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MSE 553 DEFECTS, DIFFUSION AND TRANSFORMATION IN MATERIALS

E. GIKUNOO

First Semester 2019/2020Wednesdays 2:00 – 5:00 pm 1

Lecturer: Dr. Emmanuel Gikunoo

Office : 321 New Block

Email: egikunoo.soe@knust.edu.gh

Lecture Hours: Tuesdays, 08:00 – 11:00

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Course Information

Website: https://egikunoo.wordpress.com

Objective

This course will provide students with the:

understanding of basic theory of the interaction of defects, general laws of phase transformation and mutual effects between composition, and the microstructure and macroproperties of engineering materials such as metallic alloys and ceramics;

understanding of phase transformations in materials in tailoring the microstructure and properties of various materials;

much needed underlying principles governing materials developments.3

At the end of the course the student should be able to:

discuss the relationship between atomic structure of materials and microstructure evolution on the basis of transport processes.

apply the principles of phase transformation for microstructure design and property improvements.

apply the principles of phase transformation for selection of materials and processes.

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Learning Outcomes

1 2

3 4

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Course Content

This course will introduce students to:

basic crystallography and thermodynamics;

defects classification in different crystals, defects in non-crystalline materials, mechanical, electrical, magnetic and optical properties of defects, defect characterization techniques;

Fick’s first and second laws, Intrinsic and integrated diffusion coefficient, Tracer and growth kinetics, Matano-Boltz analysis, Kirkendall effect, Darken analysis, physico-chemical approach of diffusion;

basic kinetics of transformation, interfacial and homogeneous nucleation, nucleation and growth, coarsening and the Gibbs-Thompson equation, diffusion and diffusionless transformations.

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References

W. D. Callister, Jr., Materials Science and Engineering – An Introduction, 7th Edition, John Wiley & Sons, 2006.

D. R, Askeland and P. P. Phule, The Science and Engineering of Materials, 4th Edition, Thomson Learning Vocational, 2003.

D. A. Porter, K. E. Easterling, M. Sherif, Phase Transformations in Metals and Alloys, 3rd edition, Chapman & Hall, 2009.

P. Haasen, Physical Metallurgy, 3RD edition, Cambridge University Press, 1996.

H. V. Keer, Principles of the Solid State, New Age International (P) Limited, 2005.

Presentation Slides6

7

Grading

Assignments 20

Term Papers 15

Assigned Presentation 15

Final Exam 50

Total 100

Crystal Structure – matter assumes a periodic shape

o Non-crystalline or amorphous “structures” exhibit no long range periodic shapes

o Crystal systems – not structures but potentials

FCC, BCC and HCP – common crystal structures for metals

Point, direction and planer ID’ing in crystals

X-ray diffraction and crystal structure

Crystallography

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5 6

7 8

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• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to havelower energies & thus are more stable.

Energy and PackingEnergy

r

typical neighbor

bond length

typical neighbor

bond energy

Energy

r

typical neighbor

bond length

typical neighbor

bond energy

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Crystaling Materials

Means: Periodic arrangement of atoms/ions over large atomic distances Leads to structure displaying LONG-

RANGE ORDER that is Measurable and Quantifiable

All metals, many ceramics, and some polymers exhibit this

“High Bond Energy” and a More Closely Packed Structure

STRUCTURES

Materials lacking long range order

Amorphous Materials

These less densely packed lower bond energy

“structures” can be found in some metals and are observed in almost all

ceramics, glass and many “plastics”

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11

Crystallography

12

Crystallography

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11 12

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13

Crystallography

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Crystallography

Crystal Systems

7 crystal systems of varying symmetry are known

These systems are built by changing the lattice parameters:

a, b, and c are the edge lengths, , and are interaxial angles

Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal.

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Crystal Systems

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15 16

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Rare due to low packing density (only Polonium (Po) has this structure)

Close-packed directions are cube edges.

Coordination No. = 6(# nearest neighbors) for each atom as seen

Simple Cubic Structure (SC)

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*assume hard spheres

• APF for a simple cubic structure = 0.52

APF = a3

4

3p (0.5a) 31

atoms

unit cellatom

volume

unit cell

volume

Atomic Packing Factor (APF)

close-packed directions

aR=0.5a

contains (8 x 1/8) = 1 atom/unit cell Here: a = 2 × Rat

Where Rat is the atomic radius

𝐴𝑃𝐹 = 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑖𝑛 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙

𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙

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Coordination # = 8

Atoms touch each other along cube diagonals within a unit cell.

- Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

ex: Cr, W, Fe (), Tantalum, Molybdenum

Body Centered Cubic Structure (BCC)

2 atoms/unit cell: (1 center) + (8 corners x 1/8)19

Atomic Packing Factor: BCC

a

APF =

4

3p ( 3a/4)32

atomsunit cell atom

volume

a3unit cell

volume

length = 4R =Close-packed directions:

3 a

APF for a body-centered cubic structure = 0.68

aR

a2

a3

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Coordination # = 12

Face Centered Cubic Structure (FCC)

4 atoms/unit cell: (6 face x ½) + (8 corners x 1/8)

Atoms touch each other along face diagonals. ABCABC... Stacking Sequence

- Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

ex: Al, Cu, Au, Pb, Ni, Pt, Ag

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APF for a face-centered cubic structure = 0.74

Atomic Packing Factor: FCC

The maximum achievable APF!

APF =

4

3p ( 2a/4)34

atomsunit cell atom

volume

a3unit cell

volume

Close-packed directions: length = 4R = a√2

Unit cell contains:6 x 1/2 + 8 x 1/8

= 4 atoms/unit cell

a

2 a

a = 22 R

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Coordination # = 12 APF = 0.74

3D Projection2D Projection

Hexagonal Close-Packed Structure (HCP)

6 atoms/unit cell

c/a = 1.633 (ideal)

c

a

A sites

B sites

A sitesBottom layer

Middle layer

Top layer

Atoms touch each other along face diagonals. ABAB... Stacking Sequence

ex: Cd, Mg, Ti, Zn

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We find that both

FCC & HCP are

highest density

packing schemes

(APF = .74) – this

illustration shows

their differences as

the closest packed

planes are “built-

up”.

HCP and FCC Stalking

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21 22

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Theoretical Density, r

where n = number of atoms/unit cellA = atomic weight VC = Volume of unit cell = a3 for

cubicNA = Avogadro’s number

= 6.023 x 1023 atoms/mol

Density = r =

r =

aR

r = a3

52.002

atoms

unit cellmol

g

unit cell

volume atoms

mol

6.023x1023

rtheoretical

ractual

= 7.18 g/cm3

= 7.19 g/cm3

Ex: Cr (BCC) A = 52.00 g/mol , R = 0.125 nm,

n = 2

a = √

= 0.2883 nm25 26

Basic Crystallography

Repetition = Symmetry

Types of repetitiono Rotationo Translation

Rotationo What is rotational symmetry

This object is obviously symmetric. Can be rotated 90˚ without detection.o Symmetry is doing something that

looks like nothing has been done!

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Basic Crystallography A figure has rotational symmetry if

it can be rotated by an angle between 0˚ and 360˚ so that the image coincides with the preimage.

The angle of rotational symmetry is the smallest angle for which the figure can be rotated to coincide with itself.

The order of symmetry is the number of times the figure coincides with itself as it rotates through 360˚.28

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Locations in Lattices: Point Coordinates

Point coordinates for unit cell center are

a/2, b/2, c/2 ½ ½ ½

Point coordinates for unit cell (body diagonal) corner are 111

Translation: integer multiple of lattice constants identical position in another unit cell

z

x

ya b

c

000

111

y

z

2c

b

b

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Crystallographic Directions

1. Vector is repositioned (if necessary) to pass through the Unit Cell origin.

2. Read off line projections (to principal axes of unit cell) in terms of unit cell dimensions a, b, and c

3. Adjust to smallest integer values4. Enclose in square brackets, no

commas [uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1 [ 111 ]=>

z

x

yx y z

Projections a/2 b 0c

Projections in terms of a, b, and c ½ 1 0

Reduction 1 2 0

Enclosure [brackets] [120]

where ‘overbar’ represents a negative index30

Linear Density – Considers Equivalence and is important in Slip

Linear Density of Atoms = LD =

ex: linear density of Al in [110] direction where a = 0.405 nm

a

[110]

# atoms

length

3.5 nm-1

a22LD ==

# atoms CENTERED on the direction of interest!Length is of the direction of interest within the Unit Cell

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Determining Angles Between Crystallographic Direction:

1 1 2 1 2 1 2

2 2 2 2 2 21 1 1 2 2 2

u u v v w wCos

u v w u v w

Where ui’s , vi’s & wi’s are the “Miller Indices” of the directions in question

– also, if a direction has the same Miller Indices as a plane, it is NORMAL to that plane

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HCP Crystallographic Directions

1. Vector repositioned (if necessary) to pass through origin.

2. Read off projections in terms of unitcell dimensions a1, a2, a3, or c

3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvtw]

[ 1120 ]ex: ½, ½, -1, 0 =>

dashed red lines indicate projections onto a1 and a2 axes

a1

a2

a3

-a3

2

a2

2

a1

-a3

a1

a2

zAlgorithm

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HCP Crystallographic Directions Hexagonal Crystals

4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') in the ‘3 space’ Bravais lattice as follows.

