Phase diagrams and Diffusion in Materials Science

Phase diagrams & Diffusion
Phase diagrams & Diffusion
Dr. George Rapheal
Dr. George Rapheal
Dept. of Automation &
Dept. of Automation &

Phase
Phase
A phase can be defined as any
A phase can be defined as any portion
portion, including the
, including the whole
whole, of a
, of a system
system that is
that is

physically homogeneous
physically homogeneous
within itself and bounded by a
within itself and bounded by a surface
surface that separates it from any other
that separates it from any other
portions.
portions.
A phase has the following characteristics:
A phase has the following characteristics:
1. the same
1. the same structure
structure or atomic arrangement throughout
or atomic arrangement throughout
2. roughly the same
2. roughly the same composition
composition and properties throughout
and properties throughout
3. a definite
3. a definite interface
interface between the phase and any surrounding or adjoining
between the phase and any surrounding or adjoining
phases.
phases.
For example, water has three phases—liquid water, solid ice, and steam.
For example, water has three phases—liquid water, solid ice, and steam.
Gibbs Phase Rule
Gibbs Phase Rule
It describes the relationship between the number of
It describes the relationship between the number of components
components and the
and the
number of
number of phases
phases for a given system in
for a given system in thermodynamic equilibrium
thermodynamic equilibrium and the
and the

Phases do not always have to be solid, liquid, and
Phases do not always have to be solid, liquid, and
gaseous forms of a material.
gaseous forms of a material.
An element, such as iron, can exist in FCC and
An element, such as iron, can exist in FCC and
BCC crystal structures. These two solid forms of
BCC crystal structures. These two solid forms of
iron are two different phases that will be stable at
iron are two different phases that will be stable at
different temperatures and pressure conditions.
different temperatures and pressure conditions.
Carbon can exist in many forms, eg. graphite or
Carbon can exist in many forms, eg. graphite or
diamond. These are only two of the many possible
diamond. These are only two of the many possible
phases of carbon.
phases of carbon.
Liquidus and Solidus Temperatures
Liquidus and Solidus Temperatures
Liquidus temperature can be defined as the
Liquidus temperature can be defined as the
temperature above which a material is completely
temperature above which a material is completely
liquid.solidus temperature is the temperature below
liquid.solidus temperature is the temperature below
which the alloy is 100% solid.
which the alloy is 100% solid.
Pure metals solidify at a fixed temperature (i.e., the freezing range is zero
Pure metals solidify at a fixed temperature (i.e., the freezing range is zero
degrees).
degrees).
Copper-nickel alloys melt and freeze over a range of temperatures between
Copper-nickel alloys melt and freeze over a range of temperatures between
the
liquidus and the solidus. The temperature difference between the liquidus
the
liquidus and the solidus. The temperature difference between the liquidus

Binary Isomorphous Systems (Cu- Ni system)
Binary Isomorphous Systems (Cu- Ni system)
L is homogeneous
L is homogeneous liquid solution
liquid solution composed
composed
of Cu and Ni. The α phase is a
of Cu and Ni. The α phase is a
substitutional
substitutional solid solution
solid solution consisting of
consisting of
Cu and Ni atoms and has an FCC crystal
Cu and Ni atoms and has an FCC crystal
structure.
structure.
Cu and Ni are mutually soluble in each
Cu and Ni are mutually soluble in each
other in the solid state for all compositions.
other in the solid state for all compositions.
This complete solubility is explained by the
This complete solubility is explained by the
fact that both Cu and Ni have the same
fact that both Cu and Ni have the same
crystal structure (FCC), nearly identical
crystal structure (FCC), nearly identical
atomic radii and electronegativities, and
atomic radii and electronegativities, and
similar valences (Hume Rothery Rules).
similar valences (Hume Rothery Rules).

Construction from cooling curves
Construction from cooling curves
The cooling curve for an isomorphous alloy during solidification is shown in
The cooling curve for an isomorphous alloy during solidification is shown in
Fig. We assume that cooling rates are low so that thermal equilibrium is
Fig. We assume that cooling rates are low so that thermal equilibrium is
maintained at each temperature. The changes in slope of the cooling curve
maintained at each temperature. The changes in slope of the cooling curve
indicate the liquidus and solidus temperatures, in this case, for a Cu-40% Ni
indicate the liquidus and solidus temperatures, in this case, for a Cu-40% Ni
alloy.
alloy.

