test of darken’s mobility assumption in au-ni & cu-ni-zn ...cecamp/tms2005/tms05... · oak...

OAK RIDGE NATIONAL LABORATORYU. S. DEPARTMENT OF ENERGY

Test of Darken’s Mobility Assumption in Au-Ni & Cu-Ni-Zn Systems

Nagraj Kulkarni: Oak Ridge National Labs.Robert T. DeHoff: University of Florida

Multicomponent Multiphase Diffusion Symposium in Honor of John E. Morral: Session II

2005 TMS Annual MeetingSan Francisco, Feb. 14, 2005

Overview

1. Introduction: Diffusion Formalisms & Simplifications

2. Intrinsic Diffusion Simulation- Model System

3. Tests of Darken Relations- A: Binary : Au-Ni- B: Ternary: Cu-Ni-Zn

4. Conclusions5. Comments on development of multicomponent

diffusion databases

1. Introduction

• Why is diffusion modeling needed?− Ability to use thermodynamic, diffusion, property databases to

predict microstructural evolutionary processes• Experimental studies: expensive, time-consuming, limited

variability− Assess existing diffusion formalisms (many indeed!)

• Examine inherent assumptions & simplifications− Examine newer formalisms and understand mechanisms

• How is diffusion modeling done?− Atomistic approaches

• Monte Carlo, molecular dynamics, phase field, etc.− Finite difference methods

• Solve Fick’s second law for appropriate boundary conditions and geometry

• Use intrinsic flux expression for different time intervals (here)

Diffusion Formalisms: A. Interdiffusion

Isothermal, isobaric system

Multicomponent version of Fick’s law

There are (n - 1) (n - 1) = (n - 1)2 coefficients

Fluxes defined in the lattice frame

B. Intrinsic Diffusion Formalism

Multicomponent version of Darken’s equation

There are n (n - 1) coefficients

Simplified Version of Formalism Intrinsic flux expression that ignores cross-terms

Mk is known as the “mobility” [Darken 1948]− One unique mobility for each component in a

multicomponent system

Multicomponent Darken theory connects mobilities with tracer diffusion coefficients

Diffusion coefficients in other frames can be obtained from mobilities (or Lkk’s) by suitable transformations

Not proven that Lkk’s or Mk’s are unique functions of composition− Need intrinsic measurements in diffusion couples with

crossing composition paths

Simplified Formalism Questions(used by DICTRA)

Darken relations in binary systems not systematically tested with experimental data− Tests in ternary systems where tracer diffusion

information is available are rare

Objectives

• Develop a versatile simulation capable of treating diffusion for different formalisms

• Test internal consistency of simulation with model systems

• Develop systematic procedure for the assessment of Darken theories− Binary: Ag-Cd, Au-Ni, Cu-Zn, Cu-Ni− Ternary: Cu-Ni-Zn

1. Initial concentration distribution for each component2. Depending upon the formalism, kinetic descriptors as a function

of composition, temperature or other variables3. Thermodynamics of phase, if needed4. Molar volume as a function of composition

Finite difference method for semi-infinite couples

• Simple yet effective algorithm

• Handles variable molar volumes

• Developed in MathCad – user-friendly

• Efficient – practical output for a single diffusion couple in less than a minute

• Versatile – adaptable to various formalisms

2. Single-Phase, Intrinsic Diffusion Simulation for Multicomponent Systems

R = 5Do = 10-10 cm2/s

Model System with Variable Molar Volume• Do = DV = constant: Error function solution for ck (x,t)

•Ratio of intrinsic diffusion coefficients constant:

Simulation Output for Model System

Concentration profiles ck (mol/cc)

Atom fraction Xk = ck V

t = 12 x 3600 s

Lattice velocity (vK)

Velocity in number frame (vN)

Phase diagram Molar volume dependence on composition (almost ideal mixing)

• Classical system often quoted for the success of the Darken relation between the tracer and interdiffusion coefficient

3. A: Tests of Darken Relations in Au-Ni at 900oC

Darken Au-Ni (contd.)

