1. The Structure and Dynamics of Solids
The Muppet’s Guide to:
The Structure and Dynamics of Solids
Phase Diagrams

2. Phase Diagrams
A phase diagram is a graphical representation of the different phases present in a material.
Commonly presented as a function of composition and temperature or pressure and temperature
Applies to elements, molecules etc. and can also be used to show magnetic, and ferroelectric behaviour (field vs. temperature) as well as structural information.

3. Components and Phases • Components: • Phases: b (lighter phase) a
The elements or compounds which are present in the mixture
(e.g., Al and Cu)
• Phases:
The physically and chemically distinct material regions
that result (e.g., a and b).
Aluminum-
Copper
Alloy b
(lighter
phase) a
(darker
phase)
Figure adapted from Callister, Materials science and engineering, 7th Ed.

4. Crossing any line results in a structural phase transition
Unary Phase Diagrams
A pressure-temperature plot showing the different phases present in H2O.
Phase Boundaries
Crossing any line results in a structural phase transition
Upon crossing one of these boundaries the phase abruptly changes from one state to another. Latent heat not shown

5. Reading Unary Phase Diagrams
Melting Point (solid → liquid)
Triple Point
(solid + liquid + gas)
Boiling Point
(liquid→ gas)
Sublimation (solid → gas)
As the pressure falls, the boiling point reduces, but the melting/freezing point remains reasonably constant.

6. Reading Unary Phase Diagrams
P=1atm
Melting Point: 0°C
Boiling Point: 100°C
P=0.1atm
Melting Point: 2°C
Boiling Point: 68°C

7. Water Ice

8. Binary Phase Diagrams Phase B Phase A
Nickel atom
Copper atom
• 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?

9. Phase Equilibria: Solubility Limit
Solutions – solid solutions, single phase
Mixtures – more than one phase
• Solubility Limit:
Max concentration for
which only a single phase solution occurs.
Sucrose/Water Phase Diagram
Pure
Sugar
Temperature (°C) 20 40 60 80
100 Co
=Composition (wt% sugar) L
(liquid solution
i.e., syrup)
Solubility
Limit
(liquid) + S
(solid
sugar) 4 6 8 10
Water
Question: What is the
solubility limit at 20°C?
Answer: 65 wt% sugar.
If Co < 65 wt% sugar: syrup
If Co > 65 wt% sugar: syrup + sugar. 65

11. Salt-Water(ice)

12. Phase Diagrams • Phase Diagram for Cu-Ni at P=1 atm.
• Indicate phases as function of T, Co, and P.
• For this course:
-binary systems: just 2 components.
-independent variables: T and Co (P = 1 atm is almost always used).
• 2 phases: L
(liquid) a
(FCC solid solution)
• 3 phase fields:
L +
wt% Ni 20 40 60 80
100
1000
1100
1200
1300
1400
1500
1600
T(°C)
L (liquid)
(FCC solid
solution) +
liquidus
solidus
• Phase
Diagram
for Cu-Ni
at P=1 atm.
Figure adapted from Callister, Materials science and engineering, 7th Ed.

13. Phase Diagrams • Phase Diagram for Cu-Ni at P=1 atm.
• Indicate phases as function of T, Co, and P.
• For this course:
-binary systems: just 2 components.
-independent variables: T and Co (P = 1 atm is almost always used).
wt% Ni 20 40 60 80
100
1000
1100
1200
1300
1400
1500
1600
T(°C)
L (liquid) a
(FCC solid
solution) L +
liquidus
solidus
Liquidus:
Separates the liquid from the mixed L+a phase
• Phase
Diagram
for Cu-Ni
at P=1 atm.
Solidus:
Separates the mixed L+a phase from the solid solution
Figure adapted from Callister, Materials science and engineering, 7th Ed.

14. Number and types of phases
• Rule 1: If we know T and Co, then we know:
- the number and types of phases present.
wt% Ni 20 40 60 80
100
1000
1100
1200
1300
1400
1500
1600
T(°C)
L (liquid) a
(FCC solid
solution) L +
liquidus
solidus
Cu-Ni
phase
diagram
• Examples: B
(1250°C,35)
A(1100°C, 60):
1 phase: a B
(1250°C, 35):
2 phases: L + a
A(1100°C,60)
Figure adapted from Callister, Materials science and engineering, 7th Ed.

