material science

1. PHASE DIAGRAM
(Part – 2/3)
Prepared by
VISHAL MEHTA
Reference : Material Science and Engineering,
An Introduction by William D. Callister, Jr

2. CONTENTS
Eutectic Mixture
Binary Eutectic System
Eutectoid Reaction
Peritectic Reaction
Basic Comparison
Congruent Phase Transformation
The Gibbs Phase Rule
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 2

3. EUTECTIC MIXTURE
A eutectic mixture is defined as a mixture of two or more
components which usually do not interact to form a new chemical
compound but, which at certain ratios, inhibit the crystallization
process of one another resulting in a system having a lower melting
point than either of the components
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 3

4. EUTECTIC MIXTURE
A eutectic mixture is defined as a mixture of two or more
components which usually do not interact to form a new chemical
compound but, which at certain ratios, inhibit the crystallization
process of one another resulting in a system having a lower melting
point than either of the components
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 4
Component 1
Melting Temp T1

5. EUTECTIC MIXTURE
A eutectic mixture is defined as a mixture of two or more
components which usually do not interact to form a new chemical
compound but, which at certain ratios, inhibit the crystallization
process of one another resulting in a system having a lower melting
point than either of the components
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 5
Component 1
Melting Temp T1
Component 2
Melting Temp T2

6. EUTECTIC MIXTURE
A eutectic mixture is defined as a mixture of two or more
components which usually do not interact to form a new chemical
compound but, which at certain ratios, inhibit the crystallization
process of one another resulting in a system having a lower melting
point than either of the components
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 6
Component 1
Melting Temp T1
Component 2
Melting Temp T2
Eutectic Mixture
of component 1&2
Melting Temp T12

7. EUTECTIC MIXTURE
A eutectic mixture is defined as a mixture of two or more
components which usually do not interact to form a new chemical
compound but, which at certain ratios, inhibit the crystallization
process of one another resulting in a system having a lower melting
point than either of the components
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 7
Component 1
Melting Temp T1
Component 2
Melting Temp T2
Eutectic Mixture
of component 1&2
Melting Temp T12
T12 < T1 & T12 < T2

8. BINARY EUTECTIC SYSTEM
•To understand binary eutectic system, example
of copper-silver system is considered.
•The phase diagram of the copper–silver system
is common and relatively simple phase
diagram found for binary alloys.
•This is known as a binary eutectic phase
diagram.
•A number of features of this phase diagram are
important and worth noting.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 8

9. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 9
BINARY EUTECTIC SYSTEM

10. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 10
BINARY EUTECTIC SYSTEM
Three
single-phase
regions
are
found.

11. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 11
BINARY EUTECTIC SYSTEM
Three
single-phase
regions
are
found.
𝜶

12. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 12
BINARY EUTECTIC SYSTEM
Three
single-phase
regions
are
found.
𝜶
𝜷

13. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 13
BINARY EUTECTIC SYSTEM
Three
single-phase
regions
are
found.
𝜶
𝜷
𝑳𝒊𝒒𝒖𝒊𝒅

14. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 14
BINARY EUTECTIC SYSTEM
The phase
is a solid
solution rich
in copper; it
has silver as
the solute
component
and an FCC
crystal
structure.

15. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 15
BINARY EUTECTIC SYSTEM
The β phase
is a solid
solution rich
in silver; it
has copper
as the solute
component
and an FCC
crystal
structure.

16. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 16
BINARY EUTECTIC SYSTEM
Pure copper
and
pure silver
are also
considered
to be and β
phases,
respectively.

17. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 17
BINARY EUTECTIC SYSTEM
Three
two-phase
regions
are
found.

18. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 18
BINARY EUTECTIC SYSTEM
Three
two-phase
regions
are
found.
𝜶 + 𝑳

19. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 19
BINARY EUTECTIC SYSTEM
Three
two-phase
regions
are
found.
𝜶 + 𝑳
𝜷 + 𝑳

20. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 20
BINARY EUTECTIC SYSTEM
Three
two-phase
regions
are
found.
𝜶 + 𝑳
𝜷 + 𝑳
𝜶 + 𝜷

21. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 21
BINARY EUTECTIC SYSTEM
• The solid solubility
limit line separating
single solid phase
region & two solid
phase region is
termed as SOLVUS
LINE.
• Here CB & HG are
solvus lines.
SOLVUS LINE

22. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 22
BINARY EUTECTIC SYSTEM
• The solubility limit
line separating single
solid phase region &
two phase solid+liquid
region is termed as
SOLIDUS LINE.
• Here BA & GF are
solidus lines.
SOLIDUS LINE

23. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 23
BINARY EUTECTIC SYSTEM
Horizontal isotherm
line BEG may also
be considered a
solidus line; it
represents the
lowest temperature
at which a liquid
phase may exist for
any copper–silver
alloy that is at
equilibrium.

24. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 24
BINARY EUTECTIC SYSTEM
• The solubility limit
line separating single
liquid phase region &
two phase solid+liquid
region is termed as
LIQUIDUS LINE.
• Here AE & EF are
liquidus lines.
LIQUIDUS LINE

25. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 25
BINARY EUTECTIC SYSTEM
1085°C is Melting
temperature of Cu
As silver is added
to copper, the
temperature at
which the alloys
become totally
liquid decreases
along the liquidus
line, line AE;
thus, the melting
temperature
of copper is
lowered by silver
additions.

26. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 26
BINARY EUTECTIC SYSTEM
961.8°C is Melting
temperature of Ag
As copper is added
to silver, the
temperature at
which the alloys
become totally
liquid decreases
along the liquidus
line, line EF; thus,
the melting
temperature
of silver is lowered
by copper additions.

27. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 27
BINARY EUTECTIC SYSTEM
• Both liquidus lines meet
at the point E on the
phase diagram, through
which also passes the
horizontal isotherm line
BEG. Point E is called
an invariant point,
which is designated by
the composition CE &
temperature TE
E is invariant point

28. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 28
BINARY EUTECTIC SYSTEM
• Both liquidus lines meet
at the point E on the
phase diagram, through
which also passes the
horizontal isotherm line
BEG. Point E is called
an invariant point,
which is designated by
the composition CE &
temperature TE
• For the copper–silver
system,
• CE is 71.9 wt%
• TE is 779°C

29. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 29
BINARY EUTECTIC SYSTEM
• For an alloy of
composition CE, upon
cooling, liquid phase is
transformed into two
solid phases & β at
temperature TE
• The opposite reaction
occurs upon heating.
• This is called a eutectic
reaction (eutectic
means easily melted)
• Horizontal solidus line
at TE is called eutectic
isotherm
Eutectic isotherm

30. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 30
BINARY EUTECTIC SYSTEM
• Eutectics Reaction
• TE is eutectic
temperature
• CE is eutectic
composition
• C E is eutectic
composition of at TE
• CβE is eutectic
composition of β at TE
Eutectic isotherm

31. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 31
BINARY EUTECTIC SYSTEM
• Eutectics Reaction for copper-silver system

32. EXAMPLE PROBLEM
•For a 40 wt% Sn–60 wt% Pb alloy at 150oC (300F),
•(1) What phase(s) is (are) present?
•(2) What is (are) the composition(s) of the phase(s)?
•(3) Calculate the relative amount of each phase
present in terms of mass fraction and (b) volume
fraction.
•(4) Calculate the relative amount of each phase
present in terms of volume fraction.
•Take the densities of Pb and Sn to be 11.23 and
7.24 g/cm3, respectively.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 32

33. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 33
Sn-Pb
system
Obtain point
B which
represent
40 wt% Sn–
60 wt% Pb
alloy at
150oC (300F)

34. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 34
Sn-Pb
system
Answer (1)
Obtained
point is in
+β region,
both & β
phases will
Coexist.

35. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 35
Answer (2)
Since two
phases are
present, it
becomes
necessary to
construct a
tie line
through
point B

36. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 36
Answer (2)
𝑆𝑜,
𝐶𝛼 𝑖𝑠
10 wt% Sn–
90 wt% Pb
&
𝐶𝛽 𝑖𝑠
98 wt%Sn–
2 wt% Pb

37. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 37
Answer (3)
Since the
alloy consists
of two
phases, it is
necessary to
employ the
lever rule.
One can
observe
C ,Cβ & C1
in diagram.

38. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 38
Answer (3)
So mass
fraction of
phase is,
𝑊
𝛼 =
𝐶𝛽 − 𝐶1
𝐶𝛽 − 𝐶𝛼
=
98 − 40
98 − 10
= 0.66

39. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 39
Answer (3)
So mass
fraction of
β phase is,
𝑊𝛽 =
𝐶1 − 𝐶𝛼
𝐶𝛽 − 𝐶𝛼
=
40 − 10
98 − 10
= 0.34

40. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 40
Answer (4)
Volume fractions of phase & β phase are,
𝑽𝜶 =
𝑾𝜶
𝝆𝜶
𝑾𝜶
𝝆𝜶
+
𝑾𝜷
𝝆𝜷
𝑽𝜷 =
𝑾𝜷
𝝆𝜷
𝑾𝜶
𝝆𝜶
+
𝑾𝜷
𝝆𝜷
But we don’t have values of 𝝆𝜶 & 𝝆𝜷

41. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 41
𝜌𝛼 =
100
𝐶𝑆𝑛(𝛼)
𝜌𝑆𝑛
+
𝐶𝑃𝑏(𝛼)
𝜌𝑃𝑏
=
100
10
7.24𝑔/𝑐𝑚3 +
90
11.23𝑔/𝑐𝑚3
= 10.64 𝑔/𝑐𝑚3
𝜌𝛽 =
100
𝐶𝑆𝑛(𝛽)
𝜌𝑆𝑛
+
𝐶𝑃𝑏(𝛽)
𝜌𝑃𝑏
=
100
98
7.24𝑔/𝑐𝑚3 +
2
11.23𝑔/𝑐𝑚3
= 7.29 𝑔/𝑐𝑚3
(values of 𝜌𝑆𝑛 & 𝜌𝑃𝑏 are given in question)
Answer (4)

42. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 42
Answer (4)
𝑽𝜶 =
𝑾𝜶
𝝆𝜶
𝑾𝜶
𝝆𝜶
+
𝑾𝜷
𝝆𝜷
=
𝟎. 𝟔𝟔
𝟏𝟎. 𝟔𝟒 𝒈/𝒄𝒎𝟑
𝟎. 𝟔𝟔
𝟏𝟎. 𝟔𝟒 𝒈/𝒄𝒎𝟑 +
𝟎. 𝟑𝟒
𝟕. 𝟐𝟗 𝒈/𝒄𝒎𝟑
∴ 𝑽𝜶= 𝟎. 𝟓𝟕

43. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 43
Answer (4)
𝑽𝜷 =
𝑾𝜷
𝝆𝜷
𝑾𝜶
𝝆𝜶
+
𝑾𝜷
𝝆𝜷
=
𝟎. 𝟑𝟒
𝟕. 𝟐𝟗 𝒈/𝒄𝒎𝟑
𝟎. 𝟔𝟔
𝟏𝟎. 𝟔𝟒 𝒈/𝒄𝒎𝟑 +
𝟎. 𝟑𝟒
𝟕. 𝟐𝟗 𝒈/𝒄𝒎𝟑
∴ 𝑽𝜶= 𝟎. 𝟒𝟑

44. • One can also obtaine answer (3) & (4) using following relations.
• So just find one value & other is obtained from this relation.
• One can observe the answers, which obtained previously
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 44
𝑾𝜶 + 𝑾𝜷 = 𝟏
𝑽𝜶 + 𝑽𝜷 = 𝟏
𝑾𝜶 = 𝟎. 𝟔𝟔
𝑽𝜶 = 𝟎. 𝟓𝟕
𝑾𝜷 = 𝟎. 𝟑𝟒
𝑽𝜷 = 𝟎. 𝟒𝟑
𝟎. 𝟔𝟔 + 𝟎. 𝟑𝟒 = 𝟏
𝟎. 𝟓𝟕 + 𝟎. 𝟒𝟑 = 𝟏

45. EUTECTOID REACTION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 45
When there is a
transformation
form one solid
phase to two
other solid phases
or reverse is
called Eutectoid
Reaction.
Cu-Zn system

46. EUTECTOID REACTION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 46
In simple words,
one can under
stand that Upon
cooling, a solid
phase transforms
into two other
solid phases
( γ and є ).
And the reverse
reaction occurs
upon heating.
It is called
Eutectic Reaction
Cu-Zn system

