msne 301 final exam - rice university

Chapter 16 -

MSNE 301 Final Exam

Sylvia Jung

Chapter 16 -

Formatting of the Exam

• 11 Questions – 1st question is 20 T/F questions without explanations – 2 questions on Crystal structures – 1 question on Defects– 2 questions on the Phase-Diagram – 1 question on Mechanical properties – 1 question on Diffusion – 1 question on Electrical Properties – 1 question on Electrochemical properties

Chapter 16 -

Tips on T/F Questions

• Skim through all the topics, even the topics that aren’t mentioned in Slide 2

Chp 2 - Types of Bonding & Characterization

Chp 3 - Unit Cells: Smallest repeating unit of a lattice

BCC FCC

EX) iron, chromium, tungsten, and niobium

EX) aluminium, copper, gold and silver

Chp 3 - HCP

● 6 net atoms (3 from the center, 1 from the basal planes, 2 from the vertices of the hexagon)

● Coordination number = 12● ABAB stacking pattern

Chp 3 - Unit Cell Structure

Chp 3 - Notation for Directions and Planes

Direction: [ ]Family of directions: < > Plane: ( )Family of Planes: { } Points:Whole NumbersNegative number notation - above #

Chp 3 - Closed-pack Directions and Directions

Chp 3 - APF and Theoretical Density Calculations

APF = Volume of atoms in unit cell/ Volume of unit cell

X-Ray DiffractionBragg’s Law

Interplanar Spacing Equation

Diffraction peak Indexing

● Diffraction peaks for BCC: (h+k+l) is even● First five peaks of BCC: (110), (200), (211), (220), (310), (222)

● Diffraction peaks for FCC: h,k,l are all even or all odd● First five peaks of FCC: (111), (200), (220), (311), (222), (400)

Chp 5 - Defects: Vacancies Vs. Self-interstitials

Chp 5 - Schottky and Frenkel Defects

0D Defects - Point Defects

Point Defects● Vacancies vs. self-interstitials● Solid solutions

● Substitutional● Substitutional solute or impurity atoms replace/ substitute host

atoms● Hume-Rothery rules for solubility (Atomic size factor, crystal

structure, electronegativity, valences)● Interstitial

● Impurity atoms fill voids or interstices among host atoms

1D Defects - Line Defects

Line Defects● Edge dislocation

● Linear defect, centered around line defined along end of extra half-plane of atoms (dislocation line)

● Screw dislocation● Helical path is traced around the linear defect (dislocation line) by the atomic planes

in the crystal lattice● Mixed dislocation

● Exhibits components of both screw and edge dislocations● Burgers vector

● Magnitude of direction of lattice distortion associated with dislocation● Edge dislocation: dislocation line perpendicular to Burgers vector● Screw dislocation: dislocation line parallel to Burgers vector

Chapter 6 - 16

DiffusionDiffusion - Mass transport by atomic motion

• Interdiffusion - diffusion of atoms of one material into another material

• Self-diffusion – atomic migration in a pure metal

• Diffusion Mechanisms- Gases & Liquids – random (Brownian) motion- Solids – vacancy diffusion and interstitial

diffusion

Chapter 6 - 17

Diffusion Mechanism I

• atoms and vacancies exchange positions • applies to host and substitutional impurity atoms • diffusion rate depends on: -- number of vacancies -- activation energy to exchange.

increasing elapsed time

Vacancy Diffusion

Chapter 6 - 18

Diffusion Mechanism II

• Small, interstitial atoms move from one interstitial position to an adjacent one

More rapid than vacancy diffusionFig. 6.3 (b), Callister & Rethwisch 5e.

Interstitial Diffusion

Chapter 6 - 19

Rate of Diffusion• Diffusion is a time-dependent process.• Rate of Diffusion - expressed as diffusion flux, J

M =mass

diffusedtime

• Measured experimentally– Use thin sheet (or membrane) – cross-sectional area A– Impose concentration gradient across sheet– Measure mass of diffusing species (M) that passes through

the sheet over time period (t)

Chapter 6 - 20

Steady-State Diffusion

Fick’s first law of diffusionC1

C2

x

C1

C2

x1 x2

D = diffusion coefficient

Rate of diffusion (or flux) independent of timeFlux (J) proportional to concentration gradient:

C = concentrationx = diffusion direction

C

Chapter 7 - 21

• Simple tension:

Δl = Fl oEAo

Δd = - ν FdoEAo

• Deflection is dependent on material, geometric, and loading parameters.• Materials with large elastic moduli deform less

Useful Linear Elastic Relationships

Ao

Adapted from Fig. 7.9, Callister & Rethwisch 5e.

