3 (g/cm )

Introduction to Materials Science and Engineering

Eun Soo Park

Office: 33‐313

Telephone: 880‐7221 Email: espark@snu.ac.kr

Office hours: by appointment

2020 Fall

09. 22. 2020

Contents for previous class

CHAPTER 4:

The Structure of Crystalline Solids

I. METALLIC CRYSTALS _ APF , CN …

- The face-centered cubic crystal structure (FCC) - The body-centered cubic crystal structure (BCC)

- The hexagonal closed-packed crystal structure (HCP)

II. Ceramic Crystal Structure (Chapter 12, 10^th edition)

- Ionic arrangement geometries

- Some common ceramic crystal structure/ Density computation - Silicate ceramics/ Carbon

I. Metallic crystal system

Planar Density of (100) Iron

At T < 912C iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

3 R 3 a  4

Adapted from Fig. 3.2(c), Callister 7e.

2D repeat unit

=

Planar Density =

a² 1 atoms

2D repeat unit

= nm²

atoms

12.1 m²

atoms

= 1.2 x 10¹⁹

1

2 3 R

3 area 4

2D repeat unit

Planar Density of (111) Iron

(111) plane

atoms in plane

atoms above plane atoms below plane

a

h 2

 3

a 2

1

= =

nm² atoms

7.0 m²

atoms 0.70 x 10¹⁹

3 ₂ 16 R

Planar Density =

atoms

2D repeat unit area

3

3 3 ²

2

3 R R 16

3

2 4 a 3 ah

2

area    

1 atom in plane/ unit surface cell

Materials-Bonding Classification

II. Ceramic Crystal Structures

금속-비금속 원소간 화합물

‘구운 것 (firing)’

7. Factors that Determine Ceramic Crystal Structure

1. Relative sizes of ions – Formation of stable structures:

--maximize the # of oppositely charged ion neighbors.

Adapted from Fig. 4.4, Callister & Rethwisch 9e.

- - - + -

unstable

- - - + -

stable

- -

- + -

stable 2. Maintenance of

Charge Neutrality :

--Net charge in ceramic should be zero.

--Reflected in chemical formula:

CaF2:

F- F-

anions

+

Am Xp

m, p values to achieve

charge neutrality ⁹

Chapter 4.6

• Coordination Number increases with

Coordination Number and Ionic Radii

2 rcation

ranion Coord.

Number

< 0.155 0.155 - 0.225 0.225 - 0.414 0.414 - 0.732 0.732 - 1.0

3 4 6 8

linear triangular tetrahedral octahedral

cubic

Adapted from Fig. 4.5, Callister & Rethwisch 9e.

Adapted from Fig. 4.6, Adapted from Fig. 4.7, Callister & Rethwisch 9e.

ZnS

(zinc blende)

NaCl (sodium chloride)

CsCl (cesium chloride)

rcation ranion

To form a stable structure, how many anions can surround around a cation?

Chapter 4.6

8. Ceramic Structure

Chapter 4.7, 4.8, 4.9

Densities of Material Classes

ρ_metals > ρ_ceramics > ρ_polymers Why?

ρ(g/cm )3

Graphite/

Ceramics/

Semicond Metals/

Alloys

Composites/

fibers Polymers

1 2 20 30

Based on data in Table B1, Callister

*GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers

in an epoxy matrix).

10

3 4 5

0.4 0.5

Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum

Graphite Silicon Glass -soda Concrete Si nitride Diamond Al oxide Zirconia

HDPE, PS PP, LDPE PC

PTFE PETPVC Silicone

Wood AFRE*

CFRE*

GFRE*

Glass fibers Carbon fibers Aramid fibers

Metals have...

• close-packing

(metallic bonding)

• often large atomic masses

Ceramics have...

• less dense packing

• often lighter elements

Polymers have...

