material science

1. PHASE DIAGRAM
(Part – 1/3)
Prepared by
VISHAL MEHTA
Reference : Material Science and Engineering,
An Introduction by William D. Callister, Jr

2. CONTENTS
Basic Concept Binary Phase Diagram
Solubility Limit Unary vs Binary Phase Diagram
Phase Isomorphous
Microstructure Binary Isomorphous System
Free Energy Interpretation of Phase Diagram
Equilibrium Lever Rule
Phase Equilibrium Mass Fraction & Volume Fraction
Non-equilibrium or Metastable State Equilibrium Cooling
Phase Diagram Non-equilibrium Cooling
Unary Phase Diagram Mechanical Properties of Isomorphous Alloys
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 2

3. BASIC CONCEPT
Following are the few terms which are used in discussion
of phase diagram.
• Component : components are pure metals and/or
compounds of which an alloy is composed. For example,
in a copper–zinc brass, the components are Cu and Zn
• System : system has two meanings.
• First, “system” may refer to a specific body of material
under consideration (e.g., a ladle of molten steel).
• Second, it may relate to the series of possible alloys
consisting of the same components, but without regard
to alloy composition (e.g., the iron–carbon system).
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 3

4. BASIC CONCEPT
• Solute : The substance which is dissolved is called
a solute.
• Solvent : The substance in which the solute is dissolved
is called a solvent.
• Solution : A solution is a homogenous mixture of two or
more substances.
• Water-sugar syrup is solution in which sugar is solute
and water is solvent
• This example will be taken many times in this topic of
phase diagram to discuss various terms & concept.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 4
Continue…

5. SOLUBILITY LIMIT
•For many alloy systems and at some specific
temperature, there is a maximum concentration of
solute atoms that may dissolve in the solvent to
form a solid solution; this is called a solubility
limit.
•The addition of solute in excess of this solubility
limit results in the formation of another solid
solution or compound that has a distinctly different
composition.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 5

6. SOLUBILITY LIMIT
• For example, consider the sugar–water system. Initially, as sugar
is added to water, a sugar–water solution or syrup forms.
• As more sugar is introduced, the solution becomes more
concentrated, until the solubility limit is reached, or the solution
becomes saturated with sugar.
• At this time the solution is not capable of dissolving any more
sugar, and further additions simply settle to the bottom of the
container.
• Thus, the system now consists of two separate substances: a
sugar–water syrup liquid solution and solid crystals of
undissolved sugar.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 6
Continue…

7. SOLUBILITY LIMIT
• The solubility
limit of sugar in
water depends on
the temperature of
the water
• Figure indicates
that solubility is
the solubility limit
increases slightly
with rising
temperature.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 7
Continue…

8. PHASE
• A phase may be defined as a homogeneous portion of a system
that has uniform physical and chemical characteristics.
• Every pure material is considered to be a phase; so also is
every solid, liquid, and gaseous solution.
• For example, the sugar–water syrup solution is one phase, and
solid sugar is another. Each has different physical properties
(one is a liquid, the other is a solid); furthermore, each is
different chemically (i.e., has a different chemical composition);
one is virtually pure sugar, the other is a solution
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 8

9. PHASE
• A phase may be defined as a homogeneous portion of a system
that has uniform physical and chemical characteristics.
• Every pure material is considered to be a phase; so also is
every solid, liquid, and gaseous solution.
• For example, the sugar–water syrup solution is one phase, and
solid sugar is another. Each has different physical properties
(one is a liquid, the other is a solid); furthermore, each is
different chemically (i.e., has a different chemical composition);
one is virtually pure sugar, the other is a solution
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 9
Continue…

10. PHASE
• Sometimes, a single-phase system is termed
“homogeneous.” Systems composed of two or more
phases are termed “mixtures” or “heterogeneous
systems.”
• Most metallic alloys and, for that matter, ceramic,
polymeric, and composite systems are heterogeneous.
• Ordinarily, the phases interact in such a way that the
property combination of the multiphase system is
different from, and more attractive than, either of the
individual phases.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 10
Continue…

11. MICROSTRUCTURE
•Many times, the physical
properties and, in particular, the
mechanical behavior of a material
depend on the microstructure.
•Microstructure is subject to direct
microscopic observation, using
optical or electron microscopes
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 11

