CHAPTER 6CHAPTER 6

Imperfections of StructuresImperfections of Structures

6-I. Introduction6-I. Introduction

• defects≡imperfections

• Engineering materials

very large numbers of atoms in small volumes

e.g., Fe: BCC , a0=0.287nm

8.5 × 1022 atoms/cm3

a high probability of mistakes in atomic arrangement.

• point defects (those associated with one or two atomic positions),

• linear (or one-dimensional) defects,

• planar (two-dimensional) defects = interfacial defects, or boundaries,

• Volume (three dimensional) defects

according to geometry or dimensionality of the defect

◎ Types of defectsTypes of defects

A. Point DefectsA. Point Defects

Vacancies ： missing atoms (a normally occupied position is vacant)

Interstitials ： atoms at wrong positions (atoms are located in normally unoccupied positions)

Impurities: foreign atoms contained in a metal (or other materials)

◎ Both types of defects increase the internal energy (U)

of the crystal :

1. Surrounding atoms are moved away from their

equilibrium positions strain energy

6-II. Imperfections in Metallic Materials6-II. Imperfections in Metallic Materials

“an imperfection that involves a few atoms at most”

F 4.1 F 4.2-1

F 2.8

2. Surrounding atoms are altered in their coordination number.

All crystalline solids contain vacancies (or other types of defects) and, in fact, it is not possible to create such a material that is free of these defects.

A-1. Criteria of Stability of Crystal with DefectsA-1. Criteria of Stability of Crystal with Defects

◎ Generally, criteria of occurrence of a process (e.g., a change of the state of a material)

1. Thermodynamics ( 熱力學：化工熱力學，材料熱力學 )

a process has a tendency to occur or not / a process has

a tendency to occur if the occurrence decreases the Gibb’s free energy ( under const T and P conditions )

G = H – TS

H= U + PV

where H: enthalpy, U: internal energy

S: entropy / a measure of the randomness ( 亂度 ) of a

system (number of distinct possible arrangements

of atoms for crystals)

2. Kinetics (動力學：化工動力學，化學反應工程，材料動力學 )

rate of the occurrence

◎ For the problem under consideration (i.e., stability of a crystal with defects) the atomic arrangement of a crystal with the lower G value will be the thermodynamically favored state

PV = const, T = const.

1. U and S G

2. U > S G

3. U < S G

◎ point defects: increase U (from strain energy) and also

greatly increase S (when the defect concentration is not too high)

presence of certain concentrations of defects

G

◎ Thermodynamics : point defects will (or other types of defects) be present

G = U + PV － TS

(the present case)

• A pure metal consisting of only one type of atom just isn’t

possible

• It is difficult to refine metals to a purity in excess of

99.9999%, 1022 to 1023 impurity atoms will be present in one

cubic meter of material.

• Most familiar metals are not highly pure; rather, they are

alloys, impurity atoms have been added intentionally to

impart specific characteristics to the material, e.g.,

mechanical strength, or corrosion resistance.

For example, sterling silver = 92.5% silver-7.5% copper alloy. This enhances the mechanical strength without appreciably depreciating the corrosion resistance

A-2. The equilibrium number of vacancies A-2. The equilibrium number of vacancies NNvv

kT

QNN

vv exp

(4.1)

N is the total number of atomic sites, Qv is the energy required for the formation of a vacancy.T is the absolute temperature in kelvins, and k is the gas or Boltzmann’s constant. The value of k is 1.38 10-23 J/atom-K, or 8.62 10-5 eV/atom-K.

For most metals, the fraction of vacancies Nv /N just below the melting temperature is on the order of 10-4; that is, one lattice site out of 10,000 will be empty.

A-3. Impurities In SolidsA-3. Impurities In Solids1. from contamination

2. intentionally added

e.g., P in Si semiconductor

C in Fe steel

solute : present in a minor concentration

solvent : present in the greatest amount (host atoms)

A-4. Solid SolutionA-4. Solid Solution

(A) Type of Solid Solutions

1. interstitial solid solution : impurities at interstitial sites

• in principle, formed only when the interstitial atoms are comparable in size to the interstitial sites, but no precise rules can be formulated.

e.g., C in BCC Fe: r( c )/r(Fe) = 0.077/0.124 = 0.62 k/r 0.155 for octahedral sites and k/r 0.291 for tetrahedral sites

F 4.3-1

2. substitutional solid solution : impurities at substitutional

sites

● requirement for the formation (Hume-Rothery rules)

(1) size difference ~15%( strain energy)

(2) EN : comparable

(3) valence : similar (4) crystal structure : same (for entire compositional range, not for dilute solutions) ● all are similar (or comparable): substitutional solid solution can be formed at all composition no solubility limit (isomorphous alloy)

● not sufficiently similar:

substitutional solid solution can be formed only at limited

compositions these exits solubility limit.

