The Muppet’s Guide to:
The Structure and Dynamics of Solids
7. Phase Diagrams
Material Properties
Dislocations & plastic deformation
• Cubic & hexagonal metals - plastic deformation by
plastic shear or slip where one plane of atoms slides
over adjacent plane by defect motion (dislocations).
• If dislocations don't move, deformation doesn't occur!
Adapted from Fig. 7.1, Callister 7e.
Edge Defect Motion
Dislocation Motion
• Dislocation moves along slip plane in slip direction
perpendicular to dislocation line
• Slip direction same direction as Burgers vector
Edge dislocation
Adapted from Fig. 7.2,
Callister 7e.
Screw dislocation
(Callister: Materials Science and Engineering)
Dislocations & Materials Classes
• Metals: Disl. motion easier.
-non-directional bonding
-close-packed directions
for slip.
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+
electron cloud
+
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ion cores
• Covalent Ceramics
(Si, diamond): Motion hard.
-directional (angular) bonding
• Ionic Ceramics (NaCl):
Motion hard.
-need to avoid ++ and - neighbours.
(Callister: Materials Science and Engineering)
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Pinning dislocations
• dislocations make metals
easier to deform
• to improve strength of metals,
need to stop dislocation motion
trap with:
- impurity atoms;
- other dislocations (work
hardening;
- grain boundaries.
(Callister: Materials Science and Engineering)
atom
trap
Modify Material Properties
Increase material strength through substitution
• Impurity atoms distort the lattice & generates stress.
• Stress can produce a barrier to dislocation motion.
• Smaller substitutional
impurity
• Larger substitutional
impurity
A
C
B
Impurity generates local stress at A
and B that opposes dislocation
motion to the right.
(Callister: Materials Science and Engineering)
D
Impurity generates local stress at C
and D that opposes dislocation
motion to the right.
Modify Material Properties
Increase material strength through reducing Grain size
• Grain boundaries are
barriers to slip.
• Barrier "strength"
increases with
Increasing angle of
miss-orientation.
• Smaller grain size:
more barriers to slip.
(Callister: Materials Science and Engineering)
Solid Solutions
Solid state mixture of one or more solutes in a solvent
Crystal structure remains unchanged on addition of
the solute to the solvent
Mixture remains in a homogenous phase
Generally composed on metals close in the periodic table
Ni/Cu, Pb/Sn etc.
Otherwise compounds tend to form
NaCl, Fe2O3 etc.
Solid Solutions
Solute atoms create strain fields which can inhibit dislocations
propagating in a material changing its properties
Hume-Rothery Rules –
Substitutional Solutions
Rules to describe how an element might dissolve in a
metal. Stable composition in equilibrium (thermodynamics)
1.
The solute and solvent
should be of a similar size.
(<15% difference)
2.
The crystal structures must
match.
3.
Both solute and solvent
should have similar
electronegativity
4.
The valence of the solvent
and solute metals should
be similar.
Metals – Ni/Cu, Pd/Sn, Ag/Au,
Mo/W
Phase Equilibria – Example
K-Na
K
Na
Crystal electroneg r (nm)
Structure
BCC
0.93
0.235
BCC
1.00
0.191
• Both have the same crystal structure (BCC) and have
similar electronegativities but different atomic radii.
• Rules suggest that NO solid solution will form.
• K and Na sodium are not miscible.
Phase Equilibria – Example
Simple solution system (e.g., Ni-Cu solution)
Ni
Cu
Crystal electroneg r (nm)
Structure
FCC
1.9
0.1246
FCC
1.8
0.1278
• Both have the same crystal structure (FCC) and have
similar electronegativities and atomic radii (W. Hume –
Rothery rules) suggesting high mutual solubility.
• Ni and Cu are totally miscible in all proportions.
Hume-Rothery Rules –
Interstitial Solution
Rules to describe how an element might dissolve in a
metal. Stable composition in equilibrium (thermodynamics)
1.
The solute must be smaller
than the interstitial sites in
the solvent lattice
2.
Solute and Solvent should
have similar electronegativities
Light elements – H,C, N and O.
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.
Components and Phases
• Components:
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).
AluminumCopper
Alloy
b (lighter
phase)
a (darker
phase)
Figure adapted from Callister, Materials science and engineering, 7th Ed.
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
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.
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
Water Ice
http://images.jupiterimages.com/common/detail/13/41/23044113.jpg,
http://www.homepages.ucl.ac.uk/~ucfbanf/ice_phase_diagram.jpg
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?
Phase Equilibria: Solubility Limit
– Solutions – solid solutions, single phase
– Mixtures – more than one phase
• Solubility Limit:
Answer: 65 wt% sugar.
If Co < 65 wt% sugar: syrup
If Co > 65 wt% sugar: syrup + sugar.
L
(liquid)
60
L
40
(liquid solution
i.e., syrup)
20
0
+
S
(solid
sugar)
20
40
6065 80
100
Co =Composition (wt% sugar)
Pure
Sugar
solubility limit at 20°C?
Solubility
Limit
80
Pure
Water
Question: What is the
100
Temperature (°C)
Max concentration for
which only a single phase
solution occurs.
Sucrose/Water Phase Diagram
Salt-Water(ice)
http://webserver.dmt.upm.es/~isidoro/bk3/c07sol/Solution%20properties_archivos/image001.gif
Ferroelectric Materials
Na½Ba½TiO3
T. Takenaka, K. Maruyama, and K. Sakata, Jpn. J. Appl. Phys. 30, 2236 (1991)
B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics (Academic Press, London, 1971).
