High Throughput Analysis of Multicomponent Diffusion Data
C. E. Campbell and W. J. Boettinger
National Institute of Standards and Technology Gaithersburg, MD 20899
J-C. Zhao
General Electric Company: Global Research Schenectady, NY
Need for Multicomponent Diffusion Data & Simulations Review of Multicomponent Diffusion Basics Structure of Diffusion Mobility Database Optimization to obtain mobility parameters: • from measured diffusion coefficients (normal approach) • from measured diffusion profiles (new work)
TMS Fall Meeting 2003: The Accelerated Implementation of Materials & Processes
November 11, 2003
This work was partially funded by the GE-led DARPA AIM program
AIM Strategy
R88 NiAl Ni Ta W
Rapid Experiments
Material Models
Diffusion Multiples Thermo-Calc
Property Models
Yield Strength UTS
g Experiments &
Characterization
DICTRA
Precipi-Calc
Grain Size Experiments & Characterization
g’ Fast Track
Grain size
Creep
Fatigue Crack Growth
AIM Precipi-Calc simulation of multi-modal g’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 rate 100 s < t < 150 s Primary g’ is formed 150 s < t < 350 s Primary g’ grows 400 s Secondary g’ precipitates 500 s Tertiary g’ precipitates
Particle Distribution
Mean
Primary g’
Temp. profile
Volume Fraction Total # Particles/m3
Multicomponent Diffusion
Review
Fick’s first law for Flux, Ji
J i D ij dx i dz
Al Co Cr
Multicomponent diffusion matrix for René-88 composition using NIST database René-88 at 1100 °C (x 10-14 m2/s)
Al 3 . 83 0 . 635 0 . 021 0 . 050 0 . 053 0 . 357 0 . 048 Co 0 . 325 0 . 462 0 . 227 0 . 018 0 . 033 0 . 163 0 . 009 Cr 0 . 845 0 . 421 0 . 649 0 . 033 0 . 082 0 . 512 0 . 018 Mo 0 . 799 0 . 296 0 . 095 0 . 414 0 . 036 0 . 497 0 . 014 Nb 0 . 896 0 . 442 0 . 669 0 . 058 1 . 57 0 . 619 0 . 040 Ti 1 . 05 0 . 515 0 . 663 0 . 011 0 . 101 1 . 83 0 . 032 W 1 . 27 0 . 468 0 . 382 0 . 026 0 . 044 0 . 578 0 . 0033
Fick’s second law
xi dt
where D ij D ij T , x i
x j D ij z j 1 z
n 1
Mo Nb Ti W
D ij
p 1
n
pi
x i x p M
p
p
x j
Simulations need to compute the diffusion matrix for each composition encountered in diffusion profile at each time step.
Approach enables efficient data storage
Multicomponent Diffusion Database Structure
Inputs:
– Calphad Thermodynamics – Diffusion experiments (unary, binary, ternary systems)
• Tracer diffusivity, • Intrinsic diffusivity, • Interdiffusion coefficients/Marker motion
D k ( x i ) RTM
*
k
( xi )
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)
Mi Mi Qi exp where Q i f c i , T and M i 1 RT RT n n n
Qi
x
p 1
p
Q i (T )
p
x
p q q 1
p
x q Ai ( T )
0
pq
Add terms
B i (T ) x i x j x k
ijk
if necessary to fit ternary data, etc.
Optimization of Experimental Diffusion Coefficients
Compare experimental and calculated D Simulate diffusion process Adjust Mobility Calculate diffusion Coefficients D = f(c,T)
Ni - Al
-1 2
D ata from Yam am oto e t. al. (1 9 8 0 ) A ss es s m en t b y E n g s tröm an d Å g ren (1 9 9 6 )
Estimate Mobility
Experimental diffusion data
Diffusion profile Diffusion Coefficient
Mobility M=f (c,T)
Log (Mobility)
T = 1150 °C
(m /s )
-1 2 .5 1 2 5 0° C -1 3 -1 3 .5
Composition
lo g D
g
T = 1050 °C
2
1 1 0 0 °C 1 0 5 0° C 1 0 0 0°C
-1 4
T = 950 °C
-1 4 .5 -1 5
Distance
Composition
0
0 .0 5
0 .1 M o le F ra c tio n A l
0 .1 5
0 .2
Mi
Mi
i j i, j i, j 2 i, j For a binary: Q i c i Q i c j Q i c i c j Ai ( c i c j ) B i ( c i c j ) C i ...