• low packing density (often amorphous)

• lighter elements (C,H,O)

In general

10. Silicate Ceramics

a) Most common elements on earth are Si & O

• SiO₂ (silica) polymorphic forms are quartz, crystobalite, & tridymite

• The strong Si-O bonds lead to a high melting temperature (1710ºC) for this material

Si⁴⁺

O^2-

Figs. 4.10 & 4.11, Callister & Rethwisch

9e crystobalite

Chapter 4.11

Bonding of adjacent SiO₄^4- accomplished by the sharing of common corners, edges, or faces b) Silicates

Mg₂SiO₄ Ca₂MgSi₂O₇

Adapted from Fig.

4.13, Callister &

Rethwisch 9e.

Presence of cations such as Ca²⁺, Mg²⁺, & Al³⁺

1. maintain charge neutrality, and

규산염은 조암 광물 중 가장 많은 양을 차지하는 광물

• Quartz is crystalline SiO2:

• Basic Unit: Glass is noncrystalline (amorphous)

• Fused silica is SiO₂ to which no impurities have been added

• Other common glasses contain impurity ions such as Na⁺, Ca²⁺, Al³⁺, and B³⁺

(soda glass)

Adapted from Fig. 4.12, Callister & Rethwisch 9e.

c) Glass Structure

Si04 tetrahedron4-

Si4+

O2-

Si4+

Na+

O2-

d) Layered Silicates

• Layered silicates (e.g., clays, mica, talc) – SiO₄ tetrahedra connected

together to form 2-D plane

• A net negative charge is associated with each (Si₂O₅)^2- unit

• Negative charge balanced by adjacent plane rich in positively charged cations

Fig. 4.14, Callister &

Rethwisch 9e.

• Kaolinite clay alternates (Si₂O₅)^2- layer with Al₂(OH)₄²⁺

layer

d) Layered Silicates (cont)

Note: Adjacent sheets of this type are loosely bound to one another by van der Waal’s forces.

Fig. 4.15, Callister &

Rethwisch 9e.

17

11. Polymorphic Forms of Carbon

Diamond

– tetrahedral bonding of carbon

• hardest material known

• very high thermal conductivity

– large single crystals – gem stones

– small crystals – used to grind/cut other materials – diamond thin films

• hard surface coatings – used for cutting tools, medical

devices, etc.

Fig. 4.17, Callister &

Rethwisch 9e.

Chapter 4.12

Polymorphic Forms of Carbon (cont)

Graphite

– layered structure – parallel hexagonal arrays of carbon atoms

– weak van der Waal’s forces between layers

– planes slide easily over one another -- good lubricant

Fig. 4.18, Callister &

Rethwisch 9e.

Contents for today’s class

CHAPTER 4: The Structure of Crystalline Solids

I. METALLIC CRYSTALS _ APF , CN … - FCC, BCC, HCP

III. Polymer Structure (Chapter 14, 10^th edition)

- Hydrocarbon molecules / Polymer molecules - The chemistry of polymer molecules

- Molecular weight / shape / structure/ configurations - Copolymer/ Polymer crystals

IV. X-ray diffraction: Determination of crystal structures

- The diffraction phenomenon

II. Ceramic Crystal Structure (Chapter 12, 10^th edition)

- Ionic arrangement geometries

- Some common ceramic crystal structure/ Density computation - Silicate ceramics/ Carbon

CHAPTER 5: Polymer Structures

1. What is a Polymer?

Poly mer

many repeat unit

Adapted from Fig. 5.2, Callister & Rethwisch 9e.