Experimental intrinsic diffusion coefficients[Van Dal et al. [J. Alloys & Comp., 2000]

Interdiffusion coefficient

• Polynomial functions were used to fit the intrinsic diffusion data

Darken Au-Ni: Assessing Experimental Data

Concentration profile (6000 iterations) Lattice shift profile

Quality of experimental intrinsic data appears excellent

Au-Ni, 900oC, 100 h

x (µm)

-200 0 200 400 600 800

Latti

ce s

hift

(µm

)

0

20

40

60

80

100

xK

AuNi

Au-Ni, 900oC, 50 h

x / ot (µm/oh)

-80 -60 -40 -20 0 20

Atom

Fra

ctio

n N

i

0.0

0.2

0.4

0.6

0.8

1.0

data of Dal et al. [Dal00a]simulation

xK

XNi

Au Ni

Darken Au-Ni: Tracer Diffusion

Tracer data for Au [Kurtz et al. , Acta Met., 1955] Tracer diffusion coefficients of Au and Ni

1/T (10-4) K-1

7 8 9 10

D* Au

(cm

2 /s)

10-11

10-10

10-9

10-8

100% Ni 80% Ni65% Ni 50% Ni35% Ni20% Ni0% Ni

100

8065

5035

20

0

0

50

Atom fraction (XNi)0.0 0.2 0.4 0.6 0.8 1.0

Trac

er d

iffus

ion

coef

ficie

nt (c

m2 /s

)

10-12

10-11

10-10

10-9

D*Au (Kurtz et al.) D*Ni (Reynolds et al.) D*Au - fit D*Ni - fit fit by Reynolds et al. fit by Reynolds et al.

D*Au

D*Ni

Darken Au-Ni: Thermodynamics

Redlich-Kister polynomial fits for activity data Thermodynamic factor (M)

Thermodynamic factor is less than unity

Au-Ni, 900oC

Atom fraction (XNi)

0.0 0.2 0.4 0.6 0.8 1.0

Activ

ity (a

Ni)

0.0

0.2

0.4

0.6

0.8

1.0

R-K fit to Bienzle's dataBienzle et al. [Bie95] R-K fit to Sellars and Maak's dataSellars and Maak [Sel66]Seigle et al. [Sei52]

Atom fraction (XNi)

0.0 0.2 0.4 0.6 0.8 1.0

Ther

mod

ynam

ic fa

ctor

(Φ)

0.0

0.2

0.4

0.6

0.8

1.0

Bienzle et al. [Bie95]Sellars and Maak [Sel66]Reynolds et al. [Rey57]

Darken Relations for Au-Ni: Predicted Intrinsic Diffusion Coefficients

Darken: DAu

Problem with predicted Au intrinsic diffusion coefficient

Au-Ni, 900oC

Atom fraction nickel (XNi)0.0 0.2 0.4 0.6 0.8 1.0

Au

diffu

sion

coe

ffici

ent (

cm2 /s

)

10-13

10-12

10-11

10-10

10-9

10-8

DAu (van Dal)DAu

Dark (Φ : Sellars)DAu

Dark (Φ : Bienzle)DAu

Dark by Reynolds (Φ : Seigle)DAu* (Kurtz)

Atom fraction nickel (XNi)0.0 0.2 0.4 0.6 0.8 1.0

Ni d

iffus

ion

coef

ficie

nt (c

m2 /s

)10-13

10-12

10-11

10-10

10-9

10-8

DNi (van Dal)DNi

Dark (Φ : Sellars)DNi

Dark (Φ : Bienzle)DNi

Dark by Reynolds (Φ : Seigle)DNi* (Reynolds)

Darken: DNi

Darken Relations for Au-Ni: Predicted Interdiffusion Coefficient

Darken: Do

Predicted interdiffusion coefficient reasonably close to data

Au-Ni, 900oC

Atom Fraction Nickel (XNi)0.0 0.2 0.4 0.6 0.8 1.0

Inte

rdiff

usio

n C

oeffi

cien

t (cm

2 /s)

10-13

10-12

10-11

10-10

10-9

10-8

van DalReynoldsDAu* DNi* Do Dark (Φ: Sellars)Do Dark (Φ: Bienzle)Do Dark by Reynolds (Φ : Seigle)

Do

Assessing Darken Relations for Au-Ni with Simulation

Concentration profile Lattice shift profile

Differences between predicted and experimental DAu results in large differences in lattice shift profiles

Au-Ni, 900oC

x / ot (µm/oh)

-80 -60 -40 -20 0 20

Atom

Fra

ctio

n N

i

0.0

0.2

0.4

0.6

0.8

1.0

experiment (Dal)intrinsic data (Dal)Darken (Φ: Sellars)Kirkendall location for intrinsic data

xK

XNi

Au Ni

Au-Ni, 900oC, 100 h

x (µm)

