METALLURGY AND MATERIAL

SCIENCE
ASSIGNMENT 1

MODULE 1 NOTES

SUBMITTED BY
ATHUL SAJ
S3, MECHANICAL ENGINEERING
SOE, CUSAT
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 METALLURGY AND MATERIAL SCIENCE
Material Science is branch of science that deals with relationship that existing
between the structure and properties of material. It is based on solid state physics
and the chemistry of structure of materials.
The extraction of metals from their ores and then refining the metals for their use
is known as metallurgy. Metals are commercially extracted from minerals at low
cost and minimum effort.

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 CONTENTS
MODULE 1
 Crystallography: crystal structure, space lattice, crystal systems, miller
indices of crystal planes and directions, atomic density of crystallographic
planes and lines, atomic packing factor, coordination number, inter planar
spacing.
 Solidification of metals: homogenous and heterogeneous nucleation,
crystal growth, grains and grain boundaries, equi-axed and columnar
grains, dendritic pattern, polymorphism.
 Crystal imperfections: point defect, line defect, edge dislocation, screw
dislocation, interaction between dislocation, planar defects, stacking faults,
grain boundary, twist and twin boundaries, volume defects.
 Diffusion: mechanism of diffusion in crystals, types of diffusion, factors
affecting diffusion, Fick’s law of diffusion, metallurgical application of
diffusion, Application of diffusion.

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1.CRYSTALLOGRAPHY
Solids are classified as crystalline or non-crystalline (Amorphous). A crystal or
crystalline solid is a solid material whose constituents, such as atoms, molecules
or ion, are arranged in a highly ordered microscopic structure, forming a crystal
lattice that extends in all directions. If the internal structure is not based on a
regular repetition pattern it is called non-crystalline (Amorphous) solids.

Fig 1.1.1 Crystalline and Non crystalline solids

1.2 CRYSTAL STRUCTURE

Fig 1.2.1 A crystalline structure

Crystal structure is ordered arrangement of atoms, ions, or molecules in a
crystalline material. Unit cell is the building block of crystal structure. When a
metal freezes from a state of fusion it crystalizes. During solidification, the atom
of the liquid metal arranges themselves in a systematic pattern. These patterns
sometimes control the external shape of the crystal. The boundary between
crystallites is called grain boundaries.
Table salt, sugar and snowflakes are common example that are crystals. Crystals
made up several gem stones like diamonds and quartz.
1.3 SPACE LATTICE
A space lattice is defined as an infinite array of points in three- dimensional space
in which each point is identically located with respect to each other. If the centres
of atoms are considered to be connected together by straight lines, then a system
will be obtained comprising a great number of parallelopipeds.

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1.4 UNIT CELL

Fig 1.4.1 Unit cell and space lattice

Unit cell is the smallest component of space lattice. It is the building block of
crystals. Space lattices of various substances differ in their size and shape of their
unit cell. The distance from one atom to another is measured along one of the
axes is called the space constant. For cubical cell it has same value in all three
dimensions.

Figure 1.4.2 Coordinate systems and lattice parameters

Figure 1.4.2 shows a unit cell in the shape of a parallelopiped having equality or
inequality of length of the unit cell edges (a, b, c) according to whether or not
angle (α, β, γ) are called lattice parameters. From lattice parameters form and
actual size of unit cell can be known

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1.5 SIMPLE CUBIC UNIT CELL

Fig 1.5.1 Simple cubic unit cell

A simple cubic unit cell has one sphere/atom at each corner of the cube. Rare due
to poor packing (only Polonium has this structure). Close-packed directions are
cube edges.
1.6 FACE CENTERED CUBIC STRUCTURE (FCC)

Fig 1.6.1 Face centred cubic structure

The crystal structures found in many metals has a unit cell of cubic geometry with
atoms located at each corner and centres of all the cube faces, then it is called
Face-centred cubic structure. Some of the metals having this structure are Copper,
Gold, Silver etc… A metal with FCC structure has four times as many atoms in
its unit cell. It is more densely packed than BCC structure. The packing factor of
FCC structure is 0.74. Metal with an FCC structure deformed critically.
Coordinate number of FCC structure is 12. Total number of atoms in FCC
structure is 4.

