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“Crystals are like people, it is the defects in them which tend to make them interesting!” - Colin Humphreys.
¾ 0D, Point defects 9 vacancies 9 interstitials 9 impurities, weight and atomic composition ¾ 1D, Dislocations 9 edge 9 screw
¾ 2D, Grain boundaries 9 tilt 9 twist ¾ 3D, Bulk or Volume defects
¾ Atomic vibrations
4.9 - 4.10 Microscopy & Grain size determination –
Not Covered / Not Tested
Chapter Outline
Introduction To Materials Science, Chapter 4, Imperfections in solids
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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 (I)
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Defects – Introduction (II)
Defects have a profound impact on the macroscopic properties of materials
Bonding
Structure
Defects
Properties
Introduction To Materials Science, Chapter 4, Imperfections in solids
4
Composition
Bonding Crystal Structure
Thermomechanical Processing
Microstructure
Defects – Introduction (III)
The processing determines the defects
defects introduction and manipulation
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The equilibrium number of vacancies formed as a result of thermal vibrations may be calculated from thermodynamics:
How many vacancies are there?
where N (^) s is the number of regular lattice sites, kB is the Boltzmann constant, Q (^) v is the energy needed to form a vacant lattice site in a perfect crystal, and T the temperature in Kelvin (note, not in oC or o^ F).
Using this equation we can estimate that at room temperature in copper there is one vacancy per 10 15 lattice atoms, whereas at high temperature, just below the melting point there is one vacancy for every 10,000 atoms.
Note, that the above equation gives the lower end estimation of the number of vacancies, a large numbers of additional (non- equilibrium) vacancies can be introduced in a growth process or as a result of further treatment (plastic deformation, quenching from high temperature to the ambient one, etc.)
kT
Q
N N exp
B
v v s
Introduction To Materials Science, Chapter 4, Imperfections in solids
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Let’s estimate the number of vacancies in Cu at room T
The Boltzmann’s constant k (^) B = 1.38 × 10 -23^ J/atom-K = 8.62 × 10 -5^ eV/atom-K
The temperature in Kelvin T = 27 o^ C + 273 = 300 K. k (^) BT = 300 K × 8.62 × 10 -5^ eV/K = 0.026 eV
The energy for vacancy formation Q (^) v = 0.9 eV/atom
The number of regular lattice sites Ns = N (^) Aρ/A (^) cu N (^) A = 6.023 × 10 23 atoms/mol ρ = 8.4 g/cm^3 A (^) cu = 63.5 g/mol
kT N Nexp Q B v s v
3
(^322) 23 s 8 10 atomscm mol
- 5 g
cm
- 4 g mol
- 02310 atoms N = ×
× × =
= × −
- 026 eVatom exp^0.^9 eVatom cm
atoms N (^) v (^810223)
= 7. 4 × 107 vacanciescm^3
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OBSERVING EQUIL. VACANCY CONC.
**- Low energy electron microscope view of a (110) surface of NiAl.
- Increasing T causes surface island of atoms to grow.
- Why? The equil. vacancy conc. increases via atom motion from the crystal to the surface, where they join the island.**
Reprinted with permission from Nature (K.F. McCarty, J.A. Nobel, and N.C. Bartelt, "Vacancies in Solids and the Stability of Surface Morphology", Nature, Vol. 412, pp. 622-625 (2001). Image is 5.75 μ m by 5.75 μ m.) Copyright (2001) Macmillan Publishers, Ltd.
Introduction To Materials Science, Chapter 4, Imperfections in solids
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Self-interstitials in metals introduce large distortions in the surrounding lattice ⇒ the energy of self-interstitial formation is ~ 3 times larger as compared to vacancies (Qi ~ 3×Qv) ⇒ equilibrium concentration of self-interstitials is very low (less than one self-interstitial per cm^3 at room T).
Self-interstitials
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2
3
4
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Other point defects: self-interstitials, impurities
Schematic representation of different point defects: (1) vacancy; (2) self-interstitial; (3) interstitial impurity; (4,5) substitutional impurities
The arrows show the local stresses introduced by the point defects.
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ALLOYING A SURFACE
- Low energy electron microscope view of **a (111) surface of Cu.
- Sn islands move along the surface and "alloy“ the Cu with Sn atoms, to make "bronze".
- The islands continually move into "unalloyed“ regions and leave tiny bronze particles in their wake.
- Eventually, the islands disappear.**
Reprinted with permission from: A.K. Schmid, N.C. Bartelt, and R.Q. Hwang, "Alloying at Surfaces by the Migration of Reactive Two- Dimensional Islands", Science, Vol. 290, No. 5496, pp. 1561-64 (2000). Field of view is 1. μ m and the temperature is 290K.