Tie line
Tie line
Since there is only one degree of freedom in a
Since there is only one degree of freedom in a
two-phase region of a binary phase diagram,
two-phase region of a binary phase diagram,
the compositions of the two phases are always
the compositions of the two phases are always
fixed when we specify the temperature.
fixed when we specify the temperature.
This is true even if the overall composition of
This is true even if the overall composition of
the alloy changes. Therefore, a tie line is used
the alloy changes. Therefore, a tie line is used
to determine the compositions of the two
to determine the compositions of the two
phases.
phases.
A tie line is a horizontal line within a two-
A tie line is a horizontal line within a two-
phase region drawn at the temperature of
phase region drawn at the temperature of
interest. In an
isomorphous system, the tie
interest. In an
isomorphous system, the tie
line connects the liquidus and solidus points
line connects the liquidus and solidus points

Problem
Problem
Determine the equilibrium
Determine the equilibrium
composition of each phase in a Cu-
composition of each phase in a Cu-
40% Ni alloy at 1300 °C, 1270 °C,
40% Ni alloy at 1300 °C, 1270 °C,
1250 °C, and 1200 °C. (See Fig.)
1250 °C, and 1200 °C. (See Fig.)

Lever Rule (Amount of Each Phase )
Lever Rule (Amount of Each Phase )
The relative amounts of each phase present in the alloy can be estimated by
The relative amounts of each phase present in the alloy can be estimated by
Lever rule. These amounts are normally expressed as weight percent (wt%).
Lever rule. These amounts are normally expressed as weight percent (wt%).
We express absolute amounts of different phases in units of mass or weight
We express absolute amounts of different phases in units of mass or weight
(grams, kilograms, pounds, etc.).
(grams, kilograms, pounds, etc.).
To calculate the amounts of liquid and solid, we construct a lever on the tie
To calculate the amounts of liquid and solid, we construct a lever on the tie
line, with the
line, with the fulcrum
fulcrum of our lever being the
of our lever being the original composition
original composition of the alloy.
of the alloy.
The leg of the lever opposite to the composition of the phase, the amount of
The leg of the lever opposite to the composition of the phase, the amount of
which we are
calculating, is divided by the total length of the lever to give the
which we are
calculating, is divided by the total length of the lever to give the
amount of that phase.
amount of that phase.
Problem
Problem
Calculate the amounts of α and L at 1250 °C
Calculate the amounts of α and L at 1250 °C
in the Cu-40% Ni alloy shown in Fig. The
in the Cu-40% Ni alloy shown in Fig. The
denominator represents the total length of the
denominator represents the total length of the
tie line and the numerator is the portion of
tie line and the numerator is the portion of
the lever that is opposite the composition of
the lever that is opposite the composition of
the phase we are trying to calculate. The
the phase we are trying to calculate. The
lever rule in general can be written as:
lever rule in general can be written as:
amount of L =
amount of L =

Solidification of Binary Isomorphous system
Solidification of Binary Isomorphous system
Solidification requires both
Solidification requires both nucleation
nucleation and
and growth
growth.
.
Heterogeneous nucleation permits little or no undercooling, so solidification
Heterogeneous nucleation permits little or no undercooling, so solidification
begins when the liquid reaches the
begins when the liquid reaches the liquidus temperature
liquidus temperature.
.
Two conditions are required for growth of the solid α. First, growth requires
Two conditions are required for growth of the solid α. First, growth requires
that the
that the latent heat of fusion
latent heat of fusion (ΔHf), which evolves as the liquid solidifies, be
(ΔHf), which evolves as the liquid solidifies, be
removed from the solid–liquid interface. The latent heat of fusion (ΔHf) is
removed from the solid–liquid interface. The latent heat of fusion (ΔHf) is
removed over a range of temperatures so that the cooling curve shows a
removed over a range of temperatures so that the cooling curve shows a
change in slope, rather than a flat plateau.
change in slope, rather than a flat plateau.
Second, unlike the case of pure metals,
Second, unlike the case of pure metals, diffusion
diffusion must occur so that the
must occur so that the
compositions of the solid and liquid phases follow the solidus and liquidus
compositions of the solid and liquid phases follow the solidus and liquidus
curves
during cooling.
curves
during cooling.

At the start of freezing, the liquid contains
At the start of freezing, the liquid contains
Cu-40% Ni, and the first solid contains
Cu-40% Ni, and the first solid contains
Cu-52% Ni.
Cu-52% Ni.
After cooling to 1250 °C, solidification
After cooling to 1250 °C, solidification
has advanced, and the phase diagram tells
has advanced, and the phase diagram tells
us that now all of the liquid must contain
us that now all of the liquid must contain
32% Ni and all of the solid must contain
32% Ni and all of the solid must contain
45% Ni. Therefore, some nickel atoms
45% Ni. Therefore, some nickel atoms
must diffuse from the first solid to the new
must diffuse from the first solid to the new
solid, reducing the nickel in the first solid.
solid, reducing the nickel in the first solid.
Additional nickel atoms diffuse from the
Additional nickel atoms diffuse from the
solidifying liquid to the new solid.
solidifying liquid to the new solid.
Meanwhile, copper atoms have
Meanwhile, copper atoms have
concentrated by diffusion into the
concentrated by diffusion into the
remaining liquid.
remaining liquid.
The change in structure of a Cu-40% Ni
The change in structure of a Cu-40% Ni
alloy during equilibrium solidification. The
alloy during equilibrium solidification. The
Ni and Cu atoms must diffuse during
Ni and Cu atoms must diffuse during
cooling in order to satisfy the phase
cooling in order to satisfy the phase
diagram and produce a uniform
diagram and produce a uniform

