CHAPTER 4:IMPERFECTIONS IN SOLIDS

ISSUES TO ADDRESS...

• What types of defects arise in solids?

• Can the number and type of defects be variedand controlled?

• How do defects affect material properties?

• Are defects undesirable?

TYPES OF IMPERFECTIONS

• Vacancy atoms• Interstitial atoms• Substitutional atoms

Point defects

Line defects

Area defects

• Dislocations

• Grain Boundaries

POINT DEFECTS

• Vacancies:-vacant atomic sites in a structure.

Vacancydistortion of planes

• Self-Interstitials:-"extra" atoms positioned between atomic sites.

self-interstitialdistortion

of planes

EQUIL. CONCENTRATION:POINT DEFECTS

• Equilibrium concentration varies with temperature!

Boltzmann's constant

(1.38 x 10-23 J/atom K)

(8.62 x 10-5 eV/atom K)

⎜ ⎟ NDN

= exp−QDkT

⎛

⎝ ⎜ ⎞

⎠ ⎟

No. of defects

No. of potential defect sites.

Activation energy

Temperature

Each lattice site is a potential vacancy site

MEASURING ACTIVATION ENERGY

⎜ ⎟ NDN

= exp−QDkT

⎛

⎝ ⎜ ⎞

⎠ ⎟ • We can get Q from

an experiment.

• Measure this... • Replot it...

1/T

N

NDln 1

-QD/k

slopeND

N

T

exponential dependence!

defect concentration

ESTIMATING VACANCY CONC.

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

ACu = 63.5g/molρ = 8.4 g/cm3

QV = 0.9eV/atom NA = 6.02 x 1023 atoms/mole

8.62 x 10-5 eV/atom-K

0.9eV/atom

1273K

⎜ ⎟ NDN

= exp−QDkT

⎛

⎝ ⎜ ⎞

⎠ ⎟

For 1m3, N =NAACu

ρ x x 1m3 = 8.0 x 1028 sites

= 2.7 · 10-4

• Answer:

ND = 2.7 · 10-4 · 8.0 x 1028 sites = 2.2x 1025 vacancies

POINT DEFECTS IN ALLOYSTwo outcomes if impurity (B) added to host (A):

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

Substitutional alloy(e.g., Cu in Ni)

Interstitial alloy(e.g., C in Fe)

OR

• Solid solution of B in A plus particles of a newphase (usually for a larger amount of B)

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

Alloy CompositionWeight Percentage

Ternary, Quaternary, …(3, 4, … components)

Binary(2 components)

10021

11 ×

+=

mmmC 100×

∑=

ii

jj m

mC

Atomic or Mole Percentage

10021

1'1 ×

+=

mm

m

nnnC 100' ×

∑=

imi

mjj n

nC

1

'1

1 Amnm =

100

100

2'21

'1

1'1

1

1221

21'1

×+

=

×+

=

ACACACC

ACACACC

Conversion formulas

Dislocations

DislocationsLine defectsMost important class of defects for ductile crystalline materials (metals!)

Burgers vector bBasic lattice deformation

Edge Dislocationsb perpendicular tothe dislocation line

Schematic atomic structure of an edge dislocation

Screw Dislocations

Schematic perspective view

View of plane ABCD

Screw dislocation line

Burgers vector bBasic lattice deformation

Screw Dislocationsb parallel tothe dislocation line

Mixed Dislocations

Schematic perspective view

View of plane ABCD

Edge

Mixed

Schematic of dislocation line

Screw

Images of Dislocations

~1 µm

Surface

GaN

Sapphire

Dislocations in GaNDislocations in Ti alloy

Interfacial DefectsTwo-dimensional defects – also referred to as planar defects

Free surfacesSurfaces are ‘defects’ – typical energies 0.1 – 1 J/cm2

Grain BoundariesBorders between crystalswith different orientation

High angle grain boundaryLarge misorientation

Low-angle grain boundarySmall misorientationConsists of walls of dislocations

Low Angle Grain Boundaries

Grain 1 Grain 2

Example of a tilt boundary

d

Low angle boundaries

θ ≈ b/d

θ = Grain-Grain misorientationb = Burgers vectord = dislocation spacing

Edge dislocations

Grain Boundary

Observation of MicrostructuresOptical Microscopy (referred to as metallography, ceramography, …)

1 mm = 1000 µm

Optical micrograph of brass

Resolution of optical microscopy ≈ λlight ≈ 500 nm

Observation of MicrostructuresOptical Microscope Transmission Electron Microscope

FEI Tecnai F30 300 kV FE TEM/STEM

(UCSB)

TEM/STEM Resolution~ 1 Å

Observation of MicrostructuresScanning Electron Microscope

Typical SEM Image

SEM Resolution~ 10 Å

Observation of Microstructures

Scanning Probe Microscopy (SPM)

Common modesScanning Tunneling Microscopy (STM)Atomic Force Microscopy (AFM)

AFM Image: GaNSPM Resolution: ~ 1 Å

CHAPTER 6: MECHANICAL PROPERTIES

ISSUES TO ADDRESS...• Stress and strain: What are they and why are

they used instead of load and deformation?

