Motivation Multicomponent Diffusion Basics Development and Structure of a Diffusion Mobility Database Validation of Database Applications using Diffusion Mobility Databases

Development and Application of Composition-Dependent Diffusion

Databases: Solid State Diffusion

Carelyn E. CampbellNational Institute of Standards and Technology

Metallurgy Division

Gaithersburg, MD 20899

Constructing Kinetic Databases

April 20, 2004

This work was partially funded by the GE-led DARPA AIM program

Types of Problems of Interest: Diffusion Controlled

Protective coating applications

Prevent diffusion during extended

service times at high temperatures.

' precipitation during non-isothermal cooling

Cooling

Heat treatmentoptimization

Distance across dendrite

Co

mp

os

itio

nC

om

po

sit

ion

Distance across dendrite

Heating

Goal: Improve processing and properties, while reducing time and cost.

Heating

Bonding applications

L

Diffusion in Binary Alloys

RT

QDD exp0

x

cD

xdt

c

dx

dcDJ

Fick’s first law for Flux, J

2

2

x

cD

t

c

Fick’s second law (conservation of mass)

If D is independent of composition

Simple solutions

Position, x

Con

cent

ratio

n, c

Heat

xDt

BAtxc

Dt

xBerfAtxc

Dt

x

Dt

BAtxc

2sin

4exp),(

2),(

2exp),(

2

2

2

What is needed to simulate multicomponent diffusion?

René-N4 René-N5

40.370.083.022.160.055.062.081.1

41.562.2357.625.694.494.435.163.1

23.050.076.026.024.053.033.068.0

31.085.074.005.2427.066.025.031.0

45.017.036.055.057.7426.0280.033.8

49.225.891.993.821.367.1337.526.4

69.910.783.155.567.525.800.1737.11

22.5351.4950.5143.4234.3483.3493.135.119

W

Ti

Ta

Nb

Mo

Cr

Co

Al

WTiTaNbMoCrCoAl

René-N4 (x10-14 m2/s) at 1293 °C

Diffusion matrix

At each grid point for each time step for a given temperature profile.

Thermodynamics

+L L

++P+B2+

+P

++B2

+B2

+B2+L

B2+L

+P+B2

René-N4

Need efficient data storage

Atomic Mobility

Activated stateInitial state Final state

Fre

e e

ne

rgy

of

the

latt

ice

(I) (A) (F)

*kVaQ

vacanciesoffraction site where

exp*

Va

kVaVa

y

kT

Qy

The probability for a constituent to jump to a neighboring vacant lattice site equals

Note: Tracer diffusivity is given by where and is not dependent on the thermodynamics

To describe the flux of atoms in the presence of a driving force (composition gradient) and assuming the vacancy concentration is in thermodynamic equilibrium:

zMyc

kT

z

kT

QyccJ k

kVaVakkkVa

Vakkklatticek

*

2 exp

Lattice fixed frame of reference

latticevacancies

n

k

latticek JJ

1

0

Volume fixed frame of reference

n

k

VkJ

1

0

1

1

~n

j

jnkj

Vk z

cDJ

knkjnkj DDD ~

m

n

i j

iiikikkj V

xMxxD

1

Chemical (interdiffusion coefficient)

kk RTMD *

RTRT

QM kVa

k

1exp

*2

where

Diffusion in Multicomponent Solids

*References: Borgenstam et. al., J. Phase Equilibria, 21 (2000) 269. Andersson and Ågren, J. Appl. Phys. 72 (1992) 1350.Ågren, J. Phys. Chem. Solids, 43 (1982) 421. Ågren, J. Phys. Chem. Solids, 43 (1982) 385.