Qi exp where Q i f c i , T and M i 1 RT RT
Examples of Optimized Binary Interactions
Ni-Al-Cr-Co-Fe-Hf-Nb-Mo-Re-Ta-Ti-W
Ni-Co Interdiffusion
-1 2
Data from Ustad and Sorum, Phys. Stat. Sol. A 285 (1973) 285. Calculated
Interdiffusion with Ni
-1 4 -1 4 .5
Ti Ta
Log (D) m2/s
-1 2 .5
T = 1000 oC
Log (D) m2/s
1400 oC
-1 3
-1 5
1325 oC
-1 3 .5
-1 5 .5
-1 6
W
1250 oC 1200 oC 1160 oC
-1 6 .5
-1 4
Re
-1 7
Data from Komai et al., Acta. Mater., 46, (1998) 4443. Data from Karunaratne et al., Mater. Sci. Eng., A281 (2000) 229.
-1 4 .5 0 20 40 60 80 10 0
-1 7 .5 0 0.02 0 .0 4 0 .06 0.0 8 0.1
Atomic Percent Ni
Weight Fraction
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-Cr-Fe Jönsson, Z. Metallkd 85 (7):502-509, 1994. Current assessments: Ni-Co, Ni-Hf,Ni-Mo, Ni-Nb,Ni-Re, Ni-Ta, Ni-Ti, Ni-W Co-Cr, Co-Mo C. E. Campbell, W. J. Boettinger, U. R. Kattner, Acta Mat, 50 (2002) 775
Optimization of Ni-W
T e m p e ra tu re ( C )
1451 -1 2 .5 1394 1340 1290 1242 1197 1155 1116 1078
o
Data from Monma et. Al., JIM, 28 (1964) 197.
-1 3 N i-1 .7W (a t.% ) -1 3 .5 N i-5 .3W (a t.% ) -1 4 N i-9 .2W (a t.% ) -1 4 .5 x(w )= .0 1 7 -1 5 x(w )= .0 5 3 x(w )= .0 9 2 -1 5 .5 5 .8 10
-4
Interdiffusion data
-1 3
1300 C
o
L o g D (F C C ,N i) (m /s)
L o g D (In te rd iffu sio n ) m /s
-1 4
1200 C
o
Tracer diffusivity data
2
2
-1 5
1100 C
o
*
-1 6
1000 C
o
-1 7
6 .4 10
-4
6 10
-4
6 .2 10
-4
6 .6 10
-4
6 .8 10
-4
7 10
-4
7 .2 10
-4
7 .4 10
-4
900 C
o
Data from Karuanaratne et al., Mater. Sci&Eng. 281 (2000) 229.
(1 /T em p e ra tu re ) (K )
T em p e ra tu re ( C ) -1 3 .5
1340 1290 1242 1197 1155 1116 1078
o
-1 8 0 0 .0 2 0 .0 4 0 .0 6 0 .0 8 0 .1 M a s s Frac tio n W
Data from Monma et. Al., JIM, 28 (1964) 197.
-1 4
Activation energies in the fcc phase
N i-1 .7 W N i-5 .3 W
L o g D * (F C C ,W ) (m /s)
2
-1 4 .5
Q Ni x Ni Q Ni xW Q Ni x Ni xW A Ni
* Ni W 0
Ni ,W
N i-9 .2 W
-1 5
Q W x Ni Q W xW Q W x Ni x W AW
* Ni W 0
x(w )= .0 1 7 x(w )= 0 .0 5 3 w (w )= .0 9 2
Ni ,W
-1 5 .5
-1 6 -4 6 .2 1 0
6 .4 1 0
-4
6 .6 1 0
-4
6 .8 1 0
-4
7 10
-4
7 .2 1 0
-4
7 .4 1 0
-4
Self activation energies Optimized parameters
(1 /T e m pe ra tu re ) (K )
Challenge: Analysis of Diffusion Multiples /Multicomponent Diffusion
Cannot determine diffusion coefficients from experimental data
R e n e -8 8 / IN -7 1 8
Diffusion Multiple
1 Sample 8 Diffusion Couples
M a s s F ra c tio n in F C C
0 .2
R88
R95 IN718 ME3 IN100
0 .1 5
Cr Co
0 .1
0 .0 5
W
Nb Mo Ti
R88/R95 R88/ME3 R95/IN718 IN718//IN100
R88/IN718 R88/IN100 R95/ME3 ME3/IN100
0 -5 00
Al
0 5 00 1 00 0
D ista n ce ( m )
Experimental data provides composition and phase fraction profiles as functions of distance.