} (similar bond characteristics)

F 4.3-2 F 4.2

● Examples of this type of solution

(1) P in Si

(2) 18% Cr and 8% Ni in FCC Fe ( with a solubility limit)

304 stainless steel

(with oxidation and corrosion resistance ; compared to

plain carbon steel, I.e., C in BCC Fe)

(3) copper and nickel alloy (no solubility limit)

0.128 and 0.125 nm ;

both have the FCC ; electronegativities : 1.9 and 1.8 ;

valences : +1 for copper and +2 for nickel.

◎ composition (or concentration)

• weight percent

(4.3)

m1 and m2 represent the weight (or mass) of elements 1 and 2

• atom percent (at%)

(4.5)

(4.4)

and A1 denote the mass (in grams) and atomic weight

10021

11

mm

mC

1

11

A

mnm

1m

(B) Specification of Composition

10021

11

mn

m

nn

nC

mass of one component per unit volume of material

(4.9a)

(4.9b)

density : g/cm3,

and : kg/m3

3

2

2

1

1

11 10

CC

CC

3

2

2

1

1

22 10

CC

CC

1C 2C

B. Dislocations – Linear DefectsB. Dislocations – Linear Defects

◎ linear defects ( dislocations, 差排 ) :

one – dimensional (linear) type of local faults in atomic

arrangement.

◎ four types of dislocation :

edge dislocation

screw dislocation

mixed dislocation

dislocation loop

B-1. Edge dislocationsB-1. Edge dislocations

◎ dislocation line (unit vector, )

the edge of an extra plane of atoms, where

coordination number is altered.

◎ there is a net increase in energy alone the

dislocation line.

◎ Burger’s vector, (or slip vector) :

the displacement distance for atoms around the

dislocation ; .b

b

F 4.3

F 7.1

F 7.2

B-2. Screw dislocationsB-2. Screw dislocations

◎ can be envisioned as forming the axis of a helical

ramp running through the crystal.

◎ extra energy is involved.

◎ .

B-3. Mixed dislocation and dislocation loop. B-3. Mixed dislocation and dislocation loop.

◎ Formation of Dislocations

˙“accidents” in the growth process.

˙internal stress associated with other defects.

˙interactions between existing dislocations that occur during plastic deformation.

b

F 4.4

F 4.5 F 4.6

small- (or low-) angle grain boundary

till boundary

twist boundary

C. Interfacial DefectsC. Interfacial Defects

◎ two dimensional defects ： external surfaces, grain boundaries, twin boundaries, stacking faults, and phase boundaries

C-1. External SurfacesC-1. External Surfaces

F 4.7 F 4.8

F 4-12

Surface atoms are not bonded to the maximum number of nearest neighbors, and are therefore in a higher energy state than interior positions.

C-2. Grain BoundariesC-2. Grain Boundaries

mirror lattice symmetry; produced from applied mechanical shear forces and also during annealing following deformation

C-3. Twin BoundariesC-3. Twin Boundaries

C-4. Miscellaneous Interfacial DefectsC-4. Miscellaneous Interfacial Defects

stacking faults, phase boundaries, and ferromagnetic domain walls

F 4.9

pores, cracks, foreign inclusions, and other phases. microstructure

E. Atomic VibrationsE. Atomic Vibrations

atomic vibrations may be thought of as imperfections or defects not all at the same frequency and amplitude, nor with the same energy in a random manner

D. Bulk Or Volume DefectsD. Bulk Or Volume Defects F 4-12

6-II. Microstructure and Microscopic Examination6-II. Microstructure and Microscopic Examination

• microstructure : grain size、 shape, grain boundary composition, pore size、 shape and distribution.

