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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

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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!!

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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 ?

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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)

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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.

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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

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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.

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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)

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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:

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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

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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

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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

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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

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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.

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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

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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