1. CHAPTER 10: PHASE DIAGRAMS
ISSUES TO ADDRESS...
• When we combine two elements...
what equilibrium state do we get?
• In particular, if we specify...
--a composition (e.g., wt%Cu - wt%Ni), and
--a temperature (T)
then...
How many phases do we get?
What is the composition of each phase?
How much of each phase do we get? 1

2. Introduction and Motivation
Phase Diagrams
Introduction and Motivation
Use Phase Diagrams to choose/design heating cycles for forming metal alloys
Phase Diagrams represent equilibrium states, but can also be used to guide the formation of materials under non-equilibrium conditions
End of lecture 1

3. Phase diagrams We will:
You are used to thinking about phase diagrams in vapor-liquid equilibrium
In this course we will use phase diagrams to describe and understand structure in metal alloys
Why? Structure  Properties
We will:
1. Learn some terminology
2. Learn how to interpret phase diagrams
3. Study some “simple” binary phase diagrams
4. Learn how structure evolves upon cooling
End of lecture 1

4. Phase diagrams – definitions
Component – pure species
System (more than one element)
Book also uses solute/solvent nomenclature (usually for binary mixtures)
Solubility limit – maximum amount of solute you can add to a solvent before the solute does not dissolve into the solvent
This often depends on temperature (Fig 10.1)
End of lecture 1

5. THE SOLUBILITY LIMIT • Solubility Limit: • Ex: Phase Diagram:
Max concentration for
which only a solution
occurs.
• Ex: Phase Diagram:
Water-Sugar System
Question: What is the
solubility limit at 20C?
Answer: 65wt% sugar.
If Co < 65wt% sugar: solution
If Co > 65wt% sugar: syrup + sugar.
Adapted from Fig. 9.1,
Callister 6e.
• Solubility limit increases with T:
e.g., if T = 100C, solubility limit = 80wt% sugar. 2

6. COMPONENTS AND PHASES • Components: • Phases:
The elements or compounds which are mixed initially
(e.g., Al and Cu)
• Phases:
The physically and chemically distinct material regions
that result (e.g., a and b).
Aluminum-
Copper
Alloy
Adapted from Fig. 9.0,
Callister 3e. 3 3

7. Phase Diagrams
Phases
Phase – a homogeneous portion of a system that has uniform physical & chemical properties (i.e. composition)
Again – you are familiar with two-phase systems (pure material – VLE)
Each phase may have different physical properties and same chemical composition (example water + ice; iron FCC+BCC).
In other cases, each phase may have different physical properties and different chemical composition (example VLE binary mixture)
End of lecture 1

8. EFFECT OF T & COMPOSITION (Co)
• Changing T can change # of phases: path A to B.
• Changing Co can change # of phases: path B to D.
• water-
sugar
system
Adapted from Fig. 9.1,
Callister 6e. 4

9. Phase Diagrams
Phase Equilibrium (10.5)
Know all about this – a system with multiple phases is at equilibrium (for a pure compound) when
The T, P, and Gibbs free energy of each phase (chemical potential) are the same
Why does this matter? The phase diagrams of many simple metals alloys are known
However when real materials are processed an equilibrium state is not reached (why would that be the case?)
End of lecture 1

10. PHASE DIAGRAMS • Tell us about phases as function of T, Co, P.
• For this course:
--binary systems: just 2 components.
--independent variables: T and Co (P = 1atm is always used).
• Phase
Diagram
for Cu-Ni
system
Adapted from Fig. 9.2(a), Callister 6e.
(Fig. 9.2(a) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH (1991). 5

11. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
Simplest place to start – two metals that are completely soluble in one another (e.g. Cu – Ni)
Three regions in the diagram – pure liquid, pure solid (a – phase), and a coexistence region
End of lecture 1

12. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
How to read this thing
Far left – pure copper, far right – pure nickel
Above the “liquidus” line a one phase liquid is obtained, below the “solidus” line a one phase solid is obtained
In between these lines there is a two-phase region; a+L
What phases are present at point B and what are their compositions?
Tie lines and lever rule
End of lecture 1

13. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
Point B – a+liquid phase
Phase compositions – use tie line
Draw vertical lines to x-axis from the intercepts with the solidus, liquidus lines
Line parallel to x-axis drawn from the point of interest (here B) to the solidus and liquidus lines
Where green lines intercept the x-axis gives the composition of the different phases
End of lecture 1

14. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
Point B – a+liquid phase
Now I know the phase compositions – how much of each phase is present?
Use the inverse lever rule
Or in words, the weight fraction of one phase is determined by taking the length of tie line from the overall alloy composition to the phase boundary for the other phase, and dividing it both the total length of the tie line
End of lecture 1

15. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
Example: what is the composition and phase fractions for point C shown in the Figure?
Liquid
a + L C A B a
CL ~ 35.5 wt% A
Ca ~ 53.5 wt% A
End of lecture 1
Aside: pp , volume fractions, etc..

