crystalline imperfections

IMPERFECTIONS/DISORDER

Properties of materials are influenced by the

presence of imperfections

Many are profoundly sensitive to deviations from

crystalline perfections

Disorder accounts for the behavior of

semiconductors, for the ductility of metals.

Disorder also permits the movement of atoms

during heating treatment so that new structures

and enhanced properties may be realized.

• Electrical Resistivity of Copper:

• Adding “impurity” atoms to Cu increases resistivity.

• Deforming Cu increases resistivity.

4

Adapted from Fig. 18.8, Callister 6e.(Fig. 18.8 adapted from: J.O. Linde,

Ann Physik 5, 219 (1932); and

C.A. Wert and R.M. Thomson,

Physics of Solids, 2nd edition,

McGraw-Hill Company, New York,

1970.)

• Space Shuttle Tiles:

--Silica fiber insulation

offers low heat conduction.

• Thermal Conductivity

of Copper:

--It decreases when

you add zinc!

5

Fig. 19.0, Callister 6e.(Courtesy of Lockheed

Missiles and Space

Company, Inc.)

Adapted from

Fig. 19.4W, Callister 6e. (Courtesy of

Lockheed Aerospace

Ceramics Systems,

Sunnyvale, CA)

(Note: "W" denotes

fig. is on CD-ROM.)

Adapted from Fig. 19.4, Callister 6e.(Fig. 19.4 is adapted from Metals Handbook: Properties and Selection: Nonferrous alloys and Pure Metals, Vol. 2, 9th ed., H. Baker,

(Managing Editor), American Society for

Metals, 1979, p. 315.)

• Magnetic Permeability

vs. Composition:

--Adding 3 atomic % Si

makes Fe a better

recording medium!

Adapted from C.R. Barrett, W.D. Nix, and

A.S. Tetelman, The Principles ofEngineering Materials, Fig. 1-7(a), p. 9,

1973. Electronically reproduced

by permission of Pearson Education, Inc.,

Upper Saddle River, New Jersey.

6

Fig. 20.18, Callister 6e.(Fig. 20.18 is from J.U. Lemke, MRS Bulletin,

Vol. XV, No. 3, p. 31, 1990.)

• Magnetic Storage:

--Recording medium

is magnetized by

recording head.

CRYSTALLINE IMPERFECTIONS

•Also known as defects

•a lattice irregularity having one or more of

its dimensions on the order of an atomic

diameter

CRYSTALLINE IMPERFECTIONS are

frequently classified according to geometry or

dimensionality of the defect.

Point defects

Line defects

Interfacial defects

Bulk or volume defects

POINT DEFECTS are associated with one

or two atomic positions.

Point Defects in Metals

a. Vacancy

• vacant lattice site or missing atom from atomic site

• may be formed due to imperfect packing during

solidification or by atomic rearrangement

• may arise from thermal vibration of atoms at

elevated temperatures

VACANCY

Vacancydistortion of planes

b. Self-Interstitial

interstitial site – a small void space which

under ordinary circumstances is not

occupied

• an atom from the crystal is positioned in

an interstitial site between the matrix

atoms

• does not generally occur naturally due to

resulting structural distortion

• can be introduced by irradiation

SELF-INTERSTITIAL

self-interstitialdistortion

of planes

POINT DEFECTS in METALS

SELF-INTERSTITIAL

VACANCY

Two-dimensional representations of a

vacancy and a self-interstitial. (From

Callister, Materials Science and

Engineering).