Is it possible to directly relate composition profiles to mobility parameters?
Example : René-88/IN-100; 1000 h at 1150 °C
20 20
Cr Cr
gg
g+g g+g´
At 1150 °C equilibrium phase fractions
• René-88: fg = 1 • IN-100: fg = 0.638 fg’ = 0.362
A to m ic P e rce n tt A to m ic P erc e n
15 15
Co Co
10 10
Al
Al 5 5 Mo Ti
Additional gg’ GE couples analyzed:
René-95/ René-88 IN100/ME3 IN100/ René-88 IN718/IN100 René-95/IN718 ME3/ René-88 ME3/IN718 U720/IN718 René-95/U720 U720/ME3 ME3/ René-95 IN100/U720
Ti Mo
W W Nb Nb
0 0 -4 00 -4 0 0
-3 00 -3 0 0
-2 0 0 -2 0 0
-10 0 -1 0 0
0 0
100 1 00
20 0 200
3 00 3 00
400 4 00
René-88 René-88 R88 R88
g g
D is tan c e ( m ) D ista nce ( m )
IN-100 IN-100
g g
g
g
g IN100 IN100
Experimental data from J. C. Zhao, GE-GR, Schenectady, NY
Diffusion Database Optimization Scheme
Change Mi and z0, Run new simulation Diffusion Mobility Database Thermodynamic Database Run DICTRA (via python)
Compare composition profiles
Input Experimental File (Composition Profiles)
C o-N i
1
Calculate Error (via Mathematica)
T = 1 1 00 C t = 10 0 0 h
0 .8
o
M o le F ra ctio n C o
0 .6
Minimize Error f(Mi)
D IC T R A -C o G E -D A T A C o
0 .4
0 .2
0 -40 0 -20 0 0 2 00 4 00
D istan ce (m )
Test Example: Binary Ni-Co
Interdiffusion Coefficient obtained by Boltzmann-Matano method
C o -N i
1
4 .0 1 0
-15
D iffu s io n C o e ffic ien t (m /s )
T = 1100 C t = 1000 h
0 .8
o
3 .5 1 0 3 .0 1 0 2 .5 1 0 2 .0 1 0 1 .5 1 0 1 .0 1 0 5 .0 1 0
-15
2
M o le F ra ctio n C o
-15
G E -E x p e rim e n ta lB M a n a lys is
0 .6
-15
-15
0 .4
-15
-15
0 .2
-16
0 -4 00 -2 00 0 200 4 00
0
0 .2
0 .4
0 .6
0 .8
1
D is ta n c e ( m )
M o le F ra c tio n C o
Programming Elements and Inputs
Error Definition:
1/ 2
C o-N i
1
T = 1 1 00 C t = 10 0 0 h
o
b n 2 1 meas cal ( z z 0 ) x i ( z ; M i ) dz b a W i z x i a i 1
M o le F ra ctio n C o
Error z 0 , M i
0 .8
0 .6
0 .4
Shift Matano interface
D IC T R A -C o G E -D A T A C o
Wi(z)= Weighting function
Currently set to equal 1
0 .2
z0 = Error associated with location of Matano plane
0 -40 0 -20 0 0 2 00 4 00
a
D istan ce (m )
b
Change selected mobility parameters
M Co x Co ( 286175 75 .08 * T ) x Ni ( 284169 67 .65 * T ) x Co x Ni ( 10787 11 .5 * T ) M Ni x Co ( 270348 87 .19 * T ) x Ni ( 287000 69 .8 * T ) x Co x Ni ( 7866 7.65 * T )
Ni-Co: Optimization Results
1
Initial Parameters
M Co x Co ( 286175 75 . 08 * T ) x Ni ( 284169 67 . 65 * T )
0 .8
Initial
x Co x Ni ( 10787 11 . 5 * T ) M
Ni
x Co ( 270348 87 . 19 * T ) x Ni ( 287000 69 . 8 * T ) x Co x Ni ( 7866 7 . 65 * T )
M o le F ra ctio n C o
0 .6
Optimized
0 .4
Optimized Parameters
M Co x Co ( 286175 75 . 08 * T ) x Ni ( 286310 67 . 65 * T ) x Co x Ni ( 9953 11 . 5 * T ) M