16. PHASE DIAGRAMS: # and types of phases
• Rule 1: If we know T and Co, then we know:
--the # and types of phases present.
• Examples:
Cu-Ni
phase
diagram
Adapted from Fig. 9.2(a), Callister 6e.
(Fig. 9.2(a) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991). 6

17. PHASE DIAGRAMS: composition of phases
• Rule 2: If we know T and Co, then we know:
--the composition of each phase.
Cu-Ni system
• Examples:
Adapted from Fig. 9.2(b), Callister 6e.
(Fig. 9.2(b) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991.) 7

18. PHASE DIAGRAMS: weight fractions of phases
• Rule 3: If we know T and Co, then we know:
--the amount of each phase (given in wt%).
Cu-Ni
system
• Examples:
= 27wt%
Adapted from Fig. 9.2(b), Callister 6e.
(Fig. 9.2(b) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991.) 8

19. THE LEVER RULE: A PROOF • Sum of weight fractions:
• Conservation of mass (Ni):
• Combine above equations:
• A geometric interpretation: 9

20. EX: COOLING IN A Cu-Ni BINARY
• Phase diagram:
Cu-Ni system.
Cu-Ni
system
• System is:
--binary
i.e., 2 components:
Cu and Ni.
--isomorphous
i.e., complete
solubility of one
component in
another; a phase
field extends from
0 to 100wt% Ni.
• Consider
Co = 35wt%Ni.
Adapted from Fig. 9.3, Callister 6e. 10

21. CORED VS EQUILIBRIUM PHASES
• Ca changes as we solidify.
• Cu-Ni case:
First a to solidify has Ca = 46wt%Ni.
Last a to solidify has Ca = 35wt%Ni.
• Fast rate of cooling:
Cored structure
• Slow rate of cooling:
Equilibrium structure 11

22. Microstructure development in isomorphous alloys
Phase Diagrams
Microstructure development in isomorphous alloys
How does structure evolve in the solid alloy as it is cooled
Initial analysis: assume cooling is sufficiently slow that phase equilibrium is maintained
Go from points a e as you cool
End of lecture 1

23. Non-equilibrium cooling
Phase Diagrams
Non-equilibrium cooling
What happens during “real” cooling conditions?
Much more complicated now
Why solid phases cannot re-equilibrate as T is lowered
See segregation within the grains
End of lecture 1

24. MECHANICAL PROPERTIES: Cu-Ni System
• Effect of solid solution strengthening on:
--Tensile strength (TS)
--Ductility (%EL,%AR)
Adapted from Fig. 9.5(a), Callister 6e.
Adapted from Fig. 9.5(b), Callister 6e.
--Peak as a function of Co
--Min. as a function of Co
Tensile strength: maximum stretching force that a material can stand
Ductility: capacity of being molded or shaped without breaking 12

25. BINARY-EUTECTIC SYSTEMS
has a special composition
with a min. melting T.
2 components
Cu-Ag
system
Adapted from Fig. 9.6,
Callister 6e. (Fig. 9.6 adapted
from Binary Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.) 13

26. Binary Eutectic mixtures
These are another very common class of bimetallic alloys
Two metals that are not totally miscible (e.g. copper-silver)
Can only dissolve about 8 wt% of each metal into one another
Now have two different “pure” solid metal phases (a and b)
End of lecture 1

27. EX: Pb-Sn EUTECTIC SYSTEM (1)
• For a 40wt%Sn-60wt%Pb alloy at 150C, find...
--the phases present:
a + b
--the compositions of
the phases:
Pb-Sn
system
Adapted from Fig. 9.7,
Callister 6e. (Fig. 9.7 adapted
from Binary Phase Diagrams, 2nd ed., Vol. 3, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.) 14