Chapter 7 - 22

Linear Elastic Properties

• Hooke's Law:σ = E ε

σ

Linear- elastic

• Modulus of Elasticity, E: (also known as Young's modulus)

E

ε

• Elastic deformation is nonpermanent and reversible! – generally valid at small deformations – linear stress strain curve

compression

tension

Units:E: [GPa] or [psi]1 GPa = 109 Pa

Chapter 7 - 23

Plastic Deformation

• Stress-strain plot for simple tension test:

stress, σ

strain, ε

Stressed into Plastic Region,Elastic + Plastic

εp

plastic strain

ElasticDeformation

Adapted from Fig. 7.10(a),Callister & Rethwisch 5e.

Stress Removed, Plastic Deformation Remains

• Plastic Deformation is permanent and nonrecoverable

Chapter 7 - 24

• Yield strength = stress at which noticeable plastic deformation has occurred

when εp = 0.002

Yield Strength

σy = yield strength

Note: for 5 cm sample

ε = 0.002 = Δz/z

Δz = 0.01 cm

Adapted from Fig. 6.10 (a),Callister & Rethwisch 9e.

σ (stress)

ε (strain)

σy

εp = 0.002

• Transition from elastic to plastic deformation is gradual

Chapter 7 - 25

Tensile Strength

• Metals: Maximum on stress-strain curve appears at the onset of noticeable necking


σy

strain

Typical response of a metal

Fracture strength

Neck – acts as stress concentrator

eng

inee

ring

TS s

tress

engineering strain

• Tensile strength (TS) = maximum stress on engineering stress-strain curve.

Chapter 7 - 26

• Ductility = amount of plastic deformation at failure:• Specification of ductility -- Percent elongation:

-- Percent reduction in area:

Ductility

lfAo Af

lo


tensile strain, ε

tensile stress, σ

low ductility

high ductility

Chapter 7 - 27

• Toughness of a material is expressed in several contexts • For this chapter, toughness = amount of energy absorbed before fracture • Approximate by area under the stress-strain curve—units of energy per unit volume

Toughness

Brittle fracture: small toughnessDuctile fracture: large toughness

very small toughness (unreinforced polymers)

tensile strain, ε

tensile stress, σ

small toughness (ceramics)

large toughness (metals)

Chapter 7 - 28

Mechanical PropertiesCeramic materials are more brittle than metals.

Why is this so?• Consider mechanism of deformation

– In crystalline, by dislocation motion– In highly ionic solids, dislocation motion is difficult

• few slip systems• resistance to motion of ions of like charge (e.g., anions) past

one another

Chapter 8 - 29

Motion of Edge and Screw Dislocations

• Direction of edge disl. line ( ) motion—in direction of applied shear stress τ.

Edge dislocation

Screw dislocation

Fig. 8.2, Callister & Rethwisch 5e.

• Direction of screw disl. line ( ) motion—perpendicular to direction of applied shear stress.

Chapter 8 - 30

Slip System—Combination of slip plane and slip direction – Slip Plane

• Crystallographic plane on which slip occurs most easily

• Plane with high planar density

– Slip Direction • Crystallographic direction along which slip occurs

most easily• Direction with high linear density

Slip Systems

Chapter 8 -

• For FCC crystal structure – slip system is– Dislocation motion on planes– Dislocation motion in directions– A total of 12 independent slip systems for FCC

31

Slip Systems (cont.)


direction

plane

• For BCC and HCP— other slip systems

Chapter 9 -

wt% Ni20 40 60 80 100

0

1000

1100

1200

1300

1400

1500

1600

T(°C)

L (liquid)

α

(FCC solidsolution)

L + αliquidussolidus

Cu-Niphasediagra

m

32

Phase Diagrams:Determination of phase(s) present

• Rule 1: If we know T and Co, then we know: -- which phase(s) is (are) present.

• Examples:A(1100°C, 60 wt% Ni): 1 phase: α

B

(1250°C, 35 wt% Ni): 2 phases: L + α

B

(125

0ºC

,35)

A(1100ºC,60)

Fig. 10.3(a), Callister & Rethwisch 5e. (Adapted from Phase Diagrams of BinaryNickel Alloys, P. Nash, Editor, 1991. Reprintedby permission of ASM International, MaterialsPark, OH.)