-200 0 200 400 600 800

Latti

ce s

hift

(µm

)

0

20

40

60

80

100

experiment (Dal)intrinsic data (Dal)Darken (Φ: Sellars)

xK

AuNi

3. B: Darken Relations in Cu-Ni-Zn at 900oC

Intrinsic fluxes in a multicomponent system (i = 1, 2, … , c):

Ji Lii−x

µidd

⋅ ci− Mi⋅x

µidd

⋅DtiR T⋅

− ci⋅x

µidd

⋅ Dti−XiV

⋅xln ai( )d

d⋅

Dti = tracer diffusion coefficient of component iµi = chemical potential of i; ai = activity of iV = molar volume; Xi = mole fraction of i

Cserhati, Ugaste, van Loo et al., Def. & Diff. For., 2001

Φ ij ln Xj( )ln ai( )d

d

Φij = thermodynamic factors obtained from Gibbs free energy of phase

JiDti−

VXi⋅

Φ iiXi x

Xidd

⋅

1 i≠

c 1−

j

Φ ijXj x

Xjdd

⋅∑=

+

⋅

Darken Relations in Cu-Ni-Zn: contd.

Intrinsic fluxes in a ternary:

Relation between thermodynamic factors

J 1Dt 1−

VX 1⋅

Φ 11X 1 x

X 1dd

⋅Φ 12X 2 x

X 2dd

⋅+

⋅

J 2Dt 2−

VX 2⋅

Φ 22X 2 x

X 2dd

⋅Φ 21X 1 x

X 1dd

⋅+

⋅

J 3Dt 3V

X 1Φ 11X 1 x

X 1dd

⋅Φ 12X 2 x

X 2dd

⋅+

⋅ X 2

Φ 22X 2 x

X 2dd

⋅Φ 21X 1 x

X 1dd

⋅+

⋅+

⋅

Φ 21X 11 X 1−( )

1 X 2−

X 2Φ 12⋅ Φ 22+ Φ 11−

⋅

Cu-Ni-Zn: Thermo Jiang, Ishida et al., J. Phys. Chem. Sol., 2004, in print

Isoactivity curves of Zn in fcc-CuNiZn at 727oC(liq. Zn ref.)

Cu-Ni-Zn: Thermodynamics contd.

∆G_α mix_xs X 1 X 2⋅ L_α 12 X 1 X 2, T,( )⋅ X 1 X 3⋅ L_α 13 X 1 X 3, T,( )⋅+ X 2 X 3⋅ L_α 23 X 2 X 3, T,( )⋅+X 1 X 2⋅ X 3⋅ L_α 123 X 1 X 2, X 3, T,( )⋅+

...

L_α 123 X 1 X 2, X 3, T,( ) X 1 L_α0 123 T( )⋅ X 2 L_α1 123 T( )⋅+ X 3 L_α2 123 T( )⋅+

L_α 12 X 1 X 2, T,( ) L_α0 12 T( ) L_α1 12 T( ) X 1 X 2−( )1⋅+ L_α2 12 T( ) X 1 X 2−( )2⋅+

Redlich-Kister model for Gibbs excess free energy:

∆Gxsk ∆G mix_xs Xk∆G mix_xs

dd

+

1

c

j

Xj Xj∆G mix_xs

dd

⋅∑=

−

Partial molal Gibbs free energy:

-11.0

-10.5

-10.0

-9.5

-9.0

-8.5

-8.0

05

1015

2025

3035

05

1015

2025

3035

log(DC

u ) (cm2/s)

XNi (At.%)

X Zn (A

t.%)

Cu-Ni-Zn: Tracer diffusion

Anusavice & DeHoff, Met. Trans. A., 1972log Dt Cu( ) 3.53− X Ni

1.16⋅ 3.6 X Zn1.02⋅+ 9.46−

log Dt Ni( ) 4.05− X Ni1.3⋅ 3.28 X Zn

1.07⋅+ 9.96−

log Dt Zn( ) 3.2− X Ni⋅ 5.21 X Zn1.25⋅+ 9−

D*Zn > D*

Cu > D*Ni

Similar composition dependence

Ni0 10 20 30 40

Zn

Cu

70

80

90

100

Wan experimentalDarken simulation

Darken relations in Cu-Ni-Zn: Composition paths at 900oC

#4,5,7,9

Wan, Ph.D. Thesis, Univ. of Florida, 1972Wan & DeHoff, Acta Met., 1977

Cu-Ni-Zn: Concentration profilesCouple #9 (Wan 13-00)

x (µm)

