LECTURE 20
Last Time
Landau theory of phase transition
First and second order phase transitions
Metastable States
Order parameter ξ: a single parameter that can describe the system
Landau free energy function:
First order phase transition
FL (ξ ,τ ) ≡ U (ξ ,τ ) − τσ (ξ ,τ )
Second order phase transition
F
F
ξ
ξ
Abrupt change
Smooth change
Today
Chapter 11 (part 1)
Binary mixtures
Binary mixtures
Alloys
Homogeneous and heterogeneous mixture
Energy and entropy of mixing
LECTURE 20
Binary mixtures
Binary: two things.
(Can have more: ternary mixtures, quaternary mixtures…)
Alloy: the two things are two different atoms.
Why an alloy?
Au-Si:
Au solidifies at 1063o C; Si at 1404oC
Au-Si (69/31) alloy solidifies at 370oC
The alloy is easier to mold.
Sometimes the free energy wins when things don’t mix, like oil and water.
Homogeneous mixture: single phase (as in solution)
Heterogeneous mixture: two distinct phases
Au-Si example
Au solidifies at 1063o C; Si at 1404oC
Au-Si (69/31) alloy solidifies at 370oC
Heterogeneous
370C
F
1063C
1404C
Any system at certain T evolves to
configuration with minimum free
energy
At T>370oC free energy of liquid
mix is lower than free energies of
solid Si and solid Au
T
Some Alloys
80/20 Al/Si
95/05 Zn/Mg
75/25 W/Co
80/20 Cu/Ag
http://www.ses.swin.edu.au/homes/hans/met3.htm
Images by H.G.Brinkies
Famous Alloys
Bronze: Cu + Sn (copper and tin)
Indonesian jewelry
from the bronze age
http://www.artareas.com/ArtAreas/home.nsf/
Brass: Cu + Zn. (67/33)
(homogeneous mixtures)
Solubility
A
A
A
A
B
B
A
B
A
A
A
B
A
A
A
A
B
B
A
A
B
B
A
A
A
A
A
A
B
B
A
B
A
A
B
A
A
A
A
A
B
B
B
A
A
A
A
A
A
A
A
A
A
B
A
A
B
A
B
A
Which structure will realize?
F ≡ U − τσ
Free energy of phase separation
A1-xBx
Number of atoms
N = N A + NB
Composition:
x = NB N
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
Assume pV = 0
B
A
A
A
A
A
Free energy per atom:
A
A
B
A
B
A
1− x = NA N
f ≡F N
f of smooth mixing
Sometimes a system has a
solubility gap: it costs
energy to mix it smoothly.
mixes
mixes
Free energy of phase separation
phase
f of smooth mixing
phase
Think:
80/20 Al-Si
20/80 Al-Si
N = N A + NB
f ≡F N
x = NB N
1− x = NA N
mixes
doesn’t mix
mixes
x
F = Nα f ( xα ) + N β f ( xβ )
Free energy of phase α+ free energy of phase β
N = Nα + N β
N B = xα Nα + xβ N β
Free energy of phase separation
phase
Think:
f of smooth mixing
phase
80/20 Al-Si
20/80 Al-Si
F = Nα f ( xα ) + N β f ( xβ )
mixes
doesn’t mix
x
f =
F
1
( xβ − x ) f ( xα ) + ( x − xα ) f ( xβ ) 
=

N xβ − xα 
mixes
Free energy of phase separation
f =
F
1
( xβ − x ) f ( xα ) + ( x − xα ) f ( xβ ) 
=

N xβ − xα 
f = f ( xα ) +
f ( xβ ) − f ( xα )
xβ − xα
f
( x − xα )
Will the system separate into two phases?
x
If f vs. concentration x is convex,
system won’t phase separate.
concentration “x”
Solubility gap
f =
F
1
( xβ − x ) f ( xα ) + ( x − xα ) f ( xβ ) 
=