'ww

t

v

u

)vu( +-

)'u'v2(3

1-

)'v'u2(3

1-

]uvtw[]'w'v'u[

Fig. 3.8(a), Callister 7e.

-a3

a1

a2

z

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Computing HCP Miller- Bravais Directional Indices (an alternative way):

-a3

a1

a2

z

We confine ourselves to the bravaisparallelopiped in the hexagon: a1-a2-Z and determine: (u’,v’w’)

Here: [1 1 0] - so now apply the models to create Miller Bravis indices

1 1 12 ' ' 2 1 1 13 3 3

1 1 12 ' ' 2 1 1 13 3 3

1 1 2 23 3 3

' 0

M-B Indices: [1120]

u u v

v v u

t u v

w w

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Defining Crystallographic Planes Miller Indices: Reciprocals of the (three) axial

intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.

Algorithm (in cubic lattices this is direct)

1. Read off intercepts of plane with axes in

terms of a, b, c2. Take reciprocals of intercepts

3. Reduce to smallest integer values

4. Enclose in parentheses, no commas i.e.,

(hkl) families {hkl}36

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Crystallographic Planes -- families

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Crystallographic Planesz

x

ya b

c

4. Miller Indices (110)

Examplea b c

z

x

ya b

c

4. Miller Indices (100)

1. Intercepts 1 1 2. Reciprocals 1/1 1/1 1/

1 1 03. Reduction 1 1 0

1. Intercepts 1/2 2. Reciprocals 1/½ 1/ 1/

2 0 03. Reduction 2 0 0

Examplea b c

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Crystallographic Planesz

x

ya b

c

4. Miller Indices (634)

Example

1. Intercepts 1/2 1 3/4a b c

2. Reciprocals 1/½ 1/1 1/¾2 1 4/3

3. Reduction 6 3 4

(001)(010),

Family of Planes {hkl}

(100), (010),(001),Ex: {100} = (100),

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x y zIntercepts Intercept in terms of lattice parametersReciprocals Reductions

Enclosure

a -b c/2 -1 1/20 -1 2

N/A

(012)

Determine the Miller indices for the plane shown in the sketch

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Crystallographic Planes (HCP)

In hexagonal unit cells the same idea is used

Example

a1 a2 a3 c

4. Miller-Bravais Indices (1011)

1. Intercepts 1 -1 12. Reciprocals 1 1/

1 0 -1-1

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3. Reduction 1 0 -1 1 a2

a3

a1

z

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Crystallographic Planes

We want to examine the atomic packing of crystallographic planes – those with the same packing are equivalent and part of families

Iron foil can be used as a catalyst. The atomic packing of the exposed planes is important.

a) Draw (100) and (111) crystallographic planes

for Fe.

b) Calculate the planar density for each of these planes.

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Planar Density of (100) IronSolution: At T < 912C iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a

2D repeat unit

= Planar Density =a2

1

atoms

2D repeat unit

= (nm)2

atoms12.1

m2

atoms= 1.2 x 1019

12

R3

34area

2D repeat unitAtoms: wholly contained and centered in/on plane within U.C., area of plane in U.C.43

Planar Density of (111) IronSolution (cont): (111) plane 1/2 atom centered on plane/ unit cell

atoms in plane

atoms above plane

atoms below plane

ah2

3

a2

3*1/6= =

nm2

atoms7.0m2

atoms0.70 x 1019Planar Density =

atoms

2D repeat unit

area

2D repeat unit

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3

R

Area 2D Unit: ½ hb = ½*[(3/2)a][(2)a]=1/2(3)a2=8R2/(3)

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CERAMIC AND OTHER UNIT CELLS

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Cesium chloride (CsCl) unit cell showing(a) ion positions and the two ions per lattice point and(b) full-size ions.

Note that the Cs+−Cl− pair associated with a given lattice point is not a molecule because the ionic bonding is nondirectional and because a given Cs+ is equally bonded to eight adjacent Cl−, and vice versa.

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Sodium chloride (NaCl) structure showing (a) ion positions in a unit cell, (b) full-size ions, and (c) many adjacent unit cells.

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Fluorite (CaF2) unit cell showing (a) ion positions and (b) full-size ions.

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SiO4

4

The cristobalite (SiO2) unit cell showing (a) ion positions(b) full-size ions and (c) the connectivity of SiO4

4- tetrahedra.

