CHAPTER

4Solidification & Crystalline Imperfections in Solids

4-1

“Crystals are like people, the defects in them tend to

make them interesting!” - Colin Humphreys.

“Crystals are like people, the defects in them tend to

make them interesting!” - Colin Humphreys.

ISSUES TO ADDRESS...

• What types of defects arise in solids?

• Can the number and type of defects be varied and

controlled?

• How do defects affect material properties?

• Are defects undesirable?

• What are the solidification mechanisms?

Solidification of MetalsSolidification of Metals

• Metals are melted to produce finished and semi-finished parts.

• Two steps of solidification Nucleation : Formation of stable nuclei. Growth of nuclei into crystals & the formation of grain structure.

• Thermal gradients define the shape of each grain. • Equiaxed grains are formed when their growth is equal in all directions.

Liquid

Nuclei

Crystals that willForm grains

Grain Boundaries

Grains

4-2Figure 4.1

Formation of Stable Nuclei

• Two main mechanisms: Homogenous and Heterogeneous.

• Homogenous Nucleation : First and simplest case. Metal itself will provide atoms to form nuclei. Metal, when significantly under-cooled, has several slow moving

atoms which bond each other to form nuclei. For a nucleus to be stable grow into crystal, it must reach a

critical size.Cluster of atoms below critical size is called embryo.Because of their instability, embryos are continuously being formed

and re-dissolved in the molten metal due to the agitation of the atoms.

If the cluster of atoms reach critical size, they grow into crystals. Cluster of atoms that are grater than critical size are called nucleus.

4-3

Energies involved in homogenous nucleation.

Volume free energy Gv

• Released by liquid to solid transformation (LST).

• The difference in the ΔGv between liquid and solid is the driving energy driving energy for the LST.

• ΔGv is change in free energy per unit volume between liquid and solid.

• Free energy change for a spherical nucleus of radius r is given by:

Surface energy Gs• Required to form new solid

surface.

• ΔGs is energy needed to create a surface.

• γ is specific surface free energy of the particle.

Then

• ΔGs is retarding energy.

2s 4G r

4-4

In homogenous nucleation, two kinds of energy changes must be considered :

1.The volume (or bulk) free energy released by the liquid-to-solid transformation.

2.The surface energy required to form the new surfaces of the solidified particles.

vV GrG 3

3

4

Total Free Energy• Total free energy is given by• in nature, a system can change spontaneously from a higher-to a lower-energy

state. In the case of the freezing of a pure metal, if the solid particles formed upon freezing have radii < Critical radius, , the energy of the system will be lowered if they re-dissolve.

23 43

4rGrG vT

Nucleus

Above criticalradius r*

Below criticalradius r*

Energy lowered by

growing intocrystals

EnergyLowered byredissolving

r*r

ΔG

+

- ΔGv

ΔGs

ΔGT

4-4

r*

Figure 4.4

Since when r=r*, d(ΔGT)/dr = 0Since when r=r*, d(ΔGT)/dr = 0

VGr

2*

*r

Critical Radius Versus Under-cooling

• Greater the degree of under-cooling, greater the change in volume free energy ΔGv

• ΔGs does not change significantly.

• As the amount of under-cooling ΔT increases, critical nucleus size decreases.

• Critical radius is related to under-cooling by relation:

TH

Tr

f

m

2*

r* = critical radius of nucleus

γ = Surface free energy

ΔHf = Latent heat of fusion

Δ T = Amount of under-cooling.

r* = critical radius of nucleus

γ = Surface free energy

ΔHf = Latent heat of fusion

Δ T = Amount of under-cooling.

4-6

Heterogeneous Nucleation(HN)• Heterogeneous Nucleation: Is nucleation that occurs in a liquid

on the surfaces of its container, insoluble impurities, and structural material.

• These structures, called nucleating agents, lower the free energy required to form stable nuclei.

• Heterogeneous Nucleation takes place on nucleating agents because to form stable nucleus is lower in this material than in the pure liquid itself (Homogenous Nucleation).

• Nucleating agents also lower the critical size.

• Smaller amount of under-cooling is required to solidify.