Chapter 9 - 33

• Rule 3: If we know T and C0, then can determine: -- the weight fraction of each phase.• Examples:

At TA: Only Liquid (L) present

WL = 1.00, Wα = 0

At TD:

Only Solid (α ) present

WL = 0, W

α = 1.00

Phase Diagrams:Determination of phase weight fractions

wt% Ni20

1200

1300

T(°C)

L (liquid)

α(solid)L + α

liquidus

solidus

30 40 50

L + α

Cu-Ni system

TAA

35C0

32CL

BTB

DTD

tie line

4Cα

3

R S

At TB:

Both α

and L present

= 0.27

WL= S

R +S

Wα= R

R +S

Consider C0 = 35 wt% Ni

Fig. 10.3(b), Callister & Rethwisch 5e. (Adapted from Phase Diagrams of BinaryNickel Alloys, P. Nash, Editor, 1991. Reprintedby permission of ASM International, MaterialsPark, OH.)

Chapter 9 - 34

• Tie line – connects the phases in equilibrium with each other – also sometimes called an isotherm

The Lever Rule

What fraction of each phase? Think of the tie line as a lever (teeter-totter)

ML Mα

R S

wt% Ni20

1200

1300

T(°C)

L (liquid)

α(solid)L + α

liquidus

solidus

30 40 50

L + αB

TB

tie line

C0CL Cα

SR

Adapted from Fig. 10.3(b), Callister & Rethwisch 5e.

Chapter 9 - 35

L+αL+β

α + β

200

T(°C)

18.3

C, wt% Sn20 60 80 1000

300

100

L (liquid)

α 183°C

61.9 97.8β

• For a 40 wt% Sn-60 wt% Pb alloy at 150°C, determine: -- the phases present Pb-Sn

system

EX 1: Pb-Sn Eutectic System

Answer: α + β-- the phase compositions

-- the relative amount of each phase

150

40C0

11Cα

99Cβ

SR

Answer: Cα = 11 wt% SnCβ = 99 wt% Sn

Wα =Cβ - C0Cβ - Cα

= 99 - 4099 - 11 = 59

88 = 0.67

SR+S =

Wβ

=C0 - CαCβ - Cα

=RR+S

= 2988

= 0.33= 40 - 1199 - 11

Answer:

Fig. 10.8, Callister & Rethwisch 5e. [Adapted from Binary Alloy Phase Diagrams, 2nd edition, Vol. 3, T. B. Massalski (Editor-in-Chief), 1990. Reprinted by permission of ASM International, Materials Park, OH.]

Chapter 9 - 36

• For alloys for which 2 wt% Sn < C0 < 18.3 wt% Sn• Result: at temperatures in α + β range -- polycrystalline with α grains and small β-phase particles


Microstructural Developments in Eutectic Systems II

Pb-Snsystem

L + α

200

T(°C)

C, wt% Sn10

18.3

200C0

300

100

L

α

30

α + β

400

(sol. limit at TE)

TE

2(sol. limit at Troom)

Lα

L: C0 wt% Sn

αβ

α: C0 wt% Sn

Chapter 9 - 37

• For alloy of composition C0 = CE • Result: Eutectic microstructure (lamellar structure) -- alternating layers (lamellae) of α and β phases.


Microstructural Developments in Eutectic Systems III

Fig. 10.14, Callister & Rethwisch 5e. (From Metals Handbook, 9th edition, Vol. 9,Metallography and Microstructures, 1985.Reproduced by permission of ASM International, Materials Park, OH.)

160 μm

Micrograph of Pb-Sn eutectic microstructure

Pb-Snsystem

L + β

α + β

200

T(°C)

C, wt% Sn20 60 80 1000

300

100

L

α

βL+ α

183°C

40

TE

18.3

α: 18.3 wt%Sn

97.8

β: 97.8 wt% Sn

CE61.9

L: C0 wt% Sn

Chapter 9 - 38

L+αL+β

α + β

200

C, wt% Sn20 60 80 1000

300

100

L

α

βTE

40

(Pb-Sn System)

Hypoeutectic & Hypereutectic


160 μmeutectic micro-constituent


hypereutectic: (illustration only)

β

βββ

β

β

Adapted from Fig. 10.17, Callister & Rethwisch 5e. (Illustration only)

(Figs. 9.14 and 9.17 from Metals Handbook, 9th ed., Vol. 9, Metallography and Microstructures, 1985.Reproduced by permission of ASM International,Materials Park, OH.)

