1. Classification of Phase Transformations

2. Functions and discontinuity in differentials
f(x)
g(x)
h(x)

3. g(x) h(x) f(x) Discontinuity in the curvature
Discontinuity in the slope
g(x)
Discontinuity in the function
h(x)
f(x)

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

5. B Reconstructive Mechanistic Displacive Diffusional → Civilian 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
E.g.: Precipitation in Al-Cu alloys
Nucleation of product
Growth
Reconstructive
Diffusional → Civilian B
Subset
Replacive → e.g. ordering
Mechanistic
Buerger, 1951
Displacive
Military
Cooperative motion of a large number of atoms
Homogenous distortion
or a combination
E.g.: Martensitic
Formation of nucleus of product
Movement of shear front at speed of sound
Shuffling of lattice planes
Static displacement wave

6. Displacive Shuffle dominated Lattice strain dominated
Magnitude of shuffle and of homogenous lattice strain
Presence of precursor mechanical instability
Structural basis
Displacive
Homogenous distortion
Shuffling of lattice planes
Static displacement wave
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. Buerger’s classification: full list
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)

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

10. Thermodynamic classification
Ehrenfest, 1933
Order of a phase transformation
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

11. n = 1 Finite discontinuity First Order
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)

12. Schematics

13. n = 2 Finite discontinuity Second Order Second derivative is CP
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
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.
Quartz

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. 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.
Phase Transformations: Examples from Ti and Zr Alloys, S. Banerjee and P. Mukhopadhyay, Elsevier, Oxford, 2007

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. Landau Equation
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
A, B, C.. = f(T, P)

19. n = 1 Not zero Note the barrier Two minima separated by a G barrier
First Order
Not zero
Note the barrier
Two minima separated by a G barrier
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

20. 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
Sharp interface between parent and product phases  Nucleation and Growth  Discrete nature of the transformation

21. n = 2
Second Order
Only even powers
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
ve curvature at  = 0 → corresponds to ve value of A
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
Manganese Bromide: Symmetrical -transitions
Quartz: Unsymmetrical -transitions

23. Homogenous (Continuous) Transitions
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
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
NO Sharp interface between parent and product phases  Continuous nature of the transformation

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. Spinodal clustering
Spinodal decomposition
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

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. Inverted image (black → white)
looks very similar!
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.

30. 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

31. Spinodal Ordering 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
Not zero

33. METASTABLE STATE
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 Tm
G →
Liquid stable
Solid stable
Solid Metastable
Liquid Metastable
For a second order transformation the free energy curve is a single continuous curve and the concept of a metastable state does not exist

34. 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)

35. Displacive transformation
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

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

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

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