Chapter 3 -1 ISSUES TO ADDRESS... What is the difference in atomic arrangement between crystalline and noncrystalline solids? What features of a metal’s/ceramic’s

Chapter 3 - 1

ISSUES TO ADDRESS...

• What is the difference in atomic arrangement between crystalline and noncrystalline solids?

• What features of a metal’s/ceramic’s atomic structure determine its density?

• Under what circumstances does a material property vary with the measurement direction?

Chapter 3: Structures of Metals & Ceramics

• How do the crystal structures of ceramic materials differ from those for metals?

Chapter 3 - 2

• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have lower energies.

Energy and PackingEnergy

r

typical neighbor bond length

typical neighbor bond energy

Energy

r

typical neighbor bond length

typical neighbor bond energy

Chapter 3 - 3

• atoms pack in periodic, 3D arraysCrystalline materials...

-metals-many ceramics-some polymers

• atoms have no periodic packingNoncrystalline materials...

-complex structures-rapid cooling

crystalline SiO2

noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.41(b), Callister & Rethwisch 4e.

Adapted from Fig. 3.41(a), Callister & Rethwisch 4e.

Materials and Packing

Si Oxygen

• typical of:

• occurs for:

Chapter 3 - 4

Metallic Crystal Structures

• How can we stack metal atoms to minimize empty space?

2-dimensions

vs.

Now stack these 2-D layers to make 3-D structures

Chapter 3 - 5

• Tend to be densely packed.

• Reasons for dense packing:- Typically, only one element is present, so all atomic radii are the same.- Metallic bonding is not directional.- Nearest neighbor distances tend to be small in order to lower bond energy.- Electron cloud shields cores from each other

• Metals have the simplest crystal structures.

We will examine three such structures...

Metallic Crystal Structures

Chapter 3 - 6

• Rare due to low packing density (only Po has this structure)• Close-packed directions are cube edges.

• Coordination # = 6 (# nearest neighbors)

Simple Cubic Structure (SC)

Click once on image to start animation

(Courtesy P.M. Anderson)

Chapter 3 - 7

• APF for a simple cubic structure = 0.52

APF = a3

4

3(0.5a) 31

atoms

unit cellatom

volume

unit cell

volume

Atomic Packing Factor (APF)

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

Adapted from Fig. 3.43, Callister & Rethwisch 4e.

close-packed directions

a

R=0.5a

contains 8 x 1/8 = 1 atom/unit cell

Chapter 3 - 8

• Coordination # = 8


• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

Body Centered Cubic Structure (BCC)

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

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

Click once on image to start animation


Chapter 3 -

VMSE Screenshot – BCC Unit Cell

9

Chapter 3 - 10

Atomic Packing Factor: BCC

APF =

4

3 ( 3a/4)32

atoms

unit cell atom

volume

a3unit cell

volume

length = 4R =Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

aRAdapted from

Fig. 3.2(a), Callister & Rethwisch 4e.

a

a 2

a 3

Chapter 3 - 11



• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing.

Face Centered Cubic Structure (FCC)

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

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8Click once on image to start animation


Chapter 3 - 12

• APF for a face-centered cubic structure = 0.74Atomic Packing Factor: FCC

maximum achievable APF

APF =

4

3( 2a/4)34

atoms

unit cell atom

volume

a3unit cell

volume

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

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

a

2 a

Adapted fromFig. 3.1(a),Callister & Rethwisch 4e.

Chapter 3 - 13

A sites

B B

B

BB

B B

C sites

C C

CA

B

B sites

• ABCABC... Stacking Sequence• 2D Projection

• FCC Unit Cell

FCC Stacking Sequence

B B

B

BB

B B

B sitesC C

CA

C C

CA

AB

C

Chapter 3 - 14


• ABAB... Stacking Sequence

• APF = 0.74

• 3D Projection • 2D Projection


Hexagonal Close-Packed Structure (HCP)

6 atoms/unit cell

ex: Cd, Mg, Ti, Zn

• c/a = 1.633

c

a

A sites

B sites

A sites Bottom layer

Middle layer

Top layer

Chapter 3 -

VMSE Screenshot – Stacking Sequence and Unit Cell for HCP

15

Chapter 3 - 16

Theoretical Density,

where n = number of atoms/unit cell A = atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.022 x 1023 atoms/mol

