Chapter Outline Defects in Solids “Crystals are like people, it is the defects in them which tend to make them interesting!” - Colin Humphreys. Defects in Solids 0D, Point defects vacancies interstitials impurities, weight and atomic composition 1D, Dislocations edge screw 2D, Grain boundaries tilt twist 3D, Bulk or Volume defects Atomic vibrations 4.9 - 4.11 Microscopy & Grain size determination – Not Covered / Not Tested Calculations of weight percent and atomic percent for each element as well as calculation of the equilibrium number of vacancies is emphasized in the tests We have discussed atomic structure, various types of the bonds between atoms, and the importance of the crystalline structure for understanding of materials properties. Crystal defects are very important in determining the properties of the solid.

Defects – Introduction (I) Real crystals are never perfect, there are always defects Schematic drawing of a poly-crystal with many defects by Helmut Föll, University of Kiel, Germany.

Defects – Introduction (II) Defects have a profound impact on the macroscopic properties of materials Bonding + Structure Defects Properties Does type of element defines type of structure? Polymorphism (allotropy). Examples of the relevance of defects in crystals for life in general or materials science in particular: When/if you buy a diamond ring, it is mostly the number and type of defects in the diamond crystal that define the amount of money you pay for a given crystal size. It is generation, accumulation, and interaction of defects in different parts of your car that define the mileage on your car when you tow it to the junk-yard. Production of advanced semiconductor devices require not only a rather perfect Si crystal as starting material, but also involve introduction of specific defects in small areas of the sample. Forging a metal tool introduces defects … and increases strength and elasticity of the tool. Note, that in this case the required properties are achieved without changes in composition of the material, but just by manipulating the crystal defects.

Defects – Introduction (III) The processing determines the defects Composition Bonding Crystal Structure Early work: Empirical - relying on experience and observation, without regard for science and theory The blacksmith recognized that beating a hot iron object into shape followed by quenching in water produced a strong and tough object. Thermomechanical Processing defects introduction and manipulation Microstructure

Types of Defects Defects may be classified into four categories depending on their dimension: 0D, Point defects: atoms missing or in irregular places in the lattice (vacancies, interstitials, impurities) 1D, Linear defects: groups of atoms in irregular positions (e.g. screw and edge dislocations) 2D, Planar defects: interfaces between homogeneous regions of the material (grain boundaries, external surfaces) 3D, Volume defects: extended defects (pores, cracks)

Point defects: vacancies & self-interstitials Vacancy Vacancy - a lattice position that is vacant because the atom is missing. Interstitial - an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity interstitial atom.

Equilibrium number of vacancies is due to thermal vibrations How many vacancies? Equilibrium number of vacancies is due to thermal vibrations Ns = number of regular lattice sites kB = Boltzmann constant Qv = energy to form a vacant lattice site in a perfect crystal T = temperature in Kelvin (note, not in oC or oF). Room temperature in copper there is one vacancy per 1015 lattice atoms. High temperature, just below the melting point, one vacancy for every 10,000 atoms. The higher is the temperature, more often atoms are jumping from one equilibrium position to another and larger number of vacancies can be found in a crystal. This gives lower bound to number of vacancies. Additional (non-equilibrium) vacancies introduced in growth process or treatment (plastic deformation, quenching, etc.)

Estimate number of vacancies in Cu at room T kB = 1.38  10-23 J/atom-K = 8.62  10-5 eV/atom-K T = 27o C + 273 = 300 K. kBT = 300 K  8.62  10-5 eV/K = 0.026 eV Qv = 0.9 eV/atom Ns = NA/Acu NA = 6.023  1023 atoms/mol  = 8.4 g/cm3 Acu = 63.5 g/mol

Point defects: self-interstitials, impurities (1) vacancies (2) self-interstitial (3)interstitial impurity (4,5)substitutional impurities Arrows  local stress introduced by defect Self-interstitials Large distortions in surrounding lattice  Energy of self-interstitial formation is ~ 3 x larger than for vacancies (Qi ~ 3Qv)  equilibrium concentration of self-interstitials is very low(< 1/ cm3 at 300K)

Impurities  atoms which differ from host All real solids are impure. Very pure metals 99.9999% - one impurity per 106 atoms May be intentional or unintentional Carbon in small amounts in iron makes steel, which is stronger. Boron in silicon change its electrical properties. Alloys - deliberate mixtures of metals Sterling silver is 92.5% silver – 7.5% copper alloy. Stronger than pure silver.

