This document summarizes modeling of diffusion at high pressure. It discusses when pressure is important for diffusion, defines activation volume, and explores correlations between activation volume and bulk material properties. It then outlines the Calphad approach for modeling diffusion, including defining necessary parameters. Finally, it assesses activation volumes for diffusion in the Fe-Ni system using experimental data, and shows good agreement between calculated and measured diffusion profiles up to 23 GPa.

1. Modelling of diffusion at high
pressure
Tomas Gomez-Acebo
CALPHAD XLI - Berkeley, 3-8 June, 2012

3. Outline
• When is pressure important for diffusion?
• Activation volume
• Correlations with bulk properties
• Calphad approach
• Assessment of Fe-Ni activation volumes
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4. When is pressure important for diffusion?
Geology Sintering
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5. ACTIVATION VOLUME
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6. Activation volume
0 2 G
D gf a exp
kBT
Geometrical factor
Correlation factor Lattice parameter
Attempt frequency
Gibbs energy of activation:
G H T S U T S P V
At ambient pressure, P V is negligible: U H
G
V Activation volume
P T
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7. Activation volume
0 2 G
D gf a exp
k BT
0 2 S U P V
g f a exp exp exp
kB k BT kBT
  
 
D0
0 2
ln D ln( g f a )
V kBT kBT
P T  P 
 
correctionterm
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8. Activation volume
1E-16
ln D
V k BT
800 K P T
V/ =0.76
ln( g f 0 a 2 )
k BT
D (m2/s)
 P 
 
correctionterm
Correction term can
198Au diffusion in Au single crystals
be neglected
[Werner 1983] =Vm
1E-17 (molar/atomic
0 200 400 600 volume)
P (MPa)
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9. Activation volume
• Is the isothermal change in the volume of a
crystal associated with a diffusive jump
– Parameter for diffusion modelling
– Provides a fingerprint of the mechanism of the
diffusion process
F M
V V V
• VF: formation volume of the defect
• VM: migration volume of the defect
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10. Formation volume, VM
=Vm Inward
relaxation Formation volume of a
(molar/atomic monovacancy
volume) VF1Va<
+ VF=+ Vrel
Inward
relaxation
Formation volume of a
divacancy
VF2Va> VF1Va
+2 VF=+2 Vrel
Outward
relaxation Formation volume of a
self-interstitial
VFI >0 in close-packed
VFI can be <0 in less
VF= +Vrel
packed (e.g. Ge in Si)
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11. Migration volume, VM
• Volume change between the equilibrium
position and the saddle-point position
Substitutional diffusion
VM=VS-V1
Caution: 1 S 2
• Jump event takes 10-12 s. Saddle point
• Atomic displacements, at
the velocity of sound. Interstitial diffusion
• A complete relaxation of
the saddle-point is
impossible.
1 S 2
Saddle point
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12. Migration volume, VM
• In close-packed metals: 1
VM is fairly small
0.8
• Values for Au:
– VM=0.15 0.6
[Emrick, 1961]
V/
– V=+0.72 to +0.75 0.4
[Mehrer, 2007]
Werner (1983)
0.2 Tm (Au)=1064.2 C
Rein (1982)
• Major part of activation Beyeler (1968)
volume: VF, formation 0
volume 500 700 900
T (K)
1100 1300
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13. Interstitial atoms
• C, N, O
• No defect formation term is required:
V VM VS V1
1 S 2
Saddle point
– C and N in bcc-Fe:
VM= 0.08 to +0.05 [Bosman, 1957, 1960]
– Interstitial diffusion is very weakly pressure
dependant
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14. Substitutional atoms (impurities)
• Diffusion mediated by vacancies
F B M ln f 2
V2 V 1Va V V kBT
2
 P
 

C2
V1VaF VB: formation volume of solute-vacancy pair
V2M: migration volume of solute-vacancy exchange
f2: correlation factor of impurity diffusion
C2: pressure dependence of correlation factor
V2M+C2: migration volume of the solute-vacancy pair
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15. V/
-0.5
0.5
1.5
-1
0
1
2
Cu self-diffusion
Ag self-diffusion
Au self-diffusion
Al self-diffusion
Ge in Al
Zn in Al
Mn in Al
Co in Al
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Na self-diffusion
Ge in Si
N in bcc-Fe
C in bcc-Fe
Activation volumes in metals
15
[Mehrer 2007]