Density = =

VC NA

n A =

Cell Unit of VolumeTotal

Cell Unit in Atomsof Mass

Chapter 3 - 17

• Ex: Cr (BCC)

A = 52.00 g/mol

R = 0.125 nm

n = 2 atoms/unit cell

theoretical

a = 4R/ 3 = 0.2887 nm

actual

aR

= a3

52.002

atoms

unit cellmol

g

unit cell

volume atoms

mol

6.022 x 1023

Theoretical Density,

= 7.18 g/cm3

= 7.19 g/cm3


Chapter 3 - 18

• Bonding: -- Can be ionic and/or covalent in character. -- % ionic character increases with difference in electronegativity of atoms.

Adapted from Fig. 2.7, Callister & Rethwisch 4e. (Fig. 2.7 is adapted from Linus Pauling, The Nature of the Chemical Bond, 3rd edition. Copyright 1939 and 1940, 3rd edition copyright © 1960 by Cornell University.

• Degree of ionic character may be large or small:

Atomic Bonding in Ceramics

SiC: small

CaF2: large

Chapter 3 - 19

Ceramic Crystal Structures

Oxide structures– oxygen anions larger than metal cations– close packed oxygen in a lattice (usually FCC)– cations fit into interstitial sites among oxygen ions

Chapter 3 - 20

Factors that Determine Crystal Structure1. Relative sizes of ions – Formation of stable structures: --maximize the # of oppositely charged ion neighbors.


- -

- -+

unstable

- -

- -+

stable

- -

- -+

stable

2. Maintenance of Charge Neutrality : --Net charge in ceramic should be zero. --Reflected in chemical formula:

CaF2: Ca2+cation

F-

F-

anions+

AmXp

m, p values to achieve charge neutrality

Chapter 3 - 21

• Coordination Number increases with

Coordination Number and Ionic Radii

Adapted from Table 3.3, Callister & Rethwisch 4e.

2

rcationranion

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




ZnS (zinc blende)

NaCl(sodium chloride)

CsCl(cesium chloride)

rcationranion

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

Chapter 3 - 22

Computation of Minimum Cation-Anion Radius Ratio

• Determine minimum rcation/ranion for an octahedral site (C.N. = 6)

a 2ranion

Chapter 3 - 23

Bond Hybridization

Bond Hybridization is possible when there is significant covalent bonding– hybrid electron orbitals form– For example for SiC

• XSi = 1.8 and XC = 2.5

• ~ 89% covalent bonding• Both Si and C prefer sp3 hybridization• Therefore, for SiC, Si atoms occupy tetrahedral sites

Chapter 3 - 24

• On the basis of ionic radii, what crystal structure would you predict for FeO?

• Answer:

based on this ratio,-- coord # = 6 because

0.414 < 0.550 < 0.732

-- crystal structure is NaCl

Data from Table 3.4, Callister & Rethwisch 4e.

Example Problem: Predicting the Crystal Structure of FeO

Ionic radius (nm)

0.053

0.077

0.069

0.100

0.140

0.181

0.133

Cation

Anion

Al3+

Fe2+

Fe3+

Ca2+

O2-

Cl-

F-

Chapter 3 - 25

Rock Salt StructureSame concepts can be applied to ionic solids in general. Example: NaCl (rock salt) structure

rNa = 0.102 nm

rNa/rCl = 0.564

cations (Na+) prefer octahedral sites


rCl = 0.181 nm

Chapter 3 - 26

MgO and FeO

O2- rO = 0.140 nm

Mg2+ rMg = 0.072 nm

rMg/rO = 0.514

cations prefer octahedral sites

So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms


MgO and FeO also have the NaCl structure

Chapter 3 - 27

AX Crystal Structures


Cesium Chloride structure:

Since 0.732 < 0.939 < 1.0, cubic sites preferred

So each Cs+ has 8 neighbor Cl-

AX–Type Crystal Structures include NaCl, CsCl, and zinc blende

Chapter 3 - 28

AX2 Crystal Structures

• Calcium Fluorite (CaF2)

• Cations in cubic sites

• UO2, ThO2, ZrO2, CeO2

• Antifluorite structure –

positions of cations and anions reversed


Fluorite structure

Chapter 3 - 29

ABX3 Crystal Structures


• Perovskite structure

Ex: complex oxide

BaTiO3

Chapter 3 -

VMSE Screenshot – Zinc Blende Unit Cell

30

Chapter 3 - 31

Density Computations for Ceramics

Number of formula units/unit cell

Volume of unit cell

Avogadro’s number

= sum of atomic weights of all anions in formula unit

= sum of atomic weights of all cations in formula unit

Chapter 3 - 32

Densities of Material Classesmetals > ceramics > polymers

Why?

