A (0001) plane for an HCP unit cell is show below. Each of the 6 perimeter atoms in this plane is shared with three other unit cells, whereas the center atom is shared with no other unit cells; this gives rise to three equivalent atoms belonging to this plane. In terms of the atomic radius R, the area of each of the 6 equilateral triangles that have been drawn is , or the total area of the plane shown is . . And the planar density for this (0001) plane is equal to or the total area of the plane shown is (b) the atomic radius for titanium is 0.145 nm. Therefore, the planar density for the (0001) plane is

Types of Imperfections • Vacancy atoms • Interstitial atoms • Substitutional atoms Point defects Line defects Area/Planar defects • Dislocations Edges, Screws, Mixed • Grain Boundaries Stacking Faults Anti-Phase and Twin Boundaries

Point Defects Self-interstitial: atom crowded in ‘holes’ Vacancy: a vacant lattice site It is not possible to create a crystal free of vacancies. About 1 out of 10,000 sites are vacant near melting. Self-interstitials are much less likely in metals, e.g., as it is hard to get big atom into small hole - there is large distortions in lattice required that costs energy. Thermodynamics (temperature and counting) provides an expression for Vacancy Concentration: Qv=vacancy formation energy kB= 1.38 x 10–23 J/atom-K = 8.62 x 10–5 eV/atom-K Defects ALWAYS cost energy!

Imperfections in Solids Conditions for substitutional solid solution (S.S.) W. Hume – Rothery rule 1. r (atomic radius) < 15% 2. Proximity in periodic table i.e., similar electronegativities 3. Same crystal structure for pure metals 4. Valency All else being equal, a metal will have a greater tendency to dissolve a metal of higher valency than one of lower valency

Imperfections in Solids Application of Hume–Rothery rules – Solid Solutions Would you predict more Al or Ag to dissolve in Zn? More Al because size is closer 7.43 % vs 8.48 % – FCC in HCP and val. Is higher – but not too Much! 2. More Zn or Al in Cu? Surely Zn since size is closer thus causing lower distortion (4% vs 12%) Element Atomic Crystal Electro- Valence Radius Structure nega- (nm) tivity Cu 0.1278 FCC 1.9 +2 C 0.071 H 0.046 O 0.060 Ag 0.1445 FCC 1.9 +1 Al 0.1431 FCC 1.5 +3 Co 0.1253 HCP 1.8 +2 Cr 0.1249 BCC 1.6 +3 Fe 0.1241 BCC 1.8 +2 Ni 0.1246 FCC 1.8 +2 Pd 0.1376 FCC 2.2 +2 Zn 0.1332 HCP 1.6 +2 Table on p. 106, Callister 7e.

Imperfections in Solids Linear Defects (Dislocations) Are one-dimensional defects around which atoms are misaligned Edge dislocation: extra half-plane of atoms inserted in a crystal structure b  to dislocation line Screw dislocation: spiral planar ramp resulting from shear deformation b  to dislocation line Burger’s vector, b: measure of lattice distortion and is measured as a distance along the close packed directions in the lattice

Imperfections in Solids Edge Dislocation

Burgers Vector. The magnitude and the direction of the displacement are defined by a vector, called the Burgers Vector. (a), starting from the point P, we go up by 4 steps, then move towards right by 5 steps, move down by 4 steps and finally move towards left by 5 steps to reach the starting point P. Now the Burgers circuit gets closed. When the same operation is performed on the defect crystal (b) we end up at Q instead of the starting point! So, we have to move an extra step to return to P, in order to close the Burgers circuit. BV= = b

Motion of Edge Dislocation • Dislocation motion requires the successive bumping of a half plane of atoms (from left to right here). • Bonds across the slipping planes are broken and remade in succession. Atomic view of edge dislocation motion from left to right as a crystal is sheared. (Courtesy P.M. Anderson)

Edge Dislocations Exiting Crystal Form Steps Burger’s Vector = b Shear stress The caterpillar or rug-moving analogy

Screw dislocation. The spiral stacking of crystal planes leads to the Burgers vector being parallel to the dislocation line.

Imperfections in Solids Screw Dislocation Screw Dislocation b Dislocation line Burgers vector b (b) (a) Adapted from Fig. 4.4, Callister 7e.

Edge, Screw, and Mixed Dislocations Adapted from Fig. 4.5, Callister 7e.

Motion – Edge dislocation Applied shear

Motion Screw Dislocation Apply shear

Imperfections in Solids Dislocations are visible in electron micrographs Adapted from Fig. 4.6, Callister 7e.

Dislocations & Crystal Structures • Structure: close-packed planes & directions are preferred. view onto two close-packed planes. close-packed directions close-packed plane (bottom) close-packed plane (top) • Comparison among crystal structures: FCC: many close-packed planes/directions; HCP: only one plane, 3 directions; BCC: none • Specimens that were tensile tested. Mg (HCP) tensile direction Al (FCC)

Planar Defects: Surfaces All defects cost energy (energy is higher than perfect crystal) Surfaces, grain, interphase and twin boundaries, stacking faults Planar Defect Energy is Energy per Unit Area (J/m2 or erg/cm2) Surfaces: missing or fewer number of optimal or preferred bonds. surface

SURFACE IMPERFECTIONS Surface imperfections arise from a change in the stacking of atomic planes on or across a boundary. The change may be one of the orientations or of the stacking sequence of atomic planes. In geometric concept, surface imperfections are two- dimensional. They are of two types external and internal surface imperfections.