N xβ − xα 
f of smooth mixing
Will the system separate
into two phases?
If f vs. concentration x is concave,
system will phase separate.
gap
System falls
apart in this region.
mixes
doesn’t mix
mixes
Free Energy of Mixing
Helmholtz free energy
Free energy per atom
F ≡ U − τσ
F
f ≡ = u −τ σ N
N
Two competing terms: energy and entropy
Energy of Mixing
Average energy per atom of the phase separated state, where
a composition A1-xBx breaks up into pure A and pure B:
Notice usep is a line.
If this is less than the energy per atom of the homogeneous
state, it helps phase separation:
u
umixed
A1-xBx
uA
uB
usep
0
x
1
The difference in energy
between the uniform (mixed) state
and the phase separated state
is the energy of mixing, umix.
umixed = usep + umix
re
ctu
e
L
05
Entropy of Mixing
ALLOY
AABBABAAA
BBAABABBB
ABBBABBAA
BBAABAAAB
Stirling!
Concentration of B
Entropy is always positive
ctu
Le
0
re
5
Entropy of Mixing
ALLOY
AABBABAAA
BBAABABBB
ABBBABBAA
BBAABAAAB
N=1000
x
Entropy of Mixing
What is a slope at x=0 and x=1?
Slope is infinite
Free Energy per Atom
f = u −τ σ N
u
umixed
A1-xBx
usep
A1-xBx
0
1
x
0
umixed = usep + umix
Note: temperature dependence!
x
1
x
Free Energy per Atom
u
umixed
A1-xBx
usep
A1-xBx
0
0
1
x
1
x
f
fo
Always mixes
at high temperature
due to entropy
of mixing
low T
high T
0
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
B
A
B
x
1
Alloy on a Lattice + Interactions
A
A
A
B
B
A
A
B
All attractive.
Relative to “zero energy”
for two atoms
separated by infinity.
What is the average energy per bond?
For an A atom:
Average energy is
?
?
A
per bond at A
?
?
For a B atom:
Average energy is
?
?
B
?
?
per bond at B
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
Alloy on a Lattice + Interactions
A
A
B
B
A
B
All attractive.
Relative to “zero energy”
for two atoms
separated by infinity.
What is the average energy per bond?
Energy per bond around each type of atom.
Average energy per bond around the average atom:
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
B
A
B
Alloy on a Lattice + Interactions
A
A
A
B
B
A
A
B
What is the average energy per atom?
There are two bonds for every site in 2D case.
One atom per site.
All attractive.
Relative to “zero energy”
for two atoms
separated by infinity.
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
“Apart”
Alloy on a Lattice + Interactions
What is the mixing energy umix?
A
A
A
A
A
B
A
A
A
A
B
B
A
A
A
A
B
B
A
A
A
A
B
B
A
A
A
A
B
B
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
What about the
energy of the interface?
Alloy on a Lattice + Interactions
What is the mixing energy umix?
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
Alloy on a Lattice + Interactions
What is the mixing energy umix?
Mixing
Energy
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
Alloy on a Lattice + Interactions
Mixing Energy
Positive Mixing Energy leads to phase separation.
(Entropy always favors mixing.)
u
Mixing costs internal energy if
Mixing gains internal energy if
x
Graph shows “positive mixing energy”
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
Alloy on a Lattice + Interactions
F ≡ U − τσ
Mixing Energy
Mixing Entropy
u
Which wins?
x
In this case, it costs energy
to mix the two.
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
But it always gains entropy
to mix the two.
Alloy on a Lattice + Interactions
Phase separation happens if
mixes
A
x
mixes
doesn’t mix
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
Alloy on a Lattice + Interactions
Phase separation happens if
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
Alloy on a Lattice + Interactions
Phase separation happens if
Watch this term
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
A
A
B
A
B
A
Alloy on a Lattice + Interactions
Phase separation happens if
x
Alloy on a Lattice + Interactions
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
Phase separation happens if
B
A
A
A
A
A
By assumption:
A
A
B
A
B
A
Mixing threshold.
For
, the system always mixes. (Mixing entropy wins)
For
, the system phase separates for some range of x.
Alloy on a Lattice + Interactions
A
B
A
A
A
B
A
A
B
B
A
A
A
B
A
A
B
A
B
A
A
A
A
A
Notice that
A
A
B
A
B
A
when
A
B
Threshold.
A
A
B
B
If mixed bonds are on average weaker than unmixed bonds,
phase separation happens below a certain threshold temperature.
Why oil and water phase separate:
Water-water bonds are strong. -- Hydrogen bonding everywhere.
Oil-oil and oil-water bonds are approximately the same strength,
and much weaker than water-water bonds.
H
H
H
O
Hydrogen bonding of
water molecules
O
H
Positive Mixing Energy
(costs energy to mix)
leads to phase separation.
(Entropy always favors mixing.)
Oil and Water
Mixing threshold:
∂2 f
>0
∂x 2
−4 ( 2u AB − u AA − u BB ) + τ
1
=0
x (1 − x )
mixed
T
Pure
Oil
separated
Pure
Water
τ > 4 ( 2u AB − u AA − u BB ) x (1 − x )
0
Phase diagram
Phase separates
into two parts.
One is mostly water,
and the other is mostly oil.
(Entropy of mixing does
cause some mixing!)
100
x = % Water
mostly oil
mostly water
Today
Entropy of Mixing
Energy of Mixing
Free Energy of Mixing
When system mix (good metallic alloys)
and when they don’t (oil and water)