Note that tracer diffusivity is not dependent on thermodynamics.

kk RTMD *

0

5 10-19

1 10-18

1.5 10-18

2 10-18

0 0.2 0.4 0.6 0.8 1

Mob

ility

Mole Fraction Ni

NiFe

T = 1200 oC

Ni

Fe

x

Ni

Ni

x

Th

erm

od

yna

mic

fa

cto

r

-2 105

-1 105

0

1 105

2 105

0 0.2 0.4 0.6 0.8 1Mole Fraction Ni

T = 1200 oC

0

5 10-15

1 10-14

1.5 10-14

2 10-14

2.5 10-14

3 10-14

0 0.2 0.4 0.6 0.8 1

Diff

usi

vity

(m

2/s

)

Mole Fraction Ni

D~

NiD

FeD

Fe Ni

FeNiNiFe DxDxD ~

Darken’s equation

Fe-Ni: T = 1200 °C

Concentration Dependent Diffusion Coefficients

Diffusion Database Development

Inputs:– Thermodynamics (CALPHAD approach) – Diffusion experiments (unary, binary, ternary systems)

• Tracer diffusivity, • Intrinsic diffusivity, • Interdiffusion coefficients/Marker motion

Optimize value of mobilities, Mi , for all binaries consistent with available data– Composition and Temperature-dependent

– Consistent with estimates of Metastable end members e.g., FCC W

– Optimized using code, DICTRA (Parrot)

, whereexp **

TcfQRT

Q

RT

MM ii

iii

0iM is exponentially dependent on composition

RT

Q

RTM i

i exp1

)exp(0iiM iii RTQQ *

and

p pq qv

pqvi

sspqvvqp

n

p

n

p

n

pq

m

r

rqp

pqi

rqp

pipi BvxxxxxAxxQxQ

1 0

*

Thermodynamic AssessmentCALPHAD Approach

Experimental phase diagramThermochemical data

Determine Gibbs EnergyG= f(x,T,P

Calculated phasediagram

A B

TE

Liquid

Tem

pera

ture

Composition

L

Gib

bs

En

erg

y

Composition

A B

TE

Liquid

Tem

pera

ture

Composition

T<TE

excessideal GGGG 0

Binaries Ternaries Quaternariesnth ordersystems

Assessment of Diffusion Mobilities

Experimental

diffusion data

Distance

Com

posi

tion

Diffusion profile Diffusion Coefficient

Mobility M=f (c,T)

Composition

Lo

g (

Mo

bili

ty) T = 1150 °C

T = 1050 °C

T = 950 °C

Calculate diffusion Coefficients D = f(c,T)

-15

-14.5

-14

-13.5

-13

-12.5

-12

0 0.05 0.1 0.15 0.2

log

D (

m2/s

)

Mole Fraction Al

1000°C

1050°C

1100°C

1250°C

Data from Yamamoto et. al. (1980)Assessment by Engström and Ågren (1996)

Ni - Al

Estimate MobilitySimulate diffusion processCompare experimental

and calculated D Adjust Mobility

, whereexp TcfQRT

Q

RT

MM ii

iii

...)()( ,2,, jiiji

jiiji

jiiji

jij

iiii CccBccAccQcQcQ

For a binary:

Examples of Fits for Binary Interactions

Ni-Al-Cr-Co-Fe-Hf-Nb-Mo-Re-Ta-Ti-W (-C)

-17.5

-17

-16.5

-16

-15.5

-15

-14.5

-14

0 0.02 0.04 0.06 0.08 0.1

Ti

Ta

W

Re

-14.5

-14

-13.5

-13

-12.5

-12

0 20 40 60 80 100

Atomic Percent Ni

1400 oC

1325 oC

1250 oC

1200 oC

1160 oC

Ni-Co Interdiffusion

Lo

g (

D)

m2 /

s

Data from Ustad and Sorum, Phys. Stat. Sol. A 285 (1973) 285.

Calculated

Lo

g (

D)

m2 /

sWeight Fraction

Interdiffusion with Ni

Data from Karunaratne et al., Mater. Sci. Eng., A281 (2000) 229.Data from Komai et al., Acta. Mater., 46, (1998) 4443.