28. EX: Pb-Sn EUTECTIC SYSTEM (2)
• For a 40wt%Sn-60wt%Pb alloy at 150C, find...
--the phases present: a + b
--the compositions of
the phases:
Ca = 11wt%Sn
Cb = 99wt%Sn
--the relative amounts
of each phase:
Pb-Sn
system
Adapted from Fig. 9.7,
Callister 6e. (Fig. 9.7 adapted
from Binary Phase Diagrams, 2nd ed., Vol. 3, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.) 15

29. MICROSTRUCTURES IN EUTECTIC SYSTEMS-I
• Co < 2wt%Sn
• Result:
--polycrystal of a grains.
Adapted from Fig. 9.9, Callister 6e. 16

30. MICROSTRUCTURES IN EUTECTIC SYSTEMS-II
• 2wt%Sn < Co < 18.3wt%Sn
• Result:
--a polycrystal with fine
b crystals.
Pb-Sn
system
Adapted from Fig. 9.10, Callister 6e. 17

31. MICROSTRUCTURES IN EUTECTIC SYSTEMS-III
Lead-rich α-solid soln (dark)
+ tin-rich β-solid soln (light)
Lamellae structure
• Co = CE
• Result: Eutectic microstructure
--alternating layers of a and b crystals.
Pb-Sn
system
Adapted from Fig. 9.12, Callister 6e. (Fig from Metals Handbook, Vol. 9, 9th ed., Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.)
Adapted from Fig. 9.11, Callister 6e. 18

32. MICROSTRUCTURES IN EUTECTIC SYSTEMS-IV
• 18.3wt%Sn < Co < 61.9wt%Sn
• Result: a crystals and a eutectic microstructure
Pb-Sn
system
Adapted from Fig. 9.14, Callister 6e. 19

33. Binary Eutectic mixtures
Phase Diagrams
Binary Eutectic mixtures
Point E is something new – this is called the invariant (or eutectic) point
What happens when you change temperature at the invariant point?
This is called a eutectic reaction
Upon cooling it is similar to solidification, however now two phases are formed
Line BG is also referred to as the eutectic isotherm
End of lecture 1

34. Microstructure development in Eutectic Alloys
Phase Diagrams
Microstructure development in Eutectic Alloys
Case 1 – low tin content (Pb-Sn alloy)
Looks pretty similar to before (what is different?)
Again, assuming slow cooling
End of lecture 1

35. Microstructure development in Eutectic Alloys
Phase Diagrams
Microstructure development in Eutectic Alloys
Case 2 – tin content above solid-solution max. tin loading
More going on here
Notice b phase formation at lower T
Equilibrium cooling
End of lecture 1

36. Microstructure development in Eutectic Alloys
Phase Diagrams
Microstructure development in Eutectic Alloys
Case 3 – Solidification at the eutectic composition
Go through eutectic point at 183 C
End of lecture 1

37. Microstructure development in Eutectic Alloys
Case 4 – Not eutectic composition, but cross eutectic isotherm
Two different types of a structural domains – why?
Eutectic microconstituents
50-50 wt % Pb-Sn alloy
Primary α (large dark regions)
+ lamellar eutectic structure
(lead-rich α+ tin-rich β)
End of lecture 1

38. Phase Diagrams
Example – calculation of phases/phase fractions in a binary eutectic alloy
Consider the phase diagram below – what are the phase fractions and compositions at C4’ at the eutectic isotherm?
You know how to solve this, it is just a bit more involved than the last example:
Phases: a, b
Microstructures: Eutectic (a,b), “primary” a
Why no “primary” b? – C4’ is fixed, and so came down through the a+L field
Fractions we need:
Fraction of eutectic
Fraction of a – both in the eutectic and “primary” phase
Fraction of b
End of lecture 1

39. Phase Diagrams
Example – calculation of phases/phase fractions in a binary eutectic alloy
Consider the phase diagram below – what are the phase fractions and compositions at C4’ at the eutectic isotherm?
Eutectic fraction
Primary a fraction
End of lecture 1
Total a fraction
Total b fraction

40. Phase Diagrams
Example – calculation of phases/phase fractions in a binary eutectic alloy
Consider the phase diagram below – what are the phase fractions and compositions at C4’ at the eutectic isotherm?
Eutectic fraction
you know C4’ !
(~32.5 wt% Sn)
= 0.33
= 0.67
= 0.82
= 0.18
Primary a fraction
End of lecture 1
Total a fraction
Total b fraction