• Used excessively in industries.

Liquid

Solid

Nucleating agent

θ

4-7

Figure 4.6

Growth of Crystals and Formation of Grain Structure

• Nuclei grow into crystals in different orientations.

• Crystal boundaries are formed when crystals join together at complete solidification.

• Crystals in solidified metals are called grains.

• Grains are separated by grain boundaries.

• The greater the

number of

nucleation sites

available, the greater

the number of

grains formed. Nuclei growing into grainsForming grain boundaries

4-8

Types of GrainsTypes of Grains

• Equiaxed Grains: Crystals, smaller in size, grow equally in all directions. Formed at the sites of high concentration of nuclei. Example:- Cold mold wall

• Columnar Grains: Long, thin, and coarse grains. Grow predominantly in one direction. Formed at the sites of slow cooling

and steep temperature gradient. Example:- Grains that are away from

the mold wall.

Columnar Grains

Equiaxed Grains

Mold

4-9

Figure 4.7a

Casting in Industries

• Molten metal is cast into either semi finished or finished parts.

4-10

Figure 4.8Direct-Chill semicontinuousCasting unit for aluminum

Figure 4.9b

Continuous castingOf steel ingots

Grain Structure in Industrial castings

• To produce cast ingots with fine grain size, grain refiners are added.

• Example:- For aluminum alloy, small amount of Titanium, Boron or Zirconium is added.

4-11

Figure 4.10(a) (b)

Grain structure ofAluminum cast without (a) and with (b)grain refiners.

After “Metals Handbook” vol. 8, 8th ed., American Society of Metals, 1973, p.164)

Solidification of Single Crystal

• For some applications (e.g, Gas turbine blades-high temperature environment), single crystals are needed.

• Single crystals have high temperature creep resistance.

• Latent heat of solidification is conducted through solidifying crystal to grow single crystal.

• Growth rate is kept slow so that temperature at solid-liquid interface is slightly below melting point.

4-12

Figure 4.12

Growth of singlecrystal for turbineairfoil.

(After Pratt and Whitney Co.)

Czochralski ProcessCzochralski Process

• This method is used to produce single crystals of silicon for electronic wafers.

• A crystal is dipped in molten silicon and rotated.

• The seed crystal is withdrawn slowly while silicon adheres to the seed crystal and grows as a single crystal.

4-13

Figure 4.13

Metallic Solid SolutionsMetallic Solid Solutions

• Alloys are used in most engineering applications.

• An Alloy is a mixture of two or more metals and nonmetals.

• Example: Cartridge brass is binary alloy of 70% Cu and 30% Zinc. Iconel is a nickel based superalloy with about 10 elements.

• Solid solution is a simple type of alloy in which elements are dispersed in a single phase.

• In general there are two types of Solid Solution :1. Substitutional Solid Solution (SSS).

2. Interstitial Solid Solution (ISS).

Solid solutions are made of a host (the solvent or matrix) which dissolves the minor component (solute). The ability to dissolve is called solubility.

4-14

Metallic Solid SolutionsMetallic Solid Solutions

Solvent: in an alloy, the element or compound present in greater amount.

Solute: in an alloy, the element or compound present in lesser amount.

Solid Solution:

homogeneous

maintain crystal structure

contain randomly dispersed impurities

(substitutional or interstitial)

Second Phase: as solute atoms are added, new compounds / structures

(New phase) are formed, or solute forms local precipitates (discussed in

Chapter 10).

Substitutional Solid Solution Substitutional Solid Solution

• Solute atoms substitute for parent solvent atom in a crystal lattice.

• The structure remains unchanged.

• Lattice might get slightly distorted due to change in diameter of the atoms.

• Solute percentage in solvent

can vary from fraction of a

percentage to 100%

Solvent atoms

Solute atoms

4-15

Figure 4.14

Substitutional Solid Solution (Cont..)

• The solubility of solids is greater if (W. Hume – Rothery rules) :

1. The diameter of atoms does not differ by more than 15%

2. Crystal structures are similar.

3. Not much difference in electronegativity of solute and solvent that should be comparable (otherwise new inter-metallic phases are encouraged) .