C C C C C C H H H H H H

H H H H H H

Polyethylene (PE)

Cl

Cl Cl

C C C C C C H H

H

H H H H H H

Poly(vinyl chloride) (PVC)

H H

H H

H H

Polypropylene (PP)

C C C C C C

CH₃ H

H

CH₃ CH₃H

repeat unit

repeat unit

repeat unit

Chapter 5: Structures of Polymers

Polymer (= many mer) : a substance composed of molecules characterized by the multiple repetition of one or more species of atoms or groups of atoms (constitutional repeating units) linked to each other in amounts sufficient to provide a set of properties that do not vary markedly with the addition of one or a few of the constitutional repeating units

Ancient Polymers

• Originally natural polymers were used

– Wood – Rubber

– Cotton – Wool

– Leather – Silk

• Oldest known uses

– Rubber balls used by Incas

– Noah used pitch (a natural polymer)

for the ark

2. Polymer Composition

Most polymers are hydrocarbons – i.e., made up of H and C

1) Saturated hydrocarbons

– Each carbon singly bonded to four other atoms – Example:

• Ethane, C₂H₆

C C H

H H H

H H

Chapter 5.2 hydrocarbon molecules

2) Unsaturated Hydrocarbons

• Double & triple bonds somewhat unstable – can form new bonds

– Double bond found in ethylene or ethene - C₂H₄

– Triple bond found in acetylene or ethyne - C₂H₂ C C

H H

H H

C C H

H

3. Isomerism

• Isomerism_이성질체 (metal/ceramic에서 Allotrope과 유사)

– two compounds with same chemical formula can have quite different structures

for example: C₈H₁₈

• normal-octane

• 2,4-dimethylhexane

C C C C C C C C H

H H

H H

H H

H H

H H

H H

H H

H H

H = H₃C CH₂ CH₂ CH₂ CH₂ CH₂ CH₂ CH₃

H₃C CH CH₃

CH₂ CH CH₂

CH₃

H₃C CH( ₂ ) CH₃

6



4. Polymerization and Polymer Chemistry

• Free radical polymerization

• Initiator: example - benzoyl peroxide ^{(과산화 벤조일)}

C H

H

O O C H

H

C H

H 2 O

C C H H H H

monomer (ethylene)

R +

free radical

R C C H H

H H

initiation

R C C H H

H H

C C H H H H

+ R C C

H H

H H

C C H H H H

propagation dimer

= 2R propagation

Chapter 5.3, 5.4

반응개시제 혹은 촉매

반응개시제 혹은 촉매

Chemistry and Structure of Polyethylene

Adapted from Fig.

5.1, Callister &

Rethwisch 9e.

Note: polyethylene is a long-chain hydrocarbon

→ paraffin wax for candles is short polyethylene

Bulk or Commodity Polymers

(폴리테트라플루오로에틸렌_상품명: TeflonTM)

Bulk or Commodity Polymers (cont)

(메틸 메타크릴레이트)

(페놀-포름알데이드_베이크라이트)

Bulk or Commodity Polymers (cont)

(폴리 엑사메틸렌 아디파미드)

(폴리 에틸렌 테레프탈레이트)

(폴리 카보네이트)

5. MOLECULAR WEIGHT

• Molecular weight, M: Mass of a mole of chains.

Low M

high M

Not all chains in a polymer are of the same length

— i.e., there is a distribution of molecular weights

Chapter 5.5

x_i = number fraction of chains in size range i

MOLECULAR WEIGHT DISTRIBUTION

Fig. 5.4, Callister & Rethwisch 9e.

w_i = weight fraction of chains in size range i

M_i = mean (middle) molecular weight of size range i

33

6. Degree of Polymerization, DP

DP = average number of repeat units per chain

C C C C C C C C H

H H

H H

H H

H H

H H

H H

H H

H H

C C C C H

H H H

H H

H H

( ) H DP = 6

7. Molecular Shape of Polymer

Molecular Shape (or Conformation

) – chain bending and twisting are

possible by rotation of carbon atoms around their chain bonds

– note: not necessary to break chain bonds to alter molecular shape

5.5, Callister &

Rethwisch 9e.

Chain End-to-End Distance, r

Schematic representation of a single polymer chain molecule that has

numerous random kinks and coils produced by chain bond rotations.

Adapted from Fig. 5.7, Callister & Rethwisch 9e.

