramics/ Polymers Composites/ Alloys fibers Semicond 30 Why? Based on data in Table B1, Callister 20 Platinum *GFRE, CFRE, & AFRE are Glass, Metals have... Gold, W Tantalum Carbon, & Aramid Fiber-Reinforced • close-packing Epoxy composites (values based on 60% volume fraction of aligned fibers (metallic bonding) 10 Silver, Mo Cu,Ni in an epoxy matrix). ρ (g/cm3) Steels • large atomic mass Tin, Zinc Zirconia Ceramics have... 5 Titanium 4 Al oxide • less dense packing Diamond Si nitride 3 (covalent bonding) Aluminum Glass-soda Concrete Glass fibers Silicon PTFE • often lighter elements 2 Magnesium Graphite GFRE* Carbon fibers Silicone CFRE* Polymers have... PVC PET Aramid fibers AFRE* PC • poor packing 1 HDPE, PS PP, LDPE (often amorphous) • lighter elements (C,H,O) 0.5 Wood Composites have... 0.4 0.3 • intermediate values
  45. 45. SUMMARY • 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). • Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but properties are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.

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