POINT DEFECTS in METALS

SELF-INTERSTITIAL in

BODY CENTERED

CRYSTAL (BCC)

(1) kT

QexpNN V

V

N = total number of atomic sites (defect-free)

Qv = energy required for the formation of a vacancy

T = absolute temperature (in K)

k = Boltzmann’s constant

(1.38 x 10-23 J/atom-K or 8.62 x 10-5 eV/atom-K)

EQUILIBRIUM NUMBER OF

VACANCIES or VACANT LATTICE

SITES for a given quantity of material

depends on and increases with temperature

according to:

T

1

k

Q

N

Nln

kT

QexpNN

VV

VV

or

EQUILIBRIUM NUMBER OF

VACANCIES or VACANT LATTICE

SITES

ln(N

v/

N)

T-1

Slope = -Qv / k

kT

QexpNN V

V

Sample Problem:

Calculate the equilibrium number of vacancies per cubic

meter for copper at 1000°C. The energy for vacancy

formation is 0.9 eV/atom; the atomic weight and density

(at 1000°C) for copper are 63.5 g/mol and 8.40 g/cm3,

respectively.

kT

Qexp

A

NN

A

NN

V

Cu

AV

Cu

A

Solution:

First, determine N (number of atoms per cm3) of a

defect-free Cu at 1000°C.

Substitute to equation the expression for N to equation1,

33

3

mvacancies25

cmvacancies19

KeV5-

moleg

cm

gmole

atoms23

V

V

Cu

AV

10 x 2.210 x 2.2Nv

K) )(1273 10 x (8.62

eV 0.9exp

) (63.5

) )(8.4010 x (6.023N

kT

Qexp

A

ρNN

Solution:

The number of vacancies at 1000°C (1273 K) is,

Sample Problem:

Calculate the energy of vacancy formation in silver, given

that the equilibrium number of vacancies at 800°C is

3.60 x 1023 m-3. The molar mass and density (at 800°C)

for silver are, respectively, 107.9 g/mol and 9.5 g/cm3.

Assuming negligible change in density, at what

temperature (in °C) will the equilibrium number of

vacancies equal to 1.07 x 1023 m-3?

Solution:

Rearrange equation 1

v

v

vv

v

v

N

NlnkTQ

kT

Q

N

Nln

eN

N

eNN

kT

Qv

kT

Qv

Solution:

Determine N (number of atoms per cm3) of a defect-free

Ag at 800°C, then solve for Qv

atomJ19

atomeV

v

m

vacancies23

mole

g

m

cm6

cm

g

moleatoms23

K-atomeV5-

v

v

v

mole

g

m

cm6

cm

g

moleatoms23

Ag

A

1.76x101.1Q

))(3.60x10(107.9

))(10)(9.5(6.02x10lnK) )(1073(8.62x10Q

N

NlnkTQ

107.9

))(10)(9.5(6.02x10

A

NN

3

3

3

3

3

3

3

Solution:

Temperature (in °C) where Nv (equilibrium number of

vacancies) equal to 1.07 x 1023 m-3

C 700K 973

))(1.07x10(107.9

))(10)(9.5(6.02x10ln)(8.62x10

1.1

Nv

Nlnk

QvT

eNN

3

3

3

3

kT

Qv

m

vacancies23

mole

g

m

cm6

cm

g

moleatoms23

K-atomeV5-

atomeV

v

T

T

Impurities in Solids

Alloys

•impure metals

•impurity atoms have been intentionally added to impart

specific characteristics to the material

•enhances mechanical properties and corrosion resistance

Solid Solutions

•result from the addition of impurity atoms to a metal, without

the formation of new structures (crystal structure is

maintained)

•they are compositionally homogeneous: the impurity atoms

are randomly and uniformly dispersed in the solid

solute – the element or compound present in minor concentration

solvent or host atom – the element or compound present in the

greatest amount

Impurity Point Defects in Solid Solutionsa) Substitutional

solute or impurity

atoms replace or

substitute for the host

atoms

b) Interstitial

impurity atoms fill the

voids or interstices

among the host atoms the

normal concentration of

interstitial impurity atoms

is low (less than 10%)