Binary Eutectic System (Sn-Pb)
Binary Eutectic System (Sn-Pb)
On this phase diagram, the α is a solid solution of Sn in Pb; however, the
On this phase diagram, the α is a solid solution of Sn in Pb; however, the
solubility of Sn in the α solid solution is limited. At 0°C, only 2% Sn can
solubility of Sn in the α solid solution is limited. At 0°C, only 2% Sn can
dissolve in α. As the
temperature increases, more Sn dissolves into the Pb
dissolve in α. As the
temperature increases, more Sn dissolves into the Pb
until, at 183°C, the solubility of Sn in Pb has increased to 18.3% Sn. This is
until, at 183°C, the solubility of Sn in Pb has increased to 18.3% Sn. This is
the maximum solubility of Sn in Pb.
the maximum solubility of Sn in Pb.
The solubility of Sn in solid Pb
The solubility of Sn in solid Pb
at any temperature is given by
at any temperature is given by
the
the solvus
solvus curve.
curve.

Solid-Solution Alloys
Solid-Solution Alloys
A vertical line on a phase diagram that shows a specific composition is known
A vertical line on a phase diagram that shows a specific composition is known
as an
as an isopleth
isopleth. Determination of reactions that occur upon the cooling of a
. Determination of reactions that occur upon the cooling of a
particular composition is known as an
particular composition is known as an isoplethal study
isoplethal study.
.
Alloys that contain 0 to 2% Sn behave
Alloys that contain 0 to 2% Sn behave
exactly like the Cu-Ni alloys; a single-phase
exactly like the Cu-Ni alloys; a single-phase
solid solution α forms during solidification.
solid solution α forms during solidification.
These alloys are strengthened by solid-
These alloys are strengthened by solid-
solution strengthening, strain hardening,
solution strengthening, strain hardening,
and controlling the solidification process to
and controlling the solidification process to
refine the grain structure.
refine the grain structure.
Solidification and microstructure of a
Solidification and microstructure of a

Alloys that Exceed the Solubility Limit
Alloys that Exceed the Solubility Limit
Alloys containing between 2% and 19% Sn also solidify to produce a single
Alloys containing between 2% and 19% Sn also solidify to produce a single
solid solution α; however, as the alloy continues to cool, a
solid solution α; however, as the alloy continues to cool, a solid-state reaction
solid-state reaction
occurs, permitting a second solid phase (β) to precipitate from the original α
occurs, permitting a second solid phase (β) to precipitate from the original α
phase. As any alloy containing between 2% and 19% Sn cools below the
phase. As any alloy containing between 2% and 19% Sn cools below the
solvus, the solubility limit is exceeded, and a small amount of β forms.
solvus, the solubility limit is exceeded, and a small amount of β forms.
We control the properties of this
We control the properties of this
type of alloy by several
type of alloy by several
techniques, including solid-
techniques, including solid-
solution strengthening of the a
solution strengthening of the a
portion of the structure,
portion of the structure,
controlling the microstructure
controlling the microstructure
produced during solidification,
produced during solidification,
and controlling the amount and
and controlling the amount and
characteristics of the β phase.
characteristics of the β phase. Solidification, precipitation, and
Solidification, precipitation, and
microstructure of a Pb-10% Sn alloy.
microstructure of a Pb-10% Sn alloy.

Eutectic Alloys
Eutectic Alloys
The alloy containing 61.9% Sn has the eutectic composition. This is the
The alloy containing 61.9% Sn has the eutectic composition. This is the
composition for which there is no freezing range (solidification of this alloy
composition for which there is no freezing range (solidification of this alloy
occurs at one temperature, 183°C in the Pb-Sn system). Above 183°C, the
occurs at one temperature, 183°C in the Pb-Sn system). Above 183°C, the
alloy is all liquid and, therefore, must contain 61.9% Sn. After the liquid
alloy is all liquid and, therefore, must contain 61.9% Sn. After the liquid
cools to 183°C, the eutectic reaction begins:
cools to 183°C, the eutectic reaction begins:
Alloys solidify over a range of
Alloys solidify over a range of
temperatures (between the liquidus and
temperatures (between the liquidus and
solidus) known as the freezing range.
solidus) known as the freezing range.
Since solidification occurs completely at
Since solidification occurs completely at
183°C, the cooling curve is similar to that
183°C, the cooling curve is similar to that
of a pure metal; that is, a thermal arrest or
of a pure metal; that is, a thermal arrest or
plateau occurs at the eutectic temperature.
plateau occurs at the eutectic temperature.
During solidification, growth of the
During solidification, growth of the
eutectic requires both removal of the latent
eutectic requires both removal of the latent
In the Pb-Sn system, the solid α and β phases grow from the liquid in a
In the Pb-Sn system, the solid α and β phases grow from the liquid in a
lamellar, or plate-like, arrangement. The lamellar structure permits the Pb
lamellar, or plate-like, arrangement. The lamellar structure permits the Pb

