1. 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
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.

2. 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.

3. 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.

4. 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

5. 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)

6. 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.

7. 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.)

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

9. 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)

10. Impurities  atoms which differ from host
All real solids are impure. Very pure metals % - 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.

11. 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

12. Substitutional Solid SolutionsNi 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%)

13. 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)

14. 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)

15. Composition Conversions
Weight % to Atomic %:
Atomic % to Weight %:
Textbook, pp

16. 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.

17. 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.

18. 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?

19. 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.

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

21. 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)

22. 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.

23. 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

24. 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

25. 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