8. Molecular Structures for Polymers

Branched Cross-Linked Network Linear

secondary

bonding

Chapter 5.6

9. Molecular Configurations for Polymers

주 사슬에 1개 이상의 측면 원자 (side atom) 또는 원자군이 결합된 폴리머에서 측면 그룹의 배열 규칙성과 대칭성을 폴리머 물성에 중요한 영향을 끼침

a) Stereoisomerism 원자들이 동일한 배열이지만 공간상의 배열이 다른 상태

B E A

D

C C

D A

E B

mirror C C

R H H

H C C

H

H H

R

or C C

H H

H R

Stereoisomers are mirror images – can’t superimpose without breaking a bond

Chapter 5.6

(입체 이성체)

Tacticity

Tacticity –

C C H

H

H

R R

H H

H

C C

R H H

H

C C

R H H

H

C C

isotactic – all R groups on same side of chain

C C H

H

H R

C C H

H

H R C C

H H

H

R R

H H

H

C C

syndiotactic – R groups alternate sides

Tacticity (cont.)

atactic – R groups randomly positioned

C C H

H

H

R R

H H

H

C C

R H H

H

C C

R H H

H

C C

cis/trans Isomerism

C C H CH₃

CH₂ CH₂

C C CH₃

CH₂

CH₂ H

cis

cis-isoprene (natural rubber)

H atom and CH₃ group on same side of chain

trans

trans-isoprene (gutta percha)

H atom and CH₃ group on opposite sides of chain

b) Geometrical isomerism

(기하 이성체)

10. Copolymers

two or more monomers polymerized together

• random – A and B randomly positioned along chain

• alternating – A and B

alternate in polymer chain

• block – large blocks of A units alternate with large blocks of B units

• graft – chains of B units grafted onto A backbone

A – B –

random

block

graft

Fig. 5.9, Callister &

Rethwisch 9e.

alternating Chapter 5.10

(공중합체)

11. Polymer Crystals

• Crystalline regions

– thin platelets with chain folds at faces – Chain folded structure

Fig. 5.11, Callister &

Rethwisch 9e.

≈ 10 nm Chapter 5.11

Polymer Crystals (cont.)

Polymers rarely 100% crystalline

• Difficult for all regions of all chains to become aligned

• Degree of crystallinity

expressed as % crystallinity.

-- Some physical properties depend on % crystallinity.

-- Heat treating causes

crystalline regions to grow and % crystallinity to

increase.

Fig. 14.11, Callister 6e.(From H.W. Hayden,

W.G. Moffatt, and J. Wulff, The Structure and Properties of

crystalline region

amorphous region

Polymer Single Crystals

• Electron micrograph – multilayered single crystals (chain-folded layers) of polyethylene

• Single crystals – only for slow and carefully controlled growth rates

Fig. 5.10, Callister &

Rethwisch 9e.

[From A. Keller, R. H. Doremus, B.

W. Roberts, and D. Turnbull (Eds.), Growth and Perfection of Crystals.

General Electric Company and John Wiley & Sons, Inc., 1958, p.

498. Reprinted with permission of John Wiley & Sons, Inc.]

1 μm

Semicrystalline Polymers

Spherulite surface

Fig. 5.12, Callister & Rethwisch 9e.

• Some semicrystalline polymers form

spherulite structures

• Alternating chain-folded crystallites and

amorphous regions

• Spherulite structure for relatively rapid growth rates

(구정)

12. X-Ray Diffraction

• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.

• Can’t resolve spacings  λ

• Spacing is the distance between parallel planes of atoms.

Chapter 4.18

a) X-ray generation

• Intense electron beam hits the electrons in the outer shells to excite electrons to other levels (higher energy state)

• Excited electrons come down to lower level generating photon (X-ray)

b) X-Ray Production

 a voltage of 35,000 volts is applied between the cathode and anode target metal in the x-ray tube.

 when the electrons released from the W-filament strike the metal target (anode), x-rays are produced.

 Note: 98% of the energy is converted into heat.