Problem:
Problem: Determine for a Pb-10% Sn
Determine for a Pb-10% Sn
alloy
alloy
(a) the solubility of tin in solid lead at
(a) the solubility of tin in solid lead at
100°C
100°C
(b) the maximum s
olubility of lead in
(b) the maximum s
olubility of lead in
solid tin.
solid tin.
Also if a Pb-10% Sn alloy is cooled to
Also if a Pb-10% Sn alloy is cooled to
0°C, determine
0°C, determine
(c) the amount of β that forms,
(c) the amount of β that forms,
(d) the mass of Sn contained in the α
(d) the mass of Sn contained in the α
and β phases
and β phases
(e) the mass of Pb contained in the α
(e) the mass of Pb contained in the α
and β phases.
and β phases.
Solution
Solution
(b) The maximum solubility of Pb in tin, which is found from the Sn-rich
(b) The maximum solubility of Pb in tin, which is found from the Sn-rich
side of the phase diagram, occurs at the eutectic temperature of 183°C and is
side of the phase diagram, occurs at the eutectic temperature of 183°C and is
97.5% Sn or 2.5% Pb.
97.5% Sn or 2.5% Pb.
(c) At 0°C, the 10% Sn alloy is in the α + β region of the phase diagram. By
(c) At 0°C, the 10% Sn alloy is in the α + β region of the phase diagram. By

Also % α would be (100 - % β) = 91.8%. For 100g alloy, β = 8.2 g and α = 91.8 g
Also % α would be (100 - % β) = 91.8%. For 100g alloy, β = 8.2 g and α = 91.8 g
(d) From above, α = 91.8 g and β = 8.2 g
(d) From above, α = 91.8 g and β = 8.2 g
Pb and Sn are distributed in two phases (α and β).
Pb and Sn are distributed in two phases (α and β).
At 0°C, α phase contains 2% Sn
At 0°C, α phase contains 2% Sn
Therefore, mass of Sn in the α phase = (2% Sn)(91.8 g of α phase) = 1.836 g.
Therefore, mass of Sn in the α phase = (2% Sn)(91.8 g of α phase) = 1.836 g.
At 0°C, β phase contains 100% Sn
At 0°C, β phase contains 100% Sn
The mass of Sn in the β phase = (100 % Sn) (8.2 g of β phase) = 8.2 g.
The mass of Sn in the β phase = (100 % Sn) (8.2 g of β phase) = 8.2 g.
Note that in this case, the β phase at 0°C is nearly pure Sn.
Note that in this case, the β phase at 0°C is nearly pure Sn.
(e)
(e)
At 0°C, α phase contains 98% Pb
At 0°C, α phase contains 98% Pb
The mass of Pb in the α phase = (98 % Pb)(91.8 g of α phase) = 89.964 g
The mass of Pb in the α phase = (98 % Pb)(91.8 g of α phase) = 89.964 g
At 0°C, β phase contains 0% Pb
At 0°C, β phase contains 0% Pb

Bismuth-Cadmium System (Bi-Cd)
Bismuth-Cadmium System (Bi-Cd)
Bi-Cd system is a simple binary (two-component) eutectic system. The mutual
Bi-Cd system is a simple binary (two-component) eutectic system. The mutual
solubilities of Bi and Cd in the solid state is extremely small and is less than
solubilities of Bi and Cd in the solid state is extremely small and is less than
0.03%. But in the molten state, Bi and Cd arc miscible in all proportions to
0.03%. But in the molten state, Bi and Cd arc miscible in all proportions to
form a homogeneous mixture.
form a homogeneous mixture.
Therefore, this system can have a maximum of three phases, namely solid Bi,
Therefore, this system can have a maximum of three phases, namely solid Bi,
solid Cd and a solution of molten Bi and Cd.
solid Cd and a solution of molten Bi and Cd.
The system can be represented by temperature composition diagram and
The system can be represented by temperature composition diagram and
according to condensed phase rule,
according to condensed phase rule,
F = C-P+ l
F = C-P+ l
Pure Bi has a melting point of 271 °C
Pure Bi has a melting point of 271 °C
and pure Cd has a melting point of
and pure Cd has a melting point of
321°C. They are miscible in the liquid
321°C. They are miscible in the liquid
phase. The addition of Cd to pure
phase. The addition of Cd to pure
liquid Bi lowers the freezing point
liquid Bi lowers the freezing point
below 271°C. Similarly, the addition
below 271°C. Similarly, the addition
of Bi to pure liquid Cd lowers the
of Bi to pure liquid Cd lowers the

The five most important three-phase reactions in binary phase diagrams.
The five most important three-phase reactions in binary phase diagrams.