• microscopy : optical, electron, and scanning probe microscopes

• photomicrograph (SEM or TEM photographs)

◎ sample preparation

(a) SEM

• surface :

exterior surface

interior surface : can be created by cutting or breaking.

A. GeneralA. General

B. microscopic techniquesB. microscopic techniques

F 4-12 F 5.3-3

F 4-10

• fractured surface ( 破裂 /斷面 ) : as is by breaking.

(without polishing)

• polishing surface ( 拋光面 ) : breaking grinding

polishing (to obtain a smooth and mirrorlike surface)

etching (washing away, depending on crystallographic

orientation, a greater extent at grain boundaries)

(b) TEM

• thin piece of sample

◎ Optical Microscopy : Optical Microscopy : upper limit : 2000 times

◎ Electron MicroscopyElectron Microscopy

• Transmission electron microscopy (TEM)

)003.0( nm

Magnifications approaching 1,000,000 are possible.

• Scanning Electron Microscopy (SEM)

The surface is scanned with an electron beam, reflected (or back-scattered) beam of electrons is collected

The surface must be electrically conductive : thin metallic surface coating

Magnifications ranging from 10 to in excess of 50,000 times are possible

Qualitative and semiquantitative analysis of the elemental

F 4-11 F 4-12

• 109 are possible; much better resolutions

• Three-dimensional images topographical

• a variety of environments (e.g., vacuum, air, liquid);

• a tiny probe with a very sharp tip controlled by piezoelectric ceramic components

• The advent of the SPMs has helped to usher in the era of nanomaterials – materials whose properties are designed by engineering atomic and molecular structures.

◎ ◎ Scanning Probe Microscopy (SPM)Scanning Probe Microscopy (SPM)

• standard comparison charts： from 1 to 10

• grain size number： the larger this number, the smaller are the grains

• A specimen must be properly prepared： 100× or a conversion has to be made.

N = 2n-1

n : grain size number N : average number of grains per square inch at a magnification

of 100×

C. Grain Size DeterminationC. Grain Size Determination

(a) American Society for Testing and Materials (ASTM)

• Straight lines all the same length are drawn through several photomicrographs.

• The grains intersected by each line segment are counted.

• the line length is then divided by an average of the number of grains intersected.

• The average line grain diameter is found by dividing this result by the linear magnification of the

photomicrograph.

(b) Intercept method

◎ point defect ： any lattice point not occupied by the proper ion

or atom

◎ linear defects ： dislocations

◎ Planar defects ： surface imperfections in polycrystalline solids, e.g., grain and twin boundaries.

◎ bulk defects ： pores, cracks, and inclusions (these defects are critical in determining the strength of ceramics).

6-III. Defects in Ceramics6-III. Defects in Ceramics

A. IntroductionA. Introduction

＊ Many of the properties are strongly affected by the presence or absence of defects, for instance ：

˙concentration of point defects : diffusion.

˙metals, but less so in ceramics (except at higher temperatures),

presence and movement of dislocations : ductility and creep.

˙grain size : mechanical strength.

˙scattering of light by pores : opacity.

＊ formation of ceramic defects preservation of electroneutrality

◎ Stoichiometric defects The ratio of the cations to anions is exactly stoichiometric.

Frenkel defect : cation-vacancy and a cation-interstitial pair Schottky defect : a cation vacancy-anion vacancy pair

◎ Nonstoichiometric defects The ratio of cations to anions deviates from the stoichiometric.

These defects form by the selective addition or loss of one (or more) of the constituents of the crystal and consequently lead to a change on crystal chemistry.

For example, an oxide annealed in a high oxygen particle pressure, number of oxygen atoms should be relatively greater than the number of cations. Conversely, if the oxygen particle pressure were very low, cation concentration to be higher (Fig-6.4).

B. Point DefectsB. Point Defects

F 4.2-312.20 F 4.2-2

Nonstoichiometry may occur when two valence (or ionic) states exist. An example : iron oxide (wustite, FeO), iron can be Fe2+ and Fe3+ states; depends on temperature and the ambient oxygen pressure, formation of one Fe2+ vacancy for every two Fe3+ ions, its chemical formula is often written as Fe1-xO

Importance of nonstoichiometry ： many physical properties suchas color, diffusivity, electrical conductivity, photoconductivity, and magnetic susceptibility can vary markedly with small changes in composition.