41. Example – calculation of phases/phase fractions in a binary eutectic alloy
Consider the phase diagram below – what are the phase fractions and compositions at C4’ at the eutectic isotherm?
Couple of points
The eutectic and primary a fractions tell you how much of the different microstructures are present
The a and b fractions tell you how much of each phase are present
At the eutectic point there are three phases (a, b, and the eutectic) present.
At the left of the Eutectic point, on the eutectic isotherm, there are 2 phases (primary α and eutectic)
There are now also different microstructures – here two types of a phases!
The a phase in the eutectic is 18.3wt% Sn, b phase in the eutectic is 97.8wt% Sn!
At the right of the Eutectic point, on the eutectic isotherm, there are 2 phases (primary β and eutectic)
End of lecture 1

42. Mixtures with intermediate phases/compounds
Phase Diagrams
Mixtures with intermediate phases/compounds
Binary isomorphous and eutectic mixtures are very simple
Have only two solid phases – these are often referred to as terminal solid solutions since they are in the “A” or “B” rich regions of the phase diagram
Many other alloys form intermediate solid solutions at other compositions besides the extremities
Good example – copper/zinc system
Six different solid solutions (diagram next page)
This is more representative of real solids
Important point – still only single- or two-phase regions present
Commercial brass – 70/30 Cu/Zn, single a-phase
End of lecture 1

43. Mixtures with intermediate phases/compounds
Phase Diagrams
Mixtures with intermediate phases/compounds
Good example – copper/zinc system
End of lecture 1

44. Mixtures with intermediate phases/compounds
Phase Diagrams
Mixtures with intermediate phases/compounds
Intermetallic compounds – in some systems discrete compounds (i.e. well-defined stoichiometries) form instead of solid solutions
Example Mg2Pb
End of lecture 1

45. Eutectoid and peritectic reactions
Phase Diagrams
Eutectoid and peritectic reactions
Besides the eutectic point, other invariant points involving three phases can be found in binary alloy systems
Go back to Zn-Cu system
One solid phase going to two solid phases -- eutectoid
Eutectoid isotherm
Difference with eutectic?
Eutectic: 1 liquid phase  2 solid phases
Eutectoid: 1 solid phase  2 solid phases
End of lecture 1

46. Eutectoid and peritectic reactions
Phase Diagrams
Eutectoid and peritectic reactions
One solid phase going to one liquid phase and one solid phase -- peritectic
Important point: eutectic, eutectoid, and peritectic are phase transitions involving three phases
It is essentially nomenclature
End of lecture 1

47. Phase transformations
Phase Diagrams
Phase transformations
What we have been talking about are called phase transformations
Congruent transformations – phase transformations in which there is no change in composition of the phases involved
These include melting of pure materials
Allotropic transformations (polymorphism, example carbon graphite-> diamond ->other forms)
Phase transformations where the composition of the phases involved change – incongruent transformations
These include eutectic/eutectoid reactions, and melting of an alloy in an isomorphous system
End of lecture 1

48. Phase transformations
Phase Diagrams
Phase transformations
Intermediate phases are often categorized based on whether they melt congruently or incongruently
Figure below shows both – what are examples of each?
End of lecture 1

49. Phase Diagrams
Label Eutectic, Eutectoid, Peritectic points and congruent phase transitions
Eutectic points
(~1450 C, 18 wt%V)
L  bHf + HfV2
(~1520 C, 39 wt%V)
L  HfV2 + sol. soln
Eutectoid points
(~1190 C, 6 wt%V)
bHf  aHf + HfV2
(~1550 C, 36 wt%V)
L  HfV2 (cooling)
End of lecture 1
Congruent melting

50. Ceramic phase diagrams
Can observe all the same phenomena for ceramics as you do for metals
Example: silica-alumina system
cristoballite
End of lecture 1
mullite: intermediate compound 3Al2O32SiO2

51. APPLICATION: REFRACTORIES
• Need a material to use in high temperature furnaces.
• Consider Silica (SiO2) - Alumina (Al2O3) system.
• Phase diagram shows:
mullite, alumina, and crystobalite (made up of SiO2)
tetrahedra as candidate refractories.
Adapted from Fig , Callister 6e. (Fig is adapted from F.J. Klug and R.H. Doremus, "Alumina Silica Phase Diagram in the Mullite Region", J. American Ceramic Society 70(10), p. 758, 1987.) 25

