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Lec 08, Page 1/16
A. K. M. B. Rashid
Professor, Department of MME
BUET, Dhaka
MME131: Lecture 8
Imperfections in atomic arrangements Part 1: 0D Defects
Today’s Topics
Occurrence and importance of crystal defects
Classification of crystal defects
Characteristics of 0D defects
References:
1. Callister. Materials Science and Engineering: An Introduction
2. Askeland. The Science and Engineering of Materials
Lec 08, Page 2/16
Real crystals are never perfect, there are always defects
Crystalline imperfections
Most atoms are in ideal locations, only small numbers are out of place
Most defects cause the periodicity of the crystal to be disturbed
over distances of several atomic diameters from the defect
“Crystals are like people, it is the defects in them
which tend to make them interesting!”
Colin Humphreys, Professor and Director Research
Department of Materials Science and metallurgy, University of Cambridge, UK
Defects have a profound impact on the macroscopic properties
of materials
Based on the bond strength, most materials should be much stronger than
they are. Why?
Strength of an ionic bond ≈ 106 psi,
but more typical strength is 40x103 psi.
Bonding +
Structure +
Defects
Properties
Materials do not usually fail by breaking bonds!!
Lec 08, Page 3/16
The processing of materials determines the nature and
amount of defects in it.
Composition
Bonding Crystal structure
Thermo-mechanical processing
Microstructure
Defect introduction and manipulation
Defects, even in very small concentrations, can have
a dramatic impact on properties of material.
Without defects:
solid-state electronic devices could not exist
metals would be much stronger
ceramics would be much tougher
crystals would have no color
Why are defects important ?
Lec 08, Page 4/16
3D, Bulk / Volume defects
porosity
crack
foreign inclusion
Types of crystal defects
0D, Point defects
vacancies
interstitials
substitutional
1D, Line defects
dislocations
2D, Surface defects
external surfaces
grain boundaries
twin boundaries
based on dimension of defects
Point Defects
All real materials have point defects. More can be added by
heating of material
processing of material
introducing impurities during materials processing
intentionally adding during alloying
Localized disruptions in the lattice involving one or more atoms.
Affect material properties by influencing
atomic movement (diffusion process)
dislocation movement (strengthening methods)
Lec 08, Page 5/16
Frenkel
Schottky In compound crystals (e.g., in ceramics)
Classes of point defects
Intrinsic defects Vacancy
Self-interstitial
Extrinsic defects (a.k.a. impurity) Substitutional
Interstitialcy
Point defects: (a) vacancy, (b) interstitial atom, (c) small substitutional atom,
(d) large substitutional atom, (e) Frenkel defect (vacancy-interstitial pair),
(f) Schottky defect (anion-cation vacancy pair).
All of these defects disrupt the perfect arrangement of the surrounding atoms.
Lec 08, Page 6/16
Vacancy
A lattice position that is vacant because the atom is missing
There are naturally occurring vacancies in all crystals
Point defects occur as a result of the periodic oscillation or thermal vibration
of atoms in the crystal structure.
How many vacancies are there?
The equilibrium concentration of
vacancies at a given temperature
can be expressed as:
NV: number of vacancies QV: activation energy for the formation of a vacancy
N: total number of atomic sites k: Boltzmann’s constant = 1.38x10-23 J/atom-K
T: temperature in Kelvin = 8.62x10-5 eV/atom-K
NV = N exp - QV
k T
Ideal gas constant, R = Boltzmann’s constant, k per mole = k NA = 8.314 J/mol-K
The concentrations of vacancies increase with:
increasing temperature, T
decreasing activation energy, QV
NV = N exp - QV
k T
NV
N
T defect
concentration
exponential
dependence
NV
N
T
slope
ln 1
- QV / k
Lec 08, Page 7/16
Example 1:
Vacancy concentrations in iron
An iron crystal has a density of 7.87 g/cm3. Determine the number of
vacancies present in the crystal. The lattice parameter of BCC iron is
2.866 10-8 cm.
SOLUTION
The expected theoretical density of iron can be calculated from the lattice
parameter and the atomic mass.
So, for this ‘x’ iron atoms, the density equation will become
Or, there should be 2.00 – 1.9971 = 0.0029 vacancies per unit cell.
