Chapter 3 Structure-of-Crystalline-Solids.pdf

Engineering Materials The Structure of Crystalline Solids 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?...

Engineering Materials The Structure of Crystalline Solids 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? 1  Energy and Packing Non dense, random packing Energy typical neighbor bond length typical neighbor r bond energy Dense, ordered packing Energy typical neighbor bond length typical neighbor r bond energy Dense, ordered packed structures tend to have lower energies. 2  TYPES OF SOLIDS In a Crystalline solid, atoms are arranged in an orderly manner. The atoms are having long range order. – Example : Iron, Copper and other metals, NaCl etc. In an Amorphous solid, atoms are not present in an orderly manner. They are randomly arranged. – Example : Glass "Amorphous" = Noncrystalline 3  Materials and Packing Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers crystalline SiO2 Adapted from Fig. 3.23(a), Callister & Rethwisch 8e. Si Oxygen Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2 Adapted from Fig. 3.23(b), Callister & Rethwisch 8e. 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  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... 6  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) Click once on image to start animation (Courtesy P.M. Anderson) 7  Atomic Packing Factor (APF) Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres APF for a simple cubic structure = 0.52 volume atoms atom a 4 unit cell 1 π (0.5a) 3 3 R=0.5a APF = a3 volume close-packed directions unit cell contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.24, Callister & Rethwisch 8e. 8  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 (α), Tantalum, Molybdenum Coordination # = 8 Adapted from Fig. 3.2, Click once on image to start animation Callister & Rethwisch 8e. (Courtesy P.M. Anderson) 2 atoms/unit cell: 1 center + 8 corners x 1/8 9  Atomic Packing Factor: BCC APF for a body-centered cubic structure = 0.68 3a a 2a Close-packed directions: Adapted from R length = 4R = 3 a Fig. 3.2(a), Callister & Rethwisch 8e. a atoms volume 4 unit cell 2 π ( 3a/4) 3 3 atom APF = 3 volume a unit cell 10  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 8e. Click once on image to start animation (Courtesy P.M. Anderson) 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 11  Atomic Packing Factor: FCC APF for a face-centered cubic structure = 0.74 maximum achievable APF Close-packed directions: length = 4R = 2 a 2a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a Adapted from Fig. 3.1(a), Callister & atoms volume Rethwisch 8e. 4 unit cell 4 π ( 2a/4) 3 3 atom APF = 3 volume a unit cell 12  FCC Stacking Sequence ABCABC... Stacking Sequence 2D Projection B B C A A sites B B B C C B sites B B C sites A FCC Unit Cell B C 13  Hexagonal Close-Packed Structure (HCP) ABAB... Stacking Sequence 3D Projection 2D Projection A sites Top layer c B sites Middle layer A sites Bottom layer a Adapted from Fig. 3.3(a), Callister & Rethwisch 8e. Coordination # = 12 6 atoms/unit cell APF = 0.74 ex: Cd, Mg, Ti, Zn c/a = 1.633 14  Theoretical Density, ρ Mass of Atoms in Unit Cell Density = ρ = Total Volume of Unit Cell nA ρ = VC NA 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 15  Theoretical Density, ρ Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2 atoms/unit cell R Adapted from Fig. 3.2(a), Callister & a a = 4R/ 3 = 0.2887 nm Rethwisch 8e. atoms g unit cell 2 52.00 ρtheoretical = 7.18 g/cm3 mol ρ= ρactual = 7.19 g/cm3 a3 6.022 x 1023 volume atoms unit cell mol 16  Densities of Material Classes In general Graphite/ ρmetals > ρceramics > ρpolymers Metals/ Composites/ Ceramics/ Polymers Alloys fibers Semicond 30 Why? Platinum Based on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, 20 Gold, W Metals have... Tantalum Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on close-packing 60% volume fraction of aligned fibers 10 Silver, Mo in an epoxy matrix). (metallic bonding) Cu,Ni Steels often large atomic masses Tin, Zinc Zirconia ρ (g/cm3 ) 5 Ceramics have... 4 Titanium Al oxide Diamond less dense packing 3 Si nitride Aluminum Glass -soda Glass fibers often lighter elements Concrete Silicon PTFE GFRE* 2 Carbon fibers Polymers have... Magnesium Graphite Silicone CFRE* Aramid fibers PVC low packing density PET PC AFRE* 1 HDPE, PS (often amorphous) PP, LDPE lighter elements (C,H,O) 0.5 Composites have... 0.4 Wood intermediate values 0.3 Data from Table B.1, Callister & Rethwisch, 8e. 17  Crystals as Building Blocks Some engineering applications require single crystals: -- diamond single -- turbine blades crystals for abrasives Fig. 8.33(c), Callister & Rethwisch 8e. (Fig. 8.33(c) (Courtesy Martin Deakins, courtesy of Pratt and GE Superabrasives, Whitney). 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. (Courtesy P.M. Anderson) 18  Polycrystals Most engineering materials are polycrystals. Anisotropic Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) 1 mm Nb-Hf-W plate with an electron beam weld. Isotropic 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). 19  Single vs Polycrystals Single Crystals E (diagonal) = 273 GPa Data from Table 3.3, -Properties vary with Callister & Rethwisch 8e. (Source of data is direction: anisotropic. R.W. Hertzberg, Deformation and -Example: the modulus Fracture Mechanics of Engineering Materials, of elasticity (E) in BCC iron: 3rd ed., John Wiley and Sons, 1989.) E (edge) = 125 GPa Polycrystals -Properties may/may not 200 µm Adapted from Fig. 4.14(b), Callister & vary with direction. Rethwisch 8e. (Fig. 4.14(b) is courtesy -If grains are randomly of L.C. Smith and C. Brady, the National oriented: isotropic. Bureau of Standards, Washington, DC [now (Epoly iron = 210 GPa) the National Institute of Standards and -If grains are textured, Technology, anisotropic. Gaithersburg, MD].) 20  Polymorphism Two or more distinct crystal structures for the same material (allotropy/polymorphism) iron system titanium liquid α, β-Ti 1538ºC BCC δ-Fe carbon diamond, graphite 1394ºC FCC γ-Fe 912ºC BCC α-Fe 21  X-Ray Diffraction Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation. Can’t resolve spacings < λ Spacing is the distance between parallel planes of atoms. 22  X-Ray Diffraction Pattern z z z c c c y (110) y y a b a b a b Intensity (relative) x x x (211) (200) Diffraction angle 2θ Diffraction pattern for polycrystalline α-iron (BCC) Adapted from Fig. 3.22, Callister 8e. 23  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). 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  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. 25