• Elastic behavior: When loads are small, how much deformation occurs? What materials deform least?

• Plastic behavior: At what point do dislocationscause permanent deformation? What materials aremost resistant to permanent deformation?

• Toughness and ductility: What are they and howdo we measure them?

ELASTIC DEFORMATION

F

δ

bonds stretch

return to initial

1. Initial 2. Small load 3. Unload

Elastic means reversible!

F

δ

Linear- elastic

Non-Linear-elastic

PLASTIC DEFORMATION (METALS)

1. Initial 2. Small load 3. Unload

Plastic means permanent!

F

δlinear elastic

linear elastic

δplastic

planes still sheared

F

δelastic + plastic

bonds stretch & planes shear

δplastic

ENGINEERING STRESS

• Tensile stress, σ: • Shear stress, τ:

Area, A

Ft

Ft

σ =FtAo

original area before loading

Area, A

Ft

Ft

Fs

F

F

Fs

τ = FsAo

Stress has units:N/m2 or lb/in2

COMMON STATES OF STRESS

• Simple tension: cable

oσ =

FA

• Simple shear: drive shaft

Ao = cross sectional

Area (when unloaded)

FF

σσ

oτ =

FsA

τ

Note: τ = M/AcR here.

Ski lift (photo courtesy P.M. Anderson)

M

M Ao

2R

FsAc

OTHER COMMON STRESS STATES (1)

Canyon Bridge, Los Alamos, NM

• Simple compression:

Ao

Balanced Rock, Arches National Park o

σ =FA

Note: compressivestructure member(σ < 0 here).

(photo courtesy P.M. Anderson)

(photo courtesy P.M. Anderson)

OTHER COMMON STRESS STATES (2)

• Bi-axial tension: • Hydrostatic compression:

Fish under waterPressurized tank

σ < 0h

(photo courtesyP.M. Anderson)

σz > 0

σθ > 0(photo courtesyP.M. Anderson)

ENGINEERING STRAIN

• Tensile strain: • Lateral strain:

• Shear strain:θ/2

π/2

π/2 - θ

θ/2

δ/2

δ/2

δL/2δL/2

Lowo

ε = δLo

εL =−δL

wo

γ = tan θ Strain is alwaysdimensionless.

STRESS-STRAIN TESTING

• Typical tensile specimen

• Other types of tests:--compression: brittle

materials (e.g., concrete)--torsion: cylindrical tubes,

shafts.

gauge length

(portion of sample with reduced cross section)=

• Typical tensiletest machine

load cell

extensometerspecimen

moving cross head

Adapted from Fig. 6.2,Callister 6e.

Adapted from Fig. 6.3, Callister 6e.(Fig. 6.3 is taken from H.W. Hayden, W.G. Moffatt, and J. Wulff, The Structure and Properties of Materials, Vol. III, Mechanical Behavior, p. 2, John Wiley and Sons, New York, 1965.)

LINEAR ELASTIC PROPERTIES

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

• Hooke's Law:

σ = E ε• Poisson's ratio, ν:

metals: ν ~ 0.33ceramics: ~0.25polymers: ~0.40

εν = − Lε

εL

ε

1-ν

F

Fsimple tension test

σ

Linear- elastic

1E

ε

Units:E: [GPa] or [psi]ν: dimensionless

OTHER ELASTIC PROPERTIES

• Elastic Shearmodulus, G:

τ

1G

γτ = G γ

• Elastic Bulkmodulus, K:

P= -K∆VVo

P

∆V

1-K Vo

• Special relations for isotropic materials:

P

P P

M

M

G =

E2(1+ ν)

K =E

3(1− 2ν)

simpletorsiontest

pressuretest: Init.vol =Vo. Vol chg.= ∆V

11

YOUNG’S MODULI: COMPARISON

0.2

8

0.6

1

Magnesium,Aluminum

Platinum

Silver, Gold

Tantalum

Zinc, Ti

Steel, NiMolybdenum

Graphite

Si crystal

Glass-soda

Concrete

Si nitrideAl oxide

PC

Wood( grain)

AFRE( fibers)*

CFRE*

GFRE*

Glass fibers only

Carbon fibers only

Aramid fibers only

Epoxy only

0.4

0.8

2

46

10

20

406080

100

200

600800

10001200

400

Tin

Cu alloys

Tungsten

<100>

<111>

Si carbide

Diamond

PTFE

HDPE

LDPE

PP

Polyester

PSPET

CFRE( fibers)*

GFRE( fibers)*

GFRE(|| fibers)*

AFRE(|| fibers)*

CFRE(|| fibers)*

MetalsAlloys

GraphiteCeramicsSemicond

PolymersComposites

/fibers

E(GPa)

Eceramics > Emetals >> Epolymers

109 Pa

Based on data in Table B2,Callister 6e.Composite data based onreinforced epoxy with 60 vol%of alignedcarbon (CFRE),aramid (AFRE), orglass (GFRE)fibers.