• Radiations used in Crystallography

 average diffraction properties of X-rays, electrons, and neutron

X-rays Electrons Neutrons

1) Charge 0 -1 e 0

2) Rest mass 0 9.11 10^-31 kg 1.67 10^-27 kg

3) Energy 10 keV 100 keV 0.03 eV

4) Wavelength 1.5 Å 0.04 Å 1.2 Å

5) Bragg angles Large 1^o Large

6) Extingtion lenrth 10 m 0.03 m 100 m

7) Absorption length 100 m 1 m 5 cm

8) Width of rocking curve 5” 0.6^o 5”

9) Refractive index n=1+

n < 1

  -1  10^-5 n >1

  +1  10^-4 n  1, n  1

   1 10^-6 10) Atomic scattering

amplitudes f 10^-3 Å 10 Å 10^-4 Å

11) Dependence of f on

the atomic number Z  Z  Z^2/3 Nonmonotonic

12) Anomalous dispersion Common - Rare

1 eV 3 eV 500 eV

• Diffraction

 occurs as the wave encounters a series of regularly spaced obstacles which scatter the wave.

 these scattering centers (atoms), have spacing comparable in magnitude to the wavelength.

 diffraction results from specific phase relationships established between the waves scattered by the scattering atoms (centers).

c) X-Rays to Determine Crystal Structure

X-ray intensity (from detector)

θ

d  n λ 2 sin θc

Measurement of critical angle, θ_c,

allows computation of planar spacing, d.

• Incoming X-rays diffract from crystal planes.

Adapted from Fig. 4.29, Callister & Rethwisch 9e.

reflections must be in phase for a detectable signal

spacing between planes d

θ θ ^λ

extra distance travelled by wave “2”

Diffraction geometry: Bragg’s Law

d  B

 ^{ 2 sin}

ML+LN = 2*MN=2*d*sin

d



Structure Determination (X-Ray)

• Reflection (Extinction) Rule

diffraction line h²+k²+l² (cubic) sc bcc fcc 100 1

110 2 2

111 3 3 200 4 4 4 210 5

211 6 6

2 2 2

2 2 2 2

2 sin

sin d d a

h k l

h k l

 





  

 

(interplanar spacing)

X-ray diffractometer

Bragg-Brentano Geometry

slit

slit

2 

slit

Lattice planes

Determination of crystal structure I

(110)

(200)

(211)

z

x

a b y

c

Diffraction angle 2θ

Diffraction pattern for polycrystalline α-iron (BCC)

Intensity (relative)

z

x

a b y

c z

x

a b y

c

 Problem: Determine the d-spacing for the (111) plane, the lattice parameter a₀ and the atomic radius for Al.

 

nm a .

r r a

; nm .

a

nm x

. nm

x .

a

l k

h d

l a k

h d a

nm sin .

nm .

d sin sin

d

) (

for

o o

o o

hkl o

o hkl

o o

o

144 4 0

2 2

410 4 0

3 237

0 1

1 1

237 0

237 19 0

2

1542 0

2 2

111 19

38 2

2 2

2

2 2

2 2

2 2

111

 



 





















 

 

 

 



 























Determination of crystal structure II

Summary

• Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor

are the same for both FCC and HCP crystal structures.

• Some materials can have more than one crystal structure.

This is referred to as polymorphism (or allotropy).

• Ceramic crystal structures are based on:

-- maintaining charge neutrality -- cation-anion radii ratios.

• Interatomic bonding in ceramics is ionic and/or covalent.

• Polymer (= many mer) is a substance composed of molecules characterized by the multiple repetition of one or more species of atoms or groups of atoms (constitutional repeating units) linked to each other in amounts sufficient to provide a set of properties. Most polymers are hydrocarbons – i.e., made up of H and C.

Introduction to Materials Science and Engineering

Eun Soo Park

Office: 33-313

Telephone: 880-7221 Email: espark@snu.ac.kr

Office hours: by appointment

2020 Fall

09. 24. 2020

• Metallic crystal structures are SC, FCC, BCC, and HCP.

Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures.

