Ingo Steinbach

Cooperative Growth of Pearlite

Acknowledgement: Katsumi Nakajima, Markus Apel,

JFE Steel Corporation for financial support

Cooperative Growth of Pearlite

„The microstructure ofsteels and cast iron“ byDurand-CharreMicrographs by Irsid-Arcelor

Introduction

αα ββ

λ(spacing)

900

950

1000

1050

1100

1150

1200

Tem

pera

ture

(K)

C (mass%)

γ

α

Fe3C

Fe-C system

Eutectoid steel

~50K

Introduction

The "Zehner-Hillert-Tiller-Jackson-Hunt“ Model1946 1957 1958 1966

0

4

8

12

16

0 0.2 0.4 0.6 0.8 1.0 1.2

v (a

.u.) large ΔT

medium ΔT

small ΔT

spacing λ(a.u.)

0

10

20

30

40

0 0.2 0.4 0.6 0.8 1.0 1.2spacing λ(a.u.)

large V

small V

medium V

ΔT

= T e

q.－

T int

. (a.u

.)

Eutectic: v= const.Pearlitic: ΔT= const.

ΔT = a1vλ + a2/λa1v = ΔT/λ - a2/λ2

Introduction

860

880

900

920

940

960

980

1000

0.0001 0.001 0.01 0.1

Tem

pera

ture

(K)

nodule velocity (mm/s)

• Grainboundary diffusion• Diffusion in Ferrite• Diffusion due to composition

dependent strain• Spinodal decomposition in

supersaturated Ferrite

• Finite interface mobility• Reduced transformation due

to transformation strain• Anisotropic interface

mobility and energy

Model:Diffusion in γ phase

ExperimentRidley 0.81CFrye 0.78C

Outline

Spacing selection in pearlite growth, diffusion in austenite only

The effect of diffusion in ferrite and growth of cementite from ferrite

The effect of composition dependent strain on diffusion and growth

The effect of transformation strain on growth

Conclusion

Selection of spacing, diffusion in Austenite only

[ato

m%

]

initial

λ= 0.15 µm λ= 0.25 µm λ= 0.4 µm

ΔT= 30K Diffusion only in γ phase

0.2 µm

Selection of spacing, diffusion in Austenite only

Max velocityOvergrow

[ato

m%

]

transition

λ= 0.15 µm λ= 0.25 µm λ= 0.4 µm

ΔT= 30K Diffusion only in γ phase

0.2 µm

Selection of spacing, diffusion in Austenite only

Max velocityOvergrow

[ato

m%

]

final

λ= 0.15 µm λ= 0.25 µm λ= 0.4 µm

ΔT= 30K Diffusion only in γ phase

0.2 µm

Selection of spacing, diffusion in Austenite only

Relationship between Spacing and Velocity

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5Lamellar Spacing λ (μm)

Nod

ule

Vel

ocity

v (μ

m/s

)

Diffusion only in γ phase

ΔT=10ΔT=30ΔT=50

Zener model

Calculation

(mobility : ∴)(mobility : finite)

Dv :diffusion coefficient in γ phase

λ0 : spacing where all energy is consumedfor formation of interfaces

fα : Sα/(Sα+Scm)fcm :Sα/(Sα+Scm)

Overgrow

v = 2Dv ⋅ 1 / (fα fcm) ⋅ (Ceγ/α – Ce

γ/cm) / (Ccm/γ – Cα/γ) ⋅ 1/λ ⋅ (1 – λ0/λ)

Diffusion in Ferrite and growth of Cementite from Ferrite

γ

α αCem Cem

Cγ/α

Cα/γCγ/cm

Cα/cm

Carbon diffusion coefficient

γ phase ~ 5.5*10-9 cm2/sα phase ~ 8.6*10-7 cm2/s

Equlibrium concentrations

γ phase ~ 3.7 at %α phase ~ 0.1 at %

“Flux potential” D*ceq

γ phase ~ 2.0 10-8 cm2/s at %α phase ~ 8.6 10-8 cm2/s at %

The diffusion path through ferrite was first proposed by Onsager 1948, butneglected later due to the lack of experimental evidence

Diffusion in Ferrite and growth of Cementite from Ferrite

Diffusion in γ and α phase Diffusion in γ phase

initial ΔT= 30K λ= 0.3 µm

3.8 4.0 3.8 4.0[atom%]

Diffusion in Ferrite and growth of Cementite from Ferrite

Diffusion in γ and α phase Diffusion in γ phase

ΔT= 30K λ= 0.3 µmtransition

3.8 4.0 3.8 4.0[atom%]

Diffusion in Ferrite and growth of Cementite from Ferrite

Diffusion in γ and α phase Diffusion in γ phase

final ΔT= 30K λ= 0.3 µm

3.8 4.0 3.8 4.0[atom%]

Diffusion in Ferrite and growth of Cementite from Ferrite

Diffusion in γ and α phasePhase Field Calculation

Ridley 0.81CFrye 0.78C

Experiment

860

880

900

920

940

960

980

1000

0.0001 0.001 0.01 0.1

Tem

pera

ture

(K)

nodule velocity (mm/s)

Model

Diffusion in γ phase

Diffusion in γ phase

Diffusion in γ

Diffusion in γ+α

Diffusion in γ phase

v = 2Dv ⋅ 1 / (fα fcm) ⋅ (Ceγ/α – Ce

γ/cm) / (Ccm/γ – Cα/γ) ⋅ 1/λ ⋅ (1 – λ0/λ)

kδDb=1.0 × 10-13cm3/s (fitting value)

k : partition coefficient

δ : interface thickness

Db: diffusion coefficient in boundary

Phase-field model coupled to elastic strain

elchi ffff ++=

∑βα

βαβααβ

αβ

αβ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

φφ+φ∇φ∇π

η

η

σ=

,2

2i 4

f

( ) ⎟⎠

⎞⎜⎝

⎛φ−μ+φ= ∑∑

ααα

ααα cccff chch

( ) ( )∑α

αααααααααα ε−ε−εε−ε−εφ= cCc21f 1*1*el

( )elchem GGIn1ff

n1

αβαβαβαββα

αβα Δ+Δ+μ=⎟⎟⎠

⎞⎜⎜⎝

⎛

δφδ

−δφδ

μ=φ ∑∑ββ

&

⎟⎟⎠

⎞⎜⎜⎝

⎛σ∇ε+∇

δδ

φ∇=δδ

∇φ∇= ααα

αα

αα

∑∑=

12

2mc

cfM

cfMc

1k&

( )ααααααα

ε−ε−εφ∇=σ∇= ∑ cC0 1*

Steinbach, Apel: Physica D 54 (2006)

The effect of composition dependent strain on diffusion and growth

hydrostatic stress [J/cm3] carbon concentration [at%]

lamellar spacing 0.3 μm, ε1austenite= 0.00215 c

The effect of composition dependent strain on diffusion and growth

carbon concentration [at%] carbon concentration [at%]

chemical diffusion only:v= 0.76 μm/s

chemical + stress driven diffusion: v= 1.6 μm/s

3.4

3.6

3.8

4.0

3.4

3.6

3.8

4.0

The effect of composition dependent strain on diffusion and growth

diffusion fluxes

Jchem = Daustenite∇ c = 8.4*10-5 [at% cm/s]

Jchem+stress = Daustenite∇ c + M *ε1 ∇ σ = 4.2*10-5 + 3.2*10-5 [at% cm/s] = 7.4*10-5 [at% cm/s]

chemical diffusion only

chemical + stress diffusion

The effect of composition dependent strain on diffusion and growth

Phase Field Calculation

Ridley 0.81CFrye 0.78C

Experiment

ModelDiffusion in γ phase

Diffusion in γ and α phase

Diffusion in γ phase

+ Diffusion due to Stress (extrapolated)

860

880

900

920

940

960

980

1000

0.0001 0.001 0.01 0.1

Tem

pera

ture

(K)

nodule velocity (mm/s)

Diffusion in γ

Diffusion in γ+α

+ stress driven diffusion(extrapolated)

The effect of transformation strain on growth

Phase distribution Hydrostatic stress Carbon concentration

Lattice constants: austenite a = 2.301 Å/atom

ferrite a = 2.298 Å/atom

cementite a = 1.98 Å/atom; b = 2.23 Å/atom; c = 2.953 Å/atom

-1 0[J/cm³] [at%]

1 2 3 -2000 -1000 0 1000 3.5 4.0 4.5 5.0

The effect of transformation strain on growth

v = 6 μm/s !!Phase distribution Hydrostatic stress [J/cm3] Carbon concentration [at%]

The effect of transformation strain on growth

860

880

900

920

940

960

980

1000

0.0001 0.001 0.01 0.1

Tem

pera

ture

(K)

nodule velocity (mm/s)

Diffusion in γ

Diffusion in γ+α

Ridley 0.81CFrye 0.78C

Experiment

ModelDiffusion in γ phase

Phase Field CalculationDiffusion in γ and α phase

Diffusion in γ phase

+ Diffusion due to Stress (extrapolated)

+ transformation strain

+ faceted anisotropy of cementite

Conclusion

Many features are nessesary to explain the kinetics of pearlite transformation• spacing selection through growth restriction by curvature of the austenite/ferrite

interface

• enhanced diffusion in ferrite

• stress driven diffusion in austenite

• faceted surfaceface structure allowes preferential growth of cementite

into austenite

Phase-field simulation offers the metodology to investigate these effects

and their interrellation (allmost) quantitatively

Steel metallurgy is not only the past of „solidification“, it is also part it‘s future

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