J. Rodel, W. Jo, K. T. P. Seifert, E. M. Anton, T. Granzow, and D. Damjanovic, J. Am. Ceram. Soc. 92, 1153 (2009).
Phase Diagrams
• 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).
T(°C)
• 2 phases:
1600
• Phase
Diagram
for Cu-Ni
at P=1 atm.
L (liquid)
a (FCC solid solution)
L (liquid)
1500
1400
1300
a
(FCC solid
solution)
1200
1100
1000
0
20
40
60
Figure adapted from Callister, Materials science and engineering, 7th Ed.
80
• 3 phase fields:
L
L+a
a
100
wt% Ni
Phase Diagrams
• 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).
T(°C)
1600
• Phase
Diagram
for Cu-Ni
at P=1 atm.
Liquidus:
Separates the liquid
from the mixed L+a
phase
L (liquid)
1500
1400
Solidus:
Separates the
mixed L+a phase
from the solid
solution
1300
a
(FCC solid
solution)
1200
1100
1000
0
20
40
60
Figure adapted from Callister, Materials science and engineering, 7th Ed.
80
100
wt% Ni
Number and types of phases
• Rule 1: If we know T and Co, then we know:
- the number and types of phases present.
• Examples:
T(°C)
1600
A(1100°C, 60):
1 phase: a
B (1250°C,35)
L (liquid)
1500
1400
B(1250°C, 35):
2 phases: L + a
1300
a
(FCC solid
solution)
1200
A(1100°C,60)
1100
1000
Cu-Ni
phase
diagram
0
20
Figure adapted from Callister, Materials science and engineering, 7th Ed.
40
60
80
100
wt% Ni
Composition of phases
• Rule 2: If we know T and Co, then we know:
--the composition of each phase.
• Examples:
T(°C)
Cu-Ni
system
A
TA
Co = 35 wt% Ni
1300 L (liquid)
At T A = 1320°C:
Only Liquid (L)
B
TB
CL = Co ( = 35 wt% Ni)
At T D = 1190°C:
1200
D
Only Solid ( a)
TD
Ca = Co ( = 35 wt% Ni)
20
3032 35
At T B = 1250°C:
CLCo
Both a and L
CL = C liquidus ( = 32 wt% Ni here)
Ca = C solidus ( = 43 wt% Ni here)
Figure adapted from Callister, Materials science and engineering, 7th Ed.
tie line
a
(solid)
4043
50
Ca wt% Ni
Cooling a Cu-Ni Binary - Composition
• Phase diagram:
Cu-Ni system.
• System is:
--binary
i.e., 2 components:
Cu and Ni.
T(°C) L (liquid)
130 0
L: 35 wt% Ni
a: 46 wt% Ni
• Consider
Co = 35 wt%Ni.
Cu-Ni
system
A
35
32
--isomorphous
i.e., complete
solubility of one
component in
another; a phase
field extends from
0 to 100 wt% Ni.
L: 35wt%Ni
B
C
46
43
D
24
L: 32 wt% Ni
36
120 0
a: 43 wt% Ni
E
L: 24 wt% Ni
a: 36 wt% Ni
a
(solid)
110 0
20
Figure adapted from Callister, Materials science and engineering, 7th Ed.
30
35
Co
40
50
wt% Ni
USE LEVER RULE
The Lever Rule – Weight %
• Tie line – connects the phases in equilibrium
with each other - essentially an isotherm
T(°C)
How much of each phase?
Think of it as a lever
tie line
1300
L (liquid)
B
TB
a
(solid)
1200
R
20
30C
S
40 C
a
L Co
R
50
wt% Ni
WL 
Ma
ML
C  C0
ML
S

 a
ML  M a R  S Ca  CL
Figure adapted from Callister, Materials science and engineering, 7th Ed.
S
M a S  M L R
Wa 
C  CL
R
 0
R  S Ca  CL
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%).
• Examples:
Cu-Ni
system
T(°C)
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 a = 100 wt%
At T B : Both a and L
43  35
 73 wt %
43  32
WL 
S
R +S

Wa 
R
R +S
= 27 wt%
Figure adapted from Callister, Materials science and engineering, 7th Ed.
A
TA
1300
TB
1200
TD
20
tie line
L (liquid)
B
S
R
D
3032 35
C LC o
a
(solid)
40 4 3
50
C a wt% Ni
Cooling a Cu-Ni Binary – wt. %
• Phase diagram:
Cu-Ni system.
• System is:
--binary
i.e., 2 components:
Cu and Ni.
T(°C) L (liquid)
130 0
L: 92 wt%
a: 8 wt%
i.e., complete
solubility of one
component in
another; a phase
field extends from
0 to 100 wt% Ni.
• Consider
Co = 35 wt%Ni.
Cu-Ni
system
A
32
--isomorphous
L: 35wt%Ni
34
B
C
46
43
D
24
L: 73 wt%
36
120 0
a: 27 wt%
E
L: 8 wt%
a: 92 wt%
a
(solid)
110 0
20
Figure adapted from Callister, Materials science and engineering, 7th Ed.
30
35
Co
40
50
wt% Ni
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
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
http://mathworld.wolfram.com/images/eps-gif/RandomWalk2D_1200.gif