• Some materials can have more than one crystal structure.

This is referred to as polymorphism (or allotropy).

• Ceramic crystal structures are based on:

-- maintaining charge neutrality -- cation-anion radii ratios.

Interatomic bonding in ceramics is ionic and/or covalent.

• Polymer (= many mer) is a substance composed of molecules characterized by the multiple repetition of one or more species of atoms or groups of atoms (constitutional repeating units) linked to each other in amounts sufficient to provide a set of properties. Most polymers are hydrocarbons – i.e., made up of H and C.

CHAPTER 3+ The Structure of Crystalline Solids

Contents for previous class

a. What is a Polymer?

Poly mer

many repeat unit

C C C C C C H H H H H H

H H H H H H

Polyethylene (PE)

Cl

Cl Cl

C C C C C C H H

H

H H H H H H

Poly(vinyl chloride) (PVC)

H H

H H

H H

Polypropylene (PP)

C C C C C C CH₃ H

H

CH₃ CH₃H

repeat unit

repeat unit

repeat unit

Chapter 14: Structures of Polymers

Polymer (= many mer) : a substance composed of molecules characterized by the multiple repetition of one or more species of atoms or groups of atoms (constitutional repeating units) linked to each other in amounts sufficient to provide a set of properties that do not vary markedly with the addition of one or a few of the constitutional repeating units

b. Molecular Shape of Polymer

Molecular Shape (or Conformation

) – chain bending and twisting are

possible by rotation of carbon atoms around their chain bonds

– note: not necessary to break chain bonds to alter molecular shape

5.5, Callister &

Rethwisch 9e.

Chain End-to-End Distance, r

Schematic representation of a single polymer chain molecule that has

numerous random kinks and coils produced by chain bond rotations.

Adapted from Fig. 5.7, Callister & Rethwisch 9e.

c. Molecular Structures for Polymers

Branched Cross-Linked Network Linear

secondary

bonding

Chapter 5.6

d. Molecular Configurations for Polymers

주 사슬에 1개 이상의 측면 원자 (side atom) 또는 원자군이 결합된 폴리머에서 측면 그룹의 배열 규칙성과 대칭성을 폴리머 물성에 중요한 영향을 끼침

a) Stereoisomerism 원자들이 동일한 배열이지만 공간상의 배열이 다른 상태

E B

A

D

C C

D A

B E

mirror plane C C

R H H

H

C C H

H H

R

or C C H H

H R

Stereoisomers are mirror images – can’t superimpose without breaking a bond

Chapter 5.6

(입체 이성체)

Tacticity

Tacticity –

C C H

H

H

R R

H

H H

C C

R H

H H

C C

R H

H H

C C

isotactic – all R groups on same side of chain

C C H

H

H

R

C C H

H

H

R C C

H H

H

R R

H H

H

C C

syndiotactic – R groups alternate sides

Tacticity (cont.)

atactic – R groups randomly positioned

C C H

H

H

R R

H

H H

C C

R

H H

H

C C

R H

H H

C C

cis/trans Isomerism

C C H CH₃

CH₂ CH₂

C C CH₃

CH₂

CH₂ H

cis

cis-isoprene (natural rubber)

H atom and CH₃ group on same side of chain

trans

trans-isoprene (gutta percha)

H atom and CH₃ group on opposite sides of chain

b) Geometrical isomerism

(기하 이성체)

e. Copolymers

two or more monomers polymerized together

• random – A and B randomly positioned along chain

• alternating – A and B

alternate in polymer chain

• block – large blocks of A units alternate with large blocks of B units

• graft – chains of B units grafted onto A backbone

A – B –

random

block

graft

Fig. 5.9, Callister &

Rethwisch 9e.

alternating Chapter 5.10

(공중합체)

(접지)

Polymer Crystals (cont.)

Polymers rarely 100% crystalline

• Difficult for all regions of all chains to become aligned

• Degree of crystallinity

expressed as % crystallinity. -- Some physical properties

depend on % crystallinity.

