ENS 205
Materials Science I
Chapter 4: Crystal Defects – Imperfection
1
Strength of Materials
• Based on the bond strength most materials
should be much stronger than they are
• From Chapter one: the strength for an ionic
bond should be about 106 psi
• More typical strength is 40*103 psi
• Why do we have three orders of magnitude
difference?
2
Crystalline Imperfections
• Real materials are never perfect and contain various types of
imperfections, which affect many of their properties.
• Some properties affected by imperfections include tensile and
ultimate strengths, thermal conductivity, electrical conductivity,
photonic generation and conductivity, magnetic properties, etc. For
example, Point defects ↑ ionic conductivity, but grain boundaries ↓
ionic conductivity
• Crystal lattice imperfections are classified according to their
geometry and shape.
• Point defects are zero dimension
• Line defects are one dimensional
• Planar defects are two dimensional and comprise of free surfaces
and grain boundaries
3
Dimensional scale of defects
significant effect on mechanical properties
4
Atomic Point Defects
Point defects are localized disruptions of the lattice involving one or several atoms
– Vacancy: When an atomic position in the lattice is vacant, atom missing from a
normal (Bravais) lattice position.
– Interstitial point defect: When an atom occupies and interstitial position. If
occupant atom
• the same of the material: self-interstitial
• Foreign: interstitial impurity
– Substitutional point defect: When a regular position occupied by a foreign
atom
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Chemical Impurity- foreign atom
• Size usually dictates the site and if
– substitutional
– Interstitial
• May be intentional or unintentional
– Examples: carbon added in small amounts to iron
makes steel, which is stronger than pure iron. Boron
added to silicon change its electrical properties.
• If foreign atoms are incorporated into the crystal (matrix)
 solid solutions (general)
 alloy (deliberate mixtures of metals)
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Chemical Impurity - Solid Solution
• Solid solutions are made of a host (the solvent or
matrix) which dissolves the minor component (solute).
The ability to dissolve is called solubility.
– Solvent: in an alloy, the element or compound present in
greater amount
– Solute: in an alloy, the element or compound present in
lesser amount
– Solid Solution:
• "homogeneous
• "maintain crystal structure
• "contain randomly dispersed impurities (substitutional or
interstitial)
Example: sterling silver is 92.5% silver – 7.5% copper alloy.
Stronger than pure silver.
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Solid Solution
•
•
Factors for high solubility: Hume-Rothery Rules
– Atomic size factor - atoms need to “fit” solute and solvent atomic radii should
be within ~ 15%
– Crystal structures of solute and solvent should be the same
– Electronegativities of solute and solvent should be comparable (otherwise
new inter-metallic phases are encouraged)
– The same valence
When one or more violated, partial solubility
8
Random & ordered solid solution
9
Interstitial Solid Solution
•
When atom sizes differ greatly,
instead of substitution, fit into one of
the spaces: The C atom is small
enough to fit, after introducing some
strain into the BCC lattice. –
interstitial solid solution
•
For fcc, bcc, hcp structures the holes
(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)
•
•
10
The principals of substitutional solid-solution formation also
apply to compounds. An additional rule is the maintenance of
charge neutrality
11
When a divalent cation replaces a
monovalent cation, a second
monovalent cation must also be
removed, creating a vacancy.
Only two Al3+ ions fill every three
Mg2+ sites which leaves one Mg2+
sites vacancy for each two Al3+
12
Vacancies
• Usually introduced during solidification, at high T
• The vacancy concentration in pure elements is very
low at low temperatures.
