Chapter 4

IMPERFECTION IN CRYSTAL

PHY351

Mechanical Deformation

Solidification of Metal

There are several steps of solidification:

Nucleation : Formation of stable nuclei (Fig a)

Growth of nuclei : Formation of grain structure (Fig b)

Formation of grain structure (Fig c)

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Liquid

Nuclei

Crystals that will

Form grains

Grain Boundaries

Grains

(a) (b) (c)

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Solidification of Single Crystals - Czochralski Process

For some applications, single crystals are needed due to single

crystals have high temperature and creep resistance.

This method is used to produce single crystal of silicon for

electronic wafers.

A seed crystal is dipped in molten silicon and rotated.

The seed crystal is withdrawn slowly while silicon adheres to seed

crystal and grows as a single crystal.

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Metallic Solid Slutions- 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.

Solvent atoms

Solute atoms

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Metallic Solid Slutions- Interstitial Solid Solution

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

Carbon atoms r = 0.077nm

Iron atoms r = 0.124nm

Crystalline Imperfections

The arrangement of the atoms or ions in engineered materials are

never perfect and contains various types of imperfections and

defects.

Imperfections affect mechanical properties, chemical properties

and electrical properties of materials.

These imperfections only represent defects or deviation from the

perfect or ideal atomic or ionic arrangements expected in a given

crystal structure.

Note:

The materials is not considered defective from an application

viewpoint.

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Crystal lattice imperfections can be classified according to their

geometry and shape .

There are several basic types of imperfection as following:

Zero dimension point defects.

One dimension / line defects (dislocations).

Two dimension defects.

Three dimension defects (cracks).

Zero Dimension Point Defects

Point defects are localized disruption in otherwise perfect atomic

or ionic arrangements in a crystal structure.

The disruption affects a region involving several atoms or ions or

pair of atoms or ions only.

These imperfection may be introduced by movement of the atoms

or ions when they gain energy by:

Heating

during processing of the material

Introduction of impurities

doping

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Point Defects – Vacancy

The simplest point defect is the vacancy,

which formed due to a MISSING atom from

its normal site in the crystal structure.

Vacancy is produced during solidification resulting:

a local disturbance during crystallization

atomic arrangements in an existing crystal due to atomic

mobility.

Vacancies also caused due to plastic deformation, rapid cooling or

particle bombardment.

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Vacancies can MOVE by exchanging positions with their neighbors.

This process is important in the migration or diffusion of atoms in

the solid state, particularly at high temperature where atomic

mobility is greater.

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Point Defects – Interstitial Defects

An interstitial defects is formed when an extra atom or ion is

INSERTED into the crystal structure at interstitial site which a

normally unoccupied position.

This defects not occur naturally. It is due irradiation and causing a

structural distortion.

Interstitial atoms :

- are often present

as impurities

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Point Defects – Substitutional Defects

A substitutional defects is introduced when one atom or ion is

REPLACED by a different type of atom or ion.

The substitutional atoms or ions occupy the normal lattice site.

The substitutional atoms or ion can either be larger or smaller than

normal atoms or ions in the crystal structure.

Substitutional defects can be introduced either as an impurity or as

a deliberate alloying addition.

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A Schottky defects is uniques to ionic materials and is commonly

found in many ceramic materials.

In this defects, vacancies occur when TWO appositely charged

particles are MISSING form the ionic crystal due to need to

maintain electrical neutrality.

Point Defects in Ionic Crystal

If a positive cation MOVES into an interstitial site in an ionic

crystal, a cation vacancy is created in the normal ion site. This

vacancy-interstitialcy pair is called Frenkel imperfection.

Although this is described for an ionic material, a Frenkel defect

can occur in metals and covalently bonded materials.

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The present of these defects in ionic crystal increases their

electrical conductivity.

Impurity atoms are also considered as point defects.

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Many processes involved in the production of engineering materials

are concerned with the RATE at which atoms move in the solid

state.

In this processes, a reactions occur in the solid state which involve

the spontaneous rearrangement of atoms into a new and more

stable arrangements.

Reacting atoms must have sufficient energy to overcome activation

energy barrier.

The energy required which above the average energy of the atom is

called ACTIVATION ENERGY, * . (Unit :Joules/mole).

Thermal Production

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At any temperature, only a fraction of the molecules or atoms in a

system will have sufficient energy to reach the activation energy

level of E*.

As temperature increases, more and more atoms acquire activation

energy level.

PROBABILITY of finding an atom/molecule with energy E*

GREATER than average energy, E of all atoms/ molecules is given

by

Where;

K = Boltzman’s Constant = 1.38 x 10-23 J/(atom K)

T = temperature in Kelvin

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Probability e –(E* - E) /KT

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The FRACTION of atoms having energies greater than E* in a system

(when E* is greater than average energy E) is given by

Where;

n = Number of molecules greater than energy E*

Ntotal = Total number of molecules

k = Boltzman’s Constant

C = Constant

T = Temperature in Kelvin.