A few more examples of a nonstoichiometric compound ：• Mno at 1000k MnO is stable between a Po2 of 10-345 atm (below which Mn is the stable phase and the stable phase and O/M=1.18). MnO is considered a nonstoichiometric oxide.• Fe-O system is FeO and Fe3O4 are nonstoichiometric, Fe2O3 is stoichiometric. • Transition metal oxides are more likely to be nonstoichiometric than stoichiometric: loss of oxygen to the environment and the corresponding adjustments in the crystal are much easier when the cations can readily change their oxidation states.

◎ Extrinsic defects presence of impurities in the host crystal.

T 6.1 F 6.7 F 6.8 12.20

1. Vacancies : sites where an atom is missing.

2. Interstitial atoms : atoms found on sites that are normally unoccupied.

3. Misplaced atoms : types of atoms found at a site normally occupied by other types.

4. Free electrons : electrons that are in the conduction band of the crystal.

5. Electron holes : positive mobile electronic carriers that are present in the valence band of the crystal.

6. Interstitial and substitutional impruities.

C. Types of Point DefectsC. Types of Point Defects

F 6.1{Intrinsic defect ： not related to impurities.

extrinsic defect ： due to incorporation of impurities.

• Main symbol : the species involved, i.e., chemical symbol of an element, or the letter, I, for interstitial.• Subscript : crystallographic position, or I for interstitial.• Superscript : effective electric charge, defined as the difference, a prime for each negative charge, a dot for every positive charge, an x for zero effective charge.

D. Notation for Point DefectsD. Notation for Point Defects ：： Kroger- Vink notatioKroger- Vink notationn

Example 1 Example 4

E. Defect ReactionsE. Defect Reactions

• Mass balance; Mass cannot be created or destroyed. Vacancies have zero mass.

• Electroneutrality or charge balance: Charges cannot be created

or destroyed.

• Preservation of regular site ratio: for an MaXb compound, a(Xx+Vx)=b(MM+VM)

The formation of the various point defects is best described by chemical reactions for which the following rules have to be followed

b

a

VX

VM

XX

MM

＊ Note that this does not imply that the number of atoms or ions has to maintain that ratio but only the number of sites.

◎ Stoichiometric defect reactions

the chemistry of the crystal does not change as a result of the reactions.

(A) Schottky defects, for example ：

˙ MO (e.g., MgO)

Null (or prefect crystal)

˙M2O3

Null (or prefect crystal)

OM VV

2 3M OV V

F 6.3˙MaOb

Null (or prefect crystal) a

Ob

M bVaV

Intrinsic Defects

a vacancy is created by having an ion in a regular lattice site migrate into an interstitial site.

For instance, for a trivalent cation

iMXM VM M (6.12)

on the oxygen sublattice

XO i OO O V (6.13)

One type of atom is found at a site normally occupied by another, does not occur in ionic ceramic, but covalent ceramics like SiC :

SiCSiC CSiSiC

(B) Frenkel defects

(C) Antistructure disorder or misplaced atoms

◎ Nonstoichiometric defects the composition of the crystal changes as a result of the reaction.

XOg

XO VOO )(22

1(6.15)

the oxygen has to leave as a neutral species, it has to leave two electrons (the ones that belonged to the cations in the first place!) behind.

the electrons in this configuration are usually weakly bound to the defect site and are easily excited into the conduction band; i.e., acts as a donor. The ionization reaction

XOV

eVV

eVV

OO

OX

O

the net reaction reads

eVOO OgX

O 22

1)(2 (6.16)

(A) One of the more common nonstoichiometric reactions that occurs at low oxygen partial pressures

the oxygen vacancy is said to be doubly ionized (Fig. 6.4c)and carries an effective charge of +2

(B) Anther possible nonstoichiometric defect reaction : At high oxygen partial pressures, oxygen is incorporated into the crystal interstitially, i.e.,

Xig OO )(22

1 (6.17)

Ionization can also occur in this case, creating holes in the

valence band (i.e., the defect acts as an acceptor)

hOO

hOO

ii

iX

i

net reaction being

hOO ig 22

1)(2 (6.18)

