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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