-800 -600 -400 -200 0 200 400 600 800

Atom

%

0

20

40

60

80

100

Experiment (Wan 895oC)Darken simulationDarken: Kirkendall composition

Cu

Zn

Ni

xK


x (µm)

-600 -400 -200 0 200 400 600 800 1000

Atom

%

0

20

40

60

80

100

Experiment (Wan 897oC)Darken simulationDarken: Kirkendall composition

Cu

Zn

Ni

xK

Ni0 10 20 30 40

Zn

Cu

70

80

90

100


Composition paths Cu-Ni-Zn contd.

#6,8,11


x (µm)

-600 -400 -200 0 200 400 600 800

Atom

%

0

20

40

60

80

100

Experiment (Wan)Darken simulationDarken: Kirkendall composition

Cu

Zn

Ni

xK

Ni0 10 20 30 40

Zn

Cu

70

80

90

100


Composition paths Cu-Ni-Zn contd.

#1,2,3

Cu-Ni-Zn: Concentration profiles

Couple #2 (Wan 33-10)

x (µm)

-600 -400 -200 0 200 400 600 800

Atom

%

0

20

40

60

80

100

Experiment (Wan 900oC)Darken simulation

Cu

Zn

Ni

xK

Ni

Zn

Cu

Kirkendall Shift Comparison

Kirkendall shifts

Experimental (Wan)0 20 40 60 80 100

Pre

dict

ed (D

arke

n)

0

20

40

60

80

100

4. Conclusions• In the binary Au-Ni and the ternary Cu-Ni-Zn system, Darken

relations result in− Reasonable concentration profiles; however− Kirkendall shifts are noticeably underestimated

Manning relations in Cu-Ni-Zn to be explored

• Intrinsic simulation powerful tool for testing quality of experimental data & various diffusion formalisms with their associated assumptions− Need to explore intermetallic phases with broad solubility

ranges where vacancy concentrations & fluxes are likely to be important

Alternate approachA kinetic approach based on tracer jump frequencies has also been found to have the same level of success in predicting diffusion paths in Cu-Ni-Zn using the same simulation

However the Kirkendall shifts are overestimatedBut no thermodynamics necessary; tracer diffusion coefficients sufficient

5. On development of multicomponentdiffusion databases

• Tracer diffusion data vital for high-quality databases

• More experimental efforts in this direction needed− But need not use radioactive isotopes− Stable isotopes (except Al) combined with mass

spectrometry may be used instead (errors?)*

• Explore use of combinatorial techniques for sample preparation and automated analysis of tracer (stable) isotope profiles

* Schmidt, Schmalzried et al., “Self-diffusion of B in TiB2,” J. App. Phys., 2003

John Morral’s other contributions…(thermodynamics, oxidation, solidification,

phase diagrams, coarsening, etc.)• Morral & Davies, “Thermodynamics of Isolated Miscibility Gaps,” J. Chimie Phys.

Physico-Chimie Bio, 1997.• Morral & Purdy, “Thermodynamics of Particle Coarsening,” J. Alloys & Comp.,

1995.• Hennessey & Morral, “Oxidation & Nitridation of Niobium in Air above 1150oC,”

Oxid. Met., 1992. • Morral & Gupta, “Phase-boundary, ZPF, and Topological Lines on Phase

Diagrams,” Scripta Met., 1991.• Gupta, Morral & Nowotny, “Constructing Multicomponent Phase Diagrams by

Overlapping ZPF Lines,” Scripta Met., 1986.• Pati & Morral, “The Effect of Quenching on Mushy Zone Microstructures,” Met.

Trans. A, 1986.• Cameron & Morral, “The Solubility of Boron in Iron,” Met. Trans. A, 1986.• Ohriner & Morral, “Precipitate Distribution in Sub-Scales,” Scripta Met., 1979.• Morral & Asby, “Dislocated Cellular Structures,” Acta Met., 1974.• Morral & Cahn, “Spinodal Decomposition in Ternary Systems,” Acta Met., 1971.• Morral, “Characterizing Stability Limits in Ternary Systems,” Acta Met., 1972.

… and many more

test of darken’s mobility assumption in au-ni & cu-ni-zn ...cecamp/tms2005/tms05... · oak...

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