material and its characterization

8/13/2019 material and its characterization

1/16


2/16

Northeastern University

Thermodynamics of Phase Transformation

2

For phase transformations (constant T & P) relative stability of the systemis defined by its Gibbs free energy (G).

B ActivatedState

A

dG=0

dG=0

! G a

! G

G Gibbs free energy of a system:

G=H-TS Criterion for stability:

dG=0

Criterion for phase transformation:

! G= G A-GB < 0

But How fast does the phase transformation occur ?


3/16


Kinetics of Phase Transformation

3

Phase transformations in metals/alloys occur bynucleation and growth.

Nucleation: New phase ( ! ) appears at certain sites within the metastable parent(" ) phase.

Homogeneous Nucleation: Occurs spontaneously & randomly withoutpreferential nucleation site.

Heterogeneous Nucleation: Occurs at preferential sites such as grainboundaries, dislocations or impurities.

Growth: Nuclei grows into the surrounding matrix.

(Transformations between crystallographic & non-crystallographic states)

LIQUID

SOLID

Example: Solidification , L S


4/16


Driving force for solidification

Example: Solidification , L S

At a temperature T: GL = H L - TS L ; G S = H S - TS S

! G = G L G S = ! H T ! S At the equilibrium melting point (T m ):

! G = ! H T m! S = 0 ! H = L (Latent heat of fusion)

For small undercoolings ( ! T) :

! G " L ! TTm

GL

GS

TMT

! T

! G

F r e e e n e r g y

( G )

Temperature

Decrease in free energy ( ! G) provides the driving force for solidification

Driving Force forsolidification


5/16


6/16


6

! G hom for a given undercooling ( ! T)

Interfacial energy ! r 2

Volume freeenergy ! r 3

! G*hom

r *

r = "

GL

GS

TM

T

! T

r=r*

! G=2 /r *

Note : Both r* and ! G* depend onundercooling ( ! T).

! T

GS


7/16


7

Critical Undercooling for Nucleation Assumptions:

Liquid with nuclei is an ideal solution of various size clusters.

Each size cluster contains i atoms or molecules.

C ritical undercooling for nucleation

Homogeneous nucleation occurs onlywhen liquid is undercooled by T N


8/16


8

Rate of Homogeneous Nucleation

For a given undercooling:Note: C 0 , Atoms per unit volume in the liquid.

C*, Number of atoms that have reached critical size.

Addition of one more atom, converts the clusters to a stable nuclei.

If this happens with a frequency of f 0:

clusters/m3

Nuclei / m-3

S-1

Nuclei / m-3

S-1

No nuclei is formed until ! T N is reached !!! T

N

! TN


9/16


9

Heterogeneous Nucleation In practice, homogeneous nucleation is rarely observed.

Sources of nucleation sites:

Dislocations

Grain boundaries

Dust particles

Secondary phase particles

Mould walls & cracks ! Ghet = V(G s G L) + A SL ! SL + A SM ! SM - A SM ! ML

=

where,

S( " ) ! 1 is a function of the wetting angle


10/16


10

! G het for a given undercooling ( ! T)

! G hom

! G *hom

! G het

! G *het

r *r

! G

Note:

r* depends only on # T.

# G*het depends of S( $ ) & # T

# G*het < # G*hom


11/16


12/16


12

Avrami Model for Growth

Assumptions: Nucleation occurs randomly and homogeneously Growth rate does not depend on the extent of transformation Growth occurs at the same rate in all directions

Nuclei

Parent phase

New secondary phase

Ref: www.wikipedia.com


13/16


14/16


15/16


15

Thermal hysteresis

Nucleation ofPhase #2

Complete transformation ofPhase #2

Onset of Phase #1

Complete transformationof Phase #1

Example: Hypothesized FeRh nanoparticles in Cu matrix.

T~130

K

Type II AFM FM

Phase #1: AFM ???

P HASE #2: FM ???


16/16


Extension of Avrami Equation

Minor thermal hysteresis loops during heating & cooling

Temperature dependance of area of minor loops

Reference: Manekar and Roy, J. Phys.: Condens. Matter 20 (2008

material and its characterization

Documents