Data from Table B.1, Callister & Rethwisch, 4e.

(g

/cm

)3

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

1

2

20

30Based 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

0.3

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

PET PVC 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)

Composites have... • intermediate values

In general

Chapter 3 - 33

Silicate CeramicsMost 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+

O2-

Adapted from Figs. 3.10-11, Callister & Rethwisch 4e crystobalite

Chapter 3 - 34

Bonding of adjacent SiO44- accomplished by the sharing

of common corners, edges, or faces

Silicates

Mg2SiO4 Ca2MgSi2O7


Presence of cations such as Ca2+, Mg2+, & Al3+ 1. maintain charge neutrality, and 2. ionically bond SiO4

4- to one another

Chapter 3 - 35

• 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)


Glass Structure

Si04 tetrahedron4-

Si4+

O2-

Si4+

Na+

O2-

Chapter 3 - 36

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


Chapter 3 - 37

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

layer

Layered Silicates (cont)

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


Chapter 3 - 38

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.


Chapter 3 - 39

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


Chapter 3 - 40

Polymorphic Forms of Carbon (cont) Fullerenes and Nanotubes

• Fullerenes – spherical cluster of 60 carbon atoms, C60

– Like a soccer ball • Carbon nanotubes – sheet of graphite rolled into a tube

– Ends capped with fullerene hemispheres

Adapted from Figs. 3.18 & 3.19, Callister & Rethwisch 4e.

Chapter 3 - 41

• 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.

-- diamond single crystals for abrasives

-- turbine blades

Fig. 9.40(c), Callister & Rethwisch 4e. (Fig. 9.40(c) courtesy of Pratt and Whitney).

(Courtesy Martin Deakins,GE Superabrasives, Worthington, OH. Used with permission.)

Crystals as Building Blocks

Chapter 3 - 42

• 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 typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

Adapted from Fig. K, color inset pages of Callister 5e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

Polycrystals

Isotropic

Anisotropic

Chapter 3 - 43

• Single Crystals-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

Data from Table 3.7, Callister & Rethwisch 4e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

• Polycrystals

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

200 m Adapted from Fig. 5.19(b), Callister & Rethwisch 4e.(Fig. 5.19(b) is 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

Chapter 3 - 44

Polymorphism

• Two or more distinct crystal structures for the same material (allotropy/polymorphism) titanium

, -Ti

carbon

diamond, graphite

BCC

FCC

BCC

1538ºC

1394ºC

912ºC

-Fe

-Fe

-Fe

liquid

iron system

Chapter 3 - 45

Fig. 3.20, Callister & Rethwisch 4e.

Crystal Systems

7 crystal systems

14 crystal lattices

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

a, b, and c are the lattice constants

Chapter 3 - 46

Point CoordinatesPoint coordinates for unit cell

center are

a/2, b/2, c/2 ½ ½ ½

Point coordinates for unit cell 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

Chapter 3 - 47

Crystallographic Directions

1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of unit cell dimensions a, b, and c3. Adjust to smallest integer values4. Enclose in square brackets, no commas

[uvw]

ex: 1, 0, ½ => 2, 0, 1 => [ 201 ]

-1, 1, 1

families of directions <uvw>

z

x

Algorithm

where overbar represents a negative index

[ 111 ]=>

y

Chapter 3 -

VMSE Screenshot – [101] Direction

48

Chapter 3 - 49

ex: linear density of Al in [110] direction

a = 0.405 nm

Linear Density

• Linear Density of Atoms LD =

a

[110]

Adapted fromFig. 3.1(a),Callister & Rethwisch 4e.

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD

Chapter 3 - 50

Drawing HCP Crystallographic Directions (i)

1. Remove brackets 2. Divide by largest integer so all values

are ≤ 13. Multiply terms by appropriate unit cell

dimension a (for a1, a2, and a3 axes) or c (for z-axis) to produce projections 4. Construct vector by stepping off these projections

Algorithm (Miller-Bravais coordinates)

Adapted from Figure 3.25, Callister & Rethwisch 4e.

