material and its characterization
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Thermodynamics of Phase Transformation
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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 ?
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Kinetics of Phase Transformation
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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
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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
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! 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
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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
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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
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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
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! 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
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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
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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 ???
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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