1.
Classification of Phase Transformations

2.
1 0
( ) 0 0
1 0
x
f x x
x
 


 

 

Functions and discontinuity in differentials
f(x)
g(x)
h(x)
1
1
( ) ( 1) [ ] ( 1)
x
x
g x dt t x


      

0
x 
0
x 
0
x 
( ) 1 ( )
g x by definition
 
 
0
0
1 0
1 0
( ) ( 1) (1) [ ] [ ] ( 1 ) ( 1)
x
x
g x dt dt t t x x


          
 
2 2 2
1 1
1 1
( ) ( 1) 1
2 2 2 2 2
x
x
t x x
h x t dt t x x
 
     
 
              
   
   
 
     

0
x 
0
x 
0
x 
1
(0) ( )
2
h by definition
 
0
0 2 2 2 2
1 0 1 0
1 1
( ) ( 1) ( 1) 1
2 2 2 2 2 2
x
x
t t x x
h x t dt t dt t t x x
 
   
       
 
   
 
                  
   
         
 
 
 
 
       
 
 
 
 

3.
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
f(x)
g(x)
h(x)
Discontinuity in
the function
Discontinuity in
the slope
Discontinuity in
the curvature

4.
Classification of
Phase
Transformations
Mechanism
Kinetics
Thermodynamics
Based
on
Mechanistic
Ehrenfest, 1933
Buerger, 1951
le Chatelier, (Roy 1973)
Order of a phase transformation
C
B
A

5.
Mechanistic
Displacive
Reconstructive Diffusional → Civilian
Cooperative motion of a large number of atoms
Military
Homogenous distortion
Shuffling of lattice planes
Static displacement wave
or a combination
E.g.: Martensitic
 Formation of nucleus of
product
 Movement of shear front
at speed of sound
Replacive → e.g. ordering
Subset
 Breaking of bonds and formation of new ones
 Atom movements from parent to product by diffusional jumps
 Nearest neighbour bonds broken at the transformation front and the product structure is reconstructed by
placing the incoming atoms in correct positions → growth of product lattice
 Diffusional transformation
 Even in case chemical composition same (parent & product) + strict orientation relation
→ Still lattice correspondence not present
Buerger, 1951
B
E.g.: Precipitation in
Al-Cu alloys
 Nucleation of product
 Growth

6.
Displacive
Homogenous distortion
Shuffling of lattice planes
Static displacement wave
Magnitude of shuffle and of homogenous lattice strain
Presence of precursor mechanical instability
Structural basis
Shuffle dominated
Lattice strain dominated
 Coordination between neighbours retained in the product lattice (though bond angles change)
 Atomistic coordination inherited → chemical order in parent structure is fully retained in the product
structure
 similar correspondence of crystallographic planes
 Lines → lines; planes → planes (vector, plane, unit cell correspondence)
 AFFINE TRANSFORMATION
In general, an affine transform is composed of linear transformations (rotation, scaling or shear) and a translation (or "shift").

7.
 NOTE:
Lattice correspondence does NOT imply ORIENTATION RELATION
 as phase transformations may involve rigid body rotations

8.
 Transformation involving first coordination
 Reconstructive (sluggish)
 Dilatational (rapid)
 Transformation involving second coordination
 Reconstructive (sluggish)
 Displacive (rapid)
 Transformations involving disorder
 Substitutional (sluggish)
 Rotational (rapid)
 Transformations involving bond type (sluggish)
Buerger’s classification: full list

9.
Kinetic
Quenchable
Non-quenchable Athermal → Rapid
Thermal → Sluggish
Replacive → e.g. ordering
Subset
* Usually Martensitic transformations are athermal- however there are instances of they being isothermal
le Chatelier, (Roy 1973)
C

10.
Thermodynamic classification Ehrenfest, 1933
Order of a phase transformation
1
1
(thermodynamic Variable)
(External Variable)
n
n


 

 

 
The lowest derivative (n) which shows a
discontinuity at the transition point
 Can be used for equilibrium transitions of single component systems
 There are cases of mixed order transformations
 In thermal transformations: usually the high-T form is of higher symmetry and
higher disorder
1
1
(G)
0
(T)
C
n
n
T T



 

 
 

 
(G)
0
(T)
n
n
P
 

 
 

 
A

11.
n = 1 First Order
0
0
(G)
0
(T)
C
T T

 

 
 

 
1
1
(G)
(T) C
P
H
S
T
 
 
   
 

 
G H T S
    
dG VdP SdT
 
0
G
 
(G)
(T) P
S
 

 
 