4. Have same valence. All else being equal, a metal will have a greater tendency to dissolve a metal of higher valency than one of lower valency.

• Examples:-

System

Atomic radius

Difference

Electro-negativity difference

Solid

Solubility

degree

Cu-Zn 3.9% 0.1 38.3% high

Cu-Pb 36.7% 0.2 0.17% Very low

Cu-Ni 2.3% 0 100% Very high

4-16

Interstitial Solid Solution

• Solute atoms fit in between the voids (interstices) of solvent atoms.

• Solvent atoms in this case should be much larger than solute atoms.

• An interstitial solution can only occur for small impurity atoms.

• Even then there is some strain, so typically maximum concentrations are small.

• Example:- between 912 and 13940C, interstitial solid solution of carbon in γ

iron (FCC) is formed.

• A maximum of 2.8%

of carbon can dissolve

interstitially in iron.

Carbon atoms r=0.075nm

Iron atoms r00.129nm

4-17Figure 4.15a

Examples

Examples

Examples

Specification of composition / ConcentrationSpecification of composition / Concentration

Composition can be expressed in:

1.Weight percent, useful when making the solution

2.Atom percent, useful when trying to understand the material at the atomic level

Weight percent (wt %): weight of a particular element relative to the total alloy

weight. For two-component system, concentration of element 1 in (wt. %) is:

weight percent:

m1 = mass of component 1

Atom percent (at %): number of moles (atoms) of a particular element relative to the

total number of moles (atoms) in alloy. For two-component system, concentration of

element 1 in at. % is:

atom percent:

nm1 = number of moles of component 1, , : is weight in grams of element 1,

is atomic weight of element 1).

100x 21

11 mm

mC

100x 21

1'1

mm

m

nn

nC

'1m

1

'1

1 A

mnm

1A

Composition Conversions

ExampleExample

ExampleExample

Crystalline ImperfectionsCrystalline Imperfections• Defect: deviations from the ideal are called imperfection or defects.

• Note that even if we were able to build a perfect crystal, unless we could keep it at absolute zero (T=0K), defects would appear.

• Even if they didn’t, vibrations would ensure that the crystal is not perfect

• In some casesIn some cases, it is desirable to have crystals as perfect as possible - e.g., crystals for optoelectronics;optoelectronics; in other cases, imperfections imperfections are deliberate - e.g., alloys of two randomly mixed metals for greater strengthgreater strength; doping of semiconductors to achieve specific electrical propertieselectrical properties

• In either case defect control control is important • No crystal is perfect.

• Materials are often stronger when they have defects.

• Imperfections affect mechanical properties, chemical properties, and electrical properties.

• Imperfections can be classified as 1. Zero dimensional (or point defects).

2. One dimensional (or line defects (dislocations)).

3. Two dimensional(Area) defects.

4. Three dimensional (bulk)defects (cracks).

4-18

Crystalline ImperfectionsCrystalline Imperfections

Point Defects – Vacancy

• Vacancy is formed due to a missing atom.

• Vacancy is formed (≈ one vacancy in 10000 atoms) during crystallization or

mobility of atoms.

• Energy of formation is 1 eV.

• Mobility of vacancy results in cluster of vacancies.

• Also caused due to plastic -deformation,

rapid cooling or particle bombardment.

4-19

Figure: Vacancies moving to form vacancy vacancy clustercluster

Point Defects - Interstitially

• Atom in a crystal, sometimes, occupies interstitial site.

• This does not occur naturally.

• Can be induced by irradiation.

• This defects caused structural distortion.

4-20

Figure 4.16b

Vacancies , Self/Impurity-Interstitials, and Substitutional atoms

• A vacancy is a lattice position that is vacant because the

atom is missing. It is created when the solid is formed.

• They occur naturally as a result of thermal vibrations.

• An interstitial is an atom that occupies a place outside

the normal lattice position. It may be the same type of

atom as the others (self interstitial) or an impurity atom.

• Substitutional atoms.• In the case of vacancies and interstitials, there is a change

in the coordination of atoms around the defect. This means that the forces are not balanced in the same way as for other atoms in the solid (lattice distortion).(lattice distortion).