T = 1000 oC

Previous assessments: Ni-Al-Cr Engström and Ågren, Z. Metallkd. 87 (1996) 92.,

Ni-Al-Ti Matan et al., Acta mater., 46 (1998) 4587 Ni-Fe-Cr Jönsson, Z. Metallkde. 86 (1995) 686.

Ni-Fe-Cr-C Jönsson, Z. Metallkde. 85 (1994) 502.

Current assessments: Ni-Co, Ni-Hf, Ni-Mo, Ni-Nb, Ni-Re, Ni-Ta, Ni-Ti,Ni-W, Co-Cr, Co-Mo, Fe-Al, Fe-Co C. E. Campbell, W. J. Boettinger, U. R. Kattner, Acta Mat, 50 (2002) 775,

C. E. Campbell, J-C. Zhao, M. Hnery, J. Phase Equilibria & Diffusion, 25, (2004) 6.

Assessment of Ni-W

-15.5

-15

-14.5

-14

-13.5

-13

-12.5

5.8 10-4 6 10-4 6.2 10-4 6.4 10-4 6.6 10-4 6.8 10-4 7 10-4 7.2 10-4 7.4 10-4

x(w)=.017

x(w)=.053

x(w)=.092

Lo

g D

* (F

CC

,Ni)

(m2 /s

)

(1/Temperature) (K)

Ni-1.7W (at.%)

Ni-5.3W (at.%)

Ni-9.2W (at.%)

1340 1290 1242 1197 1155 1116 107813941451

Temperature ( oC)

-16

-15.5

-15

-14.5

-14

-13.5

6.2 10-4 6.4 10-4 6.6 10-4 6.8 10-4 7 10-4 7.2 10-4 7.4 10-4

x(w)=.017

x(w)=0.053

w(w)=.092

Lo

g D

* (F

CC

,W)

(m2 /s

)

(1/Temperature) (K)

Ni-1.7 W

Ni-5.3 W

Ni-9.2 W

1340 1290 1242 1197 1155 1116 1078

Temperature ( oC)

-18

-17

-16

-15

-14

-13

0 0.02 0.04 0.06 0.08 0.1

Lo

g D

(In

terd

iffu

sio

n)

m2 /s

Mass Fraction W

900 oC

1100 oC

1000 oC

1200 oC

1300 oC

WNiNiWNi

WNiW

NiNiNiNi AxxQxQxQ ,0*

WNiWWNi

WWW

NiWNiW AxxQxQxQ ,0*

Activation energies in the fcc phase

Self activation energies

Optimized parameters

Tra

cer

diff

usiv

ity d

ata

Interdiffusion data

Diffusion Correlation at Melting Temperature

18

MRT

Q For a pure metal

Element Crystal Structure

TM, K TM, K (fcc)

TM, K (fcc)

(Kaufman)

Activation

Energy

(J/mole)

-Q/RTM (fcc)

(SGTE)

-Q/RTM (fcc)

(Kaufman)

Ni fcc 1728 1728 1725 -287000 20.0 20.0

Al fcc 933.5 933.5 931 -142000 18.3 18.4

Cr bcc 2133 1475 860 -235000 19.2 32.9

Co hcp 1770 1768 1768 -286175 19.5 19.5

Hf hcp 2504 1952 2076 -235350 14.5 13.6

Mo bcc 2895 1740 1530 -254975 17.6 20.0

Nb bcc 2468 1300 1170 -274328 25.4 28.2

Re hcp 3459 3084 2830 -382950 14.9 16.3

Ta bcc 3296 1416 1540 -268253 22.8 20.9

Ti hcp 1946 900 1421 -256900 34.3 21.7

W bcc 3695 2229 2230 -311420 16.8 16.8

Comparison with Ni-Co-Cr-Mo Data at 1300 oC*

Composition (Atomic Percent)