The number of vacancies per cm3 is:
This density is reduced to 7.8 g/cm3 due to the presence of vacancies. Let us
assume that, instead of 2 atoms, there are ‘x’ iron atoms present in the unit cell.
Lec 08, Page 8/16
Number of regular atomic sites per m3 for copper
N = NA r
ACu
(6.023x1023 # atom/mol) x (8.4 g/cm3) x (106 cm3/m3)
(63.5 g/mol) =
= 7.96x1028 # atom/m3
Example 2:
The effect of temperature on vacancy concentrations
Calculate the equilibrium number of vacancies per cubic meter for copper
at 27 and 1000 C. The energy for vacancy formation is 0.9 eV/atom;
the atomic weight and density for copper are 63.5 g/mol and 8.4 g/cm3,
respectively.
The number of vacancies at 27 C (= 300 K):
NV = N exp (-QV / kT)
= (7.96x1028 # atoms/m3) - 0.9 eV/atom
(8.62x10-5 eV/K) x (300 K) x exp
= 6.11x1013 vacancies/m3
one vacancy in every 1000 trillion (1015) lattice atoms
The number of vacancies at 1000 C (= 1273 K):
= (7.96x1028 # atoms/m3) - 0.9 eV/atom
(8.62x10-5 eV/K) x (1273 K) x exp
= 2.18x1025 vacancies/m3
one vacancy in every 3000 lattice atoms
NV = N exp (-QV / kT)
Lec 08, Page 9/16
In FCC iron, carbon atoms are located at
octahedral sites at the center of each edge
of the unit cell (1/2, 0, 0) and at the center
of the unit cell (1/2, 1/2, 1/2).
In BCC iron, carbon atoms enter the
tetrahedral sites, such as 1/4, 1/2, 0.
The lattice parameter is 0.3571 nm for FCC
iron and 0.2866 nm for BCC iron.
Assuming that carbon atoms have a radius
of 0.071 nm, would we expect a greater
distortion of the crystal by an interstitial
carbon atom in FCC or BCC iron?
Example 3:
Sites for carbon in iron
tetrahedral sites in BCC iron
octahedral sites at the edge centre
in FCC iron
SOLUTION
BCC iron atom
Interstitial site at the (1/4, 1/2, 0) location:
Atomic radius:
Lec 08, Page 10/16
FCC iron atom
The interstitial site in BCC iron (0.0361 nm) is smaller than the interstitial site
in FCC iron (0.0522 nm).
Although both are smaller than the size of carbon atom (0.071 nm),
carbon distorts the BCC crystal structure more than the FCC crystal.
As a result, fewer carbon atoms are expected to enter the interstitial positions
in BCC iron than in FCC iron.
Self-Interstitial
A point defect caused when
a ‘normal’ atom occupies an
interstitial site in the crystal
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 cm3 at room temperature).
self-interstitialdistortion
of planes
distortion of planes
self-interstials
Lec 08, Page 11/16
Impurity atoms are different from the host atoms.
All real solids are impure. Very pure metals are only about
99.9999% pure (i.e., ~ one impurity per 106 host atoms)
May be intentional or unintentional
Examples:
Carbon added in small amounts to iron makes steel, which is stronger
than pure iron (alloying)
Sulphur remained in steel due to difficulties in synthesis (impurity)
Impurities and alloys
The equilibrium amount of a given impurity in a solid
is determined by:
How well the impurity "fits" into the structure;
any bond distortion raises the energy requirement
Similarity of bonding type between the impurity and the host
How much of the impurity is available externally
In ionic crystals, requirements of charge neutrality
Lec 08, Page 12/16
initial geometry Ca2+ impurity resulting geometry
Ca2+
Na+
Na+
Ca2+
cation
vacancy
initial geometry O2- impurity
O2-
Cl-
anion vacancy
Cl-
resulting geometry
Example: NaCl Na+ Cl-
Impurities satisfying charge balance
Point defects in alloys
Two outcomes, when an alloying element B is added to a host A:
Form a solid solution of B in A (i.e., random distribution of point defects)
OR
Substitutional alloy
(e.g., Cu in Ni)
Interstitial alloy
(e.g., C in Fe)
Form a solid solution of B in A plus formation of new phase
second phase particle
--different composition
--often different structure
Lec 08, Page 13/16
Solid solutions
Solid solutions are made by dissolving the minor component (solute)
to a host (the solvent or matrix) material. The ability to dissolve is called
solubility.
homogeneous
maintain host crystal structure, no new structure formed
contain randomly dispersed impurities (substitutional or interstitial)
Second Phase: as solute atoms are added, new compounds / structures
are formed, or solute forms local precipitates.