Ex. Carbon in Iron

Hume-Rothery Rules for Solid Solubility:

a. Atomic Size Factor

the difference in atomic radius between the two atoms should be less

than about ± 15 %

b. Crystal Structure

the crystal structure for both metals should be the same for

appreciable solid solubility to occur

c. Electronegativity

similar electronegativity

the more electropositive one element is and the more electronegative

the other is, the greater is the tendency to form an intermetallic

compound than a substitutional solid solution

d. Valences

other factors being equal, a metal will have more of a tendency to

dissolve another metal of higher valency than one of a lower valency

Element Atomic Radius, nm Crystal

Structure

Electronegativity Valence

Copper 0.128 FCC 1.9 +1

Nickel 0.125 FCC 1.8 +2

Hume-Rothery Rules for Solid Solubility:

a. Atomic Size Factor

b. Crystal Structure

c. Electronegativity

d. Valences

b. Interstitial

impurity atoms fill the voids or interstices among the host atoms

the normal concentration of interstitial impurity atoms is low (less

than 10%)

Ex. Carbon in Iron

Sample Problem:

Which of these elements would you expect to form the following with

nickel:

a. A substitutional solid solution

having complete solubility.

b. A substitutional solid solution

having incomplete solubility.

c. An interstitial solid solution

Element R, nm Structure EN Valence

Ni 0.1246 FCC 1.8 +2

C 0.071 --- ---

H 0.046 --- ---

O 0.060 --- ---

Ag 0.1445 FCC 1.9 +1

Al 0.1431 FCC 1.5 +3

Co 0.1253 HCP 1.8 +2

Cr 0.1249 BCC 1.6 +3

Fe 0.1241 BCC 1.8 +2

Pt 0.1387 FCC 2.2 +2

Zn 0.1332 HCP 1.6 +2

POINT DEFECTS in CERAMICS

A unit cell of NaCl crystal (From

Callister, Materials Science and

Engineering).

Defect structure is often used to

designate the types and

concentrations of atomic defects in

ceramics.

Electroneutrality is the state that

exists when there are equal

numbers of positive and negative

charges from the ions.

As a consequence, defects in

ceramics do not occur alone.

Point Defects in Ceramics

c. Schottky Imperfection

• found in AX materials

• point imperfections in ionic compounds in which a

cation-anion pair are missing

• since both cations and anions have the same

charge, and since for every anion vacancy there

exists a cation vacancy, the charge neutrality of

the crystal is maintained.


c. Schottky Imperfection

Anion

Vacancy

Cation

Vacancy


d. Frenkel Defect

• point defect in ionic crystals in which a cation is

displaced from a lattice site into an interstitial site

• involves a cation-vacancy and a cation-interstitial

pair .

• there is no change in charge because the cation

maintains the same positive charge as an

interstitial.


d. Frenkel Defect

Cation Interstitial

Cation Vacancy


e. Nonstoichiometry

• may occur for some ceramic materials in which

two valence states exist for one of the ion types

(i.e. Fe2+ and Fe3+).

Fe3+

Fe2+

O2-

Fe2+ vacancy

(2) 2kT

QexpNN s

s


Qs = energy required for the formation of a Schottky defect




EQUILIBRIUM NUMBER OF CATION-

ANION VACANCY defect pair, Ns, for a

given quantity of ceramic material depends on

and increases with temperature according to:

(3) 2kT

QexpNN fr

fr


Qfr = energy required for the formation of a Frenkel defect




EQUILIBRIUM NUMBER OF CATION-

VACANCY/CATION-INTERSTITIAL

defect pair, Nfr, for a given quantity of

ceramic material depends on and increases

with temperature according to:

2kT

QexpNN s

s

Sample Problem:

Calculate the fraction of lattice sites that are Schottky

defects for cesium chloride at 534°C. Assume that the

energy of defect formation is 1.86 eV/cation-anion

vacancy pair.

LINE or ONE-DIMENSIONAL

DEFECTS

The linear size of these defects in one

dimension is considerably greater than the

atomic size, and is commensurable with it in

two other dimensions.