Ni = balance

Cr Co Mo Measured Calculated

NIST Thermotech

24.2 24.1 7.4 7.5±1.5 10.2 10.7

22.7 24.5 7.4 9.7 ± 1.9 10.1 10.6

20.8 25.0 7.4 9.9 ± 2.0 9.85 10.3

18.4 25.6 7.2 10.1 ± 2.0 9.56 10.0

15.2 25.8 7.4 8.2 ± 1.6 9.35 9.74

10.8 26.2 7.4 6.9 ± 1.4 8.95 9.25

6.4 27.1 7.7 6.4 ± 1.3 8.4 8.59

3.2 47.9 7.7 6.8 ± 1.4 4.94 5.03

26.8 1.7 6.6 8.9 ± 1.8 10.3 10.3

26.5 4.4 6.4 6.0 ± 1.2 9.61 9.78

26.3 7.4 6.6 4.8 ±1.0 8.96 9.22

25.8 19.8 7.1 3.7 ±0.7 7.01 7.53

25.8 21.4 7.1 4.2 ± 0.8 6.83 7.58

25.9 16.2 7.1 3.3 ± 0.7 7.47 7.93

22.2 3.7 6.2 -2.0 ± 0.4 -2.26 -4.69

6.5 23.9 7.6 -1.7 ± 0.3 -2.37 -2.27

NiCrCrD

~

NiCoCoD

~

CoNiCrD

~

smDNiij

21410~

*Heaney and Dayananda. Metall.

Trans. 17A (6):983-989, 1986.

Diffusion coefficients calculated using the different thermodynamic databases and a fixed diffusion mobility database.

René-N4/René-N5 at 1293 °C for 100 h

0

0.02

0.04

0.06

0.08

0.1

0.12

-1000 -500 0 500 1000

Ma

ss F

ract

ion

Distance (m)

Cr

Co

W

Ta

Al

Ti

Mo

Re

HfNb

Databases used

– Thermodynamics: Thermotech

– Diffusion mobilities: NIST Ni-mob

René-N4 René-N5

6.35 mm 6.35 mm

Double geometric grid: 200 points

Experimental work performed by T. Hansen, P. Merewether, B. Mueller, Howmet Corporation, Whitehall, MI.

Analysis of Diffusion Multiples /Multicomponent Diffusion

R88

R95

ME3 IN100

IN718

R88/R95 R88/IN718R88/ME3 R88/IN100R95/IN718 R95/ME3IN718//IN100 ME3/IN100

Diffusion Multiple

1 Sample 8 Diffusion Couples

0

0.05

0.1

0.15

0.2

-500 0 500 1000

Mas

s F

ract

ion

in F

CC

Distance (m)

Rene-88/ IN-718

Cr

Co

Nb

Mo

Ti

Al

W

Experimental data provides composition and phase fraction profiles as functions of distance.

Experimental data from J-C. Zhao, GE Global Research

René-88/IN-100; 1000 h at 1150 °C

0

0.05

0.1

0.15

0.2

-400 -200 0 200 400

Ma

ss F

ract

ion

in F

CC

(G

am

ma)

Distance (m)Rene-88 IN100

T = 1150 oC t = 1000 h

Co

Cr

WAlNb

Mo

Ti

Experimental data from J-C. Zhao, GE Global Research

At 1150 °C equilibrium

phase fractions

• René-88: f = 1

• IN-100: f = 0.64

René-88 IN-100

4 mm 4 mm

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

-500 0 500 1000 1500 2000 2500 3000

Pha

se F

ract

ion

Gam

ma

Prim

e (

Mol

e F

ract

ion)

Distance (m)Rene-88 IN100

T = 1150 oC t = 1000 h

1000 h10 h

1 h

IN-718/IN-100; 1000 h at 1150 °C

0

0.05

0.1

0.15

0.2

-1200 -800 -400 0 400 800 1200

Ma

ss F

ract

ion

in F

CC

pha

se

Distance (m)

FeCr

Co

Nb

Mo

Al

Ti

T = 1150 oCt = 1000 h

IN718 IN100

Experimental data from J-C. Zhao, GE Global Research

IN718 IN100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-1200 -800 -400 0 400 800 1200 1600 2000

