Impurity And Interdiffusion In The Magnesium-aluminum System
Phase Field Modeling of Interdiffusion Microstructures K. Wu, J. E. Morral and Y. Wang Department of...
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Transcript of Phase Field Modeling of Interdiffusion Microstructures K. Wu, J. E. Morral and Y. Wang Department of...
Phase Field Modeling of Interdiffusion Microstructures
K. Wu, J. E. Morral and Y. Wang
Department of Materials Science and EngineeringThe Ohio State University
Work Supported by NSFNIST Diffusion Workshop
September 16-17, 2003, Gaithersburg, MD
F-22
Ni-Al-Cr
Courtesy of M. Walter
Examples of Interdiffusion Microstructures
0.5 m
Disk alloy (courtesy of M.F. Henry)
Turbine blade and disk
Examples of Interdiffusion Microstructures
Pb and Pb-free solders
Co2Si/Zn diffusion couple
SnPb/Cu SnBi/Cu
SnAg/Cu Sn/Cu (HK Kim and KN Tu)
(MR Rijnders and FJJ Van Loo)
Challenges in Modeling Interdiffusion Microstructures
• Multi-component, multi-phase, multi-variant and polycrystalline
• Very complex microstructural features:– high volume fraction of precipitates– non-spherical shape and strong spatial correlation– elastic interactions among precipitates
• Interdiffusion induces both microstructure and phase instabilities.
• Effect of concentration gradient on nucleation, growth and coarsening.
• Roles of defects and coherency/thermal stress on interdiffusion and phase transformation.
• Robustness and computational efficiency of models.
Methods Available
•One-dimensional diffusion in a common matrix phase
•Precipitates are treated as point sources or sinks of solute
•Mutual interactions between microstructure and interdiffusion and corresponding effects on diffusion path and microstructural evolution are ignored
DICTRA
ErrorFunction
Limitations
You don’t see the microstructure!
AdvantagesSimple and efficient: Error Function solutions are close forms and Dictra is computationally efficient.
Advanced Microstructure Modeling
using Phase Field Method
2 m1 m
Y. Jin et. al.
QuickTime™ and aPNG decompressor
are needed to see this picture.
∂c r , t( )
∂t= ∇ D∇
δF
δ c r, t( )
⎛
⎝ ⎜
⎞
⎠ ⎟ + ξ c r, t( )
∂η p r, t( )
∂t= − L
δF
δ ηp
r, t( )+ ξ p r, t( )
M
QuickTime™ and aBMP decompressor
are needed to see this picture.
The phase field method handles well arbitrary microstructures consisting of diffusionally and elastically interacting particles and defects of high volume fraction and accounts self-consistently for topological changes such as particle coalescence.
The phase field method handles well arbitrary microstructures consisting of diffusionally and elastically interacting particles and defects of high volume fraction and accounts self-consistently for topological changes such as particle coalescence.
0.5 m
Study Interdiffusion Microstructures
• Link to multi-component diffusion and alloy thermodynamic databases
• Break the intrinsic length scale limit• Interaction between microstructure and
interdiffusion processe:- Motion of Kirkendall markers, second phase
particles and Type 0 boundary, non-linear diffusion path
- Interdiffusion microstructure in Ni-Al-Cr diffusion couples
Linking to Thermo. and Kinetic Databases
Kinetic description
ExperimentalDiffusivity DataExperimental
Diffusivity Data
ExperimentalTD Data
ExperimentalTD Data
MobilityDatabase Mobility
Database DICTRA DICTRA
TDDatabase
TDDatabase
Thermo_Calc
or PANDAT Thermo_Calc
or PANDAT
Chemical Diffusivity of Ti
Thermodynamic description
Phase Field
Atomistic CalculationsAtomistic
Calculations
Atomistic CalculationsAtomistic
Calculations
CAMM
Elastic PropertiesElastic
Properties
Interface Properties
Interface Properties
1173K
DICTRA
Optimizer
Linking to CALPHAD Method for Fee Energy
How to construct non-equilibrium free energy from equilibrium properties?
f
c
Δfmax
c
f
€
f = f c,η( )
∂f∂η
=0 η =η0 c( )
feq= f c,η0 c( )[ ]
= 0
=
Construction of Non-Equilibrium Free Energy
f (Xi,T,η) = fγ (Xi,T)+Δf(Xi,T,η)
Δf(Xi,T,η)=
A1(Xi,T)2
η2 −A2
4η4 +
A3
6η6
0
0
==∂Δf Xi,T,η( )
∂η=0
f (Xi,T,0) =f γ(Xi,T)
f (Xi,T,η0) =f β (Xi,T)
Equilibrium free energy cannot be incorporated directly in the expression. A1, A2, and A3 need to be fitted to thermodynamic data at different temperature in the whole composition range. It is difficult to do for multicomponent systems.