TK

E

total

CeN

n.

*

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The NUMBER OF VACANCIES at equilibrium at a particular

temperature in a metallic crystal lattice is given by

Where;

nv = Number of vacancies per m3 of metal

Ev = Activation Energy to form a vacancy

T = Absolute Temperature

k = Boltzman’s Constant

C = Constant

TK

E

vV

CeN

n.

Question 1

By assuming the energy formation of a vacancy in pure copper is

0.9 eV, C = 1 and N = 8.49 x 1028 atoms/m3, calculate

a. The equilibrium number of vacancies per cubic meter in pure

copper at 5000C

(Answer: 1.2 x 1023 vacancies/m3)

b. The vacancy fraction at 5000C in pure copper.

(Answer : 1.4 x 10-6)

Constant:

k = 8.62 x 10-5 eV/K

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Stability of Atoms and Ions

Atoms and ion in their normal positions in the crystal structures are

not stable or at rest.

Instead, the atoms or ions posses thermal energy and they will

move.

For instance, an atom may move from a normal crystal structure

location to occupy a nearby vacancy. An atom may also move from

one interstitial site to another. Atoms or ions may jump across a

grain boundary causing the grain boundary to move.

The ability of atoms or ions to diffuse increases as temperature

possess by the atoms or ions increases.

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The rate of atom or ion movement that related to temperature or thermal energy is given by Arrhenius equation.

Rate of reaction = Ce-Q/RT

Where;

Q = Activation energy J/mol

R = Molar gas constant J/mol.K

T = Temperature in Kelvin

C = Rate constant ( Independent of temperature)

Note:

Rate of reaction depends upon number of reacting molecules.

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Arrhenius equation can also be written as:

ln (rate) = ln ( C) – Q/RT

Or

Log10 (rate) = Log10 (C) – Q/2.303 RT

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Y Log10(rate)

X (1/T)

b Log10(C)

m Q/2.303R

Question 2

Suppose that interstitial atoms are found to move from one site to

another at the rates of 5 x 108 jumps/s at 5000C and 8 x 1010

jumps/s at 8000C. Calculate the activation energy for the process.

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Solid State Diffusion

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DIFFUSION is a process by which a matter is transported through

another matter.

Examples:

Movement of smoke particles in air : Very fast.

Movement of dye in water : Relatively slow.

Solid state reactions : Very restricted movement due to bonding.

(A nickel sheet bonded to a cooper sheet. At high temperature,

nickel atom gradually diffuse in the cooper and cooper migrate

into the nickel)

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There are two main mechanisms of diffusion of atoms in a

crystalline lattice:

The vacancy or substitutional mechanism

The interstitial mechanism

Diffusion Mechanism

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Vacancy or Substitutional Diffusion

Atoms diffuse in solids IF:

Vacancies or other crystal defects are present

There is enough activation energy

Atoms move into the vacancies present.

More vacancies are created at higher temperature.

Diffusion rate is higher at high temperatures.

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Example:

If atom ‘A’ has sufficient activation energy, it

moves into the vacancy self diffusion.

As the melting point increases, activation energy also increases.

Activation

Energy of

Self diffusion

Activation

Energy to

form a

Vacancy

Activation

Energy to

move a

vacancy

= +

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Interstitial Diffusion mechanism

Atoms MOVE from one interstitial site to another.

The atoms that move must be much smaller than the matrix atom.

Interstitial atomsMatrix

atoms

Steady State Diffusion

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There is NO CHANGE in concentration of solute atoms at

different planes in a system, over a period of time.

No chemical reaction occurs. Only net flow of atoms.

The rate at which atoms, ions, particles or other species diffuse

in a material can be measured by the flux, J.

The flux is defined as the number of atoms passing through a

plane of unit area per unit time.

For steady state diffusion condition, the net flow of atoms by

atomic diffusion is equal to diffusion D times the diffusion

gradient dc/dx. This is defined as Fick’s First Law.

Rate of Diffusion (Fick’s First Law)

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The flux or flow of atoms is given by:

Where;

J = Flux or net flow of atoms (Unit: atoms/m2s)

D = Diffusivity or Diffusion coefficient (Unit: m2/s)

= Concentration Gradient (Unit: atoms/m3.m)

dx

dcDJ

dx

dc

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The concentration gradient shows how the composition of the

material varies with distance: c is the difference in concentration

over the distance x.

Diffusivity depends upon:

Type of diffusion : Whether the diffusion is interstitial or substitutional.

Temperature: As the temperature increases diffusivity increases.