Ph

ase

Fra

ctio

n (G

amm

a P

rime)

Distance (m)IN718 IN100

T = 1150 oCt = 1000 h

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

-2000 -1500 -1000 -500 0 500 1000 1500 2000

Pha

se F

ract

ion

MC

ca

rbid

e

Distance (m)IN718 IN100

T = 1150 oCt = 1000 h

IN-718/IN-100; 1000 h at 1150 °C

1000 h

10 h

100 h1000 h

100 h

10 h

1 h

1 h

AIM Strategy

Rapid Experiments

Diffusion Multiples

Experiments & Characterization

Grain Size Experiments &

Characterization

Material Models

Thermo-Calc

DICTRA

Precipi-Calc

’ Fast Track

Grain size

Property Models

Yield Strength

UTS

Creep

Fatigue

Crack Growth

NiWTa

NiAl

R88

AIM Precipi-Calc simulation of multi-

modal ’size distribution for Rene-

88• Validated against GE-Interrupt

cooling experiments– GE-AE proprietary data:– Literature data: Mao (2001)

• Thermodynamics: Thermo-tech Ni-Data

• Diffusion: NIST Ni-mobility database• Thermal profile: DEFORM simulation

of blank disk

• Assume 3D spherical particle: need to add elastic energy effects

t < 100 s low nucleation rate100 s < t < 150 s Primary ’ is formed150 s < t < 350 s Primary ’ grows400 s Secondary ’ precipitates500 s Tertiary ’ precipitates

Primary ’

Particle Distribution

Mean <R>

Temp. profile

Volume Fraction

Total # Particles/m3

Transient Liquid Phase Bonding

0.00 0.04 0.08 0.120.00

0.05

0.10

0.15

0.20

LL t

L t

L t

Mole Fraction Al

Mo

le F

ract

ion

B

0.00 0.02 0.04 0.06 0.08 0.10 0.120.000

0.002

0.004

0.006

0.008

0.010

Mole Fraction Al

Mo

le F

ract

ion

B

L +

t

t = 1 st = 900 s

t = 1800 s

t = 3600 s

T = 1315 °CT = 1315 °C; Time 900 s

Ni-10.3Al/Ni-10B/Ni-10.3Al

50 m

Ni-Al Ni-Al

t

L t t

Thermodynamics &Diffusion mobilities

microstructureevolution

0

10

20

30

40

50

60

0 10 20 30 40 50 60

Rapid quench experiments

Slow cooled experiments

Prediction after isothermal hold

Prediction with cooling

Time (s)

Join

t w

idth

(m

)Lt

L

L

t

Time = 0 s

Time = 1 s

Time = 900 s

Time = 3600 s

C. E. Campbell and W. J. Boettinger, Metall. Materials Trans., 31A, 2000, 2835.

0

0.2

0.4

0.6

0.8

1

1400 1450 1500 1550 1600 1650

Solidification of Rene-N4Rene-N4 (Ni-9Cr-7Co-3.6Al-1.3Mo-0.4Nb-3.8Ta-4.1Ti-

5.75W wt.% )

Scheil

DICTRA

Lever

Solidification pathsLever L L+ + ´

ScheilL L+ L+ + ´ L+ + ´ + L+ + ´ + + BCC L+ + + BCC + L+ + + BCC + +B2 L+ + + BCC + +B2 +

DICTRA

• Cooling rate = 1 K/s;

• /2 = 20 m

• Microstructure Evolution

L

L

20 m

t = 0 s

0 s < t < 130 s

130 s < t < 135 s+´ 135 s < t < 620 s

After 620 s, phase forms

Temperature (K)

Fra

ctio

n S

olid

Thermodynamics: Ni-Data, ThermotechDiffusion: Ni-Mob, NIST

Solidification of Rene-N4:Phase Fraction Evolution

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 5 10 15 20

Ph

ase

Fra

ctio

n ´

Distance (m) Distance (m)

t = 935 s (700 K)t = 935 s (700 K)