52. Ternary phase diagrams
Can also have three-component systems
These get messy – why?
Hard to visualize (need 3D-plots)– See Phase program
Something very useful – Gibbs phase rule
After all, equilibrium phase diagrams should be governed by thermodynamics
Gibbs phase rule: for a system with C components, P phases and N non-compositional variables (T and P), the number of degrees of freedom (F) is given by:
End of lecture 1

53. Phase Diagrams
Gibbs phase rule
What does this tell you? For all the phase diagrams (binary mixtures, fixed pressure P) it says
This is 1 because P is fixed
Now assuming a 2-component system
Ok, now what? This tells you for a binary mixture that is one phase (P = 1) that two parameters must be specified (temperature and composition) to fix the position of the system on the phase diagram
For two phases (P = 2) only one need be specified – if for instance T is specified this “fixes” the composition
For three phases (P = 3) it is an invariant point (example eutectic point)
Let’s try it out…
End of lecture 1

54. Gibbs phase rule Our old friend the Cu-Ag system
Phase Diagrams
Gibbs phase rule
Our old friend the Cu-Ag system
Consider the a + L region
If we fix T then we have fixed the composition of the two phases
If we choose the composition of one phase, we have chosen the other composition as well as the temperature
End of lecture 1

55. Iron-Carbon System ferrite: BCC austenite: FCC d ferrite: BCC
This is a very important binary alloy system – steel!
Turns out carbon has a fairly low solubility in iron – this has all sorts of technological implications
ferrite: BCC
austenite: FCC
d ferrite: BCC
Note: diagram
only goes to 6.7 wt% C
– all steels/cast iron have carbon contents below this
End of lecture 1

56. Iron-Carbon System Carbon is an interstitial impurity in Fe
Phase Diagrams
Iron-Carbon System
Carbon is an interstitial impurity in Fe
Forms solid solutions with the a, g, d phases
Also forms iron carbide (Fe3C) known as cementite
Phase diagram has features we have seen before
Eutectic
Eutectoid
End of lecture 1

57. Phase Diagrams
Iron-Carbon System
Are there any eutectic points on this phase diagram?
End of lecture 1

58. Iron-Carbon System – microstructure evolution
Phase Diagrams
Iron-Carbon System – microstructure evolution
Pearlite formation
When one crosses the eutectoid point
Pearlite – alternating layers of Fe3C and a ferrite
Why do you get this structure – phases formed have different C contents – need C diffusion
End of lecture 1

59. Iron-Carbon System – microstructure evolution Hypoeutectic alloys
Phase Diagrams
Iron-Carbon System – microstructure evolution
Hypoeutectic alloys
Go from austenite  ferrite + cementite but go through the a + g 2-phase region to get there
Note that at point d the a phase has started to form
It usually nucleates at grain boundaries
As ferrite domains increase in size they coalesce (spherical  lamellar type domains)
Again see pearlite type structure at f, but now there are 2 “types” of ferrite domains
Proeutectoid ferrite
Eutectoid ferrite
End of lecture 1

60. What are the relative amounts of eutectoid and proeutectoid ferrite?
Phase Diagrams
Hypoeutectic alloys
What are the relative amounts of eutectoid and proeutectoid ferrite?
You already know how to do this – lever rule (with a twist)
Fraction of pearlite
Fraction of proeutectoid a
End of lecture 1

61. Now have two type of cementite Fe3C
Phase Diagrams
Hypereutectic alloys – go down in T from other side (i.e. C-rich) of eutectoid point
Again, note the minor phase (here Fe3C) nucleates at the grain boundaries
Now have two type of cementite Fe3C
That formed before the eutectoid (proeutectoid Fe3C)
That formed at the eutectoid
End of lecture 1

62. HYPOEUTECTIC & HYPEREUTECTIC
Adapted from Fig. 9.7,
Callister 6e. (Fig. 9.7 adapted from Binary Phase Diagrams, 2nd ed., Vol. 3, T.B. Massalski (Editor-in-Chief), ASM International, Materials Park, OH, 1990.)
(Figs and 9.15 from Metals Handbook, 9th ed.,
Vol. 9, Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.)
Adapted from
Fig. 9.15, Callister 6e.
Adapted from Fig. 9.15, Callister 6e. (Illustration only)
Adapted from Fig. 9.12, Callister 6e. 20

63. IRON-CARBON (Fe-C) PHASE DIAGRAM
(Adapted from Fig. 9.24, Callister 6e. (Fig from Metals Handbook, 9th ed., Vol. 9, Metallography and Microstructures, American Society for Metals, Materials Park, OH, 1985.)
Adapted from Fig. 9.21,Callister 6e. (Fig adapted from Binary Alloy Phase Diagrams, 2nd ed.,
Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990.) 21

