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BITSPilaniPilani Campus

BITS Pilani

presentationP. Srinivasan

Department of Mechanical Engineering

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BITSPilaniPilani Campus

ENGG ZC232 Engineering Materials(Lecture III)Crystal structure&tutorial problems.

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BITS Pilani, Pilani Campus

Rare due to poor packing (only Po has this structure)

Close-packed directionsare cube edges.

Coordination #= 6

(# nearest neighbors)

SIMPLE CUBIC STRUCTURE (SC)

3

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APF for a simple cubic structure = 0.52

ATOMIC PACKING FACTOR

4

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Coordination # = 8

Close packed directions are cube diagonals.--Note: All atoms are identical; the center atom is shaded

differently only for ease of viewing.

BODY CENTERED CUBIC STRUCTURE

(BCC)

5

Iron, chrom ium ,tung sten, tantalum etc.

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aR

APF for a body-centered cubic structure = 0.68

Unit cell c ontains:

1 + 8 x 1/8

= 2 atoms/unit cell

ATOMIC PACKING FACTOR: BCC

6

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Coordination # = 12

Close packed directions are face diagonals.

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

FACE CENTERED CUBIC STRUCTURE (FCC)

7

Aluminum, nickel, copper, gold,

silver etc.

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Unit cell c ontains:

6 x 1/2 + 8 x 1/8

= 4 atoms/unit cella

APF for a body-centered cubic structure = 0.74

ATOMIC PACKING FACTOR: FCC

8

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Coordination # = 12

ABAB... Stacking Sequence

APF = 0.74

3D Projection 2D Projection

A sites

B sites

A sites

HEXAGONAL CLOSE-PACKED STRUCTURE

(HCP)

9

Cadm ium , Cobalt, ti tanium , zinc etc.

c

a

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Element

Aluminum

ArgonBarium

Beryllium

Boron

Bromine

Cadmium

CalciumCarbon

Cesium

Chlorine

Chromium

Cobalt

CopperFlourine

Gallium

Germanium

Gold

Helium

Hydrogen

Symbol

Al

ArBa

Be

B

Br

Cd

CaC

Cs

Cl

Cr

Co

Cu F

Ga

Ge

Au

He

H

At. Weight

(amu)

26.98

39.95137.33

9.012

10.81

79.90

112.41

40.0812.011

132.91

35.45

52.00

58.93

63.55 19.00

69.72

72.59

196.97

4.003

1.008

Atomic radius

(nm)

0.143

------0.217

0.114

------

------

0.149

0.1970.071

0.265

------

0.125

0.125

0.128 ------

0.122

0.122

0.144

------

------

Density

(g/cm 3)

2.71

------3.5

1.85

2.34

------

8.65

1.552.25

1.87

------

7.19

8.9

8.94 ------

5.90

5.32

19.32

------

------

Characteristics of Selected Elements at 20C

11

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Why?

Metalshave... close-packing

(metallic bonding)

large atomic mass

Ceramicshave... less dense packing

(covalent bonding)

often lighter elements

Polymershave...

poor packing(often amorphous)

lighter elements (C,H,O)

Compositeshave... intermediate values

DENSITIES OF MATERIAL CLASSES

12

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ex: linear density of Al in [110]

direction

a= 0.405 nm

Linear Density

Linear Density of Atoms LD =

a

[110]

Unit length of direction vector

Number of atoms

# atoms

length

13.5 nma2

2LD -==

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Atomic Packing of Crystallographic Planes

We want to examine the atomic packing ofcrystallographic planes

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

for Fe.

b) Calculate the planar density for each of theseplanes.

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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 =

From Fig. 4.2(c), Callisters Materials Science

and Engineering,Adapted Version.

2D repeat unit

=Planar Density =a2

1

atoms

2D repeat unit

=

nm2

atoms12.1

m2

atoms= 1.2 x 1019

1

2

R

3

34area

2D repeat unit

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

333

2

2

R3

16R

3

42a3ah2area =

===

atoms in plane

atoms above plane

atoms below plane

ah23=

a2

1

= =

nm2

atoms7.0

m2

atoms0.70 x 1019

3 2R3

16

Planar Density =

atoms2D repeat unit

area

2D repeat unit

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ENGG ZC 232 Engineering Materials. 17

CRYSTALS AS BUILDING BLOCKS

12/6/2013 17

Someengineering applications require single crystals:

Crystal properties reveal features

of atomic structure.

(Courtesy P.M. Anderson)

--Ex: Certain crystal planes in quartz

fracture more easily than others.

--diamond single

crystals for abrasives

--turbine blades

Fig. 8.30(c), Callister 6e.

(Fig. 8.30(c) courtesy

of Pratt and Whitney).(Courtesy Martin Deakins,

GE Superabrasives,

Worthington, OH. Used with

permission.)

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POLYCRYSTALS

12/6/2013 18

Mostengineering materials are polycrystals. (Composed

of many small crystals or grains)

Nb-Hf-W plate with an electron beam weld.

Each "grain" is a single crystal. If crystals are randomly oriented,

overall component properties are not directional. Crystal 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 6e.

(Fig. K is courtesy of

Paul E. Danielson,

Teledyne Wah Chang

Albany)1 mm

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SINGLE VS POLYCRYSTALS

12/6/2013 19

Single Crystals

-Properties vary withdirection: anisotropic.

-Example: the modulus

of elasticity (E) in BCC iron:

Polycrystals

-Properties may/may not

vary with direction.

-If grains are randomly

oriented: isotropic.(Epoly iron= 210 GPa)

-If grains are textured,

anisotropic.