Each tetrahedron has a Si4+ at its center. In addition, an O2- would be at each corner of each tetrahedron and is shared with an adjacent tetrahedron. 49

Many crystallographic forms of SiO2 are stable as they are heated from room temperature to melting temperature. Each form represents a different way to connect adjacent SiO4

4- tetrahedra

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Polymorphism: Also in Metals Two or more distinct crystal structures for the same

material (allotropy/polymorphism)

titanium (HCP), (BCC)-Ti

carbon:diamond, graphite

BCC

FCC

BCC

1538ºC

1394ºC

912ºC

-Fe

-Fe

-Fe

liquid

iron system:

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The corundum (Al2O3) unit cell is shown superimposed on the repeated stacking of layers of close-packed O2− ions.

The Al3+ ions fill two-thirds of the small (octahedral) interstices between adjacent layers.

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Exploded view of the kaolinite unit cell, 2(OH)4Al2Si2O5.

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(a) An exploded view of the graphite (C) unit cell. (b) A schematic of the nature of graphite’s layered structure.

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(a) C60 molecule, or buckyball.

(b) Cylindrical array of hexagonal rings of carbon atoms, or buckytube.

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Arrangement of polymeric chains in the unit cell of polyethylene. The dark spheres are carbon atoms, and the light spheres are hydrogen atoms. The unit-cell dimensions are 0.255 nm × 0.494 nm × 0.741 nm.

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Diamond cubic unit cell showing

(a) atom positions. There are two atoms per lattice point (note the outlined example). Each atom is tetrahedrally coordinated.

(b) The actual packing of full-size atoms associated with the unit cell.

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Zinc blende (ZnS) unit cell showing

(a) ion positions. There are two ions per lattice point (note the outlined example). Compare this structure with the diamond cubic structure.

(b) The actual packing of full-size ions associated with the unit cell.

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Densities of Material Classesrmetals > rceramics > rpolymers

Why?

Data from Table B1, Callister 7e.

r(g

/cm

)3

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

1

2

20

30

*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% volume fraction of aligned fibers

in an epoxy matrix).10

3

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0.30.40.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

TantalumGold, WPlatinum

Graphite

Silicon

Glass -sodaConcrete

Si nitrideDiamondAl oxide

Zirconia

HDPE, PSPP, LDPE

PC

PTFE

PETPVCSilicone

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)

Composites have...• intermediate values

In general

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• Some engineering applications require single crystals:

• Properties of crystalline materials often related to crystal structure.Ex: Quartz fractures more easily along some crystal planes than others.

o Diamond single crystals for abrasives o turbine blades

(Courtesy of Pratt and Whitney).

(Courtesy of Martin Deakins, GE Superabrasives.)

Crystals as Building Blocks

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• Most engineering materials are polycrystals.

• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If grains are randomly oriented, overall component

properties are not directional.• Grain sizes typically ranges from 1 nm to 2 cm

(i.e., from a few to millions of atomic layers).

(Courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

Polycrystals

Isotropic

Anisotropic

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• Single Crystals-Properties vary withdirection: anisotropic.

-Example: the modulusof elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may notvary with direction.

-If grains are randomlyoriented: isotropic.(Epoly iron = 210 GPa)

-If grains are textured,anisotropic.

200 mm

Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.

courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].

Single vs PolycrystalsE (diagonal) = 273 GPa

E (edge) = 125 GPa

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Effects of Anisotropy:

Modulus of Elasticity values for several metals at various crystallographic orientations

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X-Ray Diffraction

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

Can’t resolve spacings Spacing is the distance between parallel planes of

atoms. 64

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Relationship of the Bragg angle (θ) and the experimentally measured diffraction angle (2θ).

X-ray intensity (from detector)

2c

d n2 sinc

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X-Rays to Determine Crystal Structure

X-ray intensity (from detector)

c

d n

2 sinc

Measurement of critical angle, c, allows computation of planar spacing, d.

• Incoming X-rays diffract from crystal planes.

Adapted from Fig. 3.19, Callister 7e.

reflections must be in phase for a detectable signal!

spacing between planes

d

extra distance traveled by wave “2”

2 2 2hkl

ad

h k l

For Cubic Crystals:

h, k, l are Miller Indices 66

Figure 3.34 (a) An x-ray diffractometer. (Courtesy of Scintag, Inc.) (b) A schematic of the experiment.

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X-Ray Diffraction Pattern

Adapted from Fig. 3.20, Callister 5e.

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2

Diffraction pattern for polycrystalline -iron (BCC)

Inte

nsity

(re

lativ

e)

z

x

ya b

cz

x

ya b

c

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Diffraction in Cubic Crystals:

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Atoms may assemble into crystalline or amorphous structures.

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.

We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP)

Crystallographic points, directions and planes are specified in terms of indexing schemes.Crystallographic directions and planes are related to atomic linear densities and planar densities.

SUMMARY

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Materials can be single crystals or polycrystalline. Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.

Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).

X-ray diffraction is used for crystal structure and interplanarspacing determinations.

SUMMARY

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