175 μm

α

α

α

ααα

hypoeutectic: C0 = 50 wt% Sn


T(°C)

61.9eutectic

eutectic: C0 = 61.9 wt% Sn

Chapter 9 - 39

• Eutectoid – one solid phase transforms to two other solid phasesS2 S1+S3 γ α + Fe3C (For Fe-C, 727°C, 0.76 wt% C)

intermetallic compound - cementite

coolheat

Eutectic, Eutectoid, & Peritectic• Eutectic - liquid transforms to two solid phases

L α + β (For Pb-Sn, 183°C, 61.9 wt% Sn) coolheat

coolheat

• Peritectic - liquid and one solid phase transform to a second solid phase S1 + L S2

δ + L γ (For Fe-C, 1493°C, 0.16 wt% C)

Chapter 9 - 40

Eutectoid & PeritecticCu-Zn Phase diagram


Eutectoid transformation δ γ + ε

Peritectic transformation γ + L δ

Chapter 9 - 41

Iron-Carbon (Fe-C) Phase Diagram• 2 important points

- Eutectoid (B):

γ

⇒

α+

Fe3C

- Eutectic (A):

L

⇒

γ+

Fe3C


Fe3C

(c

emen

tite)

160014001200100080060040001234566.7Lγ

(austenite)

γ

+L

γ

+Fe3C

α

+Fe3C

α+γδ(Fe)

C, wt% C

1148°C

T(°C)

α

727°C = Teutectoid

4.30Result: Pearlite = alternating

layers of α and Fe3C phases

120 μm

Fig. 10.31, Callister & Rethwisch 5e. (From Metals Handbook, Vol. 9, 9th ed.,Metallography and Microstructures, 1985.Reproduced by permission of ASM International, Materials Park, OH.)

0.76

B

γγγγ

A L+Fe3C

Fe3C (cementite-hard)

α

(ferrite-soft)

Chapter 9 - 42Fe

3C (c

emen

tite)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

γ (austenite)

γ+L

γ + Fe3C

α + Fe3C

L+Fe3C

δ

(Fe) C, wt% C

1148°C

T(°C)

α727°C

(Fe-C System)

C0

0.76

Hypoeutectoid Steel

Adapted from Figs. 10.28 and 10.33, Callister & Rethwisch 5e. [Adapted from Binary Alloy Phase Diagrams, 2nd edition, Vol. 1, T. B. Massalski (Editor-in-Chief), 1990. Reprinted by permission of ASM International, Materials Park, OH.]

Adapted from Fig. 10.34, Callister & Rethwisch 5e. (Photomicrograph courtesy of Republic Steel Corporation.)

proeutectoid ferritepearlite

100 μm Hypoeutectoidsteel

α

pearlite

γγ γ

γααα

γγγ γ

γ γγγ

Chapter 9 -Fe

3C (c

emen

tite)

1600

1400

1200

1000

800

600

4000 1 2 3 4 5 6 6.7

L

γ (austenite)

γ+L

γ + Fe3C

α + Fe3C

L+Fe3C

δ

(Fe) C, wt% C

1148°C

T(°C)

α727°C

(Fe-C System)

C0

43

Hypereutectoid Steel

0.76

C0

Fe3C

γγγ γ

γγγ γ

γγγ γ

Adapted from Fig. 10.37, Callister & Rethwisch 5e. (Copyright 1971 by United States Steel Corporation.)

proeutectoid Fe3C

60 μmHypereutectoid steel

pearlite

pearlite

Adapted from Figs. 10.28 and 10.36, Callister & Rethwisch 5e. [Adapted from Binary Alloy Phase Diagrams, 2nd edition, Vol. 1, T. B. Massalski (Editor-in-Chief), 1990. Reprinted by permission of ASM International, Materials Park, OH.]

Chapter 9 - 44

Electrical Properties• Which will have the greater resistance?

• Analogous to flow of water in a pipe• Resistance depends on sample geometry and

size.

D

2D

2

Chapter 9 - 45

Electrical Properties• Which will have the greater resistance?

• Analogous to flow of water in a pipe• Resistance depends on sample geometry and

size.

D

2D

2

Chapter 9 - 46

Extrinsic Semiconductors: Conductivity vs. Temperature

• Data for Doped Silicon: -- σ increases doping -- reason: imperfection sites lower the activation energy to produce mobile electrons.

• Comparison: intrinsic vs extrinsic conduction... -- extrinsic doping level: 1021/m3 of a n-type donor impurity (such as P). -- for T < 100 K: "freeze-out“, thermal energy insufficient to excite electrons. -- for 150 K < T < 450 K: "extrinsic" -- for T >> 450 K: "intrinsic"

Adapted from Fig. 12.17, Callister & Rethwisch 5e. (From S. M. Sze, Semiconductor Devices, Physics and Technology. Copyright © 1985 by Bell Telephone Laboratories, Inc. Reprinted by permission of John Wiley & Sons, Inc.)