-- Heat treating causes

crystalline regions to grow and % crystallinity to

increase.

Fig. 14.11, Callister 6e.(From H.W. Hayden,

W.G. Moffatt, and J. Wulff, The Structure and Properties of

crystalline region

amorphous region

f. X-Ray Diffraction

• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.

• Can’t resolve spacings < λ

• Spacing is the distance between parallel planes of atoms.

Chapter 4.18

X-Rays to Determine Crystal Structure

X-ray intensity (from detector)

θ d = nλ

2 sinθc Measurement of

critical angle, θ^c,

allows computation of planar spacing, d.

• Incoming X-rays diffract from crystal planes.

Adapted from Fig. 4.29, Callister & Rethwisch 9e.

reflections must be in phase for a detectable signal

spacing between planes d

θ θ ^λ

extra distance travelled by wave “2”

Structure Determination (X-Ray)

• Reflection (Extinction) Rule

diffraction line h²+k²+l² (cubic) sc bcc fcc 100 1

110 2 2

111 3 3 200 4 4 4 210 5

211 6 6

220 8 8 8 221 9

2 2 2

2 2 2 2

2 sin

sin d d a

h k l h k l

θ λ

θ

=

= + +

+ +



(interplanar spacing)

Determination of crystal structure I

(110)

(200)

(211)

z

x

a b y

c

Diffraction angle 2θ

Diffraction pattern for polycrystalline α-iron (BCC)

Intensity (relative)

z

x

a b y

c z

x

a b y

c

 Problem: Determine the d-spacing for the (111) plane, the lattice parameter a₀ and the atomic radius for Al.

( )

nm a .

r r a

; nm .

a

nm x

. nm

x .

a

l k

h d

a l

k h

d a

nm sin .

nm .

d sin sin

d

) (

for

o o

o o

hkl o

o hkl

o o

o

144 4 2 0

2 410 4

0

3 237

0 1

1 1

237 0

237 19 0

21542 2 0

2

111 19

38 2

2 2

2

2 2

2 2

2 2

111

⋅ =

=

⋅ ⇒

=

=

= +

+

=

+ +

⋅

= + ⇒

= +

⋅ = θ =

⋅

= λ

⇒ θ

⋅

⋅

= λ

= θ

⇒

= θ

Determination of crystal structure II

• Atomic structure & Interatomic bonding

• Crystalline Solids’ structure

• Microstructure : 결함구조

• Macrostructure : 외형

• Mechanical

• Electrical

• Magnetic

Materials Science and Engineering

Structure

Properties

Processing

Theory &

• Sintering

• Heat treatment

• Thin Film

• Melt process

• Mechanical

Chapter 4: Imperfections in Solids

19

Contents for today’s class

I. Point defects

- Point defects in metals/ceramics/polymers, impurities in solids

II. Dislocations-Linear defects

- Edge/ Screw/ Mix dislocation

III. Interfacial defects

- External surfaces/ Grain boundaries/ Phase boundaries (stacking fault)/ Twin boundaries/ domain boundaries

IV. Bulk or Volume defects

- pores/ cracks/ foreign inclusions, and other phases

V. Microscopic Examination

- Basic concepts of microscopy

- Microscopic techniques : Optical microscopy (Grain-size determination) / Electron microscopy/ Scanning probe microscopy

• Perfect and extensive ordering does not exist.

• Crystalline imperfections have a profound effect on materials behavior

• If we can control imperfections, it is possible to produce

– stronger metals and alloys – more powerful magnets

– improved transistors and solar cells – glassware of striking colors

Chapter 4: Imperfections in Solids

0-dimensional Point defects

Vacancy atoms Interstitial atoms Substitutional atoms 1-dimensional Line defects Dislocations

2-dimensional Planar (Area) defects

Surface

Grain boundary Stacking fault

Types of Imperfections

• What is a point defect?

I. Point Defects

• Vacancies:

-vacant atomic sites in a structure.