12. FREE ENERGY
•Free energy is a function of the
internal energy of a system, and also
the randomness or disorder of the
atoms or molecules (or entropy).
•In simple terms, it is energy available
to do work
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 12

13. EQUILIBRIUM
• A system is at equilibrium if its free energy is at a
minimum under some specified combination of
temperature, pressure, and composition.
• In a macroscopic sense, this means that the
characteristics of the system do not change with time
but persist indefinitely; that is, the system is stable.
• A change in temperature, pressure, and/or
composition for a system in equilibrium will result in
an increase in the free energy and in a possible
spontaneous change to another state whereby the free
energy is lowered.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 13

14. PHASE EQUILIBRIUM
• Phase equilibrium is reflected by a constancy with time in the
phase characteristics of a system.
• Sugar–syrup example illustrates the principle of phase
equilibrium using a liquid–solid system.
• Suppose that a sugar–water syrup is contained in a closed
vessel and the solution is in contact with solid sugar at 20˚C .
• If the system is at equilibrium, the composition of the syrup is
65 wt% sugar & 35 wt% water, and the amounts and
compositions of the syrup and solid sugar will remain constant
with time.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 14

15. NONEQUILIBRIUM OR METASTABLE STATE
• In some of the cases, especially in solid systems, a state of
equilibrium is never completely achieved because the rate of
approach to equilibrium is extremely slow; such a system is
said to be in a nonequilibrium or metastable state.
• A metastable state or microstructure may persist
indefinitely, experiencing only extremely slight and almost
imperceptible changes as time progresses.
• Often, metastable structures are of more practical
significance than equilibrium ones. For example, some steel
and aluminum alloys rely for their strength on the
development of metastable microstructures during carefully
designed heat treatments.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 15

16. PHASE DIAGRAMS
•Much of the information about the control of the
phase structure of a particular system is
conveniently and concisely displayed in what is
called a phase diagram, also often termed an
equilibrium diagram.
•There are three externally controllable parameters
that will affect phase structure—viz. temperature,
pressure, and composition—and phase diagrams
are constructed when various combinations of these
parameters are plotted against one another.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 16

17. ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS
• One-component system is the simplest and easiest
type of phase diagram to understand, in which
composition is held constant (i.e., the phase
diagram is for a pure substance); this means that
pressure and temperature are the variables.
• This one-component phase diagram (or unary phase
diagram) [sometimes also called a pressure–
temperature (or P–T) diagram] is represented as a two-
dimensional plot of pressure (ordinate, or vertical
axis) versus temperature (abscissa, or horizontal
axis). Most often, the pressure axis is scaled
logarithmically.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 17

18. ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 18

19. ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 19
oa, ob & oc are
Phase boundaries

20. ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 20
Solid-Liquid equilibrium/coexist
Liquid-Vapor equilibrium/coexist
Solid-Vapor equilibrium/coexist

21. ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 21
Melting
Freezing
Vaporization
Condensation
Sublimation
Deposition

22. ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 22
Freezing/Melting/Fusion Curve
Vaporization/Condensation Curve
Sublimation/Deposition Curve

23. ONE-COMPONENT (OR UNARY) PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 23
Triple Point/Invariant Point

24. TWO-COMPONENTS (OR BINARY) PHASE DIAGRAMS
•Temperature and composition are variable
parameters, and pressure is held
constant—normally 1 atm.
•Binary phase diagrams are maps that
represent the relationships between
temperature and the compositions and
quantities of phases at equilibrium, which
influence the microstructure of an alloy.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 24

25. ONE COMPONENT VS TWO COMPONENTS
PHASE DIAGRAM
(UNARY VS BINARY PHASE DIAGRAM)
One component/Unary
Phase diagram
Two components/Binary
Phase diagram
Temperature – Variable
Pressure – Variable
Composition – Constant
Temperature – Variable
Pressure – Constant
Composition – Variable
Used for only one component Used for two components
Example : Phase diagram of
𝐻2𝑂
In which only one component
𝐻2𝑂 is present
Example : Phase diagram of
Cu-Ni
In which two components Cu &
Ni are present
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 25