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1.7 BODY CENTERED CUBIC STRUCTURE (BCC)

Fig 1.7.1 Body centred cubic structure

The primitive unit cell for the body-centred cubic crystal structure contains
several fractions taken from nine atoms (if the particles in the crystal are atoms):
one on each corner of the cube and one atom in the center. Because the volume
of each of the eight corner atoms is shared between eight adjacent cells, each BCC
cell contains the equivalent volume of two atoms (one central and one on the
corner).
Number of atoms per unit cell= (8 corner atoms) (1/8) + (1 centre atom)
= 8*1/8 + 1= 2

1.8 CLOSE-PACKED HEXAGONAL STRUCTURE

A lattice structure of this type has an atom at each corner of the hexagon, one
atom each at the centres of the two hexagonal faces and one atom at the centres
of the line connecting the perpendiculars in the 3 rhombus which combine and
form hexagonal close packed structure. The atomic packing for an HCP metal is
found to equal to 0.74. This is same as that of FCC structure because of the same
coordination number 12.

Fig 1.8.1 Hexagonal close packed structure

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1.6 CRYSTAL SYSTEMS
By pure symmetry considerations, There are only fourteen independent ways of
arranging points in three dimensional space, such that each arrangement confirms
definition of space lattice. These 14 space lattices are called Bravais lattice safter
their originator. To describe basic crystal structures, seven different co ordinate
system of reference axes is required as shown in figure below.
1. The CUBIC (also called Isometric system)
2. The TETRAGONAL system
3. The HEXAGONAL system
4. The ORTHORHOMBIC system
5. The MONOCLINIC system
6. The TRICLINIC system

Table 1.6.1 Crystal systems Fig 1.6.1 Reference axes

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1.7 MILLER INDICES OF CRYSTAL PLANES
i. Important features of miller indices are listed below:
ii. Miller indices do not only define a particular plane but a set of parallel
planes
iii. A plane which is parallel to any one of the principal axes has intercept of
infinity and the miller index is therefore zero
iv. All the equally placed parallel planes with a particular orientation have a
same index number [hkl]
v. Only the ratio of indices is important
vi. The direction in spaces are represented by square brackets [ ], and the
letters xyz. Whereas Miller indices of planes are denoted by ( ) and letters
[hkl]
vii. The common inside brackets are used separately and not combined. Thus
(111) is read as one-one-one and not “one hundred eleven”.
viii. Negative indices are represented by putting a bar over the digit (0 ī 0)

Fig 1.7.1 Miller indices

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1.8 ATOMIC DENSITY OF CRYSTALLOGRAPHIC PLANES AND LINES
1.8.1 LINEAR DENSITY
Linear density (LD) is defined as the fraction of the line length in particular
crystallographic direction that passes through atom center.
𝐍𝐨.𝐨𝐟 𝐚𝐭𝐨𝐦𝐬 𝐜𝐞𝐧𝐭𝐞𝐫𝐞𝐝 𝐨𝐧 𝐝𝐢𝐫𝐞𝐜𝐭𝐢𝐨𝐧 𝐯𝐞𝐜𝐭𝐨𝐫
LD =
𝐋𝐞𝐧𝐠𝐭𝐡 𝐨𝐟 𝐝𝐢𝐫𝐞𝐜𝐭𝐢𝐨𝐧 𝐯𝐞𝐜𝐭𝐨𝐫
Unit of linear density is the reciprocal of length, nm-1 or m-1
1.8.2 PLANAR DENSITY
Planar density (PD) is simply the fraction of the total crystallographic plane area
that is occupied by atoms (atoms are represented by circles); the plane must pass
through an atom’s center for particular atom to be included.
𝐍𝐨.𝐨𝐟 𝐚𝐭𝐨𝐦𝐬 𝐜𝐞𝐧𝐭𝐞𝐫𝐞𝐝 𝐨𝐧 𝐚 𝐩𝐥𝐚𝐧𝐞
PD =
𝐚𝐫𝐞𝐚 𝐨𝐟 𝐩𝐥𝐚𝐧𝐞
Unit of planar density is the reciprocal of area, nm-2 or m-2
1.9 ATOMIC PACKING FACTOR
This is defined as the fraction of volume occupied by spherical atoms as
compared to the total available volume of the structure. It is also known as the
fraction or relative density of packing. Relation is given by
Volume of atoms in a unit cell
Atomic Packing Factor =
Volume of the unit cell

1.10 COORDINATION NUMBER

This is defined as the number of nearest atoms directly surrounding given atoms
directly surrounding a given atom. Its value is six for simple cube, eight for BCC
and twelve for FCC structures.
1.11 INTER PLANAR SPACING
Interplanar spacings are the distances between planes and represented by a
number of parts of the body diagonal of a unit cell. In the cubic system, the
interplanar spacings can be determined from the following relation:
𝑎
𝑑ℎ𝑘𝑙 =
ℎ2 + 𝑘 2 + 𝑙 2
Where a is the lattice constant and h, k, l are the indices of the planesThere are
three d111 interplanar spacings per long diagonal (body diagonal) of a unit cell in
an FCC structure.

