e RT Where; D0 = Diffusion coefficient at infinite temperature Q = Activation energy for diffusion R = Gas constant T = Absolute temperature This chapter discusses imperfections in crystals and solidification processes. There are several types of imperfections including point defects like vacancies and interstitial atoms, and line defects like dislocations. Solidification involves nucleation of stable nuclei from liquid, growth of these nuclei into a grain structure, and formation of grain boundaries. Single crystals can be grown using the Czochralski process by slowly pulling a seed crystal from a melt. Diffusion, which is temperature dependent, allows for solid state reactions and involves either vacancy or interstitial mechanisms.

1. PHY351
Chapter 4
IMPERFECTION IN CRYSTAL

2. 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)

Grains
Nuclei
Liquid
(a)
2
Crystals that will
Form grains
(b)
Grain Boundaries
(c)

3. 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The seed crystal is withdrawn slowly while silicon adheres to seed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.

3

4. 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
4

5. Metallic Solid Slutions- Interstitial Solid Solution
Solute atoms fit in between the void (interstices) of solvent atoms.

Iron atoms r = 0.124nm
Carbon atoms r = 0.077nm
5

6. 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.
6

7. 
Crystal lattice imperfections can be classified according to their
geometry and shape .

There are several basic types of imperfection as following:




7
Zero dimension point defects.
One dimension / line defects (dislocations).
Two dimension defects.
Three dimension defects (cracks).

8. 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:




8
Heating
during processing of the material
Introduction of impurities
doping

9. 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.
9

10. 
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.
10

11. 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
11

12. 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.
12

13. Point Defects in Ionic Crystal

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.
13

14. 
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.
14

15. 
The present of these defects in ionic crystal increases their
electrical conductivity.
Impurity atoms are also considered as point defects.

15

16. Thermal Production

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).
16

17. 
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
Probability  e –(E* - E) /KT
Where;
K = Boltzman’s Constant = 1.38 x 10-23 J/(atom K)
T = temperature in Kelvin
17

18. 
The FRACTION of atoms having energies greater than E* in a system
(when E* is greater than average energy E) is given by
n
N total
 E*
 Ce K .T
Where;
n
= Number of molecules greater than energy E*
Ntotal = Total number of molecules
k
= Boltzman’s Constant
C
= Constant
T
= Temperature in Kelvin.
18

19. 
The NUMBER OF VACANCIES at equilibrium at a particular
temperature in a metallic crystal lattice is given by
nv
N
 EV
 Ce K .T
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
19

20. 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
20

21. 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.
21

22. 
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.
22

23. 
Arrhenius equation can also be written as:
ln (rate)
= ln ( C)
–
Q/RT
Or
Log10 (rate) = Log10 (C) – Q/2.303 RT
Y
Log10(rate)
X
b
Log10(C)
m
23
(1/T)
Q/2.303R

24. 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.
24

25. Solid State Diffusion

DIFFUSION is a process by which a matter is transported through
another matter.

Examples:



25
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)

26. 
There are two main mechanisms of diffusion of atoms in a
crystalline lattice:


26
The vacancy or substitutional mechanism
The interstitial mechanism

27. Diffusion Mechanism
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.
27

28. 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
28
Activation
Energy to
form a
Vacancy
+
Activation
Energy to
move a
vacancy

29. Interstitial Diffusion mechanism

Atoms MOVE from one interstitial site to another.

The atoms that move must be much smaller than the matrix atom.
Matrix
atoms
29
Interstitial atoms

30. Steady State Diffusion

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.
30

31. Rate of Diffusion (Fick’s First Law)
The flux or flow of atoms is given by:

J  D
dc
dx
Where;
J = Flux or net flow of atoms (Unit: atoms/m2s)
D = Diffusivity or Diffusion coefficient (Unit: m 2/s)
dc
dx
31
= Concentration Gradient (Unit: atoms/m3.m)

32. 
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:






32
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

33. 
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:






33
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

34. 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.
ii.
34
0
The concentration gradient (Given D500 C = 4 x10-14 m2/s)
Difference in the concentration levels of cooper between the two points.

35. Non- Steady Diffusivity

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.
d  dc x 
 D
 dx 

dt
dx 

dC x
Change of concentration of solute
Atoms with change in time in different planes
35

36.  x
Cs  C x
 erf 
 2 Dt
Cs  C0

Cs = Surface concentration of
element in gas diffusing
into the surface.
C0 = Initial uniform concentration




Cs
Time = t2
Time= t1
Cx
Time = t0
of element in solid.
Cx = Concentration of element at
C0
distance x from surface at
time t1.
x = distance from surface
D = diffusivity of solute
t = time.
36
x
Distance x

37. Question 4
Consider the gas carburizing of a gear of 1020 steel at 927 0C.
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.
Erf Z
0.7112
0.75
0.7143
X
0.7421
37
Z
0.8

38. Question 5
Consider the gas carburizing of a gear of 1020 at 927 0C 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.
Erf Z
0.5000
0.5205
0.521
X
0.550
38
Z
0.5633

39. Effect of Temperature on Diffusion
Dependence of rate of diffusion on temperature is given by

Q
D  D0 e RT
ln D  ln D0 
Q
RT
log10 D  log10 D0 
Q
2.303RT
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)
39

40. Question 6
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)
40

41. 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
41

42. Edge Dislocation

An edge dislocation is created in a crystal by insertion of extra half
planes of atoms.
Positive edge
dislocation

Negative edge
dislocation
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.
Figure 4.18 : Positive edge dislocation in a crystalline lattice.
42

43. 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).
43

44. 
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.
44

45. 
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.
45

46. 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
46

47. 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.
47

48. 
The free or external surface are considered defects because:



48
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.

49. 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
49

50. 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
Plane
Twin
50

51. Small angle tilt boundary

Small angle tilt boundary:
- Array of edge dislocations tilts two regions of a crystal by < 10 0
Figure 4.24
(a) Edge dislocations in an array forming a small-angle tilt boundary
(b) Schematic of a small-angle twist boundary.
51

52. Stacking faults / Piling up faults

Stacking faults / Piling up faults :
- form during recrystallization due to collapsing.
Example:
ABCABAACBABC
52
FCC fault

53. Three dimensional imperfections

Volume or three-dimensional defects:
- produced when a cluster of point defects join to form a threedimensional void or a pore.
53

54. 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:





54
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

55. 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.
N < 3 – Coarse grained
4 < n < 6 – Medium grained
7 < n < 9 – Fine grained
N > 10 – ultrafine grained
55

56. 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.
56

57. 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)
57

58. 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
58

59. Exercise 8
Estimate the average grain diameter of a micrograph below. Given
C= 1.5 and M=200X.
(Answer :14mm)
59

60. 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.
60