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Modeling on Diffusion in Ni-base Superalloy
Tao Wang, Long-Qing Chen and Zi-Kui Liu
The Penn State University
April 19th, 2005
Microstructure Evolution of Ni-base Alloys
A real turbine blade made of Ni-base superalloys, a γ (fcc) and γ’ (L12) two phase microstructure from a phase-field simulation, and a schematic phase diagram for the γ and γ’two-phase field.
Evolution Equations in Phase-field Model
⎥⎦
⎤⎢⎣
⎡∂
∂∇∇=⎥
⎦
⎤⎢⎣
⎡∇∇=
∂∂
),(),(),(
tcfM
tcFM
ttc
mim
mim
i
rrr
δδ
Thermodynamic Database
Atomic Mobility Database
Lattice Parameter Database
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂−∇−=−=
∂
∂
),(),(
),(),( 22
tftL
tFL
tt
nnjn
njn
j
rr
rr
φφε
δφδφ
M is diffusion mobility and L is phase field mobility.
Modeling on Atomic Mobility
Ordered Phases:ordi
disii GGG ∆+∆=∆
∑∑≠
−∆=∆j jk
kjkjorderijk
ordi xxyyGG ][ βα
)lnexp(1)exp(00
RTMRTQ
RTRTQ
RTMM iiii
i+−
=−
=
)exp(1RT
GRT
M ii
∆−=
∑∑∑>
∆+∆=∆j jk
kjikj
l
lili GxxGxG ,
A-B binary system:
M: diffusion mobility in phase field
( )ABBABA MxMxxxM +=
Andersson J and Agren J, J Appl Phys, 72 (1992) 1350Helander T and Agren J, Acta Mater, 47 (1999) 1141
Thermodynamic Database
1473K
S. Zhou et al., in submission, (2005)
Diffusivity in Ni-Al fcc Phase
Ni-Al
A. Engstrom, J. Agren, Z.Metallkd., 97 (1996) 92
Diffusivity in fcc Phase
Ni-Mo Al-Mo
Diffusivity in Stoichiometric Compound (Assessment I)
Diffusivity vs Composition(Assessment I)
Diffusion Mechanism in L12
Major element (Ni)intra-lattice mechanism
Minor element (Al)six-jump cycle mechanismanti-site mechanismanti-site bridge mechanism
Diffusivity of Al (1473K)
Diffusivity of Al (973K)
Atomic Description for Diffusivity
Ni in the fcc Ni Matrix:
Ni in the Ni3Al Matrix:
Al in the fcc Ni Matrix:
Al in the Ni3Al Matrix:
NiNiNiVNi
NiNi fCaD 0
2 ω=
AlNiAlNiAlNiVAlNi
AlNiNi fCaD 333
3
30
2
32 ω=
NiNiNi
NiNi
VNiNiAl fCaD 2
3
42 ωωω
=
NiAl
AlNiAlNiAlNi
AlNiAlNi
VAlNiAlNi
Al PfCaD 33
3
33
3
32
3
42
32 ω
ωω
=
Numerical Treatment
: anti-site factor.
: reflects the impurity-vacancy and impurity-matrix interactions, assumed to be equal for the diffusion of Al in Ni3Al and fcc Ni.
NiAlAlNi
Ni
NiNi
NiAl
AlNiAl P
DD
DD
≈3
3
)/)(/( 0234 ωωωω
NiAlP
Al
NiAlNi
Al xy
P =
H. Numakura, et al., Phil. Mag. A, 77 (1998) 887
Dictra Modeling on Diffusivity
Ni in the fcc Ni Matrix:
Ni in the Ni3Al Matrix:
Al in the fcc Ni Matrix:
Al in the Ni3Al Matrix:
)exp(RTGD
disNidis
Ni∆−
=
)exp(RTGD
disAldis
Al∆−
=
)exp(3
RTGGD
ordAl
disAlAlNi
Al∆+∆
−=
)exp(3
RTGGD
ordNi
disNiAlNi
Ni∆+∆
−=
Numerical Treatment II
Where , which can be calculated from the mobility descriptions of the related disordered phase.
)'exp(3
3
RTGGG
DD
DD ord
NiordAl
AlNiNi
NiNi
NiAl
AlNiAl ∆+∆−∆
−=
disNi
NiAl
NiNi
disAl GGGGG ∆−∆−∆+∆=∆ '
Constraint from Diffusion Mechanism
NiAlAlNi
Ni
NiNi
NiAl
AlNiAl P
DD
DD
=3
3
)'exp(3
3
RTGGG
DD
DD ord
NiordAl
AlNiNi
NiNi
NiAl
AlNiAl ∆+∆−∆
−=
⇓
NiAl
ordNi
ordAl P
RTGGG
=∆+∆−∆
− )'exp(
Anti-site Factor
Consider reaction:
Anti-site factor:)( )()()1( 25.075.0325.075.023 fccAlNiAlNiLAlNi ↔
)exp(RTGf
xyP
react
fccAl
NiAlNi
Al∆
−≈≈=
reactordNi
ordAl GGGG ∆=∆+∆−∆ '
Diffusivity in Stoichiometric Compound (Assessment II)
Diffusivity vs Composition(Assessment II)
Optimized Parameters for Ni3Al
ordAlNiAlG∆ ord
NiNiAlG∆ ordAlAlNiG∆ ord
NiAlNiG∆
Assessment I 1.8804e+05 -9.3253e+04 6.0418e+05 -5.8868e+05Assessment II 1.2499e+05 -7.8982e+04 3.6406e+05 -5.3770e+05
Units: J/mole
Diffusion in L12 of the Ni-Al-Mo System
Since Mo prefers to occupy Al-sites in Ni3Al ordered phase, we assume:The effect of Al-Mo ordering can be ignored.The diffusion in the hypothetical Ni3Mo L12ordered phase is similar to that in the Ni3Al phase.The diffusion of Mo in the L12 ordered phase is similar to that of Al.
Diffusivity of Mo in Ni3Al
Phase Field Simulation vs Experimental Investigation
Alloy: Ni-7.7 at.%Al-7.9 at.%MoSystem size: 1024nm×1024nm; Temperature: 1048K; Annealing time: 67hLeft: TEM images; Right: 2D phase-field simulations
Summary
The atomic mobility modeling of Ni-Mo and Al-Mo fcc phases has been carried out based on the available experimental data. By combining them with previous work on Ni-Al, the mobility database for the fcc phase of the Ni-Al-Mo system has been developed.The effect of chemical ordering on atomic mobility is described by a phenomenological model. The available experimental data for Ni3Al are used to evaluate the model parameters.The anti-site mechanism is found to be dominant for Al diffusion in L12. The atomic mobility modeling is then refined based on the anti-site mechanism.Atomic mobility in the L12 phase of Ni-Al-Mo system is evaluated from the experimental information in the literature.Using the above atomic mobility in the fcc and L12 phases, phase field simulations for Ni-Al-Mo alloys have been performed.