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

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

Materials and Packing Si Oxygen Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Adapted from Fig. 3.40(a), Callister & Rethwisch 3e. Si Oxygen Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2 Adapted from Fig. 3.40(b), Callister & Rethwisch 3e.

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

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...

3-D Lattice

3-D lattice

3-D Lattice

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) (Courtesy P.M. Anderson)

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 Adapted from Fig. 3.42, Callister & Rethwisch 3e. close-packed directions a R=0.5a contains 8 x 1/8 = 1 atom/unit cell atom volume atoms unit cell 4 3 p (0.5a) 1 APF = 3 a unit cell volume

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 (), Tantalum, Molybdenum • Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 3e. 2 atoms/unit cell: 1 center + 8 corners x 1/8 (Courtesy P.M. Anderson)

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

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 3e. 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 (Courtesy P.M. Anderson)

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

FCC Stacking Sequence • ABCABC... Stacking Sequence • 2D Projection A sites B C sites A B sites C A C A A B C • FCC Unit Cell

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

Theoretical Density, r Density =  = n A  = VC NA Cell Unit of Volume Total in Atoms Mass Density =  = VC NA n A  = 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

Theoretical Density, r  = a R Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2 atoms/unit cell a = 4R/ 3 = 0.2887 nm Adapted from Fig. 3.2(a), Callister & Rethwisch 3e.  = a 3 52.00 2 atoms unit cell mol g volume 6.022 x 1023 theoretical = 7.18 g/cm3 ractual = 7.19 g/cm3

Atomic Bonding in Ceramics -- Can be ionic and/or covalent in character. -- % ionic character increases with difference in electronegativity of atoms. • Degree of ionic character may be large or small: CaF2: large SiC: small 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.

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

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

Coordination # and Ionic Radii cation anion • Coordination # increases with To form a stable structure, how many anions can surround around a cation? 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 r cation anion Coord # < 0.155 2 linear 0.155 - 0.225 3 triangular 0.225 - 0.414 4 tetrahedral 0.414 - 0.732 6 octahedral 0.732 - 1.0 8 cubic Adapted from Table 3.3, Callister & Rethwisch 3e.

Computation of Minimum Cation-Anion Radius Ratio Determine minimum rcation/ranion for an octahedral site (C.N. = 6) a = 2ranion

Bond Hybridization Bond Hybridization is possible when there is significant covalent bonding hybrid electron orbitals form For example for SiC XSi = 1.8 and XC = 2.5 ~ 89% covalent bonding Both Si and C prefer sp3 hybridization Therefore, for SiC, Si atoms occupy tetrahedral sites

Example Problem: Predicting the Crystal Structure of FeO • On the basis of ionic radii, what crystal structure would you predict for FeO? Cation Anion Al 3+ Fe 2 + Ca 2+ O 2- Cl - F Ionic radius (nm) 0.053 0.077 0.069 0.100 0.140 0.181 0.133 • 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.

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

MgO and FeO MgO and FeO also have the NaCl structure O2- rO = 0.140 nm Mg2+ rMg = 0.072 nm rMg/rO = 0.514 cations prefer octahedral sites Adapted from Fig. 3.5, Callister & Rethwisch 3e. So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms

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

AX2 Crystal Structures Fluorite structure Calcium Fluorite (CaF2) Cations in cubic sites UO2, ThO2, ZrO2, CeO2 Antifluorite structure – positions of cations and anions reversed Adapted from Fig. 3.8, Callister & Rethwisch 3e.

ABX3 Crystal Structures Perovskite structure Ex: complex oxide BaTiO3 Adapted from Fig. 3.9, Callister & Rethwisch 3e.

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

Densities of Material Classes In general Graphite/ r metals r ceramics r polymers Metals/ Composites/ > > Ceramics/ Polymers Alloys fibers Semicond 30 Why? B ased on data in Table B1, Callister Metals have... • close-packing (metallic bonding) • often large atomic masses 2 Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers 10 in an epoxy matrix). G raphite Silicon Glass - soda Concrete Si nitride Diamond Al oxide Zirconia Ceramics have... • less dense packing • often lighter elements 5 3 4 (g/cm ) 3 Wood AFRE * CFRE GFRE* Glass fibers Carbon fibers A ramid fibers H DPE, PS PP, LDPE PC PTFE PET PVC Silicone Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O) r 2 1 Composites have... • intermediate values 0.5 0.4 0.3 Data from Table B.1, Callister & Rethwisch, 3e.