Ph

ase

Fra

ctio

n

t = 500 s (1135 K)

t = 300 s (1335 K)

t = 135 s (1500 K)

t = 150 s (1485 K)t = 750 s (885 K)

t = 620 s (1015 K)

Center of dendrite

Center of interdendritic material

20 m

Incipient Melting Temperature

T1 (K/s)-1

Inci

pie

nt M

elti

ng

Te

mp

erat

ure

1450

1500

1550

1600

1650

0.0001 0.001 0.01 0.1 1 10 100 1000

T/´solvus

Tsolidus

Tliquidus

1000 (K/s)100 (K/s)10 (K/s)1 (K/s)

0.1 (K/s)0.01 (K/s)

Simulation Setup

• Used composition and phase fractions from solidification calculation.

• Assume linear heating rates beginning at 700 K

• Assume ’ and fraction are in equilibrium with matrix at each grid point.

• Assume incipient melting occurs at the center between dendrites.

Center of dendrite

Center of interdendritic material

20 m

Diffusion Database CenterC. E. Campbell, U.R. Kattner, C. Beauchamp, K. Dotterer, H. Gates, S.

Tobery, L. Souders

Goal: To make the NIST paper-based diffusion database center

publicly available. Convert to a searchable electronic form to be access over the internet

Task: • Need to enter 25000 bibliographic and diffusion system cards• Convert paper documents to electronic documents • Develop searchable database

Motivation • Industrial and academic support: GE $5K initiation • Center represents an unique collection summarizing the diffusion work between 1965-1980

Accomplishments •Developed database entry strategy• Entered bibliographic cards• Purchase high speed scanner and software

c

Other databases

TCS Alloy mobility database (v2.0, 1999).It is the most general alloy mobility database, within a framework of 75 elements. It can be used for steels/Fe-alloys, some Ni-based alloys, some Al-based alloys and more.

Bishop Dilute Al-alloy mobility database (v1, 1995).It contains mobility data for dilute Al-based alloys, within a framework of 30 elements.

Oikawa Dilute Fe-alloy mobility database (v1, 1995).It contains mobility data for dilute Fe-based alloys, within a framework of 28 elements.

Friberg Dilute Fe-alloy mobility database (v1, 1995). It contains mobility data for dilute Fe-based alloys, within a framework of 16 elements. It can be used for dilute Fe-alloys.

Available from ThermoCalc Software AB

Summary/Future Work

• Diffusion mobility databases Based unary, binary, and ternary tracer, intrinsic, and chemical diffusivities Dependent on available thermodynamic databases Extrapolate to n-component systems Need unstable end-members

•NIST-NiMOB (Ni (fcc) diffusion mobility database FCC phase only (Ni-rich alloys)13 components

• Applications (Ni-base superalloys, steels, Al alloys, solders) Transient Liquid Phase Bonding Heat Treatment Optimization Solidification

• Future Work Diffusion order phases Optimization of mobility parameters directly from experimental data Charged ions: Oxide phases (Höglund Thermo-Calc Software AB) Atomistic and first principle calculations of unstable phases Lateral Deformation of Diffusion Couples (3-D Kirkendall effect (Boettinger, NIST) High Throughput Analysis of Multicomponent Multiphase Diffusion Data working group (www.ctcms.nist.gov/~cecamp/workshop.html)

Diffusion in Ordered Phases

Atomic percent Al

NiAl-B2

Inte

rdiff

usio

n C

oeff

icie

nt (

cm2 /

s)

From Shankar and Seigle, Metall. Trans. 9A, (1978) 1476.

itot

ii p

N

Ny

= contribution to activation energy for component k as a result of the ordering of i-j atoms

• Expansion to Multicomponent systems Helander and Ågren, (Acta Mater., 1999, 47, 1141.)

