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 ΔGa • Criterion for stability: dG=0 dG=0 • dG=0 ΔG • Criterion for phase transformation:

• ΔG= GA-GB < 0 B Activated A State

But …… How fast does the phase transformation occur ?

2 Northeastern University 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) 3 Northeastern University Driving force for solidification

Example: Solidification , L S

• At a temperature T: L L L S S S Driving Force for • G = H - TS ; G = H - TS solidification • ΔG = GL – GS = ΔH – TΔS ΔG • At the equilibrium melting point (Tm):

• ΔG = ΔH – TmΔS = 0 GS

• ΔH = L (Latent heat of fusion) (G) energy Free

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

Tm T TM Temperature

Decrease in free energy (ΔG) provides the driving force for solidification Northeastern University Homogeneous Nucleation

• Difference in free energy:

• ΔGhom = G1 – G2 = V(Gs – GL) + AγSL • For a spherical particle:

• ΔGhom = G1 – G2 SOLID

LIQUID LIQUID

Volume Interfacial G2 G1 free energy energy

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

ΔGhom for a given undercooling (ΔT)

Interfacial energy α r2 r*

ΔG*hom

ΔT GL Volume free energy α r3 GS’ GS r=r* ΔG=2γ/r * r = ∞

ΔT Note : Both r* and ΔG* depend on undercooling (ΔT). 6 T TM Northeastern University 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 7 Northeastern University Rate of Homogeneous Nucleation

3 • For a given undercooling: clusters/m

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. -3 -1 • If this happens with a frequency of f0: Nuclei / m S

-3 -1 Nuclei / m S N

ΔT ΔT N No nuclei is formed until ΔT is reached !! N 8 Northeastern University 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* ΔGhet

Note: ΔGhom • r* depends only on ΔT.

• ΔG*het depends of S(θ) & ΔT

• ΔG*het < ΔG*hom 10 Northeastern University 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 12 Northeastern University

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 13 Northeastern University

First order magnetostructural transitions

First order magnetostrucutral transitions share common features with solidification.

Example: Bulk Fe1-xRhx (0.485 < x < 0.55)

AFM (Levitin, Soviet Physics JETP, 1966) phase Phase transition features: FM phase • Thermal hysteresis

• Tt = f(H,P) • ~ 1% volume expansion

(Kouvel and Hartelius, J. Appl. Phys ,1962) 14 Northeastern University

Thermal hysteresis

Example: Hypothesized FeRh nanoparticles in Cu matrix.

Onset of Phase #1

Complete transformation of Phase #2

Complete transformation of Phase #1 T~130 Nucleation of Phase #2 K

Type II AFM FM

Phase #1: AFM ???

PHASE #2: FM ??? 15 Northeastern University

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