Chapter 3 - 51

Drawing HCP Crystallographic Directions (ii) • Draw the direction in a hexagonal unit cell.

[1213]

4. Construct Vector

1. Remove brackets -1 -2 1 3

Algorithm a1 a2 a3 z

2. Divide by 3

3. Projections

proceed –a/3 units along a1 axis to point p

–2a/3 units parallel to a2 axis to point q

a/3 units parallel to a3 axis to point r

c units parallel to z axis to point s

p

qr

s

start at point o

Adapted from p. 62, Callister & Rethwisch 8e.

[1213] direction represented by vector from point o to point s

Chapter 3 - 52

1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of three- axis (a1, a2, and z) unit cell dimensions a and c 3. Adjust to smallest integer values4. Enclose in square brackets, no commas, for three-axis coordinates 5. Convert to four-axis Miller-Bravais lattice coordinates using equations below:

6. Adjust to smallest integer values and enclose in brackets [uvtw]


Algorithm

Determination of HCP Crystallographic Directions (ii)

Chapter 3 - 53

4. Brackets [110]

1. Reposition not needed

2. Projections a a 0c1 1 0

3. Reduction 1 1 0

Example a1 a2 z

5. Convert to 4-axis parameters

1/3, 1/3, -2/3, 0 => 1, 1, -2, 0 => [ 1120 ]

6. Reduction & Brackets


Determination of HCP Crystallographic Directions (ii)

Determine indices for green vector

Chapter 3 - 54

Crystallographic Planes


Chapter 3 - 55

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 1. Read off intercepts of plane with axes in terms of a, b, c2. Take reciprocals of intercepts3. Reduce to smallest integer values4. Enclose in parentheses, no commas i.e., (hkl)

Chapter 3 - 56

Crystallographic Planesz

x

ya b

c

4. Miller Indices (110)

example a b cz

x

ya b

c


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

example a b c

Chapter 3 - 57


z

x

ya b

c


example1. Intercepts 1/2 1 3/4

a 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),

Chapter 3 -

VMSE Screenshot – Crystallographic Planes

58

Additional practice on indexing crystallographic planes

Chapter 3 - 59

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

11

3. Reduction 1 0 -1 1

a2

a3

a1

z

Adapted from Fig. 3.24(b), Callister & Rethwisch 4e.

Chapter 3 - 60


• We want to examine the atomic packing of crystallographic planes

• 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.

Chapter 3 - 61

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

(100)

Radius of iron R = 0.1241 nm

R3

34a

Adapted from Fig. 3.2(c), Callister & Rethwisch 4e.

2D repeat unit

= Planar Density = a2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

12

R3

34area

2D repeat unit

Chapter 3 - 62

Planar Density of (111) IronSolution (cont): (111) plane 1 atom in plane/ unit surface cell

333 2

2

R3

16R

34

2a3ah2area

atoms in plane

atoms above plane

atoms below plane

ah2

3

a 2

2D re

peat

uni

t

1

= = nm2

atoms7.0m2

atoms0.70 x 1019

3 2R3

16Planar Density =

atoms

2D repeat unit

area

2D repeat unit

Chapter 3 -

VMSE Screenshot – Atomic Packing – (111) Plane for BCC

63

Chapter 3 - 64

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.

Chapter 3 - 65

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.


reflections must be in phase for a detectable signal

spacing between planes

d

incoming

X-rays

outg

oing

X-ra

ys

detector

extra distance travelled by wave “2”

“1”

“2”

“1”

“2”

Chapter 3 - 66

X-Ray Diffraction Pattern

Adapted from Fig. 3.40, Callister 4e.

(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

Chapter 3 - 67

• Atoms may assemble into crystalline or amorphous 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).

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.

• 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.

• Ceramic crystal structures are based on: -- maintaining charge neutrality -- cation-anion radii ratios.

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

Chapter 3 - 68

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

SUMMARY

• 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.

• X-ray diffraction is used for crystal structure and interplanar spacing determinations.

Chapter 3 - 69

Core Problems:

Self-help Problems:

ANNOUNCEMENTSReading:

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Chapter 3 -1 ISSUES TO ADDRESS... What is the difference in atomic arrangement between crystalline and noncrystalline solids? What features of a metal’s/ceramic’s