Materials Science and Engineering
Topic 3. Defects in crystalline structures
• Imperfections in crystals
• Point defects
• Line defects: dislocations
• Plastic deformation
• Planar defects
• Solid Solution
Topic 3. Defects in crystalline structures
Learning objectives
Learning objectives:
▪ Knowing the main network defects
▪ Performing defect concentration calculations based on temperature
▪ Performing dislocation parameter calculations based on the type of crystalline
structure
▪ Understanding the concept of solubility between two metals and the
characteristics that determine it
Topic 3. Defects in crystalline structures
Imperfections in crystals
 Perfect crystal
 All atoms in the regular lattice positions for a given structure
 All distances dhkl are the same for the given (h k l)
 No missing or extra atoms
 Only possible at 0 K
 Real crystal
Perfect crystal (NaCl)
 Atoms vibrate
 There are positions that are not occupied (vacancies)
 Atoms displaced from ideal positions
 There are defects that modify the properties
“Crystalline defect”: Disruption in the perfect
periodic arrangement of atoms/ions in a
crystalline material.
Topic 3. Defects in crystalline structures
Imperfections in crystals
Thermodynamics  Justify the presence of defects
Creation of a defect  ΔH, ΔS  ΔGcrystal
Vacancy or defect:
ΔG = ΔH-TΔS < 0
ΔH < TΔS
[Defects]eq  ΔG = 0
ΔG: Gibbs energy
ΔH: enthalpy of formation of each vacancy
ΔS: entropy of formation of each vacancy
T: absolute temperature
Configuration Entropy  S = KB ln W
N!
W=
KB= Boltzmann constant =1.380622 10-23 JK-1
( N − n)!n!
W= Way of distributing n defects in N possible random positions
In 1 mol of C+ there are NA possibilities to create 1 vacancy  W  1023
When T  [defects]
Topic 3. Defects in crystalline structures
Imperfections in crystals
Classification of defects
1. Point
Vacancies
Interstitials
Substitutional
Schottky
Frenkel
Order-Disorder
2. Linear: dislocations:
3. Planar or Extended
Edge (Taylor)
Screw (Burgers)
Mixed
4. Complex (volume defects): “clusters”
External Surfaces
Grain Boundaries
Twin Boundaries
Stacking Faults
Topic 3. Defects in crystalline structures
Point defects
Localized disruptions in an otherwise ’perfect’ atomic/ionic
arrangement in a crystal structure.
▪
Although the defect occurs at one or two sites, their presence is ‘‘felt’’ over
much larger distances in a crystalline material.
▪
Introduced by movement of the atoms when they gain energy.
▪
Point defects:
✓ Impurities: elements/ compounds present from raw materials or
processing.
✓ Dopants:
elements/compounds
concentrations).
deliberately
added
(known
Topic 3. Defects in crystalline structures
Point defects
a. Self-Interstitial
• An extra atom in the lattice in the interstitial
position
Interstitial
b. Vacancy
• An atom is missing from the regular position
• Number of vacancies at temperature T
Vacancy
nv = nº vacancies per cm3
N= nº of lattice points per cm3
Hv= Formation energy (J.mol-1)
They are produced:
▪
▪
▪
▪
During solidification (impurities, alloys)
By particle bombardment with E
During plastic deformation (processing)
When T  thermal vibrations  nv
For metals, vacancies
concentration at equilibrium
nv
» 10-4 max
N
Topic 3. Defects in crystalline structures
Point defects
c. Schottky defect
• A pair of vacancies of opposite charge
• Number of defects:
H s
ns = N exp( −
)
2 RT
HS / HF= E for Schottky/Frenkel defect
creation
N= nº of lattice positions
Ni= nº of interstitial positions
d. Frenkel defect
• Migration of an ion from a regular to an
interstitial position.
• Number of defects:
nF = NN i exp( −
H F
)
2 RT
In a crystal HS  HF
 the defect with the
lowest H will form
Topic 3. Defects in crystalline structures
Point defects
L. Smart and E. Moore. “Química
del Estado Sólido”.
Addison-Wesley Iberoamericana.
Wilmington, 1995.
Schottky defects
• NaCl (Tf=801 ºC) Hs= 3.69x10-19 J
T=300K
ns= 2.64x104 vacancies/mol
T=1000Kns= 9.38x1017 vacancies/mol
• MgO
Hs=
J
T=300K
ns= 2.12x10-32 vacancies/mol
T=1000Kns= 1.39x107 vacancies/mol
10.57x10-19
nº defects  with T
Hs(MgO) > Hs(NaCl)
ns low, more difficult to
create vacancies
Topic 3. Defects in crystalline structures
Point defects
e. Order-disorder phenomena (in substitutional solid solutions)
Solids with elements of similar electronegativities.
Topic 3. Defects in crystalline structures
Point defects
----- Solid solution ------ By mixing two different materials in the liquid state and subsequent solidification,
depending on the conditions, we can either get formation of two phases or a solid
solution
 Solid solution: crystalline solid that has two or more elements dispersed as atoms in a
structure that has just one phase. Solid solutions can be:
Substitutional solid solutions
Interstitial solid solutions
Solute atoms replace atoms of the
host
Solute atoms occupy interstitial sites
Topic 3. Defects in crystalline structures
Point defects
----- Solid solution ------Conditions for the formation of solid solutions
Hume-Rothery rules
 Substitutional solid solutions:
•
Atomic radii of solute and solvent atoms must be similar
•
The crystal structure of solute and solvent must match
•
Solute and solvent must have the same valence
•
Solute and solvent must have similar electronegativities
Interstitial solid solutions:

•
Radii of solute atoms must be smaller than those of the solvent
•
Solute and solvent must have similar electronegativities
Topic 3. Defects in crystalline structures
Point defects
e. Order-disorder phenomena (in substitutional solid solutions)
Solids with elements of similar electronegativities.
 Disorder: Atoms of one sub-lattice occupy positions of the other and vice
versa
 Order: Above certain critical temperature, disordered atoms return to
regular positions.
Example: Cu-Au alloy
T>390ºC
T<390ºC
Topic 3. Defects in crystalline structures
Linear defects:dislocations
Linear defects or dislocations are line imperfections in
an otherwise perfect crystal
Characteristics: they can be displaced in the interior of a crystal by applying
relatively low forces and can produce a complete displacement over crystalline
planes.
They explain:
- Etheoretical ( Young modulus) > Eexperimental
- Plastic deformation in metals (workability, ductility)
Formation: during solidification, upon permanent deformation, concentration
of vacancies and atomic disarrangements in solid solutions.
Types: - edge dislocation (Taylor)
- screw dislocation (Burgers)
- mixed dislocation
Topic 3. Defects in crystalline structures
Linear defects:dislocations
a) Edge dislocation (Taylor)
 Geometric modification of the lattice: extra plane of atoms
 Energetic modification (dislocations store energy)

b
2
1
A

Characterization of dislocation: Burgers vector (b )
Magnitude: distance 1-2
Direction: 1-2 (or 2-1)
b ⊥ to dislocation line
b || to the direction movement
2
E b
Dislocation line: ⊥ point A
Slipping plane: line of points
Topic 3. Defects in crystalline structures
Linear defects:dislocations
b) Screw dislocation (Burgers)
 Locally curves some atom lines. The effect is as if a shear stress is applied to
produce a distortion
D
Plane view. We can
trace a screw (spiral
ramp) around the
dislocation.
1
5
b
D
A
C
Burgers vector:
The magnitude and distance from 1-5 define b
b || to dislocation line SS´
b ⊥ to the direction of movement
C
B
Topic 3. Defects in crystalline structures
Linear defects:dislocations
c) Mixed dislocation
 Edge and screw dislocations with a transition region between them.
 Burgers vector remains the same for all portions of mixed dislocations.
Edge
B
A
C
Screw
Topic 3. Defects in crystalline structures
Plastic deformation
The capability of a metal to be plastic deformed depends on the
capability of dislocations to move.
Dislocation movement (edge
and screw) when a shear
stress is applied
Topic 3. Defects in crystalline structures
Plastic deformation
Slip → process by which the plastic deformation is produced by
dislocation motion
 Dislocation propagates along the slip direction over the slip plane
 Slip direction & slip plane form slip system
 For edge dislocations Burgers vector || to slip direction
 For screw dislocations Burgers vector
to slip direction
 Slip direction is parallel to the direction of maximum packing
Topic 3. Defects in crystalline structures
Plastic deformation
Dislocation movement or slip direction is parallel to the direction of
maximum packing since:
E used to move a dislocation= E  |b|2
Compact direction
Non compact direction
FCC [110]
FCC [100]
b
b
b = 2·R
Eb  4R2
b 2 + b 2 =(4 R ) 2  b = 2 2·R
Eb  8R2
Topic 3. Defects in crystalline structures
Plastic deformation
Most dense
atomic
packing
 Slip systems →  Ductility
Higher linear
density
Topic 3. Defects in crystalline structures
Planar defects
Planar defects
Boundaries or planes that separate a
material into regions, each having the
same crystal structure but different
orientation.
▪
▪
▪
▪
External surfaces
Grain boundaries
Twin boundaries
Stacking faults
A) External surface
It is the end of the crystal or grain structure
Coordination numbers at the surface < at the interior of the crystal 
Esurface> Einterior
Topic 3. Defects in crystalline structures
Planar defects
B) Grain boundaries
They separate crystals with different orientation.
Formation: during solidification:  crystals are formed when  nuclei grow simultaneously.
Polycrystal
Grains nucleation and growth
They limit dislocations movement in the
material
Topic 3. Defects in crystalline structures
Planar defects
Boundary width: 2-5 interatomic distances
Very energetic zones (atomic packing < in interior ):
✓ Solid state reactions
✓  Atomic diffusion
Materials properties = f (grain size)
Dislocation movement is limited when there
are many grain boundaries.
Grain size   Mechanical resistance
Topic 3. Defects in crystalline structures
Planar defects
C) Twin boundaries
 Twin boundaries are grain boundaries with a special case of mirror image
misorientation of the crystal structure
 Formation: deformation process or during thermal treatments
 The twin boundaries interfere with the slip process and increase the strength of the
material
Austenitic stainless steel
Topic 3. Defects in crystalline structures
Planar defects
C) Twin boundaries
Brass (70 Cu / 30 Zn)
Topic 3. Defects in crystalline structures
Planar defects
D) Stacking faults
HCP
A stacking fault is a one or two layer interruption
in the stacking sequence
For example, FCC structure:
A
B
A
Perfect network: ABCABC ABC
Stacking fault: ABCABABC
FCC
A
C
B
A