Iron–iron carbide phase diagram

A portion of the iron–carbon phase diagram is presented in Fig. Pure iron,
A portion of the iron–carbon phase diagram is presented in Fig. Pure iron,
upon heating, experiences two changes in crystal structure before it melts. At
upon heating, experiences two changes in crystal structure before it melts. At
room temperature, the stable form, called ferrite, or α-iron, has a BCC crystal
room temperature, the stable form, called ferrite, or α-iron, has a BCC crystal
structure.
structure.
Ferrite experiences a
Ferrite experiences a polymorphic transformation
polymorphic transformation to FCC austenite, or γ-iron,
to FCC austenite, or γ-iron,
at 912 °C. This austenite persists to 1394 °C, at which temperature the FCC
at 912 °C. This austenite persists to 1394 °C, at which temperature the FCC
austenite reverts back to a BCC phase known as δ-ferrite, which finally melts
austenite reverts back to a BCC phase known as δ-ferrite, which finally melts
at 1538 °C. Austenite is non-magnetic.
at 1538 °C. Austenite is non-magnetic.
The composition axis in Fig extends only to 6.70 wt% C; at this concentration
The composition axis in Fig extends only to 6.70 wt% C; at this concentration
the stoichiometric compound iron carbide, or cementite (Fe
the stoichiometric compound iron carbide, or cementite (Fe3
3C), is formed,
C), is formed,
which is represented by a vertical line on the phase diagram. Fe3C contains
which is represented by a vertical line on the phase diagram. Fe3C contains
6.67% C, is extremely hard and brittle (like a ceramic material), and is present
6.67% C, is extremely hard and brittle (like a ceramic material), and is present
in all commercial steels. By properly
controlling the amount, size, and shape
in all commercial steels. By properly
controlling the amount, size, and shape
of Fe3C, we control the degree of dispersion strengthening and the properties
of Fe3C, we control the degree of dispersion strengthening and the properties

Carbon is an interstitial impurity in iron and forms a solid solution with each
Carbon is an interstitial impurity in iron and forms a solid solution with each
of α- and δ-ferrites and also with austenite, as indicated by the α, δ, and γ
of α- and δ-ferrites and also with austenite, as indicated by the α, δ, and γ
single-phase fields in Fig.
single-phase fields in Fig.
In the BCC α-ferrite, only small concentrations of carbon are soluble; the
In the BCC α-ferrite, only small concentrations of carbon are soluble; the
maximum solubility is 0.022 wt% at 727 °C. The limited solubility is explained
maximum solubility is 0.022 wt% at 727 °C. The limited solubility is explained
by the shape and size of the BCC interstitial positions, which make it difficult
by the shape and size of the BCC interstitial positions, which make it difficult
to accommodate the carbon atoms. Even though present in relatively low
to accommodate the carbon atoms. Even though present in relatively low
concentrations, carbon significantly influences the mechanical properties of
concentrations, carbon significantly influences the mechanical properties of
ferrite. α-ferrite is relatively soft, may be made magnetic at temperatures below
ferrite. α-ferrite is relatively soft, may be made magnetic at temperatures below
768 °C, and has a density of 7.88 g/cm3.
768 °C, and has a density of 7.88 g/cm3.
The austenite, or γ phase, of iron, when alloyed with carbon alone, is not stable
The austenite, or γ phase, of iron, when alloyed with carbon alone, is not stable
below 727 °C. The maximum solubility of carbon in austenite, 2.14 wt%,
below 727 °C. The maximum solubility of carbon in austenite, 2.14 wt%,
occurs at 1147 °C. This solubility is approximately 100 times greater than the
occurs at 1147 °C. This solubility is approximately 100 times greater than the
maximum for BCC ferrite because the FCC octahedral sites are larger than
maximum for BCC ferrite because the FCC octahedral sites are larger than