EXTERNAL SURFACE IMPERFECTIONS They are the imperfections represented by a boundary. At the boundary the atomic bonds are terminated. The atoms on the surface cannot be compared with the atoms within the crystal. The reason is that the surface atoms have neighbors on one side only. Where as the atoms inside the crystal have neighbors on either sides. This is shown in figure. Since these surface atoms are not surrounded by others, they possess higher energy than that of internal atoms. For most metals, the energy of the surface atoms is of the order of 1J/m2.

AREA DEFECTS: GRAIN BOUNDARIES • are boundaries between crystals. • are produced by the solidification process, for example. • have a change in crystal orientation across them. • impede dislocation motion. Metal Ingot ~ 8cm grain boundaries Adapted from Fig. 4.10, Callister 6e. (Fig. 4.10 is from Metals Handbook, Vol. 9, 9th edition, Metallography and Microstructures, Am. Society for Metals, Metals Park, OH, 1985.) 15 Adapted from Fig. 4.7, Callister 6e.

Planar Defects in Solids One case is a twin boundary (plane) Essentially a reflection of atom positions across the twin plane. Stacking faults For FCC metals an error in ABCABC packing sequence Ex: ABCABABC Adapted from Fig. 4.9, Callister 7e.

STACKING FAULTS Whenever the stacking of atomic planes is not in a proper sequence throughout the crystal, the fault caused is known as stacking fault. For example, the stacking sequence in an ideal FCC crystal may be described as A-B-C-A-B-C- A-B-C-……. But the stacking fault may change the sequence to A-B-C-A-B-A-B-A-B-C. The region in which the stacking fault occurs (A-B-A-B) forms a thin region and it becomes HCP. This thin region is a surface imperfection and is called a stacking fault.

Microscopic Examination Crystallites (grains) and grain boundaries. Vary considerably in size. Can be quite large ex: Large single crystal of quartz or diamond or Si ex: Aluminum light post or garbage can - see the individual grains Crystallites (grains) can be quite small (mm or less) – necessary to observe with a microscope.

MACROSCOPIC EXAMINATION: The shape and average size or diameter of the grains for a polycrystalline specimen are large enough to observe with the unaided eye.

MICROSCOPIC EXAMINATION Applications To Examine the structural elements and defects that influence the properties of materials. Ensure that the associations between the properties and structure (and defects) are properly understood. Predict the properties of materials once these relationships have been established. Structural elements exist in ‘macroscopic’ and ‘microscopic’ dimensions

Optical Microscopy Polarized light metallographic scopes often use polarized light to increase contrast Also used for transparent samples such as polymers

Optical Microscopy • Useful up to 2000X magnification. • Polishing removes surface features (e.g., scratches) • Etching changes reflectance, depending on crystal orientation. 0.75mm crystallographic planes Adapted from Fig. 4.13(b) and (c), Callister 7e. (Fig. 4.13(c) is courtesy of J.E. Burke, General Electric Co. Micrograph of brass (a Cu-Zn alloy)

Optical Microscopy Grain boundaries... Fe-Cr alloy grain boundary surface groove polished surface (a) • are imperfections, • are more susceptible to etching, • may be revealed as dark lines, • change in crystal orientation across boundary. Adapted from Fig. 4.14(a) and (b), Callister 7e. (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].)

GRAIN SIZE DETERMINATION The grain size is often determined when the properties of a polycrystalline material are under consideration. The grain size has a significant impact of strength and response to further processing Linear Intercept method Straight lines are drawn through several photomicrographs that show the grain structure. The grains intersected by each line segment are counted The line length is then divided by an average number of grains intersected. The average grain diameter is found by dividing this result by the linear magnification of the photomicrographs.

N = 2 n-1 ASTM (American Society for testing and Materials) ASTM has prepared several standard comparison charts, all having different average grain sizes. To each is assigned a number from 1 to 10, which is termed the grain size number; the larger this number, the smaller the grains. VISUAL CHARTS (@100x) each with a number Quick and easy – used for steel N = 2 n-1 No. of grains/square inch Grain size no.

High resolutions and magnification (50,000x SEM); (TEM 1,000,000x)

Scanning Tunneling Microscopy (STM) • Atoms can be arranged and imaged! Photos produced from the work of C.P. Lutz, Zeppenfeld, and D.M. Eigler. Reprinted with permission from International Business Machines Corporation, copyright 1995. Carbon monoxide molecules arranged on a platinum (111) surface. Iron atoms arranged on a copper (111) surface. These Kanji characters represent the word “atom”.

Summary • Point, Line, and Area defects exist in solids. • The number and type of defects can be varied and controlled (e.g., T controls vacancy conc.) • Defects affect material properties (e.g., grain boundaries control crystal slip). • Defects may be desirable or undesirable (e.g., dislocations may be good or bad, depending on whether plastic deformation is desirable or not.)