• Self-Interstitials:

-"extra" atoms positioned between atomic sites.

1. Point Defects in Metals

Vacancy

distortion of planes

self-

interstitial

distortion of planes

Point defect thermodynamically predictable!

Thermodynamic arguments not only suggest that point defects may be present, but actually demand their presence and imply that it is impossible to create a stable single crystal without point defects.

2. Equilibrium Concentration:

Point Defects

1) Vacancies increase the internal energy of crystalline metal due to broken bonds formation.

2) Vacancies increase entropy because they change

the thermal vibration frequency and also the configurational entropy.

• Total entropy change is thus

V V

H H X

∆ ≅ ∆

= _A + ∆ = _A + ∆ _{V V} − ∆ _{V V} + _{V V} + − _V − _V G G G G H X T S X RT{X ln X (1 X )ln(1 X )}

The molar free energy of the crystal containing X_v mol of vacancies

∆ = ∆S S X_{V V} − R{X ln X_{V V} + −(1 X )ln(1 X )}_V − _V

With this information,

estimate the equilibrium vacancy concentration.

1.5.8. Equilibrium Vacancy Concentration ∆G = ∆H −T∆S

(Here, vacancy-vacancy interactions are ignored.)

Small change due to changes in the vibrational frequencies “Largest contribution”

G of the alloy will depend on the concentration of vacancies and will be that which gives the minimum free energy.

a) 평형에 미치는 공공의 영향

25

ordering disordering

• In practice, ∆H_V is of the order of 1 eV per atom and X_V^e

reaches a value of about 10^-4~10^-3

=

 

  =

 ^V X_V X^e_V

dG 0

dX

∆H_V − ∆T S_V + RT ln X^e_V = 0

∆ −∆

= ⋅

∆ = ∆ − ∆

= −∆

e V V

V

V V V

e V

V

S H

X exp exp

R RT

putting G H T S X exp G

RT at equilibrium

Fig. 1.37 Equilibrium vacancy concentration.

A constant ~3, independent of T Rapidly increases with increasing T

Equilibrium concentration will be that which gives the minimum free energy.

Boltzmann's constant (1.38 x 10

^-23

^-5

N

N = exp − Q

k T

No. of defects No. of potential

defect sites

Activation energy

Temperature

Each lattice site is a potential vacancy site

• Equilibrium concentration varies with temperature!

Equilibrium Concentration: Point Defects

• We can get Q_v from an experiment.

N

N = exp - Q

k T

a) Measuring Activation Energy

• Measure this...

Nv N

T

exponential dependence!

defect concentration

• Replot it...

1/T N

lnNv

-Qv /k slope

• Find the equil. # of vacancies in 1 m³ of Cu at 1000°C.

• Given:

A_Cu = 63.5 g/mol ρ = 8.4 g/cm³

Qv = 0.9 eV/atom N^A = 6.02 x 10²³ atoms/mol

b) Estimating Vacancy Concentration

For 1 m³ , N = N

A

ACu

ρ x 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

= 2.7 x 10

8.62 x 10^-5 eV/atom-K 0.9 eV/atom

1273 K

N

N = exp - Q

k T

T of Cu= 1085 °C 25 °C 의 N = 4.9 x 10¹³ vacancies 29

c) 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

Two outcomes if impurity (B) added to host (A):

• Solid solution of B in A (i.e., random dist. of point defects)

• Solid solution of B in A plus particles of a new

phase (usually for a larger amount of B) _ Precipitation!

OR

Substitutional solid soln.

(e.g., Cu in Ni)

Interstitial solid soln.

(e.g., C in Fe)

Second phase particle -- different composition -- often different structure.

3. Imperfections in Metals (i)

Solubility

• Unlimited Solubility

– Hume Rothery’ Conditions

• Similar Size

• Same Crystal Structure

• Same Valance

• Similar Electronegativity

– Implies single phase

• Limited Solubility

– Implies multiple phases

• No Solubility

- oil and water region