Ni
x Co ( 284144 87 . 19 * T ) x Ni ( 287000 69 . 8 * T ) x Co x Ni ( 7976 7 . 65 * T )
0 .2
Distance shift z0= -1.58 m
E xp e rim e ntal
0 -4 00 -20 0 0 20 0 4 00
D ista nce ( m )
Optimization Results: Diffusion Coefficient
5 .0 1 0
-15
Initial Parameters NIST MOB - Initial Optimized
M Co x Co ( 286175 75 . 08 * T ) x Ni ( 284169 67 . 65 * T ) x Co x Ni ( 10787 11 . 5 * T ) M
Ni
D iffu s io n C o e ffic ien t (m /s )
2
4 .0 1 0
-15
x Co ( 270348 87 . 19 * T ) x Ni ( 287000 69 . 8 * T ) x Co x Ni ( 7866 7 . 65 * T )
3 .0 1 0
-15
2 .0 1 0
-15
Optimized Parameters
M Co x Co ( 286175 75 . 08 * T ) x Ni ( 286310 67 . 65 * T )
1 .0 1 0
-15
GE Experimental data BM analysis
0 0 .2 0 .4 0 .6 0 .8 1
x Co x Ni ( 9953 11 . 5 * T ) M
Ni
x Co ( 284144 87 . 19 * T ) x Ni ( 287000 69 . 8 * T ) x Co x Ni ( 7976 7 . 65 * T )
Distance shift z0= -1.58 m
M o le F ra c tio n C o
Ternary Example:
Ni-5.13Al-9.77Cr/Ni-2.39Al-19.34Cr (at.%)
For a single couple cannot determine the interdiffusion coefficients using the BM method
0 .0 5 5
0 .2 0
0 .0 5 0
T = 1100 °C t = 1000 h
M o le F ra c tio n C r
0 .1 8
T = 1100 °C t = 1000 h
0 .0 4 5
M o le F ra c tio n A l
0 .1 6
0 .0 4 0
0 .1 4
0 .0 3 5
0 .1 2
0 .0 3 0
0 .0 2 5
Reference Binary interactions zeroed
-4 0 0 -2 0 0 0 200 400 600
0 .1 0
Reference Binary interactions zeroed
0 .0 8 0 -6 0 0 -4 0 0 -2 0 0 0 200 400 600
0 .0 2 0 -6 0 0
D is ta n c e ( m )
D is ta n c e ( m )
M
Al
x Al ( -142000 72 . 11 * T ) x Cr ( -235000 - 82 * T) x Ni ( -284000 - 59.82 * T ) x Al x Cr ( 335000 ) x Al x Ni ( -41300 - 91.2 * T ) x Cr x Ni ( -53200 ) x Al ( -145900 64 . 25 * T ) x Cr ( -235000 - 82 * T) x Ni ( -287000 - 69.8 * T ) x Al x Cr ( 211000 ) x Al x Ni ( -113000 65.5 * T ) x Cr x Ni ( -81000 )
M Cr x Al ( -261700 3 . 71 * T ) x Cr ( -235000 - 82 * T) x Ni ( -287000 - 64.4 * T ) x Al x Cr ( 487000 ) x Al x Ni ( -118000 ) x Cr x Ni ( -68000 ) M
Ni
Diffusion Database Optimization Scheme
Diffusion Mobility Database Thermodynamic Database Change Mi Run new simulation Run DICTRA (via python)
Compare composition profiles
Input Experimental File (Composition Profiles)
1 couple 2 profiles
Calculate Error (via Mathematica)
Minimize Error f(Mi)
Changing 9 binary interactions
Optimization Results: 9 parameters
0 .0 55
0.20
0 .0 50
T = 1100 °C t = 1000 h
M o le F ra ctio n C r
0.18
T = 1100 °C t = 1000 h
0 .0 45
M o le F ra ctio n A l
0.16
0 .0 40
0.14
0 .0 35
0.12
0 .0 30
0 .0 25
0.10
0 .0 20 -60 0
-4 0 0
-2 00
0
20 0
40 0
600
0 .0 80 -60 0
-4 0 0
-2 00
0
20 0
40 0
600
D is ta n ce ( m )
D is ta n ce ( m )
Reference
AlAlCr=335000 CrAlCr= 487000 AlAlNi=-166517 CrAlNi=-118000 AlCrNi=-53200 CrCrNi=-68000 NiAlCr=211000 NiAlNi=-23068 NiCrNi=-81000
