The Muppet’s Guide to:
The Structure and Dynamics of Solids
Phase Diagrams
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
Consider slabs of Cu and Ni.
Interface region will be a
mixed alloy (solid solution)
Interface region will grow as a
function of time
Slow Cooling in a Cu-Ni Binary
T(°C) L (liquid)
L: 35wt%Ni
Co = 35 wt%Ni.
Enough time is
allowed at each
temperature
change for atomic
diffusion to occur.
–
Thermodynamic
ground state
Each phase is
homogeneous
130 0
L: 35 wt% Ni
a: 46 wt% Ni
Cu-Ni
system
A
B
C
D
120 0
L: 32 wt% Ni
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
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.
α+L
α
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
Last a to solidify:
< 35 wt% Ni
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Uniform C a:
35 wt% Ni
Binary-Eutectic Systems – Cu/Ag
2 components
has a special composition
with a min. melting temperature
a
phase:
b
phase:
Mostly
copper
Solvus line
– the
solubility
limit
Mostly
Silver
• 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.
Binary-Eutectic Systems
Cu-Ag system
T(°C)
The Eutectic point
1200
L (liquid)
TE, Eutectic temperature, 779°C
1000
CE, eutectic composition, 71.9wt.%
a
TE
• TE
: No liquid below TE
Min. melting TE
800
E
779°C
CaE=8.0
600
CE=71.9
L +b b
CbE=91.2
a+b
400
200
• Eutectic transition
L(CE)
L+a
0
a(CaE) + b(CbE)
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Co
20
40
60 CE 80
wt% Ag in Cu/Ag alloy
100
Any other composition, Liquid
transforms to a mixed L+solid phase
Pb-Sn (Solder) Eutectic System (1)
• For a 40 wt% Sn-60 wt% Pb alloy at 150°C, find...
--the phases present: a + b
--compositions of phases:
CO = 40 wt% Sn
Ca = 11 wt% Sn
Cb = 99 wt% Sn
--the relative amount
of each phase:
Wa =
S
R+S
C -C
= b O
Cb - Ca
99 - 40
59
=
= 67 wt%
99 - 11
88
CO - Ca
Wb = R
=
Cb - Ca
R+S
40 - 11
29
=
= 33 wt%
=
99 - 11
88
Pb-Sn
system
T(°C)
300
200
L (liquid)
a
L+a
18.3
150
L +b b
183°C
61.9
R
97.8
S
100
a+ b
=
0 11 20
Ca
40
Co
60
80
C, wt% Sn
99100
Cb
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Microstructures in Eutectic
Systems: II
L: Co wt% Sn
T(°C)
• 2 wt% Sn < Co < 18.3 wt% Sn
• Result:
400
 Initially liquid → liquid + a
 then a alone
 finally two phases
 a poly-crystal
 fine b-phase inclusions
L
300
L +a
a
200
TE
a: Co wt% Sn
a
b
100
a+ b
0
10
20
Co
Co ,
2
(sol. limit at T room )
18.3
(sol. limit at TE)
Figure adapted from Callister, Materials science and engineering, 7th Ed.
L
a
Pb-Sn
system
30
wt% Sn
Microstructures
in Eutectic Systems: Co=CE
• Result: Eutectic microstructure (lamellar structure)
--alternating layers (lamellae) of a and b crystals.
T(°C)
Micrograph of Pb-Sn
eutectic
microstructure
L: Co wt% Sn
300
Pb-Sn
system
a
200
L+a
L
TE
100
a+b
0
L+b b
183°C
20
18.3
40
b: 97.8 wt% Sn
a: 18.3 wt%Sn
60
CE
61.9
80
100
97.8
C, wt% Sn
Figures adapted from Callister, Materials science and engineering, 7th Ed.
160m
Microstructures
in Eutectic Systems: Co=CE
• Result: Eutectic microstructure (lamellar structure)
--alternating layers (lamellae) of a and b crystals.
T(°C)
L: Co wt% Sn
300
Pb-Sn
system
a
200
L+a
Wa 
L+b b
183°C
TE
Pb
rich
L
Wb 
100
a+b
0
20
18.3
40
b: 97.8 wt% Sn
a: 18.3 wt%Sn
60
CE
61.9
80
100
97.8
C, wt% Sn
Figures adapted from Callister, Materials science and engineering, 7th Ed.
97.8  61.9
 45.2%
97.8  18.3
61.9  18.3
 54.8%
97.8  18.3
Sn Rich
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.
Figure adapted from Callister, Materials science and engineering, 7th Ed.
Sn
Pb
Microstructures IV
• 18.3 wt% Sn < Co < 61.9 wt% Sn
• Result: a crystals and a eutectic microstructure
L: Co wt% Sn
T(°C)
300
Pb-Sn
system
a
200
L+a
L
a
L
R
a L
L+b b
S
TE
a+ b
100
0
20
18.3
40
60
61.9
80
Co, wt% Sn
Figure adapted from Callister, Materials science and engineering, 7th Ed.
100
• Just above TE :
C a = 18.3 wt% Sn
CL = 61.9 wt% Sn
Wa= S
= 50 wt%
R +S
WL = (1- W a) = 50 wt%
Microstructures IV
• 18.3 wt% Sn < Co < 61.9 wt% Sn
• Result: a crystals and a eutectic microstructure
L: Co wt% Sn
T(°C)
300
Pb-Sn
system
a
200
L+a
TE
R
100
a+ b
L
a
L
R
a L
L+b b
S
• Just below TE :
C a = 18.3 wt% Sn
C b = 97.8 wt% Sn
Wa = S
= 73 wt%
R +S
W b = 27 wt%
S
Primary, a
0
20
18.3
40
60
61.9
80
Co, wt% Sn
Figure adapted from Callister, Materials science and engineering, 7th Ed.
100
97.8
Eutectic, a
Eutectic, b
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.
Eutectoid & Peritectic
Peritectic transition  + L
Cu-Zn Phase diagram
Eutectoid transition 
+
Figure adapted from Callister, Materials science and engineering, 7th Ed.