Q = Qdis + Qord

jijii ji

ordkij

ordk xxyyQQ

ordkijQ

Diffusion Database Optimization Scheme

0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

DICTRA -Co

GE-DATA Co

Mol

e F

ract

ion

Co

Distance (m)

Co-Ni

T = 1100 oCt = 1000 h

Input Experimental File(Composition Profiles)

Run DICTRA (via python)

Thermodynamic Database

Diffusion Mobility Database

Compare composition profiles

Calculate Error (via Mathematica)

Minimize Error f(Mi)

Cha

nge

Mi a

nd z

0, R

un n

ew s

imul

atio

n

0

0.2

0.4

0.6

0.8

1

-400 -200 0 200 400

DICTRA -Co

GE-DATA Co

Mol

e F

ract

ion

Co

Distance (m)

Co-Ni

T = 1100 oCt = 1000 h

Programming Elements and Inputs

Error Definition:

z0 = Error associated with location of Matano plane

Change selected mobility parametersMCo xCo( 286175 75.08 * T ) xNi( 284169 67.65* T ) xCoxNi(10787 11.5* T )

MNi xCo( 270348 87.19 * T) xNi( 287000 69.8* T ) xCoxNi(7866 7.65 * T)

Shift Matano interface

a b

2/1

1

2

01

0

);()(

,

n

i

b

a

icali

measii

i

dzMzxzzxzWab

MzError

Wi(z)= Weighting function

Currently set to equal 1

Mobility Description Ni-W

• FCC_A1: Mobility of Ni– MQ(FCC_A1,NI:VA;0) = -28700+69.8*T– MQ(FCC_A1,W:VA;0) = V1+R*T*LN(V2)– MQ(FCC_A1,NI,W:VA;0) = V3+V4*T

• FCC_A1: Mobility of W– MQ(FCC_A1,NI:VA;0) = V5+R*T*LN(V6)– MQ(FCC_A1,W:VA;0) = V7+R*T*LN(V8)– MQ(FCC_A1,NI,W:VA;0) = V9+V10*T

Cu-Sn Intermetallic Growth

9Cu + 6Sn(l) Cu3Sn + Cu6Sn5

Cu

Co

nce

ntr

atio

nDistance

Cu Solid Solution

Sn Solution

Cu

Co

nce

ntr

atio

n

Distance

Cu Solid Solution

Sn Solution

Time = 0 s

Time > 0 s

Cu-Sn Intermetallic

Growth

497.5

498

498.5

499

499.5

500

1 10 100 1000

Inte

rfa

ce p

osi

tion

(m

)

Time (s)

L

Cu

L/Cu3Sn

Cu6Sn

5/Cu

75 m Solder Thickness

T = 508 K

Cu3Sn/Cu

6Sn

5

0

0.2

0.4

0.6

0.8

1

498 498.5 499 499.5 500

Mol

e F

ract

ion

Sn

Distance (m)

Time = 1200 s

Time = 100 s

Time = 0 s

Time = 1 s

Liquid (Sn)

T = 508 K

Solid-State Reactions in a Binary A-B System

A BABA

ABABG 0

bABG 0

A

A

A

0ABG

AB0ABGb

A + B = AB

Diffusion control

Interface control

Relaxation control

Mixed transport interface control

DefinitionsCoefficient General Notation Mobility Notation

Tracer Diffusivity

Intrinsic

Diffusivity

(partial chemical)

Chemical

Diffusivity

(Interdiffusion)

(binary)

are related by the velocity of Kirkendall frame,

ABBA DxDxD ~

m

n

i j

iiikikkj V

xMxxD

1

N

DD ii log

log1*

DDi

~ and

MVaVJv

factorn correlatio

cubic diamondfor 1/8 and BCC and FCCfor 1

parameter lattice

frequencyvibration ~

expexp~ 2*

f

a

kT

HH

k

SSfaD

mVa

fVa

mVa

fVa

i

j

kkVakkj

i

cMcD

kk RTMD *

RTRT

GM kVa

k

1exp

*2

RT

QDD exp0