Self-interstitials in metals introduce large distortions in the surrounding lattice ⇒ the energy of self-interstitial formation is ~ 3 times larger as compared to vacancies (Qi ≈ 3×Qv) ⇒equilibrium concentration of self-interstitials is very low (less than one self-interstitial per cm3 at room T).

Self-interstitials in metals introduce large distortions in the surrounding lattice ⇒ the energy of self-interstitial formation is ~ 3 times larger as compared to vacancies (Qi ≈ 3×Qv) ⇒equilibrium concentration of self-interstitials is very low (less than one self-interstitial per cm3 at room T).

Point Defects in Ionic Crystals

• Complex as electric neutrality has to be maintained.

• If two oppositely charged particles are missing, cation-anion di-vacancy is created (a pair of oppositely charged ion vacancies). This is Scohttky defect.

• Frenkel defect is created when cation moves to interstitial site.

• Impurity atoms are also considered as point defects.

4-21Figure 4.17

Equilibrium Concentration:Equilibrium Concentration:Point DefectsPoint Defects

Measuring Activation EnergyMeasuring Activation Energy

Estimating Vacancy Concentration

Solidification of Metals

• Arrhenius equation can also be written as

ln (rate) = ln ( C) – Q/RT

Or Log10 (rate) = Log10 (C) – Q/2.303 RT

Which is similar to

Y = b + m X

Which is equation of a straight line

With Y intercept as ‘b’ and slope ‘m’.

Y Log10(rate)X (1/T)b Log10(C)m Q/2.303R

4-39

Figure 4.34 Arrhenius plot

(After J. Wulff et al., “Structure and Properties of Materials,” vol. II: “Thermodynamics of Structure,” Wiley, 1964, p.64.)

Line Defects – (Dislocations) • Dislocations: are line defects, the interatomic bonds are significantly distorted only in the immediate vicinity of the

dislocation line. This area is called the dislocation core.

• Slip between crystal planes result when dislocations move,

• Produce permanent (plastic) deformation.

• Discrepancy is due to dislocations.

• Dislocations are lattice distortions centered around a line.

• Formed during Solidification

Permanent Deformation Vacancy condensation

• Different types of line defects are Edge dislocation

Screw dislocation Mixed dislocation

4-22

before deformation after tensile elongation

slip steps

45

Imperfections in SolidsImperfections in Solids

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

• Edge dislocation:– extra half-plane of atoms inserted in a crystal structure– b perpendicular () to dislocation line

• Screw dislocation:– spiral planar ramp resulting from shear deformation– b parallel () to dislocation line

Burger’s vector, b: measure of lattice distortionBurger’s vector, b: measure of lattice distortion

Edge Dislocation

• Created by insertion of extra half planes of atoms.

• Positive edge dislocation

• Negative edge dislocation

• Burgers vector

Shows displa-

cement of

atoms (slip).

4-23

Burgers vector

Figure 4.18

After A.G. Guy , “Essentials of Materials Science,” McGraw-Hill, 1976, p.153After M. Eisenstadt, “Introduction to Mechanical Properties of Materials,” Macmillan, 1971, p.117

47

Edge Dislocation

Fig. 5.8, Callister & Rethwisch 4e.

Edge dislocation (an extra partial plane of atoms) • there will be local lattice

distortion (relaxed at long distance).

Strain fields (compression and tension).

Mathematically slip or Burger vector b is used to characterize displacement

of atoms around the dislocation b is perpendicular to the edge-dislocation line

48

• Dislocation motion requires the successive bumping of a half plane of atoms

(from left to right here).

• Bonds across the slipping planes are broken and remade in succession.

Atomic view of edgedislocation motion fromleft to right as a crystal

is sheared.

(Courtesy P.M. Anderson)

Motion of Edge Dislocation

Screw Dislocation

• Created due to shear stresses applied to regions of a perfect crystal separated by cutting plane.

• Distortion of lattice in form of a spiral ramp.

• Burgers vector is parallel to dislocation line.

