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.