16. CORRELATIONS WITH BULK
PROPERTIES
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17. Pressure dependence on diffusion
• Old experimental data [Nachtrieb 1959, 1965], for
self-diffusion of Pb and Sn:
– Diffusivity at melting temperature is independent
from pressure
d (ln D(Tm )) H ( P 0) dTm
0 V
dP Tm ( P 0) dP
– Melting point increases with pressure
– Same diffusivity at same homologous
temperature, T/TM
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18. Comparison calculated/experimental
H ( P 0) dTm
V
Tm ( P 0) dP P 0
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19. CALPHAD APPROACH
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20. Mobility parameters
0 Qi 1 mg mg
Mi M expi ; 1 for non - magnetic
RT RT
RT ln( RTMi ) RT ln M i0 Qi MQ
Qi Qi0 P V
Qi0: for 1 bar
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21. Parameters for A-B alloy
• Self-diffusion activation volume
 DV(phase&A,A)
 DV(phase&B,B)
• Impurity diffusion activation volume
 DV(phase&A,B)
 DV(phase&B,A)
• Interaction parameters (very uncommon)
 DV(phase&A,A:B) etc.
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22. PRESSURE EFFECT ON FE-NI
DIFFUSION
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23. Schematic view of the interior of Earth
1. continental crust
2. oceanic crust
3. upper mantle
4. lower mantle
5. outer core
6. inner core
A: Mohorovičić discontinuity
B: Gutenberg discontinuity
(CMB: core-mantle boundary)
C: Lehmann–Bullen discontinuity
(ICB: inner core boundary)
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24. Density, P and T distribution on Earth
8000 K
364 GPa 4500 K
330 GPa
3000 K
136 GPa
800 K
24 GPa
Upper
Inner core Outer core Mantle mantle
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25. Experimental data
• Diffusion couples Fe-Ni
– P up to 23 GPa
– T=1280 – 1700 °C
• [Goldstein, Trans. Metall. Soc. AIME, 233 (1965) 812]
• [Yunker, Earth and Planetary Sci. Lett., 254 (2007) 203]
• Thermodynamic and mobility data:
– TCFE6 and MOBFE1 (modified introducing the pressure
term in mobility)
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26. P-T diagram of Fe
Extrapolation at higher pressures
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27. Molar volume
Molar volume of Fe
and Ni at
1150, 1250, 1400, 1500
, 1600 & 1700 ºC
(SSOL2 database)
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28. Activation volume at 4 GPa
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29. Diffusion profiles at 1, 12 and 23 GPa
THERMO-CALC (2011.05.20:17.50) :
THERMO-CALC (2011.05.20:17.48) : THERMO-CALC (2011.05.20:17.50) :
100 1GPa, 1150C, 18h
100 1GPa, 1280C, 6h 100
1GPa, 1420C, 2h
90 90 1GPa, 1150C, 18h 90 12GPa, 1500C, 2h 12GPa, 1500C, 2h
12GPa, 1600C, 0.5h
80
1GPa, 1280C, 6h 80 12GPa, 1600C, 0.5h 12GPa, 1600C, 2h
80 1GPa, 1420C, 2h 12GPa, 1600C, 2h 12GPa, 1600C, 10h
2011-05-20 17:48:45.21 output by user tgacebo from PCTGACEBO
70
ATOMIC-PERCENT FE
70
ATOMIC-PERCENT FE
12GPa, 1600C, 10h
60
60 70
ATOMIC-PERCENT FE
50 50
40
60 40
-05-20 17:51:28.62 output by user tgacebo from PCTGACEBO by user tgacebo from PCTGACEBO
30 30
50
20 20
10 40 10
0 0
THERMO-CALC (2011.05.20:17.51) :
-150 -100 30
-50 0 50 100 150 200 250 -300 -200 -100 0 100 200 300 400 500
DISTANCE (um) DISTANCE (um)
