Kinetics of Phase Transformations: Northeastern University Radhika Barua

Northeastern University
Kinetics of Phase Transformations:
Nucleation & Growth
Radhika Barua
Department of Chemical Engineering
Northeastern University
Boston, MA USA
Northeastern University
Thermodynamics of Phase Transformation
For phase transformations (constant T & P) relative stability of the system
is defined by its Gibb’s free energy (G).
•  Gibb’s free energy of a system:
G
•  G=H-TS
dG=0
•  Criterion for stability:
•  dG=0
ΔGa
dG=0
ΔG
•  Criterion for phase transformation:
•  ΔG= GA-GB < 0
B
Activated
State
A
But …… How fast does the phase transformation occur ?
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Kinetics of Phase Transformation
Phase transformations in metals/alloys occur by
nucleation and growth.
•  Nucleation: New phase (β) appears at certain sites within the metastable parent
(α) phase.
•  Homogeneous Nucleation: Occurs spontaneously & randomly without
preferential nucleation site.
•  Heterogeneous Nucleation: Occurs at preferential sites such as grain
boundaries, dislocations or impurities.
•  Growth: Nuclei grows into the surrounding matrix.
SOLID
LIQUID
Example: Solidification , L
S
(Transformations between crystallographic & non-crystallographic states)
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Driving force for solidification
Example: Solidification , L
S
•  GL = HL - TSL ; GS = HS - TSS
•  ΔG = GL – GS = ΔH – TΔS
•  At the equilibrium melting point (Tm):
•  ΔG = ΔH – TmΔS = 0
•  ΔH = L (Latent heat of fusion)
Free energy (G)
•  At a temperature T:
Driving Force for
solidification
ΔG
GS
•  For small undercoolings (ΔT):
ΔT
•  ΔG ≈ L ΔT
Tm
T
GL
TM Temperature
Decrease in free energy (ΔG) provides the driving force for solidification
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Homogeneous Nucleation
•  Difference in free energy:
•  ΔGhom = G1 – G2 = V(Gs – GL) + AγSL
•  For a spherical particle:
•  ΔGhom = G1 – G2
SOLID
LIQUID
Volume
Interfacial
free energy
energy
•  Note the following:
•  Volume free energy increases as –r3
•  Interfacial free energy increases as r2
G1
LIQUID
G2
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ΔGhom for a given undercooling (ΔT)
Interfacial energy α r2
r*
ΔG*hom
ΔT
Volume free
energy α r3
GL
GS’
GS
r=r*
ΔG=2γ/r *
r=∞
ΔT
Note : Both r* and ΔG* depend on
undercooling (ΔT).
T
TM
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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.
Homogeneous nucleation occurs only
when liquid is undercooled by TN
Critical undercooling for nucleation
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Rate of Homogeneous Nucleation
3
clusters/m
•  For a given undercooling:
Note: C0 , 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 f0:
-3 -1
Nuclei / m S
-3 -1
Nuclei / m S
N
ΔTN
ΔT
No nuclei is formed until ΔTN is reached !!
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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
ΔGhet = V(Gs – GL) + ASLγSL + ASMγSM - ASMγML
=
where,
S(θ) ≤ 1 is a function of the wetting angle
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ΔGhet for a given undercooling (ΔT)
ΔG
ΔG*hom
ΔG*het
r
r*
Note:
ΔGhet
ΔGhom
•  r* depends only on ΔT.
•  ΔG*het depends of S(θ) & ΔT
•  ΔG*het < ΔG*hom
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Variation of ΔG* & nucleation rates with ΔT
Smaller undercooling is required for heterogeneous nucleation
-3 -1
Nuclei / m S
where,
f1 is the frequency factor
C1 is the # of atoms in contact with the heterogeneous
nucleation sites.
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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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Avrami Model: Derivation
NOTE:
, where G’ & N’ are the growth and nucleation rates
n = 4 ……… when growth is 3-D & N’ is constant
n = 3 ……… when growth is 3-D & nuclei are preformed
n = 1,2 …… when growth is restricted in 1-D (surface) or 2 D (edge)
2-D growth along a stepped interface
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First order magnetostructural transitions
First order magnetostrucutral transitions share common features with solidification.
Example: Bulk Fe1-xRhx (0.485 < x < 0.55)
(Levitin, Soviet Physics JETP, 1966)
AFM
phase
FM phase
Phase transition features:
•  Thermal hysteresis
•  Tt = f(H,P)
•  ~ 1% volume expansion
(Kouvel and Hartelius, J. Appl. Phys ,1962)
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Thermal hysteresis
Example: Hypothesized FeRh nanoparticles in Cu matrix.
Onset of Phase #1
Complete transformation of
Phase #2
Complete transformation
of Phase #1
Nucleation of
Phase #2
T~130
K
Type II AFM
FM
Phase #1: AFM ???
PHASE #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