 
 First order transitions are characterized by discontinuous changes in entropy, enthalpy &
specific volume.
 H → change in enthalpy corresponds to the evolution of Latent Heat of transformation
 The specific heat [J/K/mole] is thus infinite (i.e. at the transition heat is being put into the
system but the temperature is not changing)
Finite discontinuity
2
2
P
P
P
C
d G dS
dT dT T

  
1
1
(G)
0
(T) C
P
H
S
T
 
 
    
 

 

12.
Schematics
2
2
P
P C
P
d G dS C
dT dT T

    
0
G
 
1
1
(G)
0
(T) C
P
H
S
T
 
 
    
 

 

13.
n = 2 Second Order
1
1
(G)
0
(T)
C
C
T T
H
S
T

 
 
     
 

 
2
2
(G) 1
0
(T)
P
P
C C
P
H C
T T T
 
  
 
    
   
 
 
 
G H T S
    
dG VdP SdT
 
(G)
(T) P
S
 

 
 

 
 NO discontinuous changes in entropy, enthalpy & specific volume.
 NO latent heat of transformation
 High specific heat at the transition temperature
 Finite discontinuity in CP (NOT infinite)
 Lamda () Transitions (-point transitions) show infinity
Finite discontinuity
Second derivative is CP
Quartz


 Concept of a metastable phase not readily applicable to a 2 transition →
single continuous free energy curve.
 Ferromagnetic ordering, Chemical ordering are examples of 2 transitions.
 In a two component system a 2nd order transformation requires equality of
entropy and volume of two phases + identical composition of the two
phases.

14.
 2 transitions can be described by mean field descriptions
of cooperative phenomenon
 Order parameter continuously decreases to zero as T →
TC
 Any transition which can be described by a continuous
change in one or more order parameters can be treated by
a the generalized LANDAU Equation

15.
Phase Transformations: Examples from Ti and Zr Alloys, S. Banerjee and P. Mukhopadhyay, Elsevier, Oxford, 2007
Schematics
In a two component system:
 1st order transformation appears in a
phase diagram as two line bounding the
region where two phases (of different
composition) coexist.
 Second order transformation appears as a
single line.

16.
n = 3 Third Order
 There is usually no classification as third order (II and higher order are clubbed
together)
 Superconducting transition in tin at zero field & Curie points in many
ferromagnets can be considered as third order transitions

17.
Mixed Order

18.
 Close to the critical temperature:
The free energy difference (G) between finite and zero values of order
parameter () may be expanded as power series
 Practically, any physical observable quantity which varies with
temperature (or other thermodynamic variable) can be taken as a
experimental order parameter
Landau Equation
    2 3 4
0
...
finite
G G G A B C
 
  

      A, B, C.. = f(T, P)

19.
n = 1 First Order
2 3 4
...
G A B C
  
   
Not zero
C
T T
 • Two minima separated by a G barrier
C
T T
 • T slightly less than TC the system still not unstable at  = 0 (state)
(curvature remains +ve)
 a gradual transition of the system in a homogenous fashion to a the
free energy minimum at  = C (or near it) is not possible
Note the barrier

20.
Sharp interface between parent and product phases
 Nucleation and Growth  Discrete nature of the transformation
• Phase transition can initiate if localized regions are activated to cross the free energy barrier (beyond
 = *) → where phase with finite  can grow spontaneously
• Formation of localized product phase regions with  ~ C → nucleation

21.
C
T T

C
T T

C
T T

• Single equilibrium at  = 0 → corresponds to +ve value of A
• +ve curvature
• Curvature at  = 0 decreases
• System becomes unstable at
T = TC and fluctuations will lead
to lowering of energy
n = 2 Second Order
2
2
0
(G)
0
( )
Curvatureat


 
 

   

 
• ve curvature at  = 0
→ corresponds to ve value of A
2 4 6
...
G A C C
  
   
0
B 
• Only even powers
• Glass transitions, Paramagnetic-Ferromagnetic transitions

22.
 Lambda transitions
 Heat capacity tends to infinity as the transformation temperature is approached
 E.g.: Transformation in crystalline quartz
Order-disorder transition in -brass (B2 → BCC, Cu-Zn alloy)
 Symmetrical -transition → Manganese Bromide
Quartz:
Unsymmetrical -transitions
Manganese Bromide:
Symmetrical -transitions