How Are Impurities Contained in Alloy? Solid solutions Host (Solvent or Matrix) dissolves minor component (Solute). Ability to dissolve is called Solubility. Solvent: element present in greater amount Solute: element present in lesser amount Solid Solution: homogeneous maintain crystal structure randomly dispersed impurities (substitutional or interstitial) Second Phase As solute atoms added: new compounds or structures form solute forms local precipitates Whether addition of impurities results in a solid solution or second phase depends nature of impurities, concentration, temperature and pressure

Substitutional Solid Solutions Ni Cu Factors for high solubility (Solubility limit  maximum that can dissolve) Atomic size: need to “fit”  solute and solvent atomic radii should be within ~ 15% Crystal structure: solute and solvent the same Electronegativities: should be comparable (otherwise new inter-metallic phases favored) Valency: If solute has higher valency than solvent, generally more goes into solution Max solute concentration = 50% E.g. Cu-Ni (50%)

Interstitial Solid Solutions Carbon interstitial atom in BCC iron The max concentration of C in iron ~ 2% Interstitial solid solution of C in BCC Fe ( phase). C small enough to fit (some strain in BCC lattice). Factors for high solubility: FCC, BCC, HCP: void space between host (matrix) atoms are relatively small  atomic radius of solute should be significantly less than solvent Max. concentration  10%, (2% for C-Fe)

Composition / Concentration Composition expressed in weight percent (wt %): useful when making the solution wt %: weight of one element relative to total alloy weight. 2 components: concentration of element 1 in wt. % : atom percent (at %): useful when trying to understand material at the atomic level at %: number of moles (atoms) of a particular element relative to the total number of moles (atoms) in alloy. 2 component: concentration of element 1 in at. % : nm1= number density = m’1/A1 (m’1 = weight in grams of 1, A1 is atomic weight of element 1)

Composition Conversions Weight % to Atomic %: Atomic % to Weight %: Textbook, pp. 71-74

Dislocations = Linear Defects Interatomic bonds significantly distorted in immediate vicinity of dislocation line. (Creates small elastic deformations of lattice at large distances.) Area called dislocation core. Dislocations affect mechanical properties. Discovery in 1934 by Taylor, Orowan and Polyani marked beginning of our understanding of mechanical properties of materials.

Describe Dislocations Burgers Vector Burgers vector, b, describes size + direction of lattice distortion by a dislocation. Make a circuit around dislocation: go from atom to atom counting the same number of atomic distances in both directions. Vector needed to close loop is b b Burgers vector above directed perpendicular to dislocation line. These are called edge dislocations.

Screw dislocations Edge dislocation: Burgers vector perpendicular to dislocation line. Screw dislocation: parallel to the direction in which the crystal is being displaced Burgers vector parallel to dislocation line. Find the Burgers vector of a screw dislocation. How did it get its name?

Interfacial Defects External Surfaces Grain Boundaries Surface atoms  unsatisfied bonds higher energies than bulk atoms Surface energy,  (J/m2) Surface areas try to minimize (e.g. liquid drop) Solid surfaces can “reconstruct” to satisfy atomic bonds at surfaces. Grain Boundaries Polycrystalline: many small crystals or grains. Grains have different crystallographic orientation. Mismatches where grains meet. 1. Surfaces and interfaces are reactive 2. Impurities tend to segregate there. 3. Extra energy associated with interfaces  larger grains tend to grow by diffusion of atoms at expense of smaller grains, minimizing energy.

High and Low Angle Grain Boundaries Misalignments of atomic planes between grains  Distinguish low and high angle grain boundaries

Tilt and Twist Grain Boundaries Tilt boundary  Low angle grain boundary: an array of aligned edge dislocations (like joining two wedges) Transmission electron microscope image of a small angle tilt boundary in Si. The red lines mark the edge dislocations, the blue lines indicate the tilt angle Twist boundary - boundary region consisting of arrays of screw dislocations (like joining two halves of a cube and twist an angle around the cross section normal)

Bulk or Volume Defects Pores: affect optical, thermal, mechanical properties Cracks: affect mechanical properties Foreign inclusions: affect electrical, mechanical, optical properties Cluster of microcracks in a melanin granule irradiated by a short laser pulse. Computer simulation by L. V. Zhigilei and B. J. Garrison.

Atomic Vibrations Heat causes atoms to vibrate Vibration amplitude increases with temperature Melting occurs when vibrations are sufficient to rupture bonds Vibrational frequency ~ 1013 Hz Average atomic energy due to thermal excitation is of order kT

Summary Understand language and concepts: Alloy Atom percent Atomic vibration Boltzmann’s constant Burgers vector Composition Dislocation line Edge dislocation Grain size Imperfection Interstitial solid solution Microstructure Point defect Screw dislocation Self-Interstitial Solid solution Solubility limit Solute Solvent Substitutional solid solution Vacancy Weight percent

Reading for next class: Homework #1: 4.2, 4.7, 4.8, 4.18, 5.7, 5.22, 5.23 Due date: Thursday, February 8. Reading for next class: Chapter 5: Diffusion Diffusion Mechanisms (vacancy, interstitial) Steady-State Diffusion (Fick’s first law) Factors That Influence Diffusion Other Diffusion Paths Optional reading (Part that is not covered / not tested): 5.4 Nonsteady-state diffusion