Whether the addition of impurities results in formation of solid solution
or second phase depends the nature of the impurities, their concentration
and temperature, pressure…
For impurities, certain conditions favour substitutional solution formation
(Hume-Rothery rules)
Substitutional solid solution
1. R (atomic radii) <15%
2. Same crystal structures
3. Similar electronegativities (DE ≤0.6, preferably ≤ 0.4)
4. Same valence (if multivalent, must have at least one valence state in common)
all four rules must be satisfied to form a complete substitutional solid solution
Cu-Ni
Yes (2.34%)
Yes (FCC)
Yes (0.0)
Yes (2+)
Complete
solid solution
Pb-Ni
No (28.57%)
No (Pb–FCC; Ni–BCC)
Yes (0.2)
No (Pb - 2+ , 4+; Ni -1+)
Limited
solid solution
Lec 08, Page 14/16
Interstitial solid solution
Metallic materials have relatively high APF, making interstitial positions
relatively small.
An atom must be fairly tiny to fit into the interstitial holes.
The maximum allowable concentrations of interstitial impurities is ~10%
(only 2% max. C in Fe-C system) [ atomic radius of C = 0.071 nm; radius of BCC interstitial void = 0.036 nm ]
Even very small impurity atoms are larger than interstitial sites,
so all interstitial impurities introduce lattice strains on the adjacent host atoms.
Point defects in ceramic materials
Ceramic materials are usually
compounds and contain ions of at
least two kinds, and point defects
such as interstitial, substitutional, and
vacancy for each ion type may occur.
Substantial strains will be introduced
to the crystal lattice by these point
defects. Schematic representations of cation and
anion vacancies and a cation interstitial
Because the atoms exist as charged ions in ceramics, when defect structures are
considered, conditions of electroneutrality must be maintained.
As a consequence, defects in ceramics do not occur alone.
Lec 08, Page 15/16
Frenkel defect (a cation-vacancy and a cation-interstitial pair )
• a pair of point defects produced when an ion moves to create an interstitial site,
leaving behind a vacancy
Schottky defect (a cation-vacancy and a anion vacancy pair)
• a point defect in ionically bonded materials.
• In order to maintain a neutral charge, a stoichiometric number of cation and anion
vacancies must form
Assuming that we are dealing
with a AX-type ceramic crystal
(e.g., NaCl), the state of electro-
neutrality is still maintained
since the cation maintains the
same positive charge as an
interstitial, and both cations and
anions have the same charge.
When a divalent cation impurity replaces a monovalent parent cation
(e.g., Ca in place of Na), to maintain charge neutrality, a second monovalent
parent cation must also be removed, creating a vacancy
Lec 08, Page 16/16
If the ratio of cations to anions is not altered by the formation of defects, the material
is said to be “stoichiometric”.
Stoichiometry: A state for ionic compounds wherein there is the exact ratio of
cations to anions as predicted by the chemical formula.
• Fe3+ ions disrupt the electroneutrality of the crystal,
i.e., extra positive charge.
• Excess + charge offset by formation of Fe2+ vacancy.
• Formation of one Fe2+ vacancy for every two Fe3+
that are formed.
• The crystal is no longer stoichiometric because there
are more O than Fe (i.e., deficiency of Fe);
however still remains electrically neutral.
• Chemical formula becomes: Fe1-xO, where x is
some small fraction much less than 1.
A ceramic compound is nonstoichiometric if there is any deviation from the exact
cation-to-anion ratio. Nonstoichiometric may occur for some ceramic materials in
which two valence states exist for one of the ion types. Iron oxide (FeO) is one such
material, for the iron can be present in both Fe2+ and Fe3+ states; the number of each
of these ion types depends on temperature and the ambient O2 pressure.
Next Class
MME131: Lecture 9
Imperfections in atomic arrangement Part 2: 1D – 3D defects
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