IMPERFECTIONS IN SOLIDS
Week 3
1
Imperfections in Solids
• Solidification- result of casting of molten material
– 2 steps
• Nuclei form
• Nuclei grow to form crystals – grain structure
• Start with a molten material – all liquid
nuclei
liquid
crystals growing
grain structure
• Crystals grow until they meet each other
2
Polycrystalline Materials
Grain Boundaries
• regions between crystals
• transition from lattice of
one region to that of the
other
• slightly disordered
• low density in grain
boundaries
– high mobility
– high diffusivity
– high chemical reactivity
3
Solidification
Grains can be - equiaxed (roughly same size in all directions)
- columnar (elongated grains)
~ 8 cm
heat
flow
Columnar in
area with less
undercooling
Shell of
equiaxed grains
due to rapid
cooling (greater
T) near wall
Grain Refiner - added to make smaller, more uniform, equiaxed grains.
4
Imperfections in Solids
There is no such thing as a perfect crystal.
• What are these imperfections?
• Why are they important?
Many of the important properties of
materials are due to the presence of
imperfections.
Crystalline defect -> a lattice irregularity
having one or more of its dimensions on the
order of an atomic diameter
5
Types of Imperfections
• Vacancy atoms
• Interstitial atoms
• Substitutional atoms
Point defects
• Dislocations
Line defects
• Grain Boundaries
Area defects
6
• Vacancies:
Point Defects
-vacant atomic sites in a structure.
Vacancy
distortion
of planes
• Self-Interstitials:
-"extra" atoms positioned between atomic sites.
selfinterstitial
distortion
of planes
7
Equilibrium Concentration:
Point Defects
• Equilibrium concentration varies with temperature!
No. of defects
No. of potential
defect sites.
Activation energy
-Q
Nv
= exp 
 v
 kT
N



Temperature
Boltzmann's constant
(1.38 x 10 -23 J/atom-K)
(8.62 x 10 -5 eV/atom-K)
Each lattice site
is a potential
vacancy site
8
Measuring Activation Energy
• We can get Qv from
an experiment.
-Q
Nv
v


exp
=
 kT
N
• Measure this...
• Replot it...
Nv
ln
N
Nv
N



slope
-Qv /k
exponential
dependence!
T
1/T
defect concentration
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EXAMPLE PROBLEM 4.1
Calculate the equilibrium number of
vacancies per cubic meter for copper at
1000C. The energy for vacancy formation
is 0.9 eV/atom; the atomic weight and
density (at 1000C) for copper are
63.5g/mol and 8.4g/cm3, respectively
10
Estimating Vacancy Concentration
Find the equil. # of vacancies in 1m3 of Cu at 1000C.
• Given:
r = 8.4 g /cm 3
A Cu = 63.5 g/mol
Qv = 0.9 eV/atom NA = 6.02 x 1023 atoms/mol
-Q
Nv =
 v
exp 
 kT
N
r=N.Acu/V.NA
For 1
m3 ,
N= r x
NA
A Cu
0.9 eV/atom

