Classification of Phase TransformationsClassification of Phase Transformations

1 0

( ) 0 0

1 0

x

f x x

x

Functions and discontinuity in differentialsf(x)

g(x)

h(x)

1

1

( ) ( 1) [ ] ( 1)x

xg x dt t x

0x

0x

0x

( ) 1 ( )g x by definition

0

01 0

1 0

( ) ( 1) (1) [ ] [ ] ( 1 ) ( 1)x

xg x dt dt t t x x

2 2 2

1 1

1 1( ) ( 1) 1

2 2 2 2 2

xx t x xh x t dt t x x

0x

0x

0x

1(0) ( )

2h by definition

00 2 2 2 2

1 0 1 0

1 1( ) ( 1) ( 1) 1

2 2 2 2 2 2

xx t t x xh x t dt t dt t t x x

-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

Classification of Phase

TransformationsMechanism

Kinetics

ThermodynamicsB

ased

on

Mechanistic

Ehrenfest, 1933

Buerger, 1951

le Chatelier, (Roy 1973)

Order of a phase transformation

C

B

A

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.: MartensiticFormation of nucleus of

productMovement of shear front

at speed of sound

Replacive → e.g. orderingSubset

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

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").

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

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

Kinetic

Quenchable

Non-quenchable Athermal → Rapid

Thermal → Sluggish

Replacive → e.g. orderingSubset

* Usually Martensitic transformations are athermal- however there are instances of they being isothermal

le Chatelier, (Roy 1973)

C

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

n = 1 First Order

0

0

(G)0

(T)CT T

1

1

(G)

(T) CP

HS

T

G H T S

dG VdP SdT

0G

(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

2P

PP

Cd G dS

dT dT T

1

1

(G)0

(T) CP

HS

T

Schematics

2

2P

P CP

d G dS C

dT dT T

0G

1

1

(G)0

(T) CP

HS

T

n = 2 Second Order

1

1

(G)0

(T)C

CT T

HS

T

2

2

(G) 10

(T)P

PC CP

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.

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

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.

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

Mixed Order

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...

finiteG G G A B C

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

n = 1 First Order

2 3 4...G A B C Not zero

CT T • Two minima separated by a G barrier

CT 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

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

CT T

CT T

CT 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

( )Curvature at

ve curvature at = 0→ corresponds to ve value of A

2 4 6...G A C C 0B

• Only even powers

• Glass transitions, Paramagnetic-Ferromagnetic transitions

Lambda transitions

Heat capacity tends to infinity as the transformation temperature is approached

E.g.: Transformation in crystalline quartzOrder-disorder transition in -brass (B2 → BCC, Cu-Zn alloy)

Symmetrical -transition → Manganese Bromide

Quartz: Unsymmetrical -transitions

Manganese Bromide: Symmetrical -transitions

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

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

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

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

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

Inverted image (black → white)

looks very similar!

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

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

A B

L

L +

+ ’

’Ordered solid

METASTABLE STATE

Tm

G →

Liquid stableSolid 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

Enantiotropic transformationsEquilibrium 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 transformationsIrreversible (no equilibrium between parent and product phases)

A’ (metastable) → B (stable) (at T1)

Supercooled liquid (metastable) → A (stable) (at T2)

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

M.J. Buerger, Phase Transformations in Solids, John Wiley, 1951

Toy Model for Displacive Transformation

A B

L

L +

+ ’

’Ordered solid

A B

L

L +

+ ’

’Ordered solid

A B

L

L +

12

1 + 2

E.g

. Au-

Ni