Basics of Phases and Phase
Transformations
W. Püschl
University of Vienna
Content
1. Historical context
2. Classification of phase transformations
3. Graphical thermodynamics – Phase diagrams
4. Miscibility Gap – Precipitation
nucleation vs. spinodal decomposition
5. Order
6. Ising model: atomic and magnetic spin configuration
7. Martensitic transformations
Early technological application of poly-phase systems:
Damascus Steel
Aloys v. Widmannstätten
1808
Iron meteorite cut, polished,
and etched:
Intricate pattern appears
Oldest age hardening curve: Wilms Al-Cu(Mg,Mn,Fe, Si) alloy
Retarded precipitation of a disperse phase.
A scientific understanding of phases and phase transformation
begins to develop end 19th / beginning 20th centuries
physical metallurgy
Experimental:
Gustav Tammann (Göttingen)
Theoretical:
Josiah Willard Gibbs
What is a phase?
Region where intrinsic parameters have (more or less) the same value
lattice structure, composition x, degree of order , density ,…
Need not be simply (singly) connected.
Expreme example: disperse phase
and matrix phase where it is embedded (like Swiss cheese)
When is a phase thermodynamically stable?
How can we determine wihich phase is stable at a certain composition,
temperature (and pressure, magnetic field…)
What happens if this is not the case  metastability or phase transition
How can a phase transition take place?
Ehrenfest (1933)
a)
g
b)
g
Phase 1
Phase 1
Phase 2
Phase 2
Phase 2
Phase 2
Phase 1
Tu
T
1stAbb.
oder1-3
phase transition
Tu
T
2nd order (generally: higher order)
Free energy vs. order parameter according to Landau
Higher-order phase transition
1st oder phase transition
Chemical potentials gi of the components
Gibbs phase rule
f =  (n - 1) – n ( - 1) + 2 = n -  + 2
Liquid-solid transition
of a two-component
System (Ge-Si)
Excess enthalpy
and miscibility gap
Excess enthalpy
and miscibility gap
Precipitation:
alternative mechanisms
Heterophase fluctuation
corresponds to nucleation
Homophase fluctuation
corresponds to
spinodal
decomposition
Free energy of a spherical precipitate particle
Precipitation by nucleation and growth:
NV particle number, c supersaturation,
R mean particle radius
Ni36Cu9Al55
Spinodal Decomposition
Excess enthalpy
Positive: like atoms preferred: Phase separation
Negative: unlike atoms preferred: ordering
Short range order: there is (local) pair correlation

ij
nm
 1
pni pmj
pni pmj
 1
ij
Pnm
Cowley- Warren SRO parameter
pmj
Decay with distance from
reference atom
If they do not decay
 long range order
Long range order out of the fcc structure:
L12
Long range order out of the fcc structure:
L12
ordered state L12
disordered state (fcc)
fcc  L10
stoichiometry 1:1
Different long range ordered structures
in the Cu-Au phase diagram
L12
L10
L12
L12
L10
Different long range ordered structures
in the Cu-Au phase diagram
L12
L10
CuAu II (long period.)
L12
L12
L10
Statistical physics of ordered alloys
Partition function
 W ( R) 
Z ( R)   ZVr ( R) exp  r

kT


r
  0r  3 N
 
Z  exp 
 kT  j 1
r
V
1
 h rj
1  exp 
 kT





Possibly different vibration spectrum for every atom configuration
Does it really matter?
FePd: Density of phonon states g()
Mehaddene et al. 2004
L10 - ordered
fcc disordered
Simplifying almost everything:
Bragg – Williams model:
only nn pair interactions,
disregard pair correlations
R long range order parameter
FC  
NkT
2 ln 2  (1  R) ln(1  R)  (1  R) ln(1  R)  W0  NzV R 2
2
4
FC
0
R
<1
>1
tanh R/
Different levels of approximation in calculating
internal interaction energy
Experiment
Bragg-Williams
Quasi-chemical
Experiment
quasi-chemical
Bragg-Williams
Ising model (Lenz + Ising 1925)
1




H  n    J nm n m  h n
2 n ,m
n
Hamiltonian for alloy (pair interaction model)
 
 
i
1
~
ij
H pni  H pni    i  pni   Vnm
pni pmj    i  pni p n atom
occupation function
2 n ,m
i
n
i
n
Can be brought to Ising form by identifying (for nn interaction)
1
p n  1   n 
2
h

 
1 A
1
   B  V AA  V BB
2
4

1
J nm  V V  V AA  V BB  2V AB
4
Idea of mean field model: treat a few local interactions explicitly,
environment of similar cells is averaged and exerts a
mean field of interaction
mean field
Local interaction only 1 atom  Bragg- Williams – model
Correspondences:
Phase-separating
-----
ferromagnetic
Long range ordering
-----
antiferromagnetic
ferromagnetic
Structure on polished surface
after martensitic transformation:
roof-like, but no steps.
A scratched line remains continuous
Martensite
morphologies
Homogeneous distorsion by a martensitic transformation
First step :Transformation into a new lattice type:
Bain transformation
Second step: Misfit is accomodated by a
complementary transformation: twinning or dislocation glide
Thermoelastic Martensites:
Four symmetric variants per
glide plane:
Can be transformed into
one another by twinning
Shape Memory effect
Final remarks:
As the number of components grows
and interaction mechanisma are added,
phase transformations can gain considerable complexity
For instance: Phase separation and ordering
(opposites in simple systems)
may happen at the same time.
I have completely omitted many interesting topics, for instance
Gas-to-liquid or liquid-to-liquid transformations
The role of quantum phenomena at low-temperature phases
Dynamical phase transformations, self-organized phases
far from equilibrium