modelling diffusion at high pressure

Modelling of diffusion at high pressure

Tomas Gomez-Acebo

CALPHAD XLI - Berkeley, 3-8 June, 2012

• When is pressure important for diffusion?

• Activation volume

• Correlations with bulk properties

• Calphad approach

• Assessment of Fe-Ni activation volumes

Outline

CALPHAD XLI - Berkeley, 3-8 June, 2012 3

When is pressure important for diffusion?

4CALPHAD XLI - Berkeley, 3-8 June, 2012

Geology Sintering

ACTIVATION VOLUME


Activation volume

VPSTUSTHG

TP

GV Activation volume


At ambient pressure, P V is negligible: U H

Tk

GafgD

B

20 exp

Geometrical factor

Correlation factor

Attempt frequency

Lattice parameter

Gibbs energy of activation:

Activation volume


Tk

VP

Tk

U

k

Safg

Tk

GafgD

D

BBB

20

B

20

expexpexp

exp

0

termcorrection

20

BB

)ln(ln

P

afgTk

P

DTkV

T

Activation volume


1E-17

1E-16

0 200 400 600

D (

m2/s

)

P (MPa)

198Au diffusion in Au single crystals

800 KV/ =0.76

termcorrection

20

B

B

)ln(

ln

P

afgTk

P

DTkV

T

Correction term can be neglected

[Werner 1983] =Vm

(molar/atomic 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

• VF: formation volume of the defect

• VM: migration volume of the defect

Activation volume


MF VVV

Formation volume, VM


Inward

relaxation

+ VF=+ Vrel

Inward

relaxation

+2 VF=+2 Vrel

Outward

relaxation

VF= +Vrel

Formation volume of a

monovacancy


divacancy


self-interstitial

=Vm

(molar/atomic volume) VF

1Va<

VF2Va> VF

1Va

VFI >0 in close-packed

VFI can be <0 in less

packed (e.g. Ge in Si)

• Volume change between the equilibrium position and the saddle-point position

Migration volume, VM


VM=VS-V1

1 S

Saddle point2

1 S

Saddle point2

Substitutional diffusion

Interstitial diffusion

Caution: • Jump event takes 10-12 s.• Atomic displacements, at

the velocity of sound.• A complete relaxation of

the saddle-point is impossible.

• In close-packed metals: VM is fairly small

• Values for Au:– VM=0.15

[Emrick, 1961]

– V=+0.72 to +0.75[Mehrer, 2007]

• Major part of activation volume: VF, formation volume

Migration volume, VM


0

0.2

0.4

0.6

0.8

1

500 700 900 1100 1300

V/

T (K)

Werner (1983)

Rein (1982)

Beyeler (1968)

Tm (Au)=1064.2 C

• C, N, O

• No defect formation term is required:

– C and N in bcc-Fe: VM= 0.08 to +0.05 [Bosman, 1957, 1960]

– Interstitial diffusion is very weakly pressure dependant

Interstitial atoms


1VVVV S

M

1 S

Saddle point2

• Diffusion mediated by vacancies

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

Substitutional atoms (impurities)


2

2B2Va12

ln

C

MBF

P

fTkVVVV

Activation volumes in metals


-1

-0.5

0

0.5

1

1.5

2

Cu

sel

f-d

iffu

sio

n

Ag

self

-dif

fusi

on

Au

sel

f-d

iffu

sio

n

Al s

elf-

dif

fusi

on

Ge

in A

l

Zn in

Al

Mn

in A

l

Co

in A

l

Na

self

-dif

fusi

on

Ge

in S

i

N in

bcc

-Fe

C in

bcc

-Fe

V/

[Mehrer 2007]

CORRELATIONS WITH BULK PROPERTIES


• Old experimental data [Nachtrieb 1959, 1965], for self-diffusion of Pb and Sn:

– Diffusivity at melting temperature is independent from pressure

– Melting point increases with pressure

– Same diffusivity at same homologous temperature, T/TM

Pressure dependence on diffusion


dP

dT

PT

PHV

dP

TDd m

m

m

)0(

)0(0

))((ln

Comparison calculated/experimental


0)0(

)0(

P

m

m dP

dT

PT

PHV

CALPHAD APPROACH


Mobility parameters


Qi0: for 1 bar

VPQQ

QMRTRTMRT

RTRT

QMM

ii

iii

mgmgiii

0

0

0

MQln)ln(

magnetic-nonfor 1;1

exp

• 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.

Parameters for A-B alloy


PRESSURE EFFECT ON FE-NI DIFFUSION


Schematic view of the interior of Earth

23

1. continental crust 2. oceanic crust3. upper mantle 4. lower mantle 5. outer core6. inner core

A: Mohorovičić discontinuityB: Gutenberg discontinuity

(CMB: core-mantle boundary)C: Lehmann–Bullen discontinuity

(ICB: inner core boundary)


Density, P and T distribution on Earth

24

8000 K364 GPa 4500 K

330 GPa

3000 K136 GPa

800 K24 GPa

Inner core Outer core MantleUppermantle


• 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)

Experimental data


P-T diagram of Fe


Extrapolation at higher pressures

Molar volume


Molar volume of Fe and Ni at 1150, 1250, 1400, 1500, 1600 & 1700 ºC (SSOL2 database)

Activation volume at 4 GPa


• Assessment results:

– DV(fcc&Fe,Fe)=5.37E-06 m3/mol

– DV(fcc&Fe,Ni)=3.83E-06 m3/mol

– DV(fcc&Ni,Fe)=3.11E-06 m3/mol

– DV(fcc&Ni,Ni)=5.91E-06 m3/mol

Diffusion profiles at 1, 12 and 23 GPa

29

0

10

20

30

40

50

60

70

80

90

100

AT

OM

IC-P

ER

CE

NT

FE

-300 -200 -100 0 100 200 300 400 500

DISTANCE (um)

THERMO-CALC (2011.05.20:17.50) :

12GPa, 1600C, 2h

12GPa, 1500C, 2h

12GPa, 1600C, 10h

12GPa, 1600C, 0.5h

2011-05-20 17:50:25.82 output by user tgacebo from PCTGACEBO

0

10

20

30

40

50

60

70

80

90

100

AT

OM

IC-P

ER

CE

NT

FE

-150 -100 -50 0 50 100 150 200 250

DISTANCE (um)

THERMO-CALC (2011.05.20:17.51) :

23GPa, 1600C, 6h

23GPa, 1700C, 6h


0

10

20

30

40

50

60

70

80

90

100

AT

OM

IC-P

ER

CE

NT

FE

-150 -100 -50 0 50 100 150 200 250

DISTANCE (um)

THERMO-CALC (2011.05.20:17.48) :

1GPa, 1280C, 6h

1GPa, 1150C, 18h

1GPa, 1420C, 2h


0

10

20

30

40

50

60

70

80

90

100

AT

OM

IC-P

ER

CE

NT

FE

-150 -100 -50 0 50 100 150 200 250

DISTANCE (um)

THERMO-CALC (2011.05.20:17.48) :

1GPa, 1280C, 6h

1GPa, 1150C, 18h

1GPa, 1420C, 2h


0

10

20

30

40

50

60

70

80

90

100

AT

OM

IC-P

ER

CE

NT

FE

-300 -200 -100 0 100 200 300 400 500

DISTANCE (um)

THERMO-CALC (2011.05.20:17.50) :

12GPa, 1600C, 2h

12GPa, 1500C, 2h

12GPa, 1600C, 10h

12GPa, 1600C, 0.5h


0

10

20

30

40

50

60

70

80

90

100

AT

OM

IC-P

ER

CE

NT

FE

-150 -100 -50 0 50 100 150 200 250

DISTANCE (um)

THERMO-CALC (2011.05.20:17.51) :

23GPa, 1600C, 6h

23GPa, 1700C, 6h



Interdiffusion coefficients


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:17.59) :

DATABASE:USER

N=1, P=1E9, T=1427;

4GPa, 1233C, 4h [1965Gol]

4GPa, 1154C, 8h [1965Gol]


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:17.59) :

DATABASE:USER

N=1., P=1E9, T=1693;

1GPa, 1280C, 6h [2007Yun]

1GPa, 1150C, 18h [2007Yun]

1GPa, 1420C, 2h [2007Yun]


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:18.00) :

DATABASE:USER

N=1., P=1.2E10, T=1873;

12GPa, 1600C, 2h [2007Yun]

12GPa, 1500C, 2h [2007Yun]

12GPa, 1600C, 10h [2007Yun]

12GPa, 1600C, 0.5h [2007Yun]


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:18.00) :

DATABASE:USER

N=1, P=2.3E10, T=1973;

23GPa, 1600C, 6h [2007Yun]

23GPa, 1700C, 6h [2007Yun]


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:17.59) :

DATABASE:USER

N=1., P=1E9, T=1693;

1GPa, 1280C, 6h [2007Yun]

1GPa, 1150C, 18h [2007Yun]

1GPa, 1420C, 2h [2007Yun]


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:17.59) :

DATABASE:USER

N=1, P=1E9, T=1427;

4GPa, 1233C, 4h [1965Gol]

4GPa, 1154C, 8h [1965Gol]


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:18.00) :

DATABASE:USER

N=1., P=1.2E10, T=1873;

12GPa, 1600C, 2h [2007Yun]

12GPa, 1500C, 2h [2007Yun]

12GPa, 1600C, 10h [2007Yun]

12GPa, 1600C, 0.5h [2007Yun]


-16.0

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0

LO

GD

C(F

CC

,NI,

NI,

FE

)

0 10 20 30 40 50 60 70 80 90 100

MOLE_PERCENT NI

THERMO-CALC (2011.12.07:18.00) :

DATABASE:USER

N=1, P=2.3E10, T=1973;

23GPa, 1600C, 6h [2007Yun]

23GPa, 1700C, 6h [2007Yun]


D @ 1600 C, 10 at.-% Ni


-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

-12.0L

OG

DC

(FC

C,N

I,N

I,F

E)

0 5 10 15 20 25

FUNCTION PGPA

THERMO-CALC (2011.12.09:16.42) :

DATABASE:USER

N=1, T=1873.15, X(NI)=0.1;

Goldstein et al. (1965)

Yunker et al. (2007)


• 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

Summary


THANK YOU!(and see you at Calphad 2013

in San Sebastian, Spain)

modelling diffusion at high pressure

Technology

activation volumecorrelations

formation volume

diffusion modellingprovides

migration volume

modelling of diffusion

diffusion process vf

defectactivation volumecalphad

ambient pressure