47. EUTECTOID REACTION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 47
In simple words,
one can under
stand that Upon
cooling, a solid
phase transforms
into two other
solid phases
( γ and є ).
And the reverse
reaction occurs
upon heating.
It is called
Eutectic Reaction
Cu-Zn system

48. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 48
Cu-Zn system

49. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 49
Cu-Zn system

50. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 50
Cu-Zn system
Eutectoid Reaction

51. PERITECTIC REACTION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 51
When there is a
transformation
form solid &
liquid phase to
other different
solid phase or
reverse is called
Peritectic
Reaction.
Cu-Zn system

52. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 52
In simple words,
one can under
stand that Upon
cooling, a + L
phase transforms
into two other
solid phase є.
And the reverse
reaction occurs
upon heating.
It is called
Peritectic Reaction
Cu-Zn system
PERITECTIC REACTION

53. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 53
Cu-Zn system
PERITECTIC REACTION
In simple words,
one can under
stand that Upon
cooling, a + L
phase transforms
into two other
solid phase є.
And the reverse
reaction occurs
upon heating.
It is called
Peritectic Reaction

54. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 54
Cu-Zn system

55. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 55
Cu-Zn system

56. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 56
Cu-Zn system
Peritectic Reaction

57. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 57
Which one is
Eutectic Reaction ?
What is
Eutectic Reaction?

58. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 58
Which one is
Eutectic Reaction ?
What is
Eutectic Reaction?

59. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 59
Which one is
Eutectic Reaction ?
What is
Eutectic Reaction?
Which one is
Eutectoid Reaction ?
What is
Eutectoid Reaction?
How to
identify the
Reaction ?

60. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 60
Which one is
Eutectic Reaction ?
What is
Eutectic Reaction?
Which one is
Eutectoid Reaction ?
What is
Eutectoid Reaction?
Which one is
Peritectic Reaction ?
What is
Peritectic Reaction?
How to
identify the
Reaction ?

61. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 61
Which one is
Eutectic Reaction ?
What is
Eutectic Reaction?
Which one is
Eutectoid Reaction ?
What is
Eutectoid Reaction?
Which one is
Peritectic Reaction ?
What is
Peritectic Reaction?
How to
identify the
Reaction ?

62. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 62
Which one is
Eutectic Reaction ?
What is
Eutectic Reaction?
Which one is
Eutectoid Reaction ?
What is
Eutectoid Reaction?
Which one is
Peritectic Reaction ?
What is
Peritectic Reaction?
How to
identify the
Reaction ?
Confused ?

63. BASIC COMPARISION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 63
EUTECTIC REACTION EUTECTOID REACTION PERITECTIC REACTION
Liquid phase
Type-1
solid phase
Type-2
solid phase
Type-1 solid phase
Type-2
solid phase
Type-3
solid phase
Type-1
solid phase
Liquid
phase
Type-2 solid phase
Note : There is no specific meaning of type 1, 2 & 3. It is just randomly used to differentiate the solid phases.

64. CONGRUENT PHASE TRANSFORMATION
•Phase transformations may be classified according
to whether or not there is any change in
composition for the phases involved.
•Those for which there are no compositional
alterations are said to be congruent
transformations.
•It means phase changes without any change in
wt% composition.
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 64

65. CONGRUENT PHASE TRANSFORMATION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 65
Type – 1 phase
with
composition
X wt% metal A
Y wt% metal B

66. CONGRUENT PHASE TRANSFORMATION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 66
Type – 1 phase
with
composition
X wt% metal A
Y wt% metal B
Congruent
Phase
Transformation

67. CONGRUENT PHASE TRANSFORMATION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 67
Type – 1 phase
with
composition
X wt% metal A
Y wt% metal B
Type – 2 phase
with same
composition
X wt% metal A
Y wt% metal B
Congruent
Phase
Transformation

68. CONGRUENT PHASE
TRANSFORMATION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 68
A portion of the nickel–titanium
phase diagram on which is
shown a congruent melting point
for the γ - phase solid solution at
1310OC and 44.9 wt% Ti.

69. CONGRUENT PHASE
TRANSFORMATION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 69
A portion of the nickel–titanium
phase diagram on which is
shown a congruent melting point
for the γ - phase solid solution at
1310OC and 44.9 wt% Ti.

