Energy and Packing Non dense, random packing Energy typical neighbor bond length typical neighbor bond energy r Dense, regular packing Energy typical neighbor bond length typical neighbor bond energy r Dense, regular-packed structures tend to have lower energy. 24 Materials and Packing Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures -rapid cooling Si crystalline SiO2 Oxygen "Amorphous" = Noncrystalline noncrystalline SiO2 25
Crystal Systems Crystal System Axial Relationships Interaxial Angles Cubic a = b = c α = β = γ = 90 Hexagonal a = b c α = β = 90, γ = 120 Tetragonal a = b c α = β = γ = 90 Rhombohedral a = b = c α = β = γ 90 Orthorhombic a b c α = β = γ = 90 a, b, c, α, β, γ = lattice parameters representing a unit cell Monoclinic Triclinic a b c a b c α = γ = 90 β α β γ 90 26 asic types of unit cells: Simple (Primitive) ody-centered Face-centered ase-centered ravais Lattices 27
Characteristics of Crystal Structure Coordination Number (CN): number of nearestneighbor or touching atoms Atomic Packing Factor (APF): fraction of solid atoms volume in a unit cell APF = volume of atoms in a unit cell * total unit cell volume * assume hard sphere 28 Simple Cubic Structure (SC) Rare due to poor packing (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) 29
Atomic Packing Factor: SC APF for a simple cubic structure = 0.52 a close-packed directions contains 8 x 1/8 = 1 atom/unit cell R=0.5a atoms unit cell APF = 1 volume 4 atom 3 π (0.5a)3 a 3 volume unit cell 30 ody Centered Cubic (CC) Close packed directions are cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. Coordination # = 8 31
Atomic Packing Factor: CC APF for a body-centered cubic structure = 0.68 R a atoms unit cell APF = 2 4 3 π ( 3a/4)3 a 3 volume unit cell volume atom Close - packed direction : length = 4R = 3 a Unit cell contains: 1 + 8 ⅛ = 2 atoms/unit cell 32 Face Centered Cubic (FCC) Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. Coordination # = 12 33
Atomic Packing Factor: FCC APF for a face-centered cubic structure = 0.74 a atoms unit cell APF = 4 4 3 π ( 2a/4)3 a 3 volume unit cell volume atom Close - packed direction : length = 4R = 2 a Unit cell contains: 6 ½ + 8 ⅛ = 4 atoms/unit cell 34 FCC Stacking Sequence ACAC... Stacking Sequence A sites sites C sites A A C C C FCC Unit Cell A C 35
Hexagonal Close-Packed (HCP) AA... Stacking Sequence A sites sites A sites 3D Projection Top layer Middle layer ottom layer 2D Projection Coordination # = 12 APF = 0.74 36 Example: Copper Theoretical Density, ρ # atoms/unit cell Atomic weight (g/mol) Volume/unit cell (cm 3 /unit cell) ρ= n A V c N A Avogadro's number (6.023 x 10 23 atoms/mol) crystal structure = FCC: 4 atoms/unit cell atomic weight = 63.55 g/mol atomic radius R = 0.128 nm V C = a 3 ; for FCC, a = 4R/ 2 V C = 4.75 x 10-23 cm 3 Result: theoretical, ρ Cu = 8.89 g/cm 3 compare to actual, ρ Cu = 8.94 g/cm 3 37
Densities of Material Classes ρ Metals > ρ Ceramics > ρ Polymers Metals have... close-packing (metallic bonding) large atomic mass Ceramics have... less dense packing (covalent bonding) often lighter elements Polymers have... poor packing (often amorphous) lighter elements (C,H,O) Composites have... intermediate values ρ (g/cm 3 ) 30 20 10 5 4 3 2 1 0.5 0.4 0.3 Metals/ Alloys Platinum Gold, W Tantalum Silver, Mo Cu,Ni Steels Tin, Zinc Titanium Aluminum Magnesium Graphite/ Ceramics/ Semicond Polymers Composites/ fibers ased on data in Table 1, 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 Graphite PTFE Silicone PVC PET PC HDPE, PS PP, LDPE Glass fibers GFRE* Carbon fibers CFRE * Aramid fibers AFRE * Wood 38 Point Coordinates 39
Crystallographic Directions Directions are vectors. 1. Pass through the origin of the coordinate system. 2. Measure length in term of a, b, and c. 3. Reduce to the smallest integer values. 4. Enclosed in square brackets: [uvw] 40 Example: Direction (-1, + 1, -1/6) [66 1] 41
Crystallographic Planes Miller indices: 1. Measure length of planar intercept for each axis. 2. Reciprocate numbers. 3. Reduce to the smallest integer values. 4. Enclosed in parentheses: (hkl) 42 Example: Plane Intercepts: x=-1/2, y=-3/4, z=1/2 Reciprocals: x=-2, y=-4/3, z=2 Reductions: x=-6, y=-4, z=6 Miller indices : (6 4 6) 43
Example: Crystallography Direction1: [012] _ Direction2 : [112] Plane1: (020) _ Plane2 :(221) 44 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. 45
Single vs. Polycrystals Single Crystals Properties vary with direction: anisotropic. Ex. : the modulus of elasticity (E) in CC iron: E [111] E (diagonal) = 213 GPa = 273 GPa 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 (edge) = 125 GPa [100] = 125 GPa 46 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, CC, 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. 47