F 6.4

For oxides in which the cations can exist in more than one oxidation state, the electronic defects can, change the oxidation state of the cation. To illustrate, magnetite, Fe3O4, a spinel structure, two-thirds of the Fe ions in the +3 state and one-third in the +2 state :

2( )

12

2X

g O FeO O V h

a net reaction

FeX

Og VOFeFeO 32)(2 22

2

1

(6-16) ： oxygen-deficient oxide ； (6-18) ： oxygen-rich oxide

2 32 2 2Fe h Fe

◎ ◎ Extrinsic defects Extrinsic defects

to consider impurity incorporation reactions

• Impurities usually substitute for the host ion of electronegativity nearest their own, even if the sizes of the ions differ

• Cations substitute for cation and anions for anions, irrespective of size differences

• For example, in NaCl, Ca and O would be expected to occupy the cation and anion sites, repspectively. In more covalent compounds where the electronegativities may be similar, size may play a more importatn role.

• Most interstitial atoms are small, but even large atoms are sometimes found in interstitial sites.

Ca Cl V Cl

Na Cl Na Cl

Cl Na Cl Na

(a)

Na Cl Na Cl

Cl Na Cl Na

Ca Cl

Cli Host crystal

(b)

Al Al V O O O

Mg Mg Mg O O O

Mg Mg Mg O O O

(c)

Mg Mg V O O

Al Al O O O

Al Al O O O

(d)

Host crystal

Figure 6.5

Bookkeeping technique for impurity incorporation reactions. (a) CaCl2 in Na Cl leaves a vacancy on cation sublattice. (b) An alternate reaction is for the extra Cl ion to go interstital. This reaction is unlikely, however, given the large size of the Cl ion. (c) Al2O3 in MgO creates a vacancy on the cation sublattice. (d) MgO in Al2O3 creates a vacancy on the anion sublattice. Fuel

EXAMPLE 1. Incorporate CaCl2 into NaCl. From Fig 6.5a, it is immediately obvious that one possible incorporation reaction is

A second perfectly legitimate incorporation reaction is shown in Fig. 6.5b, for which the corresponding defect reaction is

Note that in both cases the overriding concern was the preservation of the regular site rations of the host crystal. In the first case, two Cl lattice sites where created by the introduction of the dopant, and hence the same number of lattice sites had to be created on the cation sublattice. But since only one Ca cation was available, a vacancy on the Na sublattice had to be created. In the second case (Fig. 6.5b), there is no need to create vacancies because the number of lattice sites created does not change the regular site rations of the host crystal (interstitial sites are not considered regular sites).

xClNaNa

2NaCl

2ClVCaCaCl

xCliNa

NaCl2 CllCCaCaCl

E. Electronic Defects E. Electronic Defects

• In a perfect semiconductor or insulating crystal at 0 K, free electrons and holes do not exist. At finite temperatures, some of these electrons get knocked loose into the conduction band as a result of lattice vibrations.

• The intrinsic electronic defect reaction can be written as

heNull ' (6.19)

Given that the energy required to excite an electron from the valence to the conduction band is the band gap energy Eg

F. Defect Equilibria and Kroger-Vink DiagramsF. Defect Equilibria and Kroger-Vink DiagramsOne of the aims is to relate the concentration of defects to temperature and other externally imposed thermodynamic parameters such as oxygen partial pressure

This is accomplished by considering defects to be structural elements which possess a chemical potential and hence activity and expressing their equilibrium concentrations by a mass action expression similar to Eq (5.30)

Here ideality has been assumed and the activities have been replaced by the mole fractions xi

To illustrate, consider an MO oxide subjected to the following oxygen partial pressure regimes:

Low oxygen partial pressure

Assume that oxygen vacancies will form according to

redxo ggoeoo

)(2

1'2V 2 (I)

(6.23)

Kred = exp(-Δgred/kT)

As long as van << Nan

1)/(][ anananxo VNNo

Intermediate oxygen partial pressure

Assumed that Schottky equilibrium dominates

X XM O M O sM O V V g

(II)

red

xo

21

o

2

KO

pn oV2

exp( )

[ ][ ]M O s

SX XM O

v V gK

M O kT

(6.24)

1][ MX

O MO

High oxygen partial preessure

A possible defect reaction

oxidMxo gvhogo 2)(

2

12

(III)

oxid

O

MX

O KP

PVO

2/1

2

2

(6.25)