23.
NO Sharp interface between parent and product phases
 Continuous nature of the transformation
USUALLY
• First order transitions are discrete (For T > Ti )
→ Nucleation and Growth
• Higher order transitions are homogenous
→ parent and product phase cannot be sharply demarcated at any stage of the transition
 Parent phase gradually evolves into the product phase without creating a localized sharp
change in the thermodynamic properties and structure in any part of the system
 The system becomes unstable with respect to small (infinitesimal) fluctuations → leading
to the transition
 The free energy of the system continuously decreases with amplification of such
fluctuations
Homogenous (Continuous) Transitions

24.
For T < Ti
• First order transitions are can proceed in a continuous mode
• Not all first order transitions have a instability temperature
• Examples of first order continuous transitions (conditions far from equilibrium):
 Spinodal clustering
 Spinodal ordering
 Displacement ordering

25.
 Phase diagrams showing miscibility gap correspond to solid solutions which
exhibit clustering tendency
 Within the miscibility gap the decomposition can take place by either
 Nucleation and Growth (First order) or by
 Spinodal Mechanism (First order)
 If the second phase is not coherent with the parent then the region of the spinodal
is called the chemical spinodal
 If the second phase is coherent with the parent phase then the spinodal
mechanism is operative only inside the coherent spinodal domain
 As coherent second phases cost additional strain energy to produce (as compared
to a incoherent second phase – only interfacial energy involved) → this requires
additional undercooling for it to occur
Spinodal clustering Spinodal decomposition

26.
 Spinodal decomposition is not limited to systems containing a miscibility gap
 Other xamples are in binary solid solutions and glasses
 All systems in which GP zones form (e.g.) contain a metastable coherent miscibility gap
→ THE GP ZONE SOLVUS
 Thus at high supersaturations it is GP zones can form by spinodal mechanism

27.
A coarsened spinodal microstructure in Al-22.5 at.% Zn-0.1 at.% Mg solution treated 2h at 400C and aged 20h at
100C. TEM micrograph at 314 kX. (K.B. Rundman, Metals Handbook, 8th edn. Vol.8, ASM, 1973, p.184.

28.
Nucleation & Growth Spinodal
The composition of the second phase remains
unaltered with time
A continuous change of composition occurs
until the equilibrium values are attained
The interfaces between the nucleating phase
and the matrix is sharp
The interface is initially very diffuse but
eventually sharpens
There is a marked tendency for random
distribution of both sizes and positions of the
equilibrium phases
A regularity- though not simple- exists both in
sizes and distribution of phases
Particles of separated phases tend to be
spherical with low connectivity
The separated phases are generally non-
spherical and posses a high degree of
connectivity

29.
 Ordering leads to the formation of a superlattice
 Ordering can take place in Second Order or First Order (in continuous mode
below Ti) modes
 Any change in the lattice dimensions due to ordering introduces a third order
term in the Landau equation
 Continuous ordering as a first order transformation requires a finite supercooling
below the Coherent Phase Boundary to the Coherent Instability (Ti) boundary
 These (continuous ordering) 1st order transitions are possible in cases where the
symmetry elements of the ordered structure form a subset of the parent
disordered structure
Spinodal Ordering
2 3 4
...
G A B C
  
   
Not zero

30.
A B
L

L + 
 + ’
’
Ordered solid

31.
METASTABLE STATE
Tm
G
→
Liquid stable
Solid stable
Solid Metastable
Liquid Metastable
For a first order transformation the free energy curve can be extrapolated (beyond the
stability of the phase) to obtain a G curve for the metastable state
For a second order transformation the free energy curve is a single continuous curve
and the concept of a metastable state does not exist

32.
Enantiotropic transformations
Equilibrium transitions: Reversible and governed by classical thermodynamics
 L → A (at the melting point: Tm = TL/A)
 A → B (at the equilibrium transformation T: TL/A)
 A → A’ (transformation between two metastable phases)
Monotropic transformations
Irreversible (no equilibrium between parent and product phases)
 A’ (metastable) → B (stable) (at T1)
 Supercooled liquid (metastable) → A (stable) (at T2)

33.
 Changes in higher coordination effected by a distortion of the primary bond
 Smaller changes in energy
 Usually Fast
 High temperature form → more open, higher specific volume, specific heat, symmetry
 E.g.:  high-low transformations of quartz (843K), tridymite (433K & 378K), cristobalite
(523K)  SrTiO3
Displacive transformation

34.
M.J. Buerger, Phase Transformations in Solids, John Wiley, 1951
Toy Model for Displacive Transformation

35.
A B
L

L + 
 + ’
’
Ordered solid
A B
L

L + 
 + ’
’
Ordered solid

36.
A B
L

L + 
1
2
1 + 2
E.g.
Au-Ni