![19
• Single Crystals
-Properties vary with
direction: anisotropic.
-Example: the modulus
of elasticity (E) in BCC iron:
Data from Table 3.3,
Callister & Rethwisch
8e. (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.
(Epoly iron = 210 GPa)
-If grains are textured,
anisotropic.
200 m Adapted from Fig.
4.14(b), Callister &
Rethwisch 8e.
(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].)
Single vs Polycrystals
E (diagonal) = 273 GPa
E (edge) = 125 GPa](https://image.slidesharecdn.com/week-3-day-1-structure-of-crystalline-solids1-240805043624-55ea2f62/85/Week-3-Day-1-Structure-of-Crystalline-Solids-1-ppt-19-320.jpg)
Week-3-Day-1-Structure-of-Crystalline-Solids (1).ppt
- 1. 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? The Structure of Crystalline Solids
- 2. 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. 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 SiO2 noncrystalline SiO2 "Amorphous" = Noncrystalline Adapted from Fig. 3.23(b), Callister & Rethwisch 8e. Adapted from Fig. 3.23(a), Callister & Rethwisch 8e. Materials and Packing Si Oxygen • typical of: • occurs for:
- 4. 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. 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. 6 • Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) Simple Cubic Structure (SC) Click once on image to start animation (Courtesy P.M. Anderson)
- 7. 7 • APF for a simple cubic structure = 0.52 APF = a3 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. 3.24, Callister & Rethwisch 8e. close-packed directions a R=0.5a contains 8 x 1/8 = 1 atom/unit cell
- 8. 8 • Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 8e. • 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 Click once on image to start animation (Courtesy P.M. Anderson)
- 9. 9 Atomic Packing Factor: BCC a APF = 4 3 ( 3a/4)3 2 atoms unit cell atom volume a3 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 8e. a 2 a 3
- 10. 10 • Coordination # = 12 Adapted from Fig. 3.1, Callister & Rethwisch 8e. • 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 Click once on image to start animation (Courtesy P.M. Anderson)
- 11. 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 a3 unit cell volume Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x1/2 + 8 x1/8 = 4 atoms/unit cell a 2 a Adapted from Fig. 3.1(a), Callister & Rethwisch 8e.
- 12. 12 A sites B B B B B B B C sites C C C A B B sites • ABCABC... Stacking Sequence • 2D Projection • FCC Unit Cell FCC Stacking Sequence B B B B B B B B sites C C C A C C C A A B C
- 13. 13 • Coordination # = 12 • ABAB... Stacking Sequence • APF = 0.74 • 3D Projection • 2D Projection Adapted from Fig. 3.3(a), Callister & Rethwisch 8e. 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. 14 Theoretical Density, where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.022 x 1023 atoms/mol Density = = VC NA n A = Cell Unit of Volume Total Cell Unit in Atoms of Mass
- 15. 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 = a3 52.00 2 atoms unit cell mol g unit cell volume atoms mol 6.022x1023 Theoretical Density, = 7.18 g/cm3 = 7.19 g/cm3 Adapted from Fig. 3.2(a), Callister & Rethwisch 8e.
- 16. 16 Densities of Material Classes metals > ceramics > polymers Why? Data from Table B.1, Callister & Rethwisch, 8e. (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 Carbon fibers 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
- 17. 17 • 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. 8.33(c), Callister & Rethwisch 8e. (Fig. 8.33(c) courtesy of Pratt and Whitney). (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) Crystals as Building Blocks
- 18. 18 • 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 typically 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
- 19. 19 • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: Data from Table 3.3, Callister & Rethwisch 8e. (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. (Epoly iron = 210 GPa) -If grains are textured, anisotropic. 200 m Adapted from Fig. 4.14(b), Callister & Rethwisch 8e. (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].) Single vs Polycrystals E (diagonal) = 273 GPa E (edge) = 125 GPa
- 20. 20 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
- 21. 21 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.
- 22. 22 X-Ray Diffraction Pattern Adapted from Fig. 3.22, Callister 8e. (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
- 23. 23 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). • 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.
- 24. 24 • 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.