100
20
• Assessment results:
2011-05-20 17:48:45.21 output
90
23GPa, 1600C, 6h
80
10 23GPa, 1700C, 6h
2011-05-20 17:51:28.62 output by user tgacebo from PCTGACEBO
– DV(fcc&Fe,Fe)=5.37E-06 m3/mol
70 23GPa, 1600C, 6h
ATOMIC-PERCENT FE
0 23GPa, 1700C, 6h
60
-300 -200 -100 0 100 200 300 400 500
DISTANCE (um) – DV(fcc&Fe,Ni)=3.83E-06 m3/mol
50
40
100 150 200 250
E (um)
30
20
– DV(fcc&Ni,Fe)=3.11E-06 m3/mol
10
0
– DV(fcc&Ni,Ni)=5.91E-06 m3/mol
-150 -100 -50 0 50 100 150 200 250
DISTANCE (um)
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30. DATABASE:USER
THERMO-CALC (2011.12.07:17.59) :
DATABASE:USER
N=1, P=1E9, T=1427;
THERMO-CALC (2011.12.07:17.59) :
Interdiffusion coefficients THERMO-CALC (2011.12.07:17.59) :
DATABASE:USER
N=1., P=1E9, T=1693; N=1, P=1E9, T=1427;
-12.0 1GPa, 1420C, 2h [2007Yun]
-12.0 1GPa, 1280C, 6h [2007Yun] -12.0
1GPa, 1150C, 18h [2007Yun]
4GPa, 1233C, 4h [1965Gol] 4GPa, 1233C, 4h [1965Gol]
-12.5 -12.5 -12.5
4GPa, 1154C, 8h [1965Gol] 4GPa, 1154C, 8h [1965Gol]
-13.0 -13.0
2011-12-07 17:59:54.77 output by user tgacebo from PCTGACEBO
-13.0
LOGDC(FCC,NI,NI,FE)
LOGDC(FCC,NI,NI,FE)
2011-12-07 17:59:41.64 output by user tgacebo from PCTGACEBO
-13.5 -13.5
LOGDC(FCC,NI,NI,FE)
-13.5
-14.0 -14.0
-14.5 -14.5
-14.0
-15.0 -15.0
1GPa, 1420C, 2h [2007Yun]
-14.5 1280C, 6h [2007Yun]
1GPa,
-15.5 -15.5
1GPa, 1150C, 18h [2007Yun]
-16.0 -15.0 -16.0
THERMO-CALC10
0 (2011.12.07:18.00)40
20 30 : 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
THERMO-CALC (2011.12.07:18.00) :
DATABASE:USER MOLE_PERCENT NI DATABASE:USER MOLE_PERCENT NI
N=1., P=1.2E10, T=1873; -15.5 N=1, P=2.3E10, T=1973;
2011-12-07 17:59:54.77 output by user tgacebo from PCTGACEBO
-12.0 THERMO-CALC (2011.12.07:18.00) : -12.0
DATABASE:USER
-12.5 -16.0
N=1, P=2.3E10, T=1973; -12.5
0 10 20 30 40 12GPa, 1600C, 10h70
50 60 80
[2007Yun] 90 100
-13.0 -12.0 12GPa, 1600C, 2h [2007Yun] -13.0
2011-12-07 18:00:08.68 output by user tgacebo from PCTGACEBO
MOLE_PERCENT NI [2007Yun]
12GPa, 1600C, 0.5h
LOGDC(FCC,NI,NI,FE)
LOGDC(FCC,NI,NI,FE)
12GPa, 1500C, 2h [2007Yun]
-13.5 -13.5
-12.5 23GPa, 1700C, 6h [2007Yun]
23GPa, 1600C, 6h [2007Yun]
-14.0 -14.0
-13.0
ut by user tgacebo from PCTGACEBO
-14.5 -14.5
12GPa, 1600C, 10h [2007Yun]
LOGDC(FCC,NI,NI,FE)
-15.0 12GPa, 1600C, 2h [2007Yun] -15.0
-13.5
r tgacebo from PCTGACEBO
23GPa, 1700C, 6h [2007Yun]
12GPa, 1600C, 0.5h [2007Yun]
-15.5 -15.5 23GPa, 1600C, 6h [2007Yun]
12GPa, 1500C, 2h [2007Yun]
-14.0
-16.0 -16.0
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
60 70 80 90 100
NT NI
-14.5
MOLE_PERCENT NI CALPHAD XLI - Berkeley, 3-8 June, 2012 MOLE_PERCENT NI 30

31. D @ 1600 C, 10 at.-% Ni
THERMO-CALC (2011.12.09:16.42) :
DATABASE:USER
N=1, T=1873.15, X(NI)=0.1;
-12.0 Goldstein et al. (1965)
Yunker et al. (2007)
-12.5
LOGDC(FCC,NI,NI,FE)
-13.0
-13.5
-14.0
-14.5
-15.0
0 5 10 15 20 25
FUNCTION PGPA
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32. Summary
• Activation volume:
– Modelling parameter
– Fingerprint of diffusion mechanism
– Formation volume & migration volume
– Pressure effect: related to melting temperature
• Calphad approach to Fe-Ni diffusion at high
pressure
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33. THANK YOU!
(and see you at Calphad 2013
in San Sebastian, Spain)