15. Composition of phases • Rule 2: If we know T and Co, then we know:
--the composition of each phase.
wt% Ni 20
1200
1300
T(°C)
L (liquid) a
(solid) L +
liquidus
solidus 30 40 50
Cu-Ni system
• Examples: T A C o
= 35 wt% Ni
tie line 35 C o
At T A
= 1320°C:
Only Liquid (L) C L
= C o
( = 35 wt% Ni) B T 32 C L 4 C a 3
At T D
= 1190°C: D T
Only Solid ( a ) C
= C o
( = 35 wt% Ni
At T B
= 1250°C:
Both a
and L C L
= C
liquidus
( = 32 wt% Ni here) C a
= C
solidus
( = 43 wt% Ni here)
Figure adapted from Callister, Materials science and engineering, 7th Ed.

16. Cooling a Cu-Ni Binary - Composition
• Phase diagram:
Cu-Ni system.
wt% Ni 20
120
130 3 4 5
110
L (liquid) a
(solid) L +
T(°C) A 35 C o
L: 35wt%Ni
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 100 wt% Ni.
a: 46 wt% Ni
L: 35 wt% Ni B 46 35 C 43 32 a :
43 wt% Ni
L: 32 wt% Ni D 24 36
L: 24 wt% Ni a :
36 wt% Ni E
• Consider
Co = 35 wt%Ni.
USE LEVER RULE
Figure adapted from Callister, Materials science and engineering, 7th Ed.

17. The Lever Rule – Weight %
Tie line – connects the phases in equilibrium with each other - essentially an isotherm
wt% Ni 20
1200
1300
T(°C)
L (liquid) a
(solid) L +
liquidus
solidus 3 4 5 B T
tie line C o S R
How much of each phase? Think of it as a lever ML M R S
Figure adapted from Callister, Materials science and engineering, 7th Ed.

18. Weight fractions of phases – ‘lever rule’
• Rule 3: If we know T and Co, then we know:
--the amount of each phase (given in wt%).
wt% Ni 20
1200
1300
T(°C)
L (liquid) a
(solid) L +
liquidus
solidus 3 4 5
Cu-Ni system T A 35 C o 32 B D
tie line R S
• Examples: C o
= 35 wt% Ni
At T A
: Only Liquid (L) W L
= 100 wt%, W a
= 0
At T D :
Only Solid ( a ) W L
= 0, W
= 100 wt%
= 27 wt%
At T B :
Both a
and L WL = S R + Wa
Figure adapted from Callister, Materials science and engineering, 7th Ed.

19. Cooling a Cu-Ni Binary – wt. %
• Phase diagram:
Cu-Ni system.
wt% Ni 20
120
130 3 4 5
110
L (liquid) a
(solid) L +
T(°C) A 35 C o
L: 35wt%Ni
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 100 wt% Ni.
a: 8 wt%
L: 92 wt% B 46 34 C 43 32 a :
27 wt%
L: 73 wt% D 24 36
L: 8 wt% a :
92 wt% E
• Consider
Co = 35 wt%Ni.
Figure adapted from Callister, Materials science and engineering, 7th Ed.

20. Equilibrium cooling Multiple freezing sites
Polycrystalline materials
Not the same as a single crystal
The compositions that freeze are a function of the temperature
At equilibrium, the ‘first to freeze’ composition must adjust on further cooling by solid state diffusion

21. Diffusion is not a flow
Concept behind mean free path in scattering phenomena - conductivity
Our models of diffusion are based on a random walk approach and not a net flow

22. Diffusion in 1 Dimension
Fick’s First Law
J = flux – amount of material per unit area per unit time
C = concentration
Diffusion coefficient which we expect is a function of the temperature, T

