l t t - kit2).pdf · 2012-09-17 · 03.09.2012 1 phase-field simulation of solid statefield...
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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
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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
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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
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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.
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8
12
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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
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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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www.OpenPhase.de
Precipitation of γ’ in a Ni-Al alloy
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