4-24 After M. Eisenstadt, “Introduction to Mechanical Properties of Materials,” Macmillan, 1971, p.118

Mixed Dislocation

• Most crystal have components

of both edge and screw

dislocation.

• Dislocation, since have

irregular atomic arrangement

will appear as dark lines

when observed in electron

microscope.

4-25

Figure 4.21

(After John Wolff et al., “Structure and Properties of Materials,” vol 3: “Mechanical Properties,” Wiley, 1965, p.65.(After “Metals Handbook” vol. 8, 8th ed., American Society of Metals, 1973, p.164)

Figure 4.22Dislocation structure of iron deformed14% at –1950C

• Most dislocations found in crystalline material are neither pure edge nor pure screw, but exhibit components of both types.

51

Edge, Screw, and Mixed Dislocations

Adapted from Fig. 5.10, Callister & Rethwisch 4e.

Edge

Screw

Mixed

Where do dislocations come from ?

• The number of dislocations in a material is expressed as the dislocation density - the total dislocation length per unit volume or the number of dislocations intersecting a unit area.

• Dislocation densities can vary from 105 cm-2 in carefully solidified metal crystals to 1012 cm-2 in heavily deformed metals.

• Where do dislocations come from ?

• Most crystalline materials, especially metals, have dislocations in their as-formed state, mainly as a result of stresses (mechanical, thermal...) associated with the forming process.

• The number of dislocations

increases dramatically

during plastic deformation

(Ch.8). Dislocations spawnspawn

from existing dislocations,

grain boundaries & surfaces.

53

Dislocations & Crystal Structures• Structure: close-packed

planes & directions are preferred.

view onto twoclose-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 that were tensile

tested.

Mg (HCP)

Al (FCC)tensile direction

Planar Defects

• Grain boundaries: twins, low/high angle

boundaries, twists and stacking faults

• Free surface is also a defect : Bonded to

atoms on only one side and hence has higher

state of energy Highly reactive

• Nano-materials have small clusters of atoms

and hence are highly reactive.

• In the previous slides, we dealt with point defects (e.g. vacancy, interstitials,

etc.) and line defects (dislocations).

• There is another class of defects called interfacial or planar defects:

– They occupy an area or surface and are therefore bi-dimensional.

– They are of great importance in mechanical metallurgy.

• Examples of these form of defects include:

– grain boundaries

– twin boundaries

– Tilt boundaries

– anti-phase boundaries & free surface of materials.

• Of all these, the grain boundaries are the most important from the

mechanical properties point of view.

Planar Defects

Grain Boundaries

• Grain boundaries separate grains.

• Formed due to simultaneously growing crystals meeting each other.

• Width = 2-5 atomic diameters.

• Some atoms in grain boundaries have higher energy.

• Restrict plastic flow and prevent dislocation movement.

4-27

Figure 4.25

3D view ofgrains

Grain BoundariesIn 1018 steel

(After A.G. Guy, “ Essentials of materials Science,” McGraw-Hill, 1976.)

At the grain boundary, there is a disturbance in the atomic packing.

• When the misorientation is large (high-angle grain boundary), more complicated structures are involved (as in a

configuration of soap bubbles simulating the atomic planes in crystal lattices).

• The grain boundaries are therefore:– where grains meet in a solid.

– transition regions between the neighboring crystals.

– Where there is a disturbance in the atomic packing.

• These transition regions (grain boundaries) may consist of various kinds of dislocation arrangements.

Tilt and Twist Boundaries

• The simplest grain boundary consists of a configuration of edge dislocations between two grains.

• The misfit in the orientation of the two grains (one on each side of the boundary) is accommodated by a perturbation of the regular arrangement of crystals in the boundary region.

• Figure below shows some vertical atomic planes termination in the boundary and each termination is represented by an edge dislocation.

Twin Boundaries

• Twin: the boundary region consisting of arrays of screw dislocations (consider joint of two halves of a cube and twist an angle around the cross section normal).

• A boundary consisting entirely of screw dislocations is called twist boundary, because the misorientation can be described by a relative rotation of two grains about an axis.

• A region in which mirror image of structure exists across a boundary is called Twin boundary.

• Formed during plastic deformation and recrystallization.