23. Diffusion cont…. BUT Requires the solution of the continuity equation:
The change in concentration as a function of time in a volume is balanced by how much material flows in per time unit minus how much flows out – the change in flux, J:
giving Fick’s second law (with D being constant):
BUT

24. Solution of Ficks’ Laws
For a semi-infinite sample the solution to Ficks’ Law gives an error function distribution whose width increases with time
t = 0 C Co C x
t = t

25. Consider slabs of Cu and Ni.
Interface region will be a mixed alloy (solid solution)
Interface region will grow as a function of time

26. Slow Cooling in a Cu-Ni Binary
wt% Ni 20
120
130 3 4 5
110
L (liquid) a
(solid) L +
T(°C) A 35 C o
L: 35wt%Ni
Cu-Ni
system
Co = 35 wt%Ni.
a: 46 wt% Ni
L: 35 wt% Ni
Enough time is allowed at each temperature change for atomic diffusion to occur. – Thermodynamic ground state B C a :
43 wt% Ni
L: 32 wt% Ni D
L: 24 wt% Ni a :
36 wt% Ni E
Each phase is homogeneous
Figure adapted from Callister, Materials science and engineering, 7th Ed.

27. Non – equilibrium coolingα L
α + L
Non – equilibrium cooling
No-longer in the thermodynamic ground state
Reduces the melting temperature
Figure adapted from Callister, Materials science and engineering, 7th Ed.

28. Cored vs Equilibrium Phases
• Ca changes as we solidify.
• Cu-Ni case:
First a to solidify has Ca = 46 wt% Ni.
Last a to solidify has Ca = 35 wt% Ni.
• Fast rate of cooling:
Cored structure
• Slow rate of cooling:
Equilibrium structure
First a
to solidify:
46 wt% Ni
Uniform C :
35 wt% Ni
Last
< 35 wt% Ni
Figure adapted from Callister, Materials science and engineering, 7th Ed.

29. Binary-Eutectic Systems – Cu/Ag
has a special composition
with a min. melting temperature
2 components
a phase:
Mostly copper
b phase:
Mostly Silver
Solvus line – the solubility limit
• Limited solubility – mixed phases
• 3 phases regions, L, a and b and 6 phase fields - L, a and b, L+a, L+b, a+b
Figure adapted from Callister, Materials science and engineering, 7th Ed.

30. Binary-Eutectic Systems
Cu-Ag system
L (liquid) a L + b Co
wt% Ag in Cu/Ag alloy 20 40 60 80
100
200
1200
T(°C)
400
600
800
1000 CE TE
CaE=8.0
CE=71.9
CbE=91.2
779°C
The Eutectic point
TE, Eutectic temperature, 779°C
CE, eutectic composition, 71.9wt.% E
• TE
: No liquid below TE
Min. melting TE
• Eutectic transition
L(CE) (CE) + (CE)
Any other composition, Liquid transforms to a mixed L+solid phase
Figure adapted from Callister, Materials science and engineering, 7th Ed.

31. Pb-Sn (Solder) Eutectic System (1)
• For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find...
--the phases present:
a + b
Pb-Sn
system
--compositions of phases: L + a b
200
T(°C)
18.3
C, wt% Sn 20 60 80
100
300
L (liquid)
183°C
61.9
97.8
CO = 40 wt% Sn
Ca = 11 wt% Sn
Cb = 99 wt% Sn
--the relative amount of each phase: W a =
C - CO
C - C 59 88
= 67 wt% S
R+S
150 R 11 C 40 Co S 99 C W  =
CO - C
C - C R
R+S 29 88
= 33 wt%
Figure adapted from Callister, Materials science and engineering, 7th Ed.

32. Microstructures in Eutectic Systems: II
L: Co wt% Sn
• 2 wt% Sn < Co < 18.3 wt% Sn
• Result:
Initially liquid → liquid + 
then  alone
finally two phases
a poly-crystal
fine -phase inclusions
Pb-Sn
system L + a
200
T(°C) Co ,
wt% Sn 10
18.3 20
300
100 30 b
400
(sol. limit at TE) TE 2
(sol. limit at T
room ) L a
a: Co wt% Sn a b
Figure adapted from Callister, Materials science and engineering, 7th Ed.