DISLOCATION

• Characterized by an open Burgers circuit

DISCLINATION

• Characterized by a closed Burgers circuit

DISLOCATION is a linear or one-dimensional defect

around which some of the atoms are misaligned.

• significant in plastic deformation by shear

stresses

• may increase the strength of a material when

present in large numbers

a. Edge Dislocation

• occurs when the edge of an extra plane of atoms,

a half plane, terminates within the crystal

Slip - process of plastic deformation by dislocation motion

Slip Plane – crystallographic plane where dislocation traverses

SLIP PLANE

b. Screw Dislocation

• may be thought of as being formed by a shear

stress that is applied to produce the distortion

SLIP PLANE

c. Mixed Dislocation

• a combination of a screw and edge dislocation

INTERFACIAL DEFECTS are

boundaries that have two

dimensions and normally separate

regions of the materials that have

different crystal structures.

a. External surfaces

b. Grain Boundaries

c. Twin Boundaries

a. External surfaces

• where the crystal structure terminates

• have a higher energy state than the atoms at the

interior positions since they are not bonded to the

maximum number of nearest neighbors

b. Grain Boundaries

• a boundary separating two small regions in

crystals having different crystallographic

orientations in polycrystalline materials

• areas of higher energy due to the presence of

interfacial or grain boundary energy

b. Grain BoundariesAngle of misalignment

High-angle grain

boundary

Low-angle grain

boundary

(a) Small crystallite

nuclei. (b) Growth

of the crystallites;

(c) grains having

irregular shapes

have formed. (d)

The grain structure

as it would appear

under the

microscope; dark

lines are the grain

boundaries. (Adapted from W. Rosenhain, An Introduction to

the Study of Physical Metallurgy, 2nd edition,

Constable & Company Ltd., London, 1915.)

OPTICAL

MICROSCOPY

Polished and etched

grains

Etching

characteristics

resulting surface

texture varying from

grain to grain

because of the

difference in

crystallographic

orientation

(Adapted from Callister , 2007, Materials

Science and Engineering – An Introduction.

MICROSTRUCTURE

• Geometric arrangement of grains (single

crystals) and phases in a material.

• Variables: relative amount, size, shape, and

distribution of the grains and phases present.

• Microscope (optical or electron) is

necessary to observe these structural

features

• Micrometer (10-6 m) up to nanometer (10-9

m) in dimensions

GRAIN SIZE is often determined when the

properties of a polycrystalline material are under

consideration.

TECHNIQUES

1. Intercept Method

2. Average number of grains per area

3. ASTM method

INTERCEPT METHOD

Straight lines are drawn through several pictographs

that show the grain structure

Grains intersected by each lines are counted

Average grain diameter, Dave, is computed as:

where

L = total length of the lines drawn,

nG = equivalent number of grains intersected,

M = linear magnification of the micrograph.

M

L

Mn

LD

G

ave

1

Magnification: 100X

the grain size can be quantified by the ASTM (American

Society for Testing and Materials) Grain Size Number, n

where N is the average number of grains per square inch

at a magnification of 100X

12 nN

1 43

65 87

2

ASTM COMPARATIVE METHOD

Grain size number, n, is determined by comparing the

pictograph of the structure obtained under 100X magnification

with a series of graded standard grain size pictographs.

The network charts above are essentially idealized

grain boundaries in a hypothetical hexagonal

network.

c. Twin Boundaries

special types of grain boundary across which there is a

specific mirror lattice symmetry

d. Stacking Faults

interruptions in the stacking sequence of close-packed

planes in the FCC structure

FCC HCP

e. Phase Boundaries

boundaries between phases in multiphase materials where

there is a sudden change in physical and/or chemical

characteristics

Bulk or Volume Defects

-three dimensional defects

-normally introduced during processing and fabrication of

the material

ex. pores, cracks, foreign inclusions and other phases-

crystalline imperfections

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