Strictly speaking, cementite is only metastable; that is, it remains as a
Strictly speaking, cementite is only metastable; that is, it remains as a
compound indefinitely at room temperature. However, if heated to between
compound indefinitely at room temperature. However, if heated to between
650 °C and 700 °C for several years, it gradually changes or transforms into
650 °C and 700 °C for several years, it gradually changes or transforms into
α-iron and carbon, in the form of graphite. Thus, the phase diagram is not a
α-iron and carbon, in the form of graphite. Thus, the phase diagram is not a
true equilibrium one because cementite is not an equilibrium compound.
true equilibrium one because cementite is not an equilibrium compound.
However, because the decomposition rate of cementite is extremely sluggish,
However, because the decomposition rate of cementite is extremely sluggish,
virtually all the carbon in steel is as Fe
virtually all the carbon in steel is as Fe3
3C instead of graphite, and the iron–
C instead of graphite, and the iron–
iron carbide phase diagram is, for all practical purposes, valid. Addition of
iron carbide phase diagram is, for all practical purposes, valid. Addition of
silicon to cast irons greatly accelerates this cementite decomposition reaction
silicon to cast irons greatly accelerates this cementite decomposition reaction
to form graphite.
to form graphite.
A
A eutectic
eutectic exists for the iron–iron carbide system, at 4.30 wt% C and 1147 °C;
exists for the iron–iron carbide system, at 4.30 wt% C and 1147 °C;
for this eutectic reaction the liquid solidifies to form austenite and cementite
for this eutectic reaction the liquid solidifies to form austenite and cementite
phases.
phases.

The two phases that form have different compositions, so atoms must diffuse
The two phases that form have different compositions, so atoms must diffuse
during the reaction. Most of the carbon in the austenite diffuses to the Fe
during the reaction. Most of the carbon in the austenite diffuses to the Fe3
3C,
C,
and iron atoms diffuse to the α. This redistribution of atoms is easiest if the
and iron atoms diffuse to the α. This redistribution of atoms is easiest if the
diffusion distances are short, which is the case when the α and Fe
diffusion distances are short, which is the case when the α and Fe3
3C grow as
C grow as
thin lamellae, or plates. The eutectoid phase changes are very important,
thin lamellae, or plates. The eutectoid phase changes are very important,
being fundamental to the heat treatment of steels. On the Fe-Fe
being fundamental to the heat treatment of steels. On the Fe-Fe3
3C diagram,
C diagram,
the eutectoid temperature is known as the A
the eutectoid temperature is known as the A1
1 temperature. The boundary
temperature. The boundary
between austenite (γ) and the two-phase field consisting of ferrite (α) and
between austenite (γ) and the two-phase field consisting of ferrite (α) and
austenite is known as the A
austenite is known as the A3
3. The boundary between austenite (γ) and the two-
. The boundary between austenite (γ) and the two-
phase field consisting of cementite (Fe
phase field consisting of cementite (Fe3
3C) and austenite is known as the A
C) and austenite is known as the Acm
cm.
.
Pearlite
Pearlite
The lamellar structure of α and Fe
The lamellar structure of α and Fe3
3C that develops in the iron-carbon system
C that develops in the iron-carbon system
is called pearlite, which is a microconstituent in steel. This was so named
is called pearlite, which is a microconstituent in steel. This was so named
because a
polished and etched pearlite shows the colorfulness of mother-of-
because a
polished and etched pearlite shows the colorfulness of mother-of-

Growth and structure of pearlite:
Growth and structure of pearlite:
(a) redistribution of carbon and iron, and
(a) redistribution of carbon and iron, and
(b) micrograph of the pearlite lamellae (× 2000)
(b) micrograph of the pearlite lamellae (× 2000)

Ferrous alloys are those in which iron is the prime component, but carbon as
Ferrous alloys are those in which iron is the prime component, but carbon as
well as other alloying elements may be present. In the classification scheme of
well as other alloying elements may be present. In the classification scheme of
ferrous alloys based on carbon content, there are three types: iron, steel, and
ferrous alloys based on carbon content, there are three types: iron, steel, and
cast iron.
cast iron.
Commercially pure iron contains less than 0.008 wt% C and, from the phase
Commercially pure iron contains less than 0.008 wt% C and, from the phase
diagram, is composed almost exclusively of the ferrite phase at room
diagram, is composed almost exclusively of the ferrite phase at room
temperature. The iron–carbon alloys that contain between 0.008 and 2.14 wt%
temperature. The iron–carbon alloys that contain between 0.008 and 2.14 wt%
C are classified as steels. In most steels, the microstructure consists of both α
C are classified as steels. In most steels, the microstructure consists of both α
and Fe
and Fe3
3C phases.
C phases.
Upon cooling to room temperature, an alloy within this composition range
Upon cooling to room temperature, an alloy within this composition range
must pass through at least a portion of the γ- phase field; distinctive
must pass through at least a portion of the γ- phase field; distinctive
microstructures are produced during heat treatment.
microstructures are produced during heat treatment.

Diffusion
Diffusion
Diffusion refers to the net flux of any species, such as ions, atoms, electrons,
Diffusion refers to the net flux of any species, such as ions, atoms, electrons,
holes and molecules. The magnitude of this flux depends upon the
holes and molecules. The magnitude of this flux depends upon the
concentration gradient and temperature.
concentration gradient and temperature.
The ability of atoms and ions to diffuse increases as the temperature, or
The ability of atoms and ions to diffuse increases as the temperature, or
thermal energy possessed by the atoms and ions, increases. The rate of atom
thermal energy possessed by the atoms and ions, increases. The rate of atom
or ion movement is related to temperature or thermal energy by the Arrhenius
or ion movement is related to temperature or thermal energy by the Arrhenius
equation:
equation:
where c
where c0
0 is a constant, R is the gas constant 8.314 J/ mol. K, T is the absolute
is a constant, R is the gas constant 8.314 J/ mol. K, T is the absolute
temperature (K), and Q is the activation energy (cal/mol or J/mol) required to
temperature (K), and Q is the activation energy (cal/mol or J/mol) required to
cause one mole of atoms or ions to move.
cause one mole of atoms or ions to move.

Interdiffusion
Interdiffusion
Diffusion of different atoms in different directions is
Diffusion of different atoms in different directions is
known as interdiffusion.
known as interdiffusion.
Consider a nickel sheet bonded to a copper sheet. At
Consider a nickel sheet bonded to a copper sheet. At
high temperatures, nickel atoms gradually diffuse
high temperatures, nickel atoms gradually diffuse
into the copper, and copper atoms migrate into the
into the copper, and copper atoms migrate into the
nickel. Nickel and copper atoms eventually are
nickel. Nickel and copper atoms eventually are
uniformly distributed.
uniformly distributed.
Vacancy Diffusion
Vacancy Diffusion
In self-diffusion and diffusion involving
In self-diffusion and diffusion involving
substitutional atoms, an atom leaves its lattice site to
substitutional atoms, an atom leaves its lattice site to
fill a nearby vacancy (thus creating a new vacancy at
fill a nearby vacancy (thus creating a new vacancy at
the original lattice site). As diffusion continues, we
the original lattice site). As diffusion continues, we
have counterflows of atoms and vacancies, called
have counterflows of atoms and vacancies, called
vacancy diffusion. The number of vacancies, which
vacancy diffusion. The number of vacancies, which

Rate of Diffusion [Fick’s First Law]
The flux J is defined as the number of atoms passing through a plane of unit
area per unit time. The rate at which atoms, ions, particles or other species
diffuse in a material can be measured by the flux J. Here we are mainly
concerned with diffusion of ions or atoms. Fick’s first law explains the net
flux of atoms:
flux of atoms:
where J is the flux, D is the diffusivity or diffusion coefficient (cm2/s), and
dc/dx is the
concentration gradient (atoms/cm3.cm).
dc/dx is the
concentration gradient (atoms/cm3.cm).
Depending upon the situation, concentration may be atoms expressed as atom
Depending upon the situation, concentration may be atoms expressed as atom
percent (at%), weight percent (wt%), mole percent (mol%), atom
fraction, or
percent (at%), weight percent (wt%), mole percent (mol%), atom
fraction, or
mole fraction. The units of concentration gradient and flux will change
mole fraction. The units of concentration gradient and flux will change

flux of atoms:
flux of atoms:
dc/dx is the
concentration gradient (atoms/cm3.cm). Depending upon the
dc/dx is the
concentration gradient (atoms/cm3.cm). Depending upon the
situation, concentration may be atoms expressed as atom percent (at%),
situation, concentration may be atoms expressed as atom percent (at%),
weight percent (wt%), mole percent (mol%), atom
fraction, or mole fraction.
weight percent (wt%), mole percent (mol%), atom
fraction, or mole fraction.
The units of concentration gradient and flux will change accordingly.
The units of concentration gradient and flux will change accordingly.

Temperature and the Diffusion Coefficient
Temperature and the Diffusion Coefficient
The kinetics of diffusion are strongly dependent on temperature. The diffusion
The kinetics of diffusion are strongly dependent on temperature. The diffusion
coefficient D is related to temperature by an Arrhenius-type equation:
coefficient D is related to temperature by an Arrhenius-type equation:
where Q is the activation energy (cal/mol or J/mol) for diffusion of the species
where Q is the activation energy (cal/mol or J/mol) for diffusion of the species
under consideration (eg. Al in Si), R is the gas constant (8.314 J/mol.K) and T
under consideration (eg. Al in Si), R is the gas constant (8.314 J/mol.K) and T
is the absolute temperature (K). D
is the absolute temperature (K). D0
0 is the pre-exponential term. D
is the pre-exponential term. D0
0 is a
is a
constant for a given diffusion system and is equal to the value of the diffusion
constant for a given diffusion system and is equal to the value of the diffusion
coefficient at 1/T = 0 or T = ∞.
coefficient at 1/T = 0 or T = ∞.
When the temperature of a material increases, the diffusion coefficient D
When the temperature of a material increases, the diffusion coefficient D
increases and, therefore, the flux of atoms increases as well. At higher
increases and, therefore, the flux of atoms increases as well. At higher
temperatures, the thermal energy supplied to the diffusing atoms permits the
temperatures, the thermal energy supplied to the diffusing atoms permits the

Concentration Gradient
Concentration Gradient
A concentration gradient may be created when
A concentration gradient may be created when
two materials of different composition are placed
two materials of different composition are placed
in contact, when a gas or liquid is in contact with
in contact, when a gas or liquid is in contact with
a solid material, when
a solid material, when non-equilibrium
non-equilibrium
structures
structures are produced in a material due to
are produced in a material due to
processing, and from a host of other sources.
processing, and from a host of other sources.
The concentration gradient shows how the
The concentration gradient shows how the
composition of the material varies with distance:
composition of the material varies with distance:
Δc is the difference in concentration over the
Δc is the difference in concentration over the
distance Δx.
distance Δx.
The flux at a particular temperature is constant
The flux at a particular temperature is constant
only if the concentration gradient is also
only if the concentration gradient is also

Composition Profile [Fick’s Second Law]
Composition Profile [Fick’s Second Law]
Fick’s second law, which describes the dynamic, or non-steady state, diffusion
Fick’s second law, which describes the dynamic, or non-steady state, diffusion
of atoms, is the differential equation.
of atoms, is the differential equation.
If we assume that the diffusion coefficient D is not a function of location x
If we assume that the diffusion coefficient D is not a function of location x
and the concentration (c) of diffusing species, we can write a simplified
and the concentration (c) of diffusing species, we can write a simplified
version of Fick’s second law as follows.
version of Fick’s second law as follows.
The solution to this equation depends on the
The solution to this equation depends on the
boundary conditions
boundary conditions
for a particular situation. One solution is:
for a particular situation. One solution is:
where c
where cs
s is a constant concentration of the
is a constant concentration of the
Concentration profiles for
Concentration profiles for
non-steady state diffusion
non-steady state diffusion

The steps in diffusion bonding:
The steps in diffusion bonding:
(a) Initially the contact area is small, application of pressure deforms the surfac
(a) Initially the contact area is small, application of pressure deforms the surfac
(b) grain boundary diffusion permits voids to shrink; and
(b) grain boundary diffusion permits voids to shrink; and
Applications
Applications
1. Diffusion Bonding
1. Diffusion Bonding
The diffusion bonding process is often used for joining
The diffusion bonding process is often used for joining
reactive metals such as titanium, for joining dissimilar
reactive metals such as titanium, for joining dissimilar
metals and materials, and for joining ceramics.
metals and materials, and for joining ceramics.
Diffusion
bonding, occurs in three steps. The first step
Diffusion
bonding, occurs in three steps. The first step
forces the two surfaces together at a high temperature
forces the two surfaces together at a high temperature
and pressure, flattening the surface, fragmenting
and pressure, flattening the surface, fragmenting
impurities, and producing a high atom-to-atom contact
impurities, and producing a high atom-to-atom contact
area.
area.
As the surfaces remain pressed together at high
As the surfaces remain pressed together at high
temperatures, atoms diffuse along grain boundaries to
temperatures, atoms diffuse along grain boundaries to
the remaining voids; the atoms condense and reduce the
the remaining voids; the atoms condense and reduce the
size of any voids at the interface. Because grain
size of any voids at the interface. Because grain

2. Dopant Diffusion for Semiconductor Devices
2. Dopant Diffusion for Semiconductor Devices
The entire microelectronics industry, as we know it today, would not exist if we
The entire microelectronics industry, as we know it today, would not exist if we
did not have a very good understanding of the diffusion of different atoms into
did not have a very good understanding of the diffusion of different atoms into
silicon or other semiconductors. The creation of the p-n junction involves
silicon or other semiconductors. The creation of the p-n junction involves
diffusing dopant atoms, such as phosphorus, arsenic, antimony, boron,
diffusing dopant atoms, such as phosphorus, arsenic, antimony, boron,
aluminum, etc., into precisely defined regions of silicon wafers. Some of these
aluminum, etc., into precisely defined regions of silicon wafers. Some of these
regions are so small that they are best measured in nanometers. A p-n
regions are so small that they are best measured in nanometers. A p-n
junction is a region of the semiconductor, one side of which is doped with n-
junction is a region of the semiconductor, one side of which is doped with n-
type dopants (e.g., As in Si) and the other side is doped with p-type dopants
type dopants (e.g., As in Si) and the other side is doped with p-type dopants
(e.g., B in Si).
(e.g., B in Si).
3. Optical Fibers and Microelectronic Components
3. Optical Fibers and Microelectronic Components
Optical fibers made from silica (SiO2) are coated with polymeric materials to
Optical fibers made from silica (SiO2) are coated with polymeric materials to
prevent diffusion of water molecules. This, in turn, improves the optical and
prevent diffusion of water molecules. This, in turn, improves the optical and
mechanical properties of the fibers.
mechanical properties of the fibers.

Phase diagrams and Diffusion in Materials Science

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