70. CONGRUENT PHASE
TRANSFORMATION
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 70
A portion of the nickel–titanium
phase diagram on which is
shown a congruent melting point
for the γ - phase solid solution at
1310OC and 44.9 wt% Ti.
Congruent
Phase
Transformation
L γ

71. THE GIBBS PHASE RULE
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 71
Josiah Willard Gibbs
the nineteenth-century physicist

72. THE GIBBS PHASE RULE
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 72
Josiah Willard Gibbs
the nineteenth-century physicist

73. THE GIBBS PHASE RULE
P + F = C + N
P = number of phases present
F = number of degrees of freedom or the
number of externally controlled
variables (e.g., temperature, pressure,
composition)
C = number of components in the system
N = number of noncompositional
variables (e.g., temperature and
pressure)
• This rule represents a criterion for the
number of phases that will coexist within a
system at equilibrium
Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 73
Josiah Willard Gibbs
the nineteenth-century physicist

74. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 74
THE GIBBS PHASE RULE
Continue…

75. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 75
THE GIBBS PHASE RULE
Continue…
Consider that we want
to find the number of
degrees of freedom or
the number of
externally controlled
variables for the single
phase field of Cu-Ag
system

76. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 76
THE GIBBS PHASE RULE
Continue…
• Now Gibbs phase rule is P + F = C + N
• Here
• P = number of phases present
• As we have considered single phase field, P = 1
• F = ?
• C = number of components in system
• As it is binary system, C = 2
• N = number of noncompositional variables (e.g.,
temperature and pressure)
• Here we need to check that temperature & pressure are
variable or not.
• If both are variable, N = 2. If any one of two is variable, N = 1.

77. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 77
THE GIBBS PHASE RULE
Continue…
• It has been discussed in initial phase of this
chapter that for binary system Temperature and
composition are variable parameters, and pressure
is held constant—normally 1 atm. (slide no 17 & 24 of Part-1)
• So here only temperature is variable, N = 1.
• Gibbs phase rule is P + F = C + N
• So 1 + F = 2 + 1
• F = 2

78. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 78
THE GIBBS PHASE RULE
Continue…
• What is the meaning of F = 2 ?
• This means that to completely describe the
characteristics of any alloy that exists within one
of these phase fields, we must specify two
parameters; these are composition and
temperature, which locate, respectively, the
horizontal and vertical positions of the alloy on the
phase diagram.

79. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 79
meaning of F = 2
• In other words,
it can be said
that one can
vary two
parameters.
• It means for
any value of T,
you may get
more than one
values of C &
vice versa.

80. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 80
THE GIBBS PHASE RULE
Continue…
• Now for the same example & same system,
consider two phase field (instead of single phase).
• So, P = 2
• Gibbs phase rule is P + F = C + N
• So 2 + F = 2 + 1
• F = 1

81. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 81
THE GIBBS PHASE RULE
Continue…
• What is the meaning of F = 1 ?
• F = 1 means it is necessary to specify either
temperature or the composition of one of the
phases to completely define the system.

82. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 82
meaning of F = 1
• Number of variable parameter
is 1
• So, if you know any one value
of three, remaining two values
you can find.
• For example if you know T1,
you will get other two values of
C & Cβ.
• Or if you know C , you can
find other two values.

83. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 83
meaning of F = 1
• In other words, it can be
said that one can vary
only one parameter and
other two will be changed.
• It means for any value of
T1, you will get one specific
value of C & one specific
value of Cβ.
• (point can be any where
on tie line/isotherm)

84. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 84
THE GIBBS PHASE RULE
Continue…
• Now for the same example & same system,
consider three phase field (instead of two phase).
• So, P = 3
• Gibbs phase rule is P + F = C + N
• So 3 + F = 2 + 1
• F = 0

85. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 85
THE GIBBS PHASE RULE
Continue…
• What is the meaning of F = 0 ?
• This means that the compositions of all three
phases as well as the temperature are fixed.

86. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 86
meaning of F = 0
• Number of
variable
parameter is 0.
• It means values
of T, CL, C &
Cβ are fixed.

87. Prepared by VISHAL MEHTA || Reference : Material Science and Engineering, An Introduction by William D. Callister, Jr 87
END OF PART – 2 ….
PLEASE CHECK PART – 3 ….