• Strengthens the metal.

Twin Plane

Other Planar DefectsTilt boundary

• Boundaries consisting entirely of edge dislocations are called tilt boundaries, because the misorientation , as can be seen in next Figure, can be described in terms of a rotation about an axis normal to the plane of the paper and contained in the plane of dislocations.

Small angle tilt boundary: Low angle grain boundary is an array of aligned edge dislocations. This type of grain boundary is called tilt boundary (consider joint of two wedges). Array of edge dislocations tilts two regions of a crystal by < 100.

• It is possible to produce misorientations between grains by combined tilt and twist boundaries. In such a case, the grain boundary structure will consist of a network of edge and screw dislocations.

Tilt and Twist Boundaries

Other Planar Defects

• Stacking faults: Piling up faults during recrystallization due to collapsing.

Example: ABCABAACBABC FCC fault

• Volume defects: Cluster of point defects join to form 3-D void.

• The Bulk or Volume Defects are:

1) Pores - can greatly affect optical, thermal, mechanical properties.

2) Cracks - can greatly affect mechanical properties.

3) Foreign inclusions - can greatly affect electrical, mechanical, optical

properties.

Experimental Techniques For Identification of Microstructures and DefectsExperimental Techniques For Identification of Microstructures and Defects

Observing Grain Boundaries – Optical MetallographyObserving Grain Boundaries – Optical Metallography

• To observe grain boundaries, the metal sample must be first mounted for easy handling

• Then the sample should be ground and polished with different grades of abrasive paper and abrasive solution.

• The surface is then etched chemically. • Tiny groves are produced at grain boundaries. • Groves do not intensely reflect light. Hence observed by optical microscope.

4-28 After M. Eisenstadt, “Introduction to Mechanical Properties of Materials,” Macmillan, 1971, p.126

Figure 4.27

Grain SizeGrain Size

• Affects the mechanical properties of the material

• The smaller the grain size are more the

grain boundaries.

• More grain boundaries means higher

resistance to slip (plastic deformation occurs due

to slip).

• More grains means are more uniform the

mechanical properties.

4-30

Grain Size Measurements

Grain structure is usually specified by giving the average diameter. Grain size can be measured by two methods.

(a) ASTM Procedure: This method of measuring grain size is common in engineering applications.

(b) Lineal Intercept Technique: This is very easy and may be the preferred method for measuring grain size.

ASTM grain size number

• ASTM grain size number ‘n’ is a measure of grain size.

N = 2 n-1 N = Number of grains per

square inch of a polished

and etched specimen at 100 x.

n = ASTM grain size number.

200 X 200 X

1018 cold rolled steel, n=10 1045 cold rolled steel, n=8

4-31

N < 3 – Coarse grained4 < n < 6 – Medium grained7 < n < 9 – Fine grainedN > 10 – ultrafine grained

Example

In a grain size measurement of an

aluminum sample, it was found that there

were 56 full grains in the area, and 48

grains were cut by the circumference of the

circle of area 1 in2. Calculate ASTM grain

size number n for this sample.

Solution

The grains cut by the circumference of the circle are taken as one-half the number. Therefore,

35.7169.038.4

12ln80ln

12lnln

2802456

24856

1

n

NnBut

N

n

Lineal Intercept Technique

3 inches 5 grains.

71

• Useful up to 2000X magnification.• Polishing removes surface features (e.g., scratches)• Etching changes reflectance, depending on crystal

orientation.

Micrograph ofbrass (a Cu-Zn alloy)

0.75mm

Optical Microscopy

Adapted from Fig. 5.18(b) and (c), Callister & Rethwisch 4e. (Fig. 5.18(c) is courtesy

of J.E. Burke, General Electric Co.)

crystallographic planes

72

Grain boundaries...

• are imperfections,• are more susceptible

to etching,• may be revealed as

dark lines,• change in crystal

orientation across boundary. Adapted from Fig. 5.19(a)

and (b), Callister & Rethwisch 4e.

(Fig. 5.19(b) is courtesyof L.C. Smith and C. Brady,

the National Bureau of Standards, Washington, DC

[now the National Institute of Standards and Technology,

Gaithersburg, MD].)