Introduction To Materials Science, Chapter 4, Imperfections in solids
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Substitutional Solid Solutions
Factors for high solubility:
¾ Atomic size factor - atoms need to “fit” ⇒ solute and solvent atomic radii should be within ~ 15%
¾ Crystal structures of solute and solvent should be the same
¾ Electronegativities of solute and solvent should be comparable (otherwise new inter-metallic phases are encouraged)
¾ Generally more solute goes into solution when it has higher valency than solvent
Ni
Cu
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Interstitial Solid Solutions
Factors for high solubility:
¾ For fcc, bcc, hcp structures the voids (or interstices) between the host atoms are relatively small ⇒ atomic radius of solute should be significantly less than solvent
Normally, max. solute concentration ≤ 10%, (2% for C-Fe)
Carbon interstitial atom in BCC iron
Interstitial solid solution of C in α-Fe. The C atom is small enough to fit, after introducing some strain into the BCC lattice.
Introduction To Materials Science, Chapter 4, Imperfections in solids
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Composition / Concentration
Atom percent (at %): number of moles (atoms) of a particular element relative to the total number of moles (atoms) in alloy. For two-component system, concentration of element 1 in at. % is
Composition can be expressed in
¾ weight percent , useful when making the solution
¾ atom percent , useful when trying to understand the material at the atomic level
Weight percent (wt %): weight of a particular element relative to the total alloy weight. For two-component system, concentration of element 1 in wt. % is
m m
m
C
1 2
1
1 ×
n n
n
C
1 2
1 m m
' m
1 ×
where nm1 = m’ 1 /A 1 m’ 1 is weight in grams of element 1, A 1 is atomic weight of element 1)
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Description of Dislocations—Burgers Vector
To describe the size and the direction of the main lattice distortion caused by a dislocation we should introduce so- called Burgers vector b. To find the Burgers vector, we should make a circuit from from atom to atom counting the same number of atomic distances in all directions. If the circuit encloses a dislocation it will not close. The vector that closes the loop is the Burgers vector b.
b
Dislocations shown above have Burgers vector directed perpendicular to the dislocation line. These dislocations are called edge dislocations.
Introduction To Materials Science, Chapter 4, Imperfections in solids
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Edge and screw dislocations
There is a second basic type of dislocation, called screw dislocation. The screw dislocation is parallel to the direction in which the crystal is being displaced (Burgers vector is parallel to the dislocation line).
Dislocations shown in previous slide are edge dislocations, have Burgers vector directed perpendicular to the dislocation line.
Find the Burgers vector of a screw dislocation. How a screw dislocation got its name?
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Mixed/partial dislocations (not tested)
The exact structure of dislocations in real crystals is usually more complicated than the ones shown in this pages. Edge and screw dislocations are just extreme forms of the possible dislocation structures. Most dislocations have mixed edge/screw character.
To add to the complexity of real defect structures, dislocation are often split in "partial“ dislocations that have their cores spread out over a larger area.
Introduction To Materials Science, Chapter 4, Imperfections in solids
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Interfacial Defects
External Surfaces
Surface atoms have have unsatisfied atomic bonds, and higher energies than the bulk atoms ⇒ Surface energy, γ (J/m^2 )
- Surface areas tend to minimize (e.g. liquid drop)
- Solid surfaces can “reconstruct” to satisfy atomic bonds at surfaces.
Grain Boundaries
Polycrystalline material comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.
Surfaces and interfaces are reactive and impurities tend to segregate there. Since energy is associated with interfaces, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion (Chapter 5), which is accelerated at high temperatures.
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Twin Boundaries (not tested)
Low-energy twin boundaries with mirrored atomic positions across boundary may be produced by deformation of materials. This gives rise to shape memory metals , which can recover their original shape if heated to a high temperature. Shape-memory alloys are twinned and when deformed they untwin. At high temperature the alloy returns back to the original twin configuration and restore the original shape.
Introduction To Materials Science, Chapter 4, Imperfections in solids
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Electron Microscopy (not tested)
Dislocations in Nickel (the dark lines and loops), transmission electron microscopy image, Manchester Materials Science Center.
High-resolution Transmission Electron Microscope image of a tilt grain boundary in aluminum, Sandia National Lab.
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Bulk or Volume Defects
¾ Pores - can greatly affect optical, thermal, mechanical properties
¾ Cracks - can greatly affect mechanical properties
¾ Foreign inclusions - can greatly affect electrical, mechanical, optical properties
A cluster of microcracks in a melanin granule irradiated by a short laser pulse. Computer simulation by L. V. Zhigilei and B. J. Garrison.
Introduction To Materials Science, Chapter 4, Imperfections in solids
g (^) 28
Atomic Vibrations
¾ Heat causes atoms to vibrate
¾ Vibration amplitude increases with temperature
¾ Melting occurs when vibrations are sufficient to rupture bonds
¾ Vibrational frequency ~ 10 13 Hz
¾ Average atomic energy due to thermal excitation is of order kT