Type of crystal structure: BCC crystal has lower APF than FCC and hence

has higher diffusivity.

Type of crystal imperfection: More open structures (grain boundaries)

increases diffusion.

The concentration of diffusing species: Higher concentrations of diffusing

solute atoms will affect diffusivity

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The concentration gradient shows how the composition of the

material varies with distance: c is the difference in concentration

over the distance x.

Diffusivity depends upon:

Type of diffusion : Whether the diffusion is interstitial or substitutional.

Temperature: As the temperature increases diffusivity increases.

Type of crystal structure: BCC crystal has lower APF than FCC and hence

has higher diffusivity.

Type of crystal imperfection: More open structures (grain boundaries)

increases diffusion.

The concentration of diffusing species: Higher concentrations of diffusing

solute atoms will affect diffusivity

Question 3

a. One way to manufacture transistor which amplify electrical

signals is to diffuse impurity atoms into a semiconductor material

such as silicon (Si). Suppose a silicon wafer with 0.1 cm thick,

which originally contains one phosphorus (P) atom for every 10

million Si atoms, is treated so that there are 400 phosphorus

atoms for every 10 million Si atoms at the surface. Calculate the

concentration gradient. (Given the lattice parameter of Si is

5.4307Å)

b. The diffusion flux of cooper solute atoms in aluminium solvent

from point A to point B, 10 mm apart is 4 x 1017 atoms/m2s at

5000C. Determine

i. The concentration gradient (Given D5000C = 4 x10-14 m2/s)

ii. Difference in the concentration levels of cooper between the two points.

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Non- Steady Diffusivity

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Concentration of solute atoms at any point in metal CHANGES with

time in this case.

Ficks second law:- Rate of compositional change is equal to

diffusivity times the rate of change of concentration gradient.

dx

dcD

dx

d

dt

dC xx

Change of concentration of solute

Atoms with change in time in different planes

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Cs = Surface concentration of

element in gas diffusing

into the surface.

C0 = Initial uniform concentration

of element in solid.

Cx = Concentration of element at

distance x from surface at

time t1.

x = distance from surface

D = diffusivity of solute

t = time.

Distance x

C0

Cx

Cs Time = t2

Time= t1

Time = t0

x

Dt

xerf

CC

CC

s

xs

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Question 4

Consider the gas carburizing of a gear of 1020 steel at 9270C.

Calculate the time necessary to increase the carbon content at

0.4% at 0.5mm below the surface. Assume that the carbon content

at the surface is 0.9% and that the steel has a nominal carbon

content of 0.2%. Given D steel at 9270C = 1.28 x 10-11 m2/s.

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Erf Z Z

0.7112 0.75

0.7143 X

0.7421 0.8

Question 5

Consider the gas carburizing of a gear of 1020 at 9270C as Question

4. Only in this problem calculate the carbon content at 0.5mm

beneath the surface of the gear after 5h carburizing time. Assume

that the carbon content of the surface of the gear is 0.9% and that

the steel has a nominal carbon content of 0.2%. Given D steel at

9270C = 1.28 x 10-11 m2/s.

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Erf Z Z

0.5000 0.5205

0.521 X

0.550 0.5633

Effect of Temperature on Diffusion

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Dependence of rate of diffusion on temperature is given by

RT

QDD

RT

QDD

eDD RT

Q

303.2loglog

lnln

01010

0

0

D = Diffusivity m2/s

D0 = Proportionality constant m2/s

Q = Activation energy of diffusing species J/mol

R = Molar gas constant = 8.314 J/mol.K

T = Temperature (K)

Question 6

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1. Calculate the value of the diffusivity, D for the diffusion of

carbon in γ iron (FCC) at 9270C.

(Given D0= 2x10-5 m2/s, Q=142kJ/mol, R=8.314J/mol.k)

2. The diffusivity of silver atoms in silver is 1 x 10-17 m2/s at

5000C and 7 x 10-13 m2/s at 10000C. Calculate activation

energy, Q for the diffusion of silver in the temperature range

5000C and 10000C.

(Given R=8.314J/mol.k)

Linear/Line Defects – (Dislocations)

Line imperfection or dislocation are defects that cause lattice

distortions.

Dislocation are created during:

Solidification

Permanent deformation of crystalline solid

Vacancy condensation

Atomic mismatch in solid solution

Different types of line defects are:

Edge dislocation

Screw dislocation

Mixed dislocation

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Edge Dislocation

An edge dislocation is created in a crystal by insertion of extra half

planes of atoms.

In figure 4.18, a linear defect occurs in the region just above the

inverted T, where an extra half plane of atoms has been wedged in.

Positive edge

dislocation

Negative edge

dislocation

Figure 4.18 : Positive edge dislocation in a crystalline lattice.