Optimized
AlAlCr= 341099 CrAlCr= 397756 AlAlNi= -175812 CrAlNi=-117332 AlCrNi= -58013 CrCrNi=-62614 NiAlCr=265578 NiAlNi=-24037 NiCrNi=-89378
Binary Interactions zero
Goal: Ni/Rene-88
0 .2 0
T = 1 15 0 C t = 1 00 0 h
0 .1 5
o
Optimization strategy
Cr Co
0 .1 0
0 .0 5 0
0 .0
-1 0 0 0
-5 0 0
0
500
1000
Ni
0 .0 5 0
D is ta n c e ( m )
R e n e -8 8
Ni end-member term • Ti, Nb Ni binary interactions • Ni-Ti Ni-Nb • Ni-Cr Ni-Al Ni ternary interactions •Ni-Al-Cr •Ni-Al-Ti •Ni-Cr-Nb
M a s s F ra c tio n
T = 1150 C t = 1000 h
0 .0 4 0 Ti Mo W 0 .0 3 0 Al Nb
o
W
M a s s F ra c tio n
Mo Ti
M
Ni
x Al M
Al Ni
x Co M x Ti M
Co , Ni Ni Nb , Ni Ni
Co Ni Ti Ni
x Cr M xW M
Cr Ni W Ni
x Mo M
Mo Ni Al , Ni Ni Mo , Ni Ni W , Ni Ni
0 .0 2 0
Al
x Nb M
Nb Ni
x Al x Ni M
Nb
0 .0 1 0
x Co x Ni M x Nb x Ni M
x Cr x Ni M x Ti x Ni M
Cr , Ni Ni Ti , Ni Ni
x Mo x Ni M x W x Ni M
Al , Cr , Ni Ni
0 .0 -1 0 0 0
-5 0 0
0
500
1000
x Al x Co x Ni M
Al , Co , Ni Ni
x Al x Cr x Ni M
Ni
D is ta n c e ( m )
R e n e -8 8
Diffusion Database Optimization Scheme
Change Mi and z0, Run new simulation Diffusion Mobility Database Thermodynamic Database Run DICTRA (via python)
Compare composition profiles
Input Experimental File (Composition Profiles)
Calculate Error (via Mathematica)
1 couple 7 profiles
Minimize Error f(Mi)
Changing 2 binary end members 2 binary interactions
Ti Profile from Ni/Rene-88
0 .0 5 0
T = 1150 C t = 1000 h
0 .0 4 0
o
M a s s F ra c tio n
Experiment
0 .0 3 0
Initial Ni-MOB Ti M Ni = -386325 Ni ,Ti M Ni =-81000 =-367650 Ni ,Ti M Ti =-68000
M Ti
Ni
4 parameters optimized
0 .0 2 0
Initial Ni-MOB
0 .0 1 0
Optimized Ti M Ni = -367867 Ni ,Ti M Ni =-93125 =-327697 Ni ,Ti M Ti =-70015
M Ti
Ni
0 .0 -1 0 00
-5 0 0
0
500
1000
Ni
D is ta n ce ( m )
R e n e -88
Need to consider additional parameters: • Other binary interactions • Ternary interactions
z0= +7.5 m
Summary
Multicomponent Ni-base diffusion mobility
• Based on optimization of available diffusion coefficient data • Comparison of simulation results with experiments shows good agreement
Optimization based on composition profiles
Method
• Relates profiles to mobility parameters • Provides ability to asses error associated with mobility parameters
Examples
• Binary: Ni-Co (1 couple, fixed T,1 profile, 4 parameters, z0) • Ternary: Ni-Al-Cr (1 couple, fixed T, 2 profiles, 9 parameters) • Multicomponent: Ni/Rene88 (1 couples, fixed T, 7 profiles, 4 parameters, z0)
Improved optimization strategy needed
• Multicomponent single phase (Need to consider more than 1 couple) • Multicomponent multiphase
Programming additions needed
• Weighting functions • Other error definitions