– The probability that an atomic site is vacancy ~ 10-6 at low T
– The probability that an atomic site is vacancy ~ 10-3 at
melting T
• can affect physical and electronic structures around
them →influence properties like color, conductivity
• they play a critical role in diffusion: control the self
diffusion and substitutional diffusion rates
• Movements of atoms coupled with movements of
vacancies
13
Vacancies Equilibrium of Point Defects
• The equilibrium number of vacancies formed as a result of thermal
vibrations may be calculated from thermodynamics:
• At equilibrium, the fraction of lattices that are vacant (or vacancy
concentration) at a given temperature is given approximately by the
equation: (The equilibrium concentration of point defects)
n
G f / kT 
e
N
• where n is the number of point defects (number of vacancy sites) in N
sites and Gf is free energy of formation of the defects-vacancy,
interstitials-(the energy required to move an atom from the interior of a
crystal to its surface). T is the absolute temperature, k is the Boltzman
constant. (this is lower end estimation, 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.) )
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Vacancies Equilibrium of Point Defects
• EXAMPLE: Estimate the number of vacancies in
Cu at room T
– The Boltzmann’s constant kB = 1.38 × 10-23 J/atom-K =
8.62 × 10-5 eV/atom-K
– The temperature in Kelvin T = 27o C + 273 = 300 K.
– kBT = 300 K × 8.62 × 10-5 eV/K = 0.026 eV
– The energy for vacancy formation Gf = Qv= 0.9 eV/atom
– The number of regular lattice sites Ns = NAρ/Acu
– NA = 6.023 × 1023 atoms/mol
– ρ = 8.4 g/cm3
– Acu = 63.5 g/mol
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Vacancies Equilibrium of Point Defects
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Vacancy vs interstitial
• Atoms that take up position between regular lattice sites
• Certain crystal structures will have certain interstitial sites
available for foreign atoms to occupy.
– There are octahedral and tetrahedral cages in an FCC lattice
(unit cell)
17
Solute Interstitials
In BCC iron, carbon exists in tetrahedral interstitial sites such as the
¼, ½, 0 site.
In FCC iron, carbon exists in the cube center (½, ½, ½,) and the
edge center octahedral interstitial sites.
18
Point Defects in compounds
• In compounds (ceramics and intermetalics)- ionic
materials →point defects should maintain the charge
neutrality of the crystal and cannot occur as freely as
in metals,
Pair of vacancies (one cation
and one anion)
Vacancy-self interstitial pair
19
Point Defect induced stresses
• 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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Point Defects
n
G f / kT 
e
N
• 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 (Gf  Qi ~ 3×Qv) ⇒
equilibrium concentration of selfinterstitials is very low (less than one
self-interstitial per cm3 at room T).
21
Solute Dopants and Conductivity
When “foreign” or “solute” or “dopant” atoms enter a semiconductor material such as
Silicon (Si) or Germanium (Ge), they can add or subtract a valence electron.
Si and Ge have 4 valence electrons, which covalently bond to their neighboring atoms.
When Arsenic (As) is introduced, which has 5 valence electrons, it can bond with 4
neighboring Si or Ge atoms and 1 electron is free to conduct, which creates n-type
conductivity.
When Boron (B) is introduced, which has 3 valence electrons, it can bond with 3
neighboring Si or Ge atoms leaving 1 electron unbounded, which creates a “hole” or
positive charge and p-type conductivity.
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If a voltage is applied, then both the electron and the hole can contribute to a small current flow
Line Defects- Dislocations
• Dislocations are very important
imperfections in real materials.
• Dislocations are line imperfections in
otherwise perfect lattices.
• Dislocations are formed during
solidification or when the material is
deformed.
• Dislocations strongly affect the
mechanical, electronic and photonic
properties of materials.
• There are two basic types of
dislocations – edge and screw.
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Edge Dislocation
a)
b)
c)
The perfect crystal in a) is cut and an extra plane of atoms is inserted
in b). The bottom edge of the extra plane is an edge dislocation
(dislocation line) in c). A Burgers vector b is required to close a loop
of equal atom spacings around the edge dislocation.
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Screw Dislocation
The perfect crystal in a) is cut and sheared one atom spacing in b)
and c). The line along which the shearing occurs is a screw
dislocation. A Burgers vector b is required to close a loop of equal
atom spacing around the screw dislocation.
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Line Defects- Dislocations
• To describe the size and the direction of the main lattice
distortion dislocation line and Burgers vector b.
• Make a circuit 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.
2
2
1
1
3
4
3
4
5
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For the common metal structures (bcc, fcc, hcp) the magnitude is
the repeat distance along the highest atomic density direction
(where atoms are touching)
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Effect of Point Defects on Mechanical Properties
• Quenching interaction of dislocations and vacancies  strengthening
28
Mixed dislocations
The exact structure of dislocations in real
crystals is usually more complicated. Edge and
screw dislocations are the two pure extremes
of linear defect structures. Most dislocations
have mixed edge/screw character.
