ENGINEERING MATERIALS LECTURE #4

ENGINEERING MATERIALS LECTURE #4

Chapter 3: The Structure of Crystalline Solids Topics to Cover What is the difference in atomic arrangement between crystalline and noncrystalline solids? What features of a material s atomic structure determine its density? Under what circumstances does a material property vary with the measurement direction? 2

Energy and Packing Non dense, random packing Energy typical neighbor bond length typical neighbor bond energy r Dense, ordered packing Energy typical neighbor bond length typical neighbor bond energy r Dense, ordered packed structures tend to have lower energies. 3

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 crystalline SiO2 Adapted from Fig. 3.25(a), Callister & Rethwisch 9e. Si Oxygen "Amorphous" = Noncrystalline noncrystalline SiO2 Adapted from Fig. 3.25(b), Callister & Rethwisch 9e. 4

THE STRUCTURE OF CRYSTALLINE SOLIDS Crystalline Structure: material in which atoms are situated in a repeating or periodic array over large atomic distances. Three-dimensional geometric structure. Amorphous or non-crystalline. 5

CRYSTAL STRUCTURE Material properties depend on crystal structure of the material Atoms are thought of as being solid spheres having well-defined diameters (atomic hard sphere model, atoms touching each other) Lattice: three-dimensional array of points coinciding with atom positions (sphere centers) 6

CRYSTAL STRUCTURE CONTD. 7

UNIT CELLS Subdivide the crystal structure into small repeating entities called UNIT CELLS These cells are mainly in cubes, prisms, sixsided figures regular, repeating geometric structures Geometric symmetry The unit cell is the basic structural unit or building block of crystal structure 8

METALLIC CRYSTAL STRUCTURES No restrictions as to the number and position of nearest-neighbor atoms (dense atomic packing) Each sphere in atomic hard sphere model equates to an ion core 9

METALLIC CRYSTAL STRUCTURES CONTD. How can we stack metal atoms to minimize empty space? 2-dimensions vs. Now stack these 2-D layers to make 3-D structures 10

METALLIC CRYSTAL STRUCTURES CONTD. Four simple crystal structures are found in metals: Simple Cubic (SC) Face-Centered Cubic (FCC) Body-Centered Cubic (BCC) Hexagonal Close-Packed (HCP) 11

ANALYSIS TECNIQUES Analysis of crystal structures gives insight into properties of material such as strength, density, how the material may behave under physicals stress, etc. Analysis steps: Identify spatial geometries associated with unit cell Relate dimensions of unit cell to atomic radius Characterize/calculate required properties of crystal structure 12

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. Metals have the simplest crystal structures. We will examine four such structures... 13

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) Fig. 3.3, Callister & Rethwisch 9e. 14

ATOMIC PACKING FACTOR (APF) APF = Volume of atoms in unit cell* Volume of unit cell APF for a simple cubic structure = 0.52 a *assume hard spheres close-packed directions contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.3 (a), Callister & Rethwisch 9e. R = 0.5a atoms unit cell APF = 1 volume 4 3 π (0.5a) 3 atom a 3 volume unit cell 15

FACE-CENTERED CUBIC (FCC) The UNIT CELL is CUBIC Atoms located at each of the corners and the centers of all the cubic faces Aluminum, Copper, Gold, Lead, Nickel, Platinum, Silver 16

FACE-CENTERED CUBIC The cube edge length a and the atomic radius are related through: a = 2R 2 17

ATOMIC PACKING FACTOR: FCC APF for a face-centered cubic structure = 0.74 maximum achievable APF 2 a a Adapted from Fig. 3.1(a), Callister & Rethwisch 9e. atoms unit cell APF = Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell 4 4 3 π ( 2 a/4 ) 3 a 3 volume atom volume unit cell 18

FCC Each corner atom is shared by eight unit cells (Eight 1/8 portions per unit cell) Each face atom is shared by two unit cells (six ½ portions per unit cell) Grand total of four whole atoms per unit cell 19

COORDINATION NUMBER The number of nearest neighbor or touching atoms The coordination number for FCC is?? 20

COORDINATION NUMBER The number of nearest neighbor or touching atoms The coordination number for FCC is 12 21

ATOMIC PACKING FACTOR (APF) The fraction of solid sphere volume in a unit cell: APF = Volume of atoms in a unit cell Total unit cell volume Using the APF equation and the volume of an FCC unit cell, find the APF of a FCC crystal structure 22

BODY-CENTERED CUBIC (BCC) 23

BCC Atoms located in all eight corners and a single atom at the cube center Derive an expression for the unit cell edge length (a) using the atomic radius (R) 24

ATOMIC PACKING FACTOR: BCC APF for a body-centered cubic structure = 0.68 3 a a 2 a Adapted from Fig. 3.2(a), Callister & Rethwisch 9e. atoms R unit cell APF = a 2 4 3 π ( 3 a/4 ) 3 a 3 Close-packed directions: length = 4R = 3 a volume unit cell volume atom 25

BCC 8 x 1/8 = 1 (each corner) 1 atom in center 2 atoms per unit cell What is the coordination number and APF? 26

BCC 8 x 1/8 = 1 (each corner) 1 atom in center 2 atoms per unit cell What is the coordination number and APF? C.N. = 8 APF =.68 27

HEXAGONAL CLOSE-PACKED (HCP) Unit cell is hexagonal Six atoms form a regular hexagon and surround a single atom Another plane is situated in between top and bottom plane (provides three additional atoms) 28

HCP Six atoms altogether within unit cell 29

HCP Six atoms altogether within unit cell 1/6 x 12 (top and bottom plane portions ½ x 2 (center face atoms) 3 midplane interior atoms C.N. = 12 APF =.74 c/a = 1.633 30

OVERVIEW Crystal Structure Relationship between a and R (cubic structures) Number of atoms per unit cell Coordination Number APF FCC 4 12 0.74 BCC 2 8 0.68 HCP ------ 6 12 0.74 31

HW (DUE WEDNESDAY, 2/22/17) Problems: 3.2, 3.4, 3.9, 3.13, 3.14