64. HYPOEUTECTOID STEEL
Adapted from Figs and 9.26,Callister 6e. (Fig adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990.)
Adapted from
Fig. 9.27,Callister
6e. (Fig courtesy Republic Steel Corporation.) 22

65. HYPEREUTECTOID STEEL
Adapted from Figs and 9.29,Callister 6e. (Fig adapted from Binary Alloy Phase Diagrams, 2nd ed., Vol. 1, T.B. Massalski (Ed.-in-Chief), ASM International, Materials Park, OH, 1990.)
Adapted from
Fig. 9.30,Callister
6e. (Fig. 9.30
copyright 1971 by United States Steel Corporation.) 23

66. ALLOYING STEEL WITH MORE ELEMENTS
• Teutectoid changes:
• Ceutectoid changes:
Adapted from Fig. 9.31,Callister 6e. (Fig from Edgar C. Bain, Functions of the Alloying Elements in Steel, American Society for Metals, 1939, p. 127.)
Adapted from Fig. 9.32,Callister 6e. (Fig from Edgar C. Bain, Functions of the Alloying Elements in Steel, American Society for Metals, 1939, p. 127.) 24

67. SUMMARY • Phase diagrams are useful tools to determine:
--the number and types of phases,
--the wt% of each phase,
--and the composition of each phase
for a given T and composition of the system.
• Alloying to produce a solid solution usually
--increases the tensile strength (TS)
--decreases the ductility.
• Binary eutectics and binary eutectoids allow for
a range of microstructures. 26

68. Example Handouts
End of lecture 1

69. Phase Diagrams
Example – calculation of phases/phase fractions in a binary eutectic alloy
Consider the phase diagram below – what are the phase fractions and compositions at C4’ at the eutectic isotherm?
You know how to solve this, it is just a bit more involved than the last example:
Phases: a, b
Microstructures: Eutectic (a,b), “primary” a
Why no “primary” b? – C4’ is fixed, and so came down through the a+L field
Fractions we need:
Fraction of eutectic
Fraction of a – both in the eutectic and “primary” phase
Fraction of b
End of lecture 1

70. Phase Diagrams
Example – calculation of phases/phase fractions in a binary eutectic alloy
Consider the phase diagram below – what are the phase fractions and compositions at C4’ at the eutectic isotherm?
Eutectic fraction
Primary a fraction
End of lecture 1
Total a fraction
Total b fraction

71. Phase Diagrams
Example – calculation of phases/phase fractions in a binary eutectic alloy
Consider the phase diagram below – what are the phase fractions and compositions at C4’ at the eutectic isotherm?
Couple of points
The eutectic and primary a fractions tell you how much of the different microstructures are present
The a and b fractions tell you how much of each phase are present
Still only two phases (a,b) present!
There are now also different microstructures – here two types of a phases!
The a phase in the eutectic is 18.3wt% Sn, b phase in the eutectic is 97.8wt% Sn!
End of lecture 1

72. Phase Diagrams
Label Eutectic, Eutectoid, Peritectic points and congruent phase transitions
Eutectic points
Eutectoid points
End of lecture 1
Congruent melting

73. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
Point B – a+liquid phase
Phase compositions – use tie line
Line parallel to x-axis drawn from the point of interest (here B) to the solidus and liquidis lines
Draw vertical lines to x-axis from the intercepts with the solidus, liquidis lines
End of lecture 1
Where green lines intercept the x-axis gives the composition of the different phases

74. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
Point B – a+liquid phase
Now I know the phase compositions – how much of each phase is present?
Use the inverse lever rule
Or in words, the weight fraction of one phase is determined by taking the length of tie line from the overall alloy composition to the phase boundary for the other phase, and dividing it both the total length of the tie line
End of lecture 1

75. Binary Isomorphous Systems
Phase Diagrams
Binary Isomorphous Systems
Example: what is the composition and phase fractions for point C shown in the Figure?
Liquid
a + L C a
End of lecture 1
Aside: pp , volume fractions, etc..

76. ANNOUNCEMENTS Reading: Chapter 10 HW # 6: Due Monday March 5th
10.1; 10.5; 10.7; 10.8; 10.12; 10.13; 10.22;
10.26; 10.27; 10.32; 10.41; 10.44