A mixed dislocation showing a screw
dislocation at the front of the crystal
gradually changing to an edge
dislocation at the side of the crystal.
Note that the line direction of the
dislocation is parallel to the Burgers
vector of the screw dislocation and
perpendicular to the edge dislocation.
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Control of Dislocations
• Control of dislocations allow us to manipulate mechanical
properties and understand their temperature dependence.
• When a shear force acting in the direction of the
Burger’s vector is applied to a crystal containing a
dislocation, the dislocation can move by breaking bonds
between the atoms in one plane.
• By this process, the dislocation moves through the crystal
to produce a step on the exterior of the crystal.
• The process by which the dislocation moves and
causes a solid to deform is called slip.
30
Dislocation Slip
• Dislocations move more readily in some crystal planes and
directions than in others as we will see.
• The slip direction of an edge dislocation is in the direction of the
Burger’s vector.
• A slip plane is defined by the direction of the Burger’s vector
and the line direction of the dislocation
– The line direction of a screw dislocation is in the same
direction as its Burger’s vector.
– An edge dislocation has its Burger’s vector perpendicular to
the line direction of a dislocation
– A dislocation having a line direction not parallel or
perpendicular to the Burger’s vector is considered a mixed
dislocation.
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Dislocation Slip
• During slip the dislocation moves from one set of
surroundings to another identical set.
• The least amount of energy expenditure requires
movement in directions in which the repeat distance is
shortest, i.e., close-packed directions.
• Slip planes tend to be those planes with a high planar
packing, i.e., close-packed planes.
• Slip reduces strength but increase ductility in materials.
32
b
b
Schematic of slip line, slip plane and slip vector (Burgers vector) for
a) an edge dislocation and b) a screw dislocation. Note the
relationships between the dislocation line, slip vector and glide plane.
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When a shear stress is applied to the dislocation in a)
the atoms are displaced, causing the dislocation to
move one Burgers vector in the slip direction b).
Continued movement of the dislocation creates a step
c) and the crystal is deformed. Motion of a caterpillar
(or a fold in a rug) is analogous to the motion of a
dislocation.
Note: the slip direction is always in the direction of the
Burgers vector of the dislocation.
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35
Defects
• See the site prepared by Prof. Dr. Helmut Föll,
http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html
a) Interstitial impurity atom, b) Edge dislocation, c) Self interstitial atom, d) Vacancy, e)
Precipitate of impurity atoms, f) Vacancy type dislocation loop, g) Interstitial type
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dislocation loop, h) Substitutional impurity atom
Observing Dislocations
• We can view dislocations indirectly by etching the
surface of a material. Where the dislocation
intersects the surface, it is preferentially etched
creating a pit, which can easily be seen optically.
• Using a transmission electron microscope, we can
see the strain contrast that a dislocation makes as it
passes through a crystal, or at the atomic level we
can see the displacement of atomic columns due to
the presence of a dislocation.
37
Optical etch pits in Silicon Carbide (SiC) corresponding to the surface
a
b

 11 20  and a line
intersection of pure edge dislocations having
3
direction of [0001], which is perpendicular to the etched surface.
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Example
• The closure (burger) vector indicates the magnitude of the structural
defect. The magnitude of the burger vector for the common metal
structures (bcc, and fcc) is simply the repeat distance along the
highest atomic density direction (the directions where atoms are
touching). Repeat distance: distance between the lattice points or
atomic positions along a direction
• Calculate the magnitude of the Burgers vector for α-Fe (bcc-iron)
and for Al.
• For a bcc structure, atoms touch along [1 1 1] direction. The repeat
distance is equal to one atomic diameter. Hence,
r=2RFe=|b|=2*(0.124)= 0.248 nm
• For a fcc structure, the highest atomic density direction is along the
face diagonal of a unit cell [1 1 0]. This direction is also line of
contact for atoms in an fcc structure. Hence, r=2RAl=|b|=2*(0.143)=39
0.286 nm
Control of Slip Process – Strengthening Mechanisms
•
•
•
•
•
•
•
We can control the strength of a material by controlling the number and
type of imperfections in real materials.