Conduction electron concentration (1021/m3)T(K)

600400200

00123freeze-outextrinsicintrinsicdopedundoped

Chapter 9 - 47

DefinitionsFurther definitions

J = σ % <= another way to state Ohm’s law

J ≡ current density

% ≡ electric field potential = V/

Electron flux conductivity voltage gradient

J = σ (V/ )

Chapter 9 - 48

What is the minimum diameter (D) of the wire so that V < 1.5 V?

Example: Conductivity Problem

Cu wire I = 2.5 A- +

V

Solve to get D > 1.87 mm

< 1.5 V

2.5 A

6.07 x 107 (Ohm-m)-1

100 m

Chapter 9 - 49

Conduction & Electron Transport• Metals (Conductors):-- for metals empty energy states are adjacent to filled states.

-- two types of band structures for metals

-- thermal energy excites electrons into empty higher energy states.

- partially filled band - empty band that overlaps filled band

filled band

Energy

partly filled band

empty band GA

P

fille

d st

ates

Partially filled band

Energy

filled band

filled band

empty band

fille

d st

ates

Overlapping bands

Chapter 9 - 50

Energy Band Structures: Insulators & Semiconductors

• Insulators: -- wide band gap (> 2 eV) -- few electrons excited across band gap

Energy

filled band

filled valence band

fille

d st

ates

GAP

emptyband

conduction

• Semiconductors: -- narrow band gap (< 2 eV) -- more electrons excited across band gap

Energy

filled band

filled valence band

fille

d st

ates

GAP

?

emptyband

conduction

Chapter 9 - 51

Charge Carriers in Insulators and Semiconductors

Two types of electronic charge carriers:

Free Electron – negative charge – in conduction band

Hole – positive charge

– vacant electron state in the valence band

Fig. 12.6 (b), Callister & Rethwisch 5e.

Move at different speeds - drift velocities

Chapter 9 - 52

Intrinsic Semiconduction in Terms of Electron and Hole Migration


electric field electric field electric field

• Electrical Conductivity given by:

# electrons/m3 electron mobility

# holes/m3

hole mobility

• Concept of electrons and holes:

+-

electron hole pair creation

+-

no applied

applied

valence electron Si atom

applied

electron hole pair migration

Chapter 9 - 53

Number of Charge CarriersIntrinsic Conductivity

For GaAs ni = 4.8 x 1024 m-3

For Si ni = 1.3 x 1016 m-3

• Ex: GaAs

• for intrinsic semiconductor n = p = ni∴ σ = ni|e|(μe + μh)

Chapter 9 - 54

• Intrinsic: -- case for pure Si -- # electrons = # holes (n = p)• Extrinsic: -- electrical behavior is determined by presence of impurities that introduce excess electrons or holes -- n ≠ p

Intrinsic vs Extrinsic Conduction3+

• p-type Extrinsic: (p >> n)

no applied electric field

Boron atom

4+4+4+4+4+4+4+4+4+4+4+

hole

• n-type Extrinsic: (n >> p)

no applied electric field

5+

4+4+4+4+4+4+4+4+4+4+4+

Phosphorus atom

valence electron

Si atom

conductionelectron

Adapted from Figs. 12.12(a) & 12.14(a), Callister & Rethwisch 5e.

Chapter 9 - 55

Extrinsic Semiconductors: Conductivity vs. Temperature

• Data for Doped Silicon: -- σ increases doping -- reason: imperfection sites lower the activation energy to produce mobile electrons.

• Comparison: intrinsic vs extrinsic conduction... -- extrinsic doping level: 1021/m3 of a n-type donor impurity (such as P). -- for T < 100 K: "freeze-out“, thermal energy insufficient to excite electrons. -- for 150 K < T < 450 K: "extrinsic" -- for T >> 450 K: "intrinsic"

Adapted from Fig. 12.17, Callister & Rethwisch 5e. (From S. M. Sze, Semiconductor Devices, Physics and Technology. Copyright © 1985 by Bell Telephone Laboratories, Inc. Reprinted by permission of John Wiley & Sons, Inc.)

Conduction electron concentration (1021/m3)T(K)

600400200

00123freeze-outextrinsicintrinsicdopedundoped

msne 301 final exam - rice university

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