200 mm

Data from Table 3.3,

Callister 6e.(Source of data is R.W.

Hertzberg, Deformation

and Fracture Mechanics

of Engineering

Materials, 3rd ed., John

Wiley and Sons, 1989.)

Adapted from Fig.

4.12(b), Callister 6e.

(Fig. 4.12(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].)

SUMMARY

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Vacancy atoms Interstitial atoms

Substitutional atomsPoint defects

Types of Imperfections

Dislocations Line defects

Grain Boundaries Area defects

06/08/2012ENGG ZC232 Engineering

Materials Lecture IV (Crystal

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Vacancies:-vacant atomic sites in a structure.

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

Point Defects

Vacancy

distortion

of planes

self-interstitial

distortionof planes



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Boltzmann's constant

(1.38 x 10

-23

J/atom-K)(8.62 x 10-5 eV/atom-K)

NvN

=exp -QvkT

No. of defects

No. of potentialdefect sites.

Activation energy

Temperature

Each lattice siteis a potentialvacancy site

Equilibrium concentration varies with temperature!

Equilibrium Concentration:

Point Defects



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We can get Qv froman experiment.

NvN

= exp -QvkT

Measuring Activation Energy

Measure this...

Nv

N

T

exponential

dependence!

defect concentration

Replot it...

1/T

N

Nvln

-Qv/k

slope



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Find the equil. # of vacancies in 1 m3of Cu at 1000C. Given:

ACu= 63.5 g/mol= 8.4 g/cm3Qv = 0.9 eV/atom NA= 6.02 x 1023 atoms/mol

Estimating Vacancy Concentration

For 1 m3, N= NAACu

x x 1 m3= 8.0 x 1028sites

8.62 x 10-5 eV/atom-K

0.9 eV/atom

1273K

NvN

=exp -Qv

kT

= 2.7 x 10

-4

Answer:

Nv = (2.7 x 10-4)(8.0 x 1028) sites = 2.2 x 1025vacancies



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

Solid solutionof BinA(i.e., random dist. of point defects)

Solid solution of BinAplus particles of a new

phase (usually for a larger amount of B)

OR

Substitutionalsolid soln.

(e.g., Cuin Ni)

Interstitialsolid soln.

(e.g., Cin Fe)

Second phase particle

--different composition

--often different structure.

Point Defects in Alloys

06/08/2012ENGG ZC232 EngineeringMaterials Lecture IV (Crystal

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Imperfections in Solids

Conditions for substitutional solid solution (S.S.) W. HumeRothery rule

1. r (atomic radius) < 15%

2. Proximity in periodic table

i.e., similar electronegativities

3. Same crystal structure for pure metals

4. Valency

All else being equal, a metal will have a greater tendency

to dissolve a metal of higher valency than one of lower

valency

06/08/2012ENGG ZC232 EngineeringMaterials Lecture IV (Crystal

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Application of HumeRothery rulesSolidSolutions

1. Would you predict

more Al or Agto dissolve in Zn?

2. More Zn or Alin Cu?

Table on p. 141,

Callisters Materials Science and Engineering,

Adapted Version.

Element Atomic Crystal Electro- Valence

Radius Structure nega-

(nm) tivity

Cu 0.1278 FCC 1.9 +2

C 0.071H 0.046

O 0.060

Ag 0.1445 FCC 1.9 +1

Al 0.1431 FCC 1.5 +3

Co 0.1253 HCP 1.8 +2

Cr 0.1249 BCC 1.6 +3

Fe 0.1241 BCC 1.8 +2Ni 0.1246 FCC 1.8 +2

Pd 0.1376 FCC 2.2 +2

Zn 0.1332 HCP 1.6 +2

06/08/2012

ENGG ZC232 EngineeringMaterials Lecture IV (Crystal

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Linear Defects (Dislocations) Are one-dimensional defects around which atoms are misaligned

Edge dislocation: extra half-plane of atoms inserted in a crystal structure

bto dislocation lineScrew dislocation: spiral planar ramp resulting from shear deformation

bto dislocation line

28


Burgers vector, b:measure of lattice distortion

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Edge Dislocation

29


Fig. 5.7


Adapted Version.

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Motion of Edge Dislocation Dislocation motion requires the successive bumping

of a half plane of atoms (from left to right here).

Bonds across the slipping planes are broken andremade in succession.

Atomic view of edge

dislocation motion from

left to right as a crystalis sheared.

(Courtesy P.M. Anderson)

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Screw Dislocation

31


From Fig. 5.8


Adapted Version.

Burgers vector b

Dislocation

line

b

(a)

(b)

Screw Dislocation

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Edge, Screw, and Mixed Dislocations

From Fig. 5.9


Adapted Version.

Edge

Screw

Mixed

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Dislocations are visible in electron micrographs

33


From Fig. 5.10

Callisters Materials Science and Engineering,Adapted Version.

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Dislocations & Crystal Structures

Structure: close-packed

planes & directions

are preferred.

view onto two

close-packed

planes.

close-packed plane (bottom) close-packed plane (top)

close-packed directions

Comparison among crystal structures:FCC: many close-packed planes/directions;

HCP: only one plane, 3 directions;

BCC: none

Specimens thatwere tensile

tested.

Mg (HCP)

Al (FCC)

tensile direction

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Summary

Point, Line, andAreadefects exist in solids.

The number and type of defects can be varied

and controlled (e.g., Tcontrols vacancy conc.)

Defects affect material properties (e.g., grainboundaries control crystal slip).

Defects may be desirable or undesirable

(e.g., dislocations may be good or bad, depending

on whether plastic deformation is desirable or not.)

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