– These imperfection block the movement of dislocations making it
difficult for dislocation motion.
Five common mechanisms for increasing the strength of a material are:
Strain hardening
Solid solution strengthening
Grain refinement
Secondary phases
Dispersion hardening (age hardening)
40
Control of Slip Process – Strengthening
Mechanisms
• 1) strain hardening, which increases the number of dislocations in the
material by deforming the material. The extra dislocations block the motion
of other dislocations.
• 2) solid solution strengthening, which adds point defects consisting of
substitutional or interstitial atoms (alloying additions or impurities, i.e.,
foreign atoms). The strain around the foreign atoms blocks the motion of
dislocations.
• 3) grain size strengthening (grain refinement), which reduces grain size
where the grain boundaries block the motion of dislocations.
• 4) Secondary phases where some grains can have one type of atomic
structure, eg., bcc, and other grains will have another type of atomic
structure, eg., fcc. An example is a/b brass. Dislocations have difficulty
passing from one type of grain to the other.
• 5) Dispersion hardening (age hardening) where small precipitates within
grains are used to block the motion of dislocations.
41
Strengthening Mechanisms
These five processes are used to increase the
strength of a material.
From time to time during this course we will discuss
these five mechanisms in greater detail.
42
If the dislocation at point A moves to the left, it is blocked by the
point defect. If it moves to the right, it interacts with the
dislocation at B and farther to the right, with the grain boundary.
43
The effect of grain size on the yield strength of steel at room
temperature
44
Planar defects – Grain Boundaries
• Generally speaking, we deal with finite amount of (any)
material contained within some exterior boundary
surface.
– Boundary surface itself a disruption of the atomic stacking
arrangement of the crystal  a planar defect
• Interior boundaries
– Aside from the electronics industry, most practical engineering
materials are polycrystalline
– When crystals of different crystallographic orientations are
joined  grain boundary
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Grain Boundaries
46
Planar defects-Twin boundaries
• A twin boundary is a special type of grain boundary across which
there is a specific mirror lattice symmetry. Twin results from atomic
displacement that are produced from applied mechanical shear
forces, and also during annealing heat treatments following
deformation (annealing twins). This gives rise to shape memory
metals, which can recover their original shape if heated to a high
temperature.
47
Planar defects-Grain Boundaries
• The most important planar defect, where the region
between two adjacent single crystals or grains
• Depending on misalignments of atomic planes
between adjacent grains we can distinguish between
the low and high angle grain boundaries
48
Planar defects-Tilt Grain Boundary
• Two adjacent grains are tilted only a few
degrees relative to each other
• The tilt boundary is accommodated by a
few isolated edge dislocations –
unusually simple
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Planar defects-Tilt Grain Boundary
Calculate the separation distance (D) of
dislocations in a low-angle (=2o) tilt
boundary in Aluminum.
The highest atomic density direction in fcc
Aluminum is along the face diagonal of
the unit cell. This direction is a line of
contact for atoms for atoms in an fcc
structure. Hence, r=2RAl=|b|=2*(0.143)=
0.286 nm.
D= |b|/ = 0.286/ (2o x (1 rad/57.3))=8.19nm
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Grain-size number, G
• It is useful to have a simple index of grain size, G grain-size number (by
ASTM): N = 2G-1, where N is the number of grains observed in an area
of 1 in2 on a photomicrograph taken at a magnification of 100 times
• 21+22/2 = 32
grains in a circular
area with diameter
= 2.25 in
The area density is
N = 32/π(2.25/2)2
= 8.04 grains/in2
N = 2G-1G = 4.01
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Affect of Grain Size on Strength
• In a small grain, a dislocation gets to the boundary
and stops – slip stops
• In a large grain, the dislocation can travel farther
• Small grain size equates to more strength
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Non-crystalline Solids – 3D imperfections
53
Amorphous Structures
• If you cool a material off too fast it does
not have a chance to crystallize
• Called a glass
• It is relatively easy to make a ceramic
glass
• It is hard to make a metallic glass
• There are no slip planes in a glass!!
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