Phase TransitionsBy Saurav Chandra Sarma

CRYSTALLOGRAPHY AND IT’S APPLICATIONS

Outline

Introduction

Classification of Phase Transition

Kinetics of Phase Transition

Martensitic Transformation

BaTiO3 Phase Transition

Glass Transition

Other Examples

Conclusion

Introduction

• A phase transition is the transformation ofa thermodynamic system from one phase or state ofmatter to another one by heat transfer.

• During a phase transition of a given medium certainproperties of the medium change, often discontinuously,as a result of the change of some external condition, suchas temperature, pressure, or others

• For example, a liquid may become gas upon heating tothe boiling point, resulting in an abrupt change in volume.

Classification of Phase Transitions

Classification of

Phase

Transformations

Mechanism

Thermodynamics

Base

d o

n

Ehrenfest, 1933

Buerger, 1951

Order of a phase transformation

B

A

Ehrenfest’s ClassificationFirst order phase transition: Discontinuity in the first

derivative of Gibb’s Free Energy,G.

Second order phase transition: Continuous first derivative but discontinuity in the second derivative of G.

Lambda Transition:

Buerger’s ClassificationReconstructive Transition: Involves a major reorganization

of the crystal structure.

E.g: Graphite Diamond

Displacive Transition:Involves distortion of bond rather than their breaking and the structural changes.

E.g: Martensitic Transformation

Diffusional or Civilian

Military

Transformation involving first coordination

Reconstructive (sluggish) DiamondGraphite

Dilatational (rapid) Rock saltCsCl

Transformation involving second coordination

Reconstructive (sluggish) QuartzCristobalite

Displacive (rapid) LowHigh Quartz

Transformations involving disorder

Substitutional (sluggish) LowHigh LiFeO2

Rotational (rapid) FerroelectricParaelectric NH4H2PO4

Transformations involving bond type (sluggish)

GreyWhite Sn

Buerger’s Classification: full list

Liquid → Solid phase transformation

Solid (GS)

Liquid (GL)

Tm T →

G

→

T

G

Liquid stableSolid stable

T - Undercooling

↑ t

“For sufficient

Undercooling”

On cooling just below Tm solid becomes stable

But solidification does not start

E.g. liquid Ni can be undercooled 250 K below Tm

G → ve

G → +ve

Nucleation

of

phase

Trasformation

→

+

Growth

till

is

exhausted

=

1nd order

nucleation & growth

Kinetics of Phase Transition:

13

Phase TransformationsNucleation

• nuclei (seeds) act as templates on which crystals grow

• for nucleus to form rate of addition of atoms to nucleus must be faster than rate of loss

• once nucleated, growth proceeds until equilibrium is attained

Driving force to nucleate increases as we increase T

– supercooling (eutectic, eutectoid)

– superheating (peritectic)

Small supercooling slow nucleation rate - few nuclei - large crystals

Large supercooling rapid nucleation rate - many nuclei - small crystals

Heterogeneous nucleation

Nucleation occur at the interface between two phases or at the grain boundary.

Homogeneous nucleation

Nucleation occur without any preferential nucleation sites.

Occurs spontaneously and randomly but it requires superheating or supercooling.

An example of supercooling: Pure water freezes at −42°C rather than at its freezing temperature of 0°C. The crystallization into ice may be facilitated by adding some nucleation “seeds”: small ice particles, or simply by shaking

15

r* = critical nucleus: for r < r* nuclei shrink; for r >r* nuclei grow (to reduce energy)

Adapted from Fig.10.2(b), Callister & Rethwisch 8e.

Homogeneous Nucleation & Energy Effects

GT = Total Free Energy

= GS + GV

Surface Free Energy - destabilizes

the nuclei (it takes energy to make

an interface)

24 rGS

= surface tension

Volume (Bulk) Free Energy –

stabilizes the nuclei (releases energy)

GrGV3

3

4

volume unit

energy free volume G

16

Solidification

TH

Tr

f

m

2*

Note: Hf and are weakly dependent on T

r* decreases as T increases

For typical T r* ~ 10 nm

Hf = latent heat of solidification

Tm = melting temperature

= surface free energy

T = Tm - T = supercooling

r* = critical radius

Avirami equation:

Transformations are often observed to follow acharacteristic S-shaped, or sigmoidal.

Initial Slow rate time reqd. for forming asignificant no. of nuclei of the new phase.

Intermediate fast rate nuclei grow in sizeand cross the critical radius

Final slow rate particles already existingbegin to touch each other, forming aboundary where growth stops.