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2.SOLIDIFICATION OF METALS
Solidification is a process in which the liquid changes to solid during cooling.
The energy of liquid is less than that of the solid above the melting point, hence
liquid is stable above melting point.
Below the melting point, the energy of liquid becomes more than that of solid and
below melting point solid become more stable than liquid. Therefore, at melting
point, liquid converts to solid during cooling. This transformation of liquid to
solid below melting point is known as solidification
Thermodynamically both liquid and solid have equal energy at melting point and
therefore both equally stable at melting point. Therefore, no solidification or
melting will take place at melting point. Liquid remains liquid and solid remains
solid. Some under cooling will be essential for solidification. This transformation
occurs by nucleation and growth.
2.1.1 NUCLEATION
When a liquid metal is poured into a mould cavity, liquid metal makes contact
with the wall of the mould. The process of cooling of the liquid metal starts near
the wall surface. Thus the process of nucleation starts at the wall surface. The size
of nucleation is very tiny in form of small droplets.
There are two types (mechanisms) of nucleation
1.Homogeneous or self nucleation
2.Heterogeneous nucleation

2.1.2 HOMOGENEOUS NUCLEATION

It is occurring in perfectly homogeneous material such as pure metals. It is the

occurrence of ordered group of atoms forming small zones of higher than average
density. Nucleation depends upon two factors

 The free energy available from the solidification process, which depends
upon the volume of the particle formed. The transformation of the old
phase (liquid) to the new phase (solid) accompanies a free energy decrease.
In the case spherical particle, the free energy change (Δƒν ) per unit volume
is equals to ( - 4/3 π r³ Δƒν) and its negative because free energy decreases,
and r is the radius of particles
 The energy required to form liquid- solid interface. The creation of
a new interface is associated with free energy increase proportional
to the surface area of the particle and this free energy (4 π r²γ).
 Thus the total free energy change:-
Δƒ = ( - 4/3 π r³ Δƒν) + (4 π r²γ)

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 Gibbs free energy = Energy required to create interface + Energy
released by volume of condensing phase

Fig 2.1.1 Gibbs free energy

2.1.3 HETEROGENEOUS NUCLEATION

It is a type of nucleation in which nucleation due to the random motion of liquid

atoms starts at the system/wall of the mould cavity. i.e. Heterogeneous
nucleation occurs at surfaces, imperfections etc. In casting usually impurities
present which lower the liquid – solid interface energy and help in nucleation.
The basic requirement for heterogeneous nucleation lies in the ability of liquid
metal to wet the foreign particles. If the impurity is wetted by solid and liquid ,
the force equilibrium where the three surfaces; i.e. impurity (m), solid (s) and
liquid (l) meet is:

γs-L

Liquid

Soli γƒm-L
γƒm-s

Foreign
matter

Fig 2.1.2 Heterogeneous nucleation

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 γƒm.s = γƒm.l – γs.l.cos Ө

 The nucleus will form on the impurity because in this

case a smaller amount of surface energy is needed.

Δƒc (heterogeneous) < Δƒc ( homogeneous)

 There is little or no supercooling at all in heterogeneous

nucleation.

2.2 CRYSTAL GROWTH

As the temperature goes down, the liquid solidifies on the nuclei and crystals
grow. One crystal grow from one nucleus. The formed crystals are not aligned
with each other and they are oriented randomly. After the progression of
solidification, the crystals join each other and form boundaries. Each boundary
has a discontinuity in the crystal structure. The individual crystals are called
grains and the boundaries are called grain boundaries. If the solidification begins
at a large number of nuclei, the resulting solid will have many grains and it is said
to be polycrystalline. Most of the engineering materials are poly crystalline.