Silicate Ceramics Most common elements on earth are Si & O SiO2 (silica) polymorphic forms are quartz, crystobalite, & tridymite The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material Si4+ O2- Adapted from Figs. 3.10-11, Callister & Rethwisch 3e crystobalite

Silicates Bonding of adjacent SiO44- accomplished by the sharing of common corners, edges, or faces Adapted from Fig. 3.12, Callister & Rethwisch 3e. Mg2SiO4 Ca2MgSi2O7 Presence of cations such as Ca2+, Mg2+, & Al3+ 1. maintain charge neutrality, and 2. ionically bond SiO44- to one another

Glass Structure • Basic Unit: Glass is noncrystalline (amorphous) • Fused silica is SiO2 to which no impurities have been added • Other common glasses contain impurity ions such as Na+, Ca2+, Al3+, and B3+ Si0 4 tetrahedron 4- Si 4+ O 2 - • Quartz is crystalline SiO2: Si 4+ Na + O 2 - (soda glass) Adapted from Fig. 3.41, Callister & Rethwisch 3e.

Layered Silicates Layered silicates (e.g., clays, mica, talc) SiO4 tetrahedra connected together to form 2-D plane A net negative charge is associated with each (Si2O5)2- unit Negative charge balanced by adjacent plane rich in positively charged cations Adapted from Fig. 3.13, Callister & Rethwisch 3e.

Layered Silicates (cont) Kaolinite clay alternates (Si2O5)2- layer with Al2(OH)42+ layer Adapted from Fig. 3.14, Callister & Rethwisch 3e. Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces.

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.

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.

Polymorphic Forms of Carbon (cont) Fullerenes and Nanotubes Fullerenes – spherical cluster of 60 carbon atoms, C60 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.

Crystals as Building Blocks • Some engineering applications require single crystals: -- 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.) • Properties of crystalline materials often related to crystal structure. -- Ex: Quartz fractures more easily along some crystal planes than others. (Courtesy P.M. Anderson)

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

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

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

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

Point Coordinates z Point coordinates for unit cell center are c x y a b c 000 111 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 2c y b b

Crystallographic Directions z Algorithm 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] y x ex: 1, 0, ½ Lecture 2 ended here => 2, 0, 1 => [ 201 ] -1, 1, 1 where overbar represents a negative index [ 111 ] => families of directions <uvw>

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

HCP Crystallographic Directions - a3 a1 a2 z Algorithm 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvtw] dashed red lines indicate projections onto a1 and a2 axes a1 a2 a3 -a3 2 a 1 Adapted from Fig. 3.24(a), Callister & Rethwisch 3e. [ 1120 ] ex: ½, ½, -1, 0 =>

HCP Crystallographic Directions Hexagonal Crystals 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows. Fig. 3.24(a), Callister & Rethwisch 3e. - a3 a1 a2 z = ' w t v u ) ( + - 2 3 1 ] uvtw [ ®

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

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)

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

Crystallographic Planes z x y a b c example a b c 1. Intercepts 1/2 1 3/4 2. Reciprocals 1/½ 1/1 1/¾ 2 1 4/3 3. Reduction 6 3 4 4. Miller Indices (634) (001) (010), Family of Planes {hkl} (100), (001), Ex: {100} = (100),

Crystallographic Planes (HCP) In hexagonal unit cells the same idea is used a2 a3 a1 z example a1 a2 a3 c 1. Intercepts 1  -1 1 2. Reciprocals 1 1/ 1 0 -1 1 3. Reduction 1 0 -1 1 4. Miller-Bravais Indices (1011) Adapted from Fig. 3.24(b), Callister & Rethwisch 3e.

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. Draw (100) and (111) crystallographic planes for Fe. b) Calculate the planar density for each of these planes.

Planar Density of (100) Iron Solution:  At T < 912C iron has the BCC structure. 2D repeat unit R 3 4 a = (100) Radius of iron R = 0.1241 nm Adapted from Fig. 3.2(c), Callister & Rethwisch 3e. = Planar Density = a 2 1 atoms 2D repeat unit nm2 12.1 m2 = 1.2 x 1019 R 3 4 area

Planar Density of (111) Iron Solution (cont):  (111) plane 1 atom in plane/ unit surface cell 2 a atoms in plane atoms above plane atoms below plane 2D repeat unit 3 h = a 2 3 2 R 16 4 a ah area = ÷ ø ö ç è æ 1 = nm2 atoms 7.0 m2 0.70 x 1019 3 2 R 16 Planar Density = 2D repeat unit area

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.  

X-Rays to Determine Crystal Structure • 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 q l extra distance travelled by wave “2” “1” “2” X-ray intensity (from detector) q c d = n l 2 sin Measurement of critical angle, qc, allows computation of planar spacing, d.

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

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). • Interatomic bonding in ceramics is ionic and/or covalent. • Ceramic crystal structures are based on: -- maintaining charge neutrality -- cation-anion radii ratios. • 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.

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.