The parameter ‘n’ depends on shape of β-phase particles (the Dimension):

Spherical→ n=3 (3D) Disk-shaped→ n=2 (2D) Rod-shaped→ n=1 (1D)

Rate of Phase Transformation

Avrami equation => y = 1- exp (-kt n)

• k & n are transformation specific parameters

transformation complete

log t

Fra

ctio

n tra

nsfo

rme

d, y

Fixed T

fraction

transformed

time

0.5

By convention rate = 1 / t0.5

Adapted from

Fig. 10.10,

Callister &

Rethwisch 8e.

maximum rate reached – now amount unconverted decreases so rate slows

t0.5

rate increases as surface area increases

& nuclei grow

Temperature Dependence of Transformation Rate

• For the recrystallization of Cu, since

rate = 1/t0.5

rate increases with increasing temperature

• Rate often so slow that attainment of equilibrium

state not possible!

Adapted from Fig.

10.11, Callister &

Rethwisch 8e.

(Fig. 10.11 adapted

from B.F. Decker and

D. Harker,

"Recrystallization in

Rolled Copper", Trans

AIME, 188, 1950, p.

888.)

135C 119C 113C 102C 88C 43C

1 10 102 104

The martensitic transformation occurs without composition change

The transformation occurs by shear without need for diffusion

The atomic movements required are only a fraction of the interatomic

spacing

The shear changes the shape of the transforming region

→ results in considerable amount of shear energy

→ plate-like shape of Martensite

The amount of martensite formed is a function of the temperature to

which the sample is quenched and not of time

Hardness of martensite is a function of the carbon content

→ but high hardness steel is very brittle as martensite is brittle

1) Martensitic Transformation:

Example??

Martensite

FCCAustenite

FCCAustenite

Alternate choice of Cell

Tetragonal

Martensite

Austenite to Martensite → 4.3 % volume increase

Possible positions of

Carbon atoms

Only a fraction of

the sites occupied

20% contraction of c-axis

12% expansion of a-axis

In Pure Fe after

the Matensitic transformation

c = a

C along the c-axis

obstructs the contraction

C

BCT

C

FCCQuench

% 8.0

)( '

% 8.0

)(

What happens actually???

Martensite Austenite

2) BaTiO3 Phase transition

>120oC

Click me

Cubic Structure(Paraelectric)

Tetragonal Strucure(Ferroelectric)

Experimental Techniques:• DSC• EXAFS (Extended X-ray absorption fine structure)• XANES (X-ray absorption near-edge structure)• PDF (Pair Distribution Function)- To undertand local structure distortion.

Change in hysteresis loop pattern

Source: Ferroelectricity, domain structure and phase transition of Barium Titanate, Reviews of Modern Physics, 22,3 (1950)

Source: Ferroelectricity, domain structure and phase transition of Barium Titanate, Reviews of Modern Physics, 22,3 (1950)

In the high-symmetry cubicphase, no reflections are split.In the tetragonal phase, (222)remains a single peak whereasthe (400) reflection is dividedinto (400/ 040) and (004) peakswith an intensity ratio of 2:1

Source: J. AM. CHEM. SOC.,130, 22, (2008)

Tetragonal system: 10 Raman active

modes but 18 observed due to LO-TO splitting.

Cubic system: Should be Raman inactive

but 2 modes observed due to displace Ti

position

Raman Spectra:

Glass forming liquids are those that are able to “by-pass” the melting point, Tm

Liquid may have a high viscosity that makes it difficult for atoms of the liquid to diffuse (rearrange) into the crystalline structure

Liquid maybe cooled so fast that it does not have enough time to crystallize

Temperature

Mola

r V

olu

me

liquidglass

2) Glass transition:

Examples of Poor Glass Formers:

Why is water, H2O, found to be a very “weak” glass former

Requires cooling the liquid faster than 1,000,000 oC/min

300 to 150K in 9 milliseconds

H2O

No bonding between

molecules and molecules

can easily flow by each

other

Examples of “Good” Glass Formers:

Why is silica, SiO2, found to be a very “strong” glass former?

Can be cooled at

10-10C/min and still by-pass Tm without crystallizing

2,000 oC to 1,000 oC in 20 million years!!

SiO2

Each Si is tetrahedrally bonded to O,

each O is bonded to two Si. Si and O

atoms cannot move unless other

neighboring atoms also move

Typical DSC thermogram

Determination of Glass Transition temperature by dilatometry

Other Examples:

Tetragonal Orthorhombic

Conclusion

• Experimental techniques used to understand the phase transitiondepends on the type of phase transition.

• Depending on the type of transition, it shows various type ofcomplexity.

• Understanding of phase diagram is must to deal with phasetransition.