Fig 2.2.1 Crystal growth

2.3 GRAINS AND GRAIN BOUNDARIES

Fig 2.3.1 Grain and grain boundary

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A grain boundary (GB) is the interface between two grains or crystallites, in a
polycrystalline material. Grain boundaries are defects in the crystal structure and
tend to decrease the electrical and thermal conductivity of material.
A grain boundary is a general planar defect that separates region of different
crystalline orientation (such as grains) within a polycrystalline solid. Grain
boundaries are usually the result of uneven growth when the solid is crystallizing.
Grain sizes vary from 1µ - 1mm
2.4 EQUI-AXED AND COLUMNAR GRAINS

Fig 2.4.1 equi-axed columnar grains

Equiaxed grains:
 Crystals, smaller in size, grow equally in all directions
 Formed at the sites of high concentration of nuclei
 E.g.: Cold mould wall
Columnar Grains:
 Long thin and coarse
 Grow predominantly in one direction
 Formed at the sites of slow cooling and steep temperature gradient
 e.g. : Grains that are away from mould wall

2.5 DENDRITIC PATTERN

Fig 2.5.1 dendritic pattern

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In the case of dendritic crystal growth, heat is not dissipated across the crystal as
is the case with polygonal crystal growth, but over the melt. This will be the case
when the melt is severely supercooled and the crystal forms freely in the melt.
This results in a negative temperature gradient in the direction of the melt. The
crystal now grows in an area with a lower temperature than itself. Any (advance)
branching that may form will continue to grow very quickly, as the cooler melt
crystallises rapidly at the branching. The crystal forms branches that are very
reminiscent of a fir-tree like structure – these are called dendrites

2.6 POLYMORPHISM

Fig 2.6.1 polymorphism

Polymorphism, in crystallography, the condition in which a solid chemical

compound exists in more than one crystalline form; the forms differ somewhat in
physical and, sometimes, chemical properties, although their solutions and
vapours are identical. The existence of different crystalline or molecular forms of
elements is called allotropy, although it has been suggested that the meaning of
allotropy should be restricted to different molecular forms of an element, such as
oxygen (O2) and ozone (O3), and that polymorphism be applied to different
crystalline forms of the same species, whether a compound or an element.

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3.1 CRYSTAL IMPERFECTION
Defects and lattice imperfections are found in most engineering alloys. These
imperfections affect the properties of crystals, such as mechanical strength,
chemical reactions, electric properties, etc. to a great extend
Major imperfections in the crystal structure of metals
1. Point defects
(i) Vacancies
(ii) Interstitial atoms
(iii) Impurities
2. Line defects: Dislocations
(i) Edge dislocation
(ii) Screw dislocation
3. Surface or Grain boundaries defects
(i) Grain boundaries
(ii) Tilt boundaries
(iii) Twin boundaries
4. Volume defects: Stacking faults

3.2 POINT DEFECT

3.2.1 VACANCY
Simple Point defect is a vacancy which simply involves a missing atom within a
metal. Such defects can be a result of imperfect packing during original
crystallisation or may arise due to increased thermal energy causing individual
atoms to jump out of their position of lowest energy. Vacancies exist in a certain
proportion in a crystal at thermal equilibrium, leading to an increase in
randomness of the structure.

Fig 3.2.1 vacancy

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3.2.2 IMPURITY
Impurities may produce compositional defects in the crystal structure. When
impurities in the form of foreign atoms occupy lattice sites where regular atoms
are missing, They produce substitutional impurity. Slag inclusion in metals
having atoms of smaller radius than the host atoms, will produce interstitial defect

Fig 3.2.2 impurity

3.2.3 FRANKEL DEFECT

Frankel defect is closely related to interstices. An ion displaced from the lattice
site into an interstitial site is called a Frankel defect. Closed packed structure have
fewer interstices and Frankel defects because additional energy is required to
force the atom into a new position

Fig 3.2.3 Frankel defect

3.2.4 SCHOTTKY DEFECT
This closely related to vacancies and is obtained when an atom or ion is removed
from a normal lattice site and replaced by an ion on the surface of crystal. Both
vacancies and Schottky defects facilitate atomic diffusion

Fig 3.2.4 Schottky defect

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3.3 LINE DEFECT
A linear disturbance linear disturbance of the atomic arrangement, which can very
easily occur on the slip plane through the crystal, is known as dislocation.
Dislocation is a two-dimensional line defects and is a very important crystal
imperfection.

3.4 EDGE DISLOCATION

An edge dislocation is formed by adding an extra partial plane of atoms to the
crystal. Near the dislocation, the crystal is distorted due to the presence of zones
compression and tension in the crystal lattice. The dislocation line is a region of
higher energy than the rest of the crystal.