33. Microstructures in Eutectic Systems: Co=CE
• Result: Eutectic microstructure (lamellar structure)
--alternating layers (lamellae) of a and b crystals.
Pb-Sn
system
L
a
200
T(°C)
C, wt% Sn 20 60 80
100
300 L a b +
183°C 40 TE
160 m
Micrograph of Pb-Sn
eutectic
microstructure
L: Co wt% Sn CE
61.9
18.3
: 18.3 wt%Sn
97.8
: 97.8 wt% Sn
Figures adapted from Callister, Materials science and engineering, 7th Ed.

34. Microstructures in Eutectic Systems: Co=CE
• Result: Eutectic microstructure (lamellar structure)
--alternating layers (lamellae) of a and b crystals.
Pb-Sn
system
L
a
200
T(°C)
C, wt% Sn 20 60 80
100
300 L a b +
183°C 40 TE
L: Co wt% Sn CE
61.9
18.3
: 18.3 wt%Sn
97.8
: 97.8 wt% Sn
Pb rich
Sn Rich
Figures adapted from Callister, Materials science and engineering, 7th Ed.

35. Lamellar Eutectic Structure
At interface, Pb moves to a-phase and Sn migrates to b- phase
Lamellar form to minimise diffusion distance – expect spatial extent to depend on D and cooling rates. Sn Pb
Figure adapted from Callister, Materials science and engineering, 7th Ed.

36. Microstructures IV • 18.3 wt% Sn < Co < 61.9 wt% Sn
• Result: a crystals and a eutectic microstructure WL
= (1- W a )
= 50 wt% C
= 18.3 wt% Sn CL
= 61.9 wt% Sn S R + =
• Just above TE :
T(°C) L a
L: Co
wt% Sn L a
300 L
Pb-Sn
system L + a a
200
18.3
61.9 S R L + b b TE
100 a + b 20 40 60 80
100
Co, wt% Sn
Figure adapted from Callister, Materials science and engineering, 7th Ed.

37. Microstructures IV • 18.3 wt% Sn < Co < 61.9 wt% Sn
• Result: a crystals and a eutectic microstructure
• Just below TE : C a
= 18.3 wt% Sn b
= 97.8 wt% Sn S R + W =
= 73 wt%
= 27 wt%
T(°C) L a
L: Co
wt% Sn L a
300 L
Pb-Sn
system L + a a b
200
18.3
61.9 S R L + b TE
97.8 S R
100 a + b
Primary, a 20 40 60 80
100
Eutectic, a
Eutectic, b
Co, wt% Sn
Figure adapted from Callister, Materials science and engineering, 7th Ed.

38. Intermetallic Compounds
a phase:
Mostly Mg
Mg2Pb
b phase:
Mostly Lead
Note: intermetallic compound forms a line - not an area - because stoichiometry (i.e. composition) is exact.
Figure adapted from Callister, Materials science and engineering, 7th Ed.

39. Eutectoid & Peritectic
Peritectic transition  + L 
Cu-Zn Phase diagram
mixed liquid and solid to single solid transition
Eutectoid transition   + 
Solid to solid ‘eutectic’ type transition
Figure adapted from Callister, Materials science and engineering, 7th Ed.

40. Iron-Carbon (Fe-C) Phase Diagram
• 2 important
T(°C)
points
1600 d
-Eutectic (A): L g +
Fe3C L
1400 g +L g A S R
4.30
1200
L+Fe3C
-Eutectoid (B): g a +
Fe3C
1148°C
(austenite)
1000 g g
+Fe3C
120 mm a
0.76 C
eutectoid B +
Fe3C (cementite)
800 a g
727°C = T
eutectoid R S
Fe3C (cementite-hard) a
(ferrite-soft)
600 a
+Fe3C
400 1 2 3 4 5 6
6.7
(Fe)
Co, wt% C
Result: Pearlite =
alternating layers of a
and Fe3C phases
Figure adapted from Callister, Materials science and engineering, 7th Ed.

41. Iron-Carbon