Indirect Coupling: Landau polynomial expansion for the free energy:
Construction of Non-Equilibrium Free Energy
Equilibrium free energy can be incorporated directly into the non-equilibrium free energy formulation for any multi-component system without any extra effort.
f (Xi,T,η) = fγ (Xi,T)+p(η) f β (Xi,T)−f γ(Xi,T)[ ]+q(η)
)61510()( 23 +−=p 22 )1()( ω −=q
1
0
==
∂f Xi,T,η( )
∂η=0
f (Xi,T, 0) =f γ(Xi,T)
f (Xi,T, 1) =f β (Xi,T)o
Direct Coupling: following the same approach as phase field modeling of solidification (e.g.,WBM or S-SK model)
Wang et. al, Physica D,1993; 69:189
Application I: Model System
• Elements A and B form ideal solution while elements A and C or B and C form regular solutions
A B
C
’
Acta Mater., 49(2001), 3401-3408
Free energy model
Gm =RT XA lnXA +XB lnXB +XC lnXc( )+I XAXC +XBXC( )
Wu et. al. Acta mater. 2001;49:3401Wu et. al. Acta mater, preprint.
Phase Field Equations
∂XB
∂t=∇ M11∇ μB −μA( )[ ]+∇ M12∇ μC −μA( )[ ]
∂XC
∂t=∇ M21∇ μB −μA( )[ ]+∇ M22∇ μC −μA( )[ ]
μB −μA =μBB −μA
B −2κ11∇2XB −2κ12∇
2XC
μC −μA =μCB −μA
B −2κ21∇2XB −2κ22∇
2XC
Mij - chemical mobilities
ij - gradient coefficients
I - atomic mobilities
- molar density
Diffusion equations
Thermodynamic parameters
Kinetics parametersM11 =ρXB 1−XB( )2βB +XBXCβC +XBXAβA[ ]
M12 =M21=ρXBXC −1−XB( )βB − 1−XC( )βC +XAβA[ ]
M22 =ρXC XBXCβB + 1−XC( )2βC +XCXAβA[ ]
= 0
= 100
= 2000
Interaction between Microstructure and Interdiffusion
4608x64 size simulation, 1024x256 size output
1=1.0 2=5.0 3=10.0
• Ppt and Type 0 boundary migrate as a results of Kirkendall effect
• Type 0 boundary becomes diffuse
• Kirkendall markers move along curved path and marker plane bends around precipitates
• Diffusion path differs significantly from 1D calcul.
• Ppt and Type 0 boundary migrate as a results of Kirkendall effect
• Type 0 boundary becomes diffuse
• Kirkendall markers move along curved path and marker plane bends around precipitates
• Diffusion path differs significantly from 1D calcul.
Size and position changes during
interdiffusion
Diffusion path: comparison with 1D simulation
Wu et. al. Acta mater. 2001;49:3401Wu et. al. Acta mater, preprint.
Diffusion Couple of Different Matrix
Phases
=0.0
=2000.0
B=10.0 C=5.0 A=1.0
XB = 25 at%, XC = 40 at% XB = 32 at%, XC = 60 at%
JA
JB
net flux of (A+B)
• Kirkendall Effect: Boundary migrates and particles drift due to difference in atomic mobility
• Moving direction opposite to the net flux of A+B
• Non-linear diffusion path penetrating into single phase regions
• Kirkendall Effect: Boundary migrates and particles drift due to difference in atomic mobility
• Moving direction opposite to the net flux of A+B
• Non-linear diffusion path penetrating into single phase regions
<<<
Effect of Mobility Variation in Different Phases
= 0
M:1=1.0 2=5.0 3=10.0Ppt1=1.0 2=5.0 3=5.0
Exp. Observation by Nesbitt and Heckel
Interdiffusion Microstructure in NI-Al-Cr Diffusion Couple
0 hr
4 hr
25 hr
320m
100 hrat 1200oC
Ni-Al-Cr at 1200oC
• Free energy data from Huang and Chang
• Mobilities in from A.EngstrÖm and J.Ågren
• Diffusivities in from Hopfe, Son, Morral and Roming
• Free energy data from Huang and Chang
• Mobilities in from A.EngstrÖm and J.Ågren
• Diffusivities in from Hopfe, Son, Morral and Roming
200m
XCr =0.25, XAl=0.001
Annealing time: 25 hours
(a)
(b)
(c)
Effect of Cr content on interface migration Effect of Cr content on interface migration
320m
Ni-Al-Cr at 1200oC
(d)
b ca d
Exp. measurement by Nesbitt and Heckel
Diffusion path and recess rate - comparison with experiment
Diffusion path and recess rate - comparison with experiment
Annealing time: 25 hours
(a)
(b)
(c)
Effect of Al content on interface migration
Effect of Al content on interface migration
320m
a
cb
c
Annealing time: 25 hours
(a)
(b)
Effect of Al content on interface migration
Effect of Al content on interface migration
320m
ab
Diffusion Path in the Two-Phase Region
t = 0
t = 25h
t = 100h
A.EngstrÖm, J. E. Morral and J.ÅgrenActa mater. 1997
500m
t = 0
t = 25h
t = 100h
Diffusion Path - Comparison with DICTRA
Two-Phase/Two-Phase Diffusion Couplet = 0
10h
35h
Two-Phase/Two-Phase Diffusion Couple
t = 0
t = 100h
t = 200h
Summary
• Computational methods based on phase field approach and linking to CALPHAD and DICTRA databases have been developed to investigate for the first time effects of microstructure on interdiffusion and interdiffusion on microstructural evolution in the interdiffusion zone.
• Realistic simulations of complicated interdiffusion microstructures in multi-component and multi-phase diffusion couples have been demonstrated.
• Many non-trivial results have been predicted, which are significantly different from earlier work based on 1D models.