Fig 3.4 edge dislocation

3.5 SCREW DISLOCATION
Screw dislocation is shown in figure. A perfect crystal and a plane cutting part
way through it are also shown. In this atoms are displaced in two separate planes
perpendicular to each other and the distortion follows a helical or screw path, both
right handed and left hand screws are possible. In this shear stresses are associated
with adjacent atoms and extra energy is involved along the dislocation.

Fig 3.5 Screw Dislocation

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3.7 PLANAR DEFECTS
These defects are two dimensional and are due to a change in the stacking of
atomic planes on cross a boundary. Such imperfections may include grain
boundary, tilt boundary, twin boundary, stacking faults etc
3.7.1 STACKING FAULTS
This type of imperfection may arise where there is only a small dissimilarity
between the stacking sequences of close packed planes in FCC and HCP metals.
It is possible for one atoms of the layers above and below, giving a fault

Fig 3.7.1 Stalking faults

3.7.2 GRAIN BOUNDARY
These imperfections separate crystals or grains of different orientation in
polycrystalline aggregates during crystallization. In grain boundaries the atomic
packing is imperfect and between to adjacent grains, there is a transition zone that
is not aligned with either grain.

Fig 3.7.2 Grain Boundary

3.7.3 TWIST AND TWIN BOUNDARIES
Twin boundary is a special type of crystal defect. It has a specific mirror lattice
symmetry. Which means atoms on one of the boundary are located in mirror
image position to the other side atoms. Twinning may result during the crystal
growth or deformation of materials. In the process of mechanical working like
recrystallisation or as a result of annealing, the twin deformation formed are
known as mechanical twins and annealing twins respectively

Fig 3.7.3 twist and twin boundary

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3.7.4 VOLUME DEFECTS
Volume defects are two dimensional defects. Volume defects such as cracks may
arise in crystals during process of crystal growth. While growing any possible
electrostatic dissimilarity between the stacking layers may result in crack. A large
vacancy may arise due to missing of cluster of atoms which is a volume defect.
Using optical microscope and interferometric techniques volume defect can be
detected

4.1 DIFFUSION
It’s the motion of atoms, ions, or vacancies through a material. Inhomogeneous
material can become homogeneous by diffusion. For an active diffusion to
occur, the temperature should be high enough to overcome energy barriers to
atomic motion

Fig 4.1 diffusion

4.2 MECHANISM OF DIFFUSION IN CRYSTALS
In order to explain diffusion process, several mechanisms have been proposed.
All of them are based on the vibrational energy of atoms in a solid.

Some of the common diffusion mechanisms are:

1. Vacancy mechanism.

2. Interstitial mechanism.

3. Direct interchange mechanism.

1. Vacancy Mechanism:

Diffusion is possible only if the atoms can shift and rearrange themselves in
the lattice. If it is assumed that the lattice sites are all occupied, the atoms
shall not be able to travel easily, particularly in substitutional solid solution
alloys. The lattice of most metals and alloys contains a large number of
unoccupied sites, called vacancies, and the diffusion is possible due to their
existence.

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Over a period of time such diffusion produces concentration changes.
Vacancies are continually being created and destroyed at the surface, grain
boundaries and suitable interior positions such as dislocations. The rate of
diffusion, therefore, increases rapidly with increasing temperature.

If the solid is composed of a single element (pure metal), the movement of

atoms is called self-diffusion because the moving atom and the solid are the
same chemical element.

Cu and Ni are mutually soluble in all proportions in the solid state and form
substitutional solid solutions, e.g., plating of Ni on Cu.

2. Interstitial Mechanism:

When a solid is composed of two or more elements whose atomic radii differ
significantly, interstitial solutions may occur. The large atoms occupy lattice
sites while the small ones fit into the voids created by the large atoms. These
voids are called interstices. The diffusion mechanism in this case is similar to
vacancy diffusion except that the interstitial atoms stay on interstitial sites.

With interstitial diffusion, an activation energy is associated, because to arrive

at the vacant site, it must squeeze past neighbouring atoms with energy
supplied by the vibrational energy of moving atoms. Thus, interstitial
diffusion is a thermally activated process.

This diffusion mechanism is important in two cases-

(i) The presence of very small atoms in the interstices of the lattice greatly
affects the mechanical properties of metals;

(ii) O2, N, and H, can be diffused in metals easily at low temperatures.

3. Direct Interchange Mechanism:

Two or more adjacent atoms jump past each other and exchange positions, but
the number of sites remains constant. This may be two-atoms or four-atoms
(Zenner ring) interchange (for BCC). Direct interchange mechanism entails
following shortcomings /objections-

(i) Severe local distortion results due to the displacement of the atoms
surrounding the jumping pair. (ii) A number of diffusion couples of different
compositions are produced. This is also called Kirke Dall’s effect. (The
inequality of diffusion was first shown by Kirkendall).

