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Kinetics of Phase Transformations: Nucleation & Growth

Radhika Barua

Department of Chemical Engineering Northeastern University

Boston, MA USA

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Thermodynamics of Phase Transformation

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For phase transformations (constant T & P) relative stability of the system is defined by its Gibb’s free energy (G).

B Activated State

A

dG=0 dG=0

ΔGa

ΔG

G • Gibb’s free energy of a system:

• G=H-TS

•  Criterion for stability:

•  dG=0

•  Criterion for phase transformation:

•  ΔG= GA-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 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.

(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 = 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)

•  For small undercoolings (ΔT): •  ΔG ≈ L ΔT Tm

GL

GS

TM T

ΔT

ΔG

Free

ene

rgy

(G)

Temperature

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

Driving Force for solidification

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Homogeneous Nucleation

LIQUID LIQUID

SOLID

G1 G2

•  Difference in free energy: •  ΔGhom = G1

– G2 = V(Gs – GL) + AγSL •  For a spherical particle:

•  ΔGhom = G1 – G2

•  Note the following: •  Volume free energy increases as –r3 •  Interfacial free energy increases as r2

Volume free energy

Interfacial energy

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ΔGhom for a given undercooling (ΔT)

Interfacial energy α r2

Volume free energy α r3

ΔG*hom

r*

r = ∞

GL

GS

TM T

ΔT

r=r*

ΔG=2γ/r *

Note : Both r* and ΔG* depend on undercooling (ΔT).

ΔT

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.

Critical undercooling for nucleation

Homogeneous nucleation occurs only when liquid is undercooled by TN

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Rate of Homogeneous Nucleation

•  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:

clusters/m3

Nuclei / m-3

S-1

Nuclei / m-3

S-1

No nuclei is formed until ΔTN is reached !! ΔT

N

ΔTN

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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)

ΔGhom

ΔG*hom

ΔGhet

ΔG*het

r* r

ΔG

Note:

•  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

Nuclei / m-3

S-1

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)

Phase transition features: •  Thermal hysteresis

•  Tt = f(H,P)

•  ~ 1% volume expansion

AFM phase

FM phase

(Kouvel and Hartelius, J. Appl. Phys ,1962)

(Levitin, Soviet Physics JETP, 1966)

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Thermal hysteresis

Nucleation of Phase #2

Complete transformation of Phase #2

Onset of Phase #1

Complete transformation of Phase #1

Example: Hypothesized FeRh nanoparticles in Cu matrix.

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