 = 2.7 x 10-4

1273K
8.62 x 10-5 eV/atom-K
x 1 m3 = 8.0 x 1028 sites
• Answer:
Nv = (2.7 x 10-4)(8.0 x 1028) sites = 2.2 x 1025 vacancies
11
Observing Equilibrium Vacancy Concentration.
• 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.
Island grows/shrinks to maintain
equil. vancancy conc. in the bulk.
12
IMPURITIES IN SOLIDS
• Impurity or foreign atoms will always be
present, and some will exist as crystalline
point defects
• Alloys -> impurity atoms have been added
intentionally to impart specific characteristics
to the material
• Alloying with copper significantly enhances
the mechanical strength without depreciating
the corrosion resistance appreciably
• The addition of impurity atoms to a metal will
result in the formation of a solid solution
13
Point Defects in Alloys
Two outcomes if impurity (B) added to host (A):
• Solid solution of B in A (i.e., random distribution of point defects)
OR
Substitutional solid soln.
(e.g., Cu in Ni)
Interstitial solid soln.
(e.g., C in Fe)
• Solid solution of B in A plus particles of a new
phase (usually for a larger amount of B)
Second phase particle
--different composition
--often different structure.
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Imperfections in Solids
Conditions for substitutional solid solution (S.S.)
• W. Hume – Rothery rule
– 1. r (atomic radius) < 15%
– 2. Proximity in periodic table
• i.e., similar electronegativities
– 3. Same crystal structure for pure metals
– 4. Valency
• All else being equal, a metal will have a greater tendency
to dissolve a metal of higher valency than one of lower
valency
15
Substitutional Solid Solution – Cu-Ni
• The atomic radii for copper and nickel
are 0.128 and 0.125nm, respectively
• Both have the FCC crystal structure
• Their electronegativities are 1.9 and 1.8
• Valencies for Cu and Ni are +2
16
Imperfections in Solids
Application of Hume–Rothery rules – Solid
Solutions
Element
Atomic Crystal
ElectroRadius Structure
(nm)
1. Would you predict
more Al or Ag
to dissolve in Zn?
2. More Zn or Al
in Cu?
Cu
C
H
O
Ag
Al
Co
Cr
Fe
Ni
Pd
Zn
0.1278
0.071
0.046
0.060
0.1445
0.1431
0.1253
0.1249
0.1241
0.1246
0.1376
0.1332
Valence
negativity
FCC
1.9
+2
FCC
FCC
HCP
BCC
BCC
FCC
FCC
HCP
1.9
1.5
1.8
1.6
1.8
1.8
2.2
1.6
+1
+3
+2
+3
+2
+2
+2
+2
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Conditions for Interstitial Impurity
• The atomic diameter of an interstitial impurity
must be substantially smaller than that of the
host atoms
• The maximum allowable concentration of
interstitial impurity atoms is low
• Even very small impurity atoms are ordinarily
larger than the interstitial sites, and as a
consequence they introduce some lattice
strains on the adjacent host atoms
• Different crystal structures can fill interstitials
18
Specification of Composition
– weight percent
m1
C1 =
x 100
m1  m2
m1 = mass of component 1
– atom percent
nm1
C =
x 100
nm1  nm 2
'
1
nm1 = number of moles of component 1
19
Dislocations - Line Defects
Dislocations:
• are line defects,
• slip between crystal planes result when dislocations move,
• produce permanent (plastic) deformation.
Schematic of Zinc (HCP):
• before deformation
• after tensile elongation
slip steps
20
Imperfections in Solids
Linear Defects (Dislocations)
– Are one-dimensional defects around which atoms are misaligned
Burgers vector (b) represents the magnitude and direction of the
distortion of dislocation in a crystal lattice
Dislocation Line -> A curve running along the center of a
dislocation.
• Edge dislocation:
– extra half-plane of atoms inserted in a crystal structure
– b  to dislocation line
• Screw dislocation:
– spiral planar ramp resulting from shear deformation
– b  to dislocation line
21
Imperfections in Solids
Edge Dislocation
22
Motion of Edge Dislocation
• Dislocation motion requires the successive bumping
of a half plane of atoms (from left to right here).
• Bonds across the slipping planes are broken and
remade in succession.
Atomic view of edge
dislocation motion from
left to right as a crystal
is sheared.
23
Imperfections in Solids
Screw Dislocation
Screw Dislocation
b
Dislocation
line
Burgers vector b
(b)
(a)
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Edge, Screw, and Mixed Dislocations
Mixed
Edge
Screw
25
Imperfections in Solids
Dislocations are visible in electron micrographs
51,450 magnified
26
Planar Defects in Solids
• One case is a twin boundary (plane)
– Essentially a reflection of atom positions across the twin plane.
• A twin boundary is a special type of grain boundary
across which there is a specific mirror lattice symmetry
• Annealing twins are typically found in metals that have
the FCC crystal structure, while mechanical twins are
observed in BCC and HCP metals
27
Planar Defects in Solids
• Stack Fault are found in FCC metals when there is an
interruption in the ABCABCABC . . . stacking sequence
of close-packed planes
• Phase boundaries exist in multiphase materials across
which there is a sudden change in physical and/or
chemical characteristics
28
Bulk or Volume Defects
• Other defects exist in all solid materials
that are much larger than those
discussed
• These include pores, cracks, foreign
inclusions, and other phases
• They are normally introduced during
processing and fabrication steps.
29
Numerical Problems
• Problems 4.1 to 4.5 and 4.7 to 4.25
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