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4.3 TYPES OF DIFFUSION
1. Self-diffusion: Atoms jumping in pure metals
2. Inter diffusion: Observed in binary metal alloys such as the Cu-Ni system
3. Volume diffusion: atomic movement in bulk materials
4. Grain boundary diffusion: atomic movement along the grain boundaries
alone
5. Surface diffusion: atomic movement along the surface of a phase
The different types of diffusion are:

1. Self-diffusion

2. Inter-diffusion

3. Volume diffusion

4. Grain boundary diffusion, and

5. Surface diffusion.

1. Self-Diffusion:

Self-diffusion is the migration of atoms in pure materials. In a pure substance,

a particular atom does not remain at one equilibrium site indefinitely, rather
it moves from place to place in the material. Self-diffusion in a pure material
can be detected experimentally by radioactive tracers.

2. Inter-Diffusion:

It occurs in binary metallic alloys. Observed in binary metal alloys such as

Cu-Ni system.

Fig 4.3 Interdiffusion

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3. Volume Diffusion:

Volume diffusion means atomic migration through the bulk of the material.

4. Grain Boundary Diffusion:

It implies atomic movement along the grain boundaries alone. The activation
energy for grain boundary diffusion is lower than for volume diffusion.

5. Surface Diffusion: It implies atomic movement along the surface of a

phase. Example: Solid-vapour interface.

4.4 FACTORS AFFECTING DIFFUSION

Fig 4.4 Diffusion factors

 Size of particles: At a given temperature, smaller particle moves or diffuses

faster than larger one
 Temperature: As the temperature increases particles gain energy and
moves faster, thus rate of diffusion is increased
 Concentration difference: The greater the concentration difference
between two regions, the faster the substance will diffuse
 Diffusion distance: At a given temperature it takes longer for particle to
diffuse a farther distance, thus slower the rate of diffusion
 Surface area: The greater surface area greater the diffusion
 Permeability: The more the permeable the separating surface is the faster
the substance can diffuse through it

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4.5 FICK’S LAW OF DIFFUSION
Fick’s laws are used to describe solid state diffusion.
Fick’s first law describes the rate at which diffusion occurs. It states
ⅆ𝐶
𝑑𝑛 = −𝐷 𝑎 𝑑𝑡
ⅆ𝑥

Where, 𝑑𝑛 = Amount of metal in kg that crosses a plane normal to direction of

diffusion.
ⅆ𝐶
=slope of concentration gradient
ⅆ𝑥

D= diffusion co efficient
A= Area of plane across which diffusion takes place
Dt= Duration of diffusion
J= The flux or the numbers of atoms moving from unit area of one plane to unit
area of another per unit time. It is proportional to the concentration gradient.
The equation becomes:
ⅆ𝑛 ⅆ𝐶
= −𝐷 𝑎
ⅆ𝑡 ⅆ𝑥
1 𝑑𝑛 𝑑𝐶
𝐽=− = −𝐷
𝑎 𝑑𝑡 𝑑𝑥
or
1 𝑑𝑚
𝐽=
𝑎 𝑑𝑡
The negative sign indicates the flow occurs down the concentration gradient.
4.5.1 FICK’S SECOND LAW
The time dependence of concentration is given by fick’s second law
It states that
ⅆ𝐶 ⅆ ⅆ2 𝐶
= [𝐷(ⅆ𝐶
ⅆ2 )] = 𝐷
̅̅̅̅
ⅆ𝑡 ⅆ𝑥 ⅆ𝑥 2

Where, D is the diffusion coefficient. This shows diffusion rate becomes slower
as the diffusion process progresses.

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4.6 APPLICATION OF DIFFUSION
 Diffusion is the fundamental of phase changes eg: γ to α iron
 Joining of material by diffusion bonding , e. g. Welding, Brazing,
Soldering, Galvanising, Metal cladding
 Important in heat treatment like homogenising of castings, recovery,
recrystallisation, and precipitation of phases
 Production of strong bodies by powder metallurgy ( sintering)
 Case hardening process of steel
 Doping of semiconductors
 Oxidation of metals

REFERENCES
 Material Science, G K Narula
 Callister’s Material Science and Engineering
 Britanica
 Engineering notes MT EDU

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