mixed liquid and solid to
single solid transition
Solid to solid ‘eutectic’ type
transition
Iron-Carbon (Fe-C) Phase Diagram
L
 + Fe3C
-Eutectoid (B):

a + Fe3C
T(°C)
1600

L
1400
1200
 +L

(austenite)
 
 
1000
800
a
600
Result: Pearlite =
alternating layers of
a and Fe3C phases
S
 +Fe3C
727°C = T eutectoid
R
S
1
0.76
C eutectoid
120 m
0
(Fe)
Figure adapted from Callister, Materials science and engineering, 7th Ed.
L+Fe3C
R
B
400
A
1148°C
Fe3C (cementite)
• 2 important
points
-Eutectic (A):
2
3
a+Fe3C
4
5
6
6.7
4.30
Co, wt% C
Fe3C (cementite-hard)
a (ferrite-soft)
Iron-Carbon
http://www.azom.com/work/pAkmxBcSVBfns037Q0LN_files/image003.gif
Characterisation
Over the course so far we have seen how thermodynamics plays an
important role in defining the basic minimum energy structure of a
solid.
Small changes in the structure (such as the perovskites) can produce
changes in the physical properties of materials
Kinetics and diffusion also play a role and give rise to different metastable structures of the same materials – allotropes / polymorphs
Alloys and mixtures undergo multiple phase changes as a function of
temperature and composition
BUT how do we
characterise samples?
Probes
Resolution better than the inter-atomic spacings
• Electromagnetic Radiation
• Neutrons
• Electrons
Probes
Treat all probes as if they were waves:
Wave-number, k:
Momentum, p:
k  k  2
p  k;
Photons

p  mv  h
‘Massive’ objects

Xavier the X-ray
Speed of Light
E  hc
Planck’s constant

Wavelength
Ex(keV)=1.2398/(nm)
Elastic scattering as Ex>>kBT
X-ray Sources
Norbert the Neutron
De Broglie equation:
  h mv
mass
velocity
2
2
1
h
Kinetic Energy: En  2 mv 
2mn  2
En(meV)=0.8178/2(nm)
Strong inelastic scattering as En~kBT
Fission
ILL: Flux Density of
1.5x1013 neutrons/s/mm2
at thermal power of 62MW
Thermal Neutron
235U
2.5 neutrons + heavy elements + 200MeV heat
Spallation
ISIS - Rutherford
Appleton Lab.
(Oxford)
Pulsed Source - 50Hz
800 MeV Protons chop pieces off heavy nucleus
Protons, muons, pions .... & 25 neutrons
Eric the Electron
• Eric’s rest mass: 9.11 × 10−31 kg.
•
electric charge: −1.602 × 10−19 C
• No substructure – point particle
De Broglie equation:
  h mv
mass
velocity
Ee depends on accelerating voltage :–
Range of Energies from 0 to MeV
Probes
Resolution better than the interatomic spacings
Absorption low – we want a ‘bulk’ probe
• Electrons - Eric
• quite surface sensitive
• Electromagnetic Radiation - Xavier
• Optical – spectroscopy
• X-rays :
• VUV and soft (spectroscopic and surfaces)
• Hard (bulk like)
• Neutrons - Norbert