Optical Microscopy

ASTM grain size number

N = 2 n -1

number of grains/in2 at 100x magnification

Fe-Cr alloy

(b)

grain boundary

surface groove

polished surface

(a)

Effect of Etching

Unetched Steel200 X

Etched Steel200 X

Unetched Brass200 X

Etched Brass200 X

4-32

Figure 4.28

Grain Size Determination Problem

• (a) Using the intercept method, determine the average grain size, in

millimeters, of the specimen whose microstructure is shown in Figure

4.14(b); us e at least seven straight-line segments.

• (b) Estimate the ASTM grain size number for this material.

• Solution:

• (a) Below is shown the photomicrograph of Figure 4.14(b), on which

seven straight line segments, each of which is 60 mm long has been

constructed; these lines are labeled “1” through “7”.

• In order to determine the average grain diameter, it is necessary to

count the number of grains intersected by each of these line segments.

These data are tabulated below.

Line Number

No. Grains Intersected

1 11

2 10

3 9

4 8.5

5 7

6 10

7 8

The average number of grain boundary intersections for these lines was 9.1. Therefore, the average line length intersected is just: hence, the average grain diameter, d,

is

• (b) This portion of the problem calls for us to estimate the ASTM grain size number for this

same material.

• The average grain size number, n, is related to the number of grains per square inch, N, at a

magnification of 100× according to Equation 4.16. Inasmuch as the magnification is 100×, the

value of N is measured directly from the micrograph. The photomicrograph on which has been

constructed a square 1 in. on a side is shown below.

•The total number of complete grains within this square is

approximately 10 (taking into account grain fractions).

•Now, in order to solve for n in Equation 4.16, it is first

necessary to take logarithms as:

log N = (n − 1) log 2

From which n equals:

Transmission Electron Microscope (TEM)Transmission Electron Microscope (TEM)

• Electron produced by heated tungsten filament.

• Accelerated by high voltage (75 - 120 KV)

• Electron beam passes through very thin specimen.

• Difference in atomic arrangement change directions of electrons.

• Beam is enlarged and focused on fluorescent screen.

4-26

Figure 4.24

Collagen Fibrilsof ligament asseen in TEM

(After L.E. Murr, “ Electron and Ion Microscopy and Microanalysis, “ Marcel Decker, 1982, p.105)

TEM (..Cont)

• TEM needs complex sample preparation• Very thin specimen needed ( several hundred

nanometers)• High resolution TEM (HRTEM) allows

resolution of 0.1 nm.• 2-D projections of a crystal with accompanying

defects can be observed. Low angle boundaryAs seenIn HTREM

The Scanning Electron Microscope (SEM)The Scanning Electron Microscope (SEM)

• Electron source generates electrons.

• Electrons hit the surface and secondary electrons are produced.

• The secondary electrons are collected to produce the signal.

• The signal

is used to

produce

the image.

TEM of fractured metal end4-33

Figure 4.31

After V.A. Phillips, “Modern Photographic techniques and Their Applications,” Wiley, 1971, p.425

Scanning Probe Microscopy

• Scanning Tunneling Microscope (STM) and Atomic Force Microscope (AFM).

• Sub-nanometer magnification.• Atomic scale topographic map of surface.• STM uses extremely sharp tip.• Tungsten, nickel, platinum

- iridium or carbon nanotubes

are used for tips.

Scanning Tunneling Microscope

• Tip placed one atom diameter from surface.• Voltage applied across tip and surface.• Electrons tunnel the gap and produce current.• Current produced is proportional to change in

gap.• Can be used only for conductive materials.

Constant height and current modes Surface of platinum with defects

Atomic Force Microscope

• Similar to STM but tip attached to cantilever beam.

• When tip interacts with surface, van der waals forces deflect the beam.

• Deflection detected by laser

and photodetector.• Non-conductive materials

can be scanned.• Used in DNA research and

polymer coating technique.

Thanks For Your Attention…..!?

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Chapter Chapter

55

DIFFIUSION DIFFIUSION

Dr. Khalil Alhatab