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Screw Dislocation

The screw dislocation can be formed in a perfect crystal by

applying upward and downward shear stresses to regions of a

perfect crystal that have been separated by a cutting plane as

shown in Figure 4.20a.

Figure 4.20a

Formation of a screw dislocation:

A perfect crystal is sliced by a cutting plane, and up and

down shear stresses are applied parallel to the cutting

plane to form the screw dislocation as in (b).

These shear stresses introduce a region of distorted crystal lattice

in the form of a spiral ramp of distorted atoms or screw dislocation

as Figure 4.20b.

Figure 4.20b

Formation of a screw dislocation:

A screw dislocation is shown with its slip or Burgers

vector b parallel to the dislocation.

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The region of distorted crystal is not well defined and is at least

several atoms in diameter.

A region of shear strain is created around the screw dislocation in

which energy stored.

The slip or Burgers vector of the screw dislocation is parallel to

the dislocation line as shown in Figure 4.20b.

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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.

Dislocation structure of iron deformed

14% at –1950C

Planar Defects

Planar defects including:

Grain boundaries

Twins / twin boundaries

low/high angle boundaries

Stacking faults / pilling-up fault

Grain boundaries are most effective in strengthening a metal

compared to twin boundaries, low/high boundaries and stacking

faults.

The free or external surface of any material is also a defect and is

the most common type of planar defect.

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The free or external surface are considered defects because:

Atom on the surface are bonded to atoms on only one side.

Therefore, the surface atoms have a lower number of neighbors.

As a result, these atoms have higher state of energy when

compared to the atoms positioned inside the crystal with an

optimal number of neighbors.

The higher energy associated with the atoms on the surface of a

material makes the surface susceptible to erosion and reaction

with elements in the environment.

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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.

3D view of

grains

Grain Boundaries

In 1018 steel

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Twin Boundaries

Twins:

- A region in which mirror image pf structure exists across a

boundary.

Formed during plastic deformation and recrystallization.

Strengthens the metal.

Twin

Twin

Plane

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Small angle tilt boundary

Small angle tilt boundary:

- Array of edge dislocations tilts two regions of a crystal by < 100

Figure 4.24

(a) Edge dislocations in an array forming a small-angle tilt boundary

(b) Schematic of a small-angle twist boundary.

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Stacking faults / Piling up faults

Stacking faults / Piling up faults :

- form during recrystallization due to collapsing.

Example:

ABCABAACBABC FCC fault

Three dimensional imperfections

Volume or three-dimensional defects:

- produced when a cluster of point defects join to form a three-

dimensional void or a pore.

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Observing Grain Boundaries

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 such as:

Transmission Electron Microscope (TEM)

Scanning electron microscope (SEM)

High Resolution Transmission Electron Microscope (HRTEM)

Scanning Tunneling Microscope (STM) – scanning probe microscope

Atomic Force Microscope (AFM) – scanning probe microscope

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Grain Size

Affects the mechanical properties of the material.

The smaller the grain size, more are the grain boundaries.

More grain boundaries means higher resistance to slip (plastic

deformation occurs due to slip).

More grains means more uniform the mechanical properties are.

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N < 3 – Coarse grained

4 < n < 6 – Medium grained

7 < n < 9 – Fine grained

N > 10 – ultrafine grained

Measuring Grain Size

ASTM grain size number ‘n’ is a measure of grain size:

N = 2 n-1

N = Number of grains per square 2.54 x 10-2m2 polished and etched

specimen at 100 x.

n = ASTM grain size number.

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Exercise 7

1. An ASTM grains size determination is being made from a

photomicrograph of a metal at magnification of 100X. What is the

ASTM grain-size number of the metal if there are 64 grains per

square 2.54 x 10-2 m ?

(Answer : 7)

2. If there are 60 grains per square 2.54 x 10-2 m on a

photomicrograph of a metal at 200X, what is the ASTM grain-size

number of the metal?

(Answer : 8.91)

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Average Grain Diameter

Average grain diameter more directly represents grain size.

Random line of known length is drawn on photomicrograph.

Number of grains intersected is counted.

Ratio of number of grains intersected to length of line, nL is determined.

d = C/nLM

d = average grain diameter

C = constant, typically 1.5

M = magnification

nL = number of grains intersected to line per length of line

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Exercise 8

Estimate the average grain diameter of a micrograph below. Given

C= 1.5 and M=200X.

(Answer :14mm)

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References

A.G. Guy (1972) Introduction to Material Science, McGraw Hill.

J.F. Shackelford (2000). Introduction to Material Science for Engineers, (5th Edition), Prentice Hall.

W.F. Smith (1996). Priciple to Material Science and Engineering, (3rd Edition), McGraw Hill.

W.D. Callister Jr. (1997) Material Science and Engineering: An Introduction, (4th Edition) John Wiley.

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