03.09.2012

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Phase-field simulation of solid statePhase field simulation of solid state transformation

with thermo-mechanical interaction

Ingo Steinbach

Thermal dendrites: Ryo Kobayashi 1994

),(2 Tf

2

pcLTT 2

γ‘ precipitates in Ni-base superalloys:Y. Wang, L.Q. Chen, A.G. Khachaturyan 1995

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Ni-base super alloy: alligned cuboids in crystallograpic directions

Grain boundary structure

Y. Ikuhara, N. Shibata, T. Watanabe, F. Oba, T. Yamamoto and T. Sakuma,

TEM Picture<110> tilt grain boundary

in ZrO2

2nm2nm2nm

Ann. Chim. Sci. Mat., 2002,27 Suppl 1:S21-30

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Grain boundaries in ferrite

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Curvature driven flow

vPhenomenological picture

How does an atom close on one side of a grain boundary know its mean curvature?

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G. Gottstein, Physical Foundation ofMaterials Science, Springer

Curvature driven flow

Mechanistic picture: The atoms at grain boundaries jump between latticesites of the adjacent crystals

Characteristic function Ф indicates the belonging of one atom to one crystal respectivelybelonging of one atom to one crystal respectively

The belonging to one crystal is favourable

2

6

The balance between both actionsleads to a stable interface

)5.0)(1(

Outline

Basics of “phase-field” models

Elastic interactions Elastic interactions

Examples: Pearlite Strain stabilized precipitation

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Diffuse interface with steep concentration profile

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Concentration profile through the interface

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Phase-field model in the Gibbs-Thomson limit

2223

„Order parameter“ , the „Phase-Field“ variable

σ: interface energy Gv

02223 fm1

2xdF

c1cc~Gc,cG172xdF 022223

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σ: interface energyμ: interface mobilityη: interface thickness∆Gαβ: Gibbs energy difference

: diffusion potential~

Gv

Caginalp et Fife 1986, Wheeler et al. 1992

The phase-field functional

"A functional is a unequivocal mapping of a space of functions (x) on to real numbers"

0 TS-UF

10

t

Thermodynamic principle

Material specific Model of

th F f ti l

Background of “phase-field": The common model philosophy

F

T,c,f~dvF the Free energy functional

Mathematical rigorous derivation

of the "thin interface limit"

Numerical sound solution

v

G

821

1 2i

2

iji

i

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µ: interface mobility; : interface energy;: interface thickness : interface curvature;G: thermodynamic undercooling; x: numerical discretization

x1 22ii

Multi-phase-field model coupled to solute diffusion

cf),,,(K)n,n(

dF,

kikm n

kik

mi cD

cFMc

FFn1

1k 11k

12

c 1k 11k

Steinbach MSMSE 2009

03.09.2012

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Free boundary problem of crystallization: the „Stephan problem“

temperatursolid liquid

TTcp 0L1.)1+2)

solsolliqliq0 TTvL

finteq TTv

energy ballance

Gibbs-Thomson equation

v : velocity of the interface : curvature

: heat conductivity

2)

3)

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finteqcp: specific heatL0: latent heat: Gibbs-Thomson coeffizient

)TT()( finteq21

2

22

23‘)

)

Outline

Basics of “phase-field” models

Elastic interactions Elastic interactions

Examples: Pearlite Strain stabilized precipitation

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Motivation I: Phase distribution in solid state

Ni-base super alloyPearlite, courtesy Benteler

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BainiteNiTi precipitates, courtesy Eggeler

Ferrite

Controlled TransformationControlled TransformationHot RollingHot Rolling

Motivation II: Processing of multiphase steels

Bainite

Ms

Tem

pera

ture

TRIP steel

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Time

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Mechanisms of Microstructure Formation

i f

=

microstructure formation

transport / diffusion(time scale)

interfaces(length scale)

+

thermodynamics(energy scale)

+ +

mechanicalequilibrium

(energy scale)

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OpenPhaseTQ interface from TCS phase field methodstress/strain solver diffusion solver

Homogenization of stress/strain in the interface region

**el C21f

discontinuity of strain

or 2

discontinuity of stress σ σ

WBM 1994: cL = cS = c

Multi-phase-field: c = L • cL + S • cS

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Tiaden 1969 cL = kcS

Kim 1999 µL(cL) = µS(cS)

Reuss limit of continuous stress

**el C21f discontinuity of strain

continuity of stress between the phases! continuity of stress between the phases

el! f ** ˆ21 Cf el

C1

**C

!

**

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;C

C

C

Hooks matrix is a homogenous mixture between phases

;

Reuss limit of continuous stress

**el C21f

111

**** ˆ11

21ˆ

CCC

CGf elel

20

elG

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Verification: growth of a single particle in a matrix

Colour coding: бyy

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Elasticity moduli: C11 = 280GPa, C12 = 120GPa, Eigenstrain = 1%

Thermodynamic data of Fe-C0.463at%-Mn0.496at%, TCFe3 database

Test against analytical solution (Eshelby)

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Isothermal transformation: Growth of a single ferrite precipitatewithout stress

ribut

ion

with stress

carb

on d

istr

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Evolution of the driving forces during transformation

1090K 1085K

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Dependence of equilibrium fraction on internal stress

including stress

no stress

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/ Transformation

Phase distribution van Mises stress

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[MPa]

Outline

Basics of “phase-field” models

Elastic interactions Elastic interactions

Examples: Pearlite Strain stabilized precipitation

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Pearlite

1100

1150

1200

ratu

re(K

)

Fe3C

Fe-C systemEutectoid

steel

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900

950

1000

1050

Tem

per

0.8 C (wt%)

~50K

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Pearlite

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Phase-field calculation of diffusion controlled growth of pearlite with different spacings (Nakajima et al. 2006)

Pearlite

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

Eutectic: v= const.Pearlitic: T= const.