26. ISOMORPHOUS
•In crystallography crystals are described
as isomorphous if they are closely similar in
shape.
•For example, Cu & Ni both have same FCC
crystal structure. So they are called
isomorphous.
(visit site http://www.periodictable.com/Properties/A/CrystalStructure.html to find out
crystal structure of various elements)
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 26

27. BINARY ISOMORPHOUS SYSTEM
•It is a system of two components which are
isomorphous.
•To understand in detail, example of Cu-Ni
phase diagram is considered.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 27

28. BINARY ISOMORPHOUS SYSTEM
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 28
• L means liquid region/field
• L represents homogeneous liquid
solution composed of both copper
and nickel
• means solid region/field
• represents solid solution
• for metallic alloys, solid solutions
are commonly designated by
lowercase Greek letters
• +L means two phase
region/field
• +L represents solution in which
both solid and liquid phases are
present.

29. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 29

30. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 30
Cu 100%
Ni 0%

31. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 31
Cu 100%
Ni 0%
Cu 0%
Ni 100%

32. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 32
Cu 100%
Ni 0%
Cu 0%
Ni 100%
1085°C is the melting
point of Cu

33. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 33
Cu 100%
Ni 0%
Cu 0%
Ni 100%
1085°C is the melting
point of Cu
1453°C is the melting
point of Ni

34. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 34
Cu 100%
Ni 0%
Cu 0%
Ni 100%
1085°C is the melting
point of Cu
1453°C is the melting
point of Ni
At temperatures below
about 1080°C copper
and nickel are
mutually soluble in
each other in the solid
state for all
compositions.

35. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 35
Cu 100%
Ni 0%
Cu 0%
Ni 100%
1085°C is the melting
point of Cu
1453°C is the melting
point of Ni
At temperatures below
about 1080°C copper
and nickel are
mutually soluble in
each other in the solid
state for all
compositions.
Complete solubility of
Cu & Ni is explained
by the fact that both
Cu and Ni have the
same crystal
structure (FCC), nearly
identical atomic radii
and electronegativities,
and similar valences.
The copper–nickel
system is termed
isomorphous
because of this
complete liquid and
solid solubility of the
two components.

36. INTERPRETATION OF PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 36

37. INTERPRETATION OF PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 37
You have Phase
Diagram.
So what ?

38. INTERPRETATION OF PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 38
You have Phase
Diagram.
So what ?
What to do
with Phase
diagram ?

39. INTERPRETATION OF PHASE DIAGRAMS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 39
You have Phase
Diagram.
So what ?
What to do
with Phase
diagram ?

40. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 40
INTERPRETATION OF PHASE DIAGRAMS

41. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 41
INTERPRETATION OF PHASE DIAGRAMS
At least three kinds of information available from Phase Diagram

42. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 42
INTERPRETATION OF PHASE DIAGRAMS
At least three kinds of information available from Phase Diagram
1.
The phases that
are present

43. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 43
INTERPRETATION OF PHASE DIAGRAMS
At least three kinds of information available from Phase Diagram
2.
The compositions
of these phases
1.
The phases that
are present

44. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 44
INTERPRETATION OF PHASE DIAGRAMS
At least three kinds of information available from Phase Diagram
2.
The compositions
of these phases
1.
The phases that
are present
3.
the percentages or
fractions of the
phases

45. 1. PHASES PRESENT
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 45
Which phase is
present ?

46. 1. PHASES PRESENT
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 46
Which phase is
present ?
Solid ?

47. 1. PHASES PRESENT
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 47
Which phase is
present ?
Liquid ?
Solid ?

48. 1. PHASES PRESENT
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 48
Which phase is
present ?
Liquid ?
Solid ?

49. 1. PHASES PRESENT
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 49
Which phase is
present ?
Liquid ?
Solid ?

50. 1. PHASES PRESENT
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 50
Which phase is
present ?
How to use
given
composition &
temperature ?
Continue…

51. 1. PHASES PRESENT
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 51
Continue…
Which phase is present,
if composition is 60 wt% Ni-40 wt% Cu
& temperature is 1100°C ?
Find graphically with
the help of Cu-Ni
phase diagram.

52. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 52
Continue…
1. PHASES PRESENT
Answer :

53. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 53
Continue…
1. PHASES PRESENT
Composition is
60 wt% Ni-40 wt% Cu
Answer :

54. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 54
Continue…
1. PHASES PRESENT
Composition is
60 wt% Ni-40 wt% Cu
Answer :
Temperature is 1100°C

55. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 55
Continue…
1. PHASES PRESENT
Composition is
60 wt% Ni-40 wt% Cu
Answer :
Temperature is 1100°C
Obtained point is in Solid region
So, answer is Solid phase

56. 2. DETERMINATION OF PHASE COMPOSITION
• In previous topic, it is discussed to find the present phase (i.e.
solid, liquid or both) with the help of obtained point.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 56
Now the
question
is
At particular obtained point
What is the wt% of Ni as Solid phase ?
What is the wt% of Cu as Solid phase ?
What is the wt% of Ni as Liquid phase ?
What is the wt% of Cu as Liquid phase ?

57. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 57
It the obtained point is in
-solid region
One can easily find the
wt% composition
by taking projection on X-axis.
Because there is no liquid
phase
Here for given point, phase composition is
60 wt% Ni as solid phase
40 wt% Cu as solid phase
There is no liquid phase.

58. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 58
It the obtained point is in
L-liquid region
One can easily find the
wt% composition
by taking projection on X-axis.
Because there is no solid phase
Here for given point, phase composition is
20 wt% Ni as liquid phase
80 wt% Cu as liquid phase
There is no solid phase.

59. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 59
The problem occurs when the
obtained point is in
+L region
Now what to
do with this
point ? How to find wt%
composition as
solid phase &
liquid phase?

60. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 60
Methodology to obtain wt% composition

61. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 61
• Draw horizontal line (i.e.
parallel to X-axis) passing
from the obtained point.
• Extend in both direction
such a way that it
intersects with liquidus &
solidus line
• This horizontal line is
called Tie Line or
Isotherm
• From the both intersection
points, take projections on
X-axis.

62. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 62

63. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 63
Represents
the
composition
of liquid
phase

64. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 64
Represents
the
composition
of liquid
phase
31.5 wt% Ni
68.5 wt% Cu
as liquid
phase

65. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 65
Represents
the
composition
of liquid
phase
Represents
the
composition
of solid
phase
31.5 wt% Ni
68.5 wt% Cu
as liquid
phase

66. 2. DETERMINATION OF PHASE COMPOSITION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 66
Represents
the
composition
of liquid
phase
Represents
the
composition
of solid
phase
31.5 wt% Ni
68.5 wt% Cu
as liquid
phase
42.5 wt% Ni
57.5 wt% Cu
as solid
phase

67. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 67

68. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 68
In previous
two topics,
it is
discussed
that…

69. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 69
In previous
two topics,
it is
discussed
that…
Which type
of phase is
present ?
Solid, liquid
or both ?

70. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 70
In previous
two topics,
it is
discussed
that…
Which type
of phase is
present ?
Solid, liquid
or both ?
What is the wt%
composition of
components as
solid & liquid
phase ?

71. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 71
In previous
two topics,
it is
discussed
that…
Which type
of phase is
present ?
Solid, liquid
or both ?
What is the wt%
composition of
components as
solid & liquid
phase ?
And we know that
problem occurs only when
both phases are present.

72. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 72

73. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 73
Assume that you have obtained a point in Cu-Ni binary phase
diagram. And this obtained point is in +L region. It means both
solid & liquid phases are present.

74. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 74
Assume that you have obtained a point in Cu-Ni binary phase
diagram. And this obtained point is in +L region. It means both
solid & liquid phases are present.
X
percent or
fraction of
solid phase
Y
percent or
fraction of
liquid phase

75. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 75
Assume that you have obtained a point in Cu-Ni binary phase
diagram. And this obtained point is in +L region. It means both
solid & liquid phases are present.
X
percent or
fraction of
solid phase
Y
percent or
fraction of
liquid phase
a wt% Cu
b wt% Ni
as solid
phase
c wt% Cu
d wt% Ni
as solid
phase
e wt% Cu
f wt% Ni
as liquid
phase
g wt% Cu
h wt% Ni
as liquid
phase

76. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 76
Assume that you have obtained a point in Cu-Ni binary phase
diagram. And this obtained point is in +L region. It means both
solid & liquid phases are present.
X
percent or
fraction of
solid phase
Y
percent or
fraction of
liquid phase
a wt% Cu
b wt% Ni
as solid
phase
c wt% Cu
d wt% Ni
as solid
phase
e wt% Cu
f wt% Ni
as liquid
phase
g wt% Cu
h wt% Ni
as liquid
phase
One can find
a, b, c, ,d ,e, ,f, g, h
with help of
Tie Line

77. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 77
Assume that you have obtained a point in Cu-Ni binary phase
diagram. And this obtained point is in +L region. It means both
solid & liquid phases are present.
X
percent or
fraction of
solid phase
Y
percent or
fraction of
liquid phase
a wt% Cu
b wt% Ni
as solid
phase
c wt% Cu
d wt% Ni
as solid
phase
e wt% Cu
f wt% Ni
as liquid
phase
g wt% Cu
h wt% Ni
as liquid
phase
But the problem is,
how to find X & Y
percent or fraction.

78. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 78
How to find X & Y
percent or
fraction ?

79. 3. DETERMINATION OF PHASE AMOUNTS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 79
How to find X & Y
percent or
fraction ?
LEVER RULE

80. LEVER RULE or
THE INVERSE LEVER RULE
• To understand lever rule, we will
consider an example.
• Assume that our obtained point is
B. So both solid & liquid phases are
present.
• Consider that
• WL is the mass fraction of liquid
phase &
• W is mass faction of solid phase.
• Total mass fraction must be 1.
• So, WL + W = 1 ……..(Eq.1)
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 80

81. LEVER RULE or
THE INVERSE LEVER RULE
• the mass of one of the
components (either Cu or
Ni) that is present in both
of the phases must be
equal to the mass of that
component in the total
alloy,
• So,
WLCL+ W C = C0 ……..(Eq.2)
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 81

82. LEVER RULE or
THE INVERSE LEVER RULE
• After solving Eq.1 & Eq.2
𝑾𝑳 =
𝑪𝜶 − 𝑪𝟎
𝑪𝜶 − 𝑪𝑳
=
𝑺
𝑹 + 𝑺
𝑾𝜶 =
𝑪𝟎 − 𝑪𝑳
𝑪𝜶 − 𝑪𝑳
=
𝑹
𝑹 + 𝑺
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 82
Expression of lever rule
(For this particular situation of binary isomorphous system)

83. LEVER RULE or
THE INVERSE LEVER RULE
• Now for our point B
• C = 42.5 wt%
• C0 = 35 wt%
• CL = 31.5 wt%
𝑾𝑳 =
𝑪𝜶 − 𝑪𝟎
𝑪𝜶 − 𝑪𝑳
=
𝟒𝟐. 𝟓 − 𝟑𝟓
𝟒𝟐. 𝟓 − 𝟑𝟏. 𝟓
𝑾𝑳 = 0.68
Same way
𝑾𝜶 = 0.32
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 83

84. VOLUME FRACTION
• For multiphase alloys, it is often more convenient to specify relative
phase amount in terms of volume fraction rather than mass
fraction. Phase volume fractions are preferred because they (rather
than mass fractions) may be determined from examination of the
microstructure; furthermore, the properties of a multiphase alloy
may be estimated on the basis of volume fractions.
• For an alloy consisting of and β phases, the volume fraction of
the phase, is defined as
𝑽𝜶=
𝒗𝜶
𝒗𝜶 + 𝒗𝜷
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 84
𝒗𝜶 𝐢𝐬 𝐯𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝜶 𝒑𝒉𝒂𝒔𝒆 𝒊𝒏 𝒂𝒍𝒍𝒐𝒚
𝒗𝜷 𝐢𝐬 𝐯𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝜷 𝒑𝒉𝒂𝒔𝒆 𝒊𝒏 𝒂𝒍𝒍𝒐𝒚

85. CONVERSION BETWEEN
MASS FRACTION & VOLUME FRACTION
𝑽𝜶 =
𝑾𝜶
𝝆𝜶
𝑾𝜶
𝝆𝜶
+
𝑾𝜷
𝝆𝜷
𝑽𝜷 =
𝑾𝜷
𝝆𝜷
𝑾𝜶
𝝆𝜶
+
𝑾𝜷
𝝆𝜷
𝑾𝜶 =
𝑽𝜶𝝆𝜶
𝑽𝜶𝝆𝜶 + 𝑽𝜷𝝆𝜷
𝑾𝜶 =
𝑽𝜶𝝆𝜶
𝑽𝜶𝝆𝜶 + 𝑽𝜷𝝆𝜷
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 85
𝝆𝜶 𝒊𝒔 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝜶 𝒑𝒉𝒂𝒔𝒆
𝝆𝜷 𝒊𝒔 𝒅𝒆𝒏𝒔𝒊𝒕𝒚 𝒐𝒇 𝜷 𝒑𝒉𝒂𝒔𝒆

86. EQUILIBRIUM COOLING
•It is the condition or situation in which
the cooling occurs very slowly, in such
a way that phase equilibrium is
continuously maintained.
•Sufficient time must be allowed to
achieve equilibrium cooling.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 86

87. EQUILIBRIUM COOLING
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 87
• In equilibrium cooling, there is
no change in microstructure.
• In L (liquid) region, only one
kind of liquid phase is there in
which wt% Ni is same.
• In +L region, two phases are
there, in which wt% Ni is same
for whole individual phases.
• Same way is (solid) region,
only one kind of solid phase is
there in which wt% Ni is same
for whole solid phase.

88. •As discussed in previous topic, to
achieve equilibrium cooling, sufficient
time must be allowed and cooling rate
is very slow.
•Now if the sufficient time is not allowed
or in other words say cooling rate is
high, the obtained cooling is called
Nonequilibrium cooling.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 88
NONEQUILIBRIUM COOLING

89. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 89
NONEQUILIBRIUM COOLING
• In equilibrium, it is observed
that in solid phase, there is
only one kind of solid
phase, in which wt% Ni is
same throughout the phase.
• Here one can observe that
on solid phase, there are
more than one kinds of
different phases are present
with different-different wt%
Ni.

90. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 90
NONEQUILIBRIUM COOLING
• Consider point c’ at which two
kinds of phases are present in
which wt% Ni are 46 & 40
respectively.
• To understand the degree of
displacement, approx. average
composition is considered as 42
wt% Ni & plotted in the diagram.
• Same way for other points, approx.
average composition is considered.
• One can see the degree of
displacement indicated with
dashed line.

91. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 91
NONEQUILIBRIUM COOLING
• The slower the cooling rate, the smaller
the displacement; that is, the
difference between the equilibrium
solidus and average solid composition
is lower.
• One can observe that in
nonequilibrium cooling the distribution
of the two elements within the grains is
nonuniform, a phenomenon termed
segregation; that is, concentration
gradients are established across the
grains that are represented by the
insets.

92. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 92
NONEQUILIBRIUM COOLING
• The center of each grain,
which is the first part to freeze,
is rich in the high-melting
element (e.g., nickel for this
Cu–Ni system), whereas the
concentration of the low-
melting element increases with
position from this region to the
grain boundary. This is termed
a cored structure, which
gives rise to less than the
optimal properties.

93. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 93
Equilibrium
NonEquilibrium

94. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 94
NonEquilibrium
Equilibrium

95. MECHANICAL PROPERTIES OF
ISOMORPHOUS ALLOYS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 95

96. MECHANICAL PROPERTIES OF
ISOMORPHOUS ALLOYS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 96
What this figure indicate ?

97. MECHANICAL PROPERTIES OF
ISOMORPHOUS ALLOYS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 97
The given figure indicates
that Tensile strength
increases with addition of
second component.
What this figure indicates ?

98. MECHANICAL PROPERTIES OF
ISOMORPHOUS ALLOYS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 98

99. MECHANICAL PROPERTIES OF
ISOMORPHOUS ALLOYS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 99
Now what about this figure ?

100. MECHANICAL PROPERTIES OF
ISOMORPHOUS ALLOYS
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 100
This given figure indicates
that ductility decreases with
addition of second
component.
Now what about this figure ?

101. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 101
END OF PART – 1 ….
PLEASE CHECK PART – 2 ….