4

8

12

16

v (a

.u.) large ΔT

medium ΔT

small ΔT10

20

30

40large V

small V

medium V

ΔT

= T e

q.－

T int

. (a.

u.)

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00 0.2 0.4 0.6 0.8 1.0 1.2

spacing λ(a.u.)

00 0.2 0.4 0.6 0.8 1.0 1.2

spacing λ(a.u.)

T = a1vλ + a2/λa1v = T/λ - a2/λ2

Transformation strain and concentration dependent strainLattice 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/atom1000 Experiment

900

920

940

960

980

Tem

pera

ture

(K)

Model:Diffusion in phase

Ridley 0.81CFrye 0.78C

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860

880

0.0001 0.001 0.01 0.1nodule velocity

(mm/s)

Phase-field model coupled to elastic strain

elchi ffff

(..)(..)),(),(21),(),( 1*1* cCtxctxtxtxf el

2

elchem GGIn

ffn

11

12

2

cc

fMcfMc

m

32

1 cc k

cCf 1*0

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The effect of transformation strain on growth

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Phase distribution Hydrostatic stress Carbon concentration

The effect of transformation strain on growth

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The effect of transformation strain on growth

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staggered growth mechanism of pearlite

quenched pearlite front (Darken,Fisher 1960)

The effect of transformation strain on growth

960

980

1000 Ridley 0.81CFrye 0.78C

Experiment

880

900

920

940

960

Tem

pera

ture

(K)

Diffusion in

Diffusion in +

Phase Field CalculationDiffusion in phase

Diffusion in and phase

+ Diffusion due to Stress (extrapolated)

+ transformation strain

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8600.0001 0.001 0.01 0.1

nodule velocity (mm/s)Model

Diffusion in phase

+ faceted anisotropy of cementite

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Mats Hillert IS John Ågren

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Stokholm 2009

Outline

Basics of “phase-field” models

Elastic interactions Elastic interactions

Examples: Pearlite Strain stabilized precipitation

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Experiment:Yang, Schryvers,

Intern. Journ. Of Appl. Electromagn. And

Precipitation in Ni-Ti shape memory alloys: experimental observations

pp gMech. 23 (2006) p. 17

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Martensite Start Temperature

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Otsuka, RenProgress in Material Science 50 (2005), p. 511

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Ni4-Ti3 precipitate in NiTi

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Ni4-Ti3 precipitate in NiTi

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Ni4Ti3 precipitates in the Ni oversaturated B2 matrix• CB2,0 = 51.0 at.%• CNi4Ti3

= 57.1 at.%

Initial conditions

4 3

Diffusion controlled precipitate growth• Diffusion only in the matrix, no diffusion in the stoichiometric

phase

Lattice mismatchA i t i i t i

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• Anisotropic eigenstrains:

The phase diagram

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Simulation: Strain (zz)

Results: Strain in the matrixSimulation: Guo et al.,

Acta Mater. 59 (2011) p. 3287

Strain (zz)

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Strain (zz)

Experiment: Tirry, Schryvers; Nature Mater 8 (2009) 752

Results concentration in the mattrix

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Phase field

Ni Concentration

Concentration Dependent Elastic Coefficients

Larche , Cahn 1982: The effect of self-stress on diffusion in solids

mech

i

chemik

i

mech

i

chemik

iiki

fccfM

cf

cfM

cfMc

2

2

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Elastic coefficients: Pure elements and the alloy

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Elastic Properties of B2 Ni-Ti with Defects• Perform full relaxation of B2 NiTi supercells

with vacancies and substitutions to compare changes in elastic energy

• Obtain DFT elastic properties as a function of Ni-concentration

• Use DFT elastic properties and trends as inputs for phase-field (PF) models

Ni vacancy

1) Obtain DFT Cij as function of Ni-concentration

2) Import concentration dependent Cij into PF-model

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Cij into PF model3) Incorporate into PF mechanical

energy term

Linear coupling between the elastic coefficients and the Ni concentration:

Concentration Dependent Elastic Coefficients

Elastic contribution to the fluxes:

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= 0

The role of hydrostatic stress

= 0

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Strain (zz)

Simulations with concentration dependent elasticity (K=2e-3)

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Ni Concentration

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Growth kinetics

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Growth kinetics

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Concentration profile

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Autocatalytic nucleation: driving force for nucleation

chemical driving force elastic driving force chemical + elastic

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Favourable position for new nucleation

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Autocatalytic nucleation

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Experiments Eggeler, Bochum

Solid state transformations cannot be well understood without the consideration of stress-strain effects.

Conclusion

The phase-field method provides a thermodynamic consistent framework for the investigation of phase transformations subject to elastic effects.

Examples were presented